[{"data":1,"prerenderedAt":16786},["ShallowReactive",2],{"lesson:\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":3,"course-wordcounts":11006,"ref-card-index":11918,"nav:reinforcement-learning":16601,"tikz:cfa00e4d47193e983391d4368cc9a9bbccbb5d040bfbd8320ef50ee8217d895d":16781,"tikz:86917793b61c294421cd58ba6e0f1f12dd18b911d43ac008c06476e8f066837d":16782,"tikz:f99f989707a4fc2327854776216542d3f9d48d3282fff645ca44144e3af8b0cb":16783,"tikz:d2609622a98cae8f72e89772cbea049f3c6d3abba70568219e18b4017d275f6f":16784,"tikz:513586c20aa3160bf5c3f1f41aa9ffd2583f3740d6a21a25350e1d0733b36916":16785},{"id":4,"title":5,"blurb":6,"body":7,"brief":10973,"category":10974,"description":10975,"draft":10976,"extension":10977,"meta":10978,"module":10982,"navigation":10983,"path":10984,"practice":10985,"rawbody":10986,"readingTime":10987,"seo":10992,"sources":10993,"status":11000,"stem":11001,"summary":11002,"topics":11003,"__hash__":11005},"course\u002F07.reinforcement-learning\u002F05.modern-deep-rl\u002F09.offline-rl.md","Offline RL: The Problem and Value-Based Fixes","",{"type":8,"value":9,"toc":10960},"minimark",[10,62,117,129,134,137,473,591,877,880,886,953,960,1058,1062,1142,1201,1760,2001,2234,2360,2363,2518,2748,2752,2762,2767,2962,3246,3709,3962,4803,5853,5856,5875,5879,6206,6353,7298,7489,7492,7913,8031,8035,8123,8427,9082,9212,9215,9427,9470,9474,9481,9551,10478,10659,10662,10776,10848,10852,10858,10861,10893,10899],[11,12,13,14,18,19,45,46,50,51],"p",{},"Every method so far assumed a running environment. The agent acts, the environment\nreturns a reward and a next state, and that fresh feedback keeps the\nestimates accurate — a value that drifts too high produces actions that get tried,\nfail, and get corrected. ",[15,16,17],"strong",{},"Offline reinforcement learning"," removes exactly that\nloop. You are handed a fixed dataset ",[20,21,24],"span",{"className":22},[23],"katex",[20,25,29],{"className":26,"ariaHidden":28},[27],"katex-html","true",[20,30,33,38],{"className":31},[32],"base",[20,34],{"className":35,"style":37},[36],"strut","height:0.6833em;",[20,39,44],{"className":40,"style":43},[41,42],"mord","mathcal","margin-right:0.0278em;","D"," of transitions logged by some\nearlier policy, and you must return the best policy you can ",[47,48,49],"em",{},"without collecting a\nsingle new transition",". No exploration, no rollouts, no chance to test an\noptimistic guess against the environment.",[52,53,54],"sup",{},[55,56,61],"a",{"href":57,"ariaDescribedBy":58,"dataFootnoteRef":6,"id":60},"#user-content-fn-offline-survey",[59],"footnote-label","user-content-fnref-offline-survey","1",[11,63,64,65,69,70,89,90,93,94,109,110,113,114],{},"This is ",[55,66,68],{"href":67},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","off-policy learning","\ntaken to its limit. Off-policy methods already learn about a target policy ",[20,71,73],{"className":72},[23],[20,74,76],{"className":75,"ariaHidden":28},[27],[20,77,79,83],{"className":78},[32],[20,80],{"className":81,"style":82},[36],"height:0.4306em;",[20,84,88],{"className":85,"style":87},[41,86],"mathnormal","margin-right:0.0359em;","π","\nfrom data generated by a different behavior policy; offline RL is the special case\nwhere the behavior data is ",[47,91,92],{},"all there is"," and ",[20,95,97],{"className":96},[23],[20,98,100],{"className":99,"ariaHidden":28},[27],[20,101,103,106],{"className":102},[32],[20,104],{"className":105,"style":82},[36],[20,107,88],{"className":108,"style":87},[41,86]," can never be run to gather\nmore. The ",[55,111,112],{"href":67},"deadly triad","\n— function approximation, bootstrapping, off-policy training — is present in full,\nand one of its stabilizers, the ability to visit the states your policy actually\ninduces, is gone. The result is a failure mode sharp enough to have its own name,\nand a family of fixes that all reduce to a single instruction: ",[15,115,116],{},"be pessimistic\nabout what the data cannot verify.",[11,118,119,120,123,124,128],{},"This lesson sets up the problem — the failure mode, and how you even ",[47,121,122],{},"score"," a policy\nfrom a log — then builds the two value-based families of fix: policy constraint (BCQ)\nand conservative value estimation (CQL). The implicit methods, model-based pessimism,\nand the sequence-modeling view of Decision Transformer continue in a\n",[55,125,127],{"href":126},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2","companion lesson",".",[130,131,133],"h2",{"id":132},"the-setting-and-why-it-matters","The setting and why it matters",[11,135,136],{},"Formally, offline RL fixes a static dataset",[20,138,141],{"className":139},[140],"katex-display",[20,142,144],{"className":143},[23],[20,145,147,169],{"className":146,"ariaHidden":28},[27],[20,148,150,153,156,161,166],{"className":149},[32],[20,151],{"className":152,"style":37},[36],[20,154,44],{"className":155,"style":43},[41,42],[20,157],{"className":158,"style":160},[159],"mspace","margin-right:0.2778em;",[20,162,165],{"className":163},[164],"mrel","=",[20,167],{"className":168,"style":160},[159],[20,170,172,176,181,239,244,248,288,291,294,336,339,342,401,406,470],{"className":171},[32],[20,173],{"className":174,"style":175},[36],"height:1.1413em;vertical-align:-0.25em;",[20,177,180],{"className":178},[179],"mopen","{(",[20,182,184,188],{"className":183},[41],[20,185,187],{"className":186},[41,86],"s",[20,189,192],{"className":190},[191],"msupsub",[20,193,197,230],{"className":194},[195,196],"vlist-t","vlist-t2",[20,198,201,225],{"className":199},[200],"vlist-r",[20,202,206],{"className":203,"style":205},[204],"vlist","height:0.3117em;",[20,207,209,214],{"style":208},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[20,210],{"className":211,"style":213},[212],"pstrut","height:2.7em;",[20,215,221],{"className":216},[217,218,219,220],"sizing","reset-size6","size3","mtight",[20,222,224],{"className":223},[41,86,220],"i",[20,226,229],{"className":227},[228],"vlist-s","​",[20,231,233],{"className":232},[200],[20,234,237],{"className":235,"style":236},[204],"height:0.15em;",[20,238],{},[20,240,243],{"className":241},[242],"mpunct",",",[20,245],{"className":246,"style":247},[159],"margin-right:0.1667em;",[20,249,251,254],{"className":250},[41],[20,252,55],{"className":253},[41,86],[20,255,257],{"className":256},[191],[20,258,260,280],{"className":259},[195,196],[20,261,263,277],{"className":262},[200],[20,264,266],{"className":265,"style":205},[204],[20,267,268,271],{"style":208},[20,269],{"className":270,"style":213},[212],[20,272,274],{"className":273},[217,218,219,220],[20,275,224],{"className":276},[41,86,220],[20,278,229],{"className":279},[228],[20,281,283],{"className":282},[200],[20,284,286],{"className":285,"style":236},[204],[20,287],{},[20,289,243],{"className":290},[242],[20,292],{"className":293,"style":247},[159],[20,295,297,301],{"className":296},[41],[20,298,300],{"className":299,"style":43},[41,86],"r",[20,302,304],{"className":303},[191],[20,305,307,328],{"className":306},[195,196],[20,308,310,325],{"className":309},[200],[20,311,313],{"className":312,"style":205},[204],[20,314,316,319],{"style":315},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[20,317],{"className":318,"style":213},[212],[20,320,322],{"className":321},[217,218,219,220],[20,323,224],{"className":324},[41,86,220],[20,326,229],{"className":327},[228],[20,329,331],{"className":330},[200],[20,332,334],{"className":333,"style":236},[204],[20,335],{},[20,337,243],{"className":338},[242],[20,340],{"className":341,"style":247},[159],[20,343,345,348],{"className":344},[41],[20,346,187],{"className":347},[41,86],[20,349,351],{"className":350},[191],[20,352,354,392],{"className":353},[195,196],[20,355,357,389],{"className":356},[200],[20,358,361,373],{"className":359,"style":360},[204],"height:0.8019em;",[20,362,364,367],{"style":363},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[20,365],{"className":366,"style":213},[212],[20,368,370],{"className":369},[217,218,219,220],[20,371,224],{"className":372},[41,86,220],[20,374,376,379],{"style":375},"top:-3.113em;margin-right:0.05em;",[20,377],{"className":378,"style":213},[212],[20,380,382],{"className":381},[217,218,219,220],[20,383,385],{"className":384},[41,220],[20,386,388],{"className":387},[41,220],"′",[20,390,229],{"className":391},[228],[20,393,395],{"className":394},[200],[20,396,399],{"className":397,"style":398},[204],"height:0.247em;",[20,400],{},[20,402,405],{"className":403},[404],"mclose",")",[20,407,409,413],{"className":408},[404],[20,410,412],{"className":411},[404],"}",[20,414,416],{"className":415},[191],[20,417,419,462],{"className":418},[195,196],[20,420,422,459],{"className":421},[200],[20,423,426,446],{"className":424,"style":425},[204],"height:0.8913em;",[20,427,428,431],{"style":363},[20,429],{"className":430,"style":213},[212],[20,432,434],{"className":433},[217,218,219,220],[20,435,437,440,443],{"className":436},[41,220],[20,438,224],{"className":439},[41,86,220],[20,441,165],{"className":442},[164,220],[20,444,61],{"className":445},[41,220],[20,447,448,451],{"style":375},[20,449],{"className":450,"style":213},[212],[20,452,454],{"className":453},[217,218,219,220],[20,455,458],{"className":456,"style":457},[41,86,220],"margin-right:0.109em;","N",[20,460,229],{"className":461},[228],[20,463,465],{"className":464},[200],[20,466,468],{"className":467,"style":398},[204],[20,469],{},[20,471,243],{"className":472},[242],[11,474,475,476,479,480,538,539,554,555,570,571,586,587,590],{},"collected by a ",[15,477,478],{},"behavior policy"," ",[20,481,483],{"className":482},[23],[20,484,486],{"className":485,"ariaHidden":28},[27],[20,487,489,493],{"className":488},[32],[20,490],{"className":491,"style":492},[36],"height:0.7167em;vertical-align:-0.2861em;",[20,494,496,499],{"className":495},[41],[20,497,88],{"className":498,"style":87},[41,86],[20,500,502],{"className":501},[191],[20,503,505,529],{"className":504},[195,196],[20,506,508,526],{"className":507},[200],[20,509,512],{"className":510,"style":511},[204],"height:0.3361em;",[20,513,515,518],{"style":514},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[20,516],{"className":517,"style":213},[212],[20,519,521],{"className":520},[217,218,219,220],[20,522,525],{"className":523,"style":524},[41,86,220],"margin-right:0.0528em;","β",[20,527,229],{"className":528},[228],[20,530,532],{"className":531},[200],[20,533,536],{"className":534,"style":535},[204],"height:0.2861em;",[20,537],{}," (possibly a mixture of several\npolicies, or logged human operators). The learner sees ",[20,540,542],{"className":541},[23],[20,543,545],{"className":544,"ariaHidden":28},[27],[20,546,548,551],{"className":547},[32],[20,549],{"className":550,"style":37},[36],[20,552,44],{"className":553,"style":43},[41,42]," once, fits a\npolicy ",[20,556,558],{"className":557},[23],[20,559,561],{"className":560,"ariaHidden":28},[27],[20,562,564,567],{"className":563},[32],[20,565],{"className":566,"style":82},[36],[20,568,88],{"className":569,"style":87},[41,86],", and is evaluated by deploying ",[20,572,574],{"className":573},[23],[20,575,577],{"className":576,"ariaHidden":28},[27],[20,578,580,583],{"className":579},[32],[20,581],{"className":582,"style":82},[36],[20,584,88],{"className":585,"style":87},[41,86]," in the true environment. Between\nfitting and deployment there is no interaction. The empirical ",[15,588,589],{},"state-action\ndistribution"," of the data,",[20,592,594],{"className":593},[140],[20,595,597],{"className":596},[23],[20,598,600,718,862],{"className":599,"ariaHidden":28},[27],[20,601,603,607,684,688,691,694,697,700,703,706,709,712,715],{"className":602},[32],[20,604],{"className":605,"style":606},[36],"height:1em;vertical-align:-0.25em;",[20,608,610,614],{"className":609},[41],[20,611,613],{"className":612},[41,86],"d",[20,615,617],{"className":616},[191],[20,618,620],{"className":619},[195],[20,621,623],{"className":622},[200],[20,624,627],{"className":625,"style":626},[204],"height:0.7144em;",[20,628,629,632],{"style":375},[20,630],{"className":631,"style":213},[212],[20,633,635],{"className":634},[217,218,219,220],[20,636,638],{"className":637},[41,220],[20,639,641,644],{"className":640},[41,220],[20,642,88],{"className":643,"style":87},[41,86,220],[20,645,647],{"className":646},[191],[20,648,650,675],{"className":649},[195,196],[20,651,653,672],{"className":652},[200],[20,654,657],{"className":655,"style":656},[204],"height:0.3448em;",[20,658,660,664],{"style":659},"top:-2.3488em;margin-left:-0.0359em;margin-right:0.0714em;",[20,661],{"className":662,"style":663},[212],"height:2.5em;",[20,665,669],{"className":666},[217,667,668,220],"reset-size3","size1",[20,670,525],{"className":671,"style":524},[41,86,220],[20,673,229],{"className":674},[228],[20,676,678],{"className":677},[200],[20,679,682],{"className":680,"style":681},[204],"height:0.2901em;",[20,683],{},[20,685,687],{"className":686},[179],"(",[20,689,187],{"className":690},[41,86],[20,692,243],{"className":693},[242],[20,695],{"className":696,"style":247},[159],[20,698,55],{"className":699},[41,86],[20,701,405],{"className":702},[404],[20,704],{"className":705,"style":160},[159],[20,707],{"className":708,"style":160},[159],[20,710,165],{"className":711},[164],[20,713],{"className":714,"style":160},[159],[20,716],{"className":717,"style":160},[159],[20,719,721,725,794,797,800,803,806,846,849,852,855,859],{"className":720},[32],[20,722],{"className":723,"style":724},[36],"height:1.0361em;vertical-align:-0.2861em;",[20,726,728,731],{"className":727},[41],[20,729,613],{"className":730},[41,86],[20,732,734],{"className":733},[191],[20,735,737],{"className":736},[195],[20,738,740],{"className":739},[200],[20,741,743],{"className":742,"style":626},[204],[20,744,745,748],{"style":375},[20,746],{"className":747,"style":213},[212],[20,749,751],{"className":750},[217,218,219,220],[20,752,754],{"className":753},[41,220],[20,755,757,760],{"className":756},[41,220],[20,758,88],{"className":759,"style":87},[41,86,220],[20,761,763],{"className":762},[191],[20,764,766,786],{"className":765},[195,196],[20,767,769,783],{"className":768},[200],[20,770,772],{"className":771,"style":656},[204],[20,773,774,777],{"style":659},[20,775],{"className":776,"style":663},[212],[20,778,780],{"className":779},[217,667,668,220],[20,781,525],{"className":782,"style":524},[41,86,220],[20,784,229],{"className":785},[228],[20,787,789],{"className":788},[200],[20,790,792],{"className":791,"style":681},[204],[20,793],{},[20,795,687],{"className":796},[179],[20,798,187],{"className":799},[41,86],[20,801,405],{"className":802},[404],[20,804],{"className":805,"style":247},[159],[20,807,809,812],{"className":808},[41],[20,810,88],{"className":811,"style":87},[41,86],[20,813,815],{"className":814},[191],[20,816,818,838],{"className":817},[195,196],[20,819,821,835],{"className":820},[200],[20,822,824],{"className":823,"style":511},[204],[20,825,826,829],{"style":514},[20,827],{"className":828,"style":213},[212],[20,830,832],{"className":831},[217,218,219,220],[20,833,525],{"className":834,"style":524},[41,86,220],[20,836,229],{"className":837},[228],[20,839,841],{"className":840},[200],[20,842,844],{"className":843,"style":535},[204],[20,845],{},[20,847,687],{"className":848},[179],[20,850,55],{"className":851},[41,86],[20,853],{"className":854,"style":160},[159],[20,856,858],{"className":857},[164],"∣",[20,860],{"className":861,"style":160},[159],[20,863,865,868,871,874],{"className":864},[32],[20,866],{"className":867,"style":606},[36],[20,869,187],{"className":870},[41,86],[20,872,405],{"className":873},[404],[20,875,243],{"className":876},[242],[11,878,879],{},"is the only region of the world the learner has ground truth for. Anything outside\nits support is a guess.",[11,881,882,883,128],{},"The motivation is that in the domains where RL would help most, online interaction\nis expensive, slow, or dangerous, while ",[47,884,885],{},"logged data already exists in bulk",[887,888,889,905],"table",{},[890,891,892],"thead",{},[893,894,895,899,902],"tr",{},[896,897,898],"th",{},"Domain",[896,900,901],{},"Why online interaction is blocked",[896,903,904],{},"What logged data exists",[906,907,908,920,931,942],"tbody",{},[893,909,910,914,917],{},[911,912,913],"td",{},"Medicine",[911,915,916],{},"a bad treatment policy harms patients",[911,918,919],{},"years of electronic health records, dosing logs",[893,921,922,925,928],{},[911,923,924],{},"Autonomous driving \u002F robotics",[911,926,927],{},"a bad action crashes hardware",[911,929,930],{},"fleet-scale driving logs, teleoperation traces",[893,932,933,936,939],{},[911,934,935],{},"Recommendation",[911,937,938],{},"live experiments cost revenue and annoy users",[911,940,941],{},"billions of impression-click histories",[893,943,944,947,950],{},[911,945,946],{},"Industrial control",[911,948,949],{},"exploration can damage the plant",[911,951,952],{},"sensor logs from existing controllers",[11,954,955,956,959],{},"In each case the promise is the same one supervised learning delivered on\nperception: turn a large static dataset into a decision-making system, with all the\nlearning done offline and only the final policy deployed. The obstacle is that a\npolicy is not a label. To evaluate a candidate action the value function must\nestimate outcomes for actions ",[47,957,958],{},"the dataset may never have taken",", and estimation\nwithout feedback is where offline RL fails.",[961,962,964],"callout",{"type":963},"definition",[11,965,966,969,970,985,986,1001,1002,1054,1055,128],{},[15,967,968],{},"Definition (Offline \u002F batch RL)."," The problem of learning a policy ",[20,971,973],{"className":972},[23],[20,974,976],{"className":975,"ariaHidden":28},[27],[20,977,979,982],{"className":978},[32],[20,980],{"className":981,"style":82},[36],[20,983,88],{"className":984,"style":87},[41,86]," to\nmaximize expected return using only a fixed dataset ",[20,987,989],{"className":988},[23],[20,990,992],{"className":991,"ariaHidden":28},[27],[20,993,995,998],{"className":994},[32],[20,996],{"className":997,"style":37},[36],[20,999,44],{"className":1000,"style":43},[41,42]," of transitions\ncollected by a behavior policy ",[20,1003,1005],{"className":1004},[23],[20,1006,1008],{"className":1007,"ariaHidden":28},[27],[20,1009,1011,1014],{"className":1010},[32],[20,1012],{"className":1013,"style":492},[36],[20,1015,1017,1020],{"className":1016},[41],[20,1018,88],{"className":1019,"style":87},[41,86],[20,1021,1023],{"className":1022},[191],[20,1024,1026,1046],{"className":1025},[195,196],[20,1027,1029,1043],{"className":1028},[200],[20,1030,1032],{"className":1031,"style":511},[204],[20,1033,1034,1037],{"style":514},[20,1035],{"className":1036,"style":213},[212],[20,1038,1040],{"className":1039},[217,218,219,220],[20,1041,525],{"className":1042,"style":524},[41,86,220],[20,1044,229],{"className":1045},[228],[20,1047,1049],{"className":1048},[200],[20,1050,1052],{"className":1051,"style":535},[204],[20,1053],{},", with no additional environment\ninteraction during or after training. Also called ",[47,1056,1057],{},"batch RL",[130,1059,1061],{"id":1060},"the-core-failure-distributional-shift-and-extrapolation-error","The core failure: distributional shift and extrapolation error",[11,1063,1064,1065,1069,1070,1122,1123,1138,1139,128],{},"Run an ordinary off-policy algorithm — Q-learning, or an actor-critic like\n",[55,1066,1068],{"href":1067},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control","DDPG"," — on a fixed\ndataset and it typically diverges or returns a policy far worse than ",[20,1071,1073],{"className":1072},[23],[20,1074,1076],{"className":1075,"ariaHidden":28},[27],[20,1077,1079,1082],{"className":1078},[32],[20,1080],{"className":1081,"style":492},[36],[20,1083,1085,1088],{"className":1084},[41],[20,1086,88],{"className":1087,"style":87},[41,86],[20,1089,1091],{"className":1090},[191],[20,1092,1094,1114],{"className":1093},[195,196],[20,1095,1097,1111],{"className":1096},[200],[20,1098,1100],{"className":1099,"style":511},[204],[20,1101,1102,1105],{"style":514},[20,1103],{"className":1104,"style":213},[212],[20,1106,1108],{"className":1107},[217,218,219,220],[20,1109,525],{"className":1110,"style":524},[41,86,220],[20,1112,229],{"className":1113},[228],[20,1115,1117],{"className":1116},[200],[20,1118,1120],{"className":1119,"style":535},[204],[20,1121],{},",\neven when ",[20,1124,1126],{"className":1125},[23],[20,1127,1129],{"className":1128,"ariaHidden":28},[27],[20,1130,1132,1135],{"className":1131},[32],[20,1133],{"className":1134,"style":37},[36],[20,1136,44],{"className":1137,"style":43},[41,42]," is large and the network has ample capacity. The culprit is\nnot optimization but the ",[15,1140,1141],{},"bootstrapping target",[11,1143,1144,1145,1200],{},"Q-learning fits ",[20,1146,1148],{"className":1147},[23],[20,1149,1151],{"className":1150,"ariaHidden":28},[27],[20,1152,1154,1158],{"className":1153},[32],[20,1155],{"className":1156,"style":1157},[36],"height:0.8778em;vertical-align:-0.1944em;",[20,1159,1161,1165],{"className":1160},[41],[20,1162,1164],{"className":1163},[41,86],"Q",[20,1166,1168],{"className":1167},[191],[20,1169,1171,1192],{"className":1170},[195,196],[20,1172,1174,1189],{"className":1173},[200],[20,1175,1177],{"className":1176,"style":511},[204],[20,1178,1179,1182],{"style":208},[20,1180],{"className":1181,"style":213},[212],[20,1183,1185],{"className":1184},[217,218,219,220],[20,1186,1188],{"className":1187,"style":43},[41,86,220],"θ",[20,1190,229],{"className":1191},[228],[20,1193,1195],{"className":1194},[200],[20,1196,1198],{"className":1197,"style":236},[204],[20,1199],{}," to the Bellman target",[20,1202,1204],{"className":1203},[140],[20,1205,1207],{"className":1206},[23],[20,1208,1210,1236,1258,1563,1581],{"className":1209,"ariaHidden":28},[27],[20,1211,1213,1217,1221,1224,1227,1230,1233],{"className":1212},[32],[20,1214],{"className":1215,"style":1216},[36],"height:0.625em;vertical-align:-0.1944em;",[20,1218,1220],{"className":1219,"style":87},[41,86],"y",[20,1222],{"className":1223,"style":160},[159],[20,1225],{"className":1226,"style":160},[159],[20,1228,165],{"className":1229},[164],[20,1231],{"className":1232,"style":160},[159],[20,1234],{"className":1235,"style":160},[159],[20,1237,1239,1243,1246,1250,1255],{"className":1238},[32],[20,1240],{"className":1241,"style":1242},[36],"height:0.6667em;vertical-align:-0.0833em;",[20,1244,300],{"className":1245,"style":43},[41,86],[20,1247],{"className":1248,"style":1249},[159],"margin-right:0.2222em;",[20,1251,1254],{"className":1252},[1253],"mbin","+",[20,1256],{"className":1257,"style":1249},[159],[20,1259,1261,1265,1270,1273,1276,1365,1368,1448,1451,1483,1486,1489,1521,1524,1527,1531,1534,1542,1545,1548,1551,1554,1557,1560],{"className":1260},[32],[20,1262],{"className":1263,"style":1264},[36],"height:1.5459em;vertical-align:-0.744em;",[20,1266,1269],{"className":1267,"style":1268},[41,86],"margin-right:0.0556em;","γ",[20,1271],{"className":1272,"style":247},[159],[20,1274],{"className":1275,"style":247},[159],[20,1277,1281],{"className":1278},[1279,1280],"mop","op-limits",[20,1282,1284,1356],{"className":1283},[195,196],[20,1285,1287,1353],{"className":1286},[200],[20,1288,1290,1337],{"className":1289,"style":82},[204],[20,1291,1293,1297],{"style":1292},"top:-2.356em;margin-left:0em;",[20,1294],{"className":1295,"style":1296},[212],"height:3em;",[20,1298,1300],{"className":1299},[217,218,219,220],[20,1301,1303],{"className":1302},[41,220],[20,1304,1306,1309],{"className":1305},[41,220],[20,1307,55],{"className":1308},[41,86,220],[20,1310,1312],{"className":1311},[191],[20,1313,1315],{"className":1314},[195],[20,1316,1318],{"className":1317},[200],[20,1319,1322],{"className":1320,"style":1321},[204],"height:0.6828em;",[20,1323,1325,1328],{"style":1324},"top:-2.786em;margin-right:0.0714em;",[20,1326],{"className":1327,"style":663},[212],[20,1329,1331],{"className":1330},[217,667,668,220],[20,1332,1334],{"className":1333},[41,220],[20,1335,388],{"className":1336},[41,220],[20,1338,1340,1343],{"style":1339},"top:-3em;",[20,1341],{"className":1342,"style":1296},[212],[20,1344,1345],{},[20,1346,1348],{"className":1347},[1279],[20,1349,1352],{"className":1350},[41,1351],"mathrm","max",[20,1354,229],{"className":1355},[228],[20,1357,1359],{"className":1358},[200],[20,1360,1363],{"className":1361,"style":1362},[204],"height:0.744em;",[20,1364],{},[20,1366],{"className":1367,"style":247},[159],[20,1369,1371,1374],{"className":1370},[41],[20,1372,1164],{"className":1373},[41,86],[20,1375,1377],{"className":1376},[191],[20,1378,1380,1439],{"className":1379},[195,196],[20,1381,1383,1436],{"className":1382},[200],[20,1384,1386],{"className":1385,"style":656},[204],[20,1387,1389,1392],{"style":1388},"top:-2.4629em;margin-left:0em;margin-right:0.05em;",[20,1390],{"className":1391,"style":213},[212],[20,1393,1395],{"className":1394},[217,218,219,220],[20,1396,1398],{"className":1397},[41,220],[20,1399,1402],{"className":1400},[41,1401,220],"accent",[20,1403,1405],{"className":1404},[195],[20,1406,1408],{"className":1407},[200],[20,1409,1412,1421],{"className":1410,"style":1411},[204],"height:0.8312em;",[20,1413,1415,1418],{"style":1414},"top:-2.7em;",[20,1416],{"className":1417,"style":213},[212],[20,1419,1188],{"className":1420,"style":43},[41,86,220],[20,1422,1424,1427],{"style":1423},"top:-2.9634em;",[20,1425],{"className":1426,"style":213},[212],[20,1428,1432],{"className":1429,"style":1431},[1430],"accent-body","left:-0.1667em;",[20,1433,1435],{"className":1434},[41,220],"ˉ",[20,1437,229],{"className":1438},[228],[20,1440,1442],{"className":1441},[200],[20,1443,1446],{"className":1444,"style":1445},[204],"height:0.2371em;",[20,1447],{},[20,1449,687],{"className":1450},[179],[20,1452,1454,1457],{"className":1453},[41],[20,1455,187],{"className":1456},[41,86],[20,1458,1460],{"className":1459},[191],[20,1461,1463],{"className":1462},[195],[20,1464,1466],{"className":1465},[200],[20,1467,1469],{"className":1468,"style":360},[204],[20,1470,1471,1474],{"style":375},[20,1472],{"className":1473,"style":213},[212],[20,1475,1477],{"className":1476},[217,218,219,220],[20,1478,1480],{"className":1479},[41,220],[20,1481,388],{"className":1482},[41,220],[20,1484,243],{"className":1485},[242],[20,1487],{"className":1488,"style":247},[159],[20,1490,1492,1495],{"className":1491},[41],[20,1493,55],{"className":1494},[41,86],[20,1496,1498],{"className":1497},[191],[20,1499,1501],{"className":1500},[195],[20,1502,1504],{"className":1503},[200],[20,1505,1507],{"className":1506,"style":360},[204],[20,1508,1509,1512],{"style":375},[20,1510],{"className":1511,"style":213},[212],[20,1513,1515],{"className":1514},[217,218,219,220],[20,1516,1518],{"className":1517},[41,220],[20,1519,388],{"className":1520},[41,220],[20,1522,405],{"className":1523},[404],[20,1525,243],{"className":1526},[242],[20,1528],{"className":1529,"style":1530},[159],"margin-right:2em;",[20,1532],{"className":1533,"style":247},[159],[20,1535,1538],{"className":1536},[41,1537],"text",[20,1539,1541],{"className":1540},[41],"or for actor-critic",[20,1543],{"className":1544,"style":1530},[159],[20,1546,1220],{"className":1547,"style":87},[41,86],[20,1549],{"className":1550,"style":160},[159],[20,1552],{"className":1553,"style":160},[159],[20,1555,165],{"className":1556},[164],[20,1558],{"className":1559,"style":160},[159],[20,1561],{"className":1562,"style":160},[159],[20,1564,1566,1569,1572,1575,1578],{"className":1565},[32],[20,1567],{"className":1568,"style":1242},[36],[20,1570,300],{"className":1571,"style":43},[41,86],[20,1573],{"className":1574,"style":1249},[159],[20,1576,1254],{"className":1577},[1253],[20,1579],{"className":1580,"style":1249},[159],[20,1582,1584,1588,1591,1594,1665,1672,1704,1707,1710,1713,1716,1748,1751,1757],{"className":1583},[32],[20,1585],{"className":1586,"style":1587},[36],"height:1.2em;vertical-align:-0.35em;",[20,1589,1269],{"className":1590,"style":1268},[41,86],[20,1592],{"className":1593,"style":247},[159],[20,1595,1597,1600],{"className":1596},[41],[20,1598,1164],{"className":1599},[41,86],[20,1601,1603],{"className":1602},[191],[20,1604,1606,1657],{"className":1605},[195,196],[20,1607,1609,1654],{"className":1608},[200],[20,1610,1612],{"className":1611,"style":656},[204],[20,1613,1614,1617],{"style":1388},[20,1615],{"className":1616,"style":213},[212],[20,1618,1620],{"className":1619},[217,218,219,220],[20,1621,1623],{"className":1622},[41,220],[20,1624,1626],{"className":1625},[41,1401,220],[20,1627,1629],{"className":1628},[195],[20,1630,1632],{"className":1631},[200],[20,1633,1635,1643],{"className":1634,"style":1411},[204],[20,1636,1637,1640],{"style":1414},[20,1638],{"className":1639,"style":213},[212],[20,1641,1188],{"className":1642,"style":43},[41,86,220],[20,1644,1645,1648],{"style":1423},[20,1646],{"className":1647,"style":213},[212],[20,1649,1651],{"className":1650,"style":1431},[1430],[20,1652,1435],{"className":1653},[41,220],[20,1655,229],{"className":1656},[228],[20,1658,1660],{"className":1659},[200],[20,1661,1663],{"className":1662,"style":1445},[204],[20,1664],{},[20,1666,1668],{"className":1667},[179],[20,1669,687],{"className":1670},[1671,668],"delimsizing",[20,1673,1675,1678],{"className":1674},[41],[20,1676,187],{"className":1677},[41,86],[20,1679,1681],{"className":1680},[191],[20,1682,1684],{"className":1683},[195],[20,1685,1687],{"className":1686},[200],[20,1688,1690],{"className":1689,"style":360},[204],[20,1691,1692,1695],{"style":375},[20,1693],{"className":1694,"style":213},[212],[20,1696,1698],{"className":1697},[217,218,219,220],[20,1699,1701],{"className":1700},[41,220],[20,1702,388],{"className":1703},[41,220],[20,1705,243],{"className":1706},[242],[20,1708],{"className":1709,"style":247},[159],[20,1711,88],{"className":1712,"style":87},[41,86],[20,1714,687],{"className":1715},[179],[20,1717,1719,1722],{"className":1718},[41],[20,1720,187],{"className":1721},[41,86],[20,1723,1725],{"className":1724},[191],[20,1726,1728],{"className":1727},[195],[20,1729,1731],{"className":1730},[200],[20,1732,1734],{"className":1733,"style":360},[204],[20,1735,1736,1739],{"style":375},[20,1737],{"className":1738,"style":213},[212],[20,1740,1742],{"className":1741},[217,218,219,220],[20,1743,1745],{"className":1744},[41,220],[20,1746,388],{"className":1747},[41,220],[20,1749,405],{"className":1750},[404],[20,1752,1754],{"className":1753},[404],[20,1755,405],{"className":1756},[1671,668],[20,1758,128],{"className":1759},[41],[11,1761,1762,1763,1778,1779,1825,1826,1829,1830,1845,1846,1935,1936,1980,1981,1996,1997,2000],{},"The maximization — or the actor ",[20,1764,1766],{"className":1765},[23],[20,1767,1769],{"className":1768,"ariaHidden":28},[27],[20,1770,1772,1775],{"className":1771},[32],[20,1773],{"className":1774,"style":82},[36],[20,1776,88],{"className":1777,"style":87},[41,86]," chasing high value — deliberately seeks the\naction ",[20,1780,1782],{"className":1781},[23],[20,1783,1785],{"className":1784,"ariaHidden":28},[27],[20,1786,1788,1792],{"className":1787},[32],[20,1789],{"className":1790,"style":1791},[36],"height:0.7519em;",[20,1793,1795,1798],{"className":1794},[41],[20,1796,55],{"className":1797},[41,86],[20,1799,1801],{"className":1800},[191],[20,1802,1804],{"className":1803},[195],[20,1805,1807],{"className":1806},[200],[20,1808,1810],{"className":1809,"style":1791},[204],[20,1811,1813,1816],{"style":1812},"top:-3.063em;margin-right:0.05em;",[20,1814],{"className":1815,"style":213},[212],[20,1817,1819],{"className":1818},[217,218,219,220],[20,1820,1822],{"className":1821},[41,220],[20,1823,388],{"className":1824},[41,220]," that ",[47,1827,1828],{},"maximizes"," the current ",[20,1831,1833],{"className":1832},[23],[20,1834,1836],{"className":1835,"ariaHidden":28},[27],[20,1837,1839,1842],{"className":1838},[32],[20,1840],{"className":1841,"style":1157},[36],[20,1843,1164],{"className":1844},[41,86],". On a full environment that is fine:\nif the max lands on an overvalued action, the agent tries it, observes the true low\nreward, and the estimate is corrected. Offline, the transition ",[20,1847,1849],{"className":1848},[23],[20,1850,1852],{"className":1851,"ariaHidden":28},[27],[20,1853,1855,1859,1862,1894,1897,1900,1932],{"className":1854},[32],[20,1856],{"className":1857,"style":1858},[36],"height:1.0019em;vertical-align:-0.25em;",[20,1860,687],{"className":1861},[179],[20,1863,1865,1868],{"className":1864},[41],[20,1866,187],{"className":1867},[41,86],[20,1869,1871],{"className":1870},[191],[20,1872,1874],{"className":1873},[195],[20,1875,1877],{"className":1876},[200],[20,1878,1880],{"className":1879,"style":1791},[204],[20,1881,1882,1885],{"style":1812},[20,1883],{"className":1884,"style":213},[212],[20,1886,1888],{"className":1887},[217,218,219,220],[20,1889,1891],{"className":1890},[41,220],[20,1892,388],{"className":1893},[41,220],[20,1895,243],{"className":1896},[242],[20,1898],{"className":1899,"style":247},[159],[20,1901,1903,1906],{"className":1902},[41],[20,1904,55],{"className":1905},[41,86],[20,1907,1909],{"className":1908},[191],[20,1910,1912],{"className":1911},[195],[20,1913,1915],{"className":1914},[200],[20,1916,1918],{"className":1917,"style":1791},[204],[20,1919,1920,1923],{"style":1812},[20,1921],{"className":1922,"style":213},[212],[20,1924,1926],{"className":1925},[217,218,219,220],[20,1927,1929],{"className":1928},[41,220],[20,1930,388],{"className":1931},[41,220],[20,1933,405],{"className":1934},[404]," for that\nchosen ",[20,1937,1939],{"className":1938},[23],[20,1940,1942],{"className":1941,"ariaHidden":28},[27],[20,1943,1945,1948],{"className":1944},[32],[20,1946],{"className":1947,"style":1791},[36],[20,1949,1951,1954],{"className":1950},[41],[20,1952,55],{"className":1953},[41,86],[20,1955,1957],{"className":1956},[191],[20,1958,1960],{"className":1959},[195],[20,1961,1963],{"className":1962},[200],[20,1964,1966],{"className":1965,"style":1791},[204],[20,1967,1968,1971],{"style":1812},[20,1969],{"className":1970,"style":213},[212],[20,1972,1974],{"className":1973},[217,218,219,220],[20,1975,1977],{"className":1976},[41,220],[20,1978,388],{"className":1979},[41,220]," may simply not be in ",[20,1982,1984],{"className":1983},[23],[20,1985,1987],{"className":1986,"ariaHidden":28},[27],[20,1988,1990,1993],{"className":1989},[32],[20,1991],{"className":1992,"style":37},[36],[20,1994,44],{"className":1995,"style":43},[41,42],". The network's value there is pure\n",[15,1998,1999],{},"extrapolation",", and the argmax preferentially selects wherever the\nextrapolation happens to overshoot.",[961,2002,2003],{"type":963},[11,2004,2005,2008,2009,2024,2025,2040,2041,2162,2163,2233],{},[15,2006,2007],{},"Definition (Out-of-distribution action)."," At a state ",[20,2010,2012],{"className":2011},[23],[20,2013,2015],{"className":2014,"ariaHidden":28},[27],[20,2016,2018,2021],{"className":2017},[32],[20,2019],{"className":2020,"style":82},[36],[20,2022,187],{"className":2023},[41,86],", an action ",[20,2026,2028],{"className":2027},[23],[20,2029,2031],{"className":2030,"ariaHidden":28},[27],[20,2032,2034,2037],{"className":2033},[32],[20,2035],{"className":2036,"style":82},[36],[20,2038,55],{"className":2039},[41,86]," with\nlittle or no support under the data distribution — ",[20,2042,2044],{"className":2043},[23],[20,2045,2047,2151],{"className":2046,"ariaHidden":28},[27],[20,2048,2050,2053,2123,2126,2129,2132,2135,2138,2141,2144,2148],{"className":2049},[32],[20,2051],{"className":2052,"style":606},[36],[20,2054,2056,2059],{"className":2055},[41],[20,2057,613],{"className":2058},[41,86],[20,2060,2062],{"className":2061},[191],[20,2063,2065],{"className":2064},[195],[20,2066,2068],{"className":2067},[200],[20,2069,2072],{"className":2070,"style":2071},[204],"height:0.6644em;",[20,2073,2074,2077],{"style":1812},[20,2075],{"className":2076,"style":213},[212],[20,2078,2080],{"className":2079},[217,218,219,220],[20,2081,2083],{"className":2082},[41,220],[20,2084,2086,2089],{"className":2085},[41,220],[20,2087,88],{"className":2088,"style":87},[41,86,220],[20,2090,2092],{"className":2091},[191],[20,2093,2095,2115],{"className":2094},[195,196],[20,2096,2098,2112],{"className":2097},[200],[20,2099,2101],{"className":2100,"style":656},[204],[20,2102,2103,2106],{"style":659},[20,2104],{"className":2105,"style":663},[212],[20,2107,2109],{"className":2108},[217,667,668,220],[20,2110,525],{"className":2111,"style":524},[41,86,220],[20,2113,229],{"className":2114},[228],[20,2116,2118],{"className":2117},[200],[20,2119,2121],{"className":2120,"style":681},[204],[20,2122],{},[20,2124,687],{"className":2125},[179],[20,2127,187],{"className":2128},[41,86],[20,2130,243],{"className":2131},[242],[20,2133],{"className":2134,"style":247},[159],[20,2136,55],{"className":2137},[41,86],[20,2139,405],{"className":2140},[404],[20,2142],{"className":2143,"style":160},[159],[20,2145,2147],{"className":2146},[164],"≈",[20,2149],{"className":2150,"style":160},[159],[20,2152,2154,2158],{"className":2153},[32],[20,2155],{"className":2156,"style":2157},[36],"height:0.6444em;",[20,2159,2161],{"className":2160},[41],"0",".\nThe value ",[20,2164,2166],{"className":2165},[23],[20,2167,2169],{"className":2168,"ariaHidden":28},[27],[20,2170,2172,2175,2215,2218,2221,2224,2227,2230],{"className":2171},[32],[20,2173],{"className":2174,"style":606},[36],[20,2176,2178,2181],{"className":2177},[41],[20,2179,1164],{"className":2180},[41,86],[20,2182,2184],{"className":2183},[191],[20,2185,2187,2207],{"className":2186},[195,196],[20,2188,2190,2204],{"className":2189},[200],[20,2191,2193],{"className":2192,"style":511},[204],[20,2194,2195,2198],{"style":208},[20,2196],{"className":2197,"style":213},[212],[20,2199,2201],{"className":2200},[217,218,219,220],[20,2202,1188],{"className":2203,"style":43},[41,86,220],[20,2205,229],{"className":2206},[228],[20,2208,2210],{"className":2209},[200],[20,2211,2213],{"className":2212,"style":236},[204],[20,2214],{},[20,2216,687],{"className":2217},[179],[20,2219,187],{"className":2220},[41,86],[20,2222,243],{"className":2223},[242],[20,2225],{"className":2226,"style":247},[159],[20,2228,55],{"className":2229},[41,86],[20,2231,405],{"className":2232},[404]," at such a point is unconstrained by any observed\nreturn and is fit only by the network's generalization.",[961,2235,2236],{"type":963},[11,2237,2238,2241,2242,2312,2313,2343,2344,2359],{},[15,2239,2240],{},"Definition (Extrapolation error)."," The error in ",[20,2243,2245],{"className":2244},[23],[20,2246,2248],{"className":2247,"ariaHidden":28},[27],[20,2249,2251,2254,2294,2297,2300,2303,2306,2309],{"className":2250},[32],[20,2252],{"className":2253,"style":606},[36],[20,2255,2257,2260],{"className":2256},[41],[20,2258,1164],{"className":2259},[41,86],[20,2261,2263],{"className":2262},[191],[20,2264,2266,2286],{"className":2265},[195,196],[20,2267,2269,2283],{"className":2268},[200],[20,2270,2272],{"className":2271,"style":511},[204],[20,2273,2274,2277],{"style":208},[20,2275],{"className":2276,"style":213},[212],[20,2278,2280],{"className":2279},[217,218,219,220],[20,2281,1188],{"className":2282,"style":43},[41,86,220],[20,2284,229],{"className":2285},[228],[20,2287,2289],{"className":2288},[200],[20,2290,2292],{"className":2291,"style":236},[204],[20,2293],{},[20,2295,687],{"className":2296},[179],[20,2298,187],{"className":2299},[41,86],[20,2301,243],{"className":2302},[242],[20,2304],{"className":2305,"style":247},[159],[20,2307,55],{"className":2308},[41,86],[20,2310,405],{"className":2311},[404]," at\nout-of-distribution ",[20,2314,2316],{"className":2315},[23],[20,2317,2319],{"className":2318,"ariaHidden":28},[27],[20,2320,2322,2325,2328,2331,2334,2337,2340],{"className":2321},[32],[20,2323],{"className":2324,"style":606},[36],[20,2326,687],{"className":2327},[179],[20,2329,187],{"className":2330},[41,86],[20,2332,243],{"className":2333},[242],[20,2335],{"className":2336,"style":247},[159],[20,2338,55],{"className":2339},[41,86],[20,2341,405],{"className":2342},[404],", arising because no transition in ",[20,2345,2347],{"className":2346},[23],[20,2348,2350],{"className":2349,"ariaHidden":28},[27],[20,2351,2353,2356],{"className":2352},[32],[20,2354],{"className":2355,"style":37},[36],[20,2357,44],{"className":2358,"style":43},[41,42],"\nconstrains the estimate there. Unlike ordinary approximation error, it is not\nreduced by more data of the same distribution.",[11,2361,2362],{},"Two properties make this fatal offline rather than merely inconvenient.",[11,2364,2365,2366,2369,2370,2422,2423,2513,2514,2517],{},"The first is ",[15,2367,2368],{},"selection bias toward overestimation",". Errors in ",[20,2371,2373],{"className":2372},[23],[20,2374,2376],{"className":2375,"ariaHidden":28},[27],[20,2377,2379,2382],{"className":2378},[32],[20,2380],{"className":2381,"style":1157},[36],[20,2383,2385,2388],{"className":2384},[41],[20,2386,1164],{"className":2387},[41,86],[20,2389,2391],{"className":2390},[191],[20,2392,2394,2414],{"className":2393},[195,196],[20,2395,2397,2411],{"className":2396},[200],[20,2398,2400],{"className":2399,"style":511},[204],[20,2401,2402,2405],{"style":208},[20,2403],{"className":2404,"style":213},[212],[20,2406,2408],{"className":2407},[217,218,219,220],[20,2409,1188],{"className":2410,"style":43},[41,86,220],[20,2412,229],{"className":2413},[228],[20,2415,2417],{"className":2416},[200],[20,2418,2420],{"className":2419,"style":236},[204],[20,2421],{}," are\nroughly symmetric — some out-of-distribution actions read too high, some too low —\nbut the ",[20,2424,2426],{"className":2425},[23],[20,2427,2429],{"className":2428,"ariaHidden":28},[27],[20,2430,2432,2436],{"className":2431},[32],[20,2433],{"className":2434,"style":2435},[36],"height:0.5806em;vertical-align:-0.15em;",[20,2437,2439,2445],{"className":2438},[1279],[20,2440,2442],{"className":2441},[1279],[20,2443,1352],{"className":2444},[41,1351],[20,2446,2448],{"className":2447},[191],[20,2449,2451,2505],{"className":2450},[195,196],[20,2452,2454,2502],{"className":2453},[200],[20,2455,2458],{"className":2456,"style":2457},[204],"height:0.328em;",[20,2459,2461,2464],{"style":2460},"top:-2.55em;margin-right:0.05em;",[20,2462],{"className":2463,"style":213},[212],[20,2465,2467],{"className":2466},[217,218,219,220],[20,2468,2470],{"className":2469},[41,220],[20,2471,2473,2476],{"className":2472},[41,220],[20,2474,55],{"className":2475},[41,86,220],[20,2477,2479],{"className":2478},[191],[20,2480,2482],{"className":2481},[195],[20,2483,2485],{"className":2484},[200],[20,2486,2488],{"className":2487,"style":1321},[204],[20,2489,2490,2493],{"style":1324},[20,2491],{"className":2492,"style":663},[212],[20,2494,2496],{"className":2495},[217,667,668,220],[20,2497,2499],{"className":2498},[41,220],[20,2500,388],{"className":2501},[41,220],[20,2503,229],{"className":2504},[228],[20,2506,2508],{"className":2507},[200],[20,2509,2511],{"className":2510,"style":236},[204],[20,2512],{}," (or a greedy actor) is a systematic search for the ",[47,2515,2516],{},"highest","\nvalue, so it lands on the positive errors by construction. This is the same\noverestimation that motivated Double Q-learning, but without any online experience\nto eventually reveal the truth.",[11,2519,2520,2521,2524,2525,2613,2614,2629,2630,2660,2661,2731,2732,2747],{},"The second is ",[15,2522,2523],{},"compounding with no corrective feedback",". Bootstrapping feeds the\ninflated target back in: an overestimate at ",[20,2526,2528],{"className":2527},[23],[20,2529,2531],{"className":2530,"ariaHidden":28},[27],[20,2532,2534,2537,2540,2572,2575,2578,2610],{"className":2533},[32],[20,2535],{"className":2536,"style":1858},[36],[20,2538,687],{"className":2539},[179],[20,2541,2543,2546],{"className":2542},[41],[20,2544,187],{"className":2545},[41,86],[20,2547,2549],{"className":2548},[191],[20,2550,2552],{"className":2551},[195],[20,2553,2555],{"className":2554},[200],[20,2556,2558],{"className":2557,"style":1791},[204],[20,2559,2560,2563],{"style":1812},[20,2561],{"className":2562,"style":213},[212],[20,2564,2566],{"className":2565},[217,218,219,220],[20,2567,2569],{"className":2568},[41,220],[20,2570,388],{"className":2571},[41,220],[20,2573,243],{"className":2574},[242],[20,2576],{"className":2577,"style":247},[159],[20,2579,2581,2584],{"className":2580},[41],[20,2582,55],{"className":2583},[41,86],[20,2585,2587],{"className":2586},[191],[20,2588,2590],{"className":2589},[195],[20,2591,2593],{"className":2592},[200],[20,2594,2596],{"className":2595,"style":1791},[204],[20,2597,2598,2601],{"style":1812},[20,2599],{"className":2600,"style":213},[212],[20,2602,2604],{"className":2603},[217,218,219,220],[20,2605,2607],{"className":2606},[41,220],[20,2608,388],{"className":2609},[41,220],[20,2611,405],{"className":2612},[404]," raises the target ",[20,2615,2617],{"className":2616},[23],[20,2618,2620],{"className":2619,"ariaHidden":28},[27],[20,2621,2623,2626],{"className":2622},[32],[20,2624],{"className":2625,"style":1216},[36],[20,2627,1220],{"className":2628,"style":87},[41,86]," for\n",[20,2631,2633],{"className":2632},[23],[20,2634,2636],{"className":2635,"ariaHidden":28},[27],[20,2637,2639,2642,2645,2648,2651,2654,2657],{"className":2638},[32],[20,2640],{"className":2641,"style":606},[36],[20,2643,687],{"className":2644},[179],[20,2646,187],{"className":2647},[41,86],[20,2649,243],{"className":2650},[242],[20,2652],{"className":2653,"style":247},[159],[20,2655,55],{"className":2656},[41,86],[20,2658,405],{"className":2659},[404],", which raises ",[20,2662,2664],{"className":2663},[23],[20,2665,2667],{"className":2666,"ariaHidden":28},[27],[20,2668,2670,2673,2713,2716,2719,2722,2725,2728],{"className":2669},[32],[20,2671],{"className":2672,"style":606},[36],[20,2674,2676,2679],{"className":2675},[41],[20,2677,1164],{"className":2678},[41,86],[20,2680,2682],{"className":2681},[191],[20,2683,2685,2705],{"className":2684},[195,196],[20,2686,2688,2702],{"className":2687},[200],[20,2689,2691],{"className":2690,"style":511},[204],[20,2692,2693,2696],{"style":208},[20,2694],{"className":2695,"style":213},[212],[20,2697,2699],{"className":2698},[217,218,219,220],[20,2700,1188],{"className":2701,"style":43},[41,86,220],[20,2703,229],{"className":2704},[228],[20,2706,2708],{"className":2707},[200],[20,2709,2711],{"className":2710,"style":236},[204],[20,2712],{},[20,2714,687],{"className":2715},[179],[20,2717,187],{"className":2718},[41,86],[20,2720,243],{"className":2721},[242],[20,2723],{"className":2724,"style":247},[159],[20,2726,55],{"className":2727},[41,86],[20,2729,405],{"className":2730},[404],", which raises the target for whatever\ntransitions bootstrap off ",[20,2733,2735],{"className":2734},[23],[20,2736,2738],{"className":2737,"ariaHidden":28},[27],[20,2739,2741,2744],{"className":2740},[32],[20,2742],{"className":2743,"style":82},[36],[20,2745,187],{"className":2746},[41,86],", and so on around the Bellman backup. Online, a\nrollout would eventually visit the offending action and puncture the bubble.\nOffline, nothing punctures it. The overestimation propagates and amplifies until\nthe value function is dominated by fictitious high-value actions and the induced\npolicy is garbage.",[2749,2750],"tikz-figure",{"hash":2751},"cfa00e4d47193e983391d4368cc9a9bbccbb5d040bfbd8320ef50ee8217d895d",[11,2753,2754,2755,2758,2759],{},"The diagnosis determines the fix. The value function is trustworthy ",[47,2756,2757],{},"inside"," the data\nsupport and unreliable outside it, and the standard Bellman backup does the one\nthing guaranteed to exploit the unreliable region. Every offline method below\nenforces the same rule a different way: ",[15,2760,2761],{},"do not trust, or do not query, values off\nthe data.",[2763,2764,2766],"h3",{"id":2765},"a-worked-example-how-the-bubble-inflates","A worked example: how the bubble inflates",[11,2768,2769,2770,93,2823,2875,2876,2910,2911,2945,2946,2961],{},"Take a two-state chain: states\n",[20,2771,2773],{"className":2772},[23],[20,2774,2776],{"className":2775,"ariaHidden":28},[27],[20,2777,2779,2782],{"className":2778},[32],[20,2780],{"className":2781,"style":2435},[36],[20,2783,2785,2788],{"className":2784},[41],[20,2786,187],{"className":2787},[41,86],[20,2789,2791],{"className":2790},[191],[20,2792,2794,2815],{"className":2793},[195,196],[20,2795,2797,2812],{"className":2796},[200],[20,2798,2801],{"className":2799,"style":2800},[204],"height:0.3011em;",[20,2802,2803,2806],{"style":208},[20,2804],{"className":2805,"style":213},[212],[20,2807,2809],{"className":2808},[217,218,219,220],[20,2810,2161],{"className":2811},[41,220],[20,2813,229],{"className":2814},[228],[20,2816,2818],{"className":2817},[200],[20,2819,2821],{"className":2820,"style":236},[204],[20,2822],{},[20,2824,2826],{"className":2825},[23],[20,2827,2829],{"className":2828,"ariaHidden":28},[27],[20,2830,2832,2835],{"className":2831},[32],[20,2833],{"className":2834,"style":2435},[36],[20,2836,2838,2841],{"className":2837},[41],[20,2839,187],{"className":2840},[41,86],[20,2842,2844],{"className":2843},[191],[20,2845,2847,2867],{"className":2846},[195,196],[20,2848,2850,2864],{"className":2849},[200],[20,2851,2853],{"className":2852,"style":2800},[204],[20,2854,2855,2858],{"style":208},[20,2856],{"className":2857,"style":213},[212],[20,2859,2861],{"className":2860},[217,218,219,220],[20,2862,61],{"className":2863},[41,220],[20,2865,229],{"className":2866},[228],[20,2868,2870],{"className":2869},[200],[20,2871,2873],{"className":2872,"style":236},[204],[20,2874],{},", with two actions ",[20,2877,2879],{"className":2878},[23],[20,2880,2882],{"className":2881,"ariaHidden":28},[27],[20,2883,2885,2888,2892,2896,2899,2902,2907],{"className":2884},[32],[20,2886],{"className":2887,"style":606},[36],[20,2889,2891],{"className":2890},[179],"{",[20,2893,2895],{"className":2894},[41,86],"L",[20,2897,243],{"className":2898},[242],[20,2900],{"className":2901,"style":247},[159],[20,2903,2906],{"className":2904,"style":2905},[41,86],"margin-right:0.0077em;","R",[20,2908,412],{"className":2909},[404],", discount ",[20,2912,2914],{"className":2913},[23],[20,2915,2917,2935],{"className":2916,"ariaHidden":28},[27],[20,2918,2920,2923,2926,2929,2932],{"className":2919},[32],[20,2921],{"className":2922,"style":1216},[36],[20,2924,1269],{"className":2925,"style":1268},[41,86],[20,2927],{"className":2928,"style":160},[159],[20,2930,165],{"className":2931},[164],[20,2933],{"className":2934,"style":160},[159],[20,2936,2938,2941],{"className":2937},[32],[20,2939],{"className":2940,"style":2157},[36],[20,2942,2944],{"className":2943},[41],"0.9",", and a\ntiny logged dataset that only ever took ",[20,2947,2949],{"className":2948},[23],[20,2950,2952],{"className":2951,"ariaHidden":28},[27],[20,2953,2955,2958],{"className":2954},[32],[20,2956],{"className":2957,"style":37},[36],[20,2959,2906],{"className":2960,"style":2905},[41,86],":",[20,2963,2965],{"className":2964},[140],[20,2966,2968],{"className":2967},[23],[20,2969,2971,2989],{"className":2970,"ariaHidden":28},[27],[20,2972,2974,2977,2980,2983,2986],{"className":2973},[32],[20,2975],{"className":2976,"style":37},[36],[20,2978,44],{"className":2979,"style":43},[41,42],[20,2981],{"className":2982,"style":160},[159],[20,2984,165],{"className":2985},[164],[20,2987],{"className":2988,"style":160},[159],[20,2990,2992,2995,3001,3004,3007,3047,3050,3053,3056,3059,3062,3065,3068,3071,3111,3114,3117,3121,3124,3127,3167,3170,3173,3176,3179,3182,3185,3188,3191,3231,3234,3237,3243],{"className":2991},[32],[20,2993],{"className":2994,"style":1587},[36],[20,2996,2998],{"className":2997},[179],[20,2999,2891],{"className":3000},[1671,668],[20,3002],{"className":3003,"style":247},[159],[20,3005,687],{"className":3006},[179],[20,3008,3010,3013],{"className":3009},[41],[20,3011,187],{"className":3012},[41,86],[20,3014,3016],{"className":3015},[191],[20,3017,3019,3039],{"className":3018},[195,196],[20,3020,3022,3036],{"className":3021},[200],[20,3023,3025],{"className":3024,"style":2800},[204],[20,3026,3027,3030],{"style":208},[20,3028],{"className":3029,"style":213},[212],[20,3031,3033],{"className":3032},[217,218,219,220],[20,3034,2161],{"className":3035},[41,220],[20,3037,229],{"className":3038},[228],[20,3040,3042],{"className":3041},[200],[20,3043,3045],{"className":3044,"style":236},[204],[20,3046],{},[20,3048,243],{"className":3049},[242],[20,3051],{"className":3052,"style":247},[159],[20,3054,2906],{"className":3055,"style":2905},[41,86],[20,3057,243],{"className":3058},[242],[20,3060],{"className":3061,"style":247},[159],[20,3063,2161],{"className":3064},[41],[20,3066,243],{"className":3067},[242],[20,3069],{"className":3070,"style":247},[159],[20,3072,3074,3077],{"className":3073},[41],[20,3075,187],{"className":3076},[41,86],[20,3078,3080],{"className":3079},[191],[20,3081,3083,3103],{"className":3082},[195,196],[20,3084,3086,3100],{"className":3085},[200],[20,3087,3089],{"className":3088,"style":2800},[204],[20,3090,3091,3094],{"style":208},[20,3092],{"className":3093,"style":213},[212],[20,3095,3097],{"className":3096},[217,218,219,220],[20,3098,61],{"className":3099},[41,220],[20,3101,229],{"className":3102},[228],[20,3104,3106],{"className":3105},[200],[20,3107,3109],{"className":3108,"style":236},[204],[20,3110],{},[20,3112,405],{"className":3113},[404],[20,3115,243],{"className":3116},[242],[20,3118,3120],{"className":3119},[159]," ",[20,3122],{"className":3123,"style":247},[159],[20,3125,687],{"className":3126},[179],[20,3128,3130,3133],{"className":3129},[41],[20,3131,187],{"className":3132},[41,86],[20,3134,3136],{"className":3135},[191],[20,3137,3139,3159],{"className":3138},[195,196],[20,3140,3142,3156],{"className":3141},[200],[20,3143,3145],{"className":3144,"style":2800},[204],[20,3146,3147,3150],{"style":208},[20,3148],{"className":3149,"style":213},[212],[20,3151,3153],{"className":3152},[217,218,219,220],[20,3154,61],{"className":3155},[41,220],[20,3157,229],{"className":3158},[228],[20,3160,3162],{"className":3161},[200],[20,3163,3165],{"className":3164,"style":236},[204],[20,3166],{},[20,3168,243],{"className":3169},[242],[20,3171],{"className":3172,"style":247},[159],[20,3174,2906],{"className":3175,"style":2905},[41,86],[20,3177,243],{"className":3178},[242],[20,3180],{"className":3181,"style":247},[159],[20,3183,61],{"className":3184},[41],[20,3186,243],{"className":3187},[242],[20,3189],{"className":3190,"style":247},[159],[20,3192,3194,3197],{"className":3193},[41],[20,3195,187],{"className":3196},[41,86],[20,3198,3200],{"className":3199},[191],[20,3201,3203,3223],{"className":3202},[195,196],[20,3204,3206,3220],{"className":3205},[200],[20,3207,3209],{"className":3208,"style":2800},[204],[20,3210,3211,3214],{"style":208},[20,3212],{"className":3213,"style":213},[212],[20,3215,3217],{"className":3216},[217,218,219,220],[20,3218,61],{"className":3219},[41,220],[20,3221,229],{"className":3222},[228],[20,3224,3226],{"className":3225},[200],[20,3227,3229],{"className":3228,"style":236},[204],[20,3230],{},[20,3232,405],{"className":3233},[404],[20,3235],{"className":3236,"style":247},[159],[20,3238,3240],{"className":3239},[404],[20,3241,412],{"className":3242},[1671,668],[20,3244,128],{"className":3245},[41],[11,3247,3248,3249,3264,3265,3280,3281,3296,3297,3376,3377,3392,3393,93,3555,3708],{},"Action ",[20,3250,3252],{"className":3251},[23],[20,3253,3255],{"className":3254,"ariaHidden":28},[27],[20,3256,3258,3261],{"className":3257},[32],[20,3259],{"className":3260,"style":37},[36],[20,3262,2895],{"className":3263},[41,86]," is never logged from either state. Suppose the true environment is dull:\n",[20,3266,3268],{"className":3267},[23],[20,3269,3271],{"className":3270,"ariaHidden":28},[27],[20,3272,3274,3277],{"className":3273},[32],[20,3275],{"className":3276,"style":37},[36],[20,3278,2895],{"className":3279},[41,86]," from anywhere gives reward ",[20,3282,3284],{"className":3283},[23],[20,3285,3287],{"className":3286,"ariaHidden":28},[27],[20,3288,3290,3293],{"className":3289},[32],[20,3291],{"className":3292,"style":2157},[36],[20,3294,2161],{"className":3295},[41]," and self-loops, so ",[20,3298,3300],{"className":3299},[23],[20,3301,3303,3367],{"className":3302,"ariaHidden":28},[27],[20,3304,3306,3309,3340,3343,3346,3349,3352,3355,3358,3361,3364],{"className":3305},[32],[20,3307],{"className":3308,"style":606},[36],[20,3310,3312,3315],{"className":3311},[41],[20,3313,1164],{"className":3314},[41,86],[20,3316,3318],{"className":3317},[191],[20,3319,3321],{"className":3320},[195],[20,3322,3324],{"className":3323},[200],[20,3325,3328],{"className":3326,"style":3327},[204],"height:0.6887em;",[20,3329,3330,3333],{"style":1812},[20,3331],{"className":3332,"style":213},[212],[20,3334,3336],{"className":3335},[217,218,219,220],[20,3337,3339],{"className":3338},[1253,220],"∗",[20,3341,687],{"className":3342},[179],[20,3344,187],{"className":3345},[41,86],[20,3347,243],{"className":3348},[242],[20,3350],{"className":3351,"style":247},[159],[20,3353,2895],{"className":3354},[41,86],[20,3356,405],{"className":3357},[404],[20,3359],{"className":3360,"style":160},[159],[20,3362,165],{"className":3363},[164],[20,3365],{"className":3366,"style":160},[159],[20,3368,3370,3373],{"className":3369},[32],[20,3371],{"className":3372,"style":2157},[36],[20,3374,2161],{"className":3375},[41],", while ",[20,3378,3380],{"className":3379},[23],[20,3381,3383],{"className":3382,"ariaHidden":28},[27],[20,3384,3386,3389],{"className":3385},[32],[20,3387],{"className":3388,"style":37},[36],[20,3390,2906],{"className":3391,"style":2905},[41,86],"\nearns the returns ",[20,3394,3396],{"className":3395},[23],[20,3397,3399,3498,3524,3545],{"className":3398,"ariaHidden":28},[27],[20,3400,3402,3405,3434,3437,3477,3480,3483,3486,3489,3492,3495],{"className":3401},[32],[20,3403],{"className":3404,"style":606},[36],[20,3406,3408,3411],{"className":3407},[41],[20,3409,1164],{"className":3410},[41,86],[20,3412,3414],{"className":3413},[191],[20,3415,3417],{"className":3416},[195],[20,3418,3420],{"className":3419},[200],[20,3421,3423],{"className":3422,"style":3327},[204],[20,3424,3425,3428],{"style":1812},[20,3426],{"className":3427,"style":213},[212],[20,3429,3431],{"className":3430},[217,218,219,220],[20,3432,3339],{"className":3433},[1253,220],[20,3435,687],{"className":3436},[179],[20,3438,3440,3443],{"className":3439},[41],[20,3441,187],{"className":3442},[41,86],[20,3444,3446],{"className":3445},[191],[20,3447,3449,3469],{"className":3448},[195,196],[20,3450,3452,3466],{"className":3451},[200],[20,3453,3455],{"className":3454,"style":2800},[204],[20,3456,3457,3460],{"style":208},[20,3458],{"className":3459,"style":213},[212],[20,3461,3463],{"className":3462},[217,218,219,220],[20,3464,61],{"className":3465},[41,220],[20,3467,229],{"className":3468},[228],[20,3470,3472],{"className":3471},[200],[20,3473,3475],{"className":3474,"style":236},[204],[20,3476],{},[20,3478,243],{"className":3479},[242],[20,3481],{"className":3482,"style":247},[159],[20,3484,2906],{"className":3485,"style":2905},[41,86],[20,3487,405],{"className":3488},[404],[20,3490],{"className":3491,"style":160},[159],[20,3493,165],{"className":3494},[164],[20,3496],{"className":3497,"style":160},[159],[20,3499,3501,3504,3508,3511,3514,3517,3521],{"className":3500},[32],[20,3502],{"className":3503,"style":606},[36],[20,3505,3507],{"className":3506},[41],"1\u002F",[20,3509,687],{"className":3510},[179],[20,3512,61],{"className":3513},[41],[20,3515],{"className":3516,"style":1249},[159],[20,3518,3520],{"className":3519},[1253],"−",[20,3522],{"className":3523,"style":1249},[159],[20,3525,3527,3530,3533,3536,3539,3542],{"className":3526},[32],[20,3528],{"className":3529,"style":606},[36],[20,3531,1269],{"className":3532,"style":1268},[41,86],[20,3534,405],{"className":3535},[404],[20,3537],{"className":3538,"style":160},[159],[20,3540,165],{"className":3541},[164],[20,3543],{"className":3544,"style":160},[159],[20,3546,3548,3551],{"className":3547},[32],[20,3549],{"className":3550,"style":2157},[36],[20,3552,3554],{"className":3553},[41],"10",[20,3556,3558],{"className":3557},[23],[20,3559,3561,3660,3680,3698],{"className":3560,"ariaHidden":28},[27],[20,3562,3564,3567,3596,3599,3639,3642,3645,3648,3651,3654,3657],{"className":3563},[32],[20,3565],{"className":3566,"style":606},[36],[20,3568,3570,3573],{"className":3569},[41],[20,3571,1164],{"className":3572},[41,86],[20,3574,3576],{"className":3575},[191],[20,3577,3579],{"className":3578},[195],[20,3580,3582],{"className":3581},[200],[20,3583,3585],{"className":3584,"style":3327},[204],[20,3586,3587,3590],{"style":1812},[20,3588],{"className":3589,"style":213},[212],[20,3591,3593],{"className":3592},[217,218,219,220],[20,3594,3339],{"className":3595},[1253,220],[20,3597,687],{"className":3598},[179],[20,3600,3602,3605],{"className":3601},[41],[20,3603,187],{"className":3604},[41,86],[20,3606,3608],{"className":3607},[191],[20,3609,3611,3631],{"className":3610},[195,196],[20,3612,3614,3628],{"className":3613},[200],[20,3615,3617],{"className":3616,"style":2800},[204],[20,3618,3619,3622],{"style":208},[20,3620],{"className":3621,"style":213},[212],[20,3623,3625],{"className":3624},[217,218,219,220],[20,3626,2161],{"className":3627},[41,220],[20,3629,229],{"className":3630},[228],[20,3632,3634],{"className":3633},[200],[20,3635,3637],{"className":3636,"style":236},[204],[20,3638],{},[20,3640,243],{"className":3641},[242],[20,3643],{"className":3644,"style":247},[159],[20,3646,2906],{"className":3647,"style":2905},[41,86],[20,3649,405],{"className":3650},[404],[20,3652],{"className":3653,"style":160},[159],[20,3655,165],{"className":3656},[164],[20,3658],{"className":3659,"style":160},[159],[20,3661,3663,3667,3670,3673,3677],{"className":3662},[32],[20,3664],{"className":3665,"style":3666},[36],"height:0.6389em;vertical-align:-0.1944em;",[20,3668,1269],{"className":3669,"style":1268},[41,86],[20,3671],{"className":3672,"style":1249},[159],[20,3674,3676],{"className":3675},[1253],"⋅",[20,3678],{"className":3679,"style":1249},[159],[20,3681,3683,3686,3689,3692,3695],{"className":3682},[32],[20,3684],{"className":3685,"style":2157},[36],[20,3687,3554],{"className":3688},[41],[20,3690],{"className":3691,"style":160},[159],[20,3693,165],{"className":3694},[164],[20,3696],{"className":3697,"style":160},[159],[20,3699,3701,3704],{"className":3700},[32],[20,3702],{"className":3703,"style":2157},[36],[20,3705,3707],{"className":3706},[41],"9",". A well-behaved learner should recover exactly these.",[11,3710,3711,3712,3838,3839,3961],{},"Now initialize a network whose generalization happens to read the unseen action\noptimistically: say it predicts ",[20,3713,3715],{"className":3714},[23],[20,3716,3718,3828],{"className":3717,"ariaHidden":28},[27],[20,3719,3721,3724,3764,3767,3807,3810,3813,3816,3819,3822,3825],{"className":3720},[32],[20,3722],{"className":3723,"style":606},[36],[20,3725,3727,3730],{"className":3726},[41],[20,3728,1164],{"className":3729},[41,86],[20,3731,3733],{"className":3732},[191],[20,3734,3736,3756],{"className":3735},[195,196],[20,3737,3739,3753],{"className":3738},[200],[20,3740,3742],{"className":3741,"style":511},[204],[20,3743,3744,3747],{"style":208},[20,3745],{"className":3746,"style":213},[212],[20,3748,3750],{"className":3749},[217,218,219,220],[20,3751,1188],{"className":3752,"style":43},[41,86,220],[20,3754,229],{"className":3755},[228],[20,3757,3759],{"className":3758},[200],[20,3760,3762],{"className":3761,"style":236},[204],[20,3763],{},[20,3765,687],{"className":3766},[179],[20,3768,3770,3773],{"className":3769},[41],[20,3771,187],{"className":3772},[41,86],[20,3774,3776],{"className":3775},[191],[20,3777,3779,3799],{"className":3778},[195,196],[20,3780,3782,3796],{"className":3781},[200],[20,3783,3785],{"className":3784,"style":2800},[204],[20,3786,3787,3790],{"style":208},[20,3788],{"className":3789,"style":213},[212],[20,3791,3793],{"className":3792},[217,218,219,220],[20,3794,61],{"className":3795},[41,220],[20,3797,229],{"className":3798},[228],[20,3800,3802],{"className":3801},[200],[20,3803,3805],{"className":3804,"style":236},[204],[20,3806],{},[20,3808,243],{"className":3809},[242],[20,3811],{"className":3812,"style":247},[159],[20,3814,2895],{"className":3815},[41,86],[20,3817,405],{"className":3818},[404],[20,3820],{"className":3821,"style":160},[159],[20,3823,165],{"className":3824},[164],[20,3826],{"className":3827,"style":160},[159],[20,3829,3831,3834],{"className":3830},[32],[20,3832],{"className":3833,"style":2157},[36],[20,3835,3837],{"className":3836},[41],"3"," before any training (a pure\nextrapolation, off-support, positive by bad luck). Run one Bellman backup on the\nlogged ",[20,3840,3842],{"className":3841},[23],[20,3843,3845],{"className":3844,"ariaHidden":28},[27],[20,3846,3848,3851,3854,3894,3897,3900,3903,3906,3909,3912,3915,3918,3958],{"className":3847},[32],[20,3849],{"className":3850,"style":606},[36],[20,3852,687],{"className":3853},[179],[20,3855,3857,3860],{"className":3856},[41],[20,3858,187],{"className":3859},[41,86],[20,3861,3863],{"className":3862},[191],[20,3864,3866,3886],{"className":3865},[195,196],[20,3867,3869,3883],{"className":3868},[200],[20,3870,3872],{"className":3871,"style":2800},[204],[20,3873,3874,3877],{"style":208},[20,3875],{"className":3876,"style":213},[212],[20,3878,3880],{"className":3879},[217,218,219,220],[20,3881,61],{"className":3882},[41,220],[20,3884,229],{"className":3885},[228],[20,3887,3889],{"className":3888},[200],[20,3890,3892],{"className":3891,"style":236},[204],[20,3893],{},[20,3895,243],{"className":3896},[242],[20,3898],{"className":3899,"style":247},[159],[20,3901,2906],{"className":3902,"style":2905},[41,86],[20,3904,243],{"className":3905},[242],[20,3907],{"className":3908,"style":247},[159],[20,3910,61],{"className":3911},[41],[20,3913,243],{"className":3914},[242],[20,3916],{"className":3917,"style":247},[159],[20,3919,3921,3924],{"className":3920},[41],[20,3922,187],{"className":3923},[41,86],[20,3925,3927],{"className":3926},[191],[20,3928,3930,3950],{"className":3929},[195,196],[20,3931,3933,3947],{"className":3932},[200],[20,3934,3936],{"className":3935,"style":2800},[204],[20,3937,3938,3941],{"style":208},[20,3939],{"className":3940,"style":213},[212],[20,3942,3944],{"className":3943},[217,218,219,220],[20,3945,61],{"className":3946},[41,220],[20,3948,229],{"className":3949},[228],[20,3951,3953],{"className":3952},[200],[20,3954,3956],{"className":3955,"style":236},[204],[20,3957],{},[20,3959,405],{"className":3960},[404]," transition with a greedy max target:",[20,3963,3965],{"className":3964},[140],[20,3966,3968],{"className":3967},[23],[20,3969,3971,3989,4007,4234,4253,4271,4739,4757,4775,4793],{"className":3970,"ariaHidden":28},[27],[20,3972,3974,3977,3980,3983,3986],{"className":3973},[32],[20,3975],{"className":3976,"style":1216},[36],[20,3978,1220],{"className":3979,"style":87},[41,86],[20,3981],{"className":3982,"style":160},[159],[20,3984,165],{"className":3985},[164],[20,3987],{"className":3988,"style":160},[159],[20,3990,3992,3995,3998,4001,4004],{"className":3991},[32],[20,3993],{"className":3994,"style":1242},[36],[20,3996,300],{"className":3997,"style":43},[41,86],[20,3999],{"className":4000,"style":1249},[159],[20,4002,1254],{"className":4003},[1253],[20,4005],{"className":4006,"style":1249},[159],[20,4008,4010,4013,4016,4019,4098,4101,4141,4144,4184,4187,4190,4222,4225,4228,4231],{"className":4009},[32],[20,4011],{"className":4012,"style":1264},[36],[20,4014,1269],{"className":4015,"style":1268},[41,86],[20,4017],{"className":4018,"style":247},[159],[20,4020,4022],{"className":4021},[1279,1280],[20,4023,4025,4090],{"className":4024},[195,196],[20,4026,4028,4087],{"className":4027},[200],[20,4029,4031,4074],{"className":4030,"style":82},[204],[20,4032,4033,4036],{"style":1292},[20,4034],{"className":4035,"style":1296},[212],[20,4037,4039],{"className":4038},[217,218,219,220],[20,4040,4042],{"className":4041},[41,220],[20,4043,4045,4048],{"className":4044},[41,220],[20,4046,55],{"className":4047},[41,86,220],[20,4049,4051],{"className":4050},[191],[20,4052,4054],{"className":4053},[195],[20,4055,4057],{"className":4056},[200],[20,4058,4060],{"className":4059,"style":1321},[204],[20,4061,4062,4065],{"style":1324},[20,4063],{"className":4064,"style":663},[212],[20,4066,4068],{"className":4067},[217,667,668,220],[20,4069,4071],{"className":4070},[41,220],[20,4072,388],{"className":4073},[41,220],[20,4075,4076,4079],{"style":1339},[20,4077],{"className":4078,"style":1296},[212],[20,4080,4081],{},[20,4082,4084],{"className":4083},[1279],[20,4085,1352],{"className":4086},[41,1351],[20,4088,229],{"className":4089},[228],[20,4091,4093],{"className":4092},[200],[20,4094,4096],{"className":4095,"style":1362},[204],[20,4097],{},[20,4099],{"className":4100,"style":247},[159],[20,4102,4104,4107],{"className":4103},[41],[20,4105,1164],{"className":4106},[41,86],[20,4108,4110],{"className":4109},[191],[20,4111,4113,4133],{"className":4112},[195,196],[20,4114,4116,4130],{"className":4115},[200],[20,4117,4119],{"className":4118,"style":511},[204],[20,4120,4121,4124],{"style":208},[20,4122],{"className":4123,"style":213},[212],[20,4125,4127],{"className":4126},[217,218,219,220],[20,4128,1188],{"className":4129,"style":43},[41,86,220],[20,4131,229],{"className":4132},[228],[20,4134,4136],{"className":4135},[200],[20,4137,4139],{"className":4138,"style":236},[204],[20,4140],{},[20,4142,687],{"className":4143},[179],[20,4145,4147,4150],{"className":4146},[41],[20,4148,187],{"className":4149},[41,86],[20,4151,4153],{"className":4152},[191],[20,4154,4156,4176],{"className":4155},[195,196],[20,4157,4159,4173],{"className":4158},[200],[20,4160,4162],{"className":4161,"style":2800},[204],[20,4163,4164,4167],{"style":208},[20,4165],{"className":4166,"style":213},[212],[20,4168,4170],{"className":4169},[217,218,219,220],[20,4171,61],{"className":4172},[41,220],[20,4174,229],{"className":4175},[228],[20,4177,4179],{"className":4178},[200],[20,4180,4182],{"className":4181,"style":236},[204],[20,4183],{},[20,4185,243],{"className":4186},[242],[20,4188],{"className":4189,"style":247},[159],[20,4191,4193,4196],{"className":4192},[41],[20,4194,55],{"className":4195},[41,86],[20,4197,4199],{"className":4198},[191],[20,4200,4202],{"className":4201},[195],[20,4203,4205],{"className":4204},[200],[20,4206,4208],{"className":4207,"style":360},[204],[20,4209,4210,4213],{"style":375},[20,4211],{"className":4212,"style":213},[212],[20,4214,4216],{"className":4215},[217,218,219,220],[20,4217,4219],{"className":4218},[41,220],[20,4220,388],{"className":4221},[41,220],[20,4223,405],{"className":4224},[404],[20,4226],{"className":4227,"style":160},[159],[20,4229,165],{"className":4230},[164],[20,4232],{"className":4233,"style":160},[159],[20,4235,4237,4241,4244,4247,4250],{"className":4236},[32],[20,4238],{"className":4239,"style":4240},[36],"height:0.7278em;vertical-align:-0.0833em;",[20,4242,61],{"className":4243},[41],[20,4245],{"className":4246,"style":1249},[159],[20,4248,1254],{"className":4249},[1253],[20,4251],{"className":4252,"style":1249},[159],[20,4254,4256,4259,4262,4265,4268],{"className":4255},[32],[20,4257],{"className":4258,"style":2157},[36],[20,4260,2944],{"className":4261},[41],[20,4263],{"className":4264,"style":1249},[159],[20,4266,3676],{"className":4267},[1253],[20,4269],{"className":4270,"style":1249},[159],[20,4272,4274,4278,4284,4287,4514,4517,4520,4523,4526,4727,4730,4733,4736],{"className":4273},[32],[20,4275],{"className":4276,"style":4277},[36],"height:2.2991em;vertical-align:-1.5491em;",[20,4279,4281],{"className":4280},[1279],[20,4282,1352],{"className":4283},[41,1351],[20,4285,687],{"className":4286},[179],[20,4288,4292],{"className":4289},[4290,4291],"minner","munder",[20,4293,4295,4505],{"className":4294},[195,196],[20,4296,4298,4502],{"className":4297},[200],[20,4299,4302,4320],{"className":4300,"style":4301},[204],"height:0.75em;",[20,4303,4305,4308],{"style":4304},"top:-1.4509em;",[20,4306],{"className":4307,"style":1296},[212],[20,4309,4311],{"className":4310},[217,218,219,220],[20,4312,4314,4317],{"className":4313},[41,220],[20,4315,2147],{"className":4316},[164,220],[20,4318,61],{"className":4319},[41,220],[20,4321,4322,4325],{"style":1339},[20,4323],{"className":4324,"style":1296},[212],[20,4326,4328],{"className":4327},[4290,4291],[20,4329,4331,4493],{"className":4330},[195,196],[20,4332,4334,4490],{"className":4333},[200],[20,4335,4337,4387],{"className":4336,"style":4301},[204],[20,4338,4342,4345],{"className":4339,"style":4341},[4340],"svg-align","top:-2.102em;",[20,4343],{"className":4344,"style":1296},[212],[20,4346,4350,4367,4377],{"className":4347,"style":4349},[4348],"stretchy","height:0.548em;min-width:1.6em;",[20,4351,4355],{"className":4352,"style":4354},[4353],"brace-left","height:0.548em;",[4356,4357,4363],"svg",{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4362},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 0 400000 548","xMinYMin slice",[4364,4365],"path",{"d":4366},"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",[20,4368,4371],{"className":4369,"style":4354},[4370],"brace-center",[4356,4372,4374],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4373},"xMidYMin slice",[4364,4375],{"d":4376},"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",[20,4378,4381],{"className":4379,"style":4354},[4380],"brace-right",[4356,4382,4384],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4383},"xMaxYMin slice",[4364,4385],{"d":4386},"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",[20,4388,4389,4392],{"style":1339},[20,4390],{"className":4391,"style":1296},[212],[20,4393,4395,4435,4438,4478,4481,4484,4487],{"className":4394},[41],[20,4396,4398,4401],{"className":4397},[41],[20,4399,1164],{"className":4400},[41,86],[20,4402,4404],{"className":4403},[191],[20,4405,4407,4427],{"className":4406},[195,196],[20,4408,4410,4424],{"className":4409},[200],[20,4411,4413],{"className":4412,"style":511},[204],[20,4414,4415,4418],{"style":208},[20,4416],{"className":4417,"style":213},[212],[20,4419,4421],{"className":4420},[217,218,219,220],[20,4422,1188],{"className":4423,"style":43},[41,86,220],[20,4425,229],{"className":4426},[228],[20,4428,4430],{"className":4429},[200],[20,4431,4433],{"className":4432,"style":236},[204],[20,4434],{},[20,4436,687],{"className":4437},[179],[20,4439,4441,4444],{"className":4440},[41],[20,4442,187],{"className":4443},[41,86],[20,4445,4447],{"className":4446},[191],[20,4448,4450,4470],{"className":4449},[195,196],[20,4451,4453,4467],{"className":4452},[200],[20,4454,4456],{"className":4455,"style":2800},[204],[20,4457,4458,4461],{"style":208},[20,4459],{"className":4460,"style":213},[212],[20,4462,4464],{"className":4463},[217,218,219,220],[20,4465,61],{"className":4466},[41,220],[20,4468,229],{"className":4469},[228],[20,4471,4473],{"className":4472},[200],[20,4474,4476],{"className":4475,"style":236},[204],[20,4477],{},[20,4479,243],{"className":4480},[242],[20,4482],{"className":4483,"style":247},[159],[20,4485,2906],{"className":4486,"style":2905},[41,86],[20,4488,405],{"className":4489},[404],[20,4491,229],{"className":4492},[228],[20,4494,4496],{"className":4495},[200],[20,4497,4500],{"className":4498,"style":4499},[204],"height:0.898em;",[20,4501],{},[20,4503,229],{"className":4504},[228],[20,4506,4508],{"className":4507},[200],[20,4509,4512],{"className":4510,"style":4511},[204],"height:1.5491em;",[20,4513],{},[20,4515],{"className":4516,"style":247},[159],[20,4518,243],{"className":4519},[242],[20,4521,3120],{"className":4522},[159],[20,4524],{"className":4525,"style":247},[159],[20,4527,4529],{"className":4528},[4290,4291],[20,4530,4532,4719],{"className":4531},[195,196],[20,4533,4535,4716],{"className":4534},[200],[20,4536,4538,4555],{"className":4537,"style":4301},[204],[20,4539,4540,4543],{"style":4304},[20,4541],{"className":4542,"style":1296},[212],[20,4544,4546],{"className":4545},[217,218,219,220],[20,4547,4549,4552],{"className":4548},[41,220],[20,4550,165],{"className":4551},[164,220],[20,4553,3837],{"className":4554},[41,220],[20,4556,4557,4560],{"style":1339},[20,4558],{"className":4559,"style":1296},[212],[20,4561,4563],{"className":4562},[4290,4291],[20,4564,4566,4708],{"className":4565},[195,196],[20,4567,4569,4705],{"className":4568},[200],[20,4570,4572,4602],{"className":4571,"style":4301},[204],[20,4573,4575,4578],{"className":4574,"style":4341},[4340],[20,4576],{"className":4577,"style":1296},[212],[20,4579,4581,4588,4595],{"className":4580,"style":4349},[4348],[20,4582,4584],{"className":4583,"style":4354},[4353],[4356,4585,4586],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4362},[4364,4587],{"d":4366},[20,4589,4591],{"className":4590,"style":4354},[4370],[4356,4592,4593],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4373},[4364,4594],{"d":4376},[20,4596,4598],{"className":4597,"style":4354},[4380],[4356,4599,4600],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4383},[4364,4601],{"d":4386},[20,4603,4604,4607],{"style":1339},[20,4605],{"className":4606,"style":1296},[212],[20,4608,4610,4650,4653,4693,4696,4699,4702],{"className":4609},[41],[20,4611,4613,4616],{"className":4612},[41],[20,4614,1164],{"className":4615},[41,86],[20,4617,4619],{"className":4618},[191],[20,4620,4622,4642],{"className":4621},[195,196],[20,4623,4625,4639],{"className":4624},[200],[20,4626,4628],{"className":4627,"style":511},[204],[20,4629,4630,4633],{"style":208},[20,4631],{"className":4632,"style":213},[212],[20,4634,4636],{"className":4635},[217,218,219,220],[20,4637,1188],{"className":4638,"style":43},[41,86,220],[20,4640,229],{"className":4641},[228],[20,4643,4645],{"className":4644},[200],[20,4646,4648],{"className":4647,"style":236},[204],[20,4649],{},[20,4651,687],{"className":4652},[179],[20,4654,4656,4659],{"className":4655},[41],[20,4657,187],{"className":4658},[41,86],[20,4660,4662],{"className":4661},[191],[20,4663,4665,4685],{"className":4664},[195,196],[20,4666,4668,4682],{"className":4667},[200],[20,4669,4671],{"className":4670,"style":2800},[204],[20,4672,4673,4676],{"style":208},[20,4674],{"className":4675,"style":213},[212],[20,4677,4679],{"className":4678},[217,218,219,220],[20,4680,61],{"className":4681},[41,220],[20,4683,229],{"className":4684},[228],[20,4686,4688],{"className":4687},[200],[20,4689,4691],{"className":4690,"style":236},[204],[20,4692],{},[20,4694,243],{"className":4695},[242],[20,4697],{"className":4698,"style":247},[159],[20,4700,2895],{"className":4701},[41,86],[20,4703,405],{"className":4704},[404],[20,4706,229],{"className":4707},[228],[20,4709,4711],{"className":4710},[200],[20,4712,4714],{"className":4713,"style":4499},[204],[20,4715],{},[20,4717,229],{"className":4718},[228],[20,4720,4722],{"className":4721},[200],[20,4723,4725],{"className":4724,"style":4511},[204],[20,4726],{},[20,4728,405],{"className":4729},[404],[20,4731],{"className":4732,"style":160},[159],[20,4734,165],{"className":4735},[164],[20,4737],{"className":4738,"style":160},[159],[20,4740,4742,4745,4748,4751,4754],{"className":4741},[32],[20,4743],{"className":4744,"style":4240},[36],[20,4746,61],{"className":4747},[41],[20,4749],{"className":4750,"style":1249},[159],[20,4752,1254],{"className":4753},[1253],[20,4755],{"className":4756,"style":1249},[159],[20,4758,4760,4763,4766,4769,4772],{"className":4759},[32],[20,4761],{"className":4762,"style":2157},[36],[20,4764,2944],{"className":4765},[41],[20,4767],{"className":4768,"style":1249},[159],[20,4770,3676],{"className":4771},[1253],[20,4773],{"className":4774,"style":1249},[159],[20,4776,4778,4781,4784,4787,4790],{"className":4777},[32],[20,4779],{"className":4780,"style":2157},[36],[20,4782,3837],{"className":4783},[41],[20,4785],{"className":4786,"style":160},[159],[20,4788,165],{"className":4789},[164],[20,4791],{"className":4792,"style":160},[159],[20,4794,4796,4799],{"className":4795},[32],[20,4797],{"className":4798,"style":2157},[36],[20,4800,4802],{"className":4801},[41],"3.7.",[11,4804,4805,4806,4809,4810,4825,4826,4829,4830,4937,4938,4954,4955,5077,5078,5310,5311,5326,5327,5508,5509,5616,5617,5669,5670,5685,5686,5701,5702,5717,5718,5733,5734,5801,5802,5817,5818,5833,5834,5852],{},"The max reached ",[47,4807,4808],{},"off the data"," and pulled the fictitious ",[20,4811,4813],{"className":4812},[23],[20,4814,4816],{"className":4815,"ariaHidden":28},[27],[20,4817,4819,4822],{"className":4818},[32],[20,4820],{"className":4821,"style":2157},[36],[20,4823,3837],{"className":4824},[41]," into the target for a\n",[47,4827,4828],{},"logged"," transition. ",[20,4831,4833],{"className":4832},[23],[20,4834,4836],{"className":4835,"ariaHidden":28},[27],[20,4837,4839,4842,4882,4885,4925,4928,4931,4934],{"className":4838},[32],[20,4840],{"className":4841,"style":606},[36],[20,4843,4845,4848],{"className":4844},[41],[20,4846,1164],{"className":4847},[41,86],[20,4849,4851],{"className":4850},[191],[20,4852,4854,4874],{"className":4853},[195,196],[20,4855,4857,4871],{"className":4856},[200],[20,4858,4860],{"className":4859,"style":511},[204],[20,4861,4862,4865],{"style":208},[20,4863],{"className":4864,"style":213},[212],[20,4866,4868],{"className":4867},[217,218,219,220],[20,4869,1188],{"className":4870,"style":43},[41,86,220],[20,4872,229],{"className":4873},[228],[20,4875,4877],{"className":4876},[200],[20,4878,4880],{"className":4879,"style":236},[204],[20,4881],{},[20,4883,687],{"className":4884},[179],[20,4886,4888,4891],{"className":4887},[41],[20,4889,187],{"className":4890},[41,86],[20,4892,4894],{"className":4893},[191],[20,4895,4897,4917],{"className":4896},[195,196],[20,4898,4900,4914],{"className":4899},[200],[20,4901,4903],{"className":4902,"style":2800},[204],[20,4904,4905,4908],{"style":208},[20,4906],{"className":4907,"style":213},[212],[20,4909,4911],{"className":4910},[217,218,219,220],[20,4912,61],{"className":4913},[41,220],[20,4915,229],{"className":4916},[228],[20,4918,4920],{"className":4919},[200],[20,4921,4923],{"className":4922,"style":236},[204],[20,4924],{},[20,4926,243],{"className":4927},[242],[20,4929],{"className":4930,"style":247},[159],[20,4932,2906],{"className":4933,"style":2905},[41,86],[20,4935,405],{"className":4936},[404]," is now dragged toward ",[20,4939,4941],{"className":4940},[23],[20,4942,4944],{"className":4943,"ariaHidden":28},[27],[20,4945,4947,4950],{"className":4946},[32],[20,4948],{"className":4949,"style":2157},[36],[20,4951,4953],{"className":4952},[41],"3.7"," — a target the logged data never justifies — and the inflation does not stop.\nThe next backup on ",[20,4956,4958],{"className":4957},[23],[20,4959,4961],{"className":4960,"ariaHidden":28},[27],[20,4962,4964,4967,4970,5010,5013,5016,5019,5022,5025,5028,5031,5034,5074],{"className":4963},[32],[20,4965],{"className":4966,"style":606},[36],[20,4968,687],{"className":4969},[179],[20,4971,4973,4976],{"className":4972},[41],[20,4974,187],{"className":4975},[41,86],[20,4977,4979],{"className":4978},[191],[20,4980,4982,5002],{"className":4981},[195,196],[20,4983,4985,4999],{"className":4984},[200],[20,4986,4988],{"className":4987,"style":2800},[204],[20,4989,4990,4993],{"style":208},[20,4991],{"className":4992,"style":213},[212],[20,4994,4996],{"className":4995},[217,218,219,220],[20,4997,2161],{"className":4998},[41,220],[20,5000,229],{"className":5001},[228],[20,5003,5005],{"className":5004},[200],[20,5006,5008],{"className":5007,"style":236},[204],[20,5009],{},[20,5011,243],{"className":5012},[242],[20,5014],{"className":5015,"style":247},[159],[20,5017,2906],{"className":5018,"style":2905},[41,86],[20,5020,243],{"className":5021},[242],[20,5023],{"className":5024,"style":247},[159],[20,5026,2161],{"className":5027},[41],[20,5029,243],{"className":5030},[242],[20,5032],{"className":5033,"style":247},[159],[20,5035,5037,5040],{"className":5036},[41],[20,5038,187],{"className":5039},[41,86],[20,5041,5043],{"className":5042},[191],[20,5044,5046,5066],{"className":5045},[195,196],[20,5047,5049,5063],{"className":5048},[200],[20,5050,5052],{"className":5051,"style":2800},[204],[20,5053,5054,5057],{"style":208},[20,5055],{"className":5056,"style":213},[212],[20,5058,5060],{"className":5059},[217,218,219,220],[20,5061,61],{"className":5062},[41,220],[20,5064,229],{"className":5065},[228],[20,5067,5069],{"className":5068},[200],[20,5070,5072],{"className":5071,"style":236},[204],[20,5073],{},[20,5075,405],{"className":5076},[404]," reads ",[20,5079,5081],{"className":5080},[23],[20,5082,5084,5301],{"className":5083,"ariaHidden":28},[27],[20,5085,5087,5090,5165,5168,5208,5211,5251,5254,5257,5289,5292,5295,5298],{"className":5086},[32],[20,5088],{"className":5089,"style":1858},[36],[20,5091,5093,5099],{"className":5092},[1279],[20,5094,5096],{"className":5095},[1279],[20,5097,1352],{"className":5098},[41,1351],[20,5100,5102],{"className":5101},[191],[20,5103,5105,5157],{"className":5104},[195,196],[20,5106,5108,5154],{"className":5107},[200],[20,5109,5111],{"className":5110,"style":2457},[204],[20,5112,5113,5116],{"style":2460},[20,5114],{"className":5115,"style":213},[212],[20,5117,5119],{"className":5118},[217,218,219,220],[20,5120,5122],{"className":5121},[41,220],[20,5123,5125,5128],{"className":5124},[41,220],[20,5126,55],{"className":5127},[41,86,220],[20,5129,5131],{"className":5130},[191],[20,5132,5134],{"className":5133},[195],[20,5135,5137],{"className":5136},[200],[20,5138,5140],{"className":5139,"style":1321},[204],[20,5141,5142,5145],{"style":1324},[20,5143],{"className":5144,"style":663},[212],[20,5146,5148],{"className":5147},[217,667,668,220],[20,5149,5151],{"className":5150},[41,220],[20,5152,388],{"className":5153},[41,220],[20,5155,229],{"className":5156},[228],[20,5158,5160],{"className":5159},[200],[20,5161,5163],{"className":5162,"style":236},[204],[20,5164],{},[20,5166],{"className":5167,"style":247},[159],[20,5169,5171,5174],{"className":5170},[41],[20,5172,1164],{"className":5173},[41,86],[20,5175,5177],{"className":5176},[191],[20,5178,5180,5200],{"className":5179},[195,196],[20,5181,5183,5197],{"className":5182},[200],[20,5184,5186],{"className":5185,"style":511},[204],[20,5187,5188,5191],{"style":208},[20,5189],{"className":5190,"style":213},[212],[20,5192,5194],{"className":5193},[217,218,219,220],[20,5195,1188],{"className":5196,"style":43},[41,86,220],[20,5198,229],{"className":5199},[228],[20,5201,5203],{"className":5202},[200],[20,5204,5206],{"className":5205,"style":236},[204],[20,5207],{},[20,5209,687],{"className":5210},[179],[20,5212,5214,5217],{"className":5213},[41],[20,5215,187],{"className":5216},[41,86],[20,5218,5220],{"className":5219},[191],[20,5221,5223,5243],{"className":5222},[195,196],[20,5224,5226,5240],{"className":5225},[200],[20,5227,5229],{"className":5228,"style":2800},[204],[20,5230,5231,5234],{"style":208},[20,5232],{"className":5233,"style":213},[212],[20,5235,5237],{"className":5236},[217,218,219,220],[20,5238,61],{"className":5239},[41,220],[20,5241,229],{"className":5242},[228],[20,5244,5246],{"className":5245},[200],[20,5247,5249],{"className":5248,"style":236},[204],[20,5250],{},[20,5252,243],{"className":5253},[242],[20,5255],{"className":5256,"style":247},[159],[20,5258,5260,5263],{"className":5259},[41],[20,5261,55],{"className":5262},[41,86],[20,5264,5266],{"className":5265},[191],[20,5267,5269],{"className":5268},[195],[20,5270,5272],{"className":5271},[200],[20,5273,5275],{"className":5274,"style":1791},[204],[20,5276,5277,5280],{"style":1812},[20,5278],{"className":5279,"style":213},[212],[20,5281,5283],{"className":5282},[217,218,219,220],[20,5284,5286],{"className":5285},[41,220],[20,5287,388],{"className":5288},[41,220],[20,5290,405],{"className":5291},[404],[20,5293],{"className":5294,"style":160},[159],[20,5296,165],{"className":5297},[164],[20,5299],{"className":5300,"style":160},[159],[20,5302,5304,5307],{"className":5303},[32],[20,5305],{"className":5306,"style":2157},[36],[20,5308,4953],{"className":5309},[41]," (or\nhigher, if ",[20,5312,5314],{"className":5313},[23],[20,5315,5317],{"className":5316,"ariaHidden":28},[27],[20,5318,5320,5323],{"className":5319},[32],[20,5321],{"className":5322,"style":37},[36],[20,5324,2895],{"className":5325},[41,86]," still reads higher), so ",[20,5328,5330],{"className":5329},[23],[20,5331,5333,5444,5462,5480,5498],{"className":5332,"ariaHidden":28},[27],[20,5334,5336,5339,5379,5382,5422,5425,5428,5431,5434,5437,5441],{"className":5335},[32],[20,5337],{"className":5338,"style":606},[36],[20,5340,5342,5345],{"className":5341},[41],[20,5343,1164],{"className":5344},[41,86],[20,5346,5348],{"className":5347},[191],[20,5349,5351,5371],{"className":5350},[195,196],[20,5352,5354,5368],{"className":5353},[200],[20,5355,5357],{"className":5356,"style":511},[204],[20,5358,5359,5362],{"style":208},[20,5360],{"className":5361,"style":213},[212],[20,5363,5365],{"className":5364},[217,218,219,220],[20,5366,1188],{"className":5367,"style":43},[41,86,220],[20,5369,229],{"className":5370},[228],[20,5372,5374],{"className":5373},[200],[20,5375,5377],{"className":5376,"style":236},[204],[20,5378],{},[20,5380,687],{"className":5381},[179],[20,5383,5385,5388],{"className":5384},[41],[20,5386,187],{"className":5387},[41,86],[20,5389,5391],{"className":5390},[191],[20,5392,5394,5414],{"className":5393},[195,196],[20,5395,5397,5411],{"className":5396},[200],[20,5398,5400],{"className":5399,"style":2800},[204],[20,5401,5402,5405],{"style":208},[20,5403],{"className":5404,"style":213},[212],[20,5406,5408],{"className":5407},[217,218,219,220],[20,5409,2161],{"className":5410},[41,220],[20,5412,229],{"className":5413},[228],[20,5415,5417],{"className":5416},[200],[20,5418,5420],{"className":5419,"style":236},[204],[20,5421],{},[20,5423,243],{"className":5424},[242],[20,5426],{"className":5427,"style":247},[159],[20,5429,2906],{"className":5430,"style":2905},[41,86],[20,5432,405],{"className":5433},[404],[20,5435],{"className":5436,"style":160},[159],[20,5438,5440],{"className":5439},[164],"→",[20,5442],{"className":5443,"style":160},[159],[20,5445,5447,5450,5453,5456,5459],{"className":5446},[32],[20,5448],{"className":5449,"style":4240},[36],[20,5451,2161],{"className":5452},[41],[20,5454],{"className":5455,"style":1249},[159],[20,5457,1254],{"className":5458},[1253],[20,5460],{"className":5461,"style":1249},[159],[20,5463,5465,5468,5471,5474,5477],{"className":5464},[32],[20,5466],{"className":5467,"style":2157},[36],[20,5469,2944],{"className":5470},[41],[20,5472],{"className":5473,"style":1249},[159],[20,5475,3676],{"className":5476},[1253],[20,5478],{"className":5479,"style":1249},[159],[20,5481,5483,5486,5489,5492,5495],{"className":5482},[32],[20,5484],{"className":5485,"style":2157},[36],[20,5487,4953],{"className":5488},[41],[20,5490],{"className":5491,"style":160},[159],[20,5493,165],{"className":5494},[164],[20,5496],{"className":5497,"style":160},[159],[20,5499,5501,5504],{"className":5500},[32],[20,5502],{"className":5503,"style":2157},[36],[20,5505,5507],{"className":5506},[41],"3.33",", and if the network's extrapolation of ",[20,5510,5512],{"className":5511},[23],[20,5513,5515],{"className":5514,"ariaHidden":28},[27],[20,5516,5518,5521,5561,5564,5604,5607,5610,5613],{"className":5517},[32],[20,5519],{"className":5520,"style":606},[36],[20,5522,5524,5527],{"className":5523},[41],[20,5525,1164],{"className":5526},[41,86],[20,5528,5530],{"className":5529},[191],[20,5531,5533,5553],{"className":5532},[195,196],[20,5534,5536,5550],{"className":5535},[200],[20,5537,5539],{"className":5538,"style":511},[204],[20,5540,5541,5544],{"style":208},[20,5542],{"className":5543,"style":213},[212],[20,5545,5547],{"className":5546},[217,218,219,220],[20,5548,1188],{"className":5549,"style":43},[41,86,220],[20,5551,229],{"className":5552},[228],[20,5554,5556],{"className":5555},[200],[20,5557,5559],{"className":5558,"style":236},[204],[20,5560],{},[20,5562,687],{"className":5563},[179],[20,5565,5567,5570],{"className":5566},[41],[20,5568,187],{"className":5569},[41,86],[20,5571,5573],{"className":5572},[191],[20,5574,5576,5596],{"className":5575},[195,196],[20,5577,5579,5593],{"className":5578},[200],[20,5580,5582],{"className":5581,"style":2800},[204],[20,5583,5584,5587],{"style":208},[20,5585],{"className":5586,"style":213},[212],[20,5588,5590],{"className":5589},[217,218,219,220],[20,5591,2161],{"className":5592},[41,220],[20,5594,229],{"className":5595},[228],[20,5597,5599],{"className":5598},[200],[20,5600,5602],{"className":5601,"style":236},[204],[20,5603],{},[20,5605,243],{"className":5606},[242],[20,5608],{"className":5609,"style":247},[159],[20,5611,2895],{"className":5612},[41,86],[20,5614,405],{"className":5615},[404]," is also optimistic\nthe same overshoot re-enters at ",[20,5618,5620],{"className":5619},[23],[20,5621,5623],{"className":5622,"ariaHidden":28},[27],[20,5624,5626,5629],{"className":5625},[32],[20,5627],{"className":5628,"style":2435},[36],[20,5630,5632,5635],{"className":5631},[41],[20,5633,187],{"className":5634},[41,86],[20,5636,5638],{"className":5637},[191],[20,5639,5641,5661],{"className":5640},[195,196],[20,5642,5644,5658],{"className":5643},[200],[20,5645,5647],{"className":5646,"style":2800},[204],[20,5648,5649,5652],{"style":208},[20,5650],{"className":5651,"style":213},[212],[20,5653,5655],{"className":5654},[217,218,219,220],[20,5656,2161],{"className":5657},[41,220],[20,5659,229],{"className":5660},[228],[20,5662,5664],{"className":5663},[200],[20,5665,5667],{"className":5666,"style":236},[204],[20,5668],{},". Each sweep the phantom value at ",[20,5671,5673],{"className":5672},[23],[20,5674,5676],{"className":5675,"ariaHidden":28},[27],[20,5677,5679,5682],{"className":5678},[32],[20,5680],{"className":5681,"style":37},[36],[20,5683,2895],{"className":5684},[41,86]," feeds a\nhigher target, the higher target raises the logged ",[20,5687,5689],{"className":5688},[23],[20,5690,5692],{"className":5691,"ariaHidden":28},[27],[20,5693,5695,5698],{"className":5694},[32],[20,5696],{"className":5697,"style":1157},[36],[20,5699,1164],{"className":5700},[41,86],", and with no rollout ever\nexecuting ",[20,5703,5705],{"className":5704},[23],[20,5706,5708],{"className":5707,"ariaHidden":28},[27],[20,5709,5711,5714],{"className":5710},[32],[20,5712],{"className":5713,"style":37},[36],[20,5715,2895],{"className":5716},[41,86]," to observe its true ",[20,5719,5721],{"className":5720},[23],[20,5722,5724],{"className":5723,"ariaHidden":28},[27],[20,5725,5727,5730],{"className":5726},[32],[20,5728],{"className":5729,"style":2157},[36],[20,5731,2161],{"className":5732},[41]," reward, nothing pushes the estimate back\ndown. Online, one visit to ",[20,5735,5737],{"className":5736},[23],[20,5738,5740],{"className":5739,"ariaHidden":28},[27],[20,5741,5743,5746,5749,5789,5792,5795,5798],{"className":5742},[32],[20,5744],{"className":5745,"style":606},[36],[20,5747,687],{"className":5748},[179],[20,5750,5752,5755],{"className":5751},[41],[20,5753,187],{"className":5754},[41,86],[20,5756,5758],{"className":5757},[191],[20,5759,5761,5781],{"className":5760},[195,196],[20,5762,5764,5778],{"className":5763},[200],[20,5765,5767],{"className":5766,"style":2800},[204],[20,5768,5769,5772],{"style":208},[20,5770],{"className":5771,"style":213},[212],[20,5773,5775],{"className":5774},[217,218,219,220],[20,5776,61],{"className":5777},[41,220],[20,5779,229],{"className":5780},[228],[20,5782,5784],{"className":5783},[200],[20,5785,5787],{"className":5786,"style":236},[204],[20,5788],{},[20,5790,243],{"className":5791},[242],[20,5793],{"className":5794,"style":247},[159],[20,5796,2895],{"className":5797},[41,86],[20,5799,405],{"className":5800},[404]," would return reward ",[20,5803,5805],{"className":5804},[23],[20,5806,5808],{"className":5807,"ariaHidden":28},[27],[20,5809,5811,5814],{"className":5810},[32],[20,5812],{"className":5813,"style":2157},[36],[20,5815,2161],{"className":5816},[41]," and collapse the\nbubble in a single update; offline, the ",[20,5819,5821],{"className":5820},[23],[20,5822,5824],{"className":5823,"ariaHidden":28},[27],[20,5825,5827,5830],{"className":5826},[32],[20,5828],{"className":5829,"style":37},[36],[20,5831,2895],{"className":5832},[41,86],"-value is never corrected and the whole\nchain drifts up unboundedly. The induced greedy policy then ",[47,5835,5836,5837],{},"prefers ",[20,5838,5840],{"className":5839},[23],[20,5841,5843],{"className":5842,"ariaHidden":28},[27],[20,5844,5846,5849],{"className":5845},[32],[20,5847],{"className":5848,"style":37},[36],[20,5850,2895],{"className":5851},[41,86]," — an\naction the data shows nothing about and the environment rewards not at all.",[2749,5854],{"hash":5855},"86917793b61c294421cd58ba6e0f1f12dd18b911d43ac008c06476e8f066837d",[11,5857,5858,5859,5874],{},"The failure is averted by one change: a value at ",[20,5860,5862],{"className":5861},[23],[20,5863,5865],{"className":5864,"ariaHidden":28},[27],[20,5866,5868,5871],{"className":5867},[32],[20,5869],{"className":5870,"style":37},[36],[20,5872,2895],{"className":5873},[41,86]," that the learner does not\ntrust. Each family below installs precisely that.",[130,5876,5878],{"id":5877},"off-policy-evaluation-scoring-a-policy-from-the-log","Off-policy evaluation: scoring a policy from the log",[11,5880,5881,5882,5884,5885,5900,5901,5916,5917,5969,5970,6185,6186,6201,6202,6205],{},"Before improving a policy offline you often want to ",[47,5883,122],{}," one: given a candidate\n",[20,5886,5888],{"className":5887},[23],[20,5889,5891],{"className":5890,"ariaHidden":28},[27],[20,5892,5894,5897],{"className":5893},[32],[20,5895],{"className":5896,"style":82},[36],[20,5898,88],{"className":5899,"style":87},[41,86]," and only the log ",[20,5902,5904],{"className":5903},[23],[20,5905,5907],{"className":5906,"ariaHidden":28},[27],[20,5908,5910,5913],{"className":5909},[32],[20,5911],{"className":5912,"style":37},[36],[20,5914,44],{"className":5915,"style":43},[41,42]," from ",[20,5918,5920],{"className":5919},[23],[20,5921,5923],{"className":5922,"ariaHidden":28},[27],[20,5924,5926,5929],{"className":5925},[32],[20,5927],{"className":5928,"style":492},[36],[20,5930,5932,5935],{"className":5931},[41],[20,5933,88],{"className":5934,"style":87},[41,86],[20,5936,5938],{"className":5937},[191],[20,5939,5941,5961],{"className":5940},[195,196],[20,5942,5944,5958],{"className":5943},[200],[20,5945,5947],{"className":5946,"style":511},[204],[20,5948,5949,5952],{"style":514},[20,5950],{"className":5951,"style":213},[212],[20,5953,5955],{"className":5954},[217,218,219,220],[20,5956,525],{"className":5957,"style":524},[41,86,220],[20,5959,229],{"className":5960},[228],[20,5962,5964],{"className":5963},[200],[20,5965,5967],{"className":5966,"style":535},[204],[20,5968],{},", estimate ",[20,5971,5973],{"className":5972},[23],[20,5974,5976,6005],{"className":5975,"ariaHidden":28},[27],[20,5977,5979,5982,5987,5990,5993,5996,5999,6002],{"className":5978},[32],[20,5980],{"className":5981,"style":606},[36],[20,5983,5986],{"className":5984,"style":5985},[41,86],"margin-right:0.0962em;","J",[20,5988,687],{"className":5989},[179],[20,5991,88],{"className":5992,"style":87},[41,86],[20,5994,405],{"className":5995},[404],[20,5997],{"className":5998,"style":160},[159],[20,6000,165],{"className":6001},[164],[20,6003],{"className":6004,"style":160},[159],[20,6006,6008,6012,6055,6059,6107,6110,6140,6181],{"className":6007},[32],[20,6009],{"className":6010,"style":6011},[36],"height:1.0933em;vertical-align:-0.2997em;",[20,6013,6015,6020],{"className":6014},[41],[20,6016,6019],{"className":6017},[41,6018],"mathbb","E",[20,6021,6023],{"className":6022},[191],[20,6024,6026,6047],{"className":6025},[195,196],[20,6027,6029,6044],{"className":6028},[200],[20,6030,6033],{"className":6031,"style":6032},[204],"height:0.1514em;",[20,6034,6035,6038],{"style":208},[20,6036],{"className":6037,"style":213},[212],[20,6039,6041],{"className":6040},[217,218,219,220],[20,6042,88],{"className":6043,"style":87},[41,86,220],[20,6045,229],{"className":6046},[228],[20,6048,6050],{"className":6049},[200],[20,6051,6053],{"className":6052,"style":236},[204],[20,6054],{},[20,6056,6058],{"className":6057},[179],"[",[20,6060,6062,6069],{"className":6061},[1279],[20,6063,6068],{"className":6064,"style":6067},[1279,6065,6066],"op-symbol","small-op","position:relative;top:0em;","∑",[20,6070,6072],{"className":6071},[191],[20,6073,6075,6098],{"className":6074},[195,196],[20,6076,6078,6095],{"className":6077},[200],[20,6079,6082],{"className":6080,"style":6081},[204],"height:0.1308em;",[20,6083,6085,6088],{"style":6084},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[20,6086],{"className":6087,"style":213},[212],[20,6089,6091],{"className":6090},[217,218,219,220],[20,6092,6094],{"className":6093},[41,86,220],"t",[20,6096,229],{"className":6097},[228],[20,6099,6101],{"className":6100},[200],[20,6102,6105],{"className":6103,"style":6104},[204],"height:0.2997em;",[20,6106],{},[20,6108],{"className":6109,"style":247},[159],[20,6111,6113,6116],{"className":6112},[41],[20,6114,1269],{"className":6115,"style":1268},[41,86],[20,6117,6119],{"className":6118},[191],[20,6120,6122],{"className":6121},[195],[20,6123,6125],{"className":6124},[200],[20,6126,6129],{"className":6127,"style":6128},[204],"height:0.7936em;",[20,6130,6131,6134],{"style":1812},[20,6132],{"className":6133,"style":213},[212],[20,6135,6137],{"className":6136},[217,218,219,220],[20,6138,6094],{"className":6139},[41,86,220],[20,6141,6143,6146],{"className":6142},[41],[20,6144,300],{"className":6145,"style":43},[41,86],[20,6147,6149],{"className":6148},[191],[20,6150,6152,6173],{"className":6151},[195,196],[20,6153,6155,6170],{"className":6154},[200],[20,6156,6159],{"className":6157,"style":6158},[204],"height:0.2806em;",[20,6160,6161,6164],{"style":315},[20,6162],{"className":6163,"style":213},[212],[20,6165,6167],{"className":6166},[217,218,219,220],[20,6168,6094],{"className":6169},[41,86,220],[20,6171,229],{"className":6172},[228],[20,6174,6176],{"className":6175},[200],[20,6177,6179],{"className":6178,"style":236},[204],[20,6180],{},[20,6182,6184],{"className":6183},[404],"]"," without ever running ",[20,6187,6189],{"className":6188},[23],[20,6190,6192],{"className":6191,"ariaHidden":28},[27],[20,6193,6195,6198],{"className":6194},[32],[20,6196],{"className":6197,"style":82},[36],[20,6199,88],{"className":6200,"style":87},[41,86],". This is ",[15,6203,6204],{},"off-policy\nevaluation"," (OPE), and it is both a subproblem inside offline RL and the way a\npractitioner decides whether a learned policy is safe to deploy. It inherits the\nsame distributional-shift problem in a purer form.",[11,6207,6208,6209,6212,6213,6230,6231,6283,6284,6299,6300,6352],{},"The unbiased classical estimator is ",[15,6210,6211],{},"importance sampling",". A trajectory ",[20,6214,6216],{"className":6215},[23],[20,6217,6219],{"className":6218,"ariaHidden":28},[27],[20,6220,6222,6225],{"className":6221},[32],[20,6223],{"className":6224,"style":82},[36],[20,6226,6229],{"className":6227,"style":6228},[41,86],"margin-right:0.1132em;","τ","\nlogged under ",[20,6232,6234],{"className":6233},[23],[20,6235,6237],{"className":6236,"ariaHidden":28},[27],[20,6238,6240,6243],{"className":6239},[32],[20,6241],{"className":6242,"style":492},[36],[20,6244,6246,6249],{"className":6245},[41],[20,6247,88],{"className":6248,"style":87},[41,86],[20,6250,6252],{"className":6251},[191],[20,6253,6255,6275],{"className":6254},[195,196],[20,6256,6258,6272],{"className":6257},[200],[20,6259,6261],{"className":6260,"style":511},[204],[20,6262,6263,6266],{"style":514},[20,6264],{"className":6265,"style":213},[212],[20,6267,6269],{"className":6268},[217,218,219,220],[20,6270,525],{"className":6271,"style":524},[41,86,220],[20,6273,229],{"className":6274},[228],[20,6276,6278],{"className":6277},[200],[20,6279,6281],{"className":6280,"style":535},[204],[20,6282],{}," is reweighted by the ratio of the probability ",[20,6285,6287],{"className":6286},[23],[20,6288,6290],{"className":6289,"ariaHidden":28},[27],[20,6291,6293,6296],{"className":6292},[32],[20,6294],{"className":6295,"style":82},[36],[20,6297,88],{"className":6298,"style":87},[41,86]," would\nhave produced it to the probability ",[20,6301,6303],{"className":6302},[23],[20,6304,6306],{"className":6305,"ariaHidden":28},[27],[20,6307,6309,6312],{"className":6308},[32],[20,6310],{"className":6311,"style":492},[36],[20,6313,6315,6318],{"className":6314},[41],[20,6316,88],{"className":6317,"style":87},[41,86],[20,6319,6321],{"className":6320},[191],[20,6322,6324,6344],{"className":6323},[195,196],[20,6325,6327,6341],{"className":6326},[200],[20,6328,6330],{"className":6329,"style":511},[204],[20,6331,6332,6335],{"style":514},[20,6333],{"className":6334,"style":213},[212],[20,6336,6338],{"className":6337},[217,218,219,220],[20,6339,525],{"className":6340,"style":524},[41,86,220],[20,6342,229],{"className":6343},[228],[20,6345,6347],{"className":6346},[200],[20,6348,6350],{"className":6349,"style":535},[204],[20,6351],{}," did:",[20,6354,6356],{"className":6355},[140],[20,6357,6359],{"className":6358},[23],[20,6360,6362,6459],{"className":6361,"ariaHidden":28},[27],[20,6363,6365,6369,6450,6453,6456],{"className":6364},[32],[20,6366],{"className":6367,"style":6368},[36],"height:1.0968em;vertical-align:-0.15em;",[20,6370,6372,6407],{"className":6371},[41],[20,6373,6375],{"className":6374},[41,1401],[20,6376,6378],{"className":6377},[195],[20,6379,6381],{"className":6380},[200],[20,6382,6385,6393],{"className":6383,"style":6384},[204],"height:0.9468em;",[20,6386,6387,6390],{"style":1339},[20,6388],{"className":6389,"style":1296},[212],[20,6391,5986],{"className":6392,"style":5985},[41,86],[20,6394,6396,6399],{"style":6395},"top:-3.2523em;",[20,6397],{"className":6398,"style":1296},[212],[20,6400,6403],{"className":6401,"style":6402},[1430],"left:-0.0833em;",[20,6404,6406],{"className":6405},[41],"^",[20,6408,6410],{"className":6409},[191],[20,6411,6413,6442],{"className":6412},[195,196],[20,6414,6416,6439],{"className":6415},[200],[20,6417,6420],{"className":6418,"style":6419},[204],"height:0.3283em;",[20,6421,6423,6426],{"style":6422},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[20,6424],{"className":6425,"style":213},[212],[20,6427,6429],{"className":6428},[217,218,219,220],[20,6430,6432],{"className":6431},[41,220],[20,6433,6435],{"className":6434},[41,1537,220],[20,6436,6438],{"className":6437},[41,220],"IS",[20,6440,229],{"className":6441},[228],[20,6443,6445],{"className":6444},[200],[20,6446,6448],{"className":6447,"style":236},[204],[20,6449],{},[20,6451],{"className":6452,"style":160},[159],[20,6454,165],{"className":6455},[164],[20,6457],{"className":6458,"style":160},[159],[20,6460,6462,6466,6538,6541,6617,6624,7142,7148,7154,7217,7220,7249,7289,7295],{"className":6461},[32],[20,6463],{"className":6464,"style":6465},[36],"height:3.6394em;vertical-align:-1.988em;",[20,6467,6469,6473,6535],{"className":6468},[41],[20,6470],{"className":6471},[179,6472],"nulldelimiter",[20,6474,6477],{"className":6475},[6476],"mfrac",[20,6478,6480,6526],{"className":6479},[195,196],[20,6481,6483,6523],{"className":6482},[200],[20,6484,6487,6500,6511],{"className":6485,"style":6486},[204],"height:1.3214em;",[20,6488,6490,6493],{"style":6489},"top:-2.314em;",[20,6491],{"className":6492,"style":1296},[212],[20,6494,6496],{"className":6495},[41],[20,6497,6499],{"className":6498},[41,86],"m",[20,6501,6503,6506],{"style":6502},"top:-3.23em;",[20,6504],{"className":6505,"style":1296},[212],[20,6507],{"className":6508,"style":6510},[6509],"frac-line","border-bottom-width:0.04em;",[20,6512,6514,6517],{"style":6513},"top:-3.677em;",[20,6515],{"className":6516,"style":1296},[212],[20,6518,6520],{"className":6519},[41],[20,6521,61],{"className":6522},[41],[20,6524,229],{"className":6525},[228],[20,6527,6529],{"className":6528},[200],[20,6530,6533],{"className":6531,"style":6532},[204],"height:0.686em;",[20,6534],{},[20,6536],{"className":6537},[404,6472],[20,6539],{"className":6540,"style":247},[159],[20,6542,6544],{"className":6543},[1279,1280],[20,6545,6547,6608],{"className":6546},[195,196],[20,6548,6550,6605],{"className":6549},[200],[20,6551,6554,6578,6590],{"className":6552,"style":6553},[204],"height:1.6514em;",[20,6555,6557,6561],{"style":6556},"top:-1.8479em;margin-left:0em;",[20,6558],{"className":6559,"style":6560},[212],"height:3.05em;",[20,6562,6564],{"className":6563},[217,218,219,220],[20,6565,6567,6572,6575],{"className":6566},[41,220],[20,6568,6571],{"className":6569,"style":6570},[41,86,220],"margin-right:0.0315em;","k",[20,6573,165],{"className":6574},[164,220],[20,6576,61],{"className":6577},[41,220],[20,6579,6581,6584],{"style":6580},"top:-3.05em;",[20,6582],{"className":6583,"style":6560},[212],[20,6585,6586],{},[20,6587,6068],{"className":6588},[1279,6065,6589],"large-op",[20,6591,6593,6596],{"style":6592},"top:-4.3em;margin-left:0em;",[20,6594],{"className":6595,"style":6560},[212],[20,6597,6599],{"className":6598},[217,218,219,220],[20,6600,6602],{"className":6601},[41,220],[20,6603,6499],{"className":6604},[41,86,220],[20,6606,229],{"className":6607},[228],[20,6609,6611],{"className":6610},[200],[20,6612,6615],{"className":6613,"style":6614},[204],"height:1.3021em;",[20,6616],{},[20,6618,6620],{"className":6619},[179],[20,6621,687],{"className":6622},[1671,6623],"size2",[20,6625,6627],{"className":6626},[4290,4291],[20,6628,6630,7133],{"className":6629},[195,196],[20,6631,6633,7130],{"className":6632},[200],[20,6634,6637,6701],{"className":6635,"style":6636},[204],"height:1.01em;",[20,6638,6640,6644],{"style":6639},"top:-1.1278em;",[20,6641],{"className":6642,"style":6643},[212],"height:3.01em;",[20,6645,6647],{"className":6646},[217,218,219,220],[20,6648,6650,6657],{"className":6649},[41,220],[20,6651,6653],{"className":6652},[41,1537,220],[20,6654,6656],{"className":6655},[41,220],"cumulative ratio ",[20,6658,6660,6665],{"className":6659},[41,220],[20,6661,6664],{"className":6662,"style":6663},[41,86,220],"margin-right:0.0269em;","w",[20,6666,6668],{"className":6667},[191],[20,6669,6671,6692],{"className":6670},[195,196],[20,6672,6674,6689],{"className":6673},[200],[20,6675,6677],{"className":6676,"style":656},[204],[20,6678,6680,6683],{"style":6679},"top:-2.3488em;margin-left:-0.0269em;margin-right:0.0714em;",[20,6681],{"className":6682,"style":663},[212],[20,6684,6686],{"className":6685},[217,667,668,220],[20,6687,6571],{"className":6688,"style":6570},[41,86,220],[20,6690,229],{"className":6691},[228],[20,6693,6695],{"className":6694},[200],[20,6696,6699],{"className":6697,"style":6698},[204],"height:0.1512em;",[20,6700],{},[20,6702,6704,6707],{"style":6703},"top:-3.01em;",[20,6705],{"className":6706,"style":6643},[212],[20,6708,6710],{"className":6709},[4290,4291],[20,6711,6713,7121],{"className":6712},[195,196],[20,6714,6716,7118],{"className":6715},[200],[20,6717,6719,6750],{"className":6718,"style":6636},[204],[20,6720,6723,6726],{"className":6721,"style":6722},[4340],"top:-1.8139em;",[20,6724],{"className":6725,"style":6643},[212],[20,6727,6729,6736,6743],{"className":6728,"style":4349},[4348],[20,6730,6732],{"className":6731,"style":4354},[4353],[4356,6733,6734],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4362},[4364,6735],{"d":4366},[20,6737,6739],{"className":6738,"style":4354},[4370],[4356,6740,6741],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4373},[4364,6742],{"d":4376},[20,6744,6746],{"className":6745,"style":4354},[4380],[4356,6747,6748],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4383},[4364,6749],{"d":4386},[20,6751,6752,6755],{"style":6703},[20,6753],{"className":6754,"style":6643},[212],[20,6756,6758,6826,6829],{"className":6757},[41],[20,6759,6761,6765],{"className":6760},[1279],[20,6762,6764],{"className":6763,"style":6067},[1279,6065,6066],"∏",[20,6766,6768],{"className":6767},[191],[20,6769,6771,6818],{"className":6770},[195,196],[20,6772,6774,6815],{"className":6773},[200],[20,6775,6778,6798],{"className":6776,"style":6777},[204],"height:0.9812em;",[20,6779,6780,6783],{"style":6084},[20,6781],{"className":6782,"style":213},[212],[20,6784,6786],{"className":6785},[217,218,219,220],[20,6787,6789,6792,6795],{"className":6788},[41,220],[20,6790,6094],{"className":6791},[41,86,220],[20,6793,165],{"className":6794},[164,220],[20,6796,2161],{"className":6797},[41,220],[20,6799,6801,6804],{"style":6800},"top:-3.2029em;margin-right:0.05em;",[20,6802],{"className":6803,"style":213},[212],[20,6805,6807],{"className":6806},[217,218,219,220],[20,6808,6810],{"className":6809},[41,220],[20,6811,6814],{"className":6812,"style":6813},[41,86,220],"margin-right:0.1389em;","T",[20,6816,229],{"className":6817},[228],[20,6819,6821],{"className":6820},[200],[20,6822,6824],{"className":6823,"style":6104},[204],[20,6825],{},[20,6827],{"className":6828,"style":247},[159],[20,6830,6832,6835,7115],{"className":6831},[41],[20,6833],{"className":6834},[179,6472],[20,6836,6838],{"className":6837},[6476],[20,6839,6841,7106],{"className":6840},[195,196],[20,6842,6844,7103],{"className":6843},[200],[20,6845,6847,6991,6999],{"className":6846,"style":6636},[204],[20,6848,6850,6853],{"style":6849},"top:-2.655em;",[20,6851],{"className":6852,"style":1296},[212],[20,6854,6856],{"className":6855},[217,218,219,220],[20,6857,6859,6899,6902,6945,6948,6988],{"className":6858},[41,220],[20,6860,6862,6865],{"className":6861},[41,220],[20,6863,88],{"className":6864,"style":87},[41,86,220],[20,6866,6868],{"className":6867},[191],[20,6869,6871,6891],{"className":6870},[195,196],[20,6872,6874,6888],{"className":6873},[200],[20,6875,6877],{"className":6876,"style":656},[204],[20,6878,6879,6882],{"style":659},[20,6880],{"className":6881,"style":663},[212],[20,6883,6885],{"className":6884},[217,667,668,220],[20,6886,525],{"className":6887,"style":524},[41,86,220],[20,6889,229],{"className":6890},[228],[20,6892,6894],{"className":6893},[200],[20,6895,6897],{"className":6896,"style":681},[204],[20,6898],{},[20,6900,687],{"className":6901},[179,220],[20,6903,6905,6908],{"className":6904},[41,220],[20,6906,55],{"className":6907},[41,86,220],[20,6909,6911],{"className":6910},[191],[20,6912,6914,6936],{"className":6913},[195,196],[20,6915,6917,6933],{"className":6916},[200],[20,6918,6921],{"className":6919,"style":6920},[204],"height:0.2963em;",[20,6922,6924,6927],{"style":6923},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[20,6925],{"className":6926,"style":663},[212],[20,6928,6930],{"className":6929},[217,667,668,220],[20,6931,6094],{"className":6932},[41,86,220],[20,6934,229],{"className":6935},[228],[20,6937,6939],{"className":6938},[200],[20,6940,6943],{"className":6941,"style":6942},[204],"height:0.143em;",[20,6944],{},[20,6946,858],{"className":6947},[164,220],[20,6949,6951,6954],{"className":6950},[41,220],[20,6952,187],{"className":6953},[41,86,220],[20,6955,6957],{"className":6956},[191],[20,6958,6960,6980],{"className":6959},[195,196],[20,6961,6963,6977],{"className":6962},[200],[20,6964,6966],{"className":6965,"style":6920},[204],[20,6967,6968,6971],{"style":6923},[20,6969],{"className":6970,"style":663},[212],[20,6972,6974],{"className":6973},[217,667,668,220],[20,6975,6094],{"className":6976},[41,86,220],[20,6978,229],{"className":6979},[228],[20,6981,6983],{"className":6982},[200],[20,6984,6986],{"className":6985,"style":6942},[204],[20,6987],{},[20,6989,405],{"className":6990},[404,220],[20,6992,6993,6996],{"style":6502},[20,6994],{"className":6995,"style":1296},[212],[20,6997],{"className":6998,"style":6510},[6509],[20,7000,7002,7005],{"style":7001},"top:-3.485em;",[20,7003],{"className":7004,"style":1296},[212],[20,7006,7008],{"className":7007},[217,218,219,220],[20,7009,7011,7014,7017,7057,7060,7100],{"className":7010},[41,220],[20,7012,88],{"className":7013,"style":87},[41,86,220],[20,7015,687],{"className":7016},[179,220],[20,7018,7020,7023],{"className":7019},[41,220],[20,7021,55],{"className":7022},[41,86,220],[20,7024,7026],{"className":7025},[191],[20,7027,7029,7049],{"className":7028},[195,196],[20,7030,7032,7046],{"className":7031},[200],[20,7033,7035],{"className":7034,"style":6920},[204],[20,7036,7037,7040],{"style":6923},[20,7038],{"className":7039,"style":663},[212],[20,7041,7043],{"className":7042},[217,667,668,220],[20,7044,6094],{"className":7045},[41,86,220],[20,7047,229],{"className":7048},[228],[20,7050,7052],{"className":7051},[200],[20,7053,7055],{"className":7054,"style":6942},[204],[20,7056],{},[20,7058,858],{"className":7059},[164,220],[20,7061,7063,7066],{"className":7062},[41,220],[20,7064,187],{"className":7065},[41,86,220],[20,7067,7069],{"className":7068},[191],[20,7070,7072,7092],{"className":7071},[195,196],[20,7073,7075,7089],{"className":7074},[200],[20,7076,7078],{"className":7077,"style":6920},[204],[20,7079,7080,7083],{"style":6923},[20,7081],{"className":7082,"style":663},[212],[20,7084,7086],{"className":7085},[217,667,668,220],[20,7087,6094],{"className":7088},[41,86,220],[20,7090,229],{"className":7091},[228],[20,7093,7095],{"className":7094},[200],[20,7096,7098],{"className":7097,"style":6942},[204],[20,7099],{},[20,7101,405],{"className":7102},[404,220],[20,7104,229],{"className":7105},[228],[20,7107,7109],{"className":7108},[200],[20,7110,7113],{"className":7111,"style":7112},[204],"height:0.5481em;",[20,7114],{},[20,7116],{"className":7117},[404,6472],[20,7119,229],{"className":7120},[228],[20,7122,7124],{"className":7123},[200],[20,7125,7128],{"className":7126,"style":7127},[204],"height:1.1961em;",[20,7129],{},[20,7131,229],{"className":7132},[228],[20,7134,7136],{"className":7135},[200],[20,7137,7140],{"className":7138,"style":7139},[204],"height:1.988em;",[20,7141],{},[20,7143,7145],{"className":7144},[404],[20,7146,405],{"className":7147},[1671,6623],[20,7149,7151],{"className":7150},[179],[20,7152,687],{"className":7153},[1671,6623],[20,7155,7157,7160],{"className":7156},[1279],[20,7158,6068],{"className":7159,"style":6067},[1279,6065,6066],[20,7161,7163],{"className":7162},[191],[20,7164,7166,7209],{"className":7165},[195,196],[20,7167,7169,7206],{"className":7168},[200],[20,7170,7172,7192],{"className":7171,"style":6777},[204],[20,7173,7174,7177],{"style":6084},[20,7175],{"className":7176,"style":213},[212],[20,7178,7180],{"className":7179},[217,218,219,220],[20,7181,7183,7186,7189],{"className":7182},[41,220],[20,7184,6094],{"className":7185},[41,86,220],[20,7187,165],{"className":7188},[164,220],[20,7190,2161],{"className":7191},[41,220],[20,7193,7194,7197],{"style":6800},[20,7195],{"className":7196,"style":213},[212],[20,7198,7200],{"className":7199},[217,218,219,220],[20,7201,7203],{"className":7202},[41,220],[20,7204,6814],{"className":7205,"style":6813},[41,86,220],[20,7207,229],{"className":7208},[228],[20,7210,7212],{"className":7211},[200],[20,7213,7215],{"className":7214,"style":6104},[204],[20,7216],{},[20,7218],{"className":7219,"style":247},[159],[20,7221,7223,7226],{"className":7222},[41],[20,7224,1269],{"className":7225,"style":1268},[41,86],[20,7227,7229],{"className":7228},[191],[20,7230,7232],{"className":7231},[195],[20,7233,7235],{"className":7234},[200],[20,7236,7238],{"className":7237,"style":6128},[204],[20,7239,7240,7243],{"style":1812},[20,7241],{"className":7242,"style":213},[212],[20,7244,7246],{"className":7245},[217,218,219,220],[20,7247,6094],{"className":7248},[41,86,220],[20,7250,7252,7255],{"className":7251},[41],[20,7253,300],{"className":7254,"style":43},[41,86],[20,7256,7258],{"className":7257},[191],[20,7259,7261,7281],{"className":7260},[195,196],[20,7262,7264,7278],{"className":7263},[200],[20,7265,7267],{"className":7266,"style":6158},[204],[20,7268,7269,7272],{"style":315},[20,7270],{"className":7271,"style":213},[212],[20,7273,7275],{"className":7274},[217,218,219,220],[20,7276,6094],{"className":7277},[41,86,220],[20,7279,229],{"className":7280},[228],[20,7282,7284],{"className":7283},[200],[20,7285,7287],{"className":7286,"style":236},[204],[20,7288],{},[20,7290,7292],{"className":7291},[404],[20,7293,405],{"className":7294},[1671,6623],[20,7296,128],{"className":7297},[41],[11,7299,7300,7301,7316,7317,7370,7371,7396,7397,7400,7401,7416,7417,7432,7433,7485,7486,128],{},"It is unbiased, but the product of per-step ratios is its undoing: over a horizon\nof ",[20,7302,7304],{"className":7303},[23],[20,7305,7307],{"className":7306,"ariaHidden":28},[27],[20,7308,7310,7313],{"className":7309},[32],[20,7311],{"className":7312,"style":37},[36],[20,7314,6814],{"className":7315,"style":6813},[41,86]," steps the weight ",[20,7318,7320],{"className":7319},[23],[20,7321,7323],{"className":7322,"ariaHidden":28},[27],[20,7324,7326,7329],{"className":7325},[32],[20,7327],{"className":7328,"style":2435},[36],[20,7330,7332,7335],{"className":7331},[41],[20,7333,6664],{"className":7334,"style":6663},[41,86],[20,7336,7338],{"className":7337},[191],[20,7339,7341,7362],{"className":7340},[195,196],[20,7342,7344,7359],{"className":7343},[200],[20,7345,7347],{"className":7346,"style":511},[204],[20,7348,7350,7353],{"style":7349},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[20,7351],{"className":7352,"style":213},[212],[20,7354,7356],{"className":7355},[217,218,219,220],[20,7357,6571],{"className":7358,"style":6570},[41,86,220],[20,7360,229],{"className":7361},[228],[20,7363,7365],{"className":7364},[200],[20,7366,7368],{"className":7367,"style":236},[204],[20,7369],{}," is a product of ",[20,7372,7374],{"className":7373},[23],[20,7375,7377],{"className":7376,"ariaHidden":28},[27],[20,7378,7380,7384,7387,7393],{"className":7379},[32],[20,7381],{"className":7382,"style":7383},[36],"height:0.7667em;vertical-align:-0.0833em;",[20,7385,6814],{"className":7386,"style":6813},[41,86],[20,7388,7390],{"className":7389},[41],[20,7391,1254],{"className":7392},[41],[20,7394,61],{"className":7395},[41]," factors, and its variance\ngrows ",[47,7398,7399],{},"exponentially"," in ",[20,7402,7404],{"className":7403},[23],[20,7405,7407],{"className":7406,"ariaHidden":28},[27],[20,7408,7410,7413],{"className":7409},[32],[20,7411],{"className":7412,"style":37},[36],[20,7414,6814],{"className":7415,"style":6813},[41,86],". A single trajectory whose actions ",[20,7418,7420],{"className":7419},[23],[20,7421,7423],{"className":7422,"ariaHidden":28},[27],[20,7424,7426,7429],{"className":7425},[32],[20,7427],{"className":7428,"style":82},[36],[20,7430,88],{"className":7431,"style":87},[41,86]," strongly\nprefers but ",[20,7434,7436],{"className":7435},[23],[20,7437,7439],{"className":7438,"ariaHidden":28},[27],[20,7440,7442,7445],{"className":7441},[32],[20,7443],{"className":7444,"style":492},[36],[20,7446,7448,7451],{"className":7447},[41],[20,7449,88],{"className":7450,"style":87},[41,86],[20,7452,7454],{"className":7453},[191],[20,7455,7457,7477],{"className":7456},[195,196],[20,7458,7460,7474],{"className":7459},[200],[20,7461,7463],{"className":7462,"style":511},[204],[20,7464,7465,7468],{"style":514},[20,7466],{"className":7467,"style":213},[212],[20,7469,7471],{"className":7470},[217,218,219,220],[20,7472,525],{"className":7473,"style":524},[41,86,220],[20,7475,229],{"className":7476},[228],[20,7478,7480],{"className":7479},[200],[20,7481,7483],{"className":7482,"style":535},[204],[20,7484],{}," rarely took gets an astronomically large weight and\ndominates the average, so the estimate is unbiased but so noisy as to be useless\npast short horizons — the ",[15,7487,7488],{},"curse of horizon",[2749,7490],{"hash":7491},"f99f989707a4fc2327854776216542d3f9d48d3282fff645ca44144e3af8b0cb",[11,7493,7494,7495,7498,7499,7596,7597,7612,7613,7616,7617,7620,7621,479,7624,7799,7800,7852,7853,7905,7906,7909,7910,128],{},"Two standard repairs reduce the variance. ",[15,7496,7497],{},"Weighted importance sampling"," (WIS)\nself-normalizes by dividing by ",[20,7500,7502],{"className":7501},[23],[20,7503,7505],{"className":7504,"ariaHidden":28},[27],[20,7506,7508,7512,7553,7556],{"className":7507},[32],[20,7509],{"className":7510,"style":7511},[36],"height:1.0497em;vertical-align:-0.2997em;",[20,7513,7515,7518],{"className":7514},[1279],[20,7516,6068],{"className":7517,"style":6067},[1279,6065,6066],[20,7519,7521],{"className":7520},[191],[20,7522,7524,7545],{"className":7523},[195,196],[20,7525,7527,7542],{"className":7526},[200],[20,7528,7531],{"className":7529,"style":7530},[204],"height:0.1864em;",[20,7532,7533,7536],{"style":6084},[20,7534],{"className":7535,"style":213},[212],[20,7537,7539],{"className":7538},[217,218,219,220],[20,7540,6571],{"className":7541,"style":6570},[41,86,220],[20,7543,229],{"className":7544},[228],[20,7546,7548],{"className":7547},[200],[20,7549,7551],{"className":7550,"style":6104},[204],[20,7552],{},[20,7554],{"className":7555,"style":247},[159],[20,7557,7559,7562],{"className":7558},[41],[20,7560,6664],{"className":7561,"style":6663},[41,86],[20,7563,7565],{"className":7564},[191],[20,7566,7568,7588],{"className":7567},[195,196],[20,7569,7571,7585],{"className":7570},[200],[20,7572,7574],{"className":7573,"style":511},[204],[20,7575,7576,7579],{"style":7349},[20,7577],{"className":7578,"style":213},[212],[20,7580,7582],{"className":7581},[217,218,219,220],[20,7583,6571],{"className":7584,"style":6570},[41,86,220],[20,7586,229],{"className":7587},[228],[20,7589,7591],{"className":7590},[200],[20,7592,7594],{"className":7593,"style":236},[204],[20,7595],{}," instead of ",[20,7598,7600],{"className":7599},[23],[20,7601,7603],{"className":7602,"ariaHidden":28},[27],[20,7604,7606,7609],{"className":7605},[32],[20,7607],{"className":7608,"style":82},[36],[20,7610,6499],{"className":7611},[41,86],"; this introduces a small\nbias but is consistent and dramatically lower-variance in practice. ",[15,7614,7615],{},"Doubly-robust","\nestimators combine importance sampling with a learned value-function model: the\nmodel supplies a low-variance baseline, and importance sampling corrects only its\nresidual, so the estimator is unbiased if ",[47,7618,7619],{},"either"," the model or the ratios are\nright, and low-variance when the model is even roughly accurate. A third route\nabandons per-step ratios entirely and estimates the ratio of ",[47,7622,7623],{},"state-action\noccupancies",[20,7625,7627],{"className":7626},[23],[20,7628,7630],{"className":7629,"ariaHidden":28},[27],[20,7631,7633,7637,7678,7681,7684,7687,7690,7693,7696,7700,7781,7784,7787,7790,7793,7796],{"className":7632},[32],[20,7634],{"className":7635,"style":7636},[36],"height:1.1031em;vertical-align:-0.3531em;",[20,7638,7640,7644],{"className":7639},[41],[20,7641,7643],{"className":7642},[41,86],"ρ",[20,7645,7647],{"className":7646},[191],[20,7648,7650,7670],{"className":7649},[195,196],[20,7651,7653,7667],{"className":7652},[200],[20,7654,7656],{"className":7655,"style":6032},[204],[20,7657,7658,7661],{"style":208},[20,7659],{"className":7660,"style":213},[212],[20,7662,7664],{"className":7663},[217,218,219,220],[20,7665,88],{"className":7666,"style":87},[41,86,220],[20,7668,229],{"className":7669},[228],[20,7671,7673],{"className":7672},[200],[20,7674,7676],{"className":7675,"style":236},[204],[20,7677],{},[20,7679,687],{"className":7680},[179],[20,7682,187],{"className":7683},[41,86],[20,7685,243],{"className":7686},[242],[20,7688],{"className":7689,"style":247},[159],[20,7691,55],{"className":7692},[41,86],[20,7694,405],{"className":7695},[404],[20,7697,7699],{"className":7698},[41],"\u002F",[20,7701,7703,7706],{"className":7702},[41],[20,7704,7643],{"className":7705},[41,86],[20,7707,7709],{"className":7708},[191],[20,7710,7712,7772],{"className":7711},[195,196],[20,7713,7715,7769],{"className":7714},[200],[20,7716,7718],{"className":7717,"style":6032},[204],[20,7719,7720,7723],{"style":208},[20,7721],{"className":7722,"style":213},[212],[20,7724,7726],{"className":7725},[217,218,219,220],[20,7727,7729],{"className":7728},[41,220],[20,7730,7732,7735],{"className":7731},[41,220],[20,7733,88],{"className":7734,"style":87},[41,86,220],[20,7736,7738],{"className":7737},[191],[20,7739,7741,7761],{"className":7740},[195,196],[20,7742,7744,7758],{"className":7743},[200],[20,7745,7747],{"className":7746,"style":656},[204],[20,7748,7749,7752],{"style":659},[20,7750],{"className":7751,"style":663},[212],[20,7753,7755],{"className":7754},[217,667,668,220],[20,7756,525],{"className":7757,"style":524},[41,86,220],[20,7759,229],{"className":7760},[228],[20,7762,7764],{"className":7763},[200],[20,7765,7767],{"className":7766,"style":681},[204],[20,7768],{},[20,7770,229],{"className":7771},[228],[20,7773,7775],{"className":7774},[200],[20,7776,7779],{"className":7777,"style":7778},[204],"height:0.3531em;",[20,7780],{},[20,7782,687],{"className":7783},[179],[20,7785,187],{"className":7786},[41,86],[20,7788,243],{"className":7789},[242],[20,7791],{"className":7792,"style":247},[159],[20,7794,55],{"className":7795},[41,86],[20,7797,405],{"className":7798},[404]," directly, sidestepping the\nhorizon product (marginalized OPE). None escapes the basic limit: a policy far\nfrom ",[20,7801,7803],{"className":7802},[23],[20,7804,7806],{"className":7805,"ariaHidden":28},[27],[20,7807,7809,7812],{"className":7808},[32],[20,7810],{"className":7811,"style":492},[36],[20,7813,7815,7818],{"className":7814},[41],[20,7816,88],{"className":7817,"style":87},[41,86],[20,7819,7821],{"className":7820},[191],[20,7822,7824,7844],{"className":7823},[195,196],[20,7825,7827,7841],{"className":7826},[200],[20,7828,7830],{"className":7829,"style":511},[204],[20,7831,7832,7835],{"style":514},[20,7833],{"className":7834,"style":213},[212],[20,7836,7838],{"className":7837},[217,218,219,220],[20,7839,525],{"className":7840,"style":524},[41,86,220],[20,7842,229],{"className":7843},[228],[20,7845,7847],{"className":7846},[200],[20,7848,7850],{"className":7849,"style":535},[204],[20,7851],{}," cannot be evaluated reliably from ",[20,7854,7856],{"className":7855},[23],[20,7857,7859],{"className":7858,"ariaHidden":28},[27],[20,7860,7862,7865],{"className":7861},[32],[20,7863],{"className":7864,"style":492},[36],[20,7866,7868,7871],{"className":7867},[41],[20,7869,88],{"className":7870,"style":87},[41,86],[20,7872,7874],{"className":7873},[191],[20,7875,7877,7897],{"className":7876},[195,196],[20,7878,7880,7894],{"className":7879},[200],[20,7881,7883],{"className":7882,"style":511},[204],[20,7884,7885,7888],{"style":514},[20,7886],{"className":7887,"style":213},[212],[20,7889,7891],{"className":7890},[217,218,219,220],[20,7892,525],{"className":7893,"style":524},[41,86,220],[20,7895,229],{"className":7896},[228],[20,7898,7900],{"className":7899},[200],[20,7901,7903],{"className":7902,"style":535},[204],[20,7904],{},"'s data — the same\nsupport limit that constrains offline ",[47,7907,7908],{},"learning"," constrains offline ",[47,7911,7912],{},"evaluation",[961,7914,7915],{"type":963},[11,7916,7917,7920,7921,7945,7946,7961,7962,8014,8015,8030],{},[15,7918,7919],{},"Definition (Off-policy evaluation)."," Estimating the value ",[20,7922,7924],{"className":7923},[23],[20,7925,7927],{"className":7926,"ariaHidden":28},[27],[20,7928,7930,7933,7936,7939,7942],{"className":7929},[32],[20,7931],{"className":7932,"style":606},[36],[20,7934,5986],{"className":7935,"style":5985},[41,86],[20,7937,687],{"className":7938},[179],[20,7940,88],{"className":7941,"style":87},[41,86],[20,7943,405],{"className":7944},[404]," of a target\npolicy ",[20,7947,7949],{"className":7948},[23],[20,7950,7952],{"className":7951,"ariaHidden":28},[27],[20,7953,7955,7958],{"className":7954},[32],[20,7956],{"className":7957,"style":82},[36],[20,7959,88],{"className":7960,"style":87},[41,86]," from a fixed dataset generated by a behavior policy ",[20,7963,7965],{"className":7964},[23],[20,7966,7968],{"className":7967,"ariaHidden":28},[27],[20,7969,7971,7974],{"className":7970},[32],[20,7972],{"className":7973,"style":492},[36],[20,7975,7977,7980],{"className":7976},[41],[20,7978,88],{"className":7979,"style":87},[41,86],[20,7981,7983],{"className":7982},[191],[20,7984,7986,8006],{"className":7985},[195,196],[20,7987,7989,8003],{"className":7988},[200],[20,7990,7992],{"className":7991,"style":511},[204],[20,7993,7994,7997],{"style":514},[20,7995],{"className":7996,"style":213},[212],[20,7998,8000],{"className":7999},[217,218,219,220],[20,8001,525],{"className":8002,"style":524},[41,86,220],[20,8004,229],{"className":8005},[228],[20,8007,8009],{"className":8008},[200],[20,8010,8012],{"className":8011,"style":535},[204],[20,8013],{},",\nwithout executing ",[20,8016,8018],{"className":8017},[23],[20,8019,8021],{"className":8020,"ariaHidden":28},[27],[20,8022,8024,8027],{"className":8023},[32],[20,8025],{"className":8026,"style":82},[36],[20,8028,88],{"className":8029,"style":87},[41,86],". Importance sampling is unbiased but has variance\nexponential in the horizon; weighted and doubly-robust variants trade a little\nbias for much lower variance.",[130,8032,8034],{"id":8033},"fix-family-1-policy-constraint-bcq","Fix family 1 — policy constraint (BCQ)",[11,8036,8037,8038,8053,8054,8069,8070,8122],{},"The most direct reading of the diagnosis is: never let the target action stray far\nfrom actions the behavior policy would take. If ",[20,8039,8041],{"className":8040},[23],[20,8042,8044],{"className":8043,"ariaHidden":28},[27],[20,8045,8047,8050],{"className":8046},[32],[20,8048],{"className":8049,"style":82},[36],[20,8051,88],{"className":8052,"style":87},[41,86]," only ever proposes actions\nwith support in ",[20,8055,8057],{"className":8056},[23],[20,8058,8060],{"className":8059,"ariaHidden":28},[27],[20,8061,8063,8066],{"className":8062},[32],[20,8064],{"className":8065,"style":37},[36],[20,8067,44],{"className":8068,"style":43},[41,42],", then ",[20,8071,8073],{"className":8072},[23],[20,8074,8076],{"className":8075,"ariaHidden":28},[27],[20,8077,8079,8082],{"className":8078},[32],[20,8080],{"className":8081,"style":1157},[36],[20,8083,8085,8088],{"className":8084},[41],[20,8086,1164],{"className":8087},[41,86],[20,8089,8091],{"className":8090},[191],[20,8092,8094,8114],{"className":8093},[195,196],[20,8095,8097,8111],{"className":8096},[200],[20,8098,8100],{"className":8099,"style":511},[204],[20,8101,8102,8105],{"style":208},[20,8103],{"className":8104,"style":213},[212],[20,8106,8108],{"className":8107},[217,218,219,220],[20,8109,1188],{"className":8110,"style":43},[41,86,220],[20,8112,229],{"className":8113},[228],[20,8115,8117],{"className":8116},[200],[20,8118,8120],{"className":8119,"style":236},[204],[20,8121],{}," is only ever queried where it is\nconstrained by data, and extrapolation error never enters the backup.",[11,8124,8125,8128,8136,8137,8200,8201,8216,8217,8269,8270,8274,8275,8336,8337,8394,8395,8411,8412,2961],{},[15,8126,8127],{},"Batch-Constrained deep Q-learning (BCQ)",[52,8129,8130],{},[55,8131,8135],{"href":8132,"ariaDescribedBy":8133,"dataFootnoteRef":6,"id":8134},"#user-content-fn-bcq",[59],"user-content-fnref-bcq","2"," implements this with a learned\ngenerative model of the behavior policy. A conditional variational autoencoder\n",[20,8138,8140],{"className":8139},[23],[20,8141,8143],{"className":8142,"ariaHidden":28},[27],[20,8144,8146,8149,8191,8194,8197],{"className":8145},[32],[20,8147],{"className":8148,"style":606},[36],[20,8150,8152,8156],{"className":8151},[41],[20,8153,8155],{"className":8154},[41,86],"G",[20,8157,8159],{"className":8158},[191],[20,8160,8162,8183],{"className":8161},[195,196],[20,8163,8165,8180],{"className":8164},[200],[20,8166,8168],{"className":8167,"style":6032},[204],[20,8169,8170,8173],{"style":208},[20,8171],{"className":8172,"style":213},[212],[20,8174,8176],{"className":8175},[217,218,219,220],[20,8177,8179],{"className":8178,"style":87},[41,86,220],"ω",[20,8181,229],{"className":8182},[228],[20,8184,8186],{"className":8185},[200],[20,8187,8189],{"className":8188,"style":236},[204],[20,8190],{},[20,8192,687],{"className":8193},[179],[20,8195,187],{"className":8196},[41,86],[20,8198,405],{"className":8199},[404]," is trained on ",[20,8202,8204],{"className":8203},[23],[20,8205,8207],{"className":8206,"ariaHidden":28},[27],[20,8208,8210,8213],{"className":8209},[32],[20,8211],{"className":8212,"style":37},[36],[20,8214,44],{"className":8215,"style":43},[41,42]," to reproduce the actions ",[20,8218,8220],{"className":8219},[23],[20,8221,8223],{"className":8222,"ariaHidden":28},[27],[20,8224,8226,8229],{"className":8225},[32],[20,8227],{"className":8228,"style":492},[36],[20,8230,8232,8235],{"className":8231},[41],[20,8233,88],{"className":8234,"style":87},[41,86],[20,8236,8238],{"className":8237},[191],[20,8239,8241,8261],{"className":8240},[195,196],[20,8242,8244,8258],{"className":8243},[200],[20,8245,8247],{"className":8246,"style":511},[204],[20,8248,8249,8252],{"style":514},[20,8250],{"className":8251,"style":213},[212],[20,8253,8255],{"className":8254},[217,218,219,220],[20,8256,525],{"className":8257,"style":524},[41,86,220],[20,8259,229],{"className":8260},[228],[20,8262,8264],{"className":8263},[200],[20,8265,8267],{"className":8266,"style":535},[204],[20,8268],{}," took\nat each state — it is a density model of ",[8271,8272,8273],"q",{},"actions that plausibly appear in the data."," At decision time BCQ samples a handful of candidate actions from\n",[20,8276,8278],{"className":8277},[23],[20,8279,8281],{"className":8280,"ariaHidden":28},[27],[20,8282,8284,8287,8327,8330,8333],{"className":8283},[32],[20,8285],{"className":8286,"style":606},[36],[20,8288,8290,8293],{"className":8289},[41],[20,8291,8155],{"className":8292},[41,86],[20,8294,8296],{"className":8295},[191],[20,8297,8299,8319],{"className":8298},[195,196],[20,8300,8302,8316],{"className":8301},[200],[20,8303,8305],{"className":8304,"style":6032},[204],[20,8306,8307,8310],{"style":208},[20,8308],{"className":8309,"style":213},[212],[20,8311,8313],{"className":8312},[217,218,219,220],[20,8314,8179],{"className":8315,"style":87},[41,86,220],[20,8317,229],{"className":8318},[228],[20,8320,8322],{"className":8321},[200],[20,8323,8325],{"className":8324,"style":236},[204],[20,8326],{},[20,8328,687],{"className":8329},[179],[20,8331,187],{"className":8332},[41,86],[20,8334,405],{"className":8335},[404],", allows each a small learned perturbation ",[20,8338,8340],{"className":8339},[23],[20,8341,8343],{"className":8342,"ariaHidden":28},[27],[20,8344,8346,8350],{"className":8345},[32],[20,8347],{"className":8348,"style":8349},[36],"height:0.9805em;vertical-align:-0.2861em;",[20,8351,8353,8358],{"className":8352},[41],[20,8354,8357],{"className":8355,"style":8356},[41,86],"margin-right:0.046em;","ξ",[20,8359,8361],{"className":8360},[191],[20,8362,8364,8386],{"className":8363},[195,196],[20,8365,8367,8383],{"className":8366},[200],[20,8368,8370],{"className":8369,"style":511},[204],[20,8371,8373,8376],{"style":8372},"top:-2.55em;margin-left:-0.046em;margin-right:0.05em;",[20,8374],{"className":8375,"style":213},[212],[20,8377,8379],{"className":8378},[217,218,219,220],[20,8380,8382],{"className":8381},[41,86,220],"ϕ",[20,8384,229],{"className":8385},[228],[20,8387,8389],{"className":8388},[200],[20,8390,8392],{"className":8391,"style":535},[204],[20,8393],{}," bounded by a\nradius ",[20,8396,8398],{"className":8397},[23],[20,8399,8401],{"className":8400,"ariaHidden":28},[27],[20,8402,8404,8407],{"className":8403},[32],[20,8405],{"className":8406,"style":37},[36],[20,8408,8410],{"className":8409},[41],"Φ",", and picks the candidate with the highest ",[20,8413,8415],{"className":8414},[23],[20,8416,8418],{"className":8417,"ariaHidden":28},[27],[20,8419,8421,8424],{"className":8420},[32],[20,8422],{"className":8423,"style":1157},[36],[20,8425,1164],{"className":8426},[41,86],[20,8428,8430],{"className":8429},[140],[20,8431,8433],{"className":8432},[23],[20,8434,8436,8469,8783,8949,9072],{"className":8435,"ariaHidden":28},[27],[20,8437,8439,8442,8445,8448,8451,8454,8457,8460,8463,8466],{"className":8438},[32],[20,8440],{"className":8441,"style":606},[36],[20,8443,88],{"className":8444,"style":87},[41,86],[20,8446,687],{"className":8447},[179],[20,8449,187],{"className":8450},[41,86],[20,8452,405],{"className":8453},[404],[20,8455],{"className":8456,"style":160},[159],[20,8458],{"className":8459,"style":160},[159],[20,8461,165],{"className":8462},[164],[20,8464],{"className":8465,"style":160},[159],[20,8467],{"className":8468,"style":160},[159],[20,8470,8472,8476,8670,8673,8676,8716,8722,8725,8728,8731,8734,8774,8777,8780],{"className":8471},[32],[20,8473],{"className":8474,"style":8475},[36],"height:2.0385em;vertical-align:-1.1885em;",[20,8477,8479],{"className":8478},[1279,1280],[20,8480,8482,8661],{"className":8481},[195,196],[20,8483,8485,8658],{"className":8484},[200],[20,8486,8488,8637],{"className":8487,"style":82},[204],[20,8489,8491,8494],{"style":8490},"top:-2.1146em;margin-left:0em;",[20,8492],{"className":8493,"style":1296},[212],[20,8495,8497],{"className":8496},[217,218,219,220],[20,8498,8500,8541,8544,8585,8588,8591,8594,8634],{"className":8499},[41,220],[20,8501,8503,8506],{"className":8502},[41,220],[20,8504,55],{"className":8505},[41,86,220],[20,8507,8509],{"className":8508},[191],[20,8510,8512,8533],{"className":8511},[195,196],[20,8513,8515,8530],{"className":8514},[200],[20,8516,8519],{"className":8517,"style":8518},[204],"height:0.3281em;",[20,8520,8521,8524],{"style":6923},[20,8522],{"className":8523,"style":663},[212],[20,8525,8527],{"className":8526},[217,667,668,220],[20,8528,224],{"className":8529},[41,86,220],[20,8531,229],{"className":8532},[228],[20,8534,8536],{"className":8535},[200],[20,8537,8539],{"className":8538,"style":6942},[204],[20,8540],{},[20,8542,1254],{"className":8543},[1253,220],[20,8545,8547,8550],{"className":8546},[41,220],[20,8548,8357],{"className":8549,"style":8356},[41,86,220],[20,8551,8553],{"className":8552},[191],[20,8554,8556,8577],{"className":8555},[195,196],[20,8557,8559,8574],{"className":8558},[200],[20,8560,8562],{"className":8561,"style":656},[204],[20,8563,8565,8568],{"style":8564},"top:-2.3488em;margin-left:-0.046em;margin-right:0.0714em;",[20,8566],{"className":8567,"style":663},[212],[20,8569,8571],{"className":8570},[217,667,668,220],[20,8572,8382],{"className":8573},[41,86,220],[20,8575,229],{"className":8576},[228],[20,8578,8580],{"className":8579},[200],[20,8581,8583],{"className":8582,"style":681},[204],[20,8584],{},[20,8586,687],{"className":8587},[179,220],[20,8589,187],{"className":8590},[41,86,220],[20,8592,243],{"className":8593},[242,220],[20,8595,8597,8600],{"className":8596},[41,220],[20,8598,55],{"className":8599},[41,86,220],[20,8601,8603],{"className":8602},[191],[20,8604,8606,8626],{"className":8605},[195,196],[20,8607,8609,8623],{"className":8608},[200],[20,8610,8612],{"className":8611,"style":8518},[204],[20,8613,8614,8617],{"style":6923},[20,8615],{"className":8616,"style":663},[212],[20,8618,8620],{"className":8619},[217,667,668,220],[20,8621,224],{"className":8622},[41,86,220],[20,8624,229],{"className":8625},[228],[20,8627,8629],{"className":8628},[200],[20,8630,8632],{"className":8631,"style":6942},[204],[20,8633],{},[20,8635,405],{"className":8636},[404,220],[20,8638,8639,8642],{"style":1339},[20,8640],{"className":8641,"style":1296},[212],[20,8643,8644],{},[20,8645,8647,8652,8655],{"className":8646},[1279],[20,8648,8651],{"className":8649,"style":8650},[41,1351],"margin-right:0.0139em;","arg",[20,8653],{"className":8654,"style":247},[159],[20,8656,1352],{"className":8657},[41,1351],[20,8659,229],{"className":8660},[228],[20,8662,8664],{"className":8663},[200],[20,8665,8668],{"className":8666,"style":8667},[204],"height:1.1885em;",[20,8669],{},[20,8671],{"className":8672,"style":160},[159],[20,8674],{"className":8675,"style":247},[159],[20,8677,8679,8682],{"className":8678},[41],[20,8680,1164],{"className":8681},[41,86],[20,8683,8685],{"className":8684},[191],[20,8686,8688,8708],{"className":8687},[195,196],[20,8689,8691,8705],{"className":8690},[200],[20,8692,8694],{"className":8693,"style":511},[204],[20,8695,8696,8699],{"style":208},[20,8697],{"className":8698,"style":213},[212],[20,8700,8702],{"className":8701},[217,218,219,220],[20,8703,1188],{"className":8704,"style":43},[41,86,220],[20,8706,229],{"className":8707},[228],[20,8709,8711],{"className":8710},[200],[20,8712,8714],{"className":8713,"style":236},[204],[20,8715],{},[20,8717,8719],{"className":8718},[179],[20,8720,687],{"className":8721},[1671,668],[20,8723,187],{"className":8724},[41,86],[20,8726,243],{"className":8727},[242],[20,8729],{"className":8730,"style":160},[159],[20,8732],{"className":8733,"style":247},[159],[20,8735,8737,8740],{"className":8736},[41],[20,8738,55],{"className":8739},[41,86],[20,8741,8743],{"className":8742},[191],[20,8744,8746,8766],{"className":8745},[195,196],[20,8747,8749,8763],{"className":8748},[200],[20,8750,8752],{"className":8751,"style":205},[204],[20,8753,8754,8757],{"style":208},[20,8755],{"className":8756,"style":213},[212],[20,8758,8760],{"className":8759},[217,218,219,220],[20,8761,224],{"className":8762},[41,86,220],[20,8764,229],{"className":8765},[228],[20,8767,8769],{"className":8768},[200],[20,8770,8772],{"className":8771,"style":236},[204],[20,8773],{},[20,8775],{"className":8776,"style":1249},[159],[20,8778,1254],{"className":8779},[1253],[20,8781],{"className":8782,"style":1249},[159],[20,8784,8786,8789,8829,8832,8835,8838,8841,8881,8884,8890,8893,8896,8899,8939,8942,8946],{"className":8785},[32],[20,8787],{"className":8788,"style":1587},[36],[20,8790,8792,8795],{"className":8791},[41],[20,8793,8357],{"className":8794,"style":8356},[41,86],[20,8796,8798],{"className":8797},[191],[20,8799,8801,8821],{"className":8800},[195,196],[20,8802,8804,8818],{"className":8803},[200],[20,8805,8807],{"className":8806,"style":511},[204],[20,8808,8809,8812],{"style":8372},[20,8810],{"className":8811,"style":213},[212],[20,8813,8815],{"className":8814},[217,218,219,220],[20,8816,8382],{"className":8817},[41,86,220],[20,8819,229],{"className":8820},[228],[20,8822,8824],{"className":8823},[200],[20,8825,8827],{"className":8826,"style":535},[204],[20,8828],{},[20,8830,687],{"className":8831},[179],[20,8833,187],{"className":8834},[41,86],[20,8836,243],{"className":8837},[242],[20,8839],{"className":8840,"style":247},[159],[20,8842,8844,8847],{"className":8843},[41],[20,8845,55],{"className":8846},[41,86],[20,8848,8850],{"className":8849},[191],[20,8851,8853,8873],{"className":8852},[195,196],[20,8854,8856,8870],{"className":8855},[200],[20,8857,8859],{"className":8858,"style":205},[204],[20,8860,8861,8864],{"style":208},[20,8862],{"className":8863,"style":213},[212],[20,8865,8867],{"className":8866},[217,218,219,220],[20,8868,224],{"className":8869},[41,86,220],[20,8871,229],{"className":8872},[228],[20,8874,8876],{"className":8875},[200],[20,8877,8879],{"className":8878,"style":236},[204],[20,8880],{},[20,8882,405],{"className":8883},[404],[20,8885,8887],{"className":8886},[404],[20,8888,405],{"className":8889},[1671,668],[20,8891,243],{"className":8892},[242],[20,8894],{"className":8895,"style":1530},[159],[20,8897],{"className":8898,"style":247},[159],[20,8900,8902,8905],{"className":8901},[41],[20,8903,55],{"className":8904},[41,86],[20,8906,8908],{"className":8907},[191],[20,8909,8911,8931],{"className":8910},[195,196],[20,8912,8914,8928],{"className":8913},[200],[20,8915,8917],{"className":8916,"style":205},[204],[20,8918,8919,8922],{"style":208},[20,8920],{"className":8921,"style":213},[212],[20,8923,8925],{"className":8924},[217,218,219,220],[20,8926,224],{"className":8927},[41,86,220],[20,8929,229],{"className":8930},[228],[20,8932,8934],{"className":8933},[200],[20,8935,8937],{"className":8936,"style":236},[204],[20,8938],{},[20,8940],{"className":8941,"style":160},[159],[20,8943,8945],{"className":8944},[164],"∼",[20,8947],{"className":8948,"style":160},[159],[20,8950,8952,8955,8995,8998,9001,9004,9007,9010,9013,9016,9019,9059,9062,9065,9069],{"className":8951},[32],[20,8953],{"className":8954,"style":724},[36],[20,8956,8958,8961],{"className":8957},[41],[20,8959,8155],{"className":8960},[41,86],[20,8962,8964],{"className":8963},[191],[20,8965,8967,8987],{"className":8966},[195,196],[20,8968,8970,8984],{"className":8969},[200],[20,8971,8973],{"className":8972,"style":6032},[204],[20,8974,8975,8978],{"style":208},[20,8976],{"className":8977,"style":213},[212],[20,8979,8981],{"className":8980},[217,218,219,220],[20,8982,8179],{"className":8983,"style":87},[41,86,220],[20,8985,229],{"className":8986},[228],[20,8988,8990],{"className":8989},[200],[20,8991,8993],{"className":8992,"style":236},[204],[20,8994],{},[20,8996,687],{"className":8997},[179],[20,8999,187],{"className":9000},[41,86],[20,9002,405],{"className":9003},[404],[20,9005,243],{"className":9006},[242],[20,9008],{"className":9009,"style":160},[159],[20,9011],{"className":9012,"style":160},[159],[20,9014],{"className":9015,"style":247},[159],[20,9017,858],{"className":9018},[179],[20,9020,9022,9025],{"className":9021},[41],[20,9023,8357],{"className":9024,"style":8356},[41,86],[20,9026,9028],{"className":9027},[191],[20,9029,9031,9051],{"className":9030},[195,196],[20,9032,9034,9048],{"className":9033},[200],[20,9035,9037],{"className":9036,"style":511},[204],[20,9038,9039,9042],{"style":8372},[20,9040],{"className":9041,"style":213},[212],[20,9043,9045],{"className":9044},[217,218,219,220],[20,9046,8382],{"className":9047},[41,86,220],[20,9049,229],{"className":9050},[228],[20,9052,9054],{"className":9053},[200],[20,9055,9057],{"className":9056,"style":535},[204],[20,9058],{},[20,9060,858],{"className":9061},[404],[20,9063],{"className":9064,"style":160},[159],[20,9066,9068],{"className":9067},[164],"≤",[20,9070],{"className":9071,"style":160},[159],[20,9073,9075,9078],{"className":9074},[32],[20,9076],{"className":9077,"style":37},[36],[20,9079,9081],{"className":9080},[41],"Φ.",[11,9083,9084,9085,9088,9089,9104,9105,9157,9158,9191,9192,9207,9208,9211],{},"The generative model keeps the argmax ",[47,9086,9087],{},"on-support","; the bounded perturbation lets\n",[20,9090,9092],{"className":9091},[23],[20,9093,9095],{"className":9094,"ariaHidden":28},[27],[20,9096,9098,9101],{"className":9097},[32],[20,9099],{"className":9100,"style":82},[36],[20,9102,88],{"className":9103,"style":87},[41,86]," improve on ",[20,9106,9108],{"className":9107},[23],[20,9109,9111],{"className":9110,"ariaHidden":28},[27],[20,9112,9114,9117],{"className":9113},[32],[20,9115],{"className":9116,"style":492},[36],[20,9118,9120,9123],{"className":9119},[41],[20,9121,88],{"className":9122,"style":87},[41,86],[20,9124,9126],{"className":9125},[191],[20,9127,9129,9149],{"className":9128},[195,196],[20,9130,9132,9146],{"className":9131},[200],[20,9133,9135],{"className":9134,"style":511},[204],[20,9136,9137,9140],{"style":514},[20,9138],{"className":9139,"style":213},[212],[20,9141,9143],{"className":9142},[217,218,219,220],[20,9144,525],{"className":9145,"style":524},[41,86,220],[20,9147,229],{"className":9148},[228],[20,9150,9152],{"className":9151},[200],[20,9153,9155],{"className":9154,"style":535},[204],[20,9156],{}," locally without leaping to unseen actions. Setting\n",[20,9159,9161],{"className":9160},[23],[20,9162,9164,9182],{"className":9163,"ariaHidden":28},[27],[20,9165,9167,9170,9173,9176,9179],{"className":9166},[32],[20,9168],{"className":9169,"style":37},[36],[20,9171,8410],{"className":9172},[41],[20,9174],{"className":9175,"style":160},[159],[20,9177,165],{"className":9178},[164],[20,9180],{"className":9181,"style":160},[159],[20,9183,9185,9188],{"className":9184},[32],[20,9186],{"className":9187,"style":2157},[36],[20,9189,2161],{"className":9190},[41]," and one sample recovers behavior cloning; larger ",[20,9193,9195],{"className":9194},[23],[20,9196,9198],{"className":9197,"ariaHidden":28},[27],[20,9199,9201,9204],{"className":9200},[32],[20,9202],{"className":9203,"style":37},[36],[20,9205,8410],{"className":9206},[41]," and more samples\nrecover something closer to Q-learning. BCQ was the paper that named\n",[15,9209,9210],{},"extrapolation error"," and showed standard off-policy deep RL fails on data its\nown earlier version generated.",[2749,9213],{"hash":9214},"d2609622a98cae8f72e89772cbea049f3c6d3abba70568219e18b4017d275f6f",[961,9216,9217],{"type":963},[11,9218,9219,9222,9223,9238,9239,9291,9292,9426],{},[15,9220,9221],{},"Definition (Policy constraint)."," A class of offline methods that restrict the\nlearned policy ",[20,9224,9226],{"className":9225},[23],[20,9227,9229],{"className":9228,"ariaHidden":28},[27],[20,9230,9232,9235],{"className":9231},[32],[20,9233],{"className":9234,"style":82},[36],[20,9236,88],{"className":9237,"style":87},[41,86]," to stay close to the behavior policy ",[20,9240,9242],{"className":9241},[23],[20,9243,9245],{"className":9244,"ariaHidden":28},[27],[20,9246,9248,9251],{"className":9247},[32],[20,9249],{"className":9250,"style":492},[36],[20,9252,9254,9257],{"className":9253},[41],[20,9255,88],{"className":9256,"style":87},[41,86],[20,9258,9260],{"className":9259},[191],[20,9261,9263,9283],{"className":9262},[195,196],[20,9264,9266,9280],{"className":9265},[200],[20,9267,9269],{"className":9268,"style":511},[204],[20,9270,9271,9274],{"style":514},[20,9272],{"className":9273,"style":213},[212],[20,9275,9277],{"className":9276},[217,218,219,220],[20,9278,525],{"className":9279,"style":524},[41,86,220],[20,9281,229],{"className":9282},[228],[20,9284,9286],{"className":9285},[200],[20,9287,9289],{"className":9288,"style":535},[204],[20,9290],{}," — by\ngenerative sampling (BCQ), an explicit divergence penalty\n",[20,9293,9295],{"className":9294},[23],[20,9296,9298,9331,9408],{"className":9297,"ariaHidden":28},[27],[20,9299,9301,9304,9307,9313,9316,9319,9322,9325,9328],{"className":9300},[32],[20,9302],{"className":9303,"style":1587},[36],[20,9305,44],{"className":9306,"style":43},[41,86],[20,9308,9310],{"className":9309},[179],[20,9311,687],{"className":9312},[1671,668],[20,9314,88],{"className":9315,"style":87},[41,86],[20,9317,687],{"className":9318},[179],[20,9320,3676],{"className":9321},[41],[20,9323],{"className":9324,"style":160},[159],[20,9326,858],{"className":9327},[164],[20,9329],{"className":9330,"style":160},[159],[20,9332,9334,9337,9340,9343,9346,9350,9353,9393,9396,9399,9402,9405],{"className":9333},[32],[20,9335],{"className":9336,"style":724},[36],[20,9338,187],{"className":9339},[41,86],[20,9341,405],{"className":9342},[404],[20,9344],{"className":9345,"style":247},[159],[20,9347,9349],{"className":9348},[41],"∥",[20,9351],{"className":9352,"style":247},[159],[20,9354,9356,9359],{"className":9355},[41],[20,9357,88],{"className":9358,"style":87},[41,86],[20,9360,9362],{"className":9361},[191],[20,9363,9365,9385],{"className":9364},[195,196],[20,9366,9368,9382],{"className":9367},[200],[20,9369,9371],{"className":9370,"style":511},[204],[20,9372,9373,9376],{"style":514},[20,9374],{"className":9375,"style":213},[212],[20,9377,9379],{"className":9378},[217,218,219,220],[20,9380,525],{"className":9381,"style":524},[41,86,220],[20,9383,229],{"className":9384},[228],[20,9386,9388],{"className":9387},[200],[20,9389,9391],{"className":9390,"style":535},[204],[20,9392],{},[20,9394,687],{"className":9395},[179],[20,9397,3676],{"className":9398},[41],[20,9400],{"className":9401,"style":160},[159],[20,9403,858],{"className":9404},[164],[20,9406],{"className":9407,"style":160},[159],[20,9409,9411,9414,9417,9420],{"className":9410},[32],[20,9412],{"className":9413,"style":1587},[36],[20,9415,187],{"className":9416},[41,86],[20,9418,405],{"className":9419},[404],[20,9421,9423],{"className":9422},[404],[20,9424,405],{"className":9425},[1671,668]," (e.g. BEAR, BRAC),\nor a KL\u002FMMD term — so the value function is only queried on in-distribution\nactions.",[11,9428,9429,9430,9445,9446,9461,9462,9465,9466,9469],{},"The tension in policy-constraint methods is the choice of ",[20,9431,9433],{"className":9432},[23],[20,9434,9436],{"className":9435,"ariaHidden":28},[27],[20,9437,9439,9442],{"className":9438},[32],[20,9440],{"className":9441,"style":37},[36],[20,9443,8410],{"className":9444},[41]," (or the\ndivergence weight): too tight and ",[20,9447,9449],{"className":9448},[23],[20,9450,9452],{"className":9451,"ariaHidden":28},[27],[20,9453,9455,9458],{"className":9454},[32],[20,9456],{"className":9457,"style":82},[36],[20,9459,88],{"className":9460,"style":87},[41,86]," can never improve on the logged behavior;\ntoo loose and out-of-distribution actions leak back in. That tuning knob is the\nrecurring cost of constraining the ",[47,9463,9464],{},"policy"," rather than the ",[47,9467,9468],{},"value",", which is what\nthe next family attacks instead.",[130,9471,9473],{"id":9472},"fix-family-2-conservative-value-estimation-cql","Fix family 2 — conservative value estimation (CQL)",[11,9475,9476,9477,9480],{},"Instead of constraining which actions the policy may take, constrain the value\nfunction itself: deliberately ",[15,9478,9479],{},"push down"," the Q-values of out-of-distribution\nactions so the argmax has no incentive to pick them. If unseen actions look bad,\nthe greedy policy stays on the data on its own, no generative model required.",[11,9482,9483,9486,9493,9494,9546,9547,9550],{},[15,9484,9485],{},"Conservative Q-Learning (CQL)",[52,9487,9488],{},[55,9489,3837],{"href":9490,"ariaDescribedBy":9491,"dataFootnoteRef":6,"id":9492},"#user-content-fn-cql",[59],"user-content-fnref-cql"," adds one term to the standard Bellman\nerror. The term maximizes ",[20,9495,9497],{"className":9496},[23],[20,9498,9500],{"className":9499,"ariaHidden":28},[27],[20,9501,9503,9506],{"className":9502},[32],[20,9504],{"className":9505,"style":1157},[36],[20,9507,9509,9512],{"className":9508},[41],[20,9510,1164],{"className":9511},[41,86],[20,9513,9515],{"className":9514},[191],[20,9516,9518,9538],{"className":9517},[195,196],[20,9519,9521,9535],{"className":9520},[200],[20,9522,9524],{"className":9523,"style":511},[204],[20,9525,9526,9529],{"style":208},[20,9527],{"className":9528,"style":213},[212],[20,9530,9532],{"className":9531},[217,218,219,220],[20,9533,1188],{"className":9534,"style":43},[41,86,220],[20,9536,229],{"className":9537},[228],[20,9539,9541],{"className":9540},[200],[20,9542,9544],{"className":9543,"style":236},[204],[20,9545],{}," on state-action pairs ",[47,9548,9549],{},"from the data"," and\nminimizes it on actions sampled from the current policy (the ones at risk of being\nout-of-distribution overestimates). In its practical form, at each state the\npenalty pushes down a log-sum-exp over all actions and pulls up the value of the\nlogged action:",[20,9552,9554],{"className":9553},[140],[20,9555,9557],{"className":9556},[23],[20,9558,9560,10143,10327],{"className":9559,"ariaHidden":28},[27],[20,9561,9563,9567,9618,9621,9624,9629,9632,9635,10128,10131,10134,10137,10140],{"className":9562},[32],[20,9564],{"className":9565,"style":9566},[36],"height:4.4702em;vertical-align:-2.7202em;",[20,9568,9570],{"className":9569},[1279,1280],[20,9571,9573,9609],{"className":9572},[195,196],[20,9574,9576,9606],{"className":9575},[200],[20,9577,9580,9592],{"className":9578,"style":9579},[204],"height:0.6679em;",[20,9581,9583,9586],{"style":9582},"top:-2.3479em;margin-left:0em;",[20,9584],{"className":9585,"style":1296},[212],[20,9587,9589],{"className":9588},[217,218,219,220],[20,9590,1188],{"className":9591,"style":43},[41,86,220],[20,9593,9594,9597],{"style":1339},[20,9595],{"className":9596,"style":1296},[212],[20,9598,9599],{},[20,9600,9602],{"className":9601},[1279],[20,9603,9605],{"className":9604},[41,1351],"min",[20,9607,229],{"className":9608},[228],[20,9610,9612],{"className":9611},[200],[20,9613,9616],{"className":9614,"style":9615},[204],"height:0.7521em;",[20,9617],{},[20,9619],{"className":9620,"style":160},[159],[20,9622],{"className":9623,"style":247},[159],[20,9625,9628],{"className":9626,"style":9627},[41,86],"margin-right:0.0037em;","α",[20,9630],{"className":9631,"style":247},[159],[20,9633],{"className":9634,"style":247},[159],[20,9636,9638],{"className":9637},[4290,4291],[20,9639,9641,10119],{"className":9640},[195,196],[20,9642,9644,10116],{"className":9643},[200],[20,9645,9648,9668],{"className":9646,"style":9647},[204],"height:1.75em;",[20,9649,9651,9655],{"style":9650},"top:-1.1659em;",[20,9652],{"className":9653,"style":9654},[212],"height:3.75em;",[20,9656,9658],{"className":9657},[217,218,219,220],[20,9659,9661],{"className":9660},[41,220],[20,9662,9664],{"className":9663},[41,1537,220],[20,9665,9667],{"className":9666},[41,220],"conservative penalty",[20,9669,9671,9674],{"style":9670},"top:-3.75em;",[20,9672],{"className":9673,"style":9654},[212],[20,9675,9677],{"className":9676},[4290,4291],[20,9678,9680,10107],{"className":9679},[195,196],[20,9681,9683,10104],{"className":9682},[200],[20,9684,9686,9717],{"className":9685,"style":9647},[204],[20,9687,9690,9693],{"className":9688,"style":9689},[4340],"top:-1.852em;",[20,9691],{"className":9692,"style":9654},[212],[20,9694,9696,9703,9710],{"className":9695,"style":4349},[4348],[20,9697,9699],{"className":9698,"style":4354},[4353],[4356,9700,9701],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4362},[4364,9702],{"d":4366},[20,9704,9706],{"className":9705,"style":4354},[4370],[4356,9707,9708],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4373},[4364,9709],{"d":4376},[20,9711,9713],{"className":9712,"style":4354},[4380],[4356,9714,9715],{"xmlns":4358,"width":4359,"height":4360,"viewBox":4361,"preserveAspectRatio":4383},[4364,9716],{"d":4386},[20,9718,9719,9722],{"style":9670},[20,9720],{"className":9721,"style":9654},[212],[20,9723,9725,9774,9778,9781],{"className":9724},[41],[20,9726,9728,9731],{"className":9727},[41],[20,9729,6019],{"className":9730},[41,6018],[20,9732,9734],{"className":9733},[191],[20,9735,9737,9766],{"className":9736},[195,196],[20,9738,9740,9763],{"className":9739},[200],[20,9741,9743],{"className":9742,"style":6419},[204],[20,9744,9745,9748],{"style":208},[20,9746],{"className":9747,"style":213},[212],[20,9749,9751],{"className":9750},[217,218,219,220],[20,9752,9754,9757,9760],{"className":9753},[41,220],[20,9755,187],{"className":9756},[41,86,220],[20,9758,8945],{"className":9759},[164,220],[20,9761,44],{"className":9762,"style":43},[41,42,220],[20,9764,229],{"className":9765},[228],[20,9767,9769],{"className":9768},[200],[20,9770,9772],{"className":9771,"style":236},[204],[20,9773],{},[20,9775],{"className":9776,"style":9777},[159],"margin-right:-0.1667em;",[20,9779],{"className":9780,"style":247},[159],[20,9782,9784,9793,9800,9803,9806,9856,9859,9863,9866,9906,9909,9912,9915,9918,9921,9924,9927,9930,9933,9936,9939,10025,10028,10034,10074,10077,10080,10083,10086,10089,10092,10098],{"className":9783},[4290],[20,9785,9789],{"className":9786,"style":9788},[179,9787],"delimcenter","top:0em;",[20,9790,6058],{"className":9791},[1671,9792],"size4",[20,9794,9796],{"className":9795},[1279],[20,9797,9799],{"className":9798,"style":8650},[41,1351],"log",[20,9801],{"className":9802,"style":9777},[159],[20,9804],{"className":9805,"style":247},[159],[20,9807,9809],{"className":9808},[1279,1280],[20,9810,9812,9847],{"className":9811},[195,196],[20,9813,9815,9844],{"className":9814},[200],[20,9816,9819,9834],{"className":9817,"style":9818},[204],"height:1.05em;",[20,9820,9822,9825],{"style":9821},"top:-1.9em;margin-left:0em;",[20,9823],{"className":9824,"style":6560},[212],[20,9826,9828],{"className":9827},[217,218,219,220],[20,9829,9831],{"className":9830},[41,220],[20,9832,55],{"className":9833},[41,86,220],[20,9835,9836,9839],{"style":6580},[20,9837],{"className":9838,"style":6560},[212],[20,9840,9841],{},[20,9842,6068],{"className":9843},[1279,6065,6589],[20,9845,229],{"className":9846},[228],[20,9848,9850],{"className":9849},[200],[20,9851,9854],{"className":9852,"style":9853},[204],"height:1.25em;",[20,9855],{},[20,9857],{"className":9858,"style":247},[159],[20,9860,9862],{"className":9861},[1279],"exp",[20,9864],{"className":9865,"style":247},[159],[20,9867,9869,9872],{"className":9868},[41],[20,9870,1164],{"className":9871},[41,86],[20,9873,9875],{"className":9874},[191],[20,9876,9878,9898],{"className":9877},[195,196],[20,9879,9881,9895],{"className":9880},[200],[20,9882,9884],{"className":9883,"style":511},[204],[20,9885,9886,9889],{"style":208},[20,9887],{"className":9888,"style":213},[212],[20,9890,9892],{"className":9891},[217,218,219,220],[20,9893,1188],{"className":9894,"style":43},[41,86,220],[20,9896,229],{"className":9897},[228],[20,9899,9901],{"className":9900},[200],[20,9902,9904],{"className":9903,"style":236},[204],[20,9905],{},[20,9907,687],{"className":9908},[179],[20,9910,187],{"className":9911},[41,86],[20,9913,243],{"className":9914},[242],[20,9916],{"className":9917,"style":247},[159],[20,9919,55],{"className":9920},[41,86],[20,9922,405],{"className":9923},[404],[20,9925],{"className":9926,"style":160},[159],[20,9928],{"className":9929,"style":1249},[159],[20,9931,3520],{"className":9932},[1253],[20,9934],{"className":9935,"style":160},[159],[20,9937],{"className":9938,"style":1249},[159],[20,9940,9942,9945],{"className":9941},[41],[20,9943,6019],{"className":9944},[41,6018],[20,9946,9948],{"className":9947},[191],[20,9949,9951,10017],{"className":9950},[195,196],[20,9952,9954,10014],{"className":9953},[200],[20,9955,9957],{"className":9956,"style":6032},[204],[20,9958,9959,9962],{"style":208},[20,9960],{"className":9961,"style":213},[212],[20,9963,9965],{"className":9964},[217,218,219,220],[20,9966,9968,9971,9974],{"className":9967},[41,220],[20,9969,55],{"className":9970},[41,86,220],[20,9972,8945],{"className":9973},[164,220],[20,9975,9977,9980],{"className":9976},[41,220],[20,9978,88],{"className":9979,"style":87},[41,86,220],[20,9981,9983],{"className":9982},[191],[20,9984,9986,10006],{"className":9985},[195,196],[20,9987,9989,10003],{"className":9988},[200],[20,9990,9992],{"className":9991,"style":656},[204],[20,9993,9994,9997],{"style":659},[20,9995],{"className":9996,"style":663},[212],[20,9998,10000],{"className":9999},[217,667,668,220],[20,10001,525],{"className":10002,"style":524},[41,86,220],[20,10004,229],{"className":10005},[228],[20,10007,10009],{"className":10008},[200],[20,10010,10012],{"className":10011,"style":681},[204],[20,10013],{},[20,10015,229],{"className":10016},[228],[20,10018,10020],{"className":10019},[200],[20,10021,10023],{"className":10022,"style":7778},[204],[20,10024],{},[20,10026],{"className":10027,"style":9777},[159],[20,10029,10031],{"className":10030},[179],[20,10032,6058],{"className":10033},[1671,668],[20,10035,10037,10040],{"className":10036},[41],[20,10038,1164],{"className":10039},[41,86],[20,10041,10043],{"className":10042},[191],[20,10044,10046,10066],{"className":10045},[195,196],[20,10047,10049,10063],{"className":10048},[200],[20,10050,10052],{"className":10051,"style":511},[204],[20,10053,10054,10057],{"style":208},[20,10055],{"className":10056,"style":213},[212],[20,10058,10060],{"className":10059},[217,218,219,220],[20,10061,1188],{"className":10062,"style":43},[41,86,220],[20,10064,229],{"className":10065},[228],[20,10067,10069],{"className":10068},[200],[20,10070,10072],{"className":10071,"style":236},[204],[20,10073],{},[20,10075,687],{"className":10076},[179],[20,10078,187],{"className":10079},[41,86],[20,10081,243],{"className":10082},[242],[20,10084],{"className":10085,"style":247},[159],[20,10087,55],{"className":10088},[41,86],[20,10090,405],{"className":10091},[404],[20,10093,10095],{"className":10094},[404],[20,10096,6184],{"className":10097},[1671,668],[20,10099,10101],{"className":10100,"style":9788},[404,9787],[20,10102,6184],{"className":10103},[1671,9792],[20,10105,229],{"className":10106},[228],[20,10108,10110],{"className":10109},[200],[20,10111,10114],{"className":10112,"style":10113},[204],"height:1.898em;",[20,10115],{},[20,10117,229],{"className":10118},[228],[20,10120,10122],{"className":10121},[200],[20,10123,10126],{"className":10124,"style":10125},[204],"height:2.7202em;",[20,10127],{},[20,10129],{"className":10130,"style":160},[159],[20,10132],{"className":10133,"style":1249},[159],[20,10135,1254],{"className":10136},[1253],[20,10138],{"className":10139,"style":160},[159],[20,10141],{"className":10142,"style":1249},[159],[20,10144,10146,10149,10220,10223,10266,10269,10275,10278,10318,10321,10324],{"className":10145},[32],[20,10147],{"className":10148,"style":1587},[36],[20,10150,10152,10155,10217],{"className":10151},[41],[20,10153],{"className":10154},[179,6472],[20,10156,10158],{"className":10157},[6476],[20,10159,10161,10208],{"className":10160},[195,196],[20,10162,10164,10205],{"className":10163},[200],[20,10165,10168,10182,10190],{"className":10166,"style":10167},[204],"height:0.8451em;",[20,10169,10170,10173],{"style":6849},[20,10171],{"className":10172,"style":1296},[212],[20,10174,10176],{"className":10175},[217,218,219,220],[20,10177,10179],{"className":10178},[41,220],[20,10180,8135],{"className":10181},[41,220],[20,10183,10184,10187],{"style":6502},[20,10185],{"className":10186,"style":1296},[212],[20,10188],{"className":10189,"style":6510},[6509],[20,10191,10193,10196],{"style":10192},"top:-3.394em;",[20,10194],{"className":10195,"style":1296},[212],[20,10197,10199],{"className":10198},[217,218,219,220],[20,10200,10202],{"className":10201},[41,220],[20,10203,61],{"className":10204},[41,220],[20,10206,229],{"className":10207},[228],[20,10209,10211],{"className":10210},[200],[20,10212,10215],{"className":10213,"style":10214},[204],"height:0.345em;",[20,10216],{},[20,10218],{"className":10219},[404,6472],[20,10221],{"className":10222,"style":247},[159],[20,10224,10226,10229],{"className":10225},[41],[20,10227,6019],{"className":10228},[41,6018],[20,10230,10232],{"className":10231},[191],[20,10233,10235,10258],{"className":10234},[195,196],[20,10236,10238,10255],{"className":10237},[200],[20,10239,10241],{"className":10240,"style":6419},[204],[20,10242,10243,10246],{"style":208},[20,10244],{"className":10245,"style":213},[212],[20,10247,10249],{"className":10248},[217,218,219,220],[20,10250,10252],{"className":10251},[41,220],[20,10253,44],{"className":10254,"style":43},[41,42,220],[20,10256,229],{"className":10257},[228],[20,10259,10261],{"className":10260},[200],[20,10262,10264],{"className":10263,"style":236},[204],[20,10265],{},[20,10267],{"className":10268,"style":9777},[159],[20,10270,10272],{"className":10271},[179],[20,10273,6058],{"className":10274},[1671,668],[20,10276,687],{"className":10277},[179],[20,10279,10281,10284],{"className":10280},[41],[20,10282,1164],{"className":10283},[41,86],[20,10285,10287],{"className":10286},[191],[20,10288,10290,10310],{"className":10289},[195,196],[20,10291,10293,10307],{"className":10292},[200],[20,10294,10296],{"className":10295,"style":511},[204],[20,10297,10298,10301],{"style":208},[20,10299],{"className":10300,"style":213},[212],[20,10302,10304],{"className":10303},[217,218,219,220],[20,10305,1188],{"className":10306,"style":43},[41,86,220],[20,10308,229],{"className":10309},[228],[20,10311,10313],{"className":10312},[200],[20,10314,10316],{"className":10315,"style":236},[204],[20,10317],{},[20,10319],{"className":10320,"style":1249},[159],[20,10322,3520],{"className":10323},[1253],[20,10325],{"className":10326,"style":1249},[159],[20,10328,10330,10334,10368,10439,10469,10475],{"className":10329},[32],[20,10331],{"className":10332,"style":10333},[36],"height:1.2968em;vertical-align:-0.35em;",[20,10335,10337],{"className":10336},[41,1401],[20,10338,10340],{"className":10339},[195],[20,10341,10343],{"className":10342},[200],[20,10344,10346,10356],{"className":10345,"style":6384},[204],[20,10347,10348,10351],{"style":1339},[20,10349],{"className":10350,"style":1296},[212],[20,10352,10355],{"className":10353,"style":10354},[41,42],"margin-right:0.0304em;","B",[20,10357,10358,10361],{"style":6395},[20,10359],{"className":10360,"style":1296},[212],[20,10362,10365],{"className":10363,"style":10364},[1430],"left:-0.1111em;",[20,10366,6406],{"className":10367},[41],[20,10369,10371,10374],{"className":10370},[41],[20,10372,1164],{"className":10373},[41,86],[20,10375,10377],{"className":10376},[191],[20,10378,10380,10431],{"className":10379},[195,196],[20,10381,10383,10428],{"className":10382},[200],[20,10384,10386],{"className":10385,"style":656},[204],[20,10387,10388,10391],{"style":1388},[20,10389],{"className":10390,"style":213},[212],[20,10392,10394],{"className":10393},[217,218,219,220],[20,10395,10397],{"className":10396},[41,220],[20,10398,10400],{"className":10399},[41,1401,220],[20,10401,10403],{"className":10402},[195],[20,10404,10406],{"className":10405},[200],[20,10407,10409,10417],{"className":10408,"style":1411},[204],[20,10410,10411,10414],{"style":1414},[20,10412],{"className":10413,"style":213},[212],[20,10415,1188],{"className":10416,"style":43},[41,86,220],[20,10418,10419,10422],{"style":1423},[20,10420],{"className":10421,"style":213},[212],[20,10423,10425],{"className":10424,"style":1431},[1430],[20,10426,1435],{"className":10427},[41,220],[20,10429,229],{"className":10430},[228],[20,10432,10434],{"className":10433},[200],[20,10435,10437],{"className":10436,"style":1445},[204],[20,10438],{},[20,10440,10442,10445],{"className":10441},[404],[20,10443,405],{"className":10444},[404],[20,10446,10448],{"className":10447},[191],[20,10449,10451],{"className":10450},[195],[20,10452,10454],{"className":10453},[200],[20,10455,10458],{"className":10456,"style":10457},[204],"height:0.8641em;",[20,10459,10460,10463],{"style":375},[20,10461],{"className":10462,"style":213},[212],[20,10464,10466],{"className":10465},[217,218,219,220],[20,10467,8135],{"className":10468},[41,220],[20,10470,10472],{"className":10471},[404],[20,10473,6184],{"className":10474},[1671,668],[20,10476,243],{"className":10477},[242],[11,10479,10480,10481,10524,10525,10540,10541,10544,10545,1825,10597,10600,10601,10642,10643,10658],{},"where ",[20,10482,10484],{"className":10483},[23],[20,10485,10487],{"className":10486,"ariaHidden":28},[27],[20,10488,10490,10493],{"className":10489},[32],[20,10491],{"className":10492,"style":6384},[36],[20,10494,10496],{"className":10495},[41,1401],[20,10497,10499],{"className":10498},[195],[20,10500,10502],{"className":10501},[200],[20,10503,10505,10513],{"className":10504,"style":6384},[204],[20,10506,10507,10510],{"style":1339},[20,10508],{"className":10509,"style":1296},[212],[20,10511,10355],{"className":10512,"style":10354},[41,42],[20,10514,10515,10518],{"style":6395},[20,10516],{"className":10517,"style":1296},[212],[20,10519,10521],{"className":10520,"style":10364},[1430],[20,10522,6406],{"className":10523},[41]," is the empirical Bellman operator and ",[20,10526,10528],{"className":10527},[23],[20,10529,10531],{"className":10530,"ariaHidden":28},[27],[20,10532,10534,10537],{"className":10533},[32],[20,10535],{"className":10536,"style":82},[36],[20,10538,9628],{"className":10539,"style":9627},[41,86]," trades off\nconservatism against fitting the data. The log-sum-exp is high wherever ",[47,10542,10543],{},"any","\naction's value is high, so minimizing it flattens the optimistic peaks; the second\nexpectation protects the values of actions actually in the data. Kumar et al. prove\nthis learns a ",[20,10546,10548],{"className":10547},[23],[20,10549,10551],{"className":10550,"ariaHidden":28},[27],[20,10552,10554,10557],{"className":10553},[32],[20,10555],{"className":10556,"style":1157},[36],[20,10558,10560,10563],{"className":10559},[41],[20,10561,1164],{"className":10562},[41,86],[20,10564,10566],{"className":10565},[191],[20,10567,10569,10589],{"className":10568},[195,196],[20,10570,10572,10586],{"className":10571},[200],[20,10573,10575],{"className":10574,"style":511},[204],[20,10576,10577,10580],{"style":208},[20,10578],{"className":10579,"style":213},[212],[20,10581,10583],{"className":10582},[217,218,219,220],[20,10584,1188],{"className":10585,"style":43},[41,86,220],[20,10587,229],{"className":10588},[228],[20,10590,10592],{"className":10591},[200],[20,10593,10595],{"className":10594,"style":236},[204],[20,10596],{},[15,10598,10599],{},"lower-bounds"," the true ",[20,10602,10604],{"className":10603},[23],[20,10605,10607],{"className":10606,"ariaHidden":28},[27],[20,10608,10610,10613],{"className":10609},[32],[20,10611],{"className":10612,"style":1157},[36],[20,10614,10616,10619],{"className":10615},[41],[20,10617,1164],{"className":10618},[41,86],[20,10620,10622],{"className":10621},[191],[20,10623,10625],{"className":10624},[195],[20,10626,10628],{"className":10627},[200],[20,10629,10631],{"className":10630,"style":2071},[204],[20,10632,10633,10636],{"style":1812},[20,10634],{"className":10635,"style":213},[212],[20,10637,10639],{"className":10638},[217,218,219,220],[20,10640,88],{"className":10641,"style":87},[41,86,220]," (for large enough\n",[20,10644,10646],{"className":10645},[23],[20,10647,10649],{"className":10648,"ariaHidden":28},[27],[20,10650,10652,10655],{"className":10651},[32],[20,10653],{"className":10654,"style":82},[36],[20,10656,9628],{"className":10657,"style":9627},[41,86],") — the value function is provably pessimistic, so its greedy policy cannot\nbe fooled by a fictitious high-value action.",[2749,10660],{"hash":10661},"513586c20aa3160bf5c3f1f41aa9ffd2583f3740d6a21a25350e1d0733b36916",[961,10663,10664],{"type":963},[11,10665,10666,10669,10670,10722,10723,10775],{},[15,10667,10668],{},"Definition (Conservative value estimation)."," A class of offline methods that\nregularize the value function to be pessimistic — pushing down ",[20,10671,10673],{"className":10672},[23],[20,10674,10676],{"className":10675,"ariaHidden":28},[27],[20,10677,10679,10682],{"className":10678},[32],[20,10680],{"className":10681,"style":1157},[36],[20,10683,10685,10688],{"className":10684},[41],[20,10686,1164],{"className":10687},[41,86],[20,10689,10691],{"className":10690},[191],[20,10692,10694,10714],{"className":10693},[195,196],[20,10695,10697,10711],{"className":10696},[200],[20,10698,10700],{"className":10699,"style":511},[204],[20,10701,10702,10705],{"style":208},[20,10703],{"className":10704,"style":213},[212],[20,10706,10708],{"className":10707},[217,218,219,220],[20,10709,1188],{"className":10710,"style":43},[41,86,220],[20,10712,229],{"className":10713},[228],[20,10715,10717],{"className":10716},[200],[20,10718,10720],{"className":10719,"style":236},[204],[20,10721],{}," at\nout-of-distribution actions so that the learned ",[20,10724,10726],{"className":10725},[23],[20,10727,10729],{"className":10728,"ariaHidden":28},[27],[20,10730,10732,10735],{"className":10731},[32],[20,10733],{"className":10734,"style":1157},[36],[20,10736,10738,10741],{"className":10737},[41],[20,10739,1164],{"className":10740},[41,86],[20,10742,10744],{"className":10743},[191],[20,10745,10747,10767],{"className":10746},[195,196],[20,10748,10750,10764],{"className":10749},[200],[20,10751,10753],{"className":10752,"style":511},[204],[20,10754,10755,10758],{"style":208},[20,10756],{"className":10757,"style":213},[212],[20,10759,10761],{"className":10760},[217,218,219,220],[20,10762,1188],{"className":10763,"style":43},[41,86,220],[20,10765,229],{"className":10766},[228],[20,10768,10770],{"className":10769},[200],[20,10771,10773],{"className":10772,"style":236},[204],[20,10774],{}," lower-bounds the true\nvalue. CQL is the canonical instance; the penalty is added directly to the\nBellman-error objective.",[11,10777,10778,10779,10831,10832,10847],{},"CQL's advantage over policy constraint is that it needs no generative model of\n",[20,10780,10782],{"className":10781},[23],[20,10783,10785],{"className":10784,"ariaHidden":28},[27],[20,10786,10788,10791],{"className":10787},[32],[20,10789],{"className":10790,"style":492},[36],[20,10792,10794,10797],{"className":10793},[41],[20,10795,88],{"className":10796,"style":87},[41,86],[20,10798,10800],{"className":10799},[191],[20,10801,10803,10823],{"className":10802},[195,196],[20,10804,10806,10820],{"className":10805},[200],[20,10807,10809],{"className":10808,"style":511},[204],[20,10810,10811,10814],{"style":514},[20,10812],{"className":10813,"style":213},[212],[20,10815,10817],{"className":10816},[217,218,219,220],[20,10818,525],{"className":10819,"style":524},[41,86,220],[20,10821,229],{"className":10822},[228],[20,10824,10826],{"className":10825},[200],[20,10827,10829],{"className":10828,"style":535},[204],[20,10830],{}," and degrades gracefully: even a loosely tuned ",[20,10833,10835],{"className":10834},[23],[20,10836,10838],{"className":10837,"ariaHidden":28},[27],[20,10839,10841,10844],{"className":10840},[32],[20,10842],{"className":10843,"style":82},[36],[20,10845,9628],{"className":10846,"style":9627},[41,86]," still yields a\nvalue function biased toward pessimism, which is the safe direction offline.",[130,10849,10851],{"id":10850},"where-this-leaves-us","Where this leaves us",[11,10853,10854,10855,10857],{},"The setup is now in place. Offline RL fails because bootstrapping queries the value\nfunction at out-of-distribution actions the log never covers; those queries come back\noptimistic, and with no online audit the errors compound through the Bellman backup\ninto a value function detached from reality. Off-policy evaluation lets you ",[47,10856,122],{}," a\ncandidate policy from the log without deploying it, but it inherits the same shift.",[11,10859,10860],{},"Two families of fixes are now in hand, both enforcing the one rule — trust only what the\ndata confirms:",[10862,10863,10864,10871],"ul",{},[10865,10866,10867,10870],"li",{},[15,10868,10869],{},"Policy constraint (BCQ)"," keeps the learned policy close to the behavior policy, so\nthe value function is never asked about far-off actions in the first place.",[10865,10872,10873,10876,10877,10892],{},[15,10874,10875],{},"Conservative value estimation (CQL)"," lets the policy roam but pushes down the\n",[20,10878,10880],{"className":10879},[23],[20,10881,10883],{"className":10882,"ariaHidden":28},[27],[20,10884,10886,10889],{"className":10885},[32],[20,10887],{"className":10888,"style":1157},[36],[20,10890,1164],{"className":10891},[41,86],"-values of out-of-distribution actions, provably lower-bounding the true value.",[11,10894,10895,10896,128],{},"A third, subtler family never queries the value function off the data at all; a\nmodel-based branch builds pessimism into a learned MDP; and a sequence-modeling view\nsidesteps bootstrapping entirely. Those — IQL, MOPO\u002FCOMBO, and Decision Transformer,\nplus the modern fine-tuning and generative-planning lines — continue in\n",[55,10897,10898],{"href":126},"Offline RL: Implicit Methods, Sequence Models, and Beyond",[10900,10901,10904,10909],"section",{"className":10902,"dataFootnotes":6},[10903],"footnotes",[130,10905,10908],{"className":10906,"id":59},[10907],"sr-only","Footnotes",[10910,10911,10912,10930,10945],"ol",{},[10865,10913,10915,10918,10919,10922,10923],{"id":10914},"user-content-fn-offline-survey",[15,10916,10917],{},"Levine, Kumar, Tucker, Fu (2020)",", ",[8271,10920,10921],{},"Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems",", arXiv:2005.01643 — the survey that frames distributional shift as the central offline challenge and organizes the method families. ",[55,10924,10929],{"href":10925,"ariaLabel":10926,"className":10927,"dataFootnoteBackref":6},"#user-content-fnref-offline-survey","Back to reference 1",[10928],"data-footnote-backref","↩",[10865,10931,10933,10918,10936,10939,10940],{"id":10932},"user-content-fn-bcq",[15,10934,10935],{},"Fujimoto, Meger, Precup (2019)",[8271,10937,10938],{},"Off-Policy Deep Reinforcement Learning without Exploration",", ICML — introduces batch-constrained Q-learning (BCQ) and names extrapolation error; shows standard off-policy deep RL fails on fixed datasets. ",[55,10941,10929],{"href":10942,"ariaLabel":10943,"className":10944,"dataFootnoteBackref":6},"#user-content-fnref-bcq","Back to reference 2",[10928],[10865,10946,10948,10918,10951,10954,10955],{"id":10947},"user-content-fn-cql",[15,10949,10950],{},"Kumar, Zhou, Tucker, Levine (2020)",[8271,10952,10953],{},"Conservative Q-Learning for Offline Reinforcement Learning",", NeurIPS — the conservative penalty that pushes down out-of-distribution Q-values and provably lower-bounds the true value. ",[55,10956,10929],{"href":10957,"ariaLabel":10958,"className":10959,"dataFootnoteBackref":6},"#user-content-fnref-cql","Back to reference 3",[10928],{"title":6,"searchDepth":10961,"depth":10961,"links":10962},2,[10963,10964,10968,10969,10970,10971,10972],{"id":132,"depth":10961,"text":133},{"id":1060,"depth":10961,"text":1061,"children":10965},[10966],{"id":2765,"depth":10967,"text":2766},3,{"id":5877,"depth":10961,"text":5878},{"id":8033,"depth":10961,"text":8034},{"id":9472,"depth":10961,"text":9473},{"id":10850,"depth":10961,"text":10851},{"id":59,"depth":10961,"text":10908},[],"computer-science","Every method so far assumed a running environment. The agent acts, the environment\nreturns a reward and a next state, and that fresh feedback keeps the\nestimates accurate — a value that drifts too high produces actions that get tried,\nfail, and get corrected. Offline reinforcement learning removes exactly that\nloop. You are handed a fixed dataset D of transitions logged by some\nearlier policy, and you must return the best policy you can without collecting a\nsingle new transition. No exploration, no rollouts, no chance to test an\noptimistic guess against the environment.1",false,"md",{"moduleNumber":10979,"lessonNumber":10980,"order":10981},5,9,509,"Modern Deep Reinforcement Learning",true,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl",[],"---\ntitle: \"Offline RL: The Problem and Value-Based Fixes\"\nmodule: Modern Deep Reinforcement Learning\nmoduleNumber: 5\nlessonNumber: 9\norder: 509\nsummary: >\n  Offline reinforcement learning learns a policy from a fixed logged dataset with\n  no further environment interaction — off-policy learning pushed to the extreme,\n  and it breaks for the extreme version of the same reason. Bootstrapping queries\n  the value function at out-of-distribution actions the data never covers, those\n  errors are optimistic, and with no online feedback to correct them they compound\n  through the Bellman backup. This lesson sets up the failure and off-policy\n  evaluation, then builds the first two families of pessimistic fixes: policy\n  constraint (BCQ) and conservative value estimation (CQL). A companion lesson\n  takes up implicit methods, model-based offline RL, and Decision Transformer.\ntopics: [Deep RL]\nsources:\n  - book: Sutton & Barto\n    ref: \"Ch. 11 — Off-policy Methods with Approximation; §11.3 The Deadly Triad\"\n  - book: Grokking Deep RL\n    ref: \"Ch. 12 — sample-efficient and off-policy actor-critic; batch\u002Foffline learning\"\n---\n\nEvery method so far assumed a running environment. The agent acts, the environment\nreturns a reward and a next state, and that fresh feedback keeps the\nestimates accurate — a value that drifts too high produces actions that get tried,\nfail, and get corrected. **Offline reinforcement learning** removes exactly that\nloop. You are handed a fixed dataset $\\mathcal{D}$ of transitions logged by some\nearlier policy, and you must return the best policy you can _without collecting a\nsingle new transition_. No exploration, no rollouts, no chance to test an\noptimistic guess against the environment.[^offline-survey]\n\nThis is [off-policy learning](\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad)\ntaken to its limit. Off-policy methods already learn about a target policy $\\pi$\nfrom data generated by a different behavior policy; offline RL is the special case\nwhere the behavior data is _all there is_ and $\\pi$ can never be run to gather\nmore. The [deadly triad](\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad)\n— function approximation, bootstrapping, off-policy training — is present in full,\nand one of its stabilizers, the ability to visit the states your policy actually\ninduces, is gone. The result is a failure mode sharp enough to have its own name,\nand a family of fixes that all reduce to a single instruction: **be pessimistic\nabout what the data cannot verify.**\n\nThis lesson sets up the problem — the failure mode, and how you even _score_ a policy\nfrom a log — then builds the two value-based families of fix: policy constraint (BCQ)\nand conservative value estimation (CQL). The implicit methods, model-based pessimism,\nand the sequence-modeling view of Decision Transformer continue in a\n[companion lesson](\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2).\n\n## The setting and why it matters\n\nFormally, offline RL fixes a static dataset\n\n$$\n\\mathcal{D} = \\{(s_i, a_i, r_i, s_i')\\}_{i=1}^N,\n$$\n\ncollected by a **behavior policy** $\\pi_\\beta$ (possibly a mixture of several\npolicies, or logged human operators). The learner sees $\\mathcal{D}$ once, fits a\npolicy $\\pi$, and is evaluated by deploying $\\pi$ in the true environment. Between\nfitting and deployment there is no interaction. The empirical **state-action\ndistribution** of the data,\n\n$$\nd^{\\pi_\\beta}(s,a) \\;=\\; d^{\\pi_\\beta}(s)\\,\\pi_\\beta(a \\mid s),\n$$\n\nis the only region of the world the learner has ground truth for. Anything outside\nits support is a guess.\n\nThe motivation is that in the domains where RL would help most, online interaction\nis expensive, slow, or dangerous, while _logged data already exists in bulk_.\n\n| Domain | Why online interaction is blocked | What logged data exists |\n| --- | --- | --- |\n| Medicine | a bad treatment policy harms patients | years of electronic health records, dosing logs |\n| Autonomous driving \u002F robotics | a bad action crashes hardware | fleet-scale driving logs, teleoperation traces |\n| Recommendation | live experiments cost revenue and annoy users | billions of impression-click histories |\n| Industrial control | exploration can damage the plant | sensor logs from existing controllers |\n\nIn each case the promise is the same one supervised learning delivered on\nperception: turn a large static dataset into a decision-making system, with all the\nlearning done offline and only the final policy deployed. The obstacle is that a\npolicy is not a label. To evaluate a candidate action the value function must\nestimate outcomes for actions _the dataset may never have taken_, and estimation\nwithout feedback is where offline RL fails.\n\n> **Definition (Offline \u002F batch RL).** The problem of learning a policy $\\pi$ to\n> maximize expected return using only a fixed dataset $\\mathcal{D}$ of transitions\n> collected by a behavior policy $\\pi_\\beta$, with no additional environment\n> interaction during or after training. Also called _batch RL_.\n\n## The core failure: distributional shift and extrapolation error\n\nRun an ordinary off-policy algorithm — Q-learning, or an actor-critic like\n[DDPG](\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control) — on a fixed\ndataset and it typically diverges or returns a policy far worse than $\\pi_\\beta$,\neven when $\\mathcal{D}$ is large and the network has ample capacity. The culprit is\nnot optimization but the **bootstrapping target**.\n\nQ-learning fits $Q_\\theta$ to the Bellman target\n\n$$\ny \\;=\\; r + \\gamma\\, \\max_{a'} Q_{\\bar\\theta}(s', a'),\n\\qquad\\text{or for actor-critic}\\qquad\ny \\;=\\; r + \\gamma\\, Q_{\\bar\\theta}\\bigl(s', \\pi(s')\\bigr).\n$$\n\nThe maximization — or the actor $\\pi$ chasing high value — deliberately seeks the\naction $a'$ that _maximizes_ the current $Q$. On a full environment that is fine:\nif the max lands on an overvalued action, the agent tries it, observes the true low\nreward, and the estimate is corrected. Offline, the transition $(s', a')$ for that\nchosen $a'$ may simply not be in $\\mathcal{D}$. The network's value there is pure\n**extrapolation**, and the argmax preferentially selects wherever the\nextrapolation happens to overshoot.\n\n> **Definition (Out-of-distribution action).** At a state $s$, an action $a$ with\n> little or no support under the data distribution — $d^{\\pi_\\beta}(s, a)\\approx 0$.\n> The value $Q_\\theta(s, a)$ at such a point is unconstrained by any observed\n> return and is fit only by the network's generalization.\n\n> **Definition (Extrapolation error).** The error in $Q_\\theta(s, a)$ at\n> out-of-distribution $(s, a)$, arising because no transition in $\\mathcal{D}$\n> constrains the estimate there. Unlike ordinary approximation error, it is not\n> reduced by more data of the same distribution.\n\nTwo properties make this fatal offline rather than merely inconvenient.\n\nThe first is **selection bias toward overestimation**. Errors in $Q_\\theta$ are\nroughly symmetric — some out-of-distribution actions read too high, some too low —\nbut the $\\max_{a'}$ (or a greedy actor) is a systematic search for the _highest_\nvalue, so it lands on the positive errors by construction. This is the same\noverestimation that motivated Double Q-learning, but without any online experience\nto eventually reveal the truth.\n\nThe second is **compounding with no corrective feedback**. Bootstrapping feeds the\ninflated target back in: an overestimate at $(s', a')$ raises the target $y$ for\n$(s, a)$, which raises $Q_\\theta(s,a)$, which raises the target for whatever\ntransitions bootstrap off $s$, and so on around the Bellman backup. Online, a\nrollout would eventually visit the offending action and puncture the bubble.\nOffline, nothing punctures it. The overestimation propagates and amplifies until\nthe value function is dominated by fictitious high-value actions and the induced\npolicy is garbage.\n\n$$\n% caption: The out-of-distribution overestimation blowup. Within the data support\n% (shaded) the fitted $Q_\\theta$ tracks the true $Q$; outside it, the Bellman\n% $\\max$ selects wherever the extrapolation overshoots, and bootstrapping feeds\n% that inflated value back as a target, so the error compounds with no online\n% return to correct it.\n\\begin{tikzpicture}[>=stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % axes\n  \\draw[->, black] (-0.2,0) -- (9.2,0) node[anchor=north, black, font=\\footnotesize] {action $a$};\n  \\draw[->, black] (0,-0.2) -- (0,4.4) node[anchor=east, black, font=\\footnotesize] {value};\n  % data support region\n  \\fill[acc!8] (2.4,0) rectangle (5.6,4.2);\n  \\draw[acc!45, dashed] (2.4,0) -- (2.4,4.2);\n  \\draw[acc!45, dashed] (5.6,0) -- (5.6,4.2);\n  \\node[acc, font=\\footnotesize, anchor=south] at (4.0,4.2) {data support};\n  % true Q: a gentle bump peaking inside support\n  \\draw[black, thick, smooth]\n    plot coordinates {(0.4,1.0)(1.4,1.25)(2.4,1.6)(3.2,2.05)(4.0,2.2)(4.8,2.0)(5.6,1.6)(6.6,1.3)(7.6,1.15)(8.7,1.05)};\n  \\node[black, anchor=west, font=\\footnotesize] at (8.75,1.05) {true value};\n  % fitted Q: matches inside support, blows up outside on the right\n  \\draw[red, thick, smooth]\n    plot coordinates {(0.4,1.0)(1.4,1.2)(2.4,1.62)(3.2,2.05)(4.0,2.2)(4.8,2.02)(5.6,1.7)(6.4,2.5)(7.2,3.4)(8.0,4.0)(8.7,4.15)};\n  \\node[red, anchor=west, font=\\footnotesize] at (5.9,3.05) {f\\\u002Fitted value};\n  % the argmax picks the overshoot\n  \\fill[red] (8.55,4.12) circle (2.2pt);\n  \\draw[red!70, ->] (7.5,3.9) to[bend left=15] (8.5,4.08);\n  \\node[red, font=\\footnotesize, anchor=east] at (7.5,3.85) {argmax picks OOD overshoot};\n\\end{tikzpicture}\n$$\n\nThe diagnosis determines the fix. The value function is trustworthy _inside_ the data\nsupport and unreliable outside it, and the standard Bellman backup does the one\nthing guaranteed to exploit the unreliable region. Every offline method below\nenforces the same rule a different way: **do not trust, or do not query, values off\nthe data.**\n\n### A worked example: how the bubble inflates\n\nTake a two-state chain: states\n$s_0$ and $s_1$, with two actions $\\{L, R\\}$, discount $\\gamma = 0.9$, and a\ntiny logged dataset that only ever took $R$:\n\n$$\n\\mathcal{D} = \\bigl\\{\\,(s_0, R, 0, s_1),\\ (s_1, R, 1, s_1)\\,\\bigr\\}.\n$$\n\nAction $L$ is never logged from either state. Suppose the true environment is dull:\n$L$ from anywhere gives reward $0$ and self-loops, so $Q^\\ast(s, L) = 0$, while $R$\nearns the returns $Q^\\ast(s_1, R) = 1\u002F(1-\\gamma) = 10$ and $Q^*(s_0, R) = \\gamma\\cdot\n10 = 9$. A well-behaved learner should recover exactly these.\n\nNow initialize a network whose generalization happens to read the unseen action\noptimistically: say it predicts $Q_\\theta(s_1, L) = 3$ before any training (a pure\nextrapolation, off-support, positive by bad luck). Run one Bellman backup on the\nlogged $(s_1, R, 1, s_1)$ transition with a greedy max target:\n\n$$\ny = r + \\gamma \\max_{a'} Q_\\theta(s_1, a')\n  = 1 + 0.9 \\cdot \\max(\\underbrace{Q_\\theta(s_1,R)}_{\\approx 1},\\ \\underbrace{Q_\\theta(s_1,L)}_{=3})\n  = 1 + 0.9 \\cdot 3 = 3.7 .\n$$\n\nThe max reached _off the data_ and pulled the fictitious $3$ into the target for a\n_logged_ transition. $Q_\\theta(s_1, R)$ is now dragged toward $3.7$ — a target the logged data never justifies — and the inflation does not stop.\nThe next backup on $(s_0, R, 0, s_1)$ reads $\\max_{a'} Q_\\theta(s_1, a') = 3.7$ (or\nhigher, if $L$ still reads higher), so $Q_\\theta(s_0, R) \\to 0 + 0.9 \\cdot 3.7 =\n3.33$, and if the network's extrapolation of $Q_\\theta(s_0, L)$ is also optimistic\nthe same overshoot re-enters at $s_0$. Each sweep the phantom value at $L$ feeds a\nhigher target, the higher target raises the logged $Q$, and with no rollout ever\nexecuting $L$ to observe its true $0$ reward, nothing pushes the estimate back\ndown. Online, one visit to $(s_1, L)$ would return reward $0$ and collapse the\nbubble in a single update; offline, the $L$-value is never corrected and the whole\nchain drifts up unboundedly. The induced greedy policy then _prefers $L$_ — an\naction the data shows nothing about and the environment rewards not at all.\n\n$$\n% caption: The overestimation bubble on the two-state chain. A single optimistic\n% off-support value at the unseen action $L$ (dashed) is pulled into the Bellman\n% max, inflating the target for a logged transition; the inflated logged value\n% feeds the next backup, and with no rollout to execute $L$ the estimate climbs\n% each sweep while the true value (flat, low) is never queried.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, black] (-0.2,0) -- (7.4,0) node[anchor=north, black, font=\\footnotesize] {backup sweep};\n  \\draw[->, black] (0,-0.2) -- (0,4.4) node[anchor=east, black, font=\\footnotesize] {Q at unseen L};\n  % sweep ticks\n  \\foreach \\i\u002F\\x in {0\u002F0.6,1\u002F2.0,2\u002F3.4,3\u002F4.8,4\u002F6.2} \\node[black, font=\\scriptsize, anchor=north] at (\\x,0) {\\i};\n  % true value flat low\n  \\draw[black, thick] (0.4,0.7) -- (7.0,0.7);\n  \\node[black, anchor=west, font=\\scriptsize] at (5.0,0.95) {true value = 0};\n  % inflating estimate\n  \\draw[red, thick] plot coordinates {(0.6,1.6)(2.0,2.3)(3.4,3.05)(4.8,3.65)(6.2,4.1)};\n  \\foreach \\x\u002F\\y in {0.6\u002F1.6,2.0\u002F2.3,3.4\u002F3.05,4.8\u002F3.65,6.2\u002F4.1} \\fill[red] (\\x,\\y) circle (1.8pt);\n  \\node[red, anchor=east, font=\\scriptsize] at (5.9,3.75) {estimate climbs};\n  \\draw[red!70, ->] (2.3,2.3) to[bend left=12] (3.2,3.0);\n  \\node[red, anchor=north, font=\\scriptsize] at (1.7,1.5) {no rollout corrects it};\n\\end{tikzpicture}\n$$\n\nThe failure is averted by one change: a value at $L$ that the learner does not\ntrust. Each family below installs precisely that.\n\n## Off-policy evaluation: scoring a policy from the log\n\nBefore improving a policy offline you often want to _score_ one: given a candidate\n$\\pi$ and only the log $\\mathcal{D}$ from $\\pi_\\beta$, estimate $J(\\pi) =\n\\mathbb{E}_\\pi[\\sum_t \\gamma^t r_t]$ without ever running $\\pi$. This is **off-policy\nevaluation** (OPE), and it is both a subproblem inside offline RL and the way a\npractitioner decides whether a learned policy is safe to deploy. It inherits the\nsame distributional-shift problem in a purer form.\n\nThe unbiased classical estimator is **importance sampling**. A trajectory $\\tau$\nlogged under $\\pi_\\beta$ is reweighted by the ratio of the probability $\\pi$ would\nhave produced it to the probability $\\pi_\\beta$ did:\n\n$$\n\\hat J_{\\text{IS}}\n= \\frac{1}{m}\\sum_{k=1}^{m}\n\\Bigl(\\underbrace{\\textstyle\\prod_{t=0}^{T} \\frac{\\pi(a_t \\mid s_t)}{\\pi_\\beta(a_t \\mid s_t)}}_{\\text{cumulative ratio }w_k}\\Bigr)\n\\Bigl(\\textstyle\\sum_{t=0}^{T}\\gamma^t r_t\\Bigr).\n$$\n\nIt is unbiased, but the product of per-step ratios is its undoing: over a horizon\nof $T$ steps the weight $w_k$ is a product of $T{+}1$ factors, and its variance\ngrows _exponentially_ in $T$. A single trajectory whose actions $\\pi$ strongly\nprefers but $\\pi_\\beta$ rarely took gets an astronomically large weight and\ndominates the average, so the estimate is unbiased but so noisy as to be useless\npast short horizons — the **curse of horizon**.\n\n$$\n% caption: Importance-sampling weights for off-policy evaluation. Each logged\n% trajectory is reweighted by the product of per-step ratios pi over pi-beta. Most\n% weights are near zero (the target policy would rarely take those actions) while a\n% few are enormous, so a handful of trajectories dominate the estimate and its\n% variance explodes with the horizon.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, black] (-0.2,0) -- (8.6,0) node[anchor=north, black, font=\\footnotesize] {logged trajectory};\n  \\draw[->, black] (0,-0.2) -- (0,3.9) node[anchor=east, black, font=\\footnotesize] {IS weight};\n  \\foreach \\x\u002F\\h in {0.5\u002F0.15,1.1\u002F0.05,1.7\u002F0.25,2.3\u002F0.1,2.9\u002F3.4,3.5\u002F0.08,4.1\u002F0.2,4.7\u002F0.05,5.3\u002F0.12,5.9\u002F0.06,6.5\u002F2.1,7.1\u002F0.09,7.7\u002F0.15,8.3\u002F0.04} {\n    \\fill[acc!70] (\\x,0) rectangle (\\x+0.34,\\h);\n  }\n  \\node[red, anchor=south, font=\\scriptsize] at (2.9,3.4) {one weight dominates};\n  \\node[black, anchor=west, font=\\scriptsize] at (3.9,0.55) {most near zero};\n\\end{tikzpicture}\n$$\n\nTwo standard repairs reduce the variance. **Weighted importance sampling** (WIS)\nself-normalizes by dividing by $\\sum_k w_k$ instead of $m$; this introduces a small\nbias but is consistent and dramatically lower-variance in practice. **Doubly-robust**\nestimators combine importance sampling with a learned value-function model: the\nmodel supplies a low-variance baseline, and importance sampling corrects only its\nresidual, so the estimator is unbiased if _either_ the model or the ratios are\nright, and low-variance when the model is even roughly accurate. A third route\nabandons per-step ratios entirely and estimates the ratio of _state-action\noccupancies_ $\\rho_\\pi(s,a)\u002F\\rho_{\\pi_\\beta}(s,a)$ directly, sidestepping the\nhorizon product (marginalized OPE). None escapes the basic limit: a policy far\nfrom $\\pi_\\beta$ cannot be evaluated reliably from $\\pi_\\beta$'s data — the same\nsupport limit that constrains offline _learning_ constrains offline _evaluation_.\n\n> **Definition (Off-policy evaluation).** Estimating the value $J(\\pi)$ of a target\n> policy $\\pi$ from a fixed dataset generated by a behavior policy $\\pi_\\beta$,\n> without executing $\\pi$. Importance sampling is unbiased but has variance\n> exponential in the horizon; weighted and doubly-robust variants trade a little\n> bias for much lower variance.\n\n## Fix family 1 — policy constraint (BCQ)\n\nThe most direct reading of the diagnosis is: never let the target action stray far\nfrom actions the behavior policy would take. If $\\pi$ only ever proposes actions\nwith support in $\\mathcal{D}$, then $Q_\\theta$ is only ever queried where it is\nconstrained by data, and extrapolation error never enters the backup.\n\n**Batch-Constrained deep Q-learning (BCQ)**[^bcq] implements this with a learned\ngenerative model of the behavior policy. A conditional variational autoencoder\n$G_\\omega(s)$ is trained on $\\mathcal{D}$ to reproduce the actions $\\pi_\\beta$ took\nat each state — it is a density model of \"actions that plausibly appear in the\ndata.\" At decision time BCQ samples a handful of candidate actions from\n$G_\\omega(s)$, allows each a small learned perturbation $\\xi_\\phi$ bounded by a\nradius $\\Phi$, and picks the candidate with the highest $Q$:\n\n$$\n\\pi(s) \\;=\\; \\argmax_{a_i + \\xi_\\phi(s, a_i)}\\;\nQ_\\theta\\bigl(s,\\; a_i + \\xi_\\phi(s, a_i)\\bigr),\n\\qquad a_i \\sim G_\\omega(s),\\;\\; \\lvert \\xi_\\phi \\rvert \\le \\Phi.\n$$\n\nThe generative model keeps the argmax _on-support_; the bounded perturbation lets\n$\\pi$ improve on $\\pi_\\beta$ locally without leaping to unseen actions. Setting\n$\\Phi = 0$ and one sample recovers behavior cloning; larger $\\Phi$ and more samples\nrecover something closer to Q-learning. BCQ was the paper that named\n**extrapolation error** and showed standard off-policy deep RL fails on data its\nown earlier version generated.\n\n$$\n% caption: BCQ's constrained argmax. A generative model of the behavior policy\n% proposes candidate actions clustered on the data support (blue); each is allowed\n% a small bounded perturbation (arrows); the argmax picks the best perturbed\n% candidate. Because every candidate stays near the data, the value function is\n% never queried at the far-off-support actions where it overshoots (red region).\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, black] (-0.2,0) -- (9.2,0) node[anchor=north, black, font=\\footnotesize] {action $a$};\n  % data support band\n  \\fill[acc!8] (2.2,0) rectangle (5.4,3.0);\n  \\node[acc, anchor=south, font=\\scriptsize] at (3.8,3.0) {data support};\n  % off-support danger\n  \\fill[red!8] (6.4,0) rectangle (9.0,3.0);\n  \\node[red, anchor=south, font=\\scriptsize] at (7.7,3.0) {o\\\u002Ff\\\u002Ff-support: overshoot};\n  % candidate samples from generative model\n  \\foreach \\x in {2.6,3.3,3.9,4.6,5.1} \\fill[acc] (\\x,0.5) circle (2pt);\n  % bounded perturbations\n  \\foreach \\x in {2.6,3.3,3.9,4.6,5.1} \\draw[acc!70, ->] (\\x,0.5) -- (\\x+0.28,0.5);\n  \\node[acc, anchor=north, font=\\scriptsize] at (3.9,0.35) {candidates + small perturbation};\n  % chosen\n  \\fill[acc] (4.6,0.5) circle (3pt);\n  \\draw[acc, ->] (4.6,1.6) -- (4.6,0.65);\n  \\node[acc, anchor=south, font=\\scriptsize] at (4.6,1.6) {argmax (on support)};\n\\end{tikzpicture}\n$$\n\n> **Definition (Policy constraint).** A class of offline methods that restrict the\n> learned policy $\\pi$ to stay close to the behavior policy $\\pi_\\beta$ — by\n> generative sampling (BCQ), an explicit divergence penalty\n> $D\\bigl(\\pi(\\cdot\\mid s)\\,\\Vert\\,\\pi_\\beta(\\cdot\\mid s)\\bigr)$ (e.g. BEAR, BRAC),\n> or a KL\u002FMMD term — so the value function is only queried on in-distribution\n> actions.\n\nThe tension in policy-constraint methods is the choice of $\\Phi$ (or the\ndivergence weight): too tight and $\\pi$ can never improve on the logged behavior;\ntoo loose and out-of-distribution actions leak back in. That tuning knob is the\nrecurring cost of constraining the _policy_ rather than the _value_, which is what\nthe next family attacks instead.\n\n## Fix family 2 — conservative value estimation (CQL)\n\nInstead of constraining which actions the policy may take, constrain the value\nfunction itself: deliberately **push down** the Q-values of out-of-distribution\nactions so the argmax has no incentive to pick them. If unseen actions look bad,\nthe greedy policy stays on the data on its own, no generative model required.\n\n**Conservative Q-Learning (CQL)**[^cql] adds one term to the standard Bellman\nerror. The term maximizes $Q_\\theta$ on state-action pairs _from the data_ and\nminimizes it on actions sampled from the current policy (the ones at risk of being\nout-of-distribution overestimates). In its practical form, at each state the\npenalty pushes down a log-sum-exp over all actions and pulls up the value of the\nlogged action:\n\n$$\n\\min_\\theta\\;\n\\alpha\\,\n\\underbrace{\\mathbb{E}_{s\\sim\\mathcal{D}}\\!\\left[\\log\\!\\sum_{a}\\exp Q_\\theta(s,a)\n\\;-\\; \\mathbb{E}_{a\\sim\\pi_\\beta}\\!\\bigl[Q_\\theta(s,a)\\bigr]\\right]}_{\\text{conservative penalty}}\n\\;+\\;\n\\tfrac12\\,\\mathbb{E}_{\\mathcal{D}}\\!\\bigl[(Q_\\theta - \\hat{\\mathcal{B}}Q_{\\bar\\theta})^2\\bigr],\n$$\n\nwhere $\\hat{\\mathcal{B}}$ is the empirical Bellman operator and $\\alpha$ trades off\nconservatism against fitting the data. The log-sum-exp is high wherever _any_\naction's value is high, so minimizing it flattens the optimistic peaks; the second\nexpectation protects the values of actions actually in the data. Kumar et al. prove\nthis learns a $Q_\\theta$ that **lower-bounds** the true $Q^\\pi$ (for large enough\n$\\alpha$) — the value function is provably pessimistic, so its greedy policy cannot\nbe fooled by a fictitious high-value action.\n\n$$\n% caption: Conservative versus naive value landscapes at a fixed state. The naive\n% $Q_\\theta$ (dashed) fits the data in-support but extrapolates upward off it,\n% handing the argmax a fictitious peak. CQL adds a penalty that pushes down\n% out-of-distribution values, so the conservative $Q_\\theta$ (solid) lower-bounds\n% the true value and its argmax stays on the data.\n\\begin{tikzpicture}[>=stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, black] (-0.2,0) -- (9.2,0) node[anchor=north, black, font=\\footnotesize] {action $a$};\n  \\draw[->, black] (0,-0.2) -- (0,4.4) node[anchor=east, black, font=\\footnotesize] {value};\n  \\fill[acc!8] (2.6,0) rectangle (5.4,4.2);\n  \\node[acc, font=\\footnotesize, anchor=south] at (4.0,4.2) {data support};\n  % true Q\n  \\draw[black, thick, smooth]\n    plot coordinates {(0.4,1.1)(1.6,1.4)(2.6,1.9)(3.4,2.35)(4.0,2.45)(4.6,2.3)(5.4,1.85)(6.6,1.45)(7.8,1.2)(8.7,1.1)};\n  \\node[black, anchor=west, font=\\footnotesize] at (8.75,1.1) {true value};\n  % naive: overshoots off-support (dashed)\n  \\draw[red, thick, dashed, smooth]\n    plot coordinates {(0.4,2.6)(1.6,3.0)(2.6,2.05)(3.4,2.4)(4.0,2.5)(4.6,2.35)(5.4,2.1)(6.6,3.1)(7.8,3.7)(8.7,3.95)};\n  \\node[red, anchor=east, font=\\footnotesize] at (8.35,4.25) {naive value};\n  % conservative: lower bound (solid acc)\n  \\draw[acc, very thick, smooth]\n    plot coordinates {(0.4,0.5)(1.6,0.75)(2.6,1.75)(3.4,2.2)(4.0,2.3)(4.6,2.15)(5.4,1.7)(6.6,0.85)(7.8,0.55)(8.7,0.5)};\n  \\node[acc, anchor=west, font=\\footnotesize] at (5.6,0.72) {conservative value};\n  % argmax markers\n  \\fill[acc] (3.9,2.3) circle (2.2pt);\n  \\fill[red] (8.6,3.92) circle (2.2pt);\n  \\draw[acc!70, ->] (2.7,3.05) to[bend right=10] (3.85,2.42);\n  \\node[acc, font=\\footnotesize, anchor=south] at (2.4,3.0) {argmax on data};\n\\end{tikzpicture}\n$$\n\n> **Definition (Conservative value estimation).** A class of offline methods that\n> regularize the value function to be pessimistic — pushing down $Q_\\theta$ at\n> out-of-distribution actions so that the learned $Q_\\theta$ lower-bounds the true\n> value. CQL is the canonical instance; the penalty is added directly to the\n> Bellman-error objective.\n\nCQL's advantage over policy constraint is that it needs no generative model of\n$\\pi_\\beta$ and degrades gracefully: even a loosely tuned $\\alpha$ still yields a\nvalue function biased toward pessimism, which is the safe direction offline.\n\n## Where this leaves us\n\nThe setup is now in place. Offline RL fails because bootstrapping queries the value\nfunction at out-of-distribution actions the log never covers; those queries come back\noptimistic, and with no online audit the errors compound through the Bellman backup\ninto a value function detached from reality. Off-policy evaluation lets you _score_ a\ncandidate policy from the log without deploying it, but it inherits the same shift.\n\nTwo families of fixes are now in hand, both enforcing the one rule — trust only what the\ndata confirms:\n\n- **Policy constraint (BCQ)** keeps the learned policy close to the behavior policy, so\n  the value function is never asked about far-off actions in the first place.\n- **Conservative value estimation (CQL)** lets the policy roam but pushes down the\n  $Q$-values of out-of-distribution actions, provably lower-bounding the true value.\n\nA third, subtler family never queries the value function off the data at all; a\nmodel-based branch builds pessimism into a learned MDP; and a sequence-modeling view\nsidesteps bootstrapping entirely. Those — IQL, MOPO\u002FCOMBO, and Decision Transformer,\nplus the modern fine-tuning and generative-planning lines — continue in\n[Offline RL: Implicit Methods, Sequence Models, and Beyond](\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2).\n\n[^offline-survey]: **Levine, Kumar, Tucker, Fu (2020)**, \"Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems\", arXiv:2005.01643 — the survey that frames distributional shift as the central offline challenge and organizes the method families.\n[^bcq]: **Fujimoto, Meger, Precup (2019)**, \"Off-Policy Deep Reinforcement Learning without Exploration\", ICML — introduces batch-constrained Q-learning (BCQ) and names extrapolation error; shows standard off-policy deep RL fails on fixed datasets.\n[^cql]: **Kumar, Zhou, Tucker, Levine (2020)**, \"Conservative Q-Learning for Offline Reinforcement Learning\", NeurIPS — the conservative penalty that pushes down out-of-distribution Q-values and provably lower-bounds the true value.\n",{"text":10988,"minutes":10989,"time":10990,"words":10991},"12 min read",11.775,706500,2355,{"title":5,"description":10975},[10994,10997],{"book":10995,"ref":10996},"Sutton & Barto","Ch. 11 — Off-policy Methods with Approximation; §11.3 The Deadly Triad",{"book":10998,"ref":10999},"Grokking Deep RL","Ch. 12 — sample-efficient and off-policy actor-critic; batch\u002Foffline learning","available","07.reinforcement-learning\u002F05.modern-deep-rl\u002F09.offline-rl","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",[11004],"Deep RL","T70oqJ0ctyfDMzFHOCCsCGWTNQFbSiLfF530JQ5zKaU",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":11007,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":11008,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":11009,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":11010,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":11011,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":11012,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":11013,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":11014,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":11015,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":11016,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":11017,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":11018,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":11019,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":11020,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":11021,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":11022,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":11023,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":11024,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":11025,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":11026,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":11027,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":11028,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":11029,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":11030,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":11031,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":11032,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":11033,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":11034,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":11035,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":11036,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":11037,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":11038,"\u002Falgorithms\u002Fsequences\u002Ftries":11039,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":11040,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":11041,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":11042,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":11043,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":11044,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":11045,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":11046,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":11047,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":11048,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":11049,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":11050,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":11051,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":11052,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":11053,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":11054,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":11055,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":11056,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":11057,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":11058,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":11059,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":11060,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":11061,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":11062,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":11063,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":11064,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":11065,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":11066,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":11067,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":11068,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":11069,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":11070,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":11071,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":11072,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":11073,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":11074,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":11075,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":11076,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":11077,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":11078,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":11079,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":11080,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":11081,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":11082,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":11083,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":11084,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":11085,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":11086,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":11087,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":11088,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":11089,"\u002Falgorithms":11090,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":11091,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":11092,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":11093,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":11094,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":11095,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":11096,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":11097,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":11098,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":11099,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":11100,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":11101,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":11102,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":11103,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":11104,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":11105,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":11106,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":11107,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":11108,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":11109,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":11110,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":11111,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":11112,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":11113,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":11114,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":11115,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":11116,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":11117,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":11118,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":11119,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":11120,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":11121,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":11122,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":11123,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":11124,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":11105,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":11125,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":11126,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":11127,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":11095,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":11128,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":11129,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":11130,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":11131,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":11132,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":11133,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":11134,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":11135,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":11136,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":11137,"\u002Fcalculus":11138,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":11139,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":11140,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":11141,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":11142,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":11143,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":11144,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":11145,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":11146,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":11147,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":11148,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":11149,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":11150,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":11151,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":11152,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":11153,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":11154,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":11155,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":11156,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":11157,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":11158,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":11159,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":11160,"\u002Fmechanics\u002Frotation\u002Frolling-motion":11161,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":11162,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":11163,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":11164,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":11165,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":11166,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":11167,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":11168,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":11169,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":11170,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":11171,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":11172,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":11173,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":11174,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":11175,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":11176,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":11177,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":11178,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":11179,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":11180,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":11181,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":11182,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":11183,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":11184,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":11185,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":11186,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":11187,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":11188,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":11189,"\u002Fmechanics":11190,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":11191,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":11192,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":11193,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":11194,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":11195,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":11196,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":11197,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":11198,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":11199,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":11200,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":11201,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":11202,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":11179,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":11203,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":11204,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":11205,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":11175,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":11040,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":11206,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":11166,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":11207,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":11208,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":11209,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":11210,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":11211,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":11212,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":11213,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":11214,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":11215,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":11140,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":11216,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":11217,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":11218,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":11219,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":11220,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":11221,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":11222,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":11223,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":11158,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":11157,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":11224,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":11225,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":11226,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":11227,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":11228,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":11229,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":11230,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":11231,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":11232,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":11184,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":11182,"\u002Felectricity-and-magnetism":11233,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":11234,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":11235,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":11236,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":11237,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":11238,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":11239,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":11240,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":11241,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":11242,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":11092,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":11243,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":11244,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":11096,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":11245,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":11246,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":11247,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":11248,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":11249,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":11250,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":11251,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":11252,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":11253,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":11254,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":11255,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":11256,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":11257,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":11258,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":11259,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":11260,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":11261,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":11262,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":11263,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":11264,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":11131,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":11265,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":11266,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":11267,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":11268,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":11269,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":11270,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":11271,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":11272,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":11273,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":11274,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":11275,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":11276,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":11277,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":11278,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":11279,"\u002Flinear-algebra":11280,"\u002Ftheory-of-computation":11281,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":11282,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":11283,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":11284,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":11285,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":11286,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":11287,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":11288,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":11289,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":11290,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":11291,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":11292,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":11293,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":11294,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":11295,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":11296,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":11297,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":11298,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":11299,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":11300,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":11301,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":11302,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":11303,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":11304,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":11305,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":11306,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":11307,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":11308,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":11309,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":11310,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":11311,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":11312,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":11313,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":11314,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":11315,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":11316,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":11317,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":11318,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":11319,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":11320,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":11321,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":11322,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":11323,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":11324,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":11325,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":11326,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":11327,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":11328,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":11329,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":11330,"\u002Fcomputer-architecture":11281,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":11331,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":11332,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":11333,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":11096,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":11334,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":11095,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":11102,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":11335,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":11336,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":11136,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":11337,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":11338,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":11339,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":11340,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":11341,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":11342,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":11343,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":11344,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":11345,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":11346,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":11347,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":11348,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":11343,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":11349,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":11350,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":11351,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":11352,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":11353,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":11354,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":11355,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":11356,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":11357,"\u002Fdifferential-equations":11358,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":11359,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":11360,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":11361,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":11362,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":11241,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":11363,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":11364,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":11365,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":11366,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":11367,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":11368,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":11111,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":11369,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":11370,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":11371,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":11372,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":11373,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":11374,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":11375,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":11376,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":11377,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":11378,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":11333,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":11379,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":11380,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":11381,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":11382,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":11383,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":11262,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":11384,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":11385,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":11272,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":11386,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":11387,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":11388,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":11389,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":11390,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":11391,"\u002Frelativity":11392,"\u002Fphysical-computing":11281,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":11393,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":11372,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":11394,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":10991,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":11395,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":11396,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":11397,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":11398,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":11399,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":11348,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":11272,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":11400,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":11401,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":11402,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":11403,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":11399,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":11404,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":11380,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":11133,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":11405,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":11406,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":11113,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":11407,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":11408,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":11409,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":11410,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":11411,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":11412,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":11413,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":11414,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":11415,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":11372,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":11416,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":11417,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":11418,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":11405,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":11094,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":11419,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":11420,"\u002Fquantum-mechanics":11421,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":11356,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":11422,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":11423,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":11245,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":11424,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":11131,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":11425,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":11426,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":11268,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":11378,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":11427,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":11428,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":11429,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":11430,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":11431,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":11404,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":11432,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":11238,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":11433,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":11434,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":11092,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":11435,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":11436,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":11394,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":11121,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":11272,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":11437,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":11438,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":11257,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":11382,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":11439,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":11440,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":11441,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":11277,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":11442,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":11443,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":11444,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":11444,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":11445,"\u002Freal-analysis":11446,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":11447,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":11448,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":11449,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":11450,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":11451,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":11452,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":11453,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":11454,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":11455,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":11456,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":11420,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":11457,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":11449,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":11367,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":11458,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":11459,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":11460,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":11461,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":11462,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":11463,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":11464,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":11465,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":11459,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":11466,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":11435,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":11467,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":11468,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":11469,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":11470,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":11471,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":11472,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":11473,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":11474,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":11475,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":11476,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":11110,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":11477,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":11478,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":11352,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":11479,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":11480,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":11408,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":11480,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":11481,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":11482,"\u002Fabstract-algebra":11483,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":11484,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":11485,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":11486,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":11487,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":11488,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":11400,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":11489,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":11490,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":11491,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":11492,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":11493,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":11494,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":11495,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":11235,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":11364,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":11110,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":11496,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":11094,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":11497,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":11498,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":11132,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":11425,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":11499,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":11500,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":11501,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":11502,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":11503,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":11504,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":11505,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":11506,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":11507,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":11508,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":11509,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":11510,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":11511,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":11512,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":11456,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":11513,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":11514,"\u002Fatomic-physics":11515,"\u002Fdatabases":11281,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":11516,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":11517,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":11518,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":11447,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":11519,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":11520,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":11521,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":11522,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":11523,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":11524,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":11525,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":11526,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":11527,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":11528,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":11529,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":11530,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":11531,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":11532,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":11533,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":11534,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":11535,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":11536,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":11537,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":11538,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":11529,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":11539,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":11540,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":11541,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":11542,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":11543,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":11488,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":11544,"\u002Fcategory-theory":11545,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":11546,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":11547,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":11548,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":11549,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":11550,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":11551,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":11510,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":11552,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":11553,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":11554,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":11555,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":11556,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":11557,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":11558,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":11559,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":11560,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":11561,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":11562,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":11563,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":11564,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":11565,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":11566,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":11567,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":11568,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":11569,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":11570,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":11571,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":11572,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":11573,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":11574,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":11575,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":11576,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":11577,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":11578,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":11522,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":11579,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":11580,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":11581,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":11582,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":11583,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":11584,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":11585,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":11075,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":11586,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":11587,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":11311,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":11588,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":11589,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":11590,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":11591,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":11592,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":11593,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":11594,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":11595,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":11596,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":11597,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":11598,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":11599,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":11284,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":11600,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":11601,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":11602,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":11603,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":11321,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":11604,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":11605,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":11606,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":11607,"\u002Fdeep-learning":11281,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":11608,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":11405,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":11609,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":11610,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":11611,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":11612,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":11613,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":11614,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":11615,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":11616,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":11452,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":11617,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":11618,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":11619,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":11340,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":11133,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":11620,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":11621,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":11622,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":11267,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":11623,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":11624,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":11345,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":11272,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":11625,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":11241,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":11111,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":11127,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":11626,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":11627,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":11628,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":11259,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":11629,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":11121,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":11630,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":11631,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":11632,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":11633,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":11454,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":11634,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":11635,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":11636,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":11637,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":11400,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":11638,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":11379,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":11639,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":11240,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":11640,"\u002Fstatistical-mechanics":11641,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":11642,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":11106,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":11366,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":11643,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":11644,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":11645,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":11646,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":11647,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":11648,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":11239,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":11649,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":11650,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":11651,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":11652,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":11382,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":11653,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":11654,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":11655,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":11656,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":11657,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":11436,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":11381,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":11658,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":11659,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":11660,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":11661,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":11372,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":11662,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":11663,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":11608,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":11508,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":11236,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":11664,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":11102,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":11665,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":11666,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":11247,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":11667,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":11501,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":11668,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":11094,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":11669,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":11105,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":11670,"\u002Fcondensed-matter":11421,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":11671,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":11672,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":11673,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":11674,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":11119,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":11675,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":11676,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":11119,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":11677,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":11531,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":11678,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":11679,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":11680,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":11678,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":11681,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":11682,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":11683,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":11684,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":11685,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":11686,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":11687,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":11688,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":11609,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":11689,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":11682,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":11690,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":11691,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":11325,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":11692,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":11509,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":11693,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":11694,"\u002Flogic":11695,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":11696,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":11697,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":11311,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":11698,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":11699,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":11700,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":11701,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":11691,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":11702,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":11703,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":11704,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":11602,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":11705,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":11706,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":11707,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":11708,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":11709,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":11328,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":11710,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":11711,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":11712,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":11713,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":11518,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":11714,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":11690,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":11715,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":11716,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":11717,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":11339,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":11718,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":11468,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":11719,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":11720,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":11301,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":11721,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":11722,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":11723,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":11724,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":11725,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":11578,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":11726,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":11727,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":11728,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":11613,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":11729,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":11730,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":11731,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":11328,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":10991,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":11732,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":11733,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":11734,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":11735,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":11736,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":11737,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":11738,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":11739,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":11740,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":11741,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":11742,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":11743,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":11744,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":11745,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":11746,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":11747,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":11748,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":11749,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":11750,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":11751,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":11752,"\u002Freinforcement-learning":11281,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":11753,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":11754,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":11755,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":11756,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":11757,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":11758,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":11759,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":11760,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":11761,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":11762,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":11763,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":11764,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":11765,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":11607,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":11462,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":11766,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":11767,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":11768,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":11769,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":11770,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":11771,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":11589,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":11772,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":11773,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":11774,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":11775,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":11776,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":11777,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":11778,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":11779,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":11780,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":11781,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":11782,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":11783,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":11521,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":11773,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":11784,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":11062,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":11785,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":11786,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":11787,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":11788,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":11789,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":11570,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":11790,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":11791,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":11792,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":11793,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":11794,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":11795,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":11796,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":11797,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":11798,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":11799,"\u002Fartificial-intelligence":11281,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":11537,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":11800,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":11098,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":11096,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":11631,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":11801,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":11124,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":11433,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":11802,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":11803,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":11804,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":11493,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":11805,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":11806,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":11807,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":11692,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":11808,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":11809,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":11108,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":11337,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":11810,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":11437,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":11811,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":11812,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":11333,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":11420,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":11813,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":11814,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":11815,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":11103,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":11477,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":11340,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":11816,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":11388,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":11451,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":11817,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":11818,"\u002Fnuclear-physics":11819,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":11820,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":11821,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":11458,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":11822,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":11823,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":11824,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":11307,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":11825,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":11826,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":11603,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":11827,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":11772,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":11828,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":11829,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":11830,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":11831,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":11832,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":11833,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":11834,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":11285,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":11835,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":11836,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":11778,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":11837,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":11838,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":11839,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":11840,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":11841,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":11842,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":11843,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":11844,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":11702,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":11845,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":11846,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":11847,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":11848,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":11849,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":11850,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":11851,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":11852,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":11853,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":11854,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":11855,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":11856,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":11556,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":11723,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":11313,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":11857,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":11858,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":11859,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":11860,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":11570,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":11861,"\u002Fnatural-language-processing":11281,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":11862,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":11478,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":11863,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":11623,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":11864,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":11865,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":11813,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":11866,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":11867,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":11428,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":11131,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":11868,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":11383,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":11869,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":11415,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":11870,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":11669,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":11871,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":11872,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":11638,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":11873,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":11102,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":11874,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":11875,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":11876,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":11419,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":11635,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":11877,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":11878,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":11494,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":11879,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":11539,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":11880,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":11881,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":11428,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":11461,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":11882,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":11883,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":11884,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":11517,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":11885,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":11259,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":11886,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":11362,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":11887,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":11888,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":11685,"\u002Fparticle-physics":11889,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":11479,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":11633,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":11890,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":11418,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":11891,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":11478,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":11391,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":11892,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":11893,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":11894,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":11895,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":11896,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":11684,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":11615,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":11897,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":11898,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":11899,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":11900,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":11812,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":11901,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":11375,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":11437,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":11419,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":11902,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":11503,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":11120,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":11903,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":11904,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":11634,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":11905,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":11906,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":11907,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":11908,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":11101,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":11909,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":11910,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":11362,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":11911,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":11912,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":11627,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":11283,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":11913,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":11914,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":11539,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":11525,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":11537,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":11634,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":11915,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":11100,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":11292,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":11916,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":11379,"\u002Fastrophysics-cosmology":11483,"\u002Fcolophon":11917,"\u002F":11281},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":11919,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":11924,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":11928,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":11932,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":11936,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":11940,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":11944,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":11949,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":11953,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":11957,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":11961,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":11966,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":11970,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":11974,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":11978,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":11983,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":11987,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":11991,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":11995,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":11999,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":12003,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":12007,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":12011,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":12015,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":12019,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":12023,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":12027,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":12032,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":12036,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":12040,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":12044,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":12048,"\u002Falgorithms\u002Fsequences\u002Ftries":12052,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":12056,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":12060,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":12065,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":12069,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":12073,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":12077,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":12081,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":12085,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":12089,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":12093,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":12097,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":12101,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":12105,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":12109,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":12113,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":12117,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":12122,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":12126,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":12130,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":12134,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":12138,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":12143,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":12147,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":12151,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":12155,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":12159,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":12163,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":12167,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":12171,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":12175,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":12179,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":12183,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":12188,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":12192,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":12196,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":12200,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":12205,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":12209,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":12213,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":12217,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":12221,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":12225,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":12229,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":12234,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":12238,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":12242,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":12246,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":12251,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":12255,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":12259,"\u002Falgorithms":12263,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":12266,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":12271,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":12275,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":12279,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":12283,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":12288,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":12292,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":12296,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":12300,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":12305,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":12309,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":12313,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":12317,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":12322,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":12326,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":12330,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":12335,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":12339,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":12343,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":12348,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":12352,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":12356,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":12361,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":12365,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":12369,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":12373,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":12378,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":12382,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":12386,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":12391,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":12395,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":12399,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":12403,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":12407,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":12412,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":12416,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":12420,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":12424,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":12428,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":12433,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":12436,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":12440,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":12444,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":12448,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":12453,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":12457,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":12461,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":12465,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":12469,"\u002Fcalculus":12473,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":12476,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":12480,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":12484,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":12489,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":12493,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":12497,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":12501,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":12505,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":12510,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":12514,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":12518,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":12522,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":12526,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":12531,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":12535,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":12539,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":12543,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":12547,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":12552,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":12556,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":12560,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":12565,"\u002Fmechanics\u002Frotation\u002Frolling-motion":12569,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":12573,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":12577,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":12581,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":12585,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":12590,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":12594,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":12598,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":12602,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":12606,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":12610,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":12614,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":12619,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":12623,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":12627,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":12631,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":12635,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":12639,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":12643,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":12647,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":12651,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":12655,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":12659,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":12663,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":12668,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":12672,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":12676,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":12680,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":12684,"\u002Fmechanics":12688,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":12691,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":12696,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":12700,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":12704,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":12708,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":12712,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":12717,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":12721,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":12726,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":12730,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":12734,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":12738,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":12742,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":12747,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":12751,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":12755,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":12759,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":12764,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":12768,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":12772,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":12777,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":12781,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":12785,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":12789,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":12793,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":12798,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":12802,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":12806,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":12810,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":12814,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":12818,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":12823,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":12827,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":12831,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":12835,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":12839,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":12843,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":12847,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":12851,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":12856,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":12860,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":12864,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":12868,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":12872,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":12877,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":12881,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":12885,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":12889,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":12893,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":12898,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":12902,"\u002Felectricity-and-magnetism":12906,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":12909,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":12914,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":12918,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":12922,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":12926,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":12930,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":12935,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":12939,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":12943,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":12947,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":12951,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":12956,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":12960,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":12964,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":12969,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":12973,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":12977,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":12981,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":12985,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":12989,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":12993,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":12998,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":13002,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":13006,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":13010,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":13014,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":13018,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":13022,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":13027,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":13031,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":13035,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":13039,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":13043,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":13047,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":13052,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":13056,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":13060,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":13064,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":13068,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":13073,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":13077,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":13081,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":13085,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":13089,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":13093,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":13098,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":13102,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":13106,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":13110,"\u002Flinear-algebra":13114,"\u002Ftheory-of-computation":13117,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":13120,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":13124,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":13128,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":13132,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":13136,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":13140,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":13145,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":13149,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":13153,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":13157,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":13161,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":13165,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":13169,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":13174,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":13178,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":13182,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":13186,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":13190,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":13195,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":13199,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":13203,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":13207,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":13211,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":13216,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":13220,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":13224,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":13228,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":13232,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":13237,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":13241,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":13245,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":13249,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":13253,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":13258,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":13262,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":13266,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":13270,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":13274,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":13279,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":13283,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":13287,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":13292,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":13296,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":13301,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":13305,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":13309,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":13313,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":13317,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":13322,"\u002Fcomputer-architecture":13326,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":13329,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":13333,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":13337,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":13342,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":13346,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":13350,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":13354,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":13358,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":13362,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":13367,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":13371,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":13375,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":13379,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":13383,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":13387,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":13392,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":13396,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":13400,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":13405,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":13409,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":13414,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":13418,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":13422,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":13427,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":13431,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":13436,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":13440,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":13444,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":13449,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":13453,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":13457,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":13462,"\u002Fdifferential-equations":13466,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":13469,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":13474,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":13478,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":13482,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":13486,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":13490,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":13495,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":13499,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":13503,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":13507,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":13512,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":13516,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":13520,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":13524,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":13529,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":13533,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":13537,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":13541,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":13546,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":13550,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":13554,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":13558,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":13562,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":13566,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":13571,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":13575,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":13579,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":13584,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":13588,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":13592,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":13596,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":13601,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":13605,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":13609,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":13614,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":13618,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":13622,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":13627,"\u002Frelativity":13631,"\u002Fphysical-computing":13634,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":13637,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":13642,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":13646,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":13650,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":13654,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":13659,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":13663,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":13667,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":13672,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":13676,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":13680,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":13684,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":13688,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":13692,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":13697,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":13701,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":13705,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":13709,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":13713,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":13717,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":13722,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":13726,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":13730,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":13734,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":13738,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":13742,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":13746,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":13751,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":13755,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":13759,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":13764,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":13768,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":13772,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":13777,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":13781,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":13786,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":13790,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":13794,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":13798,"\u002Fquantum-mechanics":13802,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":13805,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":13810,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":13814,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":13818,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":13822,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":13827,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":13831,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":13835,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":13839,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":13843,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":13847,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":13852,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":13856,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":13860,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":13864,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":13868,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":13872,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":13876,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":13880,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":13884,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":13888,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":13892,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":13897,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":13901,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":13905,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":13909,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":13914,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":13918,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":13922,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":13925,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":13929,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":13934,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":13938,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":13942,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":13946,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":13951,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":13955,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":13959,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":13963,"\u002Freal-analysis":13967,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":13970,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":13974,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":13978,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":13983,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":13987,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":13991,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":13995,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":14000,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":14004,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":14008,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":14012,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":14016,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":14020,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":14025,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":14029,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":14033,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":14037,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":14042,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":14046,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":14050,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":14054,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":14059,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":14063,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":14067,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":14072,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":14076,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":14080,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":14084,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":14089,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":14093,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":14097,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":14101,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":14106,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":14110,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":14114,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":14119,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":14123,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":14127,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":14131,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":14136,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":14140,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":14144,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":14148,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":14152,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":14157,"\u002Fabstract-algebra":14161,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":14164,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":14169,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":14173,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":14177,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":14181,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":14185,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":14190,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":14194,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":14198,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":14202,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":14206,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":14210,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":14214,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":14219,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":14223,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":14227,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":14231,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":14236,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":14240,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":14244,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":14249,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":14253,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":14257,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":14261,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":14265,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":14269,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":14274,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":14278,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":14282,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":14287,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":14291,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":14295,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":14299,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":14304,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":14308,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":14312,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":14317,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":14321,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":14325,"\u002Fatomic-physics":14329,"\u002Fdatabases":14332,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":14335,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":14339,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":14343,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":14347,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":14351,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":14355,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":14359,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":14364,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":14368,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":14372,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":14377,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":14381,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":14385,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":14390,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":14394,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":14398,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":14402,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":14406,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":14411,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":14415,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":14419,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":14423,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":14428,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":14432,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":14436,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":14440,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":14445,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":14449,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":14453,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":14457,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":14462,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":14466,"\u002Fcategory-theory":14470,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":14473,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":14477,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":14481,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":14485,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":14488,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":14492,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":14496,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":14500,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":14505,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":14509,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":14513,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":14517,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":14521,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":14526,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":14530,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":14534,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":14538,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":14542,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":14547,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":14551,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":14555,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":14559,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":14564,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":14568,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":14572,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":14576,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":14580,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":14584,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":14588,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":14592,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":14596,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":14601,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":14605,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":14609,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":14613,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":14617,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":14622,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":14626,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":14630,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":14634,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":14638,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":14642,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":14646,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":14651,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":14655,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":14659,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":14664,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":14668,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":14672,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":14676,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":14680,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":14684,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":14688,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":14693,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":14697,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":14701,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":14705,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":14709,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":14713,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":14717,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":14721,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":14725,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":14729,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":14733,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":14738,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":14742,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":14746,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":14750,"\u002Fdeep-learning":14754,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":14757,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":14761,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":14765,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":14769,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":14773,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":14777,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":14782,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":14786,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":14790,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":14794,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":14799,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":14803,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":14807,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":14811,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":14816,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":14820,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":14824,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":14828,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":14832,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":14837,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":14841,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":14845,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":14850,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":14854,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":14858,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":14863,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":14867,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":14871,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":14875,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":14880,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":14884,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":14888,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":14892,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":14896,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":14900,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":14905,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":14909,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":14913,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":14917,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":14922,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":14926,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":14930,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":14935,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":14939,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":14943,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":14947,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":14951,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":14956,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":14960,"\u002Fstatistical-mechanics":14964,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":14967,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":14972,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":14976,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":14980,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":14984,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":14989,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":14993,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":14997,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":15001,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":15006,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":15010,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":15014,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":15018,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":15023,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":15027,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":15031,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":15035,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":15040,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":15044,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":15048,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":15052,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":15057,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":15061,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":15065,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":15069,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":15074,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":15078,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":15082,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":15086,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":15090,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":15095,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":15099,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":15104,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":15108,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":15112,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":15116,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":15121,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":15125,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":15129,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":15133,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":15137,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":15142,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":15146,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":15150,"\u002Fcondensed-matter":15154,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":15157,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":15161,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":15166,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":15170,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":15174,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":15178,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":15182,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":15186,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":15190,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":15195,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":15199,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":15203,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":15207,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":15212,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":15216,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":15220,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":15224,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":15229,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":15233,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":15237,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":15241,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":15246,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":15250,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":15254,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":15258,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":15263,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":15267,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":15271,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":15276,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":15280,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":15285,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":15289,"\u002Flogic":15293,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":15296,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":15300,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":15304,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":15308,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":15312,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":15316,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":15320,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":15324,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":15328,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":15332,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":15336,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":15340,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":15344,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":15348,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":15352,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":15356,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":15360,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":15364,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":15368,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":15373,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":15377,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":15381,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":15385,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":15388,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":15392,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":15396,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":15400,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":15404,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":15408,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":15412,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":15416,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":15420,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":15424,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":15428,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":15432,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":15436,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":15440,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":15444,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":15448,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":15452,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":15456,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":15460,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":15464,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":15467,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":15471,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":15475,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":15479,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":15483,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":15487,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":15488,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":15490,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":15494,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":15498,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":15502,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":15506,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":15510,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":15514,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":15518,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":15522,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":15526,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":15530,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":15534,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":15538,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":15543,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":15547,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":15551,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":15555,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":15559,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":15563,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":15567,"\u002Freinforcement-learning":15571,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":15573,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":15577,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":15581,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":15585,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":15589,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":15594,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":15598,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":15602,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":15606,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":15610,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":15614,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":15618,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":15622,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":15626,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":15630,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":15634,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":15638,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":15643,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":15647,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":15651,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":15655,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":15659,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":15663,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":15667,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":15671,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":15675,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":15679,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":15683,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":15687,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":15692,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":15696,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":15700,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":15704,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":15708,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":15712,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":15716,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":15719,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":15723,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":15727,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":15732,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":15736,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":15740,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":15744,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":15747,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":15751,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":15755,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":15759,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":15764,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":15768,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":15772,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":15776,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":15780,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":15784,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":15788,"\u002Fartificial-intelligence":15792,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":15795,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":15800,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":15804,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":15808,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":15812,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":15816,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":15821,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":15825,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":15829,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":15833,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":15838,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":15842,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":15846,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":15850,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":15855,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":15859,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":15864,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":15868,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":15873,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":15877,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":15881,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":15885,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":15890,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":15894,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":15898,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":15903,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":15907,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":15911,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":15916,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":15920,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":15925,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":15929,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":15933,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":15938,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":15942,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":15946,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":15950,"\u002Fnuclear-physics":15954,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":15957,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":15961,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":15965,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":15969,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":15973,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":15977,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":15982,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":15986,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":15990,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":15994,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":15999,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":16003,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":16007,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":16011,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":16015,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":16019,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":16023,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":16026,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":16029,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":16033,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":16037,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":16041,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":16046,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":16050,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":16054,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":16058,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":16062,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":16066,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":16070,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":16074,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":16078,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":16082,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":16086,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":16090,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":16094,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":16098,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":16102,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":16106,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":16110,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":16114,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":16118,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":16122,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":16126,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":16130,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":16134,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":16138,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":16142,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":16146,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":16150,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":16154,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":16159,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":16163,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":16167,"\u002Fnatural-language-processing":16171,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":16174,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":16178,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":16182,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":16186,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":16191,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":16195,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":16199,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":16203,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":16208,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":16212,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":16216,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":16220,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":16225,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":16229,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":16233,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":16237,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":16242,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":16246,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":16250,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":16255,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":16259,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":16263,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":16267,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":16272,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":16276,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":16280,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":16284,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":16289,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":16293,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":16297,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":16301,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":16306,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":16310,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":16314,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":16318,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":16322,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":16327,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":16331,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":16335,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":16340,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":16344,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":16348,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":16352,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":16356,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":16360,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":16364,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":16368,"\u002Fparticle-physics":16372,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":16375,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":16380,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":16384,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":16388,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":16393,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":16397,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":16401,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":16405,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":16410,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":16414,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":16418,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":16422,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":16427,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":16431,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":16435,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":16439,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":16444,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":16448,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":16452,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":16456,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":16461,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":16465,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":16469,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":16474,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":16478,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":16482,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":16486,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":16490,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":16494,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":16498,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":16502,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":16506,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":16511,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":16515,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":16519,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":16523,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":16528,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":16532,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":16536,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":16540,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":16544,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":16549,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":16553,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":16556,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":16560,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":16564,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":16569,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":16573,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":16577,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":16581,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":16585,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":16589,"\u002Fastrophysics-cosmology":16593,"\u002Fcolophon":16596,"\u002F":16599},{"path":11920,"title":11921,"module":11922,"summary":11923},"\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":11925,"title":11926,"module":11922,"summary":11927},"\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":11929,"title":11930,"module":11922,"summary":11931},"\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":11933,"title":11934,"module":11922,"summary":11935},"\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":11937,"title":11938,"module":11922,"summary":11939},"\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":11941,"title":11942,"module":11922,"summary":11943},"\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":11945,"title":11946,"module":11947,"summary":11948},"\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":11950,"title":11951,"module":11947,"summary":11952},"\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":11954,"title":11955,"module":11947,"summary":11956},"\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":11958,"title":11959,"module":11947,"summary":11960},"\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":11962,"title":11963,"module":11964,"summary":11965},"\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":11967,"title":11968,"module":11964,"summary":11969},"\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":11971,"title":11972,"module":11964,"summary":11973},"\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":11975,"title":11976,"module":11964,"summary":11977},"\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":11979,"title":11980,"module":11981,"summary":11982},"\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":11984,"title":11985,"module":11981,"summary":11986},"\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":11988,"title":11989,"module":11981,"summary":11990},"\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":11992,"title":11993,"module":11981,"summary":11994},"\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":11996,"title":11997,"module":11981,"summary":11998},"\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":12000,"title":12001,"module":11981,"summary":12002},"\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":12004,"title":12005,"module":11981,"summary":12006},"\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":12008,"title":12009,"module":11981,"summary":12010},"\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":12012,"title":12013,"module":11981,"summary":12014},"\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":12016,"title":12017,"module":11981,"summary":12018},"\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":12020,"title":12021,"module":11981,"summary":12022},"\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":12024,"title":12025,"module":11981,"summary":12026},"\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":12028,"title":12029,"module":12030,"summary":12031},"\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":12033,"title":12034,"module":12030,"summary":12035},"\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":12037,"title":12038,"module":12030,"summary":12039},"\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":12041,"title":12042,"module":12030,"summary":12043},"\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":12045,"title":12046,"module":12030,"summary":12047},"\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":12049,"title":12050,"module":12030,"summary":12051},"\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":12053,"title":12054,"module":12030,"summary":12055},"\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":12057,"title":12058,"module":12030,"summary":12059},"\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":12061,"title":12062,"module":12063,"summary":12064},"\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":12066,"title":12067,"module":12063,"summary":12068},"\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":12070,"title":12071,"module":12063,"summary":12072},"\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":12074,"title":12075,"module":12063,"summary":12076},"\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":12078,"title":12079,"module":12063,"summary":12080},"\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":12082,"title":12083,"module":12063,"summary":12084},"\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":12086,"title":12087,"module":12063,"summary":12088},"\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":12090,"title":12091,"module":12063,"summary":12092},"\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":12094,"title":12095,"module":12063,"summary":12096},"\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":12098,"title":12099,"module":12063,"summary":12100},"\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":12102,"title":12103,"module":12063,"summary":12104},"\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":12106,"title":12107,"module":12063,"summary":12108},"\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":12110,"title":12111,"module":12063,"summary":12112},"\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":12114,"title":12115,"module":12063,"summary":12116},"\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":12118,"title":12119,"module":12120,"summary":12121},"\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":12123,"title":12124,"module":12120,"summary":12125},"\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":12127,"title":12128,"module":12120,"summary":12129},"\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":12131,"title":12132,"module":12120,"summary":12133},"\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":12135,"title":12136,"module":12120,"summary":12137},"\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":12139,"title":12140,"module":12141,"summary":12142},"\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":12144,"title":12145,"module":12141,"summary":12146},"\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":12148,"title":12149,"module":12141,"summary":12150},"\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":12152,"title":12153,"module":12141,"summary":12154},"\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":12156,"title":12157,"module":12141,"summary":12158},"\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":12160,"title":12161,"module":12141,"summary":12162},"\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":12164,"title":12165,"module":12141,"summary":12166},"\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":12168,"title":12169,"module":12141,"summary":12170},"\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":12172,"title":12173,"module":12141,"summary":12174},"\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":12176,"title":12177,"module":12141,"summary":12178},"\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":12180,"title":12181,"module":12141,"summary":12182},"\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":12184,"title":12185,"module":12186,"summary":12187},"\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":12189,"title":12190,"module":12186,"summary":12191},"\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":12193,"title":12194,"module":12186,"summary":12195},"\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":12197,"title":12198,"module":12186,"summary":12199},"\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":12201,"title":12202,"module":12203,"summary":12204},"\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":12206,"title":12207,"module":12203,"summary":12208},"\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":12210,"title":12211,"module":12203,"summary":12212},"\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":12214,"title":12215,"module":12203,"summary":12216},"\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":12218,"title":12219,"module":12203,"summary":12220},"\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":12222,"title":12223,"module":12203,"summary":12224},"\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":12226,"title":12227,"module":12203,"summary":12228},"\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":12230,"title":12231,"module":12232,"summary":12233},"\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":12235,"title":12236,"module":12232,"summary":12237},"\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":12239,"title":12240,"module":12232,"summary":12241},"\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":12243,"title":12244,"module":12232,"summary":12245},"\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":12247,"title":12248,"module":12249,"summary":12250},"\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":12252,"title":12253,"module":12249,"summary":12254},"\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":12256,"title":12257,"module":12249,"summary":12258},"\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":12260,"title":12261,"module":12249,"summary":12262},"\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":12264,"title":12265,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":12267,"title":12268,"module":12269,"summary":12270},"\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":12272,"title":12273,"module":12269,"summary":12274},"\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":12276,"title":12277,"module":12269,"summary":12278},"\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":12280,"title":12281,"module":12269,"summary":12282},"\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":12284,"title":12285,"module":12286,"summary":12287},"\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":12289,"title":12290,"module":12286,"summary":12291},"\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":12293,"title":12294,"module":12286,"summary":12295},"\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":12297,"title":12298,"module":12286,"summary":12299},"\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":12301,"title":12302,"module":12303,"summary":12304},"\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":12306,"title":12307,"module":12303,"summary":12308},"\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":12310,"title":12311,"module":12303,"summary":12312},"\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":12314,"title":12315,"module":12303,"summary":12316},"\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":12318,"title":12319,"module":12320,"summary":12321},"\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":12323,"title":12324,"module":12320,"summary":12325},"\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":12327,"title":12328,"module":12320,"summary":12329},"\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":12331,"title":12332,"module":12333,"summary":12334},"\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":12336,"title":12337,"module":12333,"summary":12338},"\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":12340,"title":12341,"module":12333,"summary":12342},"\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":12344,"title":12345,"module":12346,"summary":12347},"\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":12349,"title":12350,"module":12346,"summary":12351},"\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":12353,"title":12354,"module":12346,"summary":12355},"\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":12357,"title":12358,"module":12359,"summary":12360},"\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":12362,"title":12363,"module":12359,"summary":12364},"\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":12366,"title":12367,"module":12359,"summary":12368},"\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":12370,"title":12371,"module":12359,"summary":12372},"\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":12374,"title":12375,"module":12376,"summary":12377},"\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":12379,"title":12380,"module":12376,"summary":12381},"\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":12383,"title":12384,"module":12376,"summary":12385},"\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":12387,"title":12388,"module":12389,"summary":12390},"\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":12392,"title":12393,"module":12389,"summary":12394},"\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":12396,"title":12397,"module":12389,"summary":12398},"\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":12400,"title":12401,"module":12389,"summary":12402},"\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":12404,"title":12405,"module":12389,"summary":12406},"\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":12408,"title":12409,"module":12410,"summary":12411},"\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":12413,"title":12414,"module":12410,"summary":12415},"\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":12417,"title":12418,"module":12410,"summary":12419},"\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":12421,"title":12422,"module":12410,"summary":12423},"\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":12425,"title":12426,"module":12410,"summary":12427},"\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":12429,"title":12430,"module":12431,"summary":12432},"\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":12434,"title":12431,"module":12431,"summary":12435},"\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":12437,"title":12438,"module":12431,"summary":12439},"\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":12441,"title":12442,"module":12431,"summary":12443},"\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":12445,"title":12446,"module":12431,"summary":12447},"\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":12449,"title":12450,"module":12451,"summary":12452},"\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":12454,"title":12455,"module":12451,"summary":12456},"\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":12458,"title":12459,"module":12451,"summary":12460},"\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":12462,"title":12463,"module":12451,"summary":12464},"\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":12466,"title":12467,"module":12451,"summary":12468},"\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":12470,"title":12471,"module":12451,"summary":12472},"\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":12474,"title":12475,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":12477,"title":12478,"module":11922,"summary":12479},"\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":12481,"title":12482,"module":11922,"summary":12483},"\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":12485,"title":12486,"module":12487,"summary":12488},"\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":12490,"title":12491,"module":12487,"summary":12492},"\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":12494,"title":12495,"module":12487,"summary":12496},"\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":12498,"title":12499,"module":12487,"summary":12500},"\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":12502,"title":12503,"module":12487,"summary":12504},"\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":12506,"title":12507,"module":12508,"summary":12509},"\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":12511,"title":12512,"module":12508,"summary":12513},"\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":12515,"title":12516,"module":12508,"summary":12517},"\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":12519,"title":12520,"module":12508,"summary":12521},"\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":12523,"title":12524,"module":12508,"summary":12525},"\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":12527,"title":12528,"module":12529,"summary":12530},"\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":12532,"title":12533,"module":12529,"summary":12534},"\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":12536,"title":12537,"module":12529,"summary":12538},"\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":12540,"title":12541,"module":12529,"summary":12542},"\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":12544,"title":12545,"module":12529,"summary":12546},"\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":12548,"title":12549,"module":12550,"summary":12551},"\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":12553,"title":12554,"module":12550,"summary":12555},"\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":12557,"title":12558,"module":12550,"summary":12559},"\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":12561,"title":12562,"module":12563,"summary":12564},"\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":12566,"title":12567,"module":12563,"summary":12568},"\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":12570,"title":12571,"module":12563,"summary":12572},"\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":12574,"title":12575,"module":12563,"summary":12576},"\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":12578,"title":12579,"module":12563,"summary":12580},"\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":12582,"title":12583,"module":12563,"summary":12584},"\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":12586,"title":12587,"module":12588,"summary":12589},"\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":12591,"title":12592,"module":12588,"summary":12593},"\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":12595,"title":12596,"module":12588,"summary":12597},"\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":12599,"title":12600,"module":12588,"summary":12601},"\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":12603,"title":12604,"module":12588,"summary":12605},"\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":12607,"title":12608,"module":12588,"summary":12609},"\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":12611,"title":12612,"module":12588,"summary":12613},"\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":12615,"title":12616,"module":12617,"summary":12618},"\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":12620,"title":12621,"module":12617,"summary":12622},"\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":12624,"title":12625,"module":12617,"summary":12626},"\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":12628,"title":12629,"module":12617,"summary":12630},"\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":12632,"title":12633,"module":12617,"summary":12634},"\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":12636,"title":12637,"module":12617,"summary":12638},"\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":12640,"title":12641,"module":12617,"summary":12642},"\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":12644,"title":12645,"module":12617,"summary":12646},"\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":12648,"title":12649,"module":12617,"summary":12650},"\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":12652,"title":12653,"module":12617,"summary":12654},"\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":12656,"title":12657,"module":12617,"summary":12658},"\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":12660,"title":12661,"module":12617,"summary":12662},"\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":12664,"title":12665,"module":12666,"summary":12667},"\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":12669,"title":12670,"module":12666,"summary":12671},"\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":12673,"title":12674,"module":12666,"summary":12675},"\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":12677,"title":12678,"module":12666,"summary":12679},"\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":12681,"title":12682,"module":12666,"summary":12683},"\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":12685,"title":12686,"module":12666,"summary":12687},"\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":12689,"title":12690,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":12692,"title":12693,"module":12694,"summary":12695},"\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":12697,"title":12698,"module":12694,"summary":12699},"\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":12701,"title":12702,"module":12694,"summary":12703},"\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":12705,"title":12706,"module":12694,"summary":12707},"\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":12709,"title":12710,"module":12694,"summary":12711},"\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":12713,"title":12714,"module":12715,"summary":12716},"\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":12718,"title":12719,"module":12715,"summary":12720},"\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":12722,"title":12723,"module":12724,"summary":12725},"\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":12727,"title":12728,"module":12724,"summary":12729},"\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":12731,"title":12732,"module":12724,"summary":12733},"\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":12735,"title":12736,"module":12724,"summary":12737},"\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":12739,"title":12740,"module":12724,"summary":12741},"\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":12743,"title":12744,"module":12745,"summary":12746},"\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":12748,"title":12749,"module":12745,"summary":12750},"\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":12752,"title":12753,"module":12745,"summary":12754},"\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":12756,"title":12757,"module":12745,"summary":12758},"\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":12760,"title":12761,"module":12762,"summary":12763},"\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":12765,"title":12766,"module":12762,"summary":12767},"\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":12769,"title":12770,"module":12762,"summary":12771},"\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":12773,"title":12774,"module":12775,"summary":12776},"\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":12778,"title":12779,"module":12775,"summary":12780},"\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":12782,"title":12783,"module":12775,"summary":12784},"\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":12786,"title":12787,"module":12775,"summary":12788},"\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":12790,"title":12791,"module":12775,"summary":12792},"\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":12794,"title":12795,"module":12796,"summary":12797},"\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":12799,"title":12800,"module":12796,"summary":12801},"\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":12803,"title":12804,"module":12796,"summary":12805},"\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":12807,"title":12808,"module":12796,"summary":12809},"\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":12811,"title":12812,"module":12796,"summary":12813},"\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":12815,"title":12816,"module":12796,"summary":12817},"\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":12819,"title":12820,"module":12821,"summary":12822},"\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":12824,"title":12825,"module":12821,"summary":12826},"\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":12828,"title":12829,"module":12821,"summary":12830},"\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":12832,"title":12833,"module":12821,"summary":12834},"\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":12836,"title":12837,"module":12821,"summary":12838},"\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":12840,"title":12841,"module":12821,"summary":12842},"\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":12844,"title":12845,"module":12821,"summary":12846},"\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":12848,"title":12849,"module":12821,"summary":12850},"\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":12852,"title":12853,"module":12854,"summary":12855},"\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":12857,"title":12858,"module":12854,"summary":12859},"\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":12861,"title":12862,"module":12854,"summary":12863},"\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":12865,"title":12866,"module":12854,"summary":12867},"\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":12869,"title":12870,"module":12854,"summary":12871},"\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":12873,"title":12874,"module":12875,"summary":12876},"\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":12878,"title":12879,"module":12875,"summary":12880},"\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":12882,"title":12883,"module":12875,"summary":12884},"\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":12886,"title":12887,"module":12875,"summary":12888},"\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":12890,"title":12891,"module":12875,"summary":12892},"\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":12894,"title":12895,"module":12896,"summary":12897},"\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":12899,"title":12900,"module":12896,"summary":12901},"\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":12903,"title":12904,"module":12896,"summary":12905},"\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":12907,"title":12908,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":12910,"title":12911,"module":12912,"summary":12913},"\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":12915,"title":12916,"module":12912,"summary":12917},"\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":12919,"title":12920,"module":12912,"summary":12921},"\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":12923,"title":12924,"module":12912,"summary":12925},"\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":12927,"title":12928,"module":12912,"summary":12929},"\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":12931,"title":12932,"module":12933,"summary":12934},"\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":12936,"title":12937,"module":12933,"summary":12938},"\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":12940,"title":12941,"module":12933,"summary":12942},"\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":12944,"title":12945,"module":12933,"summary":12946},"\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":12948,"title":12949,"module":12933,"summary":12950},"\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":12952,"title":12953,"module":12954,"summary":12955},"\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":12957,"title":12958,"module":12954,"summary":12959},"\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":12961,"title":12962,"module":12954,"summary":12963},"\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":12965,"title":12966,"module":12967,"summary":12968},"\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":12970,"title":12971,"module":12967,"summary":12972},"\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":12974,"title":12975,"module":12967,"summary":12976},"\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":12978,"title":12979,"module":12967,"summary":12980},"\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":12982,"title":12983,"module":12967,"summary":12984},"\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":12986,"title":12987,"module":12967,"summary":12988},"\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":12990,"title":12991,"module":12967,"summary":12992},"\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":12994,"title":12995,"module":12996,"summary":12997},"\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":12999,"title":13000,"module":12996,"summary":13001},"\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":13003,"title":13004,"module":12996,"summary":13005},"\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":13007,"title":13008,"module":12996,"summary":13009},"\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":13011,"title":13012,"module":12996,"summary":13013},"\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":13015,"title":13016,"module":12996,"summary":13017},"\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":13019,"title":13020,"module":12996,"summary":13021},"\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":13023,"title":13024,"module":13025,"summary":13026},"\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":13028,"title":13029,"module":13025,"summary":13030},"\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":13032,"title":13033,"module":13025,"summary":13034},"\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":13036,"title":13037,"module":13025,"summary":13038},"\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":13040,"title":13041,"module":13025,"summary":13042},"\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":13044,"title":13045,"module":13025,"summary":13046},"\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":13048,"title":13049,"module":13050,"summary":13051},"\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":13053,"title":13054,"module":13050,"summary":13055},"\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":13057,"title":13058,"module":13050,"summary":13059},"\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":13061,"title":13062,"module":13050,"summary":13063},"\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":13065,"title":13066,"module":13050,"summary":13067},"\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":13069,"title":13070,"module":13071,"summary":13072},"\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":13074,"title":13075,"module":13071,"summary":13076},"\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":13078,"title":13079,"module":13071,"summary":13080},"\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":13082,"title":13083,"module":13071,"summary":13084},"\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":13086,"title":13087,"module":13071,"summary":13088},"\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":13090,"title":13091,"module":13071,"summary":13092},"\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":13094,"title":13095,"module":13096,"summary":13097},"\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":13099,"title":13100,"module":13096,"summary":13101},"\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":13103,"title":13104,"module":13096,"summary":13105},"\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":13107,"title":13108,"module":13096,"summary":13109},"\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":13111,"title":13112,"module":13096,"summary":13113},"\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":13115,"title":13116,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":13118,"title":13119,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":13121,"title":13122,"module":11922,"summary":13123},"\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":13125,"title":13126,"module":11922,"summary":13127},"\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":13129,"title":13130,"module":11922,"summary":13131},"\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":13133,"title":13134,"module":11922,"summary":13135},"\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":13137,"title":13138,"module":11922,"summary":13139},"\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":13141,"title":13142,"module":13143,"summary":13144},"\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":13146,"title":13147,"module":13143,"summary":13148},"\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":13150,"title":13151,"module":13143,"summary":13152},"\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":13154,"title":13155,"module":13143,"summary":13156},"\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":13158,"title":13159,"module":13143,"summary":13160},"\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":13162,"title":13163,"module":13143,"summary":13164},"\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":13166,"title":13167,"module":13143,"summary":13168},"\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":13170,"title":13171,"module":13172,"summary":13173},"\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":13175,"title":13176,"module":13172,"summary":13177},"\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":13179,"title":13180,"module":13172,"summary":13181},"\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":13183,"title":13184,"module":13172,"summary":13185},"\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":13187,"title":13188,"module":13172,"summary":13189},"\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":13191,"title":13192,"module":13193,"summary":13194},"\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":13196,"title":13197,"module":13193,"summary":13198},"\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":13200,"title":13201,"module":13193,"summary":13202},"\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":13204,"title":13205,"module":13193,"summary":13206},"\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":13208,"title":13209,"module":13193,"summary":13210},"\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":13212,"title":13213,"module":13214,"summary":13215},"\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":13217,"title":13218,"module":13214,"summary":13219},"\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":13221,"title":13222,"module":13214,"summary":13223},"\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":13225,"title":13226,"module":13214,"summary":13227},"\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":13229,"title":13230,"module":13214,"summary":13231},"\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":13233,"title":13234,"module":13235,"summary":13236},"\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":13238,"title":13239,"module":13235,"summary":13240},"\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":13242,"title":13243,"module":13235,"summary":13244},"\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":13246,"title":13247,"module":13235,"summary":13248},"\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":13250,"title":13251,"module":13235,"summary":13252},"\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":13254,"title":13255,"module":13256,"summary":13257},"\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":13259,"title":13260,"module":13256,"summary":13261},"\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":13263,"title":13264,"module":13256,"summary":13265},"\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":13267,"title":13268,"module":13256,"summary":13269},"\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":13271,"title":13272,"module":13256,"summary":13273},"\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":13275,"title":13276,"module":13277,"summary":13278},"\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":13280,"title":13281,"module":13277,"summary":13282},"\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":13284,"title":13285,"module":13277,"summary":13286},"\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":13288,"title":13289,"module":13290,"summary":13291},"\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":13293,"title":13294,"module":13290,"summary":13295},"\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":13297,"title":13298,"module":13299,"summary":13300},"\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":13302,"title":13303,"module":13299,"summary":13304},"\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":13306,"title":13307,"module":13299,"summary":13308},"\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":13310,"title":13311,"module":13299,"summary":13312},"\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":13314,"title":13315,"module":13299,"summary":13316},"\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":13318,"title":13319,"module":13320,"summary":13321},"\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":13323,"title":13324,"module":13320,"summary":13325},"\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":13327,"title":13328,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":13330,"title":13331,"module":11922,"summary":13332},"\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":13334,"title":13335,"module":11922,"summary":13336},"\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":13338,"title":13339,"module":13340,"summary":13341},"\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":13343,"title":13344,"module":13340,"summary":13345},"\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":13347,"title":13348,"module":13340,"summary":13349},"\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":13351,"title":13352,"module":13340,"summary":13353},"\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":13355,"title":13356,"module":13340,"summary":13357},"\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":13359,"title":13360,"module":13340,"summary":13361},"\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":13363,"title":13364,"module":13365,"summary":13366},"\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":13368,"title":13369,"module":13365,"summary":13370},"\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":13372,"title":13373,"module":13365,"summary":13374},"\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":13376,"title":13377,"module":13365,"summary":13378},"\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":13380,"title":13381,"module":13365,"summary":13382},"\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":13384,"title":13385,"module":13365,"summary":13386},"\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":13388,"title":13389,"module":13390,"summary":13391},"\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":13393,"title":13394,"module":13390,"summary":13395},"\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":13397,"title":13398,"module":13390,"summary":13399},"\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":13401,"title":13402,"module":13403,"summary":13404},"\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":13406,"title":13407,"module":13403,"summary":13408},"\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":13410,"title":13411,"module":13412,"summary":13413},"\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":13415,"title":13416,"module":13412,"summary":13417},"\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":13419,"title":13420,"module":13412,"summary":13421},"\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":13423,"title":13424,"module":13425,"summary":13426},"\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":13428,"title":13429,"module":13425,"summary":13430},"\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":13432,"title":13433,"module":13434,"summary":13435},"\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":13437,"title":13438,"module":13434,"summary":13439},"\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":13441,"title":13442,"module":13434,"summary":13443},"\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":13445,"title":13446,"module":13447,"summary":13448},"\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":13450,"title":13451,"module":13447,"summary":13452},"\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":13454,"title":13455,"module":13447,"summary":13456},"\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":13458,"title":13459,"module":13460,"summary":13461},"\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":13463,"title":13464,"module":13460,"summary":13465},"\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":13467,"title":13468,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":13470,"title":13471,"module":13472,"summary":13473},"\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":13475,"title":13476,"module":13472,"summary":13477},"\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":13479,"title":13480,"module":13472,"summary":13481},"\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":13483,"title":13484,"module":13472,"summary":13485},"\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":13487,"title":13488,"module":13472,"summary":13489},"\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":13491,"title":13492,"module":13493,"summary":13494},"\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":13496,"title":13497,"module":13493,"summary":13498},"\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":13500,"title":13501,"module":13493,"summary":13502},"\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":13504,"title":13505,"module":13493,"summary":13506},"\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":13508,"title":13509,"module":13510,"summary":13511},"\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":13513,"title":13514,"module":13510,"summary":13515},"\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":13517,"title":13518,"module":13510,"summary":13519},"\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":13521,"title":13522,"module":13510,"summary":13523},"\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":13525,"title":13526,"module":13527,"summary":13528},"\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":13530,"title":13531,"module":13527,"summary":13532},"\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":13534,"title":13535,"module":13527,"summary":13536},"\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":13538,"title":13539,"module":13527,"summary":13540},"\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":13542,"title":13543,"module":13544,"summary":13545},"\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":13547,"title":13548,"module":13544,"summary":13549},"\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":13551,"title":13552,"module":13544,"summary":13553},"\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":13555,"title":13556,"module":13544,"summary":13557},"\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":13559,"title":13560,"module":13544,"summary":13561},"\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":13563,"title":13564,"module":13544,"summary":13565},"\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":13567,"title":13568,"module":13569,"summary":13570},"\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":13572,"title":13573,"module":13569,"summary":13574},"\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":13576,"title":13577,"module":13569,"summary":13578},"\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":13580,"title":13581,"module":13582,"summary":13583},"\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":13585,"title":13586,"module":13582,"summary":13587},"\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":13589,"title":13590,"module":13582,"summary":13591},"\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":13593,"title":13594,"module":13582,"summary":13595},"\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":13597,"title":13598,"module":13599,"summary":13600},"\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":13602,"title":13603,"module":13599,"summary":13604},"\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":13606,"title":13607,"module":13599,"summary":13608},"\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":13610,"title":13611,"module":13612,"summary":13613},"\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":13615,"title":13616,"module":13612,"summary":13617},"\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":13619,"title":13620,"module":13612,"summary":13621},"\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":13623,"title":13624,"module":13625,"summary":13626},"\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":13628,"title":13629,"module":13625,"summary":13630},"\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":13632,"title":13633,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":13635,"title":13636,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":13638,"title":13639,"module":13640,"summary":13641},"\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":13643,"title":13644,"module":13640,"summary":13645},"\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":13647,"title":13648,"module":13640,"summary":13649},"\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":13651,"title":13652,"module":13640,"summary":13653},"\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":13655,"title":13656,"module":13657,"summary":13658},"\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":13660,"title":13661,"module":13657,"summary":13662},"\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":13664,"title":13665,"module":13657,"summary":13666},"\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":13668,"title":13669,"module":13670,"summary":13671},"\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":13673,"title":13674,"module":13670,"summary":13675},"\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":13677,"title":13678,"module":13670,"summary":13679},"\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":13681,"title":13682,"module":13670,"summary":13683},"\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":13685,"title":13686,"module":13670,"summary":13687},"\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":13689,"title":13690,"module":13670,"summary":13691},"\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":13693,"title":13694,"module":13695,"summary":13696},"\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":13698,"title":13699,"module":13695,"summary":13700},"\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":13702,"title":13703,"module":13695,"summary":13704},"\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":13706,"title":13707,"module":13695,"summary":13708},"\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":13710,"title":13711,"module":13695,"summary":13712},"\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":13714,"title":13715,"module":13695,"summary":13716},"\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":13718,"title":13719,"module":13720,"summary":13721},"\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":13723,"title":13724,"module":13720,"summary":13725},"\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":13727,"title":13728,"module":13720,"summary":13729},"\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":13731,"title":13732,"module":13720,"summary":13733},"\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":13735,"title":13736,"module":12575,"summary":13737},"\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":13739,"title":13740,"module":12575,"summary":13741},"\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":13743,"title":13744,"module":12575,"summary":13745},"\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":13747,"title":13748,"module":13749,"summary":13750},"\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":13752,"title":13753,"module":13749,"summary":13754},"\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":13756,"title":13757,"module":13749,"summary":13758},"\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":13760,"title":13761,"module":13762,"summary":13763},"\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":13765,"title":13766,"module":13762,"summary":13767},"\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":13769,"title":13770,"module":13762,"summary":13771},"\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":13773,"title":13774,"module":13775,"summary":13776},"\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":13778,"title":13779,"module":13775,"summary":13780},"\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":13782,"title":13783,"module":13784,"summary":13785},"\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":13787,"title":13788,"module":13784,"summary":13789},"\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":13791,"title":13792,"module":13784,"summary":13793},"\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":13795,"title":13796,"module":13784,"summary":13797},"\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":13799,"title":13800,"module":13784,"summary":13801},"\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":13803,"title":13804,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":13806,"title":13807,"module":13808,"summary":13809},"\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":13811,"title":13812,"module":13808,"summary":13813},"\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":13815,"title":13816,"module":13808,"summary":13817},"\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":13819,"title":13820,"module":13808,"summary":13821},"\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":13823,"title":13824,"module":13825,"summary":13826},"\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":13828,"title":13829,"module":13825,"summary":13830},"\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":13832,"title":13833,"module":13825,"summary":13834},"\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":13836,"title":13837,"module":13825,"summary":13838},"\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":13840,"title":13841,"module":13825,"summary":13842},"\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":13844,"title":13845,"module":13825,"summary":13846},"\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":13848,"title":13849,"module":13850,"summary":13851},"\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":13853,"title":13854,"module":13850,"summary":13855},"\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":13857,"title":13858,"module":13850,"summary":13859},"\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":13861,"title":13862,"module":13850,"summary":13863},"\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":13865,"title":13866,"module":13850,"summary":13867},"\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":13869,"title":13870,"module":12269,"summary":13871},"\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":13873,"title":13874,"module":12269,"summary":13875},"\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":13877,"title":13878,"module":12269,"summary":13879},"\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":13881,"title":13882,"module":12269,"summary":13883},"\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":13885,"title":13886,"module":12269,"summary":13887},"\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":13889,"title":13890,"module":12269,"summary":13891},"\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":13893,"title":13894,"module":13895,"summary":13896},"\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":13898,"title":13899,"module":13895,"summary":13900},"\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":13902,"title":13903,"module":13895,"summary":13904},"\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":13906,"title":13907,"module":13895,"summary":13908},"\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":13910,"title":13911,"module":13912,"summary":13913},"\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":13915,"title":13916,"module":13912,"summary":13917},"\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":13919,"title":13920,"module":13912,"summary":13921},"\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":13923,"title":12324,"module":13912,"summary":13924},"\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":13926,"title":13927,"module":13912,"summary":13928},"\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":13930,"title":13931,"module":13932,"summary":13933},"\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":13935,"title":13936,"module":13932,"summary":13937},"\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":13939,"title":13940,"module":13932,"summary":13941},"\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":13943,"title":13944,"module":13932,"summary":13945},"\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":13947,"title":13948,"module":13949,"summary":13950},"\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":13952,"title":13953,"module":13949,"summary":13954},"\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":13956,"title":13957,"module":13949,"summary":13958},"\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":13960,"title":13961,"module":13949,"summary":13962},"\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":13964,"title":13965,"module":13949,"summary":13966},"\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":13968,"title":13969,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":13971,"title":13972,"module":11922,"summary":13973},"\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":13975,"title":13976,"module":11922,"summary":13977},"\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":13979,"title":13980,"module":13981,"summary":13982},"\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":13984,"title":13985,"module":13981,"summary":13986},"\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":13988,"title":13989,"module":13981,"summary":13990},"\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":13992,"title":13993,"module":13981,"summary":13994},"\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":13996,"title":13997,"module":13998,"summary":13999},"\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":14001,"title":14002,"module":13998,"summary":14003},"\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":14005,"title":14006,"module":13998,"summary":14007},"\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":14009,"title":14010,"module":13998,"summary":14011},"\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":14013,"title":14014,"module":13998,"summary":14015},"\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":14017,"title":14018,"module":13998,"summary":14019},"\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":14021,"title":14022,"module":14023,"summary":14024},"\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":14026,"title":14027,"module":14023,"summary":14028},"\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":14030,"title":14031,"module":14023,"summary":14032},"\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":14034,"title":14035,"module":14023,"summary":14036},"\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":14038,"title":14039,"module":14040,"summary":14041},"\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":14043,"title":14044,"module":14040,"summary":14045},"\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":14047,"title":14048,"module":14040,"summary":14049},"\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":14051,"title":14052,"module":14040,"summary":14053},"\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":14055,"title":14056,"module":14057,"summary":14058},"\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":14060,"title":14061,"module":14057,"summary":14062},"\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":14064,"title":14065,"module":14057,"summary":14066},"\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":14068,"title":14069,"module":14070,"summary":14071},"\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":14073,"title":14074,"module":14070,"summary":14075},"\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":14077,"title":14078,"module":14070,"summary":14079},"\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":14081,"title":14082,"module":14070,"summary":14083},"\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":14085,"title":14086,"module":14087,"summary":14088},"\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":14090,"title":14091,"module":14087,"summary":14092},"\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":14094,"title":14095,"module":14087,"summary":14096},"\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":14098,"title":14099,"module":14087,"summary":14100},"\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":14102,"title":14103,"module":14104,"summary":14105},"\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":14107,"title":14108,"module":14104,"summary":14109},"\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":14111,"title":14112,"module":14104,"summary":14113},"\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":14115,"title":14116,"module":14117,"summary":14118},"\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":14120,"title":14121,"module":14117,"summary":14122},"\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":14124,"title":14125,"module":14117,"summary":14126},"\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":14128,"title":14129,"module":14117,"summary":14130},"\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":14132,"title":14133,"module":14134,"summary":14135},"\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":14137,"title":14138,"module":14134,"summary":14139},"\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":14141,"title":14142,"module":14134,"summary":14143},"\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":14145,"title":14146,"module":14134,"summary":14147},"\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":14149,"title":14150,"module":14134,"summary":14151},"\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":14153,"title":14154,"module":14155,"summary":14156},"\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":14158,"title":14159,"module":14155,"summary":14160},"\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":14162,"title":14163,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":14165,"title":14166,"module":14167,"summary":14168},"\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":14170,"title":14171,"module":14167,"summary":14172},"\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":14174,"title":14175,"module":14167,"summary":14176},"\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":14178,"title":14179,"module":14167,"summary":14180},"\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":14182,"title":14183,"module":14167,"summary":14184},"\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":14186,"title":14187,"module":14188,"summary":14189},"\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":14191,"title":14192,"module":14188,"summary":14193},"\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":14195,"title":14196,"module":14188,"summary":14197},"\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":14199,"title":14200,"module":14188,"summary":14201},"\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":14203,"title":14204,"module":14188,"summary":14205},"\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":14207,"title":14208,"module":14188,"summary":14209},"\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":14211,"title":14212,"module":14188,"summary":14213},"\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":14215,"title":14216,"module":14217,"summary":14218},"\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":14220,"title":14221,"module":14217,"summary":14222},"\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":14224,"title":14225,"module":14217,"summary":14226},"\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":14228,"title":14229,"module":14217,"summary":14230},"\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":14232,"title":14233,"module":14234,"summary":14235},"\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":14237,"title":14238,"module":14234,"summary":14239},"\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":14241,"title":14242,"module":14234,"summary":14243},"\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":14245,"title":14246,"module":14247,"summary":14248},"\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":14250,"title":14251,"module":14247,"summary":14252},"\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":14254,"title":14255,"module":14247,"summary":14256},"\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":14258,"title":14259,"module":14247,"summary":14260},"\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":14262,"title":14263,"module":14247,"summary":14264},"\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":14266,"title":14267,"module":14247,"summary":14268},"\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":14270,"title":14271,"module":14272,"summary":14273},"\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":14275,"title":14276,"module":14272,"summary":14277},"\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":14279,"title":14280,"module":14272,"summary":14281},"\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":14283,"title":14284,"module":14285,"summary":14286},"\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":14288,"title":14289,"module":14285,"summary":14290},"\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":14292,"title":14293,"module":14285,"summary":14294},"\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":14296,"title":14297,"module":14285,"summary":14298},"\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":14300,"title":14301,"module":14302,"summary":14303},"\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":14305,"title":14306,"module":14302,"summary":14307},"\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":14309,"title":14310,"module":14302,"summary":14311},"\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":14313,"title":14314,"module":14315,"summary":14316},"\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":14318,"title":14319,"module":14315,"summary":14320},"\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":14322,"title":14323,"module":14315,"summary":14324},"\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":14326,"title":14327,"module":14315,"summary":14328},"\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":14330,"title":14331,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":14333,"title":14334,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":14336,"title":14337,"module":11922,"summary":14338},"\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":14340,"title":14341,"module":11922,"summary":14342},"\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":14344,"title":14345,"module":11922,"summary":14346},"\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":14348,"title":14349,"module":11922,"summary":14350},"\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":14352,"title":14353,"module":11922,"summary":14354},"\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":14356,"title":14357,"module":11922,"summary":14358},"\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":14360,"title":14361,"module":14362,"summary":14363},"\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":14365,"title":14366,"module":14362,"summary":14367},"\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":14369,"title":14370,"module":14362,"summary":14371},"\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":14373,"title":14374,"module":14375,"summary":14376},"\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":14378,"title":14379,"module":14375,"summary":14380},"\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":14382,"title":14383,"module":14375,"summary":14384},"\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":14386,"title":14387,"module":14388,"summary":14389},"\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":14391,"title":14392,"module":14388,"summary":14393},"\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":14395,"title":14396,"module":14388,"summary":14397},"\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":14399,"title":14400,"module":14388,"summary":14401},"\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":14403,"title":14404,"module":14388,"summary":14405},"\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":14407,"title":14408,"module":14409,"summary":14410},"\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":14412,"title":14413,"module":14409,"summary":14414},"\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":14416,"title":14417,"module":14409,"summary":14418},"\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":14420,"title":14421,"module":14409,"summary":14422},"\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":14424,"title":14425,"module":14426,"summary":14427},"\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":14429,"title":14430,"module":14426,"summary":14431},"\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":14433,"title":14434,"module":14426,"summary":14435},"\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":14437,"title":14438,"module":14426,"summary":14439},"\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":14441,"title":14442,"module":14443,"summary":14444},"\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":14446,"title":14447,"module":14443,"summary":14448},"\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":14450,"title":14451,"module":14443,"summary":14452},"\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":14454,"title":14455,"module":14443,"summary":14456},"\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":14458,"title":14459,"module":14460,"summary":14461},"\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":14463,"title":14464,"module":14460,"summary":14465},"\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":14467,"title":14468,"module":14460,"summary":14469},"\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":14471,"title":14472,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":14474,"title":13116,"module":14475,"summary":14476},"\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":14478,"title":14479,"module":14475,"summary":14480},"\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":14482,"title":14483,"module":14475,"summary":14484},"\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":14486,"title":12475,"module":14475,"summary":14487},"\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":14489,"title":14490,"module":11922,"summary":14491},"\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":14493,"title":14494,"module":11922,"summary":14495},"\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":14497,"title":14498,"module":11922,"summary":14499},"\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":14501,"title":14502,"module":14503,"summary":14504},"\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":14506,"title":14507,"module":14503,"summary":14508},"\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":14510,"title":14511,"module":14503,"summary":14512},"\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":14514,"title":14515,"module":14503,"summary":14516},"\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":14518,"title":14519,"module":14503,"summary":14520},"\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":14522,"title":14523,"module":14524,"summary":14525},"\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":14527,"title":14528,"module":14524,"summary":14529},"\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":14531,"title":14532,"module":14524,"summary":14533},"\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":14535,"title":14536,"module":14524,"summary":14537},"\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":14539,"title":14540,"module":14524,"summary":14541},"\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":14543,"title":14544,"module":14545,"summary":14546},"\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":14548,"title":14549,"module":14545,"summary":14550},"\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":14552,"title":14553,"module":14545,"summary":14554},"\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":14556,"title":14557,"module":14545,"summary":14558},"\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":14560,"title":14561,"module":14562,"summary":14563},"\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":14565,"title":14566,"module":14562,"summary":14567},"\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":14569,"title":14570,"module":14562,"summary":14571},"\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":14573,"title":14574,"module":14562,"summary":14575},"\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":14577,"title":14578,"module":14562,"summary":14579},"\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":14581,"title":14582,"module":14562,"summary":14583},"\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":14585,"title":14586,"module":14562,"summary":14587},"\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":14589,"title":14590,"module":14562,"summary":14591},"\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":14593,"title":14594,"module":14562,"summary":14595},"\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":14597,"title":14598,"module":14599,"summary":14600},"\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":14602,"title":14603,"module":14599,"summary":14604},"\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":14606,"title":14607,"module":14599,"summary":14608},"\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":14610,"title":14611,"module":14599,"summary":14612},"\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":14614,"title":14615,"module":14599,"summary":14616},"\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":14618,"title":14619,"module":14620,"summary":14621},"\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":14623,"title":14624,"module":14620,"summary":14625},"\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":14627,"title":14628,"module":14620,"summary":14629},"\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":14631,"title":14632,"module":14620,"summary":14633},"\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":14635,"title":14636,"module":14620,"summary":14637},"\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":14639,"title":14640,"module":14620,"summary":14641},"\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":14643,"title":14644,"module":14620,"summary":14645},"\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":14647,"title":14648,"module":14649,"summary":14650},"\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":14652,"title":14653,"module":14649,"summary":14654},"\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":14656,"title":14657,"module":14649,"summary":14658},"\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":14660,"title":14661,"module":14662,"summary":14663},"\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":14665,"title":14666,"module":14662,"summary":14667},"\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":14669,"title":14670,"module":14662,"summary":14671},"\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":14673,"title":14674,"module":14662,"summary":14675},"\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":14677,"title":14678,"module":14662,"summary":14679},"\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":14681,"title":14682,"module":14662,"summary":14683},"\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":14685,"title":14686,"module":14662,"summary":14687},"\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":14689,"title":14690,"module":14691,"summary":14692},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models","Large Language Models","Large Models & Agents","A large language model is a decoder-only Transformer trained on one objective, next-token prediction, then scaled until new behavior appears. This first part builds the object itself: the equivalence between next-token prediction and lossless compression, subword tokenization (BPE, WordPiece, Unigram, SentencePiece) worked on a real sentence, the four pretraining objectives and the attention masks that distinguish them, and the three model families (encoder-only, decoder-only, encoder--decoder) with their parameter budgets. Scaling, decoding, the KV cache, and alignment continue in part two.\n",{"path":14694,"title":14695,"module":14691,"summary":14696},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment","Scaling, Inference, and Alignment of Language Models","Once a language model is built, three questions remain: how does it improve as it grows, how is it decoded and served affordably, and how is a raw next-token predictor turned into an assistant. We derive the Kaplan power laws and the Chinchilla compute-optimal balance, trace emergent abilities and in-context learning, catalog the decoding strategies from greedy to nucleus sampling, work the KV cache that makes generation quadratic instead of cubic, cover parameter-efficient adaptation by low-rank updates (LoRA), and close on the alignment stack: instruction tuning, RLHF, and DPO.\n",{"path":14698,"title":14699,"module":14691,"summary":14700},"\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":14702,"title":14703,"module":14691,"summary":14704},"\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":14706,"title":14707,"module":14691,"summary":14708},"\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":14710,"title":14711,"module":14691,"summary":14712},"\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":14714,"title":14715,"module":14691,"summary":14716},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents","AI Agents: Tools and Reasoning","A language model that only emits text is a function from prompt to prompt; an agent closes the loop, letting that model act on an environment, read back the result, and decide again. This first part formalizes the agent as a policy over interaction histories, builds out tool calling and the executor trust boundary, the ReAct interleaving of reasoning and action (with concrete traces), and search over thoughts: chain-of-thought, self-consistency, least-to-most, and Tree of Thoughts. Memory, retrieval, reflection, and multi-agent orchestration continue in part two.\n",{"path":14718,"title":14719,"module":14691,"summary":14720},"\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":14722,"title":14723,"module":14691,"summary":14724},"\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":14726,"title":14727,"module":14691,"summary":14728},"\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":14730,"title":14731,"module":14691,"summary":14732},"\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":14734,"title":14735,"module":14736,"summary":14737},"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning","Foundations of Reinforcement Learning","Reinforcement Learning","Reinforcement learning is the third paradigm: an agent learns to act by interacting with an environment that returns rewards, not labels. We formalize the interaction as a Markov decision process, define the value functions that rank states and actions, and derive the Bellman expectation and optimality equations that every method downstream solves. Dynamic programming gives the exact answer when the model is known, and its convergence rests on a single fact: the Bellman operator is a contraction.\n",{"path":14739,"title":14740,"module":14736,"summary":14741},"\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":14743,"title":14744,"module":14736,"summary":14745},"\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":14747,"title":14748,"module":14736,"summary":14749},"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic","Policy Gradients and Actor-Critic Methods","Value-based reinforcement learning learns what each state is worth and acts greedily; policy-gradient methods skip the detour and optimize a parameterized policy directly by ascending the gradient of expected return. The policy gradient theorem makes this tractable through the log-derivative trick, turning an intractable gradient of an expectation into an expectation of a gradient. REINFORCE realizes the idea but suffers high variance; baselines, the advantage function, and actor-critic learning reduce it, and trust-region methods (TRPO, PPO) keep each update from destroying the policy it just learned.\n",{"path":14751,"title":14752,"module":14736,"summary":14753},"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback","Reinforcement Learning from Human Feedback","Many objectives we want from a model, that it be helpful and harmless, are hard to write down but easy to judge by comparison. RLHF turns that asymmetry into a training signal: fit a reward model to pairwise human preferences under the Bradley-Terry likelihood, then fine-tune the policy to maximize that reward under a KL penalty toward a reference. We derive the reward loss, the KL-regularized RL objective and its closed-form optimum, then show how DPO inverts that optimum to collapse the whole pipeline into one supervised log-sigmoid loss, and survey IPO, KTO, RLAIF, and GRPO.\n",{"path":14755,"title":14756,"module":6,"summary":6},"\u002Fdeep-learning","Deep Learning",{"path":14758,"title":14759,"module":12666,"summary":14760},"\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":14762,"title":14763,"module":12666,"summary":14764},"\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":14766,"title":14767,"module":12666,"summary":14768},"\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":14770,"title":14771,"module":12666,"summary":14772},"\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":14774,"title":14775,"module":12666,"summary":14776},"\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":14778,"title":14779,"module":14780,"summary":14781},"\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":14783,"title":14784,"module":14780,"summary":14785},"\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":14787,"title":14788,"module":14780,"summary":14789},"\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":14791,"title":14792,"module":14780,"summary":14793},"\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":14795,"title":14796,"module":14797,"summary":14798},"\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":14800,"title":14801,"module":14797,"summary":14802},"\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":14804,"title":14805,"module":14797,"summary":14806},"\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":14808,"title":14809,"module":14797,"summary":14810},"\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":14812,"title":14813,"module":14814,"summary":14815},"\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":14817,"title":14818,"module":14814,"summary":14819},"\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":14821,"title":14822,"module":14814,"summary":14823},"\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":14825,"title":14826,"module":14814,"summary":14827},"\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":14829,"title":14830,"module":14814,"summary":14831},"\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":14833,"title":14834,"module":14835,"summary":14836},"\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":14838,"title":14839,"module":14835,"summary":14840},"\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":14842,"title":14843,"module":14835,"summary":14844},"\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":14846,"title":14847,"module":14848,"summary":14849},"\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":14851,"title":14852,"module":14848,"summary":14853},"\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":14855,"title":14856,"module":14848,"summary":14857},"\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":14859,"title":14860,"module":14861,"summary":14862},"\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":14864,"title":14865,"module":14861,"summary":14866},"\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":14868,"title":14869,"module":14861,"summary":14870},"\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":14872,"title":14873,"module":14861,"summary":14874},"\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":14876,"title":14877,"module":14878,"summary":14879},"\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":14881,"title":14882,"module":14878,"summary":14883},"\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":14885,"title":14886,"module":14878,"summary":14887},"\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":14889,"title":14890,"module":14878,"summary":14891},"\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":14893,"title":14894,"module":14878,"summary":14895},"\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":14897,"title":14898,"module":14878,"summary":14899},"\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":14901,"title":14902,"module":14903,"summary":14904},"\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":14906,"title":14907,"module":14903,"summary":14908},"\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":14910,"title":14911,"module":14903,"summary":14912},"\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":14914,"title":14915,"module":14903,"summary":14916},"\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":14918,"title":14919,"module":14920,"summary":14921},"\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":14923,"title":14924,"module":14920,"summary":14925},"\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":14927,"title":14928,"module":14920,"summary":14929},"\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":14931,"title":14932,"module":14933,"summary":14934},"\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":14936,"title":14937,"module":14933,"summary":14938},"\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":14940,"title":14941,"module":14933,"summary":14942},"\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":14944,"title":14945,"module":14933,"summary":14946},"\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":14948,"title":14949,"module":14933,"summary":14950},"\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":14952,"title":14953,"module":14954,"summary":14955},"\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":14957,"title":14958,"module":14954,"summary":14959},"\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":14961,"title":14962,"module":14954,"summary":14963},"\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":14965,"title":14966,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":14968,"title":14969,"module":14970,"summary":14971},"\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":14973,"title":14974,"module":14970,"summary":14975},"\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":14977,"title":14978,"module":14970,"summary":14979},"\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":14981,"title":14982,"module":14970,"summary":14983},"\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":14985,"title":14986,"module":14987,"summary":14988},"\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":14990,"title":14991,"module":14987,"summary":14992},"\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":14994,"title":14995,"module":14987,"summary":14996},"\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":14998,"title":14999,"module":14987,"summary":15000},"\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":15002,"title":15003,"module":15004,"summary":15005},"\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":15007,"title":15008,"module":15004,"summary":15009},"\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":15011,"title":15012,"module":15004,"summary":15013},"\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":15015,"title":15016,"module":15004,"summary":15017},"\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":15019,"title":15020,"module":15021,"summary":15022},"\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":15024,"title":15025,"module":15021,"summary":15026},"\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":15028,"title":15029,"module":15021,"summary":15030},"\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":15032,"title":15033,"module":15021,"summary":15034},"\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":15036,"title":15037,"module":15038,"summary":15039},"\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":15041,"title":15042,"module":15038,"summary":15043},"\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":15045,"title":15046,"module":15038,"summary":15047},"\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":15049,"title":15050,"module":15038,"summary":15051},"\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":15053,"title":15054,"module":15055,"summary":15056},"\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":15058,"title":15059,"module":15055,"summary":15060},"\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":15062,"title":15063,"module":15055,"summary":15064},"\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":15066,"title":15067,"module":15055,"summary":15068},"\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":15070,"title":15071,"module":15072,"summary":15073},"\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":15075,"title":15076,"module":15072,"summary":15077},"\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":15079,"title":15080,"module":15072,"summary":15081},"\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":15083,"title":15084,"module":15072,"summary":15085},"\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":15087,"title":15088,"module":15072,"summary":15089},"\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":15091,"title":15092,"module":15093,"summary":15094},"\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":15096,"title":15097,"module":15093,"summary":15098},"\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":15100,"title":15101,"module":15102,"summary":15103},"\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":15105,"title":15106,"module":15102,"summary":15107},"\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":15109,"title":15110,"module":15102,"summary":15111},"\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":15113,"title":15114,"module":15102,"summary":15115},"\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":15117,"title":15118,"module":15119,"summary":15120},"\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":15122,"title":15123,"module":15119,"summary":15124},"\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":15126,"title":15127,"module":15119,"summary":15128},"\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":15130,"title":15131,"module":15119,"summary":15132},"\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":15134,"title":15135,"module":15119,"summary":15136},"\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":15138,"title":15139,"module":15140,"summary":15141},"\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":15143,"title":15144,"module":15140,"summary":15145},"\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":15147,"title":15148,"module":15140,"summary":15149},"\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":15151,"title":15152,"module":15140,"summary":15153},"\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":15155,"title":15156,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":15158,"title":15159,"module":11922,"summary":15160},"\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":15162,"title":15163,"module":15164,"summary":15165},"\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":15167,"title":15168,"module":15164,"summary":15169},"\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":15171,"title":15172,"module":15164,"summary":15173},"\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":15175,"title":15176,"module":15164,"summary":15177},"\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":15179,"title":15180,"module":15164,"summary":15181},"\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":15183,"title":15184,"module":15164,"summary":15185},"\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":15187,"title":15188,"module":15164,"summary":15189},"\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":15191,"title":15192,"module":15193,"summary":15194},"\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":15196,"title":15197,"module":15193,"summary":15198},"\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":15200,"title":15201,"module":15193,"summary":15202},"\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":15204,"title":15205,"module":15193,"summary":15206},"\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":15208,"title":15209,"module":15210,"summary":15211},"\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":15213,"title":15214,"module":15210,"summary":15215},"\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":15217,"title":15218,"module":15210,"summary":15219},"\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":15221,"title":15222,"module":15210,"summary":15223},"\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":15225,"title":15226,"module":15227,"summary":15228},"\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":15230,"title":15231,"module":15227,"summary":15232},"\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":15234,"title":15235,"module":15227,"summary":15236},"\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":15238,"title":15239,"module":15227,"summary":15240},"\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":15242,"title":15243,"module":15244,"summary":15245},"\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":15247,"title":15248,"module":15244,"summary":15249},"\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":15251,"title":15252,"module":15244,"summary":15253},"\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":15255,"title":15256,"module":15244,"summary":15257},"\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":15259,"title":15260,"module":15261,"summary":15262},"\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":15264,"title":15265,"module":15261,"summary":15266},"\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":15268,"title":15269,"module":15261,"summary":15270},"\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":15272,"title":15273,"module":15274,"summary":15275},"\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":15277,"title":15278,"module":15274,"summary":15279},"\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":15281,"title":15282,"module":15283,"summary":15284},"\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":15286,"title":15287,"module":15283,"summary":15288},"\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":15290,"title":15291,"module":15283,"summary":15292},"\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":15294,"title":15295,"module":6,"summary":6},"\u002Flogic","Logic",{"path":15297,"title":15298,"module":11922,"summary":15299},"\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":15301,"title":15302,"module":11922,"summary":15303},"\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":15305,"title":15306,"module":11922,"summary":15307},"\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":15309,"title":15310,"module":11922,"summary":15311},"\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":15313,"title":15314,"module":11922,"summary":15315},"\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":15317,"title":15318,"module":11922,"summary":15319},"\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":15321,"title":12141,"module":15322,"summary":15323},"\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":15325,"title":15326,"module":15322,"summary":15327},"\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":15329,"title":15330,"module":15322,"summary":15331},"\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":15333,"title":15334,"module":15322,"summary":15335},"\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":15337,"title":15338,"module":15322,"summary":15339},"\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":15341,"title":15342,"module":15322,"summary":15343},"\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":15345,"title":15346,"module":15322,"summary":15347},"\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":15349,"title":15350,"module":15322,"summary":15351},"\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":15353,"title":15354,"module":15322,"summary":15355},"\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":15357,"title":15358,"module":15322,"summary":15359},"\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":15361,"title":15362,"module":15322,"summary":15363},"\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":15365,"title":15366,"module":15322,"summary":15367},"\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":15369,"title":15370,"module":15371,"summary":15372},"\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":15374,"title":15375,"module":15371,"summary":15376},"\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":15378,"title":15379,"module":15371,"summary":15380},"\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":15382,"title":15383,"module":15371,"summary":15384},"\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":67,"title":15386,"module":15371,"summary":15387},"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":15389,"title":15390,"module":15371,"summary":15391},"\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":15393,"title":15394,"module":15371,"summary":15395},"\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":15397,"title":15398,"module":15371,"summary":15399},"\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":15401,"title":15402,"module":15371,"summary":15403},"\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":15405,"title":15406,"module":15371,"summary":15407},"\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":15409,"title":15410,"module":15371,"summary":15411},"\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":15413,"title":15414,"module":15371,"summary":15415},"\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":15417,"title":15418,"module":15371,"summary":15419},"\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":15421,"title":15422,"module":15371,"summary":15423},"\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":15425,"title":14744,"module":15426,"summary":15427},"\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":15429,"title":15430,"module":15426,"summary":15431},"\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":15433,"title":15434,"module":15426,"summary":15435},"\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":15437,"title":15438,"module":15426,"summary":15439},"\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":15441,"title":15442,"module":15426,"summary":15443},"\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":15445,"title":15446,"module":15426,"summary":15447},"\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":15449,"title":15450,"module":15426,"summary":15451},"\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":15453,"title":15454,"module":15426,"summary":15455},"\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":15457,"title":15458,"module":10982,"summary":15459},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow","Sharpening DQN: Improvements and the Distributional Idea","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":15461,"title":15462,"module":10982,"summary":15463},"\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":1067,"title":15465,"module":10982,"summary":15466},"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":15468,"title":15469,"module":10982,"summary":15470},"\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":15472,"title":15473,"module":10982,"summary":15474},"\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":15476,"title":15477,"module":10982,"summary":15478},"\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":15480,"title":15481,"module":10982,"summary":15482},"\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":15484,"title":15485,"module":10982,"summary":15486},"\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":10984,"title":5,"module":10982,"summary":11002},{"path":126,"title":10898,"module":10982,"summary":15489},"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":15491,"title":15492,"module":10982,"summary":15493},"\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":15495,"title":15496,"module":10982,"summary":15497},"\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":15499,"title":15500,"module":10982,"summary":15501},"\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":15503,"title":15504,"module":10982,"summary":15505},"\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":15507,"title":15508,"module":10982,"summary":15509},"\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":15511,"title":15512,"module":10982,"summary":15513},"\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":15515,"title":15516,"module":10982,"summary":15517},"\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":15519,"title":15520,"module":10982,"summary":15521},"\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":15523,"title":15524,"module":10982,"summary":15525},"\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":15527,"title":15528,"module":10982,"summary":15529},"\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":15531,"title":15532,"module":10982,"summary":15533},"\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":15535,"title":15536,"module":10982,"summary":15537},"\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":15539,"title":15540,"module":15541,"summary":15542},"\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":15544,"title":15545,"module":15541,"summary":15546},"\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":15548,"title":15549,"module":15541,"summary":15550},"\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":15552,"title":15553,"module":15541,"summary":15554},"\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":15556,"title":15557,"module":15541,"summary":15558},"\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":15560,"title":15561,"module":15541,"summary":15562},"\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":15564,"title":15565,"module":15541,"summary":15566},"\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":15568,"title":15569,"module":15541,"summary":15570},"\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":15572,"title":14736,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":15574,"title":15575,"module":11922,"summary":15576},"\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":15578,"title":15579,"module":11922,"summary":15580},"\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":15582,"title":15583,"module":11922,"summary":15584},"\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":15586,"title":15587,"module":11922,"summary":15588},"\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":15590,"title":15591,"module":15592,"summary":15593},"\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":15595,"title":15596,"module":15592,"summary":15597},"\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":15599,"title":15600,"module":15592,"summary":15601},"\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":15603,"title":15604,"module":15592,"summary":15605},"\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":15607,"title":15608,"module":15592,"summary":15609},"\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":15611,"title":15612,"module":15592,"summary":15613},"\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":15615,"title":15616,"module":15592,"summary":15617},"\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":15619,"title":15620,"module":15592,"summary":15621},"\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":15623,"title":15624,"module":15592,"summary":15625},"\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":15627,"title":15628,"module":15592,"summary":15629},"\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":15631,"title":15632,"module":15592,"summary":15633},"\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":15635,"title":15636,"module":15592,"summary":15637},"\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":15639,"title":15640,"module":15641,"summary":15642},"\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":15644,"title":15645,"module":15641,"summary":15646},"\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":15648,"title":15649,"module":15641,"summary":15650},"\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":15652,"title":15653,"module":15641,"summary":15654},"\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":15656,"title":15657,"module":15641,"summary":15658},"\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":15660,"title":15661,"module":15641,"summary":15662},"\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":15664,"title":15665,"module":15641,"summary":15666},"\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":15668,"title":15669,"module":15641,"summary":15670},"\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":15672,"title":15673,"module":15641,"summary":15674},"\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":15676,"title":15677,"module":15641,"summary":15678},"\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":15680,"title":15681,"module":15641,"summary":15682},"\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":15684,"title":15685,"module":15641,"summary":15686},"\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":15688,"title":15689,"module":15690,"summary":15691},"\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":15693,"title":15694,"module":15690,"summary":15695},"\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":15697,"title":15698,"module":15690,"summary":15699},"\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":15701,"title":15702,"module":15690,"summary":15703},"\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":15705,"title":15706,"module":15690,"summary":15707},"\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":15709,"title":15710,"module":15690,"summary":15711},"\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":15713,"title":15714,"module":15690,"summary":15715},"\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":15717,"title":15314,"module":15690,"summary":15718},"\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":15720,"title":15721,"module":15690,"summary":15722},"\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":15724,"title":15725,"module":15690,"summary":15726},"\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":15728,"title":15729,"module":15730,"summary":15731},"\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":15733,"title":15734,"module":15730,"summary":15735},"\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":15737,"title":15738,"module":15730,"summary":15739},"\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":15741,"title":15742,"module":15730,"summary":15743},"\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":15745,"title":14736,"module":15730,"summary":15746},"\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":15748,"title":15749,"module":15730,"summary":15750},"\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":15752,"title":15753,"module":15730,"summary":15754},"\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":15756,"title":15757,"module":15730,"summary":15758},"\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":15760,"title":15761,"module":15762,"summary":15763},"\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":15765,"title":15766,"module":15762,"summary":15767},"\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":15769,"title":15770,"module":15762,"summary":15771},"\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":15773,"title":15774,"module":15762,"summary":15775},"\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":15777,"title":15778,"module":15762,"summary":15779},"\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":15781,"title":15782,"module":15762,"summary":15783},"\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":15785,"title":15786,"module":15762,"summary":15787},"\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":15789,"title":15790,"module":15762,"summary":15791},"\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":15793,"title":15794,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":15796,"title":15797,"module":15798,"summary":15799},"\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":15801,"title":15802,"module":15798,"summary":15803},"\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":15805,"title":15806,"module":15798,"summary":15807},"\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":15809,"title":15810,"module":15798,"summary":15811},"\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":15813,"title":15814,"module":15798,"summary":15815},"\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":15817,"title":15818,"module":15819,"summary":15820},"\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":15822,"title":15823,"module":15819,"summary":15824},"\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":15826,"title":15827,"module":15819,"summary":15828},"\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":15830,"title":15831,"module":15819,"summary":15832},"\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":15834,"title":15835,"module":15836,"summary":15837},"\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":15839,"title":15840,"module":15836,"summary":15841},"\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":15843,"title":15844,"module":15836,"summary":15845},"\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":15847,"title":15848,"module":15836,"summary":15849},"\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":15851,"title":15852,"module":15853,"summary":15854},"\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":15856,"title":15857,"module":15853,"summary":15858},"\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":15860,"title":15861,"module":15862,"summary":15863},"\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":15865,"title":15866,"module":15862,"summary":15867},"\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":15869,"title":15870,"module":15871,"summary":15872},"\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":15874,"title":15875,"module":15871,"summary":15876},"\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":15878,"title":15879,"module":15871,"summary":15880},"\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":15882,"title":15883,"module":15871,"summary":15884},"\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":15886,"title":15887,"module":15888,"summary":15889},"\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":15891,"title":15892,"module":15888,"summary":15893},"\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":15895,"title":15896,"module":15888,"summary":15897},"\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":15899,"title":15900,"module":15901,"summary":15902},"\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":15904,"title":15905,"module":15901,"summary":15906},"\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":15908,"title":15909,"module":15901,"summary":15910},"\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":15912,"title":15913,"module":15914,"summary":15915},"\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":15917,"title":15918,"module":15914,"summary":15919},"\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":15921,"title":15922,"module":15923,"summary":15924},"\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":15926,"title":15927,"module":15923,"summary":15928},"\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":15930,"title":15931,"module":15923,"summary":15932},"\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":15934,"title":15935,"module":15936,"summary":15937},"\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":15939,"title":15940,"module":15936,"summary":15941},"\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":15943,"title":15944,"module":15936,"summary":15945},"\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":15947,"title":15948,"module":15936,"summary":15949},"\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":15951,"title":15952,"module":15936,"summary":15953},"\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":15955,"title":15956,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":15958,"title":15959,"module":11922,"summary":15960},"\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":15962,"title":15963,"module":11922,"summary":15964},"\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":15966,"title":15967,"module":11922,"summary":15968},"\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":15970,"title":15971,"module":11922,"summary":15972},"\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":15974,"title":15975,"module":11922,"summary":15976},"\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":15978,"title":15979,"module":15980,"summary":15981},"\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":15983,"title":15984,"module":15980,"summary":15985},"\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":15987,"title":15988,"module":15980,"summary":15989},"\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":15991,"title":15992,"module":15980,"summary":15993},"\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":15995,"title":15996,"module":15997,"summary":15998},"\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":16000,"title":16001,"module":15997,"summary":16002},"\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":16004,"title":16005,"module":15997,"summary":16006},"\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":16008,"title":16009,"module":12388,"summary":16010},"\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":16012,"title":16013,"module":12388,"summary":16014},"\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":16016,"title":16017,"module":12388,"summary":16018},"\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":16020,"title":16021,"module":12870,"summary":16022},"\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":16024,"title":14582,"module":12870,"summary":16025},"\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":16027,"title":14690,"module":12870,"summary":16028},"\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":16030,"title":16031,"module":12870,"summary":16032},"\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":16034,"title":16035,"module":12870,"summary":16036},"\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":16038,"title":16039,"module":12870,"summary":16040},"\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":16042,"title":16043,"module":16044,"summary":16045},"\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":16047,"title":16048,"module":16044,"summary":16049},"\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":16051,"title":16052,"module":16044,"summary":16053},"\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":16055,"title":16056,"module":16044,"summary":16057},"\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":16059,"title":16060,"module":16044,"summary":16061},"\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":16063,"title":16064,"module":16044,"summary":16065},"\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":16067,"title":16068,"module":16044,"summary":16069},"\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":16071,"title":16072,"module":16044,"summary":16073},"\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":16075,"title":16076,"module":16044,"summary":16077},"\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":16079,"title":16080,"module":16044,"summary":16081},"\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":16083,"title":16084,"module":16044,"summary":16085},"\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":16087,"title":16088,"module":16044,"summary":16089},"\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":16091,"title":16092,"module":16044,"summary":16093},"\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":16095,"title":16096,"module":16044,"summary":16097},"\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":16099,"title":16100,"module":16044,"summary":16101},"\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":16103,"title":16104,"module":16044,"summary":16105},"\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":16107,"title":16108,"module":16044,"summary":16109},"\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":16111,"title":16112,"module":16044,"summary":16113},"\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":16115,"title":16116,"module":16044,"summary":16117},"\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":16119,"title":16120,"module":16044,"summary":16121},"\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":16123,"title":16124,"module":14678,"summary":16125},"\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":16127,"title":16128,"module":14678,"summary":16129},"\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":16131,"title":16132,"module":14678,"summary":16133},"\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":16135,"title":16136,"module":14678,"summary":16137},"\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":16139,"title":16140,"module":14678,"summary":16141},"\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":16143,"title":16144,"module":14678,"summary":16145},"\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":16147,"title":16148,"module":14678,"summary":16149},"\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":16151,"title":16152,"module":14678,"summary":16153},"\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":16155,"title":16156,"module":16157,"summary":16158},"\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":16160,"title":16161,"module":16157,"summary":16162},"\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":16164,"title":16165,"module":16157,"summary":16166},"\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":16168,"title":16169,"module":16157,"summary":16170},"\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":16172,"title":16173,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":16175,"title":16176,"module":11922,"summary":16177},"\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":16179,"title":16180,"module":11922,"summary":16181},"\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":16183,"title":16184,"module":11922,"summary":16185},"\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":16187,"title":16188,"module":16189,"summary":16190},"\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":16192,"title":16193,"module":16189,"summary":16194},"\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":16196,"title":16197,"module":16189,"summary":16198},"\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":16200,"title":16201,"module":16189,"summary":16202},"\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":16204,"title":16205,"module":16206,"summary":16207},"\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":16209,"title":16210,"module":16206,"summary":16211},"\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":16213,"title":16214,"module":16206,"summary":16215},"\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":16217,"title":16218,"module":16206,"summary":16219},"\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":16221,"title":16222,"module":16223,"summary":16224},"\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":16226,"title":16227,"module":16223,"summary":16228},"\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":16230,"title":16231,"module":16223,"summary":16232},"\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":16234,"title":16235,"module":16223,"summary":16236},"\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":16238,"title":16239,"module":16240,"summary":16241},"\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":16243,"title":16244,"module":16240,"summary":16245},"\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":16247,"title":16248,"module":16240,"summary":16249},"\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":16251,"title":16252,"module":16253,"summary":16254},"\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":16256,"title":16257,"module":16253,"summary":16258},"\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":16260,"title":16261,"module":16253,"summary":16262},"\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":16264,"title":16265,"module":16253,"summary":16266},"\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":16268,"title":16269,"module":16270,"summary":16271},"\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":16273,"title":16274,"module":16270,"summary":16275},"\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":16277,"title":16278,"module":16270,"summary":16279},"\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":16281,"title":16282,"module":16270,"summary":16283},"\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":16285,"title":16286,"module":16287,"summary":16288},"\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":16290,"title":16291,"module":16287,"summary":16292},"\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":16294,"title":16295,"module":16287,"summary":16296},"\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":16298,"title":16299,"module":16287,"summary":16300},"\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":16302,"title":16303,"module":16304,"summary":16305},"\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":16307,"title":16308,"module":16304,"summary":16309},"\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":16311,"title":16312,"module":16304,"summary":16313},"\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":16315,"title":16316,"module":16304,"summary":16317},"\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":16319,"title":16320,"module":16304,"summary":16321},"\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":16323,"title":16324,"module":16325,"summary":16326},"\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":16328,"title":16329,"module":16325,"summary":16330},"\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":16332,"title":16333,"module":16325,"summary":16334},"\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":16336,"title":16337,"module":16338,"summary":16339},"\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":16341,"title":16342,"module":16338,"summary":16343},"\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":16345,"title":16346,"module":16338,"summary":16347},"\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":16349,"title":16350,"module":16350,"summary":16351},"\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":16353,"title":16354,"module":16350,"summary":16355},"\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":16357,"title":16358,"module":16350,"summary":16359},"\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":16361,"title":16362,"module":16350,"summary":16363},"\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":16365,"title":16366,"module":16350,"summary":16367},"\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":16369,"title":16370,"module":16350,"summary":16371},"\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":16373,"title":16374,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":16376,"title":16377,"module":16378,"summary":16379},"\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":16381,"title":16382,"module":16378,"summary":16383},"\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":16385,"title":16386,"module":16378,"summary":16387},"\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":16389,"title":16390,"module":16391,"summary":16392},"\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":16394,"title":16395,"module":16391,"summary":16396},"\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":16398,"title":16399,"module":16391,"summary":16400},"\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":16402,"title":16403,"module":16391,"summary":16404},"\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":16406,"title":16407,"module":16408,"summary":16409},"\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":16411,"title":16412,"module":16408,"summary":16413},"\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":16415,"title":16416,"module":16408,"summary":16417},"\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":16419,"title":16420,"module":16408,"summary":16421},"\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":16423,"title":16424,"module":16425,"summary":16426},"\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":16428,"title":16429,"module":16425,"summary":16430},"\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":16432,"title":16433,"module":16425,"summary":16434},"\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":16436,"title":16437,"module":16425,"summary":16438},"\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":16440,"title":16441,"module":16442,"summary":16443},"\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":16445,"title":16446,"module":16442,"summary":16447},"\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":16449,"title":16450,"module":16442,"summary":16451},"\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":16453,"title":16454,"module":16442,"summary":16455},"\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":16457,"title":16458,"module":16459,"summary":16460},"\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":16462,"title":16463,"module":16459,"summary":16464},"\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":16466,"title":16467,"module":16459,"summary":16468},"\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":16470,"title":16471,"module":16472,"summary":16473},"\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":16475,"title":16476,"module":16472,"summary":16477},"\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":16479,"title":16480,"module":16472,"summary":16481},"\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":16483,"title":16484,"module":16472,"summary":16485},"\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":16487,"title":14911,"module":16488,"summary":16489},"\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":16491,"title":16492,"module":16488,"summary":16493},"\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":16495,"title":16496,"module":16488,"summary":16497},"\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":16499,"title":16500,"module":16488,"summary":16501},"\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":16503,"title":16504,"module":16488,"summary":16505},"\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":16507,"title":16508,"module":16509,"summary":16510},"\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":16512,"title":16513,"module":16509,"summary":16514},"\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":16516,"title":16517,"module":16509,"summary":16518},"\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":16520,"title":16521,"module":16509,"summary":16522},"\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":16524,"title":16525,"module":16526,"summary":16527},"\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":16529,"title":16530,"module":16526,"summary":16531},"\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":16533,"title":16534,"module":16526,"summary":16535},"\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":16537,"title":16538,"module":16526,"summary":16539},"\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":16541,"title":16542,"module":16526,"summary":16543},"\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":16545,"title":16546,"module":16547,"summary":16548},"\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":16550,"title":16551,"module":16547,"summary":16552},"\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":16554,"title":13629,"module":16547,"summary":16555},"\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":16557,"title":16558,"module":16547,"summary":16559},"\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":16561,"title":16562,"module":16547,"summary":16563},"\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":16565,"title":16566,"module":16567,"summary":16568},"\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":16570,"title":16571,"module":16567,"summary":16572},"\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":16574,"title":16575,"module":16567,"summary":16576},"\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":16578,"title":16579,"module":16567,"summary":16580},"\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":16582,"title":16583,"module":16567,"summary":16584},"\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":16586,"title":16587,"module":16567,"summary":16588},"\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":16590,"title":16591,"module":16567,"summary":16592},"\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":16594,"title":16595,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":16597,"title":16598,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":7699,"title":16600,"module":6,"summary":6},"Study Notes",[16602,16620,16653,16687,16706,16761],{"module":11922,"moduleNumber":16603,"slug":16604,"lessons":16605},1,"foundations",[16606,16608,16610,16612,16615,16617],{"title":15298,"path":15297,"lessonNumber":16603,"topics":16607,"summary":15299},[11922],{"title":15302,"path":15301,"lessonNumber":10961,"topics":16609,"summary":15303},[11922],{"title":15306,"path":15305,"lessonNumber":10967,"topics":16611,"summary":15307},[11922],{"title":15310,"path":15309,"lessonNumber":16613,"topics":16614,"summary":15311},4,[11922],{"title":15314,"path":15313,"lessonNumber":10979,"topics":16616,"summary":15315},[11922],{"title":15318,"path":15317,"lessonNumber":16618,"topics":16619,"summary":15319},6,[11922],{"module":15322,"moduleNumber":10961,"slug":16621,"lessons":16622},"tabular-methods",[16623,16626,16628,16630,16632,16634,16636,16639,16642,16644,16647,16650],{"title":12141,"path":15321,"lessonNumber":16603,"topics":16624,"summary":15323},[16625],"Tabular Methods",{"title":15326,"path":15325,"lessonNumber":10961,"topics":16627,"summary":15327},[16625],{"title":15330,"path":15329,"lessonNumber":10967,"topics":16629,"summary":15331},[16625],{"title":15334,"path":15333,"lessonNumber":16613,"topics":16631,"summary":15335},[16625],{"title":15338,"path":15337,"lessonNumber":10979,"topics":16633,"summary":15339},[16625],{"title":15342,"path":15341,"lessonNumber":16618,"topics":16635,"summary":15343},[16625],{"title":15346,"path":15345,"lessonNumber":16637,"topics":16638,"summary":15347},7,[16625],{"title":15350,"path":15349,"lessonNumber":16640,"topics":16641,"summary":15351},8,[16625],{"title":15354,"path":15353,"lessonNumber":10980,"topics":16643,"summary":15355},[16625],{"title":15358,"path":15357,"lessonNumber":16645,"topics":16646,"summary":15359},10,[16625],{"title":15362,"path":15361,"lessonNumber":16648,"topics":16649,"summary":15363},11,[16625],{"title":15366,"path":15365,"lessonNumber":16651,"topics":16652,"summary":15367},12,[16625],{"module":15371,"moduleNumber":10967,"slug":16654,"lessons":16655},"approximation",[16656,16659,16661,16663,16665,16667,16669,16671,16673,16675,16677,16679,16681,16684],{"title":15370,"path":15369,"lessonNumber":16603,"topics":16657,"summary":15372},[16658],"Approximation",{"title":15375,"path":15374,"lessonNumber":10961,"topics":16660,"summary":15376},[16658],{"title":15379,"path":15378,"lessonNumber":10967,"topics":16662,"summary":15380},[16658],{"title":15383,"path":15382,"lessonNumber":16613,"topics":16664,"summary":15384},[16658],{"title":15386,"path":67,"lessonNumber":10979,"topics":16666,"summary":15387},[16658],{"title":15390,"path":15389,"lessonNumber":16618,"topics":16668,"summary":15391},[16658],{"title":15394,"path":15393,"lessonNumber":16637,"topics":16670,"summary":15395},[16658],{"title":15398,"path":15397,"lessonNumber":16640,"topics":16672,"summary":15399},[16658],{"title":15402,"path":15401,"lessonNumber":10980,"topics":16674,"summary":15403},[16658],{"title":15406,"path":15405,"lessonNumber":16645,"topics":16676,"summary":15407},[16658],{"title":15410,"path":15409,"lessonNumber":16648,"topics":16678,"summary":15411},[16658],{"title":15414,"path":15413,"lessonNumber":16651,"topics":16680,"summary":15415},[16658],{"title":15418,"path":15417,"lessonNumber":16682,"topics":16683,"summary":15419},13,[16658],{"title":15422,"path":15421,"lessonNumber":16685,"topics":16686,"summary":15423},14,[16658],{"module":15426,"moduleNumber":16613,"slug":16688,"lessons":16689},"deep-rl",[16690,16692,16694,16696,16698,16700,16702,16704],{"title":14744,"path":15425,"lessonNumber":16603,"topics":16691,"summary":15427},[11004],{"title":15430,"path":15429,"lessonNumber":10961,"topics":16693,"summary":15431},[11004],{"title":15434,"path":15433,"lessonNumber":10967,"topics":16695,"summary":15435},[11004],{"title":15438,"path":15437,"lessonNumber":16613,"topics":16697,"summary":15439},[11004],{"title":15442,"path":15441,"lessonNumber":10979,"topics":16699,"summary":15443},[11004],{"title":15446,"path":15445,"lessonNumber":16618,"topics":16701,"summary":15447},[11004],{"title":15450,"path":15449,"lessonNumber":16637,"topics":16703,"summary":15451},[11004],{"title":15454,"path":15453,"lessonNumber":16640,"topics":16705,"summary":15455},[11004],{"module":10982,"moduleNumber":10979,"slug":16707,"lessons":16708},"modern-deep-rl",[16709,16711,16713,16715,16717,16719,16721,16723,16725,16727,16729,16731,16733,16735,16737,16740,16743,16746,16749,16752,16755,16758],{"title":15458,"path":15457,"lessonNumber":16603,"topics":16710,"summary":15459},[11004],{"title":15462,"path":15461,"lessonNumber":10961,"topics":16712,"summary":15463},[11004],{"title":15465,"path":1067,"lessonNumber":10967,"topics":16714,"summary":15466},[11004],{"title":15469,"path":15468,"lessonNumber":16613,"topics":16716,"summary":15470},[11004],{"title":15473,"path":15472,"lessonNumber":10979,"topics":16718,"summary":15474},[11004],{"title":15477,"path":15476,"lessonNumber":16618,"topics":16720,"summary":15478},[11004],{"title":15481,"path":15480,"lessonNumber":16637,"topics":16722,"summary":15482},[11004],{"title":15485,"path":15484,"lessonNumber":16640,"topics":16724,"summary":15486},[11004],{"title":5,"path":10984,"lessonNumber":10980,"topics":16726,"summary":11002},[11004],{"title":10898,"path":126,"lessonNumber":16645,"topics":16728,"summary":15489},[11004],{"title":15492,"path":15491,"lessonNumber":16648,"topics":16730,"summary":15493},[11004],{"title":15496,"path":15495,"lessonNumber":16651,"topics":16732,"summary":15497},[11004],{"title":15500,"path":15499,"lessonNumber":16682,"topics":16734,"summary":15501},[11004],{"title":15504,"path":15503,"lessonNumber":16685,"topics":16736,"summary":15505},[11004],{"title":15508,"path":15507,"lessonNumber":16738,"topics":16739,"summary":15509},15,[11004],{"title":15512,"path":15511,"lessonNumber":16741,"topics":16742,"summary":15513},16,[11004],{"title":15516,"path":15515,"lessonNumber":16744,"topics":16745,"summary":15517},17,[11004],{"title":15520,"path":15519,"lessonNumber":16747,"topics":16748,"summary":15521},18,[11004],{"title":15524,"path":15523,"lessonNumber":16750,"topics":16751,"summary":15525},19,[11004],{"title":15528,"path":15527,"lessonNumber":16753,"topics":16754,"summary":15529},20,[11004],{"title":15532,"path":15531,"lessonNumber":16756,"topics":16757,"summary":15533},21,[11004],{"title":15536,"path":15535,"lessonNumber":16759,"topics":16760,"summary":15537},22,[11004],{"module":15541,"moduleNumber":16618,"slug":16762,"lessons":16763},"minds-and-brains",[16764,16767,16769,16771,16773,16775,16777,16779],{"title":15540,"path":15539,"lessonNumber":16603,"topics":16765,"summary":15542},[16766],"Minds and Brains",{"title":15545,"path":15544,"lessonNumber":10961,"topics":16768,"summary":15546},[16766],{"title":15549,"path":15548,"lessonNumber":10967,"topics":16770,"summary":15550},[16766],{"title":15553,"path":15552,"lessonNumber":16613,"topics":16772,"summary":15554},[16766],{"title":15557,"path":15556,"lessonNumber":10979,"topics":16774,"summary":15558},[16766],{"title":15561,"path":15560,"lessonNumber":16618,"topics":16776,"summary":15562},[16766],{"title":15565,"path":15564,"lessonNumber":16637,"topics":16778,"summary":15566},[16766],{"title":15569,"path":15568,"lessonNumber":16640,"topics":16780,"summary":15570},[16766],"\u003Csvg style=\"width:100%;max-width:442.083px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 331.562 151.710\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-45.303 61.208h265.456\"\u002F>\u003Cpath stroke=\"none\" d=\"m222.153 61.208-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-39.276 60.376Q-39.276 59.892-38.874 59.597Q-38.471 59.302-37.921 59.183Q-37.370 59.063-36.878 59.063L-36.878 58.774Q-36.878 58.548-36.993 58.341Q-37.108 58.134-37.305 58.015Q-37.503 57.895-37.733 57.895Q-38.159 57.895-38.444 58.001Q-38.374 58.028-38.327 58.083Q-38.280 58.138-38.255 58.208Q-38.229 58.278-38.229 58.353Q-38.229 58.458-38.280 58.550Q-38.331 58.642-38.423 58.692Q-38.514 58.743-38.620 58.743Q-38.725 58.743-38.817 58.692Q-38.909 58.642-38.960 58.550Q-39.010 58.458-39.010 58.353Q-39.010 57.935-38.622 57.788Q-38.233 57.642-37.733 57.642Q-37.401 57.642-37.048 57.772Q-36.694 57.903-36.466 58.157Q-36.237 58.411-36.237 58.759L-36.237 60.560Q-36.237 60.692-36.165 60.802Q-36.092 60.911-35.964 60.911Q-35.839 60.911-35.770 60.806Q-35.702 60.700-35.702 60.560L-35.702 60.048L-35.421 60.048L-35.421 60.560Q-35.421 60.763-35.538 60.921Q-35.655 61.079-35.837 61.163Q-36.018 61.247-36.221 61.247Q-36.452 61.247-36.604 61.075Q-36.757 60.903-36.788 60.673Q-36.948 60.954-37.257 61.120Q-37.565 61.286-37.917 61.286Q-38.428 61.286-38.852 61.063Q-39.276 60.841-39.276 60.376M-38.589 60.376Q-38.589 60.661-38.362 60.847Q-38.135 61.032-37.842 61.032Q-37.596 61.032-37.372 60.915Q-37.147 60.798-37.012 60.595Q-36.878 60.392-36.878 60.138L-36.878 59.306Q-37.143 59.306-37.428 59.360Q-37.714 59.415-37.985 59.544Q-38.257 59.673-38.423 59.880Q-38.589 60.087-38.589 60.376M-35.085 59.481Q-35.085 58.985-34.835 58.560Q-34.585 58.134-34.165 57.888Q-33.745 57.642-33.245 57.642Q-32.706 57.642-32.315 57.767Q-31.925 57.892-31.925 58.306Q-31.925 58.411-31.975 58.503Q-32.026 58.595-32.118 58.645Q-32.210 58.696-32.319 58.696Q-32.425 58.696-32.516 58.645Q-32.608 58.595-32.659 58.503Q-32.710 58.411-32.710 58.306Q-32.710 58.083-32.542 57.978Q-32.764 57.919-33.237 57.919Q-33.534 57.919-33.749 58.058Q-33.964 58.196-34.094 58.427Q-34.225 58.657-34.284 58.927Q-34.342 59.196-34.342 59.481Q-34.342 59.876-34.210 60.226Q-34.077 60.575-33.805 60.792Q-33.534 61.009-33.135 61.009Q-32.760 61.009-32.485 60.792Q-32.210 60.575-32.108 60.216Q-32.092 60.153-32.030 60.153L-31.925 60.153Q-31.889 60.153-31.864 60.181Q-31.839 60.208-31.839 60.247L-31.839 60.270Q-31.971 60.751-32.356 61.019Q-32.741 61.286-33.245 61.286Q-33.608 61.286-33.942 61.149Q-34.276 61.013-34.536 60.763Q-34.796 60.513-34.940 60.177Q-35.085 59.841-35.085 59.481M-30.725 60.247L-30.725 58.056L-31.428 58.056L-31.428 57.802Q-31.073 57.802-30.831 57.569Q-30.589 57.337-30.477 56.989Q-30.366 56.642-30.366 56.286L-30.085 56.286L-30.085 57.759L-28.909 57.759L-28.909 58.056L-30.085 58.056L-30.085 60.231Q-30.085 60.552-29.966 60.780Q-29.846 61.009-29.565 61.009Q-29.385 61.009-29.268 60.886Q-29.151 60.763-29.098 60.583Q-29.046 60.403-29.046 60.231L-29.046 59.759L-28.764 59.759L-28.764 60.247Q-28.764 60.501-28.870 60.741Q-28.975 60.981-29.173 61.134Q-29.370 61.286-29.628 61.286Q-29.944 61.286-30.196 61.163Q-30.448 61.040-30.587 60.806Q-30.725 60.571-30.725 60.247M-26.186 61.208L-27.964 61.208L-27.964 60.911Q-27.690 60.911-27.522 60.864Q-27.354 60.817-27.354 60.649L-27.354 58.513Q-27.354 58.298-27.411 58.202Q-27.467 58.106-27.581 58.085Q-27.694 58.063-27.940 58.063L-27.940 57.767L-26.741 57.681L-26.741 60.649Q-26.741 60.817-26.594 60.864Q-26.448 60.911-26.186 60.911L-26.186 61.208M-27.628 56.286Q-27.628 56.095-27.493 55.964Q-27.358 55.833-27.163 55.833Q-27.042 55.833-26.938 55.895Q-26.835 55.958-26.772 56.062Q-26.710 56.165-26.710 56.286Q-26.710 56.481-26.841 56.616Q-26.971 56.751-27.163 56.751Q-27.362 56.751-27.495 56.618Q-27.628 56.485-27.628 56.286M-25.686 59.513Q-25.686 59.009-25.430 58.577Q-25.175 58.145-24.739 57.894Q-24.303 57.642-23.803 57.642Q-23.417 57.642-23.075 57.786Q-22.733 57.931-22.471 58.192Q-22.210 58.454-22.067 58.790Q-21.925 59.126-21.925 59.513Q-21.925 60.005-22.188 60.415Q-22.452 60.825-22.882 61.056Q-23.311 61.286-23.803 61.286Q-24.296 61.286-24.729 61.054Q-25.163 60.821-25.425 60.413Q-25.686 60.005-25.686 59.513M-23.803 61.009Q-23.346 61.009-23.094 60.786Q-22.842 60.563-22.755 60.212Q-22.667 59.860-22.667 59.415Q-22.667 58.985-22.760 58.647Q-22.854 58.310-23.108 58.103Q-23.362 57.895-23.803 57.895Q-24.452 57.895-24.696 58.312Q-24.940 58.728-24.940 59.415Q-24.940 59.860-24.852 60.212Q-24.764 60.563-24.512 60.786Q-24.260 61.009-23.803 61.009M-19.510 61.208L-21.366 61.208L-21.366 60.911Q-21.092 60.911-20.925 60.864Q-20.757 60.817-20.757 60.649L-20.757 58.513Q-20.757 58.298-20.819 58.202Q-20.882 58.106-21.001 58.085Q-21.120 58.063-21.366 58.063L-21.366 57.767L-20.175 57.681L-20.175 58.415Q-20.061 58.200-19.868 58.032Q-19.675 57.864-19.436 57.772Q-19.198 57.681-18.944 57.681Q-17.776 57.681-17.776 58.759L-17.776 60.649Q-17.776 60.817-17.606 60.864Q-17.436 60.911-17.167 60.911L-17.167 61.208L-19.022 61.208L-19.022 60.911Q-18.749 60.911-18.581 60.864Q-18.413 60.817-18.413 60.649L-18.413 58.774Q-18.413 58.392-18.534 58.163Q-18.655 57.935-19.007 57.935Q-19.319 57.935-19.573 58.097Q-19.827 58.259-19.973 58.528Q-20.120 58.798-20.120 59.095L-20.120 60.649Q-20.120 60.817-19.950 60.864Q-19.780 60.911-19.510 60.911\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-12.608 61.286Q-12.964 61.286-13.233 61.106Q-13.503 60.927-13.643 60.632Q-13.784 60.337-13.784 59.978Q-13.784 59.591-13.622 59.177Q-13.460 58.763-13.176 58.425Q-12.893 58.087-12.524 57.884Q-12.155 57.681-11.753 57.681Q-11.260 57.681-10.983 58.130Q-10.952 57.997-10.848 57.915Q-10.745 57.833-10.616 57.833Q-10.503 57.833-10.423 57.903Q-10.342 57.974-10.342 58.087Q-10.342 58.114-10.358 58.177L-10.913 60.376Q-10.952 60.622-10.952 60.673Q-10.952 61.032-10.706 61.032Q-10.561 61.032-10.460 60.925Q-10.358 60.817-10.294 60.663Q-10.229 60.509-10.180 60.319Q-10.132 60.130-10.112 60.032Q-10.085 59.962-10.022 59.962L-9.921 59.962Q-9.882 59.962-9.856 59.995Q-9.831 60.028-9.831 60.056Q-9.831 60.071-9.839 60.087Q-9.952 60.579-10.151 60.933Q-10.350 61.286-10.721 61.286Q-11.003 61.286-11.229 61.144Q-11.456 61.001-11.538 60.743Q-11.760 60.985-12.034 61.136Q-12.307 61.286-12.608 61.286M-12.592 61.032Q-12.382 61.032-12.173 60.919Q-11.964 60.806-11.794 60.632Q-11.624 60.458-11.495 60.263Q-11.507 60.278-11.516 60.292Q-11.526 60.306-11.538 60.321L-11.104 58.599Q-11.143 58.419-11.229 58.269Q-11.315 58.118-11.452 58.026Q-11.589 57.935-11.768 57.935Q-12.104 57.935-12.376 58.216Q-12.647 58.497-12.807 58.872Q-12.932 59.192-13.030 59.612Q-13.128 60.032-13.128 60.313Q-13.128 60.603-12.995 60.817Q-12.862 61.032-12.592 61.032\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-39.612 66.899V-61.983\"\u002F>\u003Cpath stroke=\"none\" d=\"m-39.612-63.983-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-22.658 -122.414)\">\u003Cpath d=\"M-37.573 61.177L-38.796 58.321Q-38.878 58.145-39.022 58.101Q-39.167 58.056-39.436 58.056L-39.436 57.759L-37.725 57.759L-37.725 58.056Q-38.147 58.056-38.147 58.239Q-38.147 58.274-38.132 58.321L-37.186 60.513L-36.346 58.536Q-36.307 58.458-36.307 58.368Q-36.307 58.228-36.413 58.142Q-36.518 58.056-36.659 58.056L-36.659 57.759L-35.307 57.759L-35.307 58.056Q-35.831 58.056-36.046 58.536L-37.171 61.177Q-37.233 61.286-37.339 61.286L-37.405 61.286Q-37.518 61.286-37.573 61.177\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-22.658 -122.414)\">\u003Cpath d=\"M-35.262 60.376Q-35.262 59.892-34.860 59.597Q-34.457 59.302-33.907 59.183Q-33.356 59.063-32.864 59.063L-32.864 58.774Q-32.864 58.548-32.979 58.341Q-33.094 58.134-33.291 58.015Q-33.489 57.895-33.719 57.895Q-34.145 57.895-34.430 58.001Q-34.360 58.028-34.313 58.083Q-34.266 58.138-34.241 58.208Q-34.215 58.278-34.215 58.353Q-34.215 58.458-34.266 58.550Q-34.317 58.642-34.409 58.692Q-34.500 58.743-34.606 58.743Q-34.711 58.743-34.803 58.692Q-34.895 58.642-34.946 58.550Q-34.996 58.458-34.996 58.353Q-34.996 57.935-34.608 57.788Q-34.219 57.642-33.719 57.642Q-33.387 57.642-33.034 57.772Q-32.680 57.903-32.452 58.157Q-32.223 58.411-32.223 58.759L-32.223 60.560Q-32.223 60.692-32.151 60.802Q-32.078 60.911-31.950 60.911Q-31.825 60.911-31.756 60.806Q-31.688 60.700-31.688 60.560L-31.688 60.048L-31.407 60.048L-31.407 60.560Q-31.407 60.763-31.524 60.921Q-31.641 61.079-31.823 61.163Q-32.004 61.247-32.207 61.247Q-32.438 61.247-32.590 61.075Q-32.743 60.903-32.774 60.673Q-32.934 60.954-33.243 61.120Q-33.551 61.286-33.903 61.286Q-34.414 61.286-34.838 61.063Q-35.262 60.841-35.262 60.376M-34.575 60.376Q-34.575 60.661-34.348 60.847Q-34.121 61.032-33.828 61.032Q-33.582 61.032-33.358 60.915Q-33.133 60.798-32.998 60.595Q-32.864 60.392-32.864 60.138L-32.864 59.306Q-33.129 59.306-33.414 59.360Q-33.700 59.415-33.971 59.544Q-34.243 59.673-34.409 59.880Q-34.575 60.087-34.575 60.376M-29.200 61.208L-31.032 61.208L-31.032 60.911Q-30.758 60.911-30.590 60.864Q-30.422 60.817-30.422 60.649L-30.422 56.489Q-30.422 56.274-30.485 56.179Q-30.547 56.083-30.666 56.062Q-30.785 56.040-31.032 56.040L-31.032 55.743L-29.809 55.657L-29.809 60.649Q-29.809 60.817-29.641 60.864Q-29.473 60.911-29.200 60.911L-29.200 61.208M-28.071 60.255L-28.071 58.513Q-28.071 58.298-28.133 58.202Q-28.196 58.106-28.315 58.085Q-28.434 58.063-28.680 58.063L-28.680 57.767L-27.434 57.681L-27.434 60.231L-27.434 60.255Q-27.434 60.567-27.379 60.729Q-27.325 60.892-27.174 60.962Q-27.024 61.032-26.703 61.032Q-26.274 61.032-26 60.694Q-25.727 60.356-25.727 59.911L-25.727 58.513Q-25.727 58.298-25.789 58.202Q-25.852 58.106-25.971 58.085Q-26.090 58.063-26.336 58.063L-26.336 57.767L-25.090 57.681L-25.090 60.466Q-25.090 60.677-25.028 60.772Q-24.965 60.868-24.846 60.890Q-24.727 60.911-24.481 60.911L-24.481 61.208L-25.703 61.286L-25.703 60.665Q-25.871 60.954-26.153 61.120Q-26.434 61.286-26.754 61.286Q-28.071 61.286-28.071 60.255M-24.035 59.454Q-24.035 58.974-23.803 58.558Q-23.571 58.142-23.160 57.892Q-22.750 57.642-22.274 57.642Q-21.543 57.642-21.145 58.083Q-20.746 58.524-20.746 59.255Q-20.746 59.360-20.840 59.384L-23.289 59.384L-23.289 59.454Q-23.289 59.864-23.168 60.220Q-23.047 60.575-22.776 60.792Q-22.504 61.009-22.075 61.009Q-21.711 61.009-21.414 60.780Q-21.118 60.552-21.016 60.200Q-21.008 60.153-20.922 60.138L-20.840 60.138Q-20.746 60.165-20.746 60.247Q-20.746 60.255-20.754 60.286Q-20.817 60.513-20.955 60.696Q-21.094 60.880-21.285 61.013Q-21.477 61.145-21.696 61.216Q-21.914 61.286-22.153 61.286Q-22.524 61.286-22.862 61.149Q-23.200 61.013-23.467 60.761Q-23.735 60.509-23.885 60.169Q-24.035 59.829-24.035 59.454M-23.282 59.145L-21.321 59.145Q-21.321 58.841-21.422 58.550Q-21.524 58.259-21.741 58.077Q-21.957 57.895-22.274 57.895Q-22.575 57.895-22.805 58.083Q-23.035 58.270-23.159 58.562Q-23.282 58.853-23.282 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"none\" d=\"M28.674 61.208v-119.5h91.05v119.5Zm91.05-119.5\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-soft-accent)\" d=\"M28.674 61.208v-119.5M119.723 61.208v-119.5\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-37.557 61.286Q-38.038 61.286-38.446 61.042Q-38.854 60.798-39.092 60.384Q-39.331 59.970-39.331 59.481Q-39.331 58.989-39.073 58.573Q-38.815 58.157-38.383 57.919Q-37.952 57.681-37.460 57.681Q-36.839 57.681-36.389 58.118L-36.389 56.489Q-36.389 56.274-36.452 56.179Q-36.514 56.083-36.632 56.062Q-36.749 56.040-36.995 56.040L-36.995 55.743L-35.772 55.657L-35.772 60.466Q-35.772 60.677-35.710 60.772Q-35.647 60.868-35.530 60.890Q-35.413 60.911-35.163 60.911L-35.163 61.208L-36.413 61.286L-36.413 60.802Q-36.878 61.286-37.557 61.286M-37.491 61.032Q-37.151 61.032-36.858 60.841Q-36.565 60.649-36.413 60.353L-36.413 58.520Q-36.561 58.247-36.823 58.091Q-37.085 57.935-37.397 57.935Q-38.022 57.935-38.305 58.382Q-38.589 58.829-38.589 59.489Q-38.589 60.134-38.337 60.583Q-38.085 61.032-37.491 61.032M-34.557 60.376Q-34.557 59.892-34.155 59.597Q-33.753 59.302-33.202 59.183Q-32.651 59.063-32.159 59.063L-32.159 58.774Q-32.159 58.548-32.274 58.341Q-32.389 58.134-32.587 58.015Q-32.784 57.895-33.014 57.895Q-33.440 57.895-33.725 58.001Q-33.655 58.028-33.608 58.083Q-33.561 58.138-33.536 58.208Q-33.510 58.278-33.510 58.353Q-33.510 58.458-33.561 58.550Q-33.612 58.642-33.704 58.692Q-33.796 58.743-33.901 58.743Q-34.007 58.743-34.098 58.692Q-34.190 58.642-34.241 58.550Q-34.292 58.458-34.292 58.353Q-34.292 57.935-33.903 57.788Q-33.514 57.642-33.014 57.642Q-32.682 57.642-32.329 57.772Q-31.975 57.903-31.747 58.157Q-31.518 58.411-31.518 58.759L-31.518 60.560Q-31.518 60.692-31.446 60.802Q-31.374 60.911-31.245 60.911Q-31.120 60.911-31.051 60.806Q-30.983 60.700-30.983 60.560L-30.983 60.048L-30.702 60.048L-30.702 60.560Q-30.702 60.763-30.819 60.921Q-30.936 61.079-31.118 61.163Q-31.300 61.247-31.503 61.247Q-31.733 61.247-31.885 61.075Q-32.038 60.903-32.069 60.673Q-32.229 60.954-32.538 61.120Q-32.846 61.286-33.198 61.286Q-33.710 61.286-34.133 61.063Q-34.557 60.841-34.557 60.376M-33.870 60.376Q-33.870 60.661-33.643 60.847Q-33.417 61.032-33.124 61.032Q-32.878 61.032-32.653 60.915Q-32.428 60.798-32.294 60.595Q-32.159 60.392-32.159 60.138L-32.159 59.306Q-32.425 59.306-32.710 59.360Q-32.995 59.415-33.266 59.544Q-33.538 59.673-33.704 59.880Q-33.870 60.087-33.870 60.376M-29.784 60.247L-29.784 58.056L-30.487 58.056L-30.487 57.802Q-30.132 57.802-29.889 57.569Q-29.647 57.337-29.536 56.989Q-29.425 56.642-29.425 56.286L-29.143 56.286L-29.143 57.759L-27.967 57.759L-27.967 58.056L-29.143 58.056L-29.143 60.231Q-29.143 60.552-29.024 60.780Q-28.905 61.009-28.624 61.009Q-28.444 61.009-28.327 60.886Q-28.210 60.763-28.157 60.583Q-28.104 60.403-28.104 60.231L-28.104 59.759L-27.823 59.759L-27.823 60.247Q-27.823 60.501-27.928 60.741Q-28.034 60.981-28.231 61.134Q-28.428 61.286-28.686 61.286Q-29.003 61.286-29.255 61.163Q-29.507 61.040-29.645 60.806Q-29.784 60.571-29.784 60.247M-27.007 60.376Q-27.007 59.892-26.604 59.597Q-26.202 59.302-25.651 59.183Q-25.100 59.063-24.608 59.063L-24.608 58.774Q-24.608 58.548-24.723 58.341Q-24.839 58.134-25.036 58.015Q-25.233 57.895-25.464 57.895Q-25.889 57.895-26.175 58.001Q-26.104 58.028-26.057 58.083Q-26.010 58.138-25.985 58.208Q-25.960 58.278-25.960 58.353Q-25.960 58.458-26.010 58.550Q-26.061 58.642-26.153 58.692Q-26.245 58.743-26.350 58.743Q-26.456 58.743-26.548 58.692Q-26.639 58.642-26.690 58.550Q-26.741 58.458-26.741 58.353Q-26.741 57.935-26.352 57.788Q-25.964 57.642-25.464 57.642Q-25.132 57.642-24.778 57.772Q-24.425 57.903-24.196 58.157Q-23.967 58.411-23.967 58.759L-23.967 60.560Q-23.967 60.692-23.895 60.802Q-23.823 60.911-23.694 60.911Q-23.569 60.911-23.501 60.806Q-23.432 60.700-23.432 60.560L-23.432 60.048L-23.151 60.048L-23.151 60.560Q-23.151 60.763-23.268 60.921Q-23.385 61.079-23.567 61.163Q-23.749 61.247-23.952 61.247Q-24.182 61.247-24.335 61.075Q-24.487 60.903-24.518 60.673Q-24.678 60.954-24.987 61.120Q-25.296 61.286-25.647 61.286Q-26.159 61.286-26.583 61.063Q-27.007 60.841-27.007 60.376M-26.319 60.376Q-26.319 60.661-26.092 60.847Q-25.866 61.032-25.573 61.032Q-25.327 61.032-25.102 60.915Q-24.878 60.798-24.743 60.595Q-24.608 60.392-24.608 60.138L-24.608 59.306Q-24.874 59.306-25.159 59.360Q-25.444 59.415-25.716 59.544Q-25.987 59.673-26.153 59.880Q-26.319 60.087-26.319 60.376\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-19.970 61.200L-19.970 59.978Q-19.970 59.950-19.939 59.919Q-19.907 59.888-19.884 59.888L-19.778 59.888Q-19.708 59.888-19.692 59.950Q-19.630 60.270-19.491 60.511Q-19.353 60.751-19.120 60.892Q-18.888 61.032-18.579 61.032Q-18.341 61.032-18.132 60.972Q-17.923 60.911-17.786 60.763Q-17.649 60.614-17.649 60.368Q-17.649 60.114-17.860 59.948Q-18.071 59.782-18.341 59.728L-18.962 59.614Q-19.368 59.536-19.669 59.280Q-19.970 59.024-19.970 58.649Q-19.970 58.282-19.769 58.060Q-19.567 57.837-19.243 57.739Q-18.919 57.642-18.579 57.642Q-18.114 57.642-17.817 57.849L-17.595 57.665Q-17.571 57.642-17.540 57.642L-17.489 57.642Q-17.458 57.642-17.431 57.669Q-17.403 57.696-17.403 57.728L-17.403 58.712Q-17.403 58.743-17.429 58.772Q-17.454 58.802-17.489 58.802L-17.595 58.802Q-17.630 58.802-17.657 58.774Q-17.685 58.747-17.685 58.712Q-17.685 58.313-17.937 58.093Q-18.189 57.872-18.587 57.872Q-18.942 57.872-19.226 57.995Q-19.509 58.118-19.509 58.423Q-19.509 58.642-19.308 58.774Q-19.106 58.907-18.860 58.950L-18.235 59.063Q-17.806 59.153-17.497 59.450Q-17.189 59.747-17.189 60.161Q-17.189 60.731-17.587 61.009Q-17.985 61.286-18.579 61.286Q-19.130 61.286-19.481 60.950L-19.778 61.263Q-19.802 61.286-19.837 61.286L-19.884 61.286Q-19.907 61.286-19.939 61.255Q-19.970 61.224-19.970 61.200M-15.978 60.255L-15.978 58.513Q-15.978 58.298-16.040 58.202Q-16.103 58.106-16.222 58.085Q-16.341 58.063-16.587 58.063L-16.587 57.767L-15.341 57.681L-15.341 60.231L-15.341 60.255Q-15.341 60.567-15.286 60.729Q-15.231 60.892-15.081 60.962Q-14.931 61.032-14.610 61.032Q-14.181 61.032-13.907 60.694Q-13.634 60.356-13.634 59.911L-13.634 58.513Q-13.634 58.298-13.696 58.202Q-13.759 58.106-13.878 58.085Q-13.997 58.063-14.243 58.063L-14.243 57.767L-12.997 57.681L-12.997 60.466Q-12.997 60.677-12.935 60.772Q-12.872 60.868-12.753 60.890Q-12.634 60.911-12.388 60.911L-12.388 61.208L-13.610 61.286L-13.610 60.665Q-13.778 60.954-14.060 61.120Q-14.341 61.286-14.661 61.286Q-15.978 61.286-15.978 60.255M-10.060 62.759L-11.915 62.759L-11.915 62.466Q-11.646 62.466-11.478 62.421Q-11.310 62.376-11.310 62.200L-11.310 58.376Q-11.310 58.169-11.466 58.116Q-11.622 58.063-11.915 58.063L-11.915 57.767L-10.692 57.681L-10.692 58.145Q-10.462 57.923-10.147 57.802Q-9.833 57.681-9.493 57.681Q-9.021 57.681-8.616 57.927Q-8.212 58.173-7.980 58.589Q-7.747 59.005-7.747 59.481Q-7.747 59.856-7.896 60.185Q-8.044 60.513-8.314 60.765Q-8.583 61.017-8.927 61.151Q-9.271 61.286-9.630 61.286Q-9.919 61.286-10.190 61.165Q-10.462 61.044-10.669 60.833L-10.669 62.200Q-10.669 62.376-10.501 62.421Q-10.333 62.466-10.060 62.466L-10.060 62.759M-10.669 58.544L-10.669 60.384Q-10.517 60.673-10.255 60.853Q-9.993 61.032-9.685 61.032Q-9.399 61.032-9.177 60.894Q-8.954 60.755-8.802 60.524Q-8.649 60.294-8.571 60.022Q-8.493 59.751-8.493 59.481Q-8.493 59.149-8.618 58.792Q-8.743 58.435-8.991 58.198Q-9.239 57.962-9.587 57.962Q-9.911 57.962-10.206 58.118Q-10.501 58.274-10.669 58.544M-5.341 62.759L-7.196 62.759L-7.196 62.466Q-6.927 62.466-6.759 62.421Q-6.591 62.376-6.591 62.200L-6.591 58.376Q-6.591 58.169-6.747 58.116Q-6.903 58.063-7.196 58.063L-7.196 57.767L-5.974 57.681L-5.974 58.145Q-5.743 57.923-5.429 57.802Q-5.114 57.681-4.774 57.681Q-4.302 57.681-3.897 57.927Q-3.493 58.173-3.261 58.589Q-3.028 59.005-3.028 59.481Q-3.028 59.856-3.177 60.185Q-3.325 60.513-3.595 60.765Q-3.864 61.017-4.208 61.151Q-4.552 61.286-4.911 61.286Q-5.200 61.286-5.472 61.165Q-5.743 61.044-5.950 60.833L-5.950 62.200Q-5.950 62.376-5.782 62.421Q-5.614 62.466-5.341 62.466L-5.341 62.759M-5.950 58.544L-5.950 60.384Q-5.798 60.673-5.536 60.853Q-5.274 61.032-4.966 61.032Q-4.681 61.032-4.458 60.894Q-4.235 60.755-4.083 60.524Q-3.931 60.294-3.853 60.022Q-3.774 59.751-3.774 59.481Q-3.774 59.149-3.899 58.792Q-4.024 58.435-4.272 58.198Q-4.521 57.962-4.868 57.962Q-5.192 57.962-5.487 58.118Q-5.782 58.274-5.950 58.544\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-2.257 59.513Q-2.257 59.009-2.001 58.577Q-1.745 58.145-1.309 57.894Q-0.874 57.642-0.374 57.642Q0.013 57.642 0.355 57.786Q0.696 57.931 0.958 58.192Q1.220 58.454 1.362 58.790Q1.505 59.126 1.505 59.513Q1.505 60.005 1.241 60.415Q0.978 60.825 0.548 61.056Q0.118 61.286-0.374 61.286Q-0.866 61.286-1.300 61.054Q-1.733 60.821-1.995 60.413Q-2.257 60.005-2.257 59.513M-0.374 61.009Q0.083 61.009 0.335 60.786Q0.587 60.563 0.675 60.212Q0.763 59.860 0.763 59.415Q0.763 58.985 0.669 58.647Q0.575 58.310 0.321 58.103Q0.067 57.895-0.374 57.895Q-1.022 57.895-1.266 58.312Q-1.511 58.728-1.511 59.415Q-1.511 59.860-1.423 60.212Q-1.335 60.563-1.083 60.786Q-0.831 61.009-0.374 61.009M3.997 61.208L2.017 61.208L2.017 60.911Q2.286 60.911 2.454 60.866Q2.622 60.821 2.622 60.649L2.622 58.513Q2.622 58.298 2.560 58.202Q2.497 58.106 2.380 58.085Q2.263 58.063 2.017 58.063L2.017 57.767L3.185 57.681L3.185 58.466Q3.263 58.255 3.415 58.069Q3.567 57.884 3.767 57.782Q3.966 57.681 4.192 57.681Q4.439 57.681 4.630 57.825Q4.821 57.970 4.821 58.200Q4.821 58.356 4.716 58.466Q4.610 58.575 4.454 58.575Q4.298 58.575 4.189 58.466Q4.079 58.356 4.079 58.200Q4.079 58.040 4.185 57.935Q3.860 57.935 3.646 58.163Q3.431 58.392 3.335 58.731Q3.239 59.071 3.239 59.376L3.239 60.649Q3.239 60.817 3.466 60.864Q3.692 60.911 3.997 60.911L3.997 61.208M5.927 60.247L5.927 58.056L5.224 58.056L5.224 57.802Q5.579 57.802 5.821 57.569Q6.064 57.337 6.175 56.989Q6.286 56.642 6.286 56.286L6.567 56.286L6.567 57.759L7.743 57.759L7.743 58.056L6.567 58.056L6.567 60.231Q6.567 60.552 6.687 60.780Q6.806 61.009 7.087 61.009Q7.267 61.009 7.384 60.886Q7.501 60.763 7.554 60.583Q7.607 60.403 7.607 60.231L7.607 59.759L7.888 59.759L7.888 60.247Q7.888 60.501 7.782 60.741Q7.677 60.981 7.480 61.134Q7.282 61.286 7.025 61.286Q6.708 61.286 6.456 61.163Q6.204 61.040 6.066 60.806Q5.927 60.571 5.927 60.247\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-28.231 32.756S-7.674 28.01.222 25.643c7.895-2.37 21.346-6.8 28.452-9.96C35.78 12.527 45.12 5.25 51.437 2.88 57.753.512 67.882-1.585 74.199-1.387c6.316.197 16.446 3.321 22.762 5.69 6.317 2.369 15.656 8.618 22.762 11.38 7.106 2.764 20.558 6.76 28.453 8.537 7.896 1.776 20.163 3.281 28.453 4.268s31.298 2.845 31.298 2.845\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(252.494 -27.098)\">\u003Cpath d=\"M-38.749 60.247L-38.749 58.056L-39.452 58.056L-39.452 57.802Q-39.096 57.802-38.854 57.569Q-38.612 57.337-38.501 56.989Q-38.389 56.642-38.389 56.286L-38.108 56.286L-38.108 57.759L-36.932 57.759L-36.932 58.056L-38.108 58.056L-38.108 60.231Q-38.108 60.552-37.989 60.780Q-37.870 61.009-37.589 61.009Q-37.409 61.009-37.292 60.886Q-37.175 60.763-37.122 60.583Q-37.069 60.403-37.069 60.231L-37.069 59.759L-36.788 59.759L-36.788 60.247Q-36.788 60.501-36.893 60.741Q-36.999 60.981-37.196 61.134Q-37.393 61.286-37.651 61.286Q-37.967 61.286-38.219 61.163Q-38.471 61.040-38.610 60.806Q-38.749 60.571-38.749 60.247M-34.061 61.208L-36.042 61.208L-36.042 60.911Q-35.772 60.911-35.604 60.866Q-35.436 60.821-35.436 60.649L-35.436 58.513Q-35.436 58.298-35.499 58.202Q-35.561 58.106-35.678 58.085Q-35.796 58.063-36.042 58.063L-36.042 57.767L-34.874 57.681L-34.874 58.466Q-34.796 58.255-34.643 58.069Q-34.491 57.884-34.292 57.782Q-34.092 57.681-33.866 57.681Q-33.620 57.681-33.428 57.825Q-33.237 57.970-33.237 58.200Q-33.237 58.356-33.342 58.466Q-33.448 58.575-33.604 58.575Q-33.760 58.575-33.870 58.466Q-33.979 58.356-33.979 58.200Q-33.979 58.040-33.874 57.935Q-34.198 57.935-34.413 58.163Q-34.628 58.392-34.723 58.731Q-34.819 59.071-34.819 59.376L-34.819 60.649Q-34.819 60.817-34.592 60.864Q-34.366 60.911-34.061 60.911L-34.061 61.208M-32.073 60.255L-32.073 58.513Q-32.073 58.298-32.135 58.202Q-32.198 58.106-32.317 58.085Q-32.436 58.063-32.682 58.063L-32.682 57.767L-31.436 57.681L-31.436 60.231L-31.436 60.255Q-31.436 60.567-31.382 60.729Q-31.327 60.892-31.176 60.962Q-31.026 61.032-30.706 61.032Q-30.276 61.032-30.003 60.694Q-29.729 60.356-29.729 59.911L-29.729 58.513Q-29.729 58.298-29.792 58.202Q-29.854 58.106-29.973 58.085Q-30.092 58.063-30.339 58.063L-30.339 57.767L-29.092 57.681L-29.092 60.466Q-29.092 60.677-29.030 60.772Q-28.967 60.868-28.848 60.890Q-28.729 60.911-28.483 60.911L-28.483 61.208L-29.706 61.286L-29.706 60.665Q-29.874 60.954-30.155 61.120Q-30.436 61.286-30.757 61.286Q-32.073 61.286-32.073 60.255M-28.038 59.454Q-28.038 58.974-27.805 58.558Q-27.573 58.142-27.163 57.892Q-26.753 57.642-26.276 57.642Q-25.546 57.642-25.147 58.083Q-24.749 58.524-24.749 59.255Q-24.749 59.360-24.842 59.384L-27.292 59.384L-27.292 59.454Q-27.292 59.864-27.171 60.220Q-27.050 60.575-26.778 60.792Q-26.507 61.009-26.077 61.009Q-25.714 61.009-25.417 60.780Q-25.120 60.552-25.018 60.200Q-25.010 60.153-24.925 60.138L-24.842 60.138Q-24.749 60.165-24.749 60.247Q-24.749 60.255-24.757 60.286Q-24.819 60.513-24.958 60.696Q-25.096 60.880-25.288 61.013Q-25.479 61.145-25.698 61.216Q-25.917 61.286-26.155 61.286Q-26.526 61.286-26.864 61.149Q-27.202 61.013-27.469 60.761Q-27.737 60.509-27.887 60.169Q-28.038 59.829-28.038 59.454M-27.284 59.145L-25.323 59.145Q-25.323 58.841-25.425 58.550Q-25.526 58.259-25.743 58.077Q-25.960 57.895-26.276 57.895Q-26.577 57.895-26.807 58.083Q-27.038 58.270-27.161 58.562Q-27.284 58.853-27.284 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(252.494 -27.098)\">\u003Cpath d=\"M-19.621 61.177L-20.844 58.321Q-20.926 58.145-21.070 58.101Q-21.215 58.056-21.484 58.056L-21.484 57.759L-19.773 57.759L-19.773 58.056Q-20.195 58.056-20.195 58.239Q-20.195 58.274-20.180 58.321L-19.234 60.513L-18.394 58.536Q-18.355 58.458-18.355 58.368Q-18.355 58.228-18.461 58.142Q-18.566 58.056-18.707 58.056L-18.707 57.759L-17.355 57.759L-17.355 58.056Q-17.879 58.056-18.094 58.536L-19.219 61.177Q-19.281 61.286-19.387 61.286L-19.453 61.286Q-19.566 61.286-19.621 61.177\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(252.494 -27.098)\">\u003Cpath d=\"M-17.311 60.376Q-17.311 59.892-16.909 59.597Q-16.506 59.302-15.956 59.183Q-15.405 59.063-14.913 59.063L-14.913 58.774Q-14.913 58.548-15.028 58.341Q-15.143 58.134-15.340 58.015Q-15.538 57.895-15.768 57.895Q-16.194 57.895-16.479 58.001Q-16.409 58.028-16.362 58.083Q-16.315 58.138-16.290 58.208Q-16.264 58.278-16.264 58.353Q-16.264 58.458-16.315 58.550Q-16.366 58.642-16.458 58.692Q-16.549 58.743-16.655 58.743Q-16.760 58.743-16.852 58.692Q-16.944 58.642-16.995 58.550Q-17.045 58.458-17.045 58.353Q-17.045 57.935-16.657 57.788Q-16.268 57.642-15.768 57.642Q-15.436 57.642-15.083 57.772Q-14.729 57.903-14.501 58.157Q-14.272 58.411-14.272 58.759L-14.272 60.560Q-14.272 60.692-14.200 60.802Q-14.127 60.911-13.999 60.911Q-13.874 60.911-13.805 60.806Q-13.737 60.700-13.737 60.560L-13.737 60.048L-13.456 60.048L-13.456 60.560Q-13.456 60.763-13.573 60.921Q-13.690 61.079-13.872 61.163Q-14.053 61.247-14.256 61.247Q-14.487 61.247-14.639 61.075Q-14.792 60.903-14.823 60.673Q-14.983 60.954-15.292 61.120Q-15.600 61.286-15.952 61.286Q-16.463 61.286-16.887 61.063Q-17.311 60.841-17.311 60.376M-16.624 60.376Q-16.624 60.661-16.397 60.847Q-16.170 61.032-15.877 61.032Q-15.631 61.032-15.407 60.915Q-15.182 60.798-15.047 60.595Q-14.913 60.392-14.913 60.138L-14.913 59.306Q-15.178 59.306-15.463 59.360Q-15.749 59.415-16.020 59.544Q-16.292 59.673-16.458 59.880Q-16.624 60.087-16.624 60.376M-11.249 61.208L-13.081 61.208L-13.081 60.911Q-12.807 60.911-12.639 60.864Q-12.471 60.817-12.471 60.649L-12.471 56.489Q-12.471 56.274-12.534 56.179Q-12.596 56.083-12.715 56.062Q-12.834 56.040-13.081 56.040L-13.081 55.743L-11.858 55.657L-11.858 60.649Q-11.858 60.817-11.690 60.864Q-11.522 60.911-11.249 60.911L-11.249 61.208M-10.120 60.255L-10.120 58.513Q-10.120 58.298-10.182 58.202Q-10.245 58.106-10.364 58.085Q-10.483 58.063-10.729 58.063L-10.729 57.767L-9.483 57.681L-9.483 60.231L-9.483 60.255Q-9.483 60.567-9.428 60.729Q-9.374 60.892-9.223 60.962Q-9.073 61.032-8.752 61.032Q-8.323 61.032-8.049 60.694Q-7.776 60.356-7.776 59.911L-7.776 58.513Q-7.776 58.298-7.838 58.202Q-7.901 58.106-8.020 58.085Q-8.139 58.063-8.385 58.063L-8.385 57.767L-7.139 57.681L-7.139 60.466Q-7.139 60.677-7.077 60.772Q-7.014 60.868-6.895 60.890Q-6.776 60.911-6.530 60.911L-6.530 61.208L-7.752 61.286L-7.752 60.665Q-7.920 60.954-8.202 61.120Q-8.483 61.286-8.803 61.286Q-10.120 61.286-10.120 60.255M-6.084 59.454Q-6.084 58.974-5.852 58.558Q-5.620 58.142-5.209 57.892Q-4.799 57.642-4.323 57.642Q-3.592 57.642-3.194 58.083Q-2.795 58.524-2.795 59.255Q-2.795 59.360-2.889 59.384L-5.338 59.384L-5.338 59.454Q-5.338 59.864-5.217 60.220Q-5.096 60.575-4.825 60.792Q-4.553 61.009-4.124 61.009Q-3.760 61.009-3.463 60.780Q-3.167 60.552-3.065 60.200Q-3.057 60.153-2.971 60.138L-2.889 60.138Q-2.795 60.165-2.795 60.247Q-2.795 60.255-2.803 60.286Q-2.866 60.513-3.004 60.696Q-3.143 60.880-3.334 61.013Q-3.526 61.145-3.745 61.216Q-3.963 61.286-4.202 61.286Q-4.573 61.286-4.911 61.149Q-5.249 61.013-5.516 60.761Q-5.784 60.509-5.934 60.169Q-6.084 59.829-6.084 59.454M-5.331 59.145L-3.370 59.145Q-3.370 58.841-3.471 58.550Q-3.573 58.259-3.790 58.077Q-4.006 57.895-4.323 57.895Q-4.624 57.895-4.854 58.083Q-5.084 58.270-5.208 58.562Q-5.331 58.853-5.331 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M-28.231 32.756s20.557-3.243 28.453-5.69c7.895-2.448 21.346-8.595 28.452-11.95C35.78 11.76 45.12 5.17 51.437 2.88 57.753.59 67.882-1.506 74.199-1.387c6.316.118 16.446 3.147 22.762 5.12 6.317 1.975 16.446 11 22.762 9.106 6.317-1.895 16.446-16.051 22.762-22.762 6.317-6.712 16.446-19.686 22.763-25.608 6.316-5.921 16.84-14.11 22.762-17.071 5.921-2.961 19.917-4.268 19.917-4.268\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(171.404 -84.003)\">\u003Cpath d=\"M-37.307 61.208L-39.292 61.208L-39.292 60.911Q-39.018 60.911-38.850 60.864Q-38.682 60.817-38.682 60.649L-38.682 58.056L-39.323 58.056L-39.323 57.759L-38.682 57.759L-38.682 56.825Q-38.682 56.560-38.565 56.323Q-38.448 56.087-38.255 55.923Q-38.061 55.759-37.813 55.667Q-37.565 55.575-37.300 55.575Q-37.014 55.575-36.790 55.733Q-36.565 55.892-36.565 56.169Q-36.565 56.325-36.671 56.435Q-36.776 56.544-36.940 56.544Q-37.096 56.544-37.206 56.435Q-37.315 56.325-37.315 56.169Q-37.315 55.962-37.155 55.856Q-37.253 55.833-37.346 55.833Q-37.577 55.833-37.749 55.989Q-37.921 56.145-38.007 56.382Q-38.092 56.618-38.092 56.841L-38.092 57.759L-37.124 57.759L-37.124 58.056L-38.069 58.056L-38.069 60.649Q-38.069 60.817-37.842 60.864Q-37.616 60.911-37.307 60.911\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(171.404 -84.003)\">\u003Cpath d=\"M-34.285 61.208L-36.063 61.208L-36.063 60.911Q-35.789 60.911-35.621 60.864Q-35.453 60.817-35.453 60.649L-35.453 58.513Q-35.453 58.298-35.510 58.202Q-35.567 58.106-35.680 58.085Q-35.793 58.063-36.039 58.063L-36.039 57.767L-34.840 57.681L-34.840 60.649Q-34.840 60.817-34.694 60.864Q-34.547 60.911-34.285 60.911L-34.285 61.208M-35.727 56.286Q-35.727 56.095-35.592 55.964Q-35.457 55.833-35.262 55.833Q-35.141 55.833-35.037 55.895Q-34.934 55.958-34.871 56.062Q-34.809 56.165-34.809 56.286Q-34.809 56.481-34.940 56.616Q-35.071 56.751-35.262 56.751Q-35.461 56.751-35.594 56.618Q-35.727 56.485-35.727 56.286M-33.160 60.247L-33.160 58.056L-33.863 58.056L-33.863 57.802Q-33.508 57.802-33.266 57.569Q-33.024 57.337-32.912 56.989Q-32.801 56.642-32.801 56.286L-32.520 56.286L-32.520 57.759L-31.344 57.759L-31.344 58.056L-32.520 58.056L-32.520 60.231Q-32.520 60.552-32.401 60.780Q-32.281 61.009-32 61.009Q-31.821 61.009-31.703 60.886Q-31.586 60.763-31.533 60.583Q-31.481 60.403-31.481 60.231L-31.481 59.759L-31.199 59.759L-31.199 60.247Q-31.199 60.501-31.305 60.741Q-31.410 60.981-31.608 61.134Q-31.805 61.286-32.063 61.286Q-32.379 61.286-32.631 61.163Q-32.883 61.040-33.022 60.806Q-33.160 60.571-33.160 60.247M-29.856 60.247L-29.856 58.056L-30.559 58.056L-30.559 57.802Q-30.203 57.802-29.961 57.569Q-29.719 57.337-29.608 56.989Q-29.496 56.642-29.496 56.286L-29.215 56.286L-29.215 57.759L-28.039 57.759L-28.039 58.056L-29.215 58.056L-29.215 60.231Q-29.215 60.552-29.096 60.780Q-28.977 61.009-28.696 61.009Q-28.516 61.009-28.399 60.886Q-28.281 60.763-28.229 60.583Q-28.176 60.403-28.176 60.231L-28.176 59.759L-27.895 59.759L-27.895 60.247Q-27.895 60.501-28 60.741Q-28.106 60.981-28.303 61.134Q-28.500 61.286-28.758 61.286Q-29.074 61.286-29.326 61.163Q-29.578 61.040-29.717 60.806Q-29.856 60.571-29.856 60.247M-27.176 59.454Q-27.176 58.974-26.944 58.558Q-26.711 58.142-26.301 57.892Q-25.891 57.642-25.414 57.642Q-24.684 57.642-24.285 58.083Q-23.887 58.524-23.887 59.255Q-23.887 59.360-23.981 59.384L-26.430 59.384L-26.430 59.454Q-26.430 59.864-26.309 60.220Q-26.188 60.575-25.916 60.792Q-25.645 61.009-25.215 61.009Q-24.852 61.009-24.555 60.780Q-24.258 60.552-24.156 60.200Q-24.149 60.153-24.063 60.138L-23.981 60.138Q-23.887 60.165-23.887 60.247Q-23.887 60.255-23.895 60.286Q-23.957 60.513-24.096 60.696Q-24.235 60.880-24.426 61.013Q-24.617 61.145-24.836 61.216Q-25.055 61.286-25.293 61.286Q-25.664 61.286-26.002 61.149Q-26.340 61.013-26.608 60.761Q-26.875 60.509-27.026 60.169Q-27.176 59.829-27.176 59.454M-26.422 59.145L-24.461 59.145Q-24.461 58.841-24.563 58.550Q-24.664 58.259-24.881 58.077Q-25.098 57.895-25.414 57.895Q-25.715 57.895-25.946 58.083Q-26.176 58.270-26.299 58.562Q-26.422 58.853-26.422 59.145M-21.582 61.286Q-22.063 61.286-22.471 61.042Q-22.879 60.798-23.117 60.384Q-23.356 59.970-23.356 59.481Q-23.356 58.989-23.098 58.573Q-22.840 58.157-22.408 57.919Q-21.977 57.681-21.485 57.681Q-20.863 57.681-20.414 58.118L-20.414 56.489Q-20.414 56.274-20.477 56.179Q-20.539 56.083-20.656 56.062Q-20.774 56.040-21.020 56.040L-21.020 55.743L-19.797 55.657L-19.797 60.466Q-19.797 60.677-19.735 60.772Q-19.672 60.868-19.555 60.890Q-19.438 60.911-19.188 60.911L-19.188 61.208L-20.438 61.286L-20.438 60.802Q-20.903 61.286-21.582 61.286M-21.516 61.032Q-21.176 61.032-20.883 60.841Q-20.590 60.649-20.438 60.353L-20.438 58.520Q-20.586 58.247-20.848 58.091Q-21.110 57.935-21.422 57.935Q-22.047 57.935-22.330 58.382Q-22.613 58.829-22.613 59.489Q-22.613 60.134-22.362 60.583Q-22.110 61.032-21.516 61.032\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(171.404 -84.003)\">\u003Cpath d=\"M-14.038 61.177L-15.261 58.321Q-15.343 58.145-15.487 58.101Q-15.632 58.056-15.901 58.056L-15.901 57.759L-14.190 57.759L-14.190 58.056Q-14.612 58.056-14.612 58.239Q-14.612 58.274-14.597 58.321L-13.651 60.513L-12.811 58.536Q-12.772 58.458-12.772 58.368Q-12.772 58.228-12.878 58.142Q-12.983 58.056-13.124 58.056L-13.124 57.759L-11.772 57.759L-11.772 58.056Q-12.296 58.056-12.511 58.536L-13.636 61.177Q-13.698 61.286-13.804 61.286L-13.870 61.286Q-13.983 61.286-14.038 61.177\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(171.404 -84.003)\">\u003Cpath d=\"M-11.727 60.376Q-11.727 59.892-11.325 59.597Q-10.922 59.302-10.372 59.183Q-9.821 59.063-9.329 59.063L-9.329 58.774Q-9.329 58.548-9.444 58.341Q-9.559 58.134-9.756 58.015Q-9.954 57.895-10.184 57.895Q-10.610 57.895-10.895 58.001Q-10.825 58.028-10.778 58.083Q-10.731 58.138-10.706 58.208Q-10.680 58.278-10.680 58.353Q-10.680 58.458-10.731 58.550Q-10.782 58.642-10.874 58.692Q-10.965 58.743-11.071 58.743Q-11.176 58.743-11.268 58.692Q-11.360 58.642-11.411 58.550Q-11.461 58.458-11.461 58.353Q-11.461 57.935-11.073 57.788Q-10.684 57.642-10.184 57.642Q-9.852 57.642-9.499 57.772Q-9.145 57.903-8.917 58.157Q-8.688 58.411-8.688 58.759L-8.688 60.560Q-8.688 60.692-8.616 60.802Q-8.543 60.911-8.415 60.911Q-8.290 60.911-8.221 60.806Q-8.153 60.700-8.153 60.560L-8.153 60.048L-7.872 60.048L-7.872 60.560Q-7.872 60.763-7.989 60.921Q-8.106 61.079-8.288 61.163Q-8.469 61.247-8.672 61.247Q-8.903 61.247-9.055 61.075Q-9.208 60.903-9.239 60.673Q-9.399 60.954-9.708 61.120Q-10.016 61.286-10.368 61.286Q-10.879 61.286-11.303 61.063Q-11.727 60.841-11.727 60.376M-11.040 60.376Q-11.040 60.661-10.813 60.847Q-10.586 61.032-10.293 61.032Q-10.047 61.032-9.823 60.915Q-9.598 60.798-9.463 60.595Q-9.329 60.392-9.329 60.138L-9.329 59.306Q-9.594 59.306-9.879 59.360Q-10.165 59.415-10.436 59.544Q-10.708 59.673-10.874 59.880Q-11.040 60.087-11.040 60.376M-5.665 61.208L-7.497 61.208L-7.497 60.911Q-7.223 60.911-7.055 60.864Q-6.887 60.817-6.887 60.649L-6.887 56.489Q-6.887 56.274-6.950 56.179Q-7.012 56.083-7.131 56.062Q-7.251 56.040-7.497 56.040L-7.497 55.743L-6.274 55.657L-6.274 60.649Q-6.274 60.817-6.106 60.864Q-5.938 60.911-5.665 60.911L-5.665 61.208M-4.536 60.255L-4.536 58.513Q-4.536 58.298-4.598 58.202Q-4.661 58.106-4.780 58.085Q-4.899 58.063-5.145 58.063L-5.145 57.767L-3.899 57.681L-3.899 60.231L-3.899 60.255Q-3.899 60.567-3.844 60.729Q-3.790 60.892-3.639 60.962Q-3.489 61.032-3.168 61.032Q-2.739 61.032-2.465 60.694Q-2.192 60.356-2.192 59.911L-2.192 58.513Q-2.192 58.298-2.254 58.202Q-2.317 58.106-2.436 58.085Q-2.555 58.063-2.801 58.063L-2.801 57.767L-1.555 57.681L-1.555 60.466Q-1.555 60.677-1.493 60.772Q-1.430 60.868-1.311 60.890Q-1.192 60.911-0.946 60.911L-0.946 61.208L-2.168 61.286L-2.168 60.665Q-2.336 60.954-2.618 61.120Q-2.899 61.286-3.219 61.286Q-4.536 61.286-4.536 60.255M-0.501 59.454Q-0.501 58.974-0.268 58.558Q-0.036 58.142 0.374 57.892Q0.785 57.642 1.261 57.642Q1.992 57.642 2.390 58.083Q2.789 58.524 2.789 59.255Q2.789 59.360 2.695 59.384L0.246 59.384L0.246 59.454Q0.246 59.864 0.367 60.220Q0.488 60.575 0.759 60.792Q1.031 61.009 1.460 61.009Q1.824 61.009 2.121 60.780Q2.417 60.552 2.519 60.200Q2.527 60.153 2.613 60.138L2.695 60.138Q2.789 60.165 2.789 60.247Q2.789 60.255 2.781 60.286Q2.718 60.513 2.580 60.696Q2.441 60.880 2.249 61.013Q2.058 61.145 1.839 61.216Q1.621 61.286 1.382 61.286Q1.011 61.286 0.673 61.149Q0.335 61.013 0.068 60.761Q-0.200 60.509-0.350 60.169Q-0.501 59.829-0.501 59.454M0.253 59.145L2.214 59.145Q2.214 58.841 2.113 58.550Q2.011 58.259 1.794 58.077Q1.578 57.895 1.261 57.895Q0.960 57.895 0.730 58.083Q0.499 58.270 0.376 58.562Q0.253 58.853 0.253 59.145\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M205.859-56.017a2.2 2.2 0 1 0-4.4 0 2.2 2.2 0 0 0 4.4 0m-2.2 0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M173.783-49.757c10.239-4.819 17.177-6.068 26.46-5.289\"\u002F>\u003Cpath stroke=\"none\" d=\"m202.236-54.879-3.055-1.862 1.062 1.695-1.33 1.494\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M-39.276 60.376Q-39.276 59.892-38.874 59.597Q-38.471 59.302-37.921 59.183Q-37.370 59.063-36.878 59.063L-36.878 58.774Q-36.878 58.548-36.993 58.341Q-37.108 58.134-37.305 58.015Q-37.503 57.895-37.733 57.895Q-38.159 57.895-38.444 58.001Q-38.374 58.028-38.327 58.083Q-38.280 58.138-38.255 58.208Q-38.229 58.278-38.229 58.353Q-38.229 58.458-38.280 58.550Q-38.331 58.642-38.423 58.692Q-38.514 58.743-38.620 58.743Q-38.725 58.743-38.817 58.692Q-38.909 58.642-38.960 58.550Q-39.010 58.458-39.010 58.353Q-39.010 57.935-38.622 57.788Q-38.233 57.642-37.733 57.642Q-37.401 57.642-37.048 57.772Q-36.694 57.903-36.466 58.157Q-36.237 58.411-36.237 58.759L-36.237 60.560Q-36.237 60.692-36.165 60.802Q-36.092 60.911-35.964 60.911Q-35.839 60.911-35.770 60.806Q-35.702 60.700-35.702 60.560L-35.702 60.048L-35.421 60.048L-35.421 60.560Q-35.421 60.763-35.538 60.921Q-35.655 61.079-35.837 61.163Q-36.018 61.247-36.221 61.247Q-36.452 61.247-36.604 61.075Q-36.757 60.903-36.788 60.673Q-36.948 60.954-37.257 61.120Q-37.565 61.286-37.917 61.286Q-38.428 61.286-38.852 61.063Q-39.276 60.841-39.276 60.376M-38.589 60.376Q-38.589 60.661-38.362 60.847Q-38.135 61.032-37.842 61.032Q-37.596 61.032-37.372 60.915Q-37.147 60.798-37.012 60.595Q-36.878 60.392-36.878 60.138L-36.878 59.306Q-37.143 59.306-37.428 59.360Q-37.714 59.415-37.985 59.544Q-38.257 59.673-38.423 59.880Q-38.589 60.087-38.589 60.376M-33.120 61.208L-35.100 61.208L-35.100 60.911Q-34.831 60.911-34.663 60.866Q-34.495 60.821-34.495 60.649L-34.495 58.513Q-34.495 58.298-34.557 58.202Q-34.620 58.106-34.737 58.085Q-34.854 58.063-35.100 58.063L-35.100 57.767L-33.932 57.681L-33.932 58.466Q-33.854 58.255-33.702 58.069Q-33.550 57.884-33.350 57.782Q-33.151 57.681-32.925 57.681Q-32.678 57.681-32.487 57.825Q-32.296 57.970-32.296 58.200Q-32.296 58.356-32.401 58.466Q-32.507 58.575-32.663 58.575Q-32.819 58.575-32.928 58.466Q-33.038 58.356-33.038 58.200Q-33.038 58.040-32.932 57.935Q-33.257 57.935-33.471 58.163Q-33.686 58.392-33.782 58.731Q-33.878 59.071-33.878 59.376L-33.878 60.649Q-33.878 60.817-33.651 60.864Q-33.425 60.911-33.120 60.911L-33.120 61.208M-31.815 61.817Q-31.815 61.536-31.604 61.325Q-31.393 61.114-31.108 61.024Q-31.264 60.899-31.342 60.710Q-31.421 60.520-31.421 60.321Q-31.421 59.966-31.190 59.673Q-31.557 59.333-31.557 58.864Q-31.557 58.513-31.354 58.243Q-31.151 57.974-30.831 57.827Q-30.510 57.681-30.167 57.681Q-29.647 57.681-29.276 57.962Q-28.913 57.591-28.366 57.591Q-28.186 57.591-28.059 57.718Q-27.932 57.845-27.932 58.024Q-27.932 58.130-28.010 58.208Q-28.089 58.286-28.198 58.286Q-28.307 58.286-28.383 58.210Q-28.460 58.134-28.460 58.024Q-28.460 57.923-28.421 57.872Q-28.413 57.864-28.409 57.858Q-28.405 57.853-28.405 57.849Q-28.780 57.849-29.100 58.103Q-28.780 58.442-28.780 58.864Q-28.780 59.134-28.897 59.351Q-29.014 59.567-29.219 59.726Q-29.425 59.884-29.667 59.966Q-29.909 60.048-30.167 60.048Q-30.385 60.048-30.598 59.989Q-30.811 59.931-31.007 59.810Q-31.100 59.950-31.100 60.130Q-31.100 60.337-30.964 60.489Q-30.827 60.642-30.620 60.642L-29.925 60.642Q-29.436 60.642-29.024 60.726Q-28.612 60.810-28.333 61.067Q-28.053 61.325-28.053 61.817Q-28.053 62.181-28.374 62.413Q-28.694 62.645-29.135 62.747Q-29.577 62.849-29.932 62.849Q-30.288 62.849-30.731 62.747Q-31.175 62.645-31.495 62.413Q-31.815 62.181-31.815 61.817M-31.311 61.817Q-31.311 62.013-31.167 62.161Q-31.022 62.310-30.809 62.399Q-30.596 62.489-30.356 62.536Q-30.116 62.583-29.932 62.583Q-29.690 62.583-29.360 62.505Q-29.030 62.427-28.794 62.253Q-28.557 62.079-28.557 61.817Q-28.557 61.411-28.967 61.302Q-29.378 61.192-29.940 61.192L-30.620 61.192Q-30.889 61.192-31.100 61.370Q-31.311 61.548-31.311 61.817M-30.167 59.782Q-29.444 59.782-29.444 58.864Q-29.444 57.942-30.167 57.942Q-30.893 57.942-30.893 58.864Q-30.893 59.782-30.167 59.782M-25.639 61.208L-27.495 61.208L-27.495 60.911Q-27.221 60.911-27.053 60.864Q-26.885 60.817-26.885 60.649L-26.885 58.513Q-26.885 58.298-26.948 58.202Q-27.010 58.106-27.130 58.085Q-27.249 58.063-27.495 58.063L-27.495 57.767L-26.303 57.681L-26.303 58.415Q-26.190 58.200-25.997 58.032Q-25.803 57.864-25.565 57.772Q-25.327 57.681-25.073 57.681Q-24.112 57.681-23.936 58.392Q-23.753 58.063-23.425 57.872Q-23.096 57.681-22.717 57.681Q-21.542 57.681-21.542 58.759L-21.542 60.649Q-21.542 60.817-21.374 60.864Q-21.206 60.911-20.936 60.911L-20.936 61.208L-22.792 61.208L-22.792 60.911Q-22.518 60.911-22.350 60.866Q-22.182 60.821-22.182 60.649L-22.182 58.774Q-22.182 58.388-22.307 58.161Q-22.432 57.935-22.784 57.935Q-23.089 57.935-23.344 58.097Q-23.600 58.259-23.749 58.528Q-23.897 58.798-23.897 59.095L-23.897 60.649Q-23.897 60.817-23.727 60.864Q-23.557 60.911-23.288 60.911L-23.288 61.208L-25.143 61.208L-25.143 60.911Q-24.870 60.911-24.702 60.864Q-24.534 60.817-24.534 60.649L-24.534 58.774Q-24.534 58.388-24.659 58.161Q-24.784 57.935-25.135 57.935Q-25.440 57.935-25.696 58.097Q-25.952 58.259-26.100 58.528Q-26.249 58.798-26.249 59.095L-26.249 60.649Q-26.249 60.817-26.079 60.864Q-25.909 60.911-25.639 60.911L-25.639 61.208M-20.393 60.376Q-20.393 59.892-19.991 59.597Q-19.589 59.302-19.038 59.183Q-18.487 59.063-17.995 59.063L-17.995 58.774Q-17.995 58.548-18.110 58.341Q-18.225 58.134-18.423 58.015Q-18.620 57.895-18.850 57.895Q-19.276 57.895-19.561 58.001Q-19.491 58.028-19.444 58.083Q-19.397 58.138-19.372 58.208Q-19.346 58.278-19.346 58.353Q-19.346 58.458-19.397 58.550Q-19.448 58.642-19.540 58.692Q-19.632 58.743-19.737 58.743Q-19.842 58.743-19.934 58.692Q-20.026 58.642-20.077 58.550Q-20.128 58.458-20.128 58.353Q-20.128 57.935-19.739 57.788Q-19.350 57.642-18.850 57.642Q-18.518 57.642-18.165 57.772Q-17.811 57.903-17.583 58.157Q-17.354 58.411-17.354 58.759L-17.354 60.560Q-17.354 60.692-17.282 60.802Q-17.210 60.911-17.081 60.911Q-16.956 60.911-16.887 60.806Q-16.819 60.700-16.819 60.560L-16.819 60.048L-16.538 60.048L-16.538 60.560Q-16.538 60.763-16.655 60.921Q-16.772 61.079-16.954 61.163Q-17.135 61.247-17.339 61.247Q-17.569 61.247-17.721 61.075Q-17.874 60.903-17.905 60.673Q-18.065 60.954-18.374 61.120Q-18.682 61.286-19.034 61.286Q-19.546 61.286-19.969 61.063Q-20.393 60.841-20.393 60.376M-19.706 60.376Q-19.706 60.661-19.479 60.847Q-19.253 61.032-18.960 61.032Q-18.714 61.032-18.489 60.915Q-18.264 60.798-18.130 60.595Q-17.995 60.392-17.995 60.138L-17.995 59.306Q-18.260 59.306-18.546 59.360Q-18.831 59.415-19.102 59.544Q-19.374 59.673-19.540 59.880Q-19.706 60.087-19.706 60.376M-14.858 61.208L-16.354 61.208L-16.354 60.911Q-15.721 60.911-15.300 60.431L-14.530 59.520L-15.522 58.321Q-15.678 58.142-15.841 58.099Q-16.003 58.056-16.307 58.056L-16.307 57.759L-14.620 57.759L-14.620 58.056Q-14.714 58.056-14.790 58.099Q-14.866 58.142-14.866 58.231Q-14.866 58.274-14.835 58.321L-14.178 59.110L-13.698 58.536Q-13.581 58.399-13.581 58.263Q-13.581 58.173-13.632 58.114Q-13.682 58.056-13.764 58.056L-13.764 57.759L-12.276 57.759L-12.276 58.056Q-12.913 58.056-13.323 58.536L-14.003 59.337L-12.917 60.649Q-12.757 60.825-12.596 60.868Q-12.436 60.911-12.132 60.911L-12.132 61.208L-13.819 61.208L-13.819 60.911Q-13.729 60.911-13.651 60.868Q-13.573 60.825-13.573 60.735Q-13.573 60.712-13.604 60.649L-14.346 59.743L-14.932 60.431Q-15.050 60.567-15.050 60.704Q-15.050 60.790-14.999 60.851Q-14.948 60.911-14.858 60.911\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M-7.025 62.759L-8.880 62.759L-8.880 62.466Q-8.611 62.466-8.443 62.421Q-8.275 62.376-8.275 62.200L-8.275 58.376Q-8.275 58.169-8.431 58.116Q-8.587 58.063-8.880 58.063L-8.880 57.767L-7.658 57.681L-7.658 58.145Q-7.427 57.923-7.113 57.802Q-6.798 57.681-6.459 57.681Q-5.986 57.681-5.582 57.927Q-5.177 58.173-4.945 58.589Q-4.712 59.005-4.712 59.481Q-4.712 59.856-4.861 60.185Q-5.009 60.513-5.279 60.765Q-5.548 61.017-5.892 61.151Q-6.236 61.286-6.595 61.286Q-6.884 61.286-7.156 61.165Q-7.427 61.044-7.634 60.833L-7.634 62.200Q-7.634 62.376-7.466 62.421Q-7.298 62.466-7.025 62.466L-7.025 62.759M-7.634 58.544L-7.634 60.384Q-7.482 60.673-7.220 60.853Q-6.959 61.032-6.650 61.032Q-6.365 61.032-6.142 60.894Q-5.919 60.755-5.767 60.524Q-5.615 60.294-5.537 60.022Q-5.459 59.751-5.459 59.481Q-5.459 59.149-5.584 58.792Q-5.709 58.435-5.957 58.198Q-6.205 57.962-6.552 57.962Q-6.876 57.962-7.171 58.118Q-7.466 58.274-7.634 58.544M-2.330 61.208L-4.107 61.208L-4.107 60.911Q-3.834 60.911-3.666 60.864Q-3.498 60.817-3.498 60.649L-3.498 58.513Q-3.498 58.298-3.554 58.202Q-3.611 58.106-3.724 58.085Q-3.837 58.063-4.084 58.063L-4.084 57.767L-2.884 57.681L-2.884 60.649Q-2.884 60.817-2.738 60.864Q-2.591 60.911-2.330 60.911L-2.330 61.208M-3.771 56.286Q-3.771 56.095-3.636 55.964Q-3.501 55.833-3.306 55.833Q-3.185 55.833-3.082 55.895Q-2.978 55.958-2.916 56.062Q-2.853 56.165-2.853 56.286Q-2.853 56.481-2.984 56.616Q-3.115 56.751-3.306 56.751Q-3.505 56.751-3.638 56.618Q-3.771 56.485-3.771 56.286M-1.787 59.481Q-1.787 58.985-1.537 58.560Q-1.287 58.134-0.867 57.888Q-0.447 57.642 0.053 57.642Q0.592 57.642 0.983 57.767Q1.374 57.892 1.374 58.306Q1.374 58.411 1.323 58.503Q1.272 58.595 1.180 58.645Q1.088 58.696 0.979 58.696Q0.874 58.696 0.782 58.645Q0.690 58.595 0.639 58.503Q0.588 58.411 0.588 58.306Q0.588 58.083 0.756 57.978Q0.534 57.919 0.061 57.919Q-0.236 57.919-0.451 58.058Q-0.666 58.196-0.796 58.427Q-0.927 58.657-0.986 58.927Q-1.044 59.196-1.044 59.481Q-1.044 59.876-0.912 60.226Q-0.779 60.575-0.507 60.792Q-0.236 61.009 0.163 61.009Q0.538 61.009 0.813 60.792Q1.088 60.575 1.190 60.216Q1.206 60.153 1.268 60.153L1.374 60.153Q1.409 60.153 1.434 60.181Q1.459 60.208 1.459 60.247L1.459 60.270Q1.327 60.751 0.942 61.019Q0.557 61.286 0.053 61.286Q-0.310 61.286-0.644 61.149Q-0.978 61.013-1.238 60.763Q-1.498 60.513-1.642 60.177Q-1.787 59.841-1.787 59.481\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M3.542 61.208L1.745 61.208L1.745 60.911Q2.014 60.911 2.182 60.866Q2.350 60.821 2.350 60.649L2.350 56.489Q2.350 56.274 2.288 56.179Q2.225 56.083 2.108 56.062Q1.991 56.040 1.745 56.040L1.745 55.743L2.967 55.657L2.967 59.423L4.065 58.536Q4.272 58.356 4.272 58.208Q4.272 58.142 4.219 58.099Q4.167 58.056 4.096 58.056L4.096 57.759L5.631 57.759L5.631 58.056Q5.100 58.056 4.502 58.536L3.893 59.032L4.967 60.431Q5.104 60.606 5.211 60.714Q5.319 60.821 5.454 60.866Q5.588 60.911 5.815 60.911L5.815 61.208L4.190 61.208L4.190 60.911Q4.432 60.911 4.432 60.759Q4.432 60.681 4.389 60.610Q4.346 60.540 4.264 60.431L3.463 59.384L2.936 59.810L2.936 60.649Q2.936 60.817 3.104 60.864Q3.272 60.911 3.542 60.911L3.542 61.208M6.241 61.200L6.241 59.978Q6.241 59.950 6.272 59.919Q6.303 59.888 6.327 59.888L6.432 59.888Q6.502 59.888 6.518 59.950Q6.581 60.270 6.719 60.511Q6.858 60.751 7.090 60.892Q7.323 61.032 7.631 61.032Q7.870 61.032 8.079 60.972Q8.288 60.911 8.424 60.763Q8.561 60.614 8.561 60.368Q8.561 60.114 8.350 59.948Q8.139 59.782 7.870 59.728L7.249 59.614Q6.842 59.536 6.542 59.280Q6.241 59.024 6.241 58.649Q6.241 58.282 6.442 58.060Q6.643 57.837 6.967 57.739Q7.292 57.642 7.631 57.642Q8.096 57.642 8.393 57.849L8.616 57.665Q8.639 57.642 8.670 57.642L8.721 57.642Q8.752 57.642 8.780 57.669Q8.807 57.696 8.807 57.728L8.807 58.712Q8.807 58.743 8.782 58.772Q8.756 58.802 8.721 58.802L8.616 58.802Q8.581 58.802 8.553 58.774Q8.526 58.747 8.526 58.712Q8.526 58.313 8.274 58.093Q8.022 57.872 7.624 57.872Q7.268 57.872 6.985 57.995Q6.702 58.118 6.702 58.423Q6.702 58.642 6.903 58.774Q7.104 58.907 7.350 58.950L7.975 59.063Q8.405 59.153 8.713 59.450Q9.022 59.747 9.022 60.161Q9.022 60.731 8.624 61.009Q8.225 61.286 7.631 61.286Q7.081 61.286 6.729 60.950L6.432 61.263Q6.409 61.286 6.374 61.286L6.327 61.286Q6.303 61.286 6.272 61.255Q6.241 61.224 6.241 61.200\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M15.456 61.376Q14.874 61.376 14.356 61.149Q13.838 60.923 13.450 60.524Q13.061 60.126 12.842 59.601Q12.624 59.075 12.624 58.505Q12.624 57.735 12.999 57.058Q13.374 56.380 14.024 55.978Q14.674 55.575 15.456 55.575Q16.229 55.575 16.880 55.978Q17.530 56.380 17.905 57.058Q18.280 57.735 18.280 58.505Q18.280 59.075 18.059 59.606Q17.838 60.138 17.454 60.530Q17.069 60.923 16.551 61.149Q16.034 61.376 15.456 61.376M14.014 60.306Q14.178 60.536 14.409 60.712Q14.639 60.888 14.907 60.983Q15.174 61.079 15.456 61.079Q15.874 61.079 16.255 60.868Q16.635 60.657 16.885 60.306Q17.417 59.587 17.417 58.368Q17.417 57.739 17.202 57.161Q16.987 56.583 16.546 56.220Q16.104 55.856 15.456 55.856Q14.803 55.856 14.360 56.220Q13.917 56.583 13.702 57.161Q13.487 57.739 13.487 58.368Q13.487 59.587 14.014 60.306M22.065 61.376Q21.483 61.376 20.965 61.149Q20.448 60.923 20.059 60.524Q19.671 60.126 19.452 59.601Q19.233 59.075 19.233 58.505Q19.233 57.735 19.608 57.058Q19.983 56.380 20.633 55.978Q21.284 55.575 22.065 55.575Q22.838 55.575 23.489 55.978Q24.139 56.380 24.514 57.058Q24.889 57.735 24.889 58.505Q24.889 59.075 24.669 59.606Q24.448 60.138 24.063 60.530Q23.678 60.923 23.161 61.149Q22.643 61.376 22.065 61.376M20.624 60.306Q20.788 60.536 21.018 60.712Q21.249 60.888 21.516 60.983Q21.784 61.079 22.065 61.079Q22.483 61.079 22.864 60.868Q23.245 60.657 23.495 60.306Q24.026 59.587 24.026 58.368Q24.026 57.739 23.811 57.161Q23.596 56.583 23.155 56.220Q22.713 55.856 22.065 55.856Q21.413 55.856 20.969 56.220Q20.526 56.583 20.311 57.161Q20.096 57.739 20.096 58.368Q20.096 59.587 20.624 60.306M28.784 61.208L25.721 61.208L25.721 60.911Q26.046 60.911 26.288 60.864Q26.530 60.817 26.530 60.649L26.530 56.306Q26.530 56.134 26.288 56.087Q26.046 56.040 25.721 56.040L25.721 55.743L28.784 55.743Q29.335 55.743 29.813 55.970Q30.292 56.196 30.645 56.591Q30.999 56.985 31.188 57.485Q31.378 57.985 31.378 58.528Q31.378 59.235 31.034 59.853Q30.690 60.470 30.094 60.839Q29.499 61.208 28.784 61.208M27.272 56.306L27.272 60.649Q27.272 60.821 27.364 60.866Q27.456 60.911 27.674 60.911L28.569 60.911Q29.014 60.911 29.407 60.741Q29.799 60.571 30.071 60.247Q30.342 59.923 30.440 59.503Q30.538 59.083 30.538 58.528Q30.538 58.173 30.501 57.860Q30.463 57.548 30.358 57.253Q30.253 56.958 30.073 56.735Q29.889 56.501 29.651 56.351Q29.413 56.200 29.131 56.120Q28.850 56.040 28.569 56.040L27.674 56.040Q27.456 56.040 27.364 56.083Q27.272 56.126 27.272 56.306\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M34.934 59.513Q34.934 59.009 35.190 58.577Q35.446 58.145 35.882 57.894Q36.317 57.642 36.817 57.642Q37.204 57.642 37.546 57.786Q37.887 57.931 38.149 58.192Q38.411 58.454 38.553 58.790Q38.696 59.126 38.696 59.513Q38.696 60.005 38.432 60.415Q38.169 60.825 37.739 61.056Q37.309 61.286 36.817 61.286Q36.325 61.286 35.891 61.054Q35.458 60.821 35.196 60.413Q34.934 60.005 34.934 59.513M36.817 61.009Q37.274 61.009 37.526 60.786Q37.778 60.563 37.866 60.212Q37.954 59.860 37.954 59.415Q37.954 58.985 37.860 58.647Q37.766 58.310 37.512 58.103Q37.258 57.895 36.817 57.895Q36.169 57.895 35.925 58.312Q35.680 58.728 35.680 59.415Q35.680 59.860 35.768 60.212Q35.856 60.563 36.108 60.786Q36.360 61.009 36.817 61.009\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M40.749 61.177L39.526 58.321Q39.444 58.145 39.300 58.101Q39.155 58.056 38.886 58.056L38.886 57.759L40.597 57.759L40.597 58.056Q40.175 58.056 40.175 58.239Q40.175 58.274 40.190 58.321L41.136 60.513L41.976 58.536Q42.015 58.458 42.015 58.368Q42.015 58.228 41.909 58.142Q41.804 58.056 41.663 58.056L41.663 57.759L43.015 57.759L43.015 58.056Q42.491 58.056 42.276 58.536L41.151 61.177Q41.089 61.286 40.983 61.286L40.917 61.286Q40.804 61.286 40.749 61.177\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M43.198 59.454Q43.198 58.974 43.431 58.558Q43.663 58.142 44.073 57.892Q44.483 57.642 44.960 57.642Q45.690 57.642 46.089 58.083Q46.487 58.524 46.487 59.255Q46.487 59.360 46.394 59.384L43.944 59.384L43.944 59.454Q43.944 59.864 44.065 60.220Q44.187 60.575 44.458 60.792Q44.730 61.009 45.159 61.009Q45.523 61.009 45.819 60.780Q46.116 60.552 46.218 60.200Q46.226 60.153 46.312 60.138L46.394 60.138Q46.487 60.165 46.487 60.247Q46.487 60.255 46.480 60.286Q46.417 60.513 46.278 60.696Q46.140 60.880 45.948 61.013Q45.757 61.145 45.538 61.216Q45.319 61.286 45.081 61.286Q44.710 61.286 44.372 61.149Q44.034 61.013 43.767 60.761Q43.499 60.509 43.349 60.169Q43.198 59.829 43.198 59.454M43.952 59.145L45.913 59.145Q45.913 58.841 45.812 58.550Q45.710 58.259 45.493 58.077Q45.276 57.895 44.960 57.895Q44.659 57.895 44.429 58.083Q44.198 58.270 44.075 58.562Q43.952 58.853 43.952 59.145M48.983 61.208L47.003 61.208L47.003 60.911Q47.273 60.911 47.440 60.866Q47.608 60.821 47.608 60.649L47.608 58.513Q47.608 58.298 47.546 58.202Q47.483 58.106 47.366 58.085Q47.249 58.063 47.003 58.063L47.003 57.767L48.171 57.681L48.171 58.466Q48.249 58.255 48.401 58.069Q48.554 57.884 48.753 57.782Q48.952 57.681 49.179 57.681Q49.425 57.681 49.616 57.825Q49.808 57.970 49.808 58.200Q49.808 58.356 49.702 58.466Q49.597 58.575 49.440 58.575Q49.284 58.575 49.175 58.466Q49.065 58.356 49.065 58.200Q49.065 58.040 49.171 57.935Q48.847 57.935 48.632 58.163Q48.417 58.392 48.321 58.731Q48.226 59.071 48.226 59.376L48.226 60.649Q48.226 60.817 48.452 60.864Q48.679 60.911 48.983 60.911L48.983 61.208M50.331 61.200L50.331 59.978Q50.331 59.950 50.362 59.919Q50.394 59.888 50.417 59.888L50.523 59.888Q50.593 59.888 50.608 59.950Q50.671 60.270 50.810 60.511Q50.948 60.751 51.181 60.892Q51.413 61.032 51.722 61.032Q51.960 61.032 52.169 60.972Q52.378 60.911 52.515 60.763Q52.651 60.614 52.651 60.368Q52.651 60.114 52.440 59.948Q52.230 59.782 51.960 59.728L51.339 59.614Q50.933 59.536 50.632 59.280Q50.331 59.024 50.331 58.649Q50.331 58.282 50.532 58.060Q50.733 57.837 51.058 57.739Q51.382 57.642 51.722 57.642Q52.187 57.642 52.483 57.849L52.706 57.665Q52.730 57.642 52.761 57.642L52.812 57.642Q52.843 57.642 52.870 57.669Q52.898 57.696 52.898 57.728L52.898 58.712Q52.898 58.743 52.872 58.772Q52.847 58.802 52.812 58.802L52.706 58.802Q52.671 58.802 52.644 58.774Q52.616 58.747 52.616 58.712Q52.616 58.313 52.364 58.093Q52.112 57.872 51.714 57.872Q51.358 57.872 51.075 57.995Q50.792 58.118 50.792 58.423Q50.792 58.642 50.993 58.774Q51.194 58.907 51.440 58.950L52.065 59.063Q52.495 59.153 52.804 59.450Q53.112 59.747 53.112 60.161Q53.112 60.731 52.714 61.009Q52.315 61.286 51.722 61.286Q51.171 61.286 50.819 60.950L50.523 61.263Q50.499 61.286 50.464 61.286L50.417 61.286Q50.394 61.286 50.362 61.255Q50.331 61.224 50.331 61.200M55.569 61.208L53.714 61.208L53.714 60.911Q53.987 60.911 54.155 60.864Q54.323 60.817 54.323 60.649L54.323 56.489Q54.323 56.274 54.261 56.179Q54.198 56.083 54.079 56.062Q53.960 56.040 53.714 56.040L53.714 55.743L54.937 55.657L54.937 58.360Q55.062 58.149 55.249 57.999Q55.437 57.849 55.663 57.765Q55.890 57.681 56.136 57.681Q57.304 57.681 57.304 58.759L57.304 60.649Q57.304 60.817 57.474 60.864Q57.644 60.911 57.913 60.911L57.913 61.208L56.058 61.208L56.058 60.911Q56.331 60.911 56.499 60.864Q56.667 60.817 56.667 60.649L56.667 58.774Q56.667 58.392 56.546 58.163Q56.425 57.935 56.073 57.935Q55.761 57.935 55.507 58.097Q55.253 58.259 55.106 58.528Q54.960 58.798 54.960 59.095L54.960 60.649Q54.960 60.817 55.130 60.864Q55.300 60.911 55.569 60.911L55.569 61.208M58.358 59.513Q58.358 59.009 58.614 58.577Q58.870 58.145 59.306 57.894Q59.741 57.642 60.241 57.642Q60.628 57.642 60.970 57.786Q61.312 57.931 61.573 58.192Q61.835 58.454 61.978 58.790Q62.120 59.126 62.120 59.513Q62.120 60.005 61.856 60.415Q61.593 60.825 61.163 61.056Q60.733 61.286 60.241 61.286Q59.749 61.286 59.315 61.054Q58.882 60.821 58.620 60.413Q58.358 60.005 58.358 59.513M60.241 61.009Q60.698 61.009 60.950 60.786Q61.202 60.563 61.290 60.212Q61.378 59.860 61.378 59.415Q61.378 58.985 61.284 58.647Q61.190 58.310 60.937 58.103Q60.683 57.895 60.241 57.895Q59.593 57.895 59.349 58.312Q59.105 58.728 59.105 59.415Q59.105 59.860 59.192 60.212Q59.280 60.563 59.532 60.786Q59.784 61.009 60.241 61.009\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.083 -107.543)\">\u003Cpath d=\"M62.850 59.513Q62.850 59.009 63.106 58.577Q63.362 58.145 63.798 57.894Q64.233 57.642 64.733 57.642Q65.120 57.642 65.462 57.786Q65.803 57.931 66.065 58.192Q66.327 58.454 66.469 58.790Q66.612 59.126 66.612 59.513Q66.612 60.005 66.348 60.415Q66.085 60.825 65.655 61.056Q65.225 61.286 64.733 61.286Q64.241 61.286 63.807 61.054Q63.374 60.821 63.112 60.413Q62.850 60.005 62.850 59.513M64.733 61.009Q65.190 61.009 65.442 60.786Q65.694 60.563 65.782 60.212Q65.870 59.860 65.870 59.415Q65.870 58.985 65.776 58.647Q65.682 58.310 65.428 58.103Q65.174 57.895 64.733 57.895Q64.085 57.895 63.841 58.312Q63.596 58.728 63.596 59.415Q63.596 59.860 63.684 60.212Q63.772 60.563 64.024 60.786Q64.276 61.009 64.733 61.009M67.721 60.247L67.721 58.056L67.018 58.056L67.018 57.802Q67.374 57.802 67.616 57.569Q67.858 57.337 67.969 56.989Q68.081 56.642 68.081 56.286L68.362 56.286L68.362 57.759L69.538 57.759L69.538 58.056L68.362 58.056L68.362 60.231Q68.362 60.552 68.481 60.780Q68.600 61.009 68.882 61.009Q69.061 61.009 69.178 60.886Q69.296 60.763 69.348 60.583Q69.401 60.403 69.401 60.231L69.401 59.759L69.682 59.759L69.682 60.247Q69.682 60.501 69.577 60.741Q69.471 60.981 69.274 61.134Q69.077 61.286 68.819 61.286Q68.503 61.286 68.251 61.163Q67.999 61.040 67.860 60.806Q67.721 60.571 67.721 60.247\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The out-of-distribution overestimation blowup. Within the data support (shaded) the fitted \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">Q\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:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> tracks the true \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>; outside it, the Bellman \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">max\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> selects wherever the extrapolation overshoots, and bootstrapping feeds that inflated value back as a target, so the error compounds with no online return to correct it.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:415.830px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 311.872 152.013\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-11.377 59.766h214.24\"\u002F>\u003Cpath stroke=\"none\" d=\"m204.863 59.766-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(185.144 9.089)\">\u003Cpath d=\"M-4.535 59.766L-4.816 59.766L-4.816 55.047Q-4.816 54.832-4.878 54.737Q-4.941 54.641-5.058 54.620Q-5.175 54.598-5.421 54.598L-5.421 54.301L-4.199 54.215L-4.199 56.703Q-3.722 56.239-3.023 56.239Q-2.542 56.239-2.134 56.483Q-1.726 56.727-1.490 57.141Q-1.253 57.555-1.253 58.039Q-1.253 58.414-1.402 58.743Q-1.550 59.071-1.820 59.323Q-2.089 59.575-2.433 59.709Q-2.777 59.844-3.136 59.844Q-3.457 59.844-3.755 59.696Q-4.054 59.547-4.261 59.286L-4.535 59.766M-4.175 57.094L-4.175 58.934Q-4.023 59.231-3.763 59.411Q-3.503 59.590-3.191 59.590Q-2.765 59.590-2.498 59.371Q-2.230 59.153-2.115 58.807Q-2 58.461-2 58.039Q-2 57.391-2.248 56.942Q-2.496 56.493-3.093 56.493Q-3.429 56.493-3.718 56.651Q-4.007 56.809-4.175 57.094M-0.632 58.934Q-0.632 58.450-0.230 58.155Q0.172 57.860 0.723 57.741Q1.274 57.621 1.766 57.621L1.766 57.332Q1.766 57.106 1.651 56.899Q1.536 56.692 1.338 56.573Q1.141 56.453 0.911 56.453Q0.485 56.453 0.200 56.559Q0.270 56.586 0.317 56.641Q0.364 56.696 0.389 56.766Q0.415 56.836 0.415 56.911Q0.415 57.016 0.364 57.108Q0.313 57.200 0.221 57.250Q0.129 57.301 0.024 57.301Q-0.082 57.301-0.173 57.250Q-0.265 57.200-0.316 57.108Q-0.367 57.016-0.367 56.911Q-0.367 56.493 0.022 56.346Q0.411 56.200 0.911 56.200Q1.243 56.200 1.596 56.330Q1.950 56.461 2.178 56.715Q2.407 56.969 2.407 57.317L2.407 59.118Q2.407 59.250 2.479 59.360Q2.551 59.469 2.680 59.469Q2.805 59.469 2.874 59.364Q2.942 59.258 2.942 59.118L2.942 58.606L3.223 58.606L3.223 59.118Q3.223 59.321 3.106 59.479Q2.989 59.637 2.807 59.721Q2.625 59.805 2.422 59.805Q2.192 59.805 2.040 59.633Q1.887 59.461 1.856 59.231Q1.696 59.512 1.387 59.678Q1.079 59.844 0.727 59.844Q0.215 59.844-0.208 59.621Q-0.632 59.399-0.632 58.934M0.055 58.934Q0.055 59.219 0.282 59.405Q0.508 59.590 0.801 59.590Q1.047 59.590 1.272 59.473Q1.497 59.356 1.631 59.153Q1.766 58.950 1.766 58.696L1.766 57.864Q1.500 57.864 1.215 57.918Q0.930 57.973 0.659 58.102Q0.387 58.231 0.221 58.438Q0.055 58.645 0.055 58.934M3.559 58.039Q3.559 57.543 3.809 57.118Q4.059 56.692 4.479 56.446Q4.899 56.200 5.399 56.200Q5.938 56.200 6.329 56.325Q6.719 56.450 6.719 56.864Q6.719 56.969 6.668 57.061Q6.618 57.153 6.526 57.203Q6.434 57.254 6.325 57.254Q6.219 57.254 6.127 57.203Q6.036 57.153 5.985 57.061Q5.934 56.969 5.934 56.864Q5.934 56.641 6.102 56.536Q5.879 56.477 5.407 56.477Q5.110 56.477 4.895 56.616Q4.680 56.754 4.549 56.985Q4.418 57.215 4.360 57.485Q4.301 57.754 4.301 58.039Q4.301 58.434 4.434 58.784Q4.567 59.133 4.838 59.350Q5.110 59.567 5.508 59.567Q5.883 59.567 6.159 59.350Q6.434 59.133 6.536 58.774Q6.551 58.711 6.614 58.711L6.719 58.711Q6.754 58.711 6.780 58.739Q6.805 58.766 6.805 58.805L6.805 58.828Q6.672 59.309 6.288 59.577Q5.903 59.844 5.399 59.844Q5.036 59.844 4.702 59.707Q4.368 59.571 4.108 59.321Q3.848 59.071 3.704 58.735Q3.559 58.399 3.559 58.039\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(185.144 9.089)\">\u003Cpath d=\"M8.890 59.766L7.093 59.766L7.093 59.469Q7.362 59.469 7.530 59.424Q7.698 59.379 7.698 59.207L7.698 55.047Q7.698 54.832 7.636 54.737Q7.573 54.641 7.456 54.620Q7.339 54.598 7.093 54.598L7.093 54.301L8.315 54.215L8.315 57.981L9.413 57.094Q9.620 56.914 9.620 56.766Q9.620 56.700 9.567 56.657Q9.515 56.614 9.444 56.614L9.444 56.317L10.979 56.317L10.979 56.614Q10.448 56.614 9.850 57.094L9.241 57.590L10.315 58.989Q10.452 59.164 10.559 59.272Q10.667 59.379 10.802 59.424Q10.936 59.469 11.163 59.469L11.163 59.766L9.538 59.766L9.538 59.469Q9.780 59.469 9.780 59.317Q9.780 59.239 9.737 59.168Q9.694 59.098 9.612 58.989L8.811 57.942L8.284 58.368L8.284 59.207Q8.284 59.375 8.452 59.422Q8.620 59.469 8.890 59.469L8.890 59.766M12.229 58.813L12.229 57.071Q12.229 56.856 12.167 56.760Q12.104 56.664 11.985 56.643Q11.866 56.621 11.620 56.621L11.620 56.325L12.866 56.239L12.866 58.789L12.866 58.813Q12.866 59.125 12.921 59.287Q12.975 59.450 13.126 59.520Q13.276 59.590 13.597 59.590Q14.026 59.590 14.300 59.252Q14.573 58.914 14.573 58.469L14.573 57.071Q14.573 56.856 14.511 56.760Q14.448 56.664 14.329 56.643Q14.210 56.621 13.964 56.621L13.964 56.325L15.210 56.239L15.210 59.024Q15.210 59.235 15.272 59.330Q15.335 59.426 15.454 59.448Q15.573 59.469 15.819 59.469L15.819 59.766L14.597 59.844L14.597 59.223Q14.429 59.512 14.147 59.678Q13.866 59.844 13.546 59.844Q12.229 59.844 12.229 58.813M18.147 61.317L16.292 61.317L16.292 61.024Q16.561 61.024 16.729 60.979Q16.897 60.934 16.897 60.758L16.897 56.934Q16.897 56.727 16.741 56.674Q16.585 56.621 16.292 56.621L16.292 56.325L17.514 56.239L17.514 56.703Q17.745 56.481 18.059 56.360Q18.374 56.239 18.714 56.239Q19.186 56.239 19.591 56.485Q19.995 56.731 20.227 57.147Q20.460 57.563 20.460 58.039Q20.460 58.414 20.311 58.743Q20.163 59.071 19.893 59.323Q19.624 59.575 19.280 59.709Q18.936 59.844 18.577 59.844Q18.288 59.844 18.016 59.723Q17.745 59.602 17.538 59.391L17.538 60.758Q17.538 60.934 17.706 60.979Q17.874 61.024 18.147 61.024L18.147 61.317M17.538 57.102L17.538 58.942Q17.690 59.231 17.952 59.411Q18.214 59.590 18.522 59.590Q18.807 59.590 19.030 59.452Q19.253 59.313 19.405 59.082Q19.557 58.852 19.636 58.580Q19.714 58.309 19.714 58.039Q19.714 57.707 19.589 57.350Q19.464 56.993 19.216 56.756Q18.968 56.520 18.620 56.520Q18.296 56.520 18.001 56.676Q17.706 56.832 17.538 57.102\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(185.144 9.089)\">\u003Cpath d=\"M23.873 59.758L23.873 58.536Q23.873 58.508 23.904 58.477Q23.936 58.446 23.959 58.446L24.065 58.446Q24.135 58.446 24.151 58.508Q24.213 58.828 24.352 59.069Q24.490 59.309 24.723 59.450Q24.955 59.590 25.264 59.590Q25.502 59.590 25.711 59.530Q25.920 59.469 26.057 59.321Q26.194 59.172 26.194 58.926Q26.194 58.672 25.983 58.506Q25.772 58.340 25.502 58.286L24.881 58.172Q24.475 58.094 24.174 57.838Q23.873 57.582 23.873 57.207Q23.873 56.840 24.074 56.618Q24.276 56.395 24.600 56.297Q24.924 56.200 25.264 56.200Q25.729 56.200 26.026 56.407L26.248 56.223Q26.272 56.200 26.303 56.200L26.354 56.200Q26.385 56.200 26.412 56.227Q26.440 56.254 26.440 56.286L26.440 57.270Q26.440 57.301 26.414 57.330Q26.389 57.360 26.354 57.360L26.248 57.360Q26.213 57.360 26.186 57.332Q26.158 57.305 26.158 57.270Q26.158 56.871 25.906 56.651Q25.654 56.430 25.256 56.430Q24.901 56.430 24.617 56.553Q24.334 56.676 24.334 56.981Q24.334 57.200 24.535 57.332Q24.737 57.465 24.983 57.508L25.608 57.621Q26.037 57.711 26.346 58.008Q26.654 58.305 26.654 58.719Q26.654 59.289 26.256 59.567Q25.858 59.844 25.264 59.844Q24.713 59.844 24.362 59.508L24.065 59.821Q24.041 59.844 24.006 59.844L23.959 59.844Q23.936 59.844 23.904 59.813Q23.873 59.782 23.873 59.758M28.768 59.735L27.697 56.879Q27.631 56.700 27.500 56.657Q27.369 56.614 27.112 56.614L27.112 56.317L28.791 56.317L28.791 56.614Q28.342 56.614 28.342 56.813Q28.346 56.828 28.348 56.846Q28.350 56.864 28.350 56.879L29.143 58.973L29.854 57.063Q29.819 56.969 29.819 56.924Q29.819 56.879 29.783 56.879Q29.717 56.700 29.586 56.657Q29.455 56.614 29.201 56.614L29.201 56.317L30.791 56.317L30.791 56.614Q30.342 56.614 30.342 56.813Q30.346 56.832 30.348 56.850Q30.350 56.868 30.350 56.879L31.182 59.094L31.936 57.094Q31.959 57.036 31.959 56.965Q31.959 56.805 31.822 56.709Q31.686 56.614 31.518 56.614L31.518 56.317L32.904 56.317L32.904 56.614Q32.670 56.614 32.492 56.741Q32.315 56.868 32.233 57.094L31.248 59.735Q31.194 59.844 31.080 59.844L31.022 59.844Q30.908 59.844 30.865 59.735L30.006 57.461L29.151 59.735Q29.112 59.844 28.990 59.844L28.936 59.844Q28.822 59.844 28.768 59.735\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(185.144 9.089)\">\u003Cpath d=\"M33.085 58.012Q33.085 57.532 33.318 57.116Q33.550 56.700 33.960 56.450Q34.370 56.200 34.847 56.200Q35.577 56.200 35.976 56.641Q36.374 57.082 36.374 57.813Q36.374 57.918 36.281 57.942L33.831 57.942L33.831 58.012Q33.831 58.422 33.952 58.778Q34.074 59.133 34.345 59.350Q34.617 59.567 35.046 59.567Q35.410 59.567 35.706 59.338Q36.003 59.110 36.105 58.758Q36.113 58.711 36.199 58.696L36.281 58.696Q36.374 58.723 36.374 58.805Q36.374 58.813 36.367 58.844Q36.304 59.071 36.165 59.254Q36.027 59.438 35.835 59.571Q35.644 59.703 35.425 59.774Q35.206 59.844 34.968 59.844Q34.597 59.844 34.259 59.707Q33.921 59.571 33.654 59.319Q33.386 59.067 33.236 58.727Q33.085 58.387 33.085 58.012M33.839 57.703L35.800 57.703Q35.800 57.399 35.699 57.108Q35.597 56.817 35.380 56.635Q35.163 56.453 34.847 56.453Q34.546 56.453 34.316 56.641Q34.085 56.828 33.962 57.120Q33.839 57.411 33.839 57.703M36.863 58.012Q36.863 57.532 37.095 57.116Q37.327 56.700 37.738 56.450Q38.148 56.200 38.624 56.200Q39.355 56.200 39.753 56.641Q40.152 57.082 40.152 57.813Q40.152 57.918 40.058 57.942L37.609 57.942L37.609 58.012Q37.609 58.422 37.730 58.778Q37.851 59.133 38.122 59.350Q38.394 59.567 38.824 59.567Q39.187 59.567 39.484 59.338Q39.781 59.110 39.882 58.758Q39.890 58.711 39.976 58.696L40.058 58.696Q40.152 58.723 40.152 58.805Q40.152 58.813 40.144 58.844Q40.081 59.071 39.943 59.254Q39.804 59.438 39.613 59.571Q39.421 59.703 39.202 59.774Q38.984 59.844 38.745 59.844Q38.374 59.844 38.036 59.707Q37.699 59.571 37.431 59.319Q37.163 59.067 37.013 58.727Q36.863 58.387 36.863 58.012M37.617 57.703L39.577 57.703Q39.577 57.399 39.476 57.108Q39.374 56.817 39.158 56.635Q38.941 56.453 38.624 56.453Q38.324 56.453 38.093 56.641Q37.863 56.828 37.740 57.120Q37.617 57.411 37.617 57.703M42.523 61.317L40.667 61.317L40.667 61.024Q40.937 61.024 41.105 60.979Q41.273 60.934 41.273 60.758L41.273 56.934Q41.273 56.727 41.117 56.674Q40.960 56.621 40.667 56.621L40.667 56.325L41.890 56.239L41.890 56.703Q42.120 56.481 42.435 56.360Q42.749 56.239 43.089 56.239Q43.562 56.239 43.966 56.485Q44.370 56.731 44.603 57.147Q44.835 57.563 44.835 58.039Q44.835 58.414 44.687 58.743Q44.538 59.071 44.269 59.323Q43.999 59.575 43.656 59.709Q43.312 59.844 42.952 59.844Q42.663 59.844 42.392 59.723Q42.120 59.602 41.913 59.391L41.913 60.758Q41.913 60.934 42.081 60.979Q42.249 61.024 42.523 61.024L42.523 61.317M41.913 57.102L41.913 58.942Q42.066 59.231 42.327 59.411Q42.589 59.590 42.898 59.590Q43.183 59.590 43.406 59.452Q43.628 59.313 43.781 59.082Q43.933 58.852 44.011 58.580Q44.089 58.309 44.089 58.039Q44.089 57.707 43.964 57.350Q43.839 56.993 43.591 56.756Q43.343 56.520 42.995 56.520Q42.671 56.520 42.376 56.676Q42.081 56.832 41.913 57.102\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-5.687 65.456V-63.426\"\u002F>\u003Cpath stroke=\"none\" d=\"m-5.687-65.426-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-56.584 -123.236)\">\u003Cpath d=\"M-2.382 59.934Q-2.964 59.934-3.482 59.707Q-4 59.481-4.388 59.082Q-4.777 58.684-4.996 58.159Q-5.214 57.633-5.214 57.063Q-5.214 56.293-4.839 55.616Q-4.464 54.938-3.814 54.536Q-3.164 54.133-2.382 54.133Q-1.609 54.133-0.958 54.536Q-0.308 54.938 0.067 55.616Q0.442 56.293 0.442 57.063Q0.442 57.625 0.227 58.149Q0.012 58.672-0.378 59.078Q-0.769 59.485-1.273 59.703Q-1.105 60.063-0.937 60.258Q-0.769 60.453-0.488 60.453Q-0.289 60.453-0.115 60.362Q0.059 60.270 0.163 60.108Q0.266 59.946 0.266 59.750Q0.293 59.653 0.375 59.653Q0.430 59.653 0.459 59.688Q0.489 59.723 0.489 59.782Q0.489 60.133 0.385 60.485Q0.282 60.836 0.049 61.077Q-0.183 61.317-0.542 61.317Q-0.832 61.317-1.021 61.178Q-1.210 61.039-1.320 60.819Q-1.429 60.598-1.492 60.334Q-1.554 60.071-1.609 59.821Q-1.984 59.934-2.382 59.934M-2.847 59.231Q-2.847 59.422-2.716 59.537Q-2.585 59.653-2.382 59.653Q-2.023 59.653-1.687 59.500Q-1.777 59.184-1.935 58.991Q-2.093 58.797-2.382 58.797Q-2.570 58.797-2.708 58.920Q-2.847 59.043-2.847 59.231M-3.101 59.231Q-3.101 58.946-2.888 58.745Q-2.675 58.543-2.382 58.543Q-2.031 58.543-1.814 58.768Q-1.597 58.993-1.421 59.368Q-1.058 59.133-0.830 58.764Q-0.601 58.395-0.500 57.961Q-0.398 57.528-0.398 57.063Q-0.398 56.582-0.511 56.118Q-0.625 55.653-0.869 55.264Q-1.113 54.875-1.494 54.645Q-1.875 54.414-2.382 54.414Q-2.890 54.414-3.277 54.647Q-3.664 54.879-3.900 55.258Q-4.136 55.637-4.255 56.110Q-4.375 56.582-4.375 57.063Q-4.375 57.586-4.242 58.078Q-4.109 58.571-3.808 58.953Q-3.507 59.336-3.031 59.528Q-3.101 59.395-3.101 59.231\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-56.584 -123.236)\">\u003Cpath d=\"M4.094 58.934Q4.094 58.450 4.496 58.155Q4.899 57.860 5.449 57.741Q6 57.621 6.492 57.621L6.492 57.332Q6.492 57.106 6.377 56.899Q6.262 56.692 6.065 56.573Q5.867 56.453 5.637 56.453Q5.211 56.453 4.926 56.559Q4.996 56.586 5.043 56.641Q5.090 56.696 5.115 56.766Q5.141 56.836 5.141 56.911Q5.141 57.016 5.090 57.108Q5.039 57.200 4.947 57.250Q4.856 57.301 4.750 57.301Q4.645 57.301 4.553 57.250Q4.461 57.200 4.410 57.108Q4.360 57.016 4.360 56.911Q4.360 56.493 4.748 56.346Q5.137 56.200 5.637 56.200Q5.969 56.200 6.322 56.330Q6.676 56.461 6.904 56.715Q7.133 56.969 7.133 57.317L7.133 59.118Q7.133 59.250 7.205 59.360Q7.278 59.469 7.406 59.469Q7.531 59.469 7.600 59.364Q7.668 59.258 7.668 59.118L7.668 58.606L7.949 58.606L7.949 59.118Q7.949 59.321 7.832 59.479Q7.715 59.637 7.533 59.721Q7.352 59.805 7.149 59.805Q6.918 59.805 6.766 59.633Q6.613 59.461 6.582 59.231Q6.422 59.512 6.113 59.678Q5.805 59.844 5.453 59.844Q4.942 59.844 4.518 59.621Q4.094 59.399 4.094 58.934M4.781 58.934Q4.781 59.219 5.008 59.405Q5.235 59.590 5.528 59.590Q5.774 59.590 5.998 59.473Q6.223 59.356 6.358 59.153Q6.492 58.950 6.492 58.696L6.492 57.864Q6.227 57.864 5.942 57.918Q5.656 57.973 5.385 58.102Q5.113 58.231 4.947 58.438Q4.781 58.645 4.781 58.934M8.867 58.805L8.867 56.614L8.164 56.614L8.164 56.360Q8.520 56.360 8.762 56.127Q9.004 55.895 9.115 55.547Q9.227 55.200 9.227 54.844L9.508 54.844L9.508 56.317L10.684 56.317L10.684 56.614L9.508 56.614L9.508 58.789Q9.508 59.110 9.627 59.338Q9.746 59.567 10.028 59.567Q10.207 59.567 10.324 59.444Q10.442 59.321 10.494 59.141Q10.547 58.961 10.547 58.789L10.547 58.317L10.828 58.317L10.828 58.805Q10.828 59.059 10.723 59.299Q10.617 59.539 10.420 59.692Q10.223 59.844 9.965 59.844Q9.649 59.844 9.397 59.721Q9.145 59.598 9.006 59.364Q8.867 59.129 8.867 58.805\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-56.584 -123.236)\">\u003Cpath d=\"M15.069 58.813L15.069 57.071Q15.069 56.856 15.006 56.760Q14.944 56.664 14.825 56.643Q14.706 56.621 14.460 56.621L14.460 56.325L15.706 56.239L15.706 58.789L15.706 58.813Q15.706 59.125 15.760 59.287Q15.815 59.450 15.965 59.520Q16.116 59.590 16.436 59.590Q16.866 59.590 17.139 59.252Q17.413 58.914 17.413 58.469L17.413 57.071Q17.413 56.856 17.350 56.760Q17.288 56.664 17.168 56.643Q17.049 56.621 16.803 56.621L16.803 56.325L18.049 56.239L18.049 59.024Q18.049 59.235 18.112 59.330Q18.174 59.426 18.293 59.448Q18.413 59.469 18.659 59.469L18.659 59.766L17.436 59.844L17.436 59.223Q17.268 59.512 16.987 59.678Q16.706 59.844 16.385 59.844Q15.069 59.844 15.069 58.813M21.034 59.766L19.178 59.766L19.178 59.469Q19.452 59.469 19.620 59.422Q19.788 59.375 19.788 59.207L19.788 57.071Q19.788 56.856 19.725 56.760Q19.663 56.664 19.543 56.643Q19.424 56.621 19.178 56.621L19.178 56.325L20.370 56.239L20.370 56.973Q20.483 56.758 20.676 56.590Q20.870 56.422 21.108 56.330Q21.346 56.239 21.600 56.239Q22.768 56.239 22.768 57.317L22.768 59.207Q22.768 59.375 22.938 59.422Q23.108 59.469 23.377 59.469L23.377 59.766L21.522 59.766L21.522 59.469Q21.795 59.469 21.963 59.422Q22.131 59.375 22.131 59.207L22.131 57.332Q22.131 56.950 22.010 56.721Q21.889 56.493 21.538 56.493Q21.225 56.493 20.971 56.655Q20.717 56.817 20.571 57.086Q20.424 57.356 20.424 57.653L20.424 59.207Q20.424 59.375 20.594 59.422Q20.764 59.469 21.034 59.469L21.034 59.766M23.866 59.758L23.866 58.536Q23.866 58.508 23.897 58.477Q23.928 58.446 23.952 58.446L24.057 58.446Q24.127 58.446 24.143 58.508Q24.206 58.828 24.344 59.069Q24.483 59.309 24.715 59.450Q24.948 59.590 25.256 59.590Q25.495 59.590 25.704 59.530Q25.913 59.469 26.049 59.321Q26.186 59.172 26.186 58.926Q26.186 58.672 25.975 58.506Q25.764 58.340 25.495 58.286L24.874 58.172Q24.467 58.094 24.167 57.838Q23.866 57.582 23.866 57.207Q23.866 56.840 24.067 56.618Q24.268 56.395 24.592 56.297Q24.917 56.200 25.256 56.200Q25.721 56.200 26.018 56.407L26.241 56.223Q26.264 56.200 26.295 56.200L26.346 56.200Q26.377 56.200 26.405 56.227Q26.432 56.254 26.432 56.286L26.432 57.270Q26.432 57.301 26.407 57.330Q26.381 57.360 26.346 57.360L26.241 57.360Q26.206 57.360 26.178 57.332Q26.151 57.305 26.151 57.270Q26.151 56.871 25.899 56.651Q25.647 56.430 25.249 56.430Q24.893 56.430 24.610 56.553Q24.327 56.676 24.327 56.981Q24.327 57.200 24.528 57.332Q24.729 57.465 24.975 57.508L25.600 57.621Q26.030 57.711 26.338 58.008Q26.647 58.305 26.647 58.719Q26.647 59.289 26.249 59.567Q25.850 59.844 25.256 59.844Q24.706 59.844 24.354 59.508L24.057 59.821Q24.034 59.844 23.999 59.844L23.952 59.844Q23.928 59.844 23.897 59.813Q23.866 59.782 23.866 59.758M27.174 58.012Q27.174 57.532 27.407 57.116Q27.639 56.700 28.049 56.450Q28.459 56.200 28.936 56.200Q29.667 56.200 30.065 56.641Q30.463 57.082 30.463 57.813Q30.463 57.918 30.370 57.942L27.920 57.942L27.920 58.012Q27.920 58.422 28.042 58.778Q28.163 59.133 28.434 59.350Q28.706 59.567 29.135 59.567Q29.499 59.567 29.795 59.338Q30.092 59.110 30.194 58.758Q30.202 58.711 30.288 58.696L30.370 58.696Q30.463 58.723 30.463 58.805Q30.463 58.813 30.456 58.844Q30.393 59.071 30.254 59.254Q30.116 59.438 29.924 59.571Q29.733 59.703 29.514 59.774Q29.295 59.844 29.057 59.844Q28.686 59.844 28.348 59.707Q28.010 59.571 27.743 59.319Q27.475 59.067 27.325 58.727Q27.174 58.387 27.174 58.012M27.928 57.703L29.889 57.703Q29.889 57.399 29.788 57.108Q29.686 56.817 29.469 56.635Q29.252 56.453 28.936 56.453Q28.635 56.453 28.405 56.641Q28.174 56.828 28.051 57.120Q27.928 57.411 27.928 57.703M30.952 58.012Q30.952 57.532 31.184 57.116Q31.417 56.700 31.827 56.450Q32.237 56.200 32.713 56.200Q33.444 56.200 33.842 56.641Q34.241 57.082 34.241 57.813Q34.241 57.918 34.147 57.942L31.698 57.942L31.698 58.012Q31.698 58.422 31.819 58.778Q31.940 59.133 32.211 59.350Q32.483 59.567 32.913 59.567Q33.276 59.567 33.573 59.338Q33.870 59.110 33.971 58.758Q33.979 58.711 34.065 58.696L34.147 58.696Q34.241 58.723 34.241 58.805Q34.241 58.813 34.233 58.844Q34.170 59.071 34.032 59.254Q33.893 59.438 33.702 59.571Q33.510 59.703 33.292 59.774Q33.073 59.844 32.834 59.844Q32.463 59.844 32.126 59.707Q31.788 59.571 31.520 59.319Q31.252 59.067 31.102 58.727Q30.952 58.387 30.952 58.012M31.706 57.703L33.667 57.703Q33.667 57.399 33.565 57.108Q33.463 56.817 33.247 56.635Q33.030 56.453 32.713 56.453Q32.413 56.453 32.182 56.641Q31.952 56.828 31.829 57.120Q31.706 57.411 31.706 57.703M36.659 59.766L34.803 59.766L34.803 59.469Q35.077 59.469 35.245 59.422Q35.413 59.375 35.413 59.207L35.413 57.071Q35.413 56.856 35.350 56.760Q35.288 56.664 35.168 56.643Q35.049 56.621 34.803 56.621L34.803 56.325L35.995 56.239L35.995 56.973Q36.108 56.758 36.301 56.590Q36.495 56.422 36.733 56.330Q36.971 56.239 37.225 56.239Q38.393 56.239 38.393 57.317L38.393 59.207Q38.393 59.375 38.563 59.422Q38.733 59.469 39.002 59.469L39.002 59.766L37.147 59.766L37.147 59.469Q37.420 59.469 37.588 59.422Q37.756 59.375 37.756 59.207L37.756 57.332Q37.756 56.950 37.635 56.721Q37.514 56.493 37.163 56.493Q36.850 56.493 36.596 56.655Q36.342 56.817 36.196 57.086Q36.049 57.356 36.049 57.653L36.049 59.207Q36.049 59.375 36.219 59.422Q36.389 59.469 36.659 59.469\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-56.584 -123.236)\">\u003Cpath d=\"M46.775 59.766L42.392 59.766L42.392 59.469Q42.712 59.469 42.956 59.422Q43.201 59.375 43.201 59.207L43.201 54.864Q43.201 54.692 42.956 54.645Q42.712 54.598 42.392 54.598L42.392 54.301L44.978 54.301L44.978 54.598Q43.966 54.598 43.966 54.864L43.966 59.207Q43.966 59.379 44.058 59.424Q44.150 59.469 44.368 59.469L45.056 59.469Q45.529 59.469 45.837 59.344Q46.146 59.219 46.324 58.989Q46.501 58.758 46.593 58.432Q46.685 58.106 46.728 57.653L47.009 57.653\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(15.079 8.044)\">\u003Cpath d=\"M-3.698 59.906Q-4.333 59.906-4.697 59.561Q-5.062 59.216-5.197 58.691Q-5.332 58.166-5.332 57.541Q-5.332 56.516-4.976 55.817Q-4.621 55.118-3.698 55.118Q-2.771 55.118-2.419 55.817Q-2.067 56.516-2.067 57.541Q-2.067 58.166-2.202 58.691Q-2.337 59.216-2.700 59.561Q-3.062 59.906-3.698 59.906M-3.698 59.681Q-3.260 59.681-3.047 59.306Q-2.833 58.932-2.783 58.465Q-2.734 57.999-2.734 57.421Q-2.734 56.868-2.783 56.440Q-2.833 56.013-3.045 55.678Q-3.257 55.343-3.698 55.343Q-4.040 55.343-4.243 55.550Q-4.446 55.757-4.533 56.069Q-4.621 56.382-4.643 56.698Q-4.665 57.015-4.665 57.421Q-4.665 57.838-4.643 58.180Q-4.621 58.522-4.532 58.870Q-4.443 59.219-4.238 59.450Q-4.033 59.681-3.698 59.681\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.912 8.044)\">\u003Cpath d=\"M-2.361 59.766L-4.891 59.766L-4.891 59.486Q-3.923 59.486-3.923 59.277L-3.923 55.658Q-4.316 55.846-4.938 55.846L-4.938 55.565Q-4.521 55.565-4.157 55.464Q-3.793 55.364-3.537 55.118L-3.411 55.118Q-3.346 55.135-3.329 55.203L-3.329 59.277Q-3.329 59.486-2.361 59.486\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(94.746 8.044)\">\u003Cpath d=\"M-2.361 59.766L-5.246 59.766L-5.246 59.564Q-5.246 59.534-5.219 59.506L-3.971 58.289Q-3.899 58.214-3.857 58.172Q-3.814 58.129-3.735 58.050Q-3.322 57.637-3.091 57.279Q-2.860 56.922-2.860 56.498Q-2.860 56.266-2.939 56.063Q-3.018 55.859-3.159 55.709Q-3.301 55.558-3.496 55.478Q-3.691 55.398-3.923 55.398Q-4.234 55.398-4.492 55.557Q-4.750 55.716-4.880 55.993L-4.860 55.993Q-4.692 55.993-4.585 56.104Q-4.477 56.215-4.477 56.379Q-4.477 56.536-4.586 56.649Q-4.696 56.762-4.860 56.762Q-5.020 56.762-5.133 56.649Q-5.246 56.536-5.246 56.379Q-5.246 56.003-5.038 55.716Q-4.829 55.429-4.494 55.273Q-4.159 55.118-3.804 55.118Q-3.380 55.118-3 55.276Q-2.621 55.435-2.387 55.752Q-2.153 56.068-2.153 56.498Q-2.153 56.809-2.293 57.078Q-2.433 57.346-2.638 57.551Q-2.843 57.756-3.206 58.038Q-3.568 58.320-3.677 58.416L-4.532 59.144L-3.889 59.144Q-3.626 59.144-3.337 59.142Q-3.048 59.141-2.830 59.132Q-2.611 59.123-2.594 59.106Q-2.532 59.041-2.495 58.874Q-2.457 58.706-2.419 58.464L-2.153 58.464\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(134.58 8.044)\">\u003Cpath d=\"M-4.891 59.219Q-4.771 59.376-4.580 59.475Q-4.388 59.575-4.173 59.614Q-3.958 59.653-3.735 59.653Q-3.438 59.653-3.243 59.498Q-3.048 59.342-2.958 59.088Q-2.867 58.833-2.867 58.549Q-2.867 58.255-2.959 58.004Q-3.052 57.753-3.250 57.597Q-3.448 57.442-3.742 57.442L-4.258 57.442Q-4.286 57.442-4.311 57.416Q-4.337 57.391-4.337 57.367L-4.337 57.295Q-4.337 57.264-4.311 57.242Q-4.286 57.220-4.258 57.220L-3.817 57.189Q-3.455 57.189-3.235 56.832Q-3.014 56.474-3.014 56.085Q-3.014 55.757-3.209 55.553Q-3.404 55.350-3.735 55.350Q-4.022 55.350-4.275 55.434Q-4.528 55.517-4.692 55.705Q-4.545 55.705-4.445 55.820Q-4.344 55.934-4.344 56.085Q-4.344 56.235-4.450 56.345Q-4.556 56.454-4.713 56.454Q-4.874 56.454-4.983 56.345Q-5.092 56.235-5.092 56.085Q-5.092 55.760-4.884 55.541Q-4.675 55.323-4.359 55.220Q-4.043 55.118-3.735 55.118Q-3.417 55.118-3.089 55.222Q-2.761 55.326-2.534 55.548Q-2.307 55.770-2.307 56.085Q-2.307 56.519-2.594 56.844Q-2.881 57.168-3.315 57.315Q-3.004 57.380-2.724 57.546Q-2.443 57.712-2.266 57.970Q-2.088 58.228-2.088 58.549Q-2.088 58.959-2.332 59.269Q-2.577 59.578-2.958 59.742Q-3.339 59.906-3.735 59.906Q-4.104 59.906-4.462 59.793Q-4.819 59.681-5.063 59.431Q-5.308 59.182-5.308 58.812Q-5.308 58.641-5.191 58.529Q-5.075 58.416-4.904 58.416Q-4.795 58.416-4.704 58.467Q-4.614 58.518-4.559 58.611Q-4.504 58.703-4.504 58.812Q-4.504 58.980-4.617 59.099Q-4.730 59.219-4.891 59.219\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(174.414 8.044)\">\u003Cpath d=\"M-3.370 58.618L-5.414 58.618L-5.414 58.337L-3.083 55.165Q-3.048 55.118-2.983 55.118L-2.847 55.118Q-2.802 55.118-2.775 55.145Q-2.748 55.172-2.748 55.217L-2.748 58.337L-1.985 58.337L-1.985 58.618L-2.748 58.618L-2.748 59.277Q-2.748 59.486-1.992 59.486L-1.992 59.766L-4.125 59.766L-4.125 59.486Q-3.370 59.486-3.370 59.277L-3.370 58.618M-3.322 55.893L-5.113 58.337L-3.322 58.337\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M5.694 39.85h187.789\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(145.797 -24.6)\">\u003Cpath d=\"M-4.846 58.925L-4.846 57.028L-5.485 57.028L-5.485 56.806Q-5.167 56.806-4.950 56.596Q-4.733 56.386-4.633 56.076Q-4.532 55.767-4.532 55.459L-4.265 55.459L-4.265 56.748L-3.188 56.748L-3.188 57.028L-4.265 57.028L-4.265 58.912Q-4.265 59.188-4.161 59.387Q-4.057 59.585-3.797 59.585Q-3.640 59.585-3.534 59.481Q-3.428 59.376-3.378 59.223Q-3.329 59.069-3.329 58.912L-3.329 58.498L-3.062 58.498L-3.062 58.925Q-3.062 59.151-3.161 59.361Q-3.260 59.571-3.445 59.703Q-3.629 59.834-3.858 59.834Q-4.296 59.834-4.571 59.597Q-4.846 59.359-4.846 58.925M-0.502 59.766L-2.238 59.766L-2.238 59.486Q-2.009 59.486-1.861 59.452Q-1.712 59.417-1.712 59.277L-1.712 57.428Q-1.712 57.158-1.820 57.097Q-1.927 57.035-2.238 57.035L-2.238 56.755L-1.209 56.680L-1.209 57.387Q-1.080 57.079-0.837 56.880Q-0.594 56.680-0.276 56.680Q-0.058 56.680 0.113 56.804Q0.284 56.929 0.284 57.141Q0.284 57.278 0.185 57.377Q0.086 57.476-0.047 57.476Q-0.184 57.476-0.283 57.377Q-0.382 57.278-0.382 57.141Q-0.382 57.001-0.283 56.902Q-0.574 56.902-0.774 57.098Q-0.974 57.295-1.066 57.589Q-1.158 57.883-1.158 58.163L-1.158 59.277Q-1.158 59.486-0.502 59.486L-0.502 59.766M1.443 58.932L1.443 57.428Q1.443 57.158 1.335 57.097Q1.228 57.035 0.917 57.035L0.917 56.755L2.024 56.680L2.024 58.912L2.024 58.932Q2.024 59.212 2.075 59.356Q2.126 59.499 2.268 59.556Q2.410 59.612 2.697 59.612Q2.950 59.612 3.155 59.472Q3.360 59.332 3.477 59.106Q3.593 58.881 3.593 58.631L3.593 57.428Q3.593 57.158 3.485 57.097Q3.377 57.035 3.066 57.035L3.066 56.755L4.174 56.680L4.174 59.093Q4.174 59.284 4.227 59.366Q4.280 59.448 4.381 59.467Q4.481 59.486 4.697 59.486L4.697 59.766L3.620 59.834L3.620 59.270Q3.511 59.452 3.365 59.575Q3.220 59.698 3.034 59.766Q2.848 59.834 2.646 59.834Q1.443 59.834 1.443 58.932M5.244 58.231Q5.244 57.910 5.368 57.621Q5.493 57.332 5.719 57.109Q5.944 56.885 6.240 56.765Q6.536 56.645 6.854 56.645Q7.182 56.645 7.443 56.745Q7.705 56.844 7.881 57.026Q8.057 57.209 8.151 57.467Q8.245 57.725 8.245 58.057Q8.245 58.149 8.163 58.170L5.907 58.170L5.907 58.231Q5.907 58.819 6.190 59.202Q6.474 59.585 7.042 59.585Q7.363 59.585 7.631 59.392Q7.899 59.199 7.988 58.884Q7.995 58.843 8.070 58.829L8.163 58.829Q8.245 58.853 8.245 58.925Q8.245 58.932 8.238 58.959Q8.125 59.356 7.754 59.595Q7.383 59.834 6.959 59.834Q6.522 59.834 6.122 59.626Q5.722 59.417 5.483 59.050Q5.244 58.683 5.244 58.231M5.914 57.961L7.729 57.961Q7.729 57.684 7.631 57.432Q7.534 57.179 7.335 57.023Q7.137 56.868 6.854 56.868Q6.577 56.868 6.363 57.026Q6.149 57.185 6.032 57.440Q5.914 57.695 5.914 57.961\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(145.797 -24.6)\">\u003Cpath d=\"M13.134 59.739L12.006 57.240Q11.934 57.093 11.804 57.061Q11.674 57.028 11.445 57.028L11.445 56.748L12.959 56.748L12.959 57.028Q12.607 57.028 12.607 57.175Q12.607 57.220 12.618 57.240L13.482 59.158L14.262 57.428Q14.296 57.360 14.296 57.281Q14.296 57.168 14.212 57.098Q14.128 57.028 14.009 57.028L14.009 56.748L15.205 56.748L15.205 57.028Q14.986 57.028 14.815 57.131Q14.645 57.233 14.556 57.428L13.520 59.739Q13.472 59.834 13.366 59.834L13.288 59.834Q13.182 59.834 13.134 59.739\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(145.797 -24.6)\">\u003Cpath d=\"M15.374 59.038Q15.374 58.706 15.597 58.479Q15.821 58.252 16.165 58.124Q16.508 57.995 16.881 57.943Q17.253 57.890 17.558 57.890L17.558 57.637Q17.558 57.432 17.450 57.252Q17.342 57.073 17.161 56.970Q16.980 56.868 16.772 56.868Q16.365 56.868 16.129 56.960Q16.218 56.997 16.264 57.081Q16.310 57.165 16.310 57.267Q16.310 57.363 16.264 57.442Q16.218 57.520 16.137 57.565Q16.057 57.609 15.968 57.609Q15.818 57.609 15.717 57.512Q15.616 57.414 15.616 57.267Q15.616 56.645 16.772 56.645Q16.983 56.645 17.233 56.709Q17.482 56.772 17.684 56.891Q17.886 57.011 18.012 57.196Q18.139 57.380 18.139 57.623L18.139 59.199Q18.139 59.315 18.200 59.411Q18.262 59.506 18.375 59.506Q18.484 59.506 18.549 59.412Q18.614 59.318 18.614 59.199L18.614 58.751L18.880 58.751L18.880 59.199Q18.880 59.469 18.653 59.634Q18.426 59.800 18.146 59.800Q17.937 59.800 17.800 59.646Q17.664 59.493 17.640 59.277Q17.493 59.544 17.211 59.689Q16.929 59.834 16.604 59.834Q16.327 59.834 16.043 59.759Q15.760 59.684 15.567 59.505Q15.374 59.325 15.374 59.038M15.989 59.038Q15.989 59.212 16.090 59.342Q16.190 59.472 16.346 59.542Q16.501 59.612 16.666 59.612Q16.884 59.612 17.093 59.515Q17.301 59.417 17.429 59.236Q17.558 59.055 17.558 58.829L17.558 58.101Q17.233 58.101 16.867 58.192Q16.501 58.283 16.245 58.495Q15.989 58.706 15.989 59.038M20.965 59.766L19.362 59.766L19.362 59.486Q19.588 59.486 19.737 59.452Q19.885 59.417 19.885 59.277L19.885 55.658Q19.885 55.388 19.778 55.326Q19.670 55.265 19.362 55.265L19.362 54.984L20.439 54.909L20.439 59.277Q20.439 59.414 20.589 59.450Q20.740 59.486 20.965 59.486L20.965 59.766M22.134 58.932L22.134 57.428Q22.134 57.158 22.027 57.097Q21.919 57.035 21.608 57.035L21.608 56.755L22.715 56.680L22.715 58.912L22.715 58.932Q22.715 59.212 22.767 59.356Q22.818 59.499 22.960 59.556Q23.102 59.612 23.389 59.612Q23.642 59.612 23.847 59.472Q24.052 59.332 24.168 59.106Q24.284 58.881 24.284 58.631L24.284 57.428Q24.284 57.158 24.177 57.097Q24.069 57.035 23.758 57.035L23.758 56.755L24.865 56.680L24.865 59.093Q24.865 59.284 24.918 59.366Q24.971 59.448 25.072 59.467Q25.173 59.486 25.388 59.486L25.388 59.766L24.312 59.834L24.312 59.270Q24.202 59.452 24.057 59.575Q23.912 59.698 23.725 59.766Q23.539 59.834 23.337 59.834Q22.134 59.834 22.134 58.932M25.935 58.231Q25.935 57.910 26.060 57.621Q26.185 57.332 26.410 57.109Q26.636 56.885 26.931 56.765Q27.227 56.645 27.545 56.645Q27.873 56.645 28.135 56.745Q28.396 56.844 28.572 57.026Q28.748 57.209 28.842 57.467Q28.936 57.725 28.936 58.057Q28.936 58.149 28.854 58.170L26.598 58.170L26.598 58.231Q26.598 58.819 26.882 59.202Q27.166 59.585 27.733 59.585Q28.054 59.585 28.323 59.392Q28.591 59.199 28.680 58.884Q28.687 58.843 28.762 58.829L28.854 58.829Q28.936 58.853 28.936 58.925Q28.936 58.932 28.929 58.959Q28.816 59.356 28.446 59.595Q28.075 59.834 27.651 59.834Q27.213 59.834 26.813 59.626Q26.414 59.417 26.174 59.050Q25.935 58.683 25.935 58.231M26.605 57.961L28.420 57.961Q28.420 57.684 28.323 57.432Q28.225 57.179 28.027 57.023Q27.829 56.868 27.545 56.868Q27.268 56.868 27.054 57.026Q26.841 57.185 26.723 57.440Q26.605 57.695 26.605 57.961\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(145.797 -24.6)\">\u003Cpath d=\"M37.400 58.959L32.567 58.959Q32.499 58.949 32.453 58.903Q32.407 58.857 32.407 58.785Q32.407 58.720 32.453 58.674Q32.499 58.628 32.567 58.618L37.400 58.618Q37.469 58.628 37.515 58.674Q37.561 58.720 37.561 58.785Q37.561 58.857 37.515 58.903Q37.469 58.949 37.400 58.959M37.400 57.421L32.567 57.421Q32.499 57.411 32.453 57.365Q32.407 57.319 32.407 57.247Q32.407 57.103 32.567 57.079L37.400 57.079Q37.561 57.103 37.561 57.247Q37.561 57.319 37.515 57.365Q37.469 57.411 37.400 57.421\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(145.797 -24.6)\">\u003Cpath d=\"M42.740 59.906Q42.105 59.906 41.741 59.561Q41.376 59.216 41.241 58.691Q41.106 58.166 41.106 57.541Q41.106 56.516 41.462 55.817Q41.817 55.118 42.740 55.118Q43.667 55.118 44.019 55.817Q44.371 56.516 44.371 57.541Q44.371 58.166 44.236 58.691Q44.101 59.216 43.738 59.561Q43.376 59.906 42.740 59.906M42.740 59.681Q43.178 59.681 43.391 59.306Q43.605 58.932 43.655 58.465Q43.704 57.999 43.704 57.421Q43.704 56.868 43.655 56.440Q43.605 56.013 43.393 55.678Q43.181 55.343 42.740 55.343Q42.398 55.343 42.195 55.550Q41.992 55.757 41.905 56.069Q41.817 56.382 41.795 56.698Q41.773 57.015 41.773 57.421Q41.773 57.838 41.795 58.180Q41.817 58.522 41.906 58.870Q41.995 59.219 42.200 59.450Q42.405 59.681 42.740 59.681\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M11.385 14.241 51.22-5.675l39.834-21.34 39.834-17.071L170.72-56.89\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M13.185 14.241a1.8 1.8 0 1 0-3.6 0 1.8 1.8 0 0 0 3.6 0M53.019-5.675a1.8 1.8 0 1 0-3.6 0 1.8 1.8 0 0 0 3.6 0M92.853-27.015a1.8 1.8 0 1 0-3.6 0 1.8 1.8 0 0 0 3.6 0M132.687-44.086a1.8 1.8 0 1 0-3.6 0 1.8 1.8 0 0 0 3.6 0M172.52-56.89a1.8 1.8 0 1 0-3.6 0 1.8 1.8 0 0 0 3.6 0m-1.8 0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(110.272 -104.267)\">\u003Cpath d=\"M-5.414 58.231Q-5.414 57.910-5.289 57.621Q-5.164 57.332-4.938 57.109Q-4.713 56.885-4.417 56.765Q-4.122 56.645-3.804 56.645Q-3.476 56.645-3.214 56.745Q-2.953 56.844-2.777 57.026Q-2.601 57.209-2.507 57.467Q-2.413 57.725-2.413 58.057Q-2.413 58.149-2.495 58.170L-4.750 58.170L-4.750 58.231Q-4.750 58.819-4.467 59.202Q-4.183 59.585-3.616 59.585Q-3.294 59.585-3.026 59.392Q-2.758 59.199-2.669 58.884Q-2.662 58.843-2.587 58.829L-2.495 58.829Q-2.413 58.853-2.413 58.925Q-2.413 58.932-2.419 58.959Q-2.532 59.356-2.903 59.595Q-3.274 59.834-3.698 59.834Q-4.135 59.834-4.535 59.626Q-4.935 59.417-5.174 59.050Q-5.414 58.683-5.414 58.231M-4.744 57.961L-2.929 57.961Q-2.929 57.684-3.026 57.432Q-3.124 57.179-3.322 57.023Q-3.520 56.868-3.804 56.868Q-4.081 56.868-4.294 57.026Q-4.508 57.185-4.626 57.440Q-4.744 57.695-4.744 57.961M-1.825 59.759L-1.825 58.696Q-1.825 58.672-1.797 58.645Q-1.770 58.618-1.746 58.618L-1.637 58.618Q-1.572 58.618-1.558 58.676Q-1.462 59.110-1.216 59.361Q-0.970 59.612-0.557 59.612Q-0.215 59.612 0.038 59.479Q0.291 59.346 0.291 59.038Q0.291 58.881 0.197 58.766Q0.103 58.652-0.035 58.583Q-0.174 58.515-0.341 58.477L-0.922 58.378Q-1.278 58.310-1.551 58.089Q-1.825 57.869-1.825 57.527Q-1.825 57.278-1.714 57.103Q-1.603 56.929-1.416 56.830Q-1.230 56.731-1.015 56.688Q-0.799 56.645-0.557 56.645Q-0.143 56.645 0.137 56.827L0.353 56.652Q0.363 56.649 0.370 56.647Q0.376 56.645 0.387 56.645L0.438 56.645Q0.465 56.645 0.489 56.669Q0.513 56.693 0.513 56.721L0.513 57.568Q0.513 57.589 0.489 57.616Q0.465 57.643 0.438 57.643L0.325 57.643Q0.298 57.643 0.272 57.618Q0.247 57.592 0.247 57.568Q0.247 57.332 0.141 57.168Q0.035 57.004-0.148 56.922Q-0.331 56.840-0.563 56.840Q-0.892 56.840-1.148 56.943Q-1.404 57.045-1.404 57.322Q-1.404 57.517-1.221 57.626Q-1.039 57.736-0.810 57.777L-0.235 57.883Q0.011 57.931 0.224 58.059Q0.438 58.187 0.575 58.390Q0.711 58.594 0.711 58.843Q0.711 59.356 0.346 59.595Q-0.020 59.834-0.557 59.834Q-1.052 59.834-1.384 59.540L-1.650 59.814Q-1.671 59.834-1.698 59.834L-1.746 59.834Q-1.770 59.834-1.797 59.807Q-1.825 59.780-1.825 59.759M1.867 58.925L1.867 57.028L1.228 57.028L1.228 56.806Q1.545 56.806 1.762 56.596Q1.980 56.386 2.080 56.076Q2.181 55.767 2.181 55.459L2.448 55.459L2.448 56.748L3.524 56.748L3.524 57.028L2.448 57.028L2.448 58.912Q2.448 59.188 2.552 59.387Q2.656 59.585 2.916 59.585Q3.073 59.585 3.179 59.481Q3.285 59.376 3.335 59.223Q3.384 59.069 3.384 58.912L3.384 58.498L3.651 58.498L3.651 58.925Q3.651 59.151 3.552 59.361Q3.453 59.571 3.268 59.703Q3.084 59.834 2.855 59.834Q2.417 59.834 2.142 59.597Q1.867 59.359 1.867 58.925M6.078 59.766L4.526 59.766L4.526 59.486Q4.751 59.486 4.900 59.452Q5.049 59.417 5.049 59.277L5.049 57.428Q5.049 57.240 5.001 57.156Q4.953 57.073 4.856 57.054Q4.758 57.035 4.546 57.035L4.546 56.755L5.603 56.680L5.603 59.277Q5.603 59.417 5.734 59.452Q5.866 59.486 6.078 59.486L6.078 59.766M4.806 55.459Q4.806 55.288 4.929 55.169Q5.052 55.049 5.223 55.049Q5.391 55.049 5.514 55.169Q5.637 55.288 5.637 55.459Q5.637 55.634 5.514 55.757Q5.391 55.880 5.223 55.880Q5.052 55.880 4.929 55.757Q4.806 55.634 4.806 55.459M8.405 59.766L6.771 59.766L6.771 59.486Q7 59.486 7.149 59.452Q7.298 59.417 7.298 59.277L7.298 57.428Q7.298 57.158 7.190 57.097Q7.083 57.035 6.771 57.035L6.771 56.755L7.831 56.680L7.831 57.329Q8.002 57.021 8.306 56.850Q8.610 56.680 8.956 56.680Q9.355 56.680 9.632 56.820Q9.909 56.960 9.995 57.308Q10.162 57.015 10.461 56.847Q10.760 56.680 11.105 56.680Q11.611 56.680 11.895 56.903Q12.179 57.127 12.179 57.623L12.179 59.277Q12.179 59.414 12.327 59.450Q12.476 59.486 12.702 59.486L12.702 59.766L11.071 59.766L11.071 59.486Q11.297 59.486 11.447 59.450Q11.598 59.414 11.598 59.277L11.598 57.637Q11.598 57.302 11.478 57.102Q11.358 56.902 11.044 56.902Q10.774 56.902 10.540 57.038Q10.306 57.175 10.167 57.409Q10.029 57.643 10.029 57.917L10.029 59.277Q10.029 59.414 10.178 59.450Q10.326 59.486 10.552 59.486L10.552 59.766L8.921 59.766L8.921 59.486Q9.150 59.486 9.299 59.452Q9.448 59.417 9.448 59.277L9.448 57.637Q9.448 57.302 9.328 57.102Q9.209 56.902 8.894 56.902Q8.624 56.902 8.390 57.038Q8.156 57.175 8.017 57.409Q7.879 57.643 7.879 57.917L7.879 59.277Q7.879 59.414 8.029 59.450Q8.180 59.486 8.405 59.486L8.405 59.766M13.348 59.038Q13.348 58.706 13.572 58.479Q13.795 58.252 14.139 58.124Q14.482 57.995 14.855 57.943Q15.228 57.890 15.532 57.890L15.532 57.637Q15.532 57.432 15.424 57.252Q15.316 57.073 15.135 56.970Q14.954 56.868 14.746 56.868Q14.339 56.868 14.103 56.960Q14.192 56.997 14.238 57.081Q14.284 57.165 14.284 57.267Q14.284 57.363 14.238 57.442Q14.192 57.520 14.112 57.565Q14.031 57.609 13.942 57.609Q13.792 57.609 13.691 57.512Q13.590 57.414 13.590 57.267Q13.590 56.645 14.746 56.645Q14.958 56.645 15.207 56.709Q15.457 56.772 15.658 56.891Q15.860 57.011 15.986 57.196Q16.113 57.380 16.113 57.623L16.113 59.199Q16.113 59.315 16.174 59.411Q16.236 59.506 16.349 59.506Q16.458 59.506 16.523 59.412Q16.588 59.318 16.588 59.199L16.588 58.751L16.855 58.751L16.855 59.199Q16.855 59.469 16.627 59.634Q16.400 59.800 16.120 59.800Q15.911 59.800 15.774 59.646Q15.638 59.493 15.614 59.277Q15.467 59.544 15.185 59.689Q14.903 59.834 14.578 59.834Q14.301 59.834 14.018 59.759Q13.734 59.684 13.541 59.505Q13.348 59.325 13.348 59.038M13.963 59.038Q13.963 59.212 14.064 59.342Q14.165 59.472 14.320 59.542Q14.476 59.612 14.640 59.612Q14.858 59.612 15.067 59.515Q15.275 59.417 15.404 59.236Q15.532 59.055 15.532 58.829L15.532 58.101Q15.207 58.101 14.841 58.192Q14.476 58.283 14.219 58.495Q13.963 58.706 13.963 59.038M17.798 58.925L17.798 57.028L17.159 57.028L17.159 56.806Q17.477 56.806 17.694 56.596Q17.911 56.386 18.011 56.076Q18.112 55.767 18.112 55.459L18.379 55.459L18.379 56.748L19.456 56.748L19.456 57.028L18.379 57.028L18.379 58.912Q18.379 59.188 18.483 59.387Q18.587 59.585 18.847 59.585Q19.004 59.585 19.110 59.481Q19.216 59.376 19.266 59.223Q19.315 59.069 19.315 58.912L19.315 58.498L19.582 58.498L19.582 58.925Q19.582 59.151 19.483 59.361Q19.384 59.571 19.199 59.703Q19.015 59.834 18.786 59.834Q18.348 59.834 18.073 59.597Q17.798 59.359 17.798 58.925M20.351 58.231Q20.351 57.910 20.476 57.621Q20.601 57.332 20.826 57.109Q21.052 56.885 21.347 56.765Q21.643 56.645 21.961 56.645Q22.289 56.645 22.551 56.745Q22.812 56.844 22.988 57.026Q23.164 57.209 23.258 57.467Q23.352 57.725 23.352 58.057Q23.352 58.149 23.270 58.170L21.014 58.170L21.014 58.231Q21.014 58.819 21.298 59.202Q21.582 59.585 22.149 59.585Q22.470 59.585 22.739 59.392Q23.007 59.199 23.096 58.884Q23.103 58.843 23.178 58.829L23.270 58.829Q23.352 58.853 23.352 58.925Q23.352 58.932 23.345 58.959Q23.232 59.356 22.862 59.595Q22.491 59.834 22.067 59.834Q21.629 59.834 21.230 59.626Q20.830 59.417 20.590 59.050Q20.351 58.683 20.351 58.231M21.021 57.961L22.836 57.961Q22.836 57.684 22.739 57.432Q22.641 57.179 22.443 57.023Q22.245 56.868 21.961 56.868Q21.684 56.868 21.470 57.026Q21.257 57.185 21.139 57.440Q21.021 57.695 21.021 57.961\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(110.272 -104.267)\">\u003Cpath d=\"M26.671 58.255Q26.671 57.927 26.806 57.626Q26.941 57.326 27.177 57.105Q27.413 56.885 27.717 56.765Q28.022 56.645 28.346 56.645Q28.852 56.645 29.201 56.748Q29.549 56.850 29.549 57.226Q29.549 57.373 29.452 57.474Q29.355 57.575 29.208 57.575Q29.054 57.575 28.955 57.476Q28.856 57.377 28.856 57.226Q28.856 57.038 28.996 56.946Q28.794 56.895 28.353 56.895Q27.998 56.895 27.769 57.091Q27.540 57.288 27.439 57.597Q27.338 57.907 27.338 58.255Q27.338 58.604 27.464 58.910Q27.591 59.216 27.846 59.400Q28.100 59.585 28.456 59.585Q28.678 59.585 28.862 59.501Q29.047 59.417 29.182 59.262Q29.317 59.106 29.375 58.898Q29.389 58.843 29.443 58.843L29.556 58.843Q29.587 58.843 29.609 58.867Q29.631 58.891 29.631 58.925L29.631 58.946Q29.546 59.233 29.358 59.431Q29.170 59.629 28.905 59.732Q28.640 59.834 28.346 59.834Q27.916 59.834 27.528 59.628Q27.140 59.421 26.906 59.058Q26.671 58.696 26.671 58.255M31.887 59.766L30.284 59.766L30.284 59.486Q30.510 59.486 30.659 59.452Q30.807 59.417 30.807 59.277L30.807 55.658Q30.807 55.388 30.700 55.326Q30.592 55.265 30.284 55.265L30.284 54.984L31.361 54.909L31.361 59.277Q31.361 59.414 31.511 59.450Q31.662 59.486 31.887 59.486L31.887 59.766M34.099 59.766L32.547 59.766L32.547 59.486Q32.773 59.486 32.921 59.452Q33.070 59.417 33.070 59.277L33.070 57.428Q33.070 57.240 33.022 57.156Q32.974 57.073 32.877 57.054Q32.779 57.035 32.567 57.035L32.567 56.755L33.624 56.680L33.624 59.277Q33.624 59.417 33.755 59.452Q33.887 59.486 34.099 59.486L34.099 59.766M32.827 55.459Q32.827 55.288 32.950 55.169Q33.073 55.049 33.244 55.049Q33.412 55.049 33.535 55.169Q33.658 55.288 33.658 55.459Q33.658 55.634 33.535 55.757Q33.412 55.880 33.244 55.880Q33.073 55.880 32.950 55.757Q32.827 55.634 32.827 55.459M36.426 59.766L34.793 59.766L34.793 59.486Q35.022 59.486 35.170 59.452Q35.319 59.417 35.319 59.277L35.319 57.428Q35.319 57.158 35.211 57.097Q35.104 57.035 34.793 57.035L34.793 56.755L35.852 56.680L35.852 57.329Q36.023 57.021 36.327 56.850Q36.631 56.680 36.977 56.680Q37.377 56.680 37.653 56.820Q37.930 56.960 38.016 57.308Q38.183 57.015 38.482 56.847Q38.781 56.680 39.127 56.680Q39.632 56.680 39.916 56.903Q40.200 57.127 40.200 57.623L40.200 59.277Q40.200 59.414 40.348 59.450Q40.497 59.486 40.723 59.486L40.723 59.766L39.092 59.766L39.092 59.486Q39.318 59.486 39.468 59.450Q39.619 59.414 39.619 59.277L39.619 57.637Q39.619 57.302 39.499 57.102Q39.379 56.902 39.065 56.902Q38.795 56.902 38.561 57.038Q38.327 57.175 38.188 57.409Q38.050 57.643 38.050 57.917L38.050 59.277Q38.050 59.414 38.199 59.450Q38.347 59.486 38.573 59.486L38.573 59.766L36.942 59.766L36.942 59.486Q37.171 59.486 37.320 59.452Q37.469 59.417 37.469 59.277L37.469 57.637Q37.469 57.302 37.349 57.102Q37.230 56.902 36.915 56.902Q36.645 56.902 36.411 57.038Q36.177 57.175 36.038 57.409Q35.900 57.643 35.900 57.917L35.900 59.277Q35.900 59.414 36.050 59.450Q36.201 59.486 36.426 59.486\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(110.272 -104.267)\">\u003Cpath d=\"M41.915 59.766L41.648 59.766L41.648 55.658Q41.648 55.388 41.541 55.326Q41.433 55.265 41.122 55.265L41.122 54.984L42.202 54.909L42.202 57.079Q42.411 56.888 42.696 56.784Q42.981 56.680 43.279 56.680Q43.597 56.680 43.894 56.801Q44.191 56.922 44.414 57.138Q44.636 57.353 44.762 57.638Q44.889 57.924 44.889 58.255Q44.889 58.700 44.649 59.064Q44.410 59.428 44.017 59.631Q43.624 59.834 43.180 59.834Q42.985 59.834 42.795 59.778Q42.606 59.722 42.445 59.617Q42.284 59.513 42.144 59.352L41.915 59.766M42.230 57.421L42.230 59.038Q42.366 59.298 42.607 59.455Q42.848 59.612 43.125 59.612Q43.419 59.612 43.631 59.505Q43.843 59.397 43.976 59.205Q44.109 59.014 44.168 58.775Q44.226 58.536 44.226 58.255Q44.226 57.896 44.132 57.592Q44.038 57.288 43.810 57.095Q43.583 56.902 43.217 56.902Q42.917 56.902 42.650 57.038Q42.383 57.175 42.230 57.421M45.524 59.759L45.524 58.696Q45.524 58.672 45.552 58.645Q45.579 58.618 45.603 58.618L45.712 58.618Q45.777 58.618 45.791 58.676Q45.887 59.110 46.133 59.361Q46.379 59.612 46.793 59.612Q47.134 59.612 47.387 59.479Q47.640 59.346 47.640 59.038Q47.640 58.881 47.546 58.766Q47.452 58.652 47.314 58.583Q47.175 58.515 47.008 58.477L46.427 58.378Q46.071 58.310 45.798 58.089Q45.524 57.869 45.524 57.527Q45.524 57.278 45.636 57.103Q45.747 56.929 45.933 56.830Q46.119 56.731 46.335 56.688Q46.550 56.645 46.793 56.645Q47.206 56.645 47.486 56.827L47.702 56.652Q47.712 56.649 47.719 56.647Q47.726 56.645 47.736 56.645L47.787 56.645Q47.815 56.645 47.838 56.669Q47.862 56.693 47.862 56.721L47.862 57.568Q47.862 57.589 47.838 57.616Q47.815 57.643 47.787 57.643L47.674 57.643Q47.647 57.643 47.621 57.618Q47.596 57.592 47.596 57.568Q47.596 57.332 47.490 57.168Q47.384 57.004 47.201 56.922Q47.018 56.840 46.786 56.840Q46.458 56.840 46.201 56.943Q45.945 57.045 45.945 57.322Q45.945 57.517 46.128 57.626Q46.311 57.736 46.540 57.777L47.114 57.883Q47.360 57.931 47.574 58.059Q47.787 58.187 47.924 58.390Q48.061 58.594 48.061 58.843Q48.061 59.356 47.695 59.595Q47.329 59.834 46.793 59.834Q46.297 59.834 45.965 59.540L45.699 59.814Q45.678 59.834 45.651 59.834L45.603 59.834Q45.579 59.834 45.552 59.807Q45.524 59.780 45.524 59.759\" 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-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M59.755-5.675C67.942-15.39 73.93-20.048 83.563-24.72\"\u002F>\u003Cpath stroke=\"none\" d=\"m85.362-25.592-3.577-.043 1.778.915-.382 1.964\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(12.25 -34.285)\">\u003Cpath d=\"M-3.691 59.766L-5.325 59.766L-5.325 59.486Q-5.096 59.486-4.947 59.452Q-4.798 59.417-4.798 59.277L-4.798 57.428Q-4.798 57.158-4.906 57.097Q-5.014 57.035-5.325 57.035L-5.325 56.755L-4.265 56.680L-4.265 57.329Q-4.094 57.021-3.790 56.850Q-3.486 56.680-3.141 56.680Q-2.635 56.680-2.351 56.903Q-2.067 57.127-2.067 57.623L-2.067 59.277Q-2.067 59.414-1.919 59.450Q-1.770 59.486-1.544 59.486L-1.544 59.766L-3.175 59.766L-3.175 59.486Q-2.946 59.486-2.797 59.452Q-2.648 59.417-2.648 59.277L-2.648 57.637Q-2.648 57.302-2.768 57.102Q-2.888 56.902-3.202 56.902Q-3.472 56.902-3.706 57.038Q-3.940 57.175-4.079 57.409Q-4.217 57.643-4.217 57.917L-4.217 59.277Q-4.217 59.414-4.067 59.450Q-3.916 59.486-3.691 59.486L-3.691 59.766M-0.998 58.283Q-0.998 57.941-0.863 57.642Q-0.728 57.343-0.488 57.119Q-0.249 56.895 0.069 56.770Q0.387 56.645 0.718 56.645Q1.163 56.645 1.563 56.861Q1.962 57.076 2.197 57.454Q2.431 57.831 2.431 58.283Q2.431 58.624 2.289 58.908Q2.147 59.192 1.903 59.399Q1.658 59.605 1.349 59.720Q1.040 59.834 0.718 59.834Q0.288 59.834-0.114 59.633Q-0.516 59.431-0.757 59.079Q-0.998 58.727-0.998 58.283M0.718 59.585Q1.320 59.585 1.544 59.207Q1.768 58.829 1.768 58.197Q1.768 57.585 1.533 57.226Q1.299 56.868 0.718 56.868Q-0.334 56.868-0.334 58.197Q-0.334 58.829-0.109 59.207Q0.117 59.585 0.718 59.585\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(12.25 -34.285)\">\u003Cpath d=\"M7.475 59.766L5.739 59.766L5.739 59.486Q5.968 59.486 6.117 59.452Q6.265 59.417 6.265 59.277L6.265 57.428Q6.265 57.158 6.158 57.097Q6.050 57.035 5.739 57.035L5.739 56.755L6.768 56.680L6.768 57.387Q6.898 57.079 7.140 56.880Q7.383 56.680 7.701 56.680Q7.920 56.680 8.091 56.804Q8.262 56.929 8.262 57.141Q8.262 57.278 8.162 57.377Q8.063 57.476 7.930 57.476Q7.793 57.476 7.694 57.377Q7.595 57.278 7.595 57.141Q7.595 57.001 7.694 56.902Q7.404 56.902 7.204 57.098Q7.004 57.295 6.911 57.589Q6.819 57.883 6.819 58.163L6.819 59.277Q6.819 59.486 7.475 59.486L7.475 59.766M8.805 58.283Q8.805 57.941 8.940 57.642Q9.075 57.343 9.314 57.119Q9.554 56.895 9.871 56.770Q10.189 56.645 10.521 56.645Q10.965 56.645 11.365 56.861Q11.765 57.076 11.999 57.454Q12.233 57.831 12.233 58.283Q12.233 58.624 12.091 58.908Q11.950 59.192 11.705 59.399Q11.461 59.605 11.151 59.720Q10.842 59.834 10.521 59.834Q10.090 59.834 9.689 59.633Q9.287 59.431 9.046 59.079Q8.805 58.727 8.805 58.283M10.521 59.585Q11.122 59.585 11.346 59.207Q11.570 58.829 11.570 58.197Q11.570 57.585 11.336 57.226Q11.102 56.868 10.521 56.868Q9.468 56.868 9.468 58.197Q9.468 58.829 9.694 59.207Q9.919 59.585 10.521 59.585M14.496 59.766L12.893 59.766L12.893 59.486Q13.119 59.486 13.267 59.452Q13.416 59.417 13.416 59.277L13.416 55.658Q13.416 55.388 13.308 55.326Q13.201 55.265 12.893 55.265L12.893 54.984L13.970 54.909L13.970 59.277Q13.970 59.414 14.120 59.450Q14.270 59.486 14.496 59.486L14.496 59.766M16.759 59.766L15.156 59.766L15.156 59.486Q15.381 59.486 15.530 59.452Q15.679 59.417 15.679 59.277L15.679 55.658Q15.679 55.388 15.571 55.326Q15.463 55.265 15.156 55.265L15.156 54.984L16.232 54.909L16.232 59.277Q16.232 59.414 16.383 59.450Q16.533 59.486 16.759 59.486L16.759 59.766M17.312 58.283Q17.312 57.941 17.447 57.642Q17.582 57.343 17.822 57.119Q18.061 56.895 18.379 56.770Q18.697 56.645 19.028 56.645Q19.473 56.645 19.872 56.861Q20.272 57.076 20.506 57.454Q20.741 57.831 20.741 58.283Q20.741 58.624 20.599 58.908Q20.457 59.192 20.213 59.399Q19.968 59.605 19.659 59.720Q19.349 59.834 19.028 59.834Q18.598 59.834 18.196 59.633Q17.794 59.431 17.553 59.079Q17.312 58.727 17.312 58.283M19.028 59.585Q19.630 59.585 19.854 59.207Q20.078 58.829 20.078 58.197Q20.078 57.585 19.843 57.226Q19.609 56.868 19.028 56.868Q17.975 56.868 17.975 58.197Q17.975 58.829 18.201 59.207Q18.427 59.585 19.028 59.585M21.910 58.932L21.910 57.428Q21.910 57.158 21.802 57.097Q21.694 57.035 21.383 57.035L21.383 56.755L22.491 56.680L22.491 58.912L22.491 58.932Q22.491 59.212 22.542 59.356Q22.593 59.499 22.735 59.556Q22.877 59.612 23.164 59.612Q23.417 59.612 23.622 59.472Q23.827 59.332 23.943 59.106Q24.059 58.881 24.059 58.631L24.059 57.428Q24.059 57.158 23.952 57.097Q23.844 57.035 23.533 57.035L23.533 56.755L24.640 56.680L24.640 59.093Q24.640 59.284 24.693 59.366Q24.746 59.448 24.847 59.467Q24.948 59.486 25.163 59.486L25.163 59.766L24.087 59.834L24.087 59.270Q23.977 59.452 23.832 59.575Q23.687 59.698 23.501 59.766Q23.314 59.834 23.113 59.834Q21.910 59.834 21.910 58.932M26.278 58.925L26.278 57.028L25.639 57.028L25.639 56.806Q25.956 56.806 26.173 56.596Q26.390 56.386 26.491 56.076Q26.592 55.767 26.592 55.459L26.859 55.459L26.859 56.748L27.935 56.748L27.935 57.028L26.859 57.028L26.859 58.912Q26.859 59.188 26.963 59.387Q27.067 59.585 27.327 59.585Q27.484 59.585 27.590 59.481Q27.696 59.376 27.746 59.223Q27.795 59.069 27.795 58.912L27.795 58.498L28.062 58.498L28.062 58.925Q28.062 59.151 27.963 59.361Q27.864 59.571 27.679 59.703Q27.494 59.834 27.265 59.834Q26.828 59.834 26.553 59.597Q26.278 59.359 26.278 58.925\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(12.25 -34.285)\">\u003Cpath d=\"M31.586 58.255Q31.586 57.927 31.721 57.626Q31.856 57.326 32.092 57.105Q32.328 56.885 32.632 56.765Q32.937 56.645 33.261 56.645Q33.767 56.645 34.116 56.748Q34.464 56.850 34.464 57.226Q34.464 57.373 34.367 57.474Q34.270 57.575 34.123 57.575Q33.969 57.575 33.870 57.476Q33.771 57.377 33.771 57.226Q33.771 57.038 33.911 56.946Q33.709 56.895 33.268 56.895Q32.913 56.895 32.684 57.091Q32.455 57.288 32.354 57.597Q32.253 57.907 32.253 58.255Q32.253 58.604 32.379 58.910Q32.506 59.216 32.761 59.400Q33.015 59.585 33.371 59.585Q33.593 59.585 33.777 59.501Q33.962 59.417 34.097 59.262Q34.232 59.106 34.290 58.898Q34.304 58.843 34.358 58.843L34.471 58.843Q34.502 58.843 34.524 58.867Q34.546 58.891 34.546 58.925L34.546 58.946Q34.461 59.233 34.273 59.431Q34.085 59.629 33.820 59.732Q33.555 59.834 33.261 59.834Q32.831 59.834 32.443 59.628Q32.055 59.421 31.821 59.058Q31.586 58.696 31.586 58.255M35.093 58.283Q35.093 57.941 35.228 57.642Q35.363 57.343 35.603 57.119Q35.842 56.895 36.160 56.770Q36.478 56.645 36.809 56.645Q37.253 56.645 37.653 56.861Q38.053 57.076 38.287 57.454Q38.522 57.831 38.522 58.283Q38.522 58.624 38.380 58.908Q38.238 59.192 37.993 59.399Q37.749 59.605 37.440 59.720Q37.130 59.834 36.809 59.834Q36.378 59.834 35.977 59.633Q35.575 59.431 35.334 59.079Q35.093 58.727 35.093 58.283M36.809 59.585Q37.411 59.585 37.635 59.207Q37.858 58.829 37.858 58.197Q37.858 57.585 37.624 57.226Q37.390 56.868 36.809 56.868Q35.756 56.868 35.756 58.197Q35.756 58.829 35.982 59.207Q36.208 59.585 36.809 59.585M40.866 59.766L39.130 59.766L39.130 59.486Q39.359 59.486 39.508 59.452Q39.656 59.417 39.656 59.277L39.656 57.428Q39.656 57.158 39.549 57.097Q39.441 57.035 39.130 57.035L39.130 56.755L40.159 56.680L40.159 57.387Q40.289 57.079 40.531 56.880Q40.774 56.680 41.092 56.680Q41.311 56.680 41.481 56.804Q41.652 56.929 41.652 57.141Q41.652 57.278 41.553 57.377Q41.454 57.476 41.321 57.476Q41.184 57.476 41.085 57.377Q40.986 57.278 40.986 57.141Q40.986 57.001 41.085 56.902Q40.794 56.902 40.595 57.098Q40.395 57.295 40.302 57.589Q40.210 57.883 40.210 58.163L40.210 59.277Q40.210 59.486 40.866 59.486L40.866 59.766M43.987 59.766L42.251 59.766L42.251 59.486Q42.480 59.486 42.628 59.452Q42.777 59.417 42.777 59.277L42.777 57.428Q42.777 57.158 42.669 57.097Q42.562 57.035 42.251 57.035L42.251 56.755L43.279 56.680L43.279 57.387Q43.409 57.079 43.652 56.880Q43.895 56.680 44.212 56.680Q44.431 56.680 44.602 56.804Q44.773 56.929 44.773 57.141Q44.773 57.278 44.674 57.377Q44.575 57.476 44.441 57.476Q44.305 57.476 44.206 57.377Q44.106 57.278 44.106 57.141Q44.106 57.001 44.206 56.902Q43.915 56.902 43.715 57.098Q43.515 57.295 43.423 57.589Q43.331 57.883 43.331 58.163L43.331 59.277Q43.331 59.486 43.987 59.486L43.987 59.766M45.316 58.231Q45.316 57.910 45.441 57.621Q45.566 57.332 45.792 57.109Q46.017 56.885 46.313 56.765Q46.608 56.645 46.926 56.645Q47.254 56.645 47.516 56.745Q47.777 56.844 47.953 57.026Q48.129 57.209 48.223 57.467Q48.317 57.725 48.317 58.057Q48.317 58.149 48.235 58.170L45.980 58.170L45.980 58.231Q45.980 58.819 46.263 59.202Q46.547 59.585 47.114 59.585Q47.436 59.585 47.704 59.392Q47.972 59.199 48.061 58.884Q48.068 58.843 48.143 58.829L48.235 58.829Q48.317 58.853 48.317 58.925Q48.317 58.932 48.311 58.959Q48.198 59.356 47.827 59.595Q47.456 59.834 47.032 59.834Q46.595 59.834 46.195 59.626Q45.795 59.417 45.556 59.050Q45.316 58.683 45.316 58.231M45.986 57.961L47.801 57.961Q47.801 57.684 47.704 57.432Q47.606 57.179 47.408 57.023Q47.210 56.868 46.926 56.868Q46.649 56.868 46.436 57.026Q46.222 57.185 46.104 57.440Q45.986 57.695 45.986 57.961M48.905 58.255Q48.905 57.927 49.040 57.626Q49.175 57.326 49.411 57.105Q49.647 56.885 49.951 56.765Q50.255 56.645 50.580 56.645Q51.086 56.645 51.435 56.748Q51.783 56.850 51.783 57.226Q51.783 57.373 51.686 57.474Q51.588 57.575 51.441 57.575Q51.288 57.575 51.189 57.476Q51.089 57.377 51.089 57.226Q51.089 57.038 51.230 56.946Q51.028 56.895 50.587 56.895Q50.231 56.895 50.002 57.091Q49.773 57.288 49.673 57.597Q49.572 57.907 49.572 58.255Q49.572 58.604 49.698 58.910Q49.825 59.216 50.079 59.400Q50.334 59.585 50.689 59.585Q50.912 59.585 51.096 59.501Q51.281 59.417 51.416 59.262Q51.551 59.106 51.609 58.898Q51.623 58.843 51.677 58.843L51.790 58.843Q51.821 58.843 51.843 58.867Q51.865 58.891 51.865 58.925L51.865 58.946Q51.780 59.233 51.592 59.431Q51.404 59.629 51.139 59.732Q50.874 59.834 50.580 59.834Q50.149 59.834 49.762 59.628Q49.374 59.421 49.139 59.058Q48.905 58.696 48.905 58.255M52.980 58.925L52.980 57.028L52.340 57.028L52.340 56.806Q52.658 56.806 52.875 56.596Q53.092 56.386 53.193 56.076Q53.294 55.767 53.294 55.459L53.561 55.459L53.561 56.748L54.637 56.748L54.637 57.028L53.561 57.028L53.561 58.912Q53.561 59.188 53.665 59.387Q53.769 59.585 54.029 59.585Q54.186 59.585 54.292 59.481Q54.398 59.376 54.448 59.223Q54.497 59.069 54.497 58.912L54.497 58.498L54.764 58.498L54.764 58.925Q54.764 59.151 54.665 59.361Q54.565 59.571 54.381 59.703Q54.196 59.834 53.967 59.834Q53.530 59.834 53.255 59.597Q52.980 59.359 52.980 58.925M55.574 59.759L55.574 58.696Q55.574 58.672 55.601 58.645Q55.628 58.618 55.652 58.618L55.762 58.618Q55.827 58.618 55.840 58.676Q55.936 59.110 56.182 59.361Q56.428 59.612 56.842 59.612Q57.184 59.612 57.437 59.479Q57.689 59.346 57.689 59.038Q57.689 58.881 57.595 58.766Q57.501 58.652 57.363 58.583Q57.225 58.515 57.057 58.477L56.476 58.378Q56.121 58.310 55.847 58.089Q55.574 57.869 55.574 57.527Q55.574 57.278 55.685 57.103Q55.796 56.929 55.982 56.830Q56.168 56.731 56.384 56.688Q56.599 56.645 56.842 56.645Q57.255 56.645 57.536 56.827L57.751 56.652Q57.761 56.649 57.768 56.647Q57.775 56.645 57.785 56.645L57.836 56.645Q57.864 56.645 57.888 56.669Q57.912 56.693 57.912 56.721L57.912 57.568Q57.912 57.589 57.888 57.616Q57.864 57.643 57.836 57.643L57.724 57.643Q57.696 57.643 57.671 57.618Q57.645 57.592 57.645 57.568Q57.645 57.332 57.539 57.168Q57.433 57.004 57.250 56.922Q57.067 56.840 56.835 56.840Q56.507 56.840 56.251 56.943Q55.994 57.045 55.994 57.322Q55.994 57.517 56.177 57.626Q56.360 57.736 56.589 57.777L57.163 57.883Q57.409 57.931 57.623 58.059Q57.836 58.187 57.973 58.390Q58.110 58.594 58.110 58.843Q58.110 59.356 57.744 59.595Q57.378 59.834 56.842 59.834Q56.346 59.834 56.015 59.540L55.748 59.814Q55.728 59.834 55.700 59.834L55.652 59.834Q55.628 59.834 55.601 59.807Q55.574 59.780 55.574 59.759\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(12.25 -34.285)\">\u003Cpath d=\"M63.093 59.766L61.541 59.766L61.541 59.486Q61.767 59.486 61.916 59.452Q62.064 59.417 62.064 59.277L62.064 57.428Q62.064 57.240 62.016 57.156Q61.969 57.073 61.871 57.054Q61.774 57.035 61.562 57.035L61.562 56.755L62.618 56.680L62.618 59.277Q62.618 59.417 62.750 59.452Q62.881 59.486 63.093 59.486L63.093 59.766M61.822 55.459Q61.822 55.288 61.945 55.169Q62.068 55.049 62.239 55.049Q62.406 55.049 62.529 55.169Q62.652 55.288 62.652 55.459Q62.652 55.634 62.529 55.757Q62.406 55.880 62.239 55.880Q62.068 55.880 61.945 55.757Q61.822 55.634 61.822 55.459M64.266 58.925L64.266 57.028L63.626 57.028L63.626 56.806Q63.944 56.806 64.161 56.596Q64.378 56.386 64.479 56.076Q64.580 55.767 64.580 55.459L64.847 55.459L64.847 56.748L65.923 56.748L65.923 57.028L64.847 57.028L64.847 58.912Q64.847 59.188 64.951 59.387Q65.055 59.585 65.315 59.585Q65.472 59.585 65.578 59.481Q65.684 59.376 65.734 59.223Q65.783 59.069 65.783 58.912L65.783 58.498L66.050 58.498L66.050 58.925Q66.050 59.151 65.951 59.361Q65.851 59.571 65.667 59.703Q65.482 59.834 65.253 59.834Q64.816 59.834 64.541 59.597Q64.266 59.359 64.266 58.925\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The overestimation bubble on the two-state chain. A single optimistic off-support value at the unseen action \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> (dashed) is pulled into the Bellman max, inflating the target for a logged transition; the inflated logged value feeds the next backup, and with no rollout to execute \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> the estimate climbs each sweep while the true value (flat, low) is never queried.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:445.208px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 333.906 137.831\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-29.727 45.584h248.384\"\u002F>\u003Cpath stroke=\"none\" d=\"m220.657 45.584-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(213.047 9.089)\">\u003Cpath d=\"M-21.885 45.584L-23.717 45.584L-23.717 45.287Q-23.443 45.287-23.275 45.240Q-23.107 45.193-23.107 45.025L-23.107 40.865Q-23.107 40.650-23.170 40.555Q-23.232 40.459-23.351 40.438Q-23.471 40.416-23.717 40.416L-23.717 40.119L-22.494 40.033L-22.494 45.025Q-22.494 45.193-22.326 45.240Q-22.158 45.287-21.885 45.287L-21.885 45.584M-21.439 43.889Q-21.439 43.385-21.183 42.953Q-20.928 42.522-20.492 42.270Q-20.057 42.018-19.557 42.018Q-19.170 42.018-18.828 42.162Q-18.486 42.307-18.224 42.568Q-17.963 42.830-17.820 43.166Q-17.678 43.502-17.678 43.889Q-17.678 44.381-17.941 44.791Q-18.205 45.201-18.635 45.432Q-19.064 45.662-19.557 45.662Q-20.049 45.662-20.482 45.430Q-20.916 45.197-21.178 44.789Q-21.439 44.381-21.439 43.889M-19.557 45.385Q-19.099 45.385-18.848 45.162Q-18.596 44.939-18.508 44.588Q-18.420 44.236-18.420 43.791Q-18.420 43.361-18.514 43.023Q-18.607 42.686-18.861 42.479Q-19.115 42.272-19.557 42.272Q-20.205 42.272-20.449 42.688Q-20.693 43.104-20.693 43.791Q-20.693 44.236-20.605 44.588Q-20.517 44.939-20.266 45.162Q-20.014 45.385-19.557 45.385M-17.193 46.193Q-17.193 45.912-16.982 45.701Q-16.771 45.490-16.486 45.400Q-16.642 45.275-16.721 45.086Q-16.799 44.897-16.799 44.697Q-16.799 44.342-16.568 44.049Q-16.935 43.709-16.935 43.240Q-16.935 42.889-16.732 42.619Q-16.529 42.350-16.209 42.203Q-15.889 42.057-15.545 42.057Q-15.025 42.057-14.654 42.338Q-14.291 41.967-13.744 41.967Q-13.564 41.967-13.437 42.094Q-13.310 42.221-13.310 42.400Q-13.310 42.506-13.389 42.584Q-13.467 42.662-13.576 42.662Q-13.685 42.662-13.762 42.586Q-13.838 42.510-13.838 42.400Q-13.838 42.299-13.799 42.248Q-13.791 42.240-13.787 42.234Q-13.783 42.229-13.783 42.225Q-14.158 42.225-14.478 42.479Q-14.158 42.818-14.158 43.240Q-14.158 43.510-14.275 43.727Q-14.392 43.943-14.598 44.102Q-14.803 44.260-15.045 44.342Q-15.287 44.424-15.545 44.424Q-15.764 44.424-15.976 44.365Q-16.189 44.307-16.385 44.186Q-16.478 44.326-16.478 44.506Q-16.478 44.713-16.342 44.865Q-16.205 45.018-15.998 45.018L-15.303 45.018Q-14.814 45.018-14.402 45.102Q-13.990 45.186-13.711 45.443Q-13.432 45.701-13.432 46.193Q-13.432 46.557-13.752 46.789Q-14.072 47.022-14.514 47.123Q-14.955 47.225-15.310 47.225Q-15.666 47.225-16.109 47.123Q-16.553 47.022-16.873 46.789Q-17.193 46.557-17.193 46.193M-16.689 46.193Q-16.689 46.389-16.545 46.537Q-16.400 46.686-16.187 46.775Q-15.974 46.865-15.734 46.912Q-15.494 46.959-15.310 46.959Q-15.068 46.959-14.738 46.881Q-14.408 46.803-14.172 46.629Q-13.935 46.455-13.935 46.193Q-13.935 45.787-14.346 45.678Q-14.756 45.568-15.318 45.568L-15.998 45.568Q-16.267 45.568-16.478 45.746Q-16.689 45.924-16.689 46.193M-15.545 44.158Q-14.822 44.158-14.822 43.240Q-14.822 42.318-15.545 42.318Q-16.271 42.318-16.271 43.240Q-16.271 44.158-15.545 44.158M-12.947 46.193Q-12.947 45.912-12.736 45.701Q-12.525 45.490-12.240 45.400Q-12.396 45.275-12.474 45.086Q-12.553 44.897-12.553 44.697Q-12.553 44.342-12.322 44.049Q-12.689 43.709-12.689 43.240Q-12.689 42.889-12.486 42.619Q-12.283 42.350-11.963 42.203Q-11.642 42.057-11.299 42.057Q-10.779 42.057-10.408 42.338Q-10.045 41.967-9.498 41.967Q-9.318 41.967-9.191 42.094Q-9.064 42.221-9.064 42.400Q-9.064 42.506-9.142 42.584Q-9.221 42.662-9.330 42.662Q-9.439 42.662-9.516 42.586Q-9.592 42.510-9.592 42.400Q-9.592 42.299-9.553 42.248Q-9.545 42.240-9.541 42.234Q-9.537 42.229-9.537 42.225Q-9.912 42.225-10.232 42.479Q-9.912 42.818-9.912 43.240Q-9.912 43.510-10.029 43.727Q-10.146 43.943-10.351 44.102Q-10.557 44.260-10.799 44.342Q-11.041 44.424-11.299 44.424Q-11.517 44.424-11.730 44.365Q-11.943 44.307-12.139 44.186Q-12.232 44.326-12.232 44.506Q-12.232 44.713-12.096 44.865Q-11.959 45.018-11.752 45.018L-11.057 45.018Q-10.568 45.018-10.156 45.102Q-9.744 45.186-9.465 45.443Q-9.185 45.701-9.185 46.193Q-9.185 46.557-9.506 46.789Q-9.826 47.022-10.267 47.123Q-10.709 47.225-11.064 47.225Q-11.420 47.225-11.863 47.123Q-12.307 47.022-12.627 46.789Q-12.947 46.557-12.947 46.193M-12.443 46.193Q-12.443 46.389-12.299 46.537Q-12.154 46.686-11.941 46.775Q-11.728 46.865-11.488 46.912Q-11.248 46.959-11.064 46.959Q-10.822 46.959-10.492 46.881Q-10.162 46.803-9.926 46.629Q-9.689 46.455-9.689 46.193Q-9.689 45.787-10.099 45.678Q-10.510 45.568-11.072 45.568L-11.752 45.568Q-12.021 45.568-12.232 45.746Q-12.443 45.924-12.443 46.193M-11.299 44.158Q-10.576 44.158-10.576 43.240Q-10.576 42.318-11.299 42.318Q-12.025 42.318-12.025 43.240Q-12.025 44.158-11.299 44.158M-8.701 43.830Q-8.701 43.350-8.469 42.934Q-8.236 42.518-7.826 42.268Q-7.416 42.018-6.939 42.018Q-6.209 42.018-5.810 42.459Q-5.412 42.900-5.412 43.631Q-5.412 43.736-5.506 43.760L-7.955 43.760L-7.955 43.830Q-7.955 44.240-7.834 44.596Q-7.713 44.951-7.441 45.168Q-7.170 45.385-6.740 45.385Q-6.377 45.385-6.080 45.156Q-5.783 44.928-5.682 44.576Q-5.674 44.529-5.588 44.514L-5.506 44.514Q-5.412 44.541-5.412 44.623Q-5.412 44.631-5.420 44.662Q-5.482 44.889-5.621 45.072Q-5.760 45.256-5.951 45.389Q-6.142 45.522-6.361 45.592Q-6.580 45.662-6.818 45.662Q-7.189 45.662-7.527 45.525Q-7.865 45.389-8.133 45.137Q-8.400 44.885-8.551 44.545Q-8.701 44.205-8.701 43.830M-7.947 43.522L-5.986 43.522Q-5.986 43.217-6.088 42.926Q-6.189 42.635-6.406 42.453Q-6.623 42.272-6.939 42.272Q-7.240 42.272-7.471 42.459Q-7.701 42.647-7.824 42.938Q-7.947 43.229-7.947 43.522M-3.107 45.662Q-3.588 45.662-3.996 45.418Q-4.404 45.174-4.642 44.760Q-4.881 44.346-4.881 43.857Q-4.881 43.365-4.623 42.949Q-4.365 42.533-3.933 42.295Q-3.502 42.057-3.010 42.057Q-2.389 42.057-1.939 42.494L-1.939 40.865Q-1.939 40.650-2.002 40.555Q-2.064 40.459-2.182 40.438Q-2.299 40.416-2.545 40.416L-2.545 40.119L-1.322 40.033L-1.322 44.842Q-1.322 45.053-1.260 45.148Q-1.197 45.244-1.080 45.266Q-0.963 45.287-0.713 45.287L-0.713 45.584L-1.963 45.662L-1.963 45.178Q-2.428 45.662-3.107 45.662M-3.041 45.408Q-2.701 45.408-2.408 45.217Q-2.115 45.025-1.963 44.729L-1.963 42.897Q-2.111 42.623-2.373 42.467Q-2.635 42.311-2.947 42.311Q-3.572 42.311-3.855 42.758Q-4.139 43.205-4.139 43.865Q-4.139 44.510-3.887 44.959Q-3.635 45.408-3.041 45.408\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(213.047 9.089)\">\u003Cpath d=\"M3.271 44.623L3.271 42.432L2.568 42.432L2.568 42.178Q2.924 42.178 3.166 41.945Q3.408 41.713 3.519 41.365Q3.631 41.018 3.631 40.662L3.912 40.662L3.912 42.135L5.088 42.135L5.088 42.432L3.912 42.432L3.912 44.607Q3.912 44.928 4.031 45.156Q4.150 45.385 4.431 45.385Q4.611 45.385 4.728 45.262Q4.845 45.139 4.898 44.959Q4.951 44.779 4.951 44.607L4.951 44.135L5.232 44.135L5.232 44.623Q5.232 44.877 5.127 45.117Q5.021 45.357 4.824 45.510Q4.627 45.662 4.369 45.662Q4.053 45.662 3.801 45.539Q3.549 45.416 3.410 45.182Q3.271 44.947 3.271 44.623M7.959 45.584L5.978 45.584L5.978 45.287Q6.248 45.287 6.416 45.242Q6.584 45.197 6.584 45.025L6.584 42.889Q6.584 42.674 6.521 42.578Q6.459 42.482 6.342 42.461Q6.224 42.439 5.978 42.439L5.978 42.143L7.146 42.057L7.146 42.842Q7.224 42.631 7.377 42.445Q7.529 42.260 7.728 42.158Q7.928 42.057 8.154 42.057Q8.400 42.057 8.592 42.201Q8.783 42.346 8.783 42.576Q8.783 42.732 8.678 42.842Q8.572 42.951 8.416 42.951Q8.260 42.951 8.150 42.842Q8.041 42.732 8.041 42.576Q8.041 42.416 8.146 42.311Q7.822 42.311 7.607 42.539Q7.392 42.768 7.297 43.107Q7.201 43.447 7.201 43.752L7.201 45.025Q7.201 45.193 7.428 45.240Q7.654 45.287 7.959 45.287L7.959 45.584M9.361 44.752Q9.361 44.268 9.763 43.973Q10.166 43.678 10.717 43.559Q11.267 43.439 11.760 43.439L11.760 43.150Q11.760 42.924 11.644 42.717Q11.529 42.510 11.332 42.391Q11.135 42.272 10.904 42.272Q10.478 42.272 10.193 42.377Q10.263 42.404 10.310 42.459Q10.357 42.514 10.383 42.584Q10.408 42.654 10.408 42.729Q10.408 42.834 10.357 42.926Q10.306 43.018 10.215 43.068Q10.123 43.119 10.017 43.119Q9.912 43.119 9.820 43.068Q9.728 43.018 9.678 42.926Q9.627 42.834 9.627 42.729Q9.627 42.311 10.015 42.164Q10.404 42.018 10.904 42.018Q11.236 42.018 11.590 42.148Q11.943 42.279 12.172 42.533Q12.400 42.787 12.400 43.135L12.400 44.936Q12.400 45.068 12.472 45.178Q12.545 45.287 12.674 45.287Q12.799 45.287 12.867 45.182Q12.935 45.076 12.935 44.936L12.935 44.424L13.217 44.424L13.217 44.936Q13.217 45.139 13.099 45.297Q12.982 45.455 12.801 45.539Q12.619 45.623 12.416 45.623Q12.185 45.623 12.033 45.451Q11.881 45.279 11.849 45.049Q11.689 45.330 11.381 45.496Q11.072 45.662 10.720 45.662Q10.209 45.662 9.785 45.439Q9.361 45.217 9.361 44.752M10.049 44.752Q10.049 45.037 10.275 45.223Q10.502 45.408 10.795 45.408Q11.041 45.408 11.265 45.291Q11.490 45.174 11.625 44.971Q11.760 44.768 11.760 44.514L11.760 43.682Q11.494 43.682 11.209 43.736Q10.924 43.791 10.652 43.920Q10.381 44.049 10.215 44.256Q10.049 44.463 10.049 44.752\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(213.047 9.089)\">\u003Cpath d=\"M13.428 46.561Q13.428 46.455 13.478 46.363Q13.529 46.272 13.621 46.221Q13.713 46.170 13.818 46.170Q13.928 46.170 14.019 46.221Q14.111 46.272 14.162 46.363Q14.213 46.455 14.213 46.561Q14.213 46.775 14.021 46.897Q14.178 46.959 14.404 46.959Q14.705 46.959 14.834 46.650Q14.963 46.342 14.963 45.975L14.963 42.889Q14.963 42.584 14.816 42.512Q14.670 42.439 14.291 42.439L14.291 42.143L15.580 42.057L15.580 45.998Q15.580 46.318 15.420 46.602Q15.260 46.885 14.982 47.051Q14.705 47.217 14.389 47.217Q14.029 47.217 13.728 47.053Q13.428 46.889 13.428 46.561M14.658 40.662Q14.658 40.479 14.797 40.344Q14.935 40.209 15.123 40.209Q15.310 40.209 15.445 40.340Q15.580 40.471 15.580 40.662Q15.580 40.861 15.447 40.994Q15.314 41.127 15.123 41.127Q14.932 41.127 14.795 40.990Q14.658 40.854 14.658 40.662M16.580 43.830Q16.580 43.350 16.812 42.934Q17.045 42.518 17.455 42.268Q17.865 42.018 18.342 42.018Q19.072 42.018 19.471 42.459Q19.869 42.900 19.869 43.631Q19.869 43.736 19.775 43.760L17.326 43.760L17.326 43.830Q17.326 44.240 17.447 44.596Q17.568 44.951 17.840 45.168Q18.111 45.385 18.541 45.385Q18.904 45.385 19.201 45.156Q19.498 44.928 19.600 44.576Q19.607 44.529 19.693 44.514L19.775 44.514Q19.869 44.541 19.869 44.623Q19.869 44.631 19.861 44.662Q19.799 44.889 19.660 45.072Q19.521 45.256 19.330 45.389Q19.139 45.522 18.920 45.592Q18.701 45.662 18.463 45.662Q18.092 45.662 17.754 45.525Q17.416 45.389 17.148 45.137Q16.881 44.885 16.730 44.545Q16.580 44.205 16.580 43.830M17.334 43.522L19.295 43.522Q19.295 43.217 19.193 42.926Q19.092 42.635 18.875 42.453Q18.658 42.272 18.342 42.272Q18.041 42.272 17.810 42.459Q17.580 42.647 17.457 42.938Q17.334 43.229 17.334 43.522M20.400 43.857Q20.400 43.361 20.650 42.936Q20.900 42.510 21.320 42.264Q21.740 42.018 22.240 42.018Q22.779 42.018 23.170 42.143Q23.560 42.268 23.560 42.682Q23.560 42.787 23.510 42.879Q23.459 42.971 23.367 43.022Q23.275 43.072 23.166 43.072Q23.060 43.072 22.969 43.022Q22.877 42.971 22.826 42.879Q22.775 42.787 22.775 42.682Q22.775 42.459 22.943 42.354Q22.721 42.295 22.248 42.295Q21.951 42.295 21.736 42.434Q21.521 42.572 21.391 42.803Q21.260 43.033 21.201 43.303Q21.143 43.572 21.143 43.857Q21.143 44.252 21.275 44.602Q21.408 44.951 21.680 45.168Q21.951 45.385 22.350 45.385Q22.725 45.385 23 45.168Q23.275 44.951 23.377 44.592Q23.393 44.529 23.455 44.529L23.560 44.529Q23.596 44.529 23.621 44.557Q23.646 44.584 23.646 44.623L23.646 44.647Q23.514 45.127 23.129 45.395Q22.744 45.662 22.240 45.662Q21.877 45.662 21.543 45.525Q21.209 45.389 20.949 45.139Q20.689 44.889 20.545 44.553Q20.400 44.217 20.400 43.857M24.760 44.623L24.760 42.432L24.057 42.432L24.057 42.178Q24.412 42.178 24.654 41.945Q24.896 41.713 25.008 41.365Q25.119 41.018 25.119 40.662L25.400 40.662L25.400 42.135L26.576 42.135L26.576 42.432L25.400 42.432L25.400 44.607Q25.400 44.928 25.519 45.156Q25.639 45.385 25.920 45.385Q26.100 45.385 26.217 45.262Q26.334 45.139 26.387 44.959Q26.439 44.779 26.439 44.607L26.439 44.135L26.721 44.135L26.721 44.623Q26.721 44.877 26.615 45.117Q26.510 45.357 26.312 45.510Q26.115 45.662 25.857 45.662Q25.541 45.662 25.289 45.539Q25.037 45.416 24.898 45.182Q24.760 44.947 24.760 44.623M27.439 43.889Q27.439 43.385 27.695 42.953Q27.951 42.522 28.387 42.270Q28.822 42.018 29.322 42.018Q29.709 42.018 30.051 42.162Q30.393 42.307 30.654 42.568Q30.916 42.830 31.059 43.166Q31.201 43.502 31.201 43.889Q31.201 44.381 30.937 44.791Q30.674 45.201 30.244 45.432Q29.814 45.662 29.322 45.662Q28.830 45.662 28.396 45.430Q27.963 45.197 27.701 44.789Q27.439 44.381 27.439 43.889M29.322 45.385Q29.779 45.385 30.031 45.162Q30.283 44.939 30.371 44.588Q30.459 44.236 30.459 43.791Q30.459 43.361 30.365 43.023Q30.271 42.686 30.018 42.479Q29.764 42.272 29.322 42.272Q28.674 42.272 28.430 42.688Q28.185 43.104 28.185 43.791Q28.185 44.236 28.273 44.588Q28.361 44.939 28.613 45.162Q28.865 45.385 29.322 45.385M33.693 45.584L31.713 45.584L31.713 45.287Q31.982 45.287 32.150 45.242Q32.318 45.197 32.318 45.025L32.318 42.889Q32.318 42.674 32.256 42.578Q32.193 42.482 32.076 42.461Q31.959 42.439 31.713 42.439L31.713 42.143L32.881 42.057L32.881 42.842Q32.959 42.631 33.111 42.445Q33.264 42.260 33.463 42.158Q33.662 42.057 33.889 42.057Q34.135 42.057 34.326 42.201Q34.518 42.346 34.518 42.576Q34.518 42.732 34.412 42.842Q34.307 42.951 34.150 42.951Q33.994 42.951 33.885 42.842Q33.775 42.732 33.775 42.576Q33.775 42.416 33.881 42.311Q33.557 42.311 33.342 42.539Q33.127 42.768 33.031 43.107Q32.935 43.447 32.935 43.752L32.935 45.025Q32.935 45.193 33.162 45.240Q33.389 45.287 33.693 45.287L33.693 45.584M35.416 46.881Q35.529 46.959 35.705 46.959Q35.994 46.959 36.215 46.746Q36.435 46.533 36.560 46.232L36.850 45.584L35.576 42.697Q35.494 42.522 35.350 42.477Q35.205 42.432 34.935 42.432L34.935 42.135L36.654 42.135L36.654 42.432Q36.232 42.432 36.232 42.615Q36.232 42.627 36.248 42.697L37.185 44.822L38.018 42.912Q38.057 42.822 38.057 42.744Q38.057 42.604 37.955 42.518Q37.853 42.432 37.713 42.432L37.713 42.135L39.064 42.135L39.064 42.432Q38.810 42.432 38.617 42.557Q38.424 42.682 38.318 42.912L36.873 46.232Q36.760 46.486 36.594 46.709Q36.428 46.932 36.199 47.074Q35.971 47.217 35.705 47.217Q35.408 47.217 35.168 47.025Q34.928 46.834 34.928 46.545Q34.928 46.389 35.033 46.287Q35.139 46.186 35.287 46.186Q35.393 46.186 35.473 46.232Q35.553 46.279 35.600 46.357Q35.646 46.436 35.646 46.545Q35.646 46.666 35.586 46.754Q35.525 46.842 35.416 46.881\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-24.037 51.275V-63.382\"\u002F>\u003Cpath stroke=\"none\" d=\"m-24.037-65.381-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-38.233 -108.966)\">\u003Cpath d=\"M-21.275 45.584L-23.740 45.584L-23.740 45.287Q-23.408 45.287-23.150 45.240Q-22.892 45.193-22.892 45.025L-22.892 40.682Q-22.892 40.416-23.740 40.416L-23.740 40.119L-21.275 40.119L-21.275 40.416Q-22.127 40.416-22.127 40.682L-22.127 45.025Q-22.127 45.287-21.275 45.287L-21.275 45.584M-20.510 45.662L-20.510 43.857Q-20.510 43.830-20.478 43.799Q-20.447 43.768-20.424 43.768L-20.318 43.768Q-20.287 43.768-20.258 43.797Q-20.228 43.826-20.228 43.857Q-20.228 44.639-19.713 45.047Q-19.197 45.455-18.389 45.455Q-18.092 45.455-17.836 45.305Q-17.580 45.154-17.430 44.898Q-17.279 44.643-17.279 44.346Q-17.279 43.947-17.525 43.643Q-17.771 43.338-18.142 43.256L-19.264 42.998Q-19.603 42.924-19.891 42.703Q-20.178 42.482-20.344 42.164Q-20.510 41.846-20.510 41.494Q-20.510 41.064-20.279 40.709Q-20.049 40.354-19.668 40.152Q-19.287 39.951-18.861 39.951Q-18.611 39.951-18.365 40.010Q-18.119 40.068-17.900 40.191Q-17.682 40.314-17.517 40.494L-17.189 39.998Q-17.158 39.951-17.119 39.951L-17.072 39.951Q-17.045 39.951-17.014 39.982Q-16.982 40.014-16.982 40.041L-16.982 41.850Q-16.982 41.873-17.014 41.904Q-17.045 41.936-17.072 41.936L-17.174 41.936Q-17.205 41.936-17.234 41.906Q-17.264 41.877-17.264 41.850Q-17.264 41.717-17.307 41.531Q-17.349 41.346-17.414 41.191Q-17.478 41.037-17.578 40.879Q-17.678 40.721-17.767 40.631Q-18.197 40.225-18.861 40.225Q-19.139 40.225-19.398 40.357Q-19.658 40.490-19.816 40.725Q-19.974 40.959-19.974 41.240Q-19.974 41.596-19.734 41.867Q-19.494 42.139-19.127 42.225L-18.014 42.479Q-17.736 42.545-17.504 42.699Q-17.271 42.854-17.101 43.072Q-16.932 43.291-16.838 43.549Q-16.744 43.807-16.744 44.096Q-16.744 44.424-16.869 44.727Q-16.994 45.029-17.228 45.266Q-17.463 45.502-17.756 45.627Q-18.049 45.752-18.389 45.752Q-19.404 45.752-19.974 45.209L-20.303 45.705Q-20.334 45.752-20.373 45.752L-20.424 45.752Q-20.447 45.752-20.478 45.721Q-20.510 45.689-20.510 45.662\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.233 -108.966)\">\u003Cpath d=\"M-11.596 45.553L-12.666 42.697Q-12.732 42.518-12.863 42.475Q-12.994 42.432-13.252 42.432L-13.252 42.135L-11.572 42.135L-11.572 42.432Q-12.022 42.432-12.022 42.631Q-12.018 42.647-12.016 42.664Q-12.014 42.682-12.014 42.697L-11.221 44.791L-10.510 42.881Q-10.545 42.787-10.545 42.742Q-10.545 42.697-10.580 42.697Q-10.647 42.518-10.777 42.475Q-10.908 42.432-11.162 42.432L-11.162 42.135L-9.572 42.135L-9.572 42.432Q-10.022 42.432-10.022 42.631Q-10.018 42.650-10.016 42.668Q-10.014 42.686-10.014 42.697L-9.182 44.912L-8.428 42.912Q-8.404 42.854-8.404 42.783Q-8.404 42.623-8.541 42.527Q-8.678 42.432-8.846 42.432L-8.846 42.135L-7.459 42.135L-7.459 42.432Q-7.693 42.432-7.871 42.559Q-8.049 42.686-8.131 42.912L-9.115 45.553Q-9.170 45.662-9.283 45.662L-9.342 45.662Q-9.455 45.662-9.498 45.553L-10.357 43.279L-11.213 45.553Q-11.252 45.662-11.373 45.662L-11.428 45.662Q-11.541 45.662-11.596 45.553\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.233 -108.966)\">\u003Cpath d=\"M-7.279 43.830Q-7.279 43.350-7.046 42.934Q-6.814 42.518-6.404 42.268Q-5.994 42.018-5.517 42.018Q-4.787 42.018-4.388 42.459Q-3.990 42.900-3.990 43.631Q-3.990 43.736-4.083 43.760L-6.533 43.760L-6.533 43.830Q-6.533 44.240-6.412 44.596Q-6.290 44.951-6.019 45.168Q-5.747 45.385-5.318 45.385Q-4.955 45.385-4.658 45.156Q-4.361 44.928-4.259 44.576Q-4.251 44.529-4.165 44.514L-4.083 44.514Q-3.990 44.541-3.990 44.623Q-3.990 44.631-3.997 44.662Q-4.060 44.889-4.199 45.072Q-4.337 45.256-4.529 45.389Q-4.720 45.522-4.939 45.592Q-5.158 45.662-5.396 45.662Q-5.767 45.662-6.105 45.525Q-6.443 45.389-6.710 45.137Q-6.978 44.885-7.128 44.545Q-7.279 44.205-7.279 43.830M-6.525 43.522L-4.564 43.522Q-4.564 43.217-4.665 42.926Q-4.767 42.635-4.984 42.453Q-5.201 42.272-5.517 42.272Q-5.818 42.272-6.048 42.459Q-6.279 42.647-6.402 42.938Q-6.525 43.229-6.525 43.522M-1.642 45.584L-3.419 45.584L-3.419 45.287Q-3.146 45.287-2.978 45.240Q-2.810 45.193-2.810 45.025L-2.810 42.889Q-2.810 42.674-2.867 42.578Q-2.923 42.482-3.037 42.461Q-3.150 42.439-3.396 42.439L-3.396 42.143L-2.197 42.057L-2.197 45.025Q-2.197 45.193-2.050 45.240Q-1.904 45.287-1.642 45.287L-1.642 45.584M-3.083 40.662Q-3.083 40.471-2.949 40.340Q-2.814 40.209-2.619 40.209Q-2.497 40.209-2.394 40.272Q-2.290 40.334-2.228 40.438Q-2.165 40.541-2.165 40.662Q-2.165 40.857-2.296 40.992Q-2.427 41.127-2.619 41.127Q-2.818 41.127-2.951 40.994Q-3.083 40.861-3.083 40.662M-1.142 46.193Q-1.142 45.912-0.931 45.701Q-0.720 45.490-0.435 45.400Q-0.591 45.275-0.669 45.086Q-0.747 44.897-0.747 44.697Q-0.747 44.342-0.517 44.049Q-0.884 43.709-0.884 43.240Q-0.884 42.889-0.681 42.619Q-0.478 42.350-0.158 42.203Q0.163 42.057 0.506 42.057Q1.026 42.057 1.397 42.338Q1.760 41.967 2.307 41.967Q2.487 41.967 2.614 42.094Q2.741 42.221 2.741 42.400Q2.741 42.506 2.663 42.584Q2.585 42.662 2.475 42.662Q2.366 42.662 2.290 42.586Q2.213 42.510 2.213 42.400Q2.213 42.299 2.253 42.248Q2.260 42.240 2.264 42.234Q2.268 42.229 2.268 42.225Q1.893 42.225 1.573 42.479Q1.893 42.818 1.893 43.240Q1.893 43.510 1.776 43.727Q1.659 43.943 1.454 44.102Q1.249 44.260 1.006 44.342Q0.764 44.424 0.506 44.424Q0.288 44.424 0.075 44.365Q-0.138 44.307-0.333 44.186Q-0.427 44.326-0.427 44.506Q-0.427 44.713-0.290 44.865Q-0.154 45.018 0.053 45.018L0.749 45.018Q1.237 45.018 1.649 45.102Q2.061 45.186 2.340 45.443Q2.620 45.701 2.620 46.193Q2.620 46.557 2.299 46.789Q1.979 47.022 1.538 47.123Q1.096 47.225 0.741 47.225Q0.385 47.225-0.058 47.123Q-0.501 47.022-0.822 46.789Q-1.142 46.557-1.142 46.193M-0.638 46.193Q-0.638 46.389-0.494 46.537Q-0.349 46.686-0.136 46.775Q0.077 46.865 0.317 46.912Q0.557 46.959 0.741 46.959Q0.983 46.959 1.313 46.881Q1.643 46.803 1.879 46.629Q2.116 46.455 2.116 46.193Q2.116 45.787 1.706 45.678Q1.295 45.568 0.733 45.568L0.053 45.568Q-0.216 45.568-0.427 45.746Q-0.638 45.924-0.638 46.193M0.506 44.158Q1.229 44.158 1.229 43.240Q1.229 42.318 0.506 42.318Q-0.220 42.318-0.220 43.240Q-0.220 44.158 0.506 44.158M5.034 45.584L3.178 45.584L3.178 45.287Q3.452 45.287 3.620 45.240Q3.788 45.193 3.788 45.025L3.788 40.865Q3.788 40.650 3.725 40.555Q3.663 40.459 3.544 40.438Q3.424 40.416 3.178 40.416L3.178 40.119L4.401 40.033L4.401 42.736Q4.526 42.525 4.713 42.375Q4.901 42.225 5.128 42.141Q5.354 42.057 5.600 42.057Q6.768 42.057 6.768 43.135L6.768 45.025Q6.768 45.193 6.938 45.240Q7.108 45.287 7.378 45.287L7.378 45.584L5.522 45.584L5.522 45.287Q5.795 45.287 5.963 45.240Q6.131 45.193 6.131 45.025L6.131 43.150Q6.131 42.768 6.010 42.539Q5.889 42.311 5.538 42.311Q5.225 42.311 4.971 42.473Q4.717 42.635 4.571 42.904Q4.424 43.174 4.424 43.471L4.424 45.025Q4.424 45.193 4.594 45.240Q4.764 45.287 5.034 45.287\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.233 -108.966)\">\u003Cpath d=\"M8.221 44.623L8.221 42.432L7.518 42.432L7.518 42.178Q7.874 42.178 8.116 41.945Q8.358 41.713 8.469 41.365Q8.581 41.018 8.581 40.662L8.862 40.662L8.862 42.135L10.038 42.135L10.038 42.432L8.862 42.432L8.862 44.607Q8.862 44.928 8.981 45.156Q9.100 45.385 9.381 45.385Q9.561 45.385 9.678 45.262Q9.796 45.139 9.848 44.959Q9.901 44.779 9.901 44.607L9.901 44.135L10.182 44.135L10.182 44.623Q10.182 44.877 10.077 45.117Q9.971 45.357 9.774 45.510Q9.577 45.662 9.319 45.662Q9.003 45.662 8.751 45.539Q8.499 45.416 8.360 45.182Q8.221 44.947 8.221 44.623\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M-9.81 45.584v-4.268h9.673v4.268ZM7.261 45.584v-1.423h9.674v1.423ZM24.333 45.584v-7.113h9.674v7.113ZM41.404 45.584v-2.845h9.674v2.845ZM58.476 45.584v-96.74h9.674v96.74ZM75.548 45.584v-2.276h9.674v2.276ZM92.62 45.584v-5.69h9.673v5.69ZM109.69 45.584v-1.423h9.675v1.423ZM126.763 45.584V42.17h9.674v3.414ZM143.834 45.584v-1.707h9.674v1.707ZM160.906 45.584v-59.75h9.674v59.75ZM177.978 45.584v-2.56h9.674v2.56ZM195.05 45.584v-4.268h9.673v4.268ZM212.121 45.584v-1.138h9.674v1.138Zm9.674-1.138\"\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(44.783 -101.633)\">\u003Cpath d=\"M-23.764 44.101Q-23.764 43.759-23.629 43.460Q-23.494 43.161-23.254 42.937Q-23.015 42.713-22.697 42.588Q-22.379 42.463-22.048 42.463Q-21.603 42.463-21.204 42.679Q-20.804 42.894-20.569 43.272Q-20.335 43.649-20.335 44.101Q-20.335 44.442-20.477 44.726Q-20.619 45.010-20.863 45.217Q-21.108 45.423-21.417 45.538Q-21.726 45.652-22.048 45.652Q-22.478 45.652-22.880 45.451Q-23.282 45.249-23.523 44.897Q-23.764 44.545-23.764 44.101M-22.048 45.403Q-21.446 45.403-21.222 45.025Q-20.998 44.647-20.998 44.015Q-20.998 43.403-21.233 43.044Q-21.467 42.686-22.048 42.686Q-23.100 42.686-23.100 44.015Q-23.100 44.647-22.875 45.025Q-22.649 45.403-22.048 45.403M-18.059 45.584L-19.693 45.584L-19.693 45.304Q-19.464 45.304-19.315 45.270Q-19.166 45.235-19.166 45.095L-19.166 43.246Q-19.166 42.976-19.274 42.915Q-19.382 42.853-19.693 42.853L-19.693 42.573L-18.633 42.498L-18.633 43.147Q-18.462 42.839-18.158 42.668Q-17.854 42.498-17.509 42.498Q-17.003 42.498-16.719 42.721Q-16.435 42.945-16.435 43.441L-16.435 45.095Q-16.435 45.232-16.287 45.268Q-16.138 45.304-15.912 45.304L-15.912 45.584L-17.543 45.584L-17.543 45.304Q-17.314 45.304-17.165 45.270Q-17.016 45.235-17.016 45.095L-17.016 43.455Q-17.016 43.120-17.136 42.920Q-17.256 42.720-17.570 42.720Q-17.840 42.720-18.074 42.856Q-18.308 42.993-18.447 43.227Q-18.585 43.461-18.585 43.735L-18.585 45.095Q-18.585 45.232-18.435 45.268Q-18.285 45.304-18.059 45.304L-18.059 45.584M-15.366 44.049Q-15.366 43.728-15.241 43.439Q-15.116 43.150-14.891 42.927Q-14.665 42.703-14.369 42.583Q-14.074 42.463-13.756 42.463Q-13.428 42.463-13.166 42.563Q-12.905 42.662-12.729 42.844Q-12.553 43.027-12.459 43.285Q-12.365 43.543-12.365 43.875Q-12.365 43.967-12.447 43.988L-14.703 43.988L-14.703 44.049Q-14.703 44.637-14.419 45.020Q-14.135 45.403-13.568 45.403Q-13.246 45.403-12.978 45.210Q-12.710 45.017-12.621 44.702Q-12.614 44.661-12.539 44.647L-12.447 44.647Q-12.365 44.671-12.365 44.743Q-12.365 44.750-12.371 44.777Q-12.484 45.174-12.855 45.413Q-13.226 45.652-13.650 45.652Q-14.087 45.652-14.487 45.444Q-14.887 45.235-15.126 44.868Q-15.366 44.501-15.366 44.049M-14.696 43.779L-12.881 43.779Q-12.881 43.502-12.978 43.250Q-13.076 42.997-13.274 42.841Q-13.472 42.686-13.756 42.686Q-14.033 42.686-14.246 42.844Q-14.460 43.003-14.578 43.258Q-14.696 43.513-14.696 43.779\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.783 -101.633)\">\u003Cpath d=\"M-7.682 45.557L-8.663 43.058Q-8.724 42.915-8.842 42.880Q-8.960 42.846-9.176 42.846L-9.176 42.566L-7.696 42.566L-7.696 42.846Q-8.075 42.846-8.075 43.007Q-8.075 43.017-8.061 43.058L-7.347 44.890L-6.674 43.185Q-6.704 43.113-6.704 43.085Q-6.704 43.058-6.732 43.058Q-6.793 42.911-6.911 42.879Q-7.029 42.846-7.241 42.846L-7.241 42.566L-5.843 42.566L-5.843 42.846Q-6.219 42.846-6.219 43.007Q-6.219 43.038-6.212 43.058L-5.457 44.996L-4.770 43.246Q-4.749 43.195-4.749 43.140Q-4.749 43-4.862 42.923Q-4.975 42.846-5.115 42.846L-5.115 42.566L-3.895 42.566L-3.895 42.846Q-4.100 42.846-4.255 42.952Q-4.411 43.058-4.483 43.246L-5.388 45.557Q-5.423 45.652-5.535 45.652L-5.604 45.652Q-5.713 45.652-5.751 45.557L-6.533 43.554L-7.320 45.557Q-7.354 45.652-7.467 45.652L-7.535 45.652Q-7.644 45.652-7.682 45.557\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.783 -101.633)\">\u003Cpath d=\"M-3.618 44.049Q-3.618 43.728-3.493 43.439Q-3.368 43.150-3.142 42.927Q-2.917 42.703-2.621 42.583Q-2.326 42.463-2.008 42.463Q-1.680 42.463-1.418 42.563Q-1.157 42.662-0.981 42.844Q-0.805 43.027-0.711 43.285Q-0.617 43.543-0.617 43.875Q-0.617 43.967-0.699 43.988L-2.954 43.988L-2.954 44.049Q-2.954 44.637-2.671 45.020Q-2.387 45.403-1.820 45.403Q-1.498 45.403-1.230 45.210Q-0.962 45.017-0.873 44.702Q-0.866 44.661-0.791 44.647L-0.699 44.647Q-0.617 44.671-0.617 44.743Q-0.617 44.750-0.623 44.777Q-0.736 45.174-1.107 45.413Q-1.478 45.652-1.902 45.652Q-2.339 45.652-2.739 45.444Q-3.139 45.235-3.378 44.868Q-3.618 44.501-3.618 44.049M-2.948 43.779L-1.133 43.779Q-1.133 43.502-1.230 43.250Q-1.328 42.997-1.526 42.841Q-1.724 42.686-2.008 42.686Q-2.285 42.686-2.498 42.844Q-2.712 43.003-2.830 43.258Q-2.948 43.513-2.948 43.779M1.588 45.584L0.036 45.584L0.036 45.304Q0.262 45.304 0.411 45.270Q0.559 45.235 0.559 45.095L0.559 43.246Q0.559 43.058 0.511 42.974Q0.463 42.891 0.366 42.872Q0.269 42.853 0.057 42.853L0.057 42.573L1.113 42.498L1.113 45.095Q1.113 45.235 1.244 45.270Q1.376 45.304 1.588 45.304L1.588 45.584M0.317 41.277Q0.317 41.106 0.440 40.987Q0.563 40.867 0.734 40.867Q0.901 40.867 1.024 40.987Q1.147 41.106 1.147 41.277Q1.147 41.452 1.024 41.575Q0.901 41.698 0.734 41.698Q0.563 41.698 0.440 41.575Q0.317 41.452 0.317 41.277M2.193 46.117Q2.193 45.871 2.390 45.687Q2.586 45.502 2.842 45.423Q2.706 45.311 2.634 45.150Q2.562 44.989 2.562 44.808Q2.562 44.487 2.774 44.241Q2.439 43.943 2.439 43.533Q2.439 43.072 2.829 42.785Q3.218 42.498 3.697 42.498Q4.169 42.498 4.504 42.744Q4.678 42.590 4.888 42.508Q5.098 42.426 5.327 42.426Q5.491 42.426 5.613 42.533Q5.734 42.641 5.734 42.805Q5.734 42.901 5.662 42.973Q5.590 43.044 5.498 43.044Q5.399 43.044 5.329 42.971Q5.259 42.897 5.259 42.798Q5.259 42.744 5.273 42.713L5.279 42.699Q5.286 42.679 5.295 42.668Q5.303 42.658 5.307 42.651Q4.951 42.651 4.664 42.874Q4.951 43.167 4.951 43.533Q4.951 43.848 4.767 44.080Q4.582 44.313 4.293 44.441Q4.005 44.569 3.697 44.569Q3.495 44.569 3.304 44.519Q3.112 44.470 2.935 44.360Q2.842 44.487 2.842 44.630Q2.842 44.812 2.971 44.947Q3.099 45.082 3.283 45.082L3.916 45.082Q4.363 45.082 4.733 45.153Q5.102 45.225 5.361 45.454Q5.621 45.683 5.621 46.117Q5.621 46.438 5.326 46.640Q5.030 46.842 4.627 46.931Q4.223 47.020 3.909 47.020Q3.591 47.020 3.188 46.931Q2.784 46.842 2.489 46.640Q2.193 46.438 2.193 46.117M2.648 46.117Q2.648 46.346 2.866 46.495Q3.085 46.644 3.377 46.712Q3.670 46.780 3.909 46.780Q4.073 46.780 4.281 46.744Q4.490 46.709 4.697 46.628Q4.903 46.548 5.035 46.420Q5.167 46.292 5.167 46.117Q5.167 45.765 4.786 45.671Q4.404 45.577 3.902 45.577L3.283 45.577Q3.044 45.577 2.846 45.728Q2.648 45.878 2.648 46.117M3.697 44.330Q4.363 44.330 4.363 43.533Q4.363 42.733 3.697 42.733Q3.027 42.733 3.027 43.533Q3.027 44.330 3.697 44.330M7.898 45.584L6.264 45.584L6.264 45.304Q6.493 45.304 6.641 45.270Q6.790 45.235 6.790 45.095L6.790 41.476Q6.790 41.206 6.682 41.144Q6.575 41.083 6.264 41.083L6.264 40.802L7.344 40.727L7.344 43.113Q7.450 42.928 7.628 42.786Q7.805 42.645 8.014 42.571Q8.222 42.498 8.448 42.498Q8.954 42.498 9.237 42.721Q9.521 42.945 9.521 43.441L9.521 45.095Q9.521 45.232 9.670 45.268Q9.818 45.304 10.044 45.304L10.044 45.584L8.414 45.584L8.414 45.304Q8.643 45.304 8.791 45.270Q8.940 45.235 8.940 45.095L8.940 43.455Q8.940 43.120 8.820 42.920Q8.701 42.720 8.386 42.720Q8.116 42.720 7.882 42.856Q7.648 42.993 7.510 43.227Q7.371 43.461 7.371 43.735L7.371 45.095Q7.371 45.232 7.522 45.268Q7.672 45.304 7.898 45.304\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.783 -101.633)\">\u003Cpath d=\"M10.957 44.743L10.957 42.846L10.318 42.846L10.318 42.624Q10.636 42.624 10.853 42.414Q11.070 42.204 11.170 41.894Q11.271 41.585 11.271 41.277L11.538 41.277L11.538 42.566L12.615 42.566L12.615 42.846L11.538 42.846L11.538 44.730Q11.538 45.006 11.642 45.205Q11.746 45.403 12.006 45.403Q12.163 45.403 12.269 45.299Q12.375 45.194 12.425 45.041Q12.474 44.887 12.474 44.730L12.474 44.316L12.741 44.316L12.741 44.743Q12.741 44.969 12.642 45.179Q12.543 45.389 12.358 45.521Q12.174 45.652 11.945 45.652Q11.507 45.652 11.232 45.415Q10.957 45.177 10.957 44.743\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.783 -101.633)\">\u003Cpath d=\"M16.250 44.073Q16.250 43.735 16.391 43.444Q16.531 43.154 16.775 42.940Q17.019 42.727 17.324 42.612Q17.628 42.498 17.953 42.498Q18.223 42.498 18.486 42.597Q18.749 42.696 18.940 42.874L18.940 41.476Q18.940 41.206 18.833 41.144Q18.725 41.083 18.414 41.083L18.414 40.802L19.491 40.727L19.491 44.911Q19.491 45.099 19.545 45.182Q19.600 45.266 19.701 45.285Q19.802 45.304 20.017 45.304L20.017 45.584L18.910 45.652L18.910 45.235Q18.493 45.652 17.867 45.652Q17.436 45.652 17.064 45.440Q16.691 45.229 16.471 44.868Q16.250 44.507 16.250 44.073M17.925 45.430Q18.134 45.430 18.320 45.358Q18.506 45.287 18.660 45.150Q18.814 45.013 18.910 44.835L18.910 43.226Q18.824 43.079 18.679 42.959Q18.534 42.839 18.364 42.780Q18.195 42.720 18.014 42.720Q17.454 42.720 17.185 43.109Q16.917 43.499 16.917 44.080Q16.917 44.651 17.151 45.041Q17.385 45.430 17.925 45.430M20.625 44.101Q20.625 43.759 20.760 43.460Q20.895 43.161 21.135 42.937Q21.374 42.713 21.692 42.588Q22.010 42.463 22.341 42.463Q22.786 42.463 23.186 42.679Q23.585 42.894 23.820 43.272Q24.054 43.649 24.054 44.101Q24.054 44.442 23.912 44.726Q23.770 45.010 23.526 45.217Q23.281 45.423 22.972 45.538Q22.663 45.652 22.341 45.652Q21.911 45.652 21.509 45.451Q21.107 45.249 20.866 44.897Q20.625 44.545 20.625 44.101M22.341 45.403Q22.943 45.403 23.167 45.025Q23.391 44.647 23.391 44.015Q23.391 43.403 23.156 43.044Q22.922 42.686 22.341 42.686Q21.289 42.686 21.289 44.015Q21.289 44.647 21.514 45.025Q21.740 45.403 22.341 45.403M26.330 45.584L24.696 45.584L24.696 45.304Q24.925 45.304 25.074 45.270Q25.223 45.235 25.223 45.095L25.223 43.246Q25.223 42.976 25.115 42.915Q25.007 42.853 24.696 42.853L24.696 42.573L25.756 42.498L25.756 43.147Q25.927 42.839 26.231 42.668Q26.535 42.498 26.880 42.498Q27.280 42.498 27.557 42.638Q27.834 42.778 27.919 43.126Q28.087 42.833 28.386 42.665Q28.685 42.498 29.030 42.498Q29.536 42.498 29.820 42.721Q30.103 42.945 30.103 43.441L30.103 45.095Q30.103 45.232 30.252 45.268Q30.401 45.304 30.626 45.304L30.626 45.584L28.996 45.584L28.996 45.304Q29.222 45.304 29.372 45.268Q29.522 45.232 29.522 45.095L29.522 43.455Q29.522 43.120 29.403 42.920Q29.283 42.720 28.969 42.720Q28.699 42.720 28.465 42.856Q28.230 42.993 28.092 43.227Q27.954 43.461 27.954 43.735L27.954 45.095Q27.954 45.232 28.102 45.268Q28.251 45.304 28.477 45.304L28.477 45.584L26.846 45.584L26.846 45.304Q27.075 45.304 27.224 45.270Q27.373 45.235 27.373 45.095L27.373 43.455Q27.373 43.120 27.253 42.920Q27.133 42.720 26.819 42.720Q26.549 42.720 26.315 42.856Q26.081 42.993 25.942 43.227Q25.804 43.461 25.804 43.735L25.804 45.095Q25.804 45.232 25.954 45.268Q26.104 45.304 26.330 45.304L26.330 45.584M32.831 45.584L31.279 45.584L31.279 45.304Q31.505 45.304 31.654 45.270Q31.802 45.235 31.802 45.095L31.802 43.246Q31.802 43.058 31.754 42.974Q31.707 42.891 31.609 42.872Q31.512 42.853 31.300 42.853L31.300 42.573L32.356 42.498L32.356 45.095Q32.356 45.235 32.488 45.270Q32.619 45.304 32.831 45.304L32.831 45.584M31.560 41.277Q31.560 41.106 31.683 40.987Q31.806 40.867 31.977 40.867Q32.144 40.867 32.267 40.987Q32.390 41.106 32.390 41.277Q32.390 41.452 32.267 41.575Q32.144 41.698 31.977 41.698Q31.806 41.698 31.683 41.575Q31.560 41.452 31.560 41.277M35.159 45.584L33.525 45.584L33.525 45.304Q33.754 45.304 33.903 45.270Q34.051 45.235 34.051 45.095L34.051 43.246Q34.051 42.976 33.944 42.915Q33.836 42.853 33.525 42.853L33.525 42.573L34.584 42.498L34.584 43.147Q34.755 42.839 35.060 42.668Q35.364 42.498 35.709 42.498Q36.215 42.498 36.498 42.721Q36.782 42.945 36.782 43.441L36.782 45.095Q36.782 45.232 36.931 45.268Q37.080 45.304 37.305 45.304L37.305 45.584L35.675 45.584L35.675 45.304Q35.904 45.304 36.052 45.270Q36.201 45.235 36.201 45.095L36.201 43.455Q36.201 43.120 36.082 42.920Q35.962 42.720 35.647 42.720Q35.377 42.720 35.143 42.856Q34.909 42.993 34.771 43.227Q34.632 43.461 34.632 43.735L34.632 45.095Q34.632 45.232 34.783 45.268Q34.933 45.304 35.159 45.304L35.159 45.584M37.951 44.856Q37.951 44.524 38.175 44.297Q38.399 44.070 38.742 43.942Q39.086 43.813 39.458 43.761Q39.831 43.708 40.135 43.708L40.135 43.455Q40.135 43.250 40.028 43.070Q39.920 42.891 39.739 42.788Q39.558 42.686 39.349 42.686Q38.942 42.686 38.707 42.778Q38.795 42.815 38.842 42.899Q38.888 42.983 38.888 43.085Q38.888 43.181 38.842 43.260Q38.795 43.338 38.715 43.383Q38.635 43.427 38.546 43.427Q38.395 43.427 38.295 43.330Q38.194 43.232 38.194 43.085Q38.194 42.463 39.349 42.463Q39.561 42.463 39.811 42.527Q40.060 42.590 40.262 42.709Q40.463 42.829 40.590 43.014Q40.716 43.198 40.716 43.441L40.716 45.017Q40.716 45.133 40.778 45.229Q40.839 45.324 40.952 45.324Q41.061 45.324 41.126 45.230Q41.191 45.136 41.191 45.017L41.191 44.569L41.458 44.569L41.458 45.017Q41.458 45.287 41.231 45.452Q41.003 45.618 40.723 45.618Q40.515 45.618 40.378 45.464Q40.241 45.311 40.217 45.095Q40.070 45.362 39.788 45.507Q39.506 45.652 39.182 45.652Q38.905 45.652 38.621 45.577Q38.337 45.502 38.144 45.323Q37.951 45.143 37.951 44.856M38.566 44.856Q38.566 45.030 38.667 45.160Q38.768 45.290 38.924 45.360Q39.079 45.430 39.243 45.430Q39.462 45.430 39.670 45.333Q39.879 45.235 40.007 45.054Q40.135 44.873 40.135 44.647L40.135 43.919Q39.811 43.919 39.445 44.010Q39.079 44.101 38.823 44.313Q38.566 44.524 38.566 44.856M42.401 44.743L42.401 42.846L41.762 42.846L41.762 42.624Q42.080 42.624 42.297 42.414Q42.514 42.204 42.615 41.894Q42.716 41.585 42.716 41.277L42.982 41.277L42.982 42.566L44.059 42.566L44.059 42.846L42.982 42.846L42.982 44.730Q42.982 45.006 43.087 45.205Q43.191 45.403 43.451 45.403Q43.608 45.403 43.714 45.299Q43.820 45.194 43.869 45.041Q43.919 44.887 43.919 44.730L43.919 44.316L44.186 44.316L44.186 44.743Q44.186 44.969 44.086 45.179Q43.987 45.389 43.803 45.521Q43.618 45.652 43.389 45.652Q42.952 45.652 42.676 45.415Q42.401 45.177 42.401 44.743M44.955 44.049Q44.955 43.728 45.079 43.439Q45.204 43.150 45.430 42.927Q45.655 42.703 45.951 42.583Q46.247 42.463 46.564 42.463Q46.893 42.463 47.154 42.563Q47.415 42.662 47.592 42.844Q47.768 43.027 47.862 43.285Q47.956 43.543 47.956 43.875Q47.956 43.967 47.873 43.988L45.618 43.988L45.618 44.049Q45.618 44.637 45.901 45.020Q46.185 45.403 46.752 45.403Q47.074 45.403 47.342 45.210Q47.610 45.017 47.699 44.702Q47.706 44.661 47.781 44.647L47.873 44.647Q47.956 44.671 47.956 44.743Q47.956 44.750 47.949 44.777Q47.836 45.174 47.465 45.413Q47.094 45.652 46.670 45.652Q46.233 45.652 45.833 45.444Q45.433 45.235 45.194 44.868Q44.955 44.501 44.955 44.049M45.624 43.779L47.439 43.779Q47.439 43.502 47.342 43.250Q47.245 42.997 47.046 42.841Q46.848 42.686 46.564 42.686Q46.288 42.686 46.074 42.844Q45.860 43.003 45.742 43.258Q45.624 43.513 45.624 43.779M48.543 45.577L48.543 44.514Q48.543 44.490 48.571 44.463Q48.598 44.436 48.622 44.436L48.731 44.436Q48.796 44.436 48.810 44.494Q48.906 44.928 49.152 45.179Q49.398 45.430 49.811 45.430Q50.153 45.430 50.406 45.297Q50.659 45.164 50.659 44.856Q50.659 44.699 50.565 44.584Q50.471 44.470 50.333 44.401Q50.194 44.333 50.027 44.295L49.446 44.196Q49.090 44.128 48.817 43.907Q48.543 43.687 48.543 43.345Q48.543 43.096 48.655 42.921Q48.766 42.747 48.952 42.648Q49.138 42.549 49.353 42.506Q49.569 42.463 49.811 42.463Q50.225 42.463 50.505 42.645L50.721 42.470Q50.731 42.467 50.738 42.465Q50.745 42.463 50.755 42.463L50.806 42.463Q50.833 42.463 50.857 42.487Q50.881 42.511 50.881 42.539L50.881 43.386Q50.881 43.407 50.857 43.434Q50.833 43.461 50.806 43.461L50.693 43.461Q50.666 43.461 50.640 43.436Q50.615 43.410 50.615 43.386Q50.615 43.150 50.509 42.986Q50.403 42.822 50.220 42.740Q50.037 42.658 49.805 42.658Q49.477 42.658 49.220 42.761Q48.964 42.863 48.964 43.140Q48.964 43.335 49.147 43.444Q49.330 43.554 49.559 43.595L50.133 43.701Q50.379 43.749 50.592 43.877Q50.806 44.005 50.943 44.208Q51.080 44.412 51.080 44.661Q51.080 45.174 50.714 45.413Q50.348 45.652 49.811 45.652Q49.316 45.652 48.984 45.358L48.718 45.632Q48.697 45.652 48.670 45.652L48.622 45.652Q48.598 45.652 48.571 45.625Q48.543 45.598 48.543 45.577\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" 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(114.498 -13.496)\">\u003Cpath d=\"M-22.041 45.584L-23.675 45.584L-23.675 45.304Q-23.446 45.304-23.297 45.270Q-23.148 45.235-23.148 45.095L-23.148 43.246Q-23.148 42.976-23.256 42.915Q-23.364 42.853-23.675 42.853L-23.675 42.573L-22.615 42.498L-22.615 43.147Q-22.444 42.839-22.140 42.668Q-21.836 42.498-21.491 42.498Q-21.091 42.498-20.814 42.638Q-20.537 42.778-20.452 43.126Q-20.284 42.833-19.985 42.665Q-19.686 42.498-19.341 42.498Q-18.835 42.498-18.551 42.721Q-18.267 42.945-18.267 43.441L-18.267 45.095Q-18.267 45.232-18.119 45.268Q-17.970 45.304-17.745 45.304L-17.745 45.584L-19.375 45.584L-19.375 45.304Q-19.149 45.304-18.999 45.268Q-18.849 45.232-18.849 45.095L-18.849 43.455Q-18.849 43.120-18.968 42.920Q-19.088 42.720-19.402 42.720Q-19.672 42.720-19.906 42.856Q-20.141 42.993-20.279 43.227Q-20.417 43.461-20.417 43.735L-20.417 45.095Q-20.417 45.232-20.269 45.268Q-20.120 45.304-19.894 45.304L-19.894 45.584L-21.525 45.584L-21.525 45.304Q-21.296 45.304-21.147 45.270Q-20.998 45.235-20.998 45.095L-20.998 43.455Q-20.998 43.120-21.118 42.920Q-21.238 42.720-21.552 42.720Q-21.822 42.720-22.056 42.856Q-22.290 42.993-22.429 43.227Q-22.567 43.461-22.567 43.735L-22.567 45.095Q-22.567 45.232-22.417 45.268Q-22.266 45.304-22.041 45.304L-22.041 45.584M-17.198 44.101Q-17.198 43.759-17.063 43.460Q-16.928 43.161-16.688 42.937Q-16.449 42.713-16.131 42.588Q-15.813 42.463-15.482 42.463Q-15.037 42.463-14.638 42.679Q-14.238 42.894-14.004 43.272Q-13.769 43.649-13.769 44.101Q-13.769 44.442-13.911 44.726Q-14.053 45.010-14.297 45.217Q-14.542 45.423-14.851 45.538Q-15.161 45.652-15.482 45.652Q-15.912 45.652-16.314 45.451Q-16.716 45.249-16.957 44.897Q-17.198 44.545-17.198 44.101M-15.482 45.403Q-14.880 45.403-14.656 45.025Q-14.433 44.647-14.433 44.015Q-14.433 43.403-14.667 43.044Q-14.901 42.686-15.482 42.686Q-16.535 42.686-16.535 44.015Q-16.535 44.647-16.309 45.025Q-16.083 45.403-15.482 45.403M-13.175 45.577L-13.175 44.514Q-13.175 44.490-13.147 44.463Q-13.120 44.436-13.096 44.436L-12.987 44.436Q-12.922 44.436-12.908 44.494Q-12.812 44.928-12.566 45.179Q-12.320 45.430-11.907 45.430Q-11.565 45.430-11.312 45.297Q-11.059 45.164-11.059 44.856Q-11.059 44.699-11.153 44.584Q-11.247 44.470-11.385 44.401Q-11.524 44.333-11.691 44.295L-12.272 44.196Q-12.628 44.128-12.901 43.907Q-13.175 43.687-13.175 43.345Q-13.175 43.096-13.064 42.921Q-12.953 42.747-12.766 42.648Q-12.580 42.549-12.365 42.506Q-12.149 42.463-11.907 42.463Q-11.493 42.463-11.213 42.645L-10.997 42.470Q-10.987 42.467-10.980 42.465Q-10.974 42.463-10.963 42.463L-10.912 42.463Q-10.885 42.463-10.861 42.487Q-10.837 42.511-10.837 42.539L-10.837 43.386Q-10.837 43.407-10.861 43.434Q-10.885 43.461-10.912 43.461L-11.025 43.461Q-11.052 43.461-11.078 43.436Q-11.103 43.410-11.103 43.386Q-11.103 43.150-11.209 42.986Q-11.315 42.822-11.498 42.740Q-11.681 42.658-11.913 42.658Q-12.242 42.658-12.498 42.761Q-12.754 42.863-12.754 43.140Q-12.754 43.335-12.571 43.444Q-12.389 43.554-12.160 43.595L-11.585 43.701Q-11.339 43.749-11.126 43.877Q-10.912 44.005-10.775 44.208Q-10.639 44.412-10.639 44.661Q-10.639 45.174-11.004 45.413Q-11.370 45.652-11.907 45.652Q-12.402 45.652-12.734 45.358L-13 45.632Q-13.021 45.652-13.048 45.652L-13.096 45.652Q-13.120 45.652-13.147 45.625Q-13.175 45.598-13.175 45.577M-9.483 44.743L-9.483 42.846L-10.122 42.846L-10.122 42.624Q-9.805 42.624-9.588 42.414Q-9.370 42.204-9.270 41.894Q-9.169 41.585-9.169 41.277L-8.902 41.277L-8.902 42.566L-7.826 42.566L-7.826 42.846L-8.902 42.846L-8.902 44.730Q-8.902 45.006-8.798 45.205Q-8.694 45.403-8.434 45.403Q-8.277 45.403-8.171 45.299Q-8.065 45.194-8.015 45.041Q-7.966 44.887-7.966 44.730L-7.966 44.316L-7.699 44.316L-7.699 44.743Q-7.699 44.969-7.798 45.179Q-7.897 45.389-8.082 45.521Q-8.266 45.652-8.495 45.652Q-8.933 45.652-9.208 45.415Q-9.483 45.177-9.483 44.743\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(114.498 -13.496)\">\u003Cpath d=\"M-2.498 45.584L-4.132 45.584L-4.132 45.304Q-3.903 45.304-3.754 45.270Q-3.605 45.235-3.605 45.095L-3.605 43.246Q-3.605 42.976-3.713 42.915Q-3.821 42.853-4.132 42.853L-4.132 42.573L-3.072 42.498L-3.072 43.147Q-2.901 42.839-2.597 42.668Q-2.293 42.498-1.948 42.498Q-1.442 42.498-1.158 42.721Q-0.874 42.945-0.874 43.441L-0.874 45.095Q-0.874 45.232-0.726 45.268Q-0.577 45.304-0.351 45.304L-0.351 45.584L-1.982 45.584L-1.982 45.304Q-1.753 45.304-1.604 45.270Q-1.455 45.235-1.455 45.095L-1.455 43.455Q-1.455 43.120-1.575 42.920Q-1.695 42.720-2.009 42.720Q-2.279 42.720-2.513 42.856Q-2.747 42.993-2.886 43.227Q-3.024 43.461-3.024 43.735L-3.024 45.095Q-3.024 45.232-2.874 45.268Q-2.723 45.304-2.498 45.304L-2.498 45.584M0.195 44.049Q0.195 43.728 0.320 43.439Q0.445 43.150 0.671 42.927Q0.896 42.703 1.192 42.583Q1.487 42.463 1.805 42.463Q2.133 42.463 2.395 42.563Q2.656 42.662 2.832 42.844Q3.008 43.027 3.102 43.285Q3.196 43.543 3.196 43.875Q3.196 43.967 3.114 43.988L0.859 43.988L0.859 44.049Q0.859 44.637 1.142 45.020Q1.426 45.403 1.993 45.403Q2.315 45.403 2.583 45.210Q2.851 45.017 2.940 44.702Q2.947 44.661 3.022 44.647L3.114 44.647Q3.196 44.671 3.196 44.743Q3.196 44.750 3.190 44.777Q3.077 45.174 2.706 45.413Q2.335 45.652 1.911 45.652Q1.474 45.652 1.074 45.444Q0.674 45.235 0.435 44.868Q0.195 44.501 0.195 44.049M0.865 43.779L2.680 43.779Q2.680 43.502 2.583 43.250Q2.485 42.997 2.287 42.841Q2.089 42.686 1.805 42.686Q1.528 42.686 1.315 42.844Q1.101 43.003 0.983 43.258Q0.865 43.513 0.865 43.779M3.842 44.856Q3.842 44.524 4.066 44.297Q4.290 44.070 4.634 43.942Q4.977 43.813 5.350 43.761Q5.722 43.708 6.027 43.708L6.027 43.455Q6.027 43.250 5.919 43.070Q5.811 42.891 5.630 42.788Q5.449 42.686 5.240 42.686Q4.834 42.686 4.598 42.778Q4.687 42.815 4.733 42.899Q4.779 42.983 4.779 43.085Q4.779 43.181 4.733 43.260Q4.687 43.338 4.606 43.383Q4.526 43.427 4.437 43.427Q4.287 43.427 4.186 43.330Q4.085 43.232 4.085 43.085Q4.085 42.463 5.240 42.463Q5.452 42.463 5.702 42.527Q5.951 42.590 6.153 42.709Q6.355 42.829 6.481 43.014Q6.608 43.198 6.608 43.441L6.608 45.017Q6.608 45.133 6.669 45.229Q6.731 45.324 6.843 45.324Q6.953 45.324 7.018 45.230Q7.083 45.136 7.083 45.017L7.083 44.569L7.349 44.569L7.349 45.017Q7.349 45.287 7.122 45.452Q6.895 45.618 6.614 45.618Q6.406 45.618 6.269 45.464Q6.132 45.311 6.109 45.095Q5.962 45.362 5.680 45.507Q5.398 45.652 5.073 45.652Q4.796 45.652 4.512 45.577Q4.229 45.502 4.036 45.323Q3.842 45.143 3.842 44.856M4.458 44.856Q4.458 45.030 4.558 45.160Q4.659 45.290 4.815 45.360Q4.970 45.430 5.134 45.430Q5.353 45.430 5.562 45.333Q5.770 45.235 5.898 45.054Q6.027 44.873 6.027 44.647L6.027 43.919Q5.702 43.919 5.336 44.010Q4.970 44.101 4.714 44.313Q4.458 44.524 4.458 44.856M9.516 45.584L7.780 45.584L7.780 45.304Q8.009 45.304 8.158 45.270Q8.306 45.235 8.306 45.095L8.306 43.246Q8.306 42.976 8.199 42.915Q8.091 42.853 7.780 42.853L7.780 42.573L8.809 42.498L8.809 43.205Q8.939 42.897 9.181 42.698Q9.424 42.498 9.742 42.498Q9.961 42.498 10.131 42.622Q10.302 42.747 10.302 42.959Q10.302 43.096 10.203 43.195Q10.104 43.294 9.971 43.294Q9.834 43.294 9.735 43.195Q9.636 43.096 9.636 42.959Q9.636 42.819 9.735 42.720Q9.444 42.720 9.245 42.916Q9.045 43.113 8.952 43.407Q8.860 43.701 8.860 43.981L8.860 45.095Q8.860 45.304 9.516 45.304\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(114.498 -13.496)\">\u003Cpath d=\"M16.329 45.584L13.657 45.584Q13.612 45.584 13.585 45.557Q13.557 45.529 13.557 45.485L13.557 45.417Q13.557 45.376 13.585 45.345L15.694 42.792L15.055 42.792Q14.631 42.792 14.398 42.858Q14.166 42.925 14.046 43.135Q13.927 43.345 13.927 43.766L13.663 43.766L13.745 42.566L16.336 42.566Q16.377 42.566 16.406 42.593Q16.435 42.621 16.435 42.665L16.435 42.713Q16.435 42.757 16.411 42.785L14.306 45.331L14.986 45.331Q15.314 45.331 15.531 45.295Q15.748 45.259 15.916 45.109Q16.056 44.972 16.109 44.748Q16.162 44.524 16.189 44.196L16.456 44.196L16.329 45.584M17.105 44.049Q17.105 43.728 17.230 43.439Q17.355 43.150 17.580 42.927Q17.806 42.703 18.102 42.583Q18.397 42.463 18.715 42.463Q19.043 42.463 19.305 42.563Q19.566 42.662 19.742 42.844Q19.918 43.027 20.012 43.285Q20.106 43.543 20.106 43.875Q20.106 43.967 20.024 43.988L17.768 43.988L17.768 44.049Q17.768 44.637 18.052 45.020Q18.336 45.403 18.903 45.403Q19.224 45.403 19.493 45.210Q19.761 45.017 19.850 44.702Q19.857 44.661 19.932 44.647L20.024 44.647Q20.106 44.671 20.106 44.743Q20.106 44.750 20.099 44.777Q19.987 45.174 19.616 45.413Q19.245 45.652 18.821 45.652Q18.384 45.652 17.984 45.444Q17.584 45.235 17.345 44.868Q17.105 44.501 17.105 44.049M17.775 43.779L19.590 43.779Q19.590 43.502 19.493 43.250Q19.395 42.997 19.197 42.841Q18.999 42.686 18.715 42.686Q18.438 42.686 18.225 42.844Q18.011 43.003 17.893 43.258Q17.775 43.513 17.775 43.779M22.444 45.584L20.708 45.584L20.708 45.304Q20.937 45.304 21.086 45.270Q21.234 45.235 21.234 45.095L21.234 43.246Q21.234 42.976 21.127 42.915Q21.019 42.853 20.708 42.853L20.708 42.573L21.737 42.498L21.737 43.205Q21.867 42.897 22.109 42.698Q22.352 42.498 22.670 42.498Q22.888 42.498 23.059 42.622Q23.230 42.747 23.230 42.959Q23.230 43.096 23.131 43.195Q23.032 43.294 22.899 43.294Q22.762 43.294 22.663 43.195Q22.564 43.096 22.564 42.959Q22.564 42.819 22.663 42.720Q22.372 42.720 22.172 42.916Q21.972 43.113 21.880 43.407Q21.788 43.701 21.788 43.981L21.788 45.095Q21.788 45.304 22.444 45.304L22.444 45.584M23.774 44.101Q23.774 43.759 23.909 43.460Q24.044 43.161 24.283 42.937Q24.522 42.713 24.840 42.588Q25.158 42.463 25.490 42.463Q25.934 42.463 26.334 42.679Q26.734 42.894 26.968 43.272Q27.202 43.649 27.202 44.101Q27.202 44.442 27.060 44.726Q26.918 45.010 26.674 45.217Q26.430 45.423 26.120 45.538Q25.811 45.652 25.490 45.652Q25.059 45.652 24.657 45.451Q24.256 45.249 24.015 44.897Q23.774 44.545 23.774 44.101M25.490 45.403Q26.091 45.403 26.315 45.025Q26.539 44.647 26.539 44.015Q26.539 43.403 26.305 43.044Q26.071 42.686 25.490 42.686Q24.437 42.686 24.437 44.015Q24.437 44.647 24.662 45.025Q24.888 45.403 25.490 45.403\" 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\">Importance-sampling weights for off-policy evaluation. Each logged trajectory is reweighted by the product of per-step ratios pi over pi-beta. Most weights are near zero (the target policy would rarely take those actions) while a few are enormous, so a handful of trajectories dominate the estimate and its variance explodes with the horizon.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:398.217px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 298.663 116.677\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-65.403 26.176h265.455\"\u002F>\u003Cpath stroke=\"none\" d=\"m202.052 26.176-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-59.377 25.344Q-59.377 24.860-58.975 24.565Q-58.572 24.270-58.022 24.151Q-57.471 24.031-56.979 24.031L-56.979 23.742Q-56.979 23.516-57.094 23.309Q-57.209 23.102-57.406 22.983Q-57.604 22.863-57.834 22.863Q-58.260 22.863-58.545 22.969Q-58.475 22.996-58.428 23.051Q-58.381 23.106-58.356 23.176Q-58.330 23.246-58.330 23.321Q-58.330 23.426-58.381 23.518Q-58.432 23.610-58.524 23.660Q-58.615 23.711-58.721 23.711Q-58.826 23.711-58.918 23.660Q-59.010 23.610-59.061 23.518Q-59.111 23.426-59.111 23.321Q-59.111 22.903-58.723 22.756Q-58.334 22.610-57.834 22.610Q-57.502 22.610-57.149 22.740Q-56.795 22.871-56.567 23.125Q-56.338 23.379-56.338 23.727L-56.338 25.528Q-56.338 25.660-56.266 25.770Q-56.193 25.879-56.065 25.879Q-55.940 25.879-55.871 25.774Q-55.803 25.668-55.803 25.528L-55.803 25.016L-55.522 25.016L-55.522 25.528Q-55.522 25.731-55.639 25.889Q-55.756 26.047-55.938 26.131Q-56.119 26.215-56.322 26.215Q-56.553 26.215-56.705 26.043Q-56.858 25.871-56.889 25.641Q-57.049 25.922-57.358 26.088Q-57.666 26.254-58.018 26.254Q-58.529 26.254-58.953 26.031Q-59.377 25.809-59.377 25.344M-58.690 25.344Q-58.690 25.629-58.463 25.815Q-58.236 26-57.943 26Q-57.697 26-57.473 25.883Q-57.248 25.766-57.113 25.563Q-56.979 25.360-56.979 25.106L-56.979 24.274Q-57.244 24.274-57.529 24.328Q-57.815 24.383-58.086 24.512Q-58.358 24.641-58.524 24.848Q-58.690 25.055-58.690 25.344M-55.186 24.449Q-55.186 23.953-54.936 23.528Q-54.686 23.102-54.266 22.856Q-53.846 22.610-53.346 22.610Q-52.807 22.610-52.416 22.735Q-52.026 22.860-52.026 23.274Q-52.026 23.379-52.076 23.471Q-52.127 23.563-52.219 23.613Q-52.311 23.664-52.420 23.664Q-52.526 23.664-52.617 23.613Q-52.709 23.563-52.760 23.471Q-52.811 23.379-52.811 23.274Q-52.811 23.051-52.643 22.946Q-52.865 22.887-53.338 22.887Q-53.635 22.887-53.850 23.026Q-54.065 23.164-54.195 23.395Q-54.326 23.625-54.385 23.895Q-54.443 24.164-54.443 24.449Q-54.443 24.844-54.311 25.194Q-54.178 25.543-53.906 25.760Q-53.635 25.977-53.236 25.977Q-52.861 25.977-52.586 25.760Q-52.311 25.543-52.209 25.184Q-52.193 25.121-52.131 25.121L-52.026 25.121Q-51.990 25.121-51.965 25.149Q-51.940 25.176-51.940 25.215L-51.940 25.238Q-52.072 25.719-52.457 25.987Q-52.842 26.254-53.346 26.254Q-53.709 26.254-54.043 26.117Q-54.377 25.981-54.637 25.731Q-54.897 25.481-55.041 25.145Q-55.186 24.809-55.186 24.449M-50.826 25.215L-50.826 23.024L-51.529 23.024L-51.529 22.770Q-51.174 22.770-50.932 22.537Q-50.690 22.305-50.578 21.957Q-50.467 21.610-50.467 21.254L-50.186 21.254L-50.186 22.727L-49.010 22.727L-49.010 23.024L-50.186 23.024L-50.186 25.199Q-50.186 25.520-50.067 25.748Q-49.947 25.977-49.666 25.977Q-49.486 25.977-49.369 25.854Q-49.252 25.731-49.199 25.551Q-49.147 25.371-49.147 25.199L-49.147 24.727L-48.865 24.727L-48.865 25.215Q-48.865 25.469-48.971 25.709Q-49.076 25.949-49.274 26.102Q-49.471 26.254-49.729 26.254Q-50.045 26.254-50.297 26.131Q-50.549 26.008-50.688 25.774Q-50.826 25.539-50.826 25.215M-46.287 26.176L-48.065 26.176L-48.065 25.879Q-47.791 25.879-47.623 25.832Q-47.455 25.785-47.455 25.617L-47.455 23.481Q-47.455 23.266-47.512 23.170Q-47.568 23.074-47.682 23.053Q-47.795 23.031-48.041 23.031L-48.041 22.735L-46.842 22.649L-46.842 25.617Q-46.842 25.785-46.695 25.832Q-46.549 25.879-46.287 25.879L-46.287 26.176M-47.729 21.254Q-47.729 21.063-47.594 20.932Q-47.459 20.801-47.264 20.801Q-47.143 20.801-47.039 20.863Q-46.936 20.926-46.873 21.030Q-46.811 21.133-46.811 21.254Q-46.811 21.449-46.942 21.584Q-47.072 21.719-47.264 21.719Q-47.463 21.719-47.596 21.586Q-47.729 21.453-47.729 21.254M-45.787 24.481Q-45.787 23.977-45.531 23.545Q-45.276 23.113-44.840 22.862Q-44.404 22.610-43.904 22.610Q-43.518 22.610-43.176 22.754Q-42.834 22.899-42.572 23.160Q-42.311 23.422-42.168 23.758Q-42.026 24.094-42.026 24.481Q-42.026 24.973-42.289 25.383Q-42.553 25.793-42.983 26.024Q-43.412 26.254-43.904 26.254Q-44.397 26.254-44.830 26.022Q-45.264 25.789-45.526 25.381Q-45.787 24.973-45.787 24.481M-43.904 25.977Q-43.447 25.977-43.195 25.754Q-42.943 25.531-42.856 25.180Q-42.768 24.828-42.768 24.383Q-42.768 23.953-42.861 23.615Q-42.955 23.278-43.209 23.071Q-43.463 22.863-43.904 22.863Q-44.553 22.863-44.797 23.280Q-45.041 23.696-45.041 24.383Q-45.041 24.828-44.953 25.180Q-44.865 25.531-44.613 25.754Q-44.361 25.977-43.904 25.977M-39.611 26.176L-41.467 26.176L-41.467 25.879Q-41.193 25.879-41.026 25.832Q-40.858 25.785-40.858 25.617L-40.858 23.481Q-40.858 23.266-40.920 23.170Q-40.983 23.074-41.102 23.053Q-41.221 23.031-41.467 23.031L-41.467 22.735L-40.276 22.649L-40.276 23.383Q-40.162 23.168-39.969 23Q-39.776 22.832-39.537 22.740Q-39.299 22.649-39.045 22.649Q-37.877 22.649-37.877 23.727L-37.877 25.617Q-37.877 25.785-37.707 25.832Q-37.537 25.879-37.268 25.879L-37.268 26.176L-39.123 26.176L-39.123 25.879Q-38.850 25.879-38.682 25.832Q-38.514 25.785-38.514 25.617L-38.514 23.742Q-38.514 23.360-38.635 23.131Q-38.756 22.903-39.108 22.903Q-39.420 22.903-39.674 23.065Q-39.928 23.227-40.074 23.496Q-40.221 23.766-40.221 24.063L-40.221 25.617Q-40.221 25.785-40.051 25.832Q-39.881 25.879-39.611 25.879\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-32.709 26.254Q-33.065 26.254-33.334 26.074Q-33.604 25.895-33.744 25.600Q-33.885 25.305-33.885 24.946Q-33.885 24.559-33.723 24.145Q-33.561 23.731-33.277 23.393Q-32.994 23.055-32.625 22.852Q-32.256 22.649-31.854 22.649Q-31.361 22.649-31.084 23.098Q-31.053 22.965-30.949 22.883Q-30.846 22.801-30.717 22.801Q-30.604 22.801-30.524 22.871Q-30.443 22.942-30.443 23.055Q-30.443 23.082-30.459 23.145L-31.014 25.344Q-31.053 25.590-31.053 25.641Q-31.053 26-30.807 26Q-30.662 26-30.561 25.893Q-30.459 25.785-30.395 25.631Q-30.330 25.477-30.281 25.287Q-30.233 25.098-30.213 25Q-30.186 24.930-30.123 24.930L-30.022 24.930Q-29.983 24.930-29.957 24.963Q-29.932 24.996-29.932 25.024Q-29.932 25.039-29.940 25.055Q-30.053 25.547-30.252 25.901Q-30.451 26.254-30.822 26.254Q-31.104 26.254-31.330 26.112Q-31.557 25.969-31.639 25.711Q-31.861 25.953-32.135 26.104Q-32.408 26.254-32.709 26.254M-32.693 26Q-32.483 26-32.274 25.887Q-32.065 25.774-31.895 25.600Q-31.725 25.426-31.596 25.231Q-31.608 25.246-31.617 25.260Q-31.627 25.274-31.639 25.289L-31.205 23.567Q-31.244 23.387-31.330 23.237Q-31.416 23.086-31.553 22.994Q-31.690 22.903-31.869 22.903Q-32.205 22.903-32.477 23.184Q-32.748 23.465-32.908 23.840Q-33.033 24.160-33.131 24.580Q-33.229 25-33.229 25.281Q-33.229 25.571-33.096 25.785Q-32.963 26-32.693 26\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"none\" d=\"M2.883 26.176v-85.358h91.049v85.358Zm91.049-85.358\"\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(85.582 -90.252)\">\u003Cpath d=\"M-59.399 24.665Q-59.399 24.327-59.258 24.036Q-59.118 23.746-58.874 23.532Q-58.630 23.319-58.325 23.204Q-58.021 23.090-57.696 23.090Q-57.426 23.090-57.163 23.189Q-56.900 23.288-56.709 23.466L-56.709 22.068Q-56.709 21.798-56.816 21.736Q-56.924 21.675-57.235 21.675L-57.235 21.394L-56.158 21.319L-56.158 25.503Q-56.158 25.691-56.104 25.774Q-56.049 25.858-55.948 25.877Q-55.847 25.896-55.632 25.896L-55.632 26.176L-56.739 26.244L-56.739 25.827Q-57.156 26.244-57.782 26.244Q-58.213 26.244-58.585 26.032Q-58.958 25.821-59.178 25.460Q-59.399 25.099-59.399 24.665M-57.724 26.022Q-57.515 26.022-57.329 25.950Q-57.143 25.879-56.989 25.742Q-56.835 25.605-56.739 25.427L-56.739 23.818Q-56.825 23.671-56.970 23.551Q-57.115 23.431-57.285 23.372Q-57.454 23.312-57.635 23.312Q-58.195 23.312-58.464 23.701Q-58.732 24.091-58.732 24.672Q-58.732 25.243-58.498 25.633Q-58.264 26.022-57.724 26.022M-54.924 25.448Q-54.924 25.116-54.701 24.889Q-54.477 24.662-54.133 24.534Q-53.790 24.405-53.417 24.353Q-53.045 24.300-52.740 24.300L-52.740 24.047Q-52.740 23.842-52.848 23.662Q-52.956 23.483-53.137 23.380Q-53.318 23.278-53.526 23.278Q-53.933 23.278-54.169 23.370Q-54.080 23.407-54.034 23.491Q-53.988 23.575-53.988 23.677Q-53.988 23.773-54.034 23.852Q-54.080 23.930-54.161 23.975Q-54.241 24.019-54.330 24.019Q-54.480 24.019-54.581 23.922Q-54.682 23.824-54.682 23.677Q-54.682 23.055-53.526 23.055Q-53.315 23.055-53.065 23.119Q-52.816 23.182-52.614 23.301Q-52.412 23.421-52.286 23.606Q-52.159 23.790-52.159 24.033L-52.159 25.609Q-52.159 25.725-52.098 25.821Q-52.036 25.916-51.923 25.916Q-51.814 25.916-51.749 25.822Q-51.684 25.728-51.684 25.609L-51.684 25.161L-51.418 25.161L-51.418 25.609Q-51.418 25.879-51.645 26.044Q-51.872 26.210-52.152 26.210Q-52.361 26.210-52.498 26.056Q-52.634 25.903-52.658 25.687Q-52.805 25.954-53.087 26.099Q-53.369 26.244-53.694 26.244Q-53.971 26.244-54.255 26.169Q-54.538 26.094-54.731 25.915Q-54.924 25.735-54.924 25.448M-54.309 25.448Q-54.309 25.622-54.208 25.752Q-54.108 25.882-53.952 25.952Q-53.796 26.022-53.632 26.022Q-53.414 26.022-53.205 25.925Q-52.997 25.827-52.869 25.646Q-52.740 25.465-52.740 25.239L-52.740 24.511Q-53.065 24.511-53.431 24.602Q-53.796 24.693-54.053 24.905Q-54.309 25.116-54.309 25.448M-50.474 25.335L-50.474 23.438L-51.113 23.438L-51.113 23.216Q-50.796 23.216-50.578 23.006Q-50.361 22.796-50.261 22.486Q-50.160 22.177-50.160 21.869L-49.893 21.869L-49.893 23.158L-48.817 23.158L-48.817 23.438L-49.893 23.438L-49.893 25.322Q-49.893 25.598-49.789 25.797Q-49.685 25.995-49.425 25.995Q-49.268 25.995-49.162 25.891Q-49.056 25.786-49.006 25.633Q-48.957 25.479-48.957 25.322L-48.957 24.908L-48.690 24.908L-48.690 25.335Q-48.690 25.561-48.789 25.771Q-48.888 25.981-49.073 26.113Q-49.257 26.244-49.486 26.244Q-49.924 26.244-50.199 26.007Q-50.474 25.769-50.474 25.335M-47.822 25.448Q-47.822 25.116-47.598 24.889Q-47.374 24.662-47.031 24.534Q-46.687 24.405-46.315 24.353Q-45.942 24.300-45.638 24.300L-45.638 24.047Q-45.638 23.842-45.745 23.662Q-45.853 23.483-46.034 23.380Q-46.215 23.278-46.424 23.278Q-46.831 23.278-47.067 23.370Q-46.978 23.407-46.932 23.491Q-46.885 23.575-46.885 23.677Q-46.885 23.773-46.932 23.852Q-46.978 23.930-47.058 23.975Q-47.138 24.019-47.227 24.019Q-47.378 24.019-47.478 23.922Q-47.579 23.824-47.579 23.677Q-47.579 23.055-46.424 23.055Q-46.212 23.055-45.963 23.119Q-45.713 23.182-45.511 23.301Q-45.310 23.421-45.183 23.606Q-45.057 23.790-45.057 24.033L-45.057 25.609Q-45.057 25.725-44.995 25.821Q-44.934 25.916-44.821 25.916Q-44.712 25.916-44.647 25.822Q-44.582 25.728-44.582 25.609L-44.582 25.161L-44.315 25.161L-44.315 25.609Q-44.315 25.879-44.542 26.044Q-44.770 26.210-45.050 26.210Q-45.258 26.210-45.395 26.056Q-45.532 25.903-45.556 25.687Q-45.703 25.954-45.985 26.099Q-46.267 26.244-46.591 26.244Q-46.868 26.244-47.152 26.169Q-47.436 26.094-47.629 25.915Q-47.822 25.735-47.822 25.448M-47.207 25.448Q-47.207 25.622-47.106 25.752Q-47.005 25.882-46.849 25.952Q-46.694 26.022-46.530 26.022Q-46.311 26.022-46.103 25.925Q-45.894 25.827-45.766 25.646Q-45.638 25.465-45.638 25.239L-45.638 24.511Q-45.963 24.511-46.328 24.602Q-46.694 24.693-46.950 24.905Q-47.207 25.116-47.207 25.448\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(85.582 -90.252)\">\u003Cpath d=\"M-41.190 26.169L-41.190 25.106Q-41.190 25.082-41.162 25.055Q-41.135 25.028-41.111 25.028L-41.002 25.028Q-40.937 25.028-40.923 25.086Q-40.827 25.520-40.581 25.771Q-40.335 26.022-39.921 26.022Q-39.580 26.022-39.327 25.889Q-39.074 25.756-39.074 25.448Q-39.074 25.291-39.168 25.176Q-39.262 25.062-39.400 24.993Q-39.539 24.925-39.706 24.887L-40.287 24.788Q-40.643 24.720-40.916 24.499Q-41.190 24.279-41.190 23.937Q-41.190 23.688-41.078 23.513Q-40.967 23.339-40.781 23.240Q-40.595 23.141-40.379 23.098Q-40.164 23.055-39.921 23.055Q-39.508 23.055-39.228 23.237L-39.012 23.062Q-39.002 23.059-38.995 23.057Q-38.988 23.055-38.978 23.055L-38.927 23.055Q-38.900 23.055-38.876 23.079Q-38.852 23.103-38.852 23.131L-38.852 23.978Q-38.852 23.999-38.876 24.026Q-38.900 24.053-38.927 24.053L-39.040 24.053Q-39.067 24.053-39.093 24.028Q-39.118 24.002-39.118 23.978Q-39.118 23.742-39.224 23.578Q-39.330 23.414-39.513 23.332Q-39.696 23.250-39.928 23.250Q-40.256 23.250-40.513 23.353Q-40.769 23.455-40.769 23.732Q-40.769 23.927-40.586 24.036Q-40.403 24.146-40.174 24.187L-39.600 24.293Q-39.354 24.341-39.140 24.469Q-38.927 24.597-38.790 24.800Q-38.653 25.004-38.653 25.253Q-38.653 25.766-39.019 26.005Q-39.385 26.244-39.921 26.244Q-40.417 26.244-40.749 25.950L-41.015 26.224Q-41.036 26.244-41.063 26.244L-41.111 26.244Q-41.135 26.244-41.162 26.217Q-41.190 26.190-41.190 26.169M-37.450 25.342L-37.450 23.838Q-37.450 23.568-37.558 23.507Q-37.666 23.445-37.977 23.445L-37.977 23.165L-36.869 23.090L-36.869 25.322L-36.869 25.342Q-36.869 25.622-36.818 25.766Q-36.767 25.909-36.625 25.966Q-36.483 26.022-36.196 26.022Q-35.943 26.022-35.738 25.882Q-35.533 25.742-35.417 25.516Q-35.300 25.291-35.300 25.041L-35.300 23.838Q-35.300 23.568-35.408 23.507Q-35.516 23.445-35.827 23.445L-35.827 23.165L-34.719 23.090L-34.719 25.503Q-34.719 25.694-34.666 25.776Q-34.613 25.858-34.513 25.877Q-34.412 25.896-34.196 25.896L-34.196 26.176L-35.273 26.244L-35.273 25.680Q-35.382 25.862-35.528 25.985Q-35.673 26.108-35.859 26.176Q-36.046 26.244-36.247 26.244Q-37.450 26.244-37.450 25.342M-31.964 27.533L-33.595 27.533L-33.595 27.253Q-33.366 27.253-33.217 27.218Q-33.068 27.184-33.068 27.044L-33.068 23.698Q-33.068 23.527-33.205 23.486Q-33.342 23.445-33.595 23.445L-33.595 23.165L-32.515 23.090L-32.515 23.496Q-32.293 23.295-32.005 23.192Q-31.718 23.090-31.411 23.090Q-30.983 23.090-30.619 23.303Q-30.255 23.517-30.042 23.881Q-29.828 24.245-29.828 24.665Q-29.828 25.110-30.067 25.474Q-30.307 25.838-30.700 26.041Q-31.093 26.244-31.537 26.244Q-31.804 26.244-32.052 26.144Q-32.299 26.043-32.487 25.862L-32.487 27.044Q-32.487 27.181-32.339 27.217Q-32.190 27.253-31.964 27.253L-31.964 27.533M-32.487 23.845L-32.487 25.455Q-32.354 25.708-32.111 25.865Q-31.869 26.022-31.592 26.022Q-31.264 26.022-31.011 25.821Q-30.758 25.619-30.625 25.301Q-30.491 24.983-30.491 24.665Q-30.491 24.436-30.556 24.207Q-30.621 23.978-30.749 23.780Q-30.878 23.582-31.072 23.462Q-31.267 23.343-31.500 23.343Q-31.794 23.343-32.062 23.472Q-32.330 23.602-32.487 23.845M-27.548 27.533L-29.179 27.533L-29.179 27.253Q-28.950 27.253-28.801 27.218Q-28.652 27.184-28.652 27.044L-28.652 23.698Q-28.652 23.527-28.789 23.486Q-28.926 23.445-29.179 23.445L-29.179 23.165L-28.099 23.090L-28.099 23.496Q-27.877 23.295-27.589 23.192Q-27.302 23.090-26.995 23.090Q-26.567 23.090-26.203 23.303Q-25.839 23.517-25.626 23.881Q-25.412 24.245-25.412 24.665Q-25.412 25.110-25.651 25.474Q-25.891 25.838-26.284 26.041Q-26.677 26.244-27.121 26.244Q-27.388 26.244-27.636 26.144Q-27.883 26.043-28.071 25.862L-28.071 27.044Q-28.071 27.181-27.923 27.217Q-27.774 27.253-27.548 27.253L-27.548 27.533M-28.071 23.845L-28.071 25.455Q-27.938 25.708-27.695 25.865Q-27.453 26.022-27.176 26.022Q-26.848 26.022-26.595 25.821Q-26.342 25.619-26.209 25.301Q-26.075 24.983-26.075 24.665Q-26.075 24.436-26.140 24.207Q-26.205 23.978-26.333 23.780Q-26.462 23.582-26.656 23.462Q-26.851 23.343-27.084 23.343Q-27.378 23.343-27.646 23.472Q-27.914 23.602-28.071 23.845\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(85.582 -90.252)\">\u003Cpath d=\"M-24.598 24.693Q-24.598 24.351-24.463 24.052Q-24.328 23.753-24.088 23.529Q-23.849 23.305-23.531 23.180Q-23.213 23.055-22.882 23.055Q-22.437 23.055-22.038 23.271Q-21.638 23.486-21.403 23.864Q-21.169 24.241-21.169 24.693Q-21.169 25.034-21.311 25.318Q-21.453 25.602-21.697 25.809Q-21.942 26.015-22.251 26.130Q-22.560 26.244-22.882 26.244Q-23.312 26.244-23.714 26.043Q-24.116 25.841-24.357 25.489Q-24.598 25.137-24.598 24.693M-22.882 25.995Q-22.280 25.995-22.056 25.617Q-21.832 25.239-21.832 24.607Q-21.832 23.995-22.067 23.636Q-22.301 23.278-22.882 23.278Q-23.934 23.278-23.934 24.607Q-23.934 25.239-23.709 25.617Q-23.483 25.995-22.882 25.995M-18.825 26.176L-20.561 26.176L-20.561 25.896Q-20.332 25.896-20.183 25.862Q-20.035 25.827-20.035 25.687L-20.035 23.838Q-20.035 23.568-20.142 23.507Q-20.250 23.445-20.561 23.445L-20.561 23.165L-19.532 23.090L-19.532 23.797Q-19.402 23.489-19.160 23.290Q-18.917 23.090-18.599 23.090Q-18.380 23.090-18.209 23.214Q-18.038 23.339-18.038 23.551Q-18.038 23.688-18.138 23.787Q-18.237 23.886-18.370 23.886Q-18.507 23.886-18.606 23.787Q-18.705 23.688-18.705 23.551Q-18.705 23.411-18.606 23.312Q-18.896 23.312-19.096 23.508Q-19.296 23.705-19.389 23.999Q-19.481 24.293-19.481 24.573L-19.481 25.687Q-19.481 25.896-18.825 25.896L-18.825 26.176M-16.928 25.335L-16.928 23.438L-17.567 23.438L-17.567 23.216Q-17.249 23.216-17.032 23.006Q-16.815 22.796-16.714 22.486Q-16.613 22.177-16.613 21.869L-16.347 21.869L-16.347 23.158L-15.270 23.158L-15.270 23.438L-16.347 23.438L-16.347 25.322Q-16.347 25.598-16.242 25.797Q-16.138 25.995-15.878 25.995Q-15.721 25.995-15.615 25.891Q-15.509 25.786-15.460 25.633Q-15.410 25.479-15.410 25.322L-15.410 24.908L-15.143 24.908L-15.143 25.335Q-15.143 25.561-15.243 25.771Q-15.342 25.981-15.526 26.113Q-15.711 26.244-15.940 26.244Q-16.377 26.244-16.652 26.007Q-16.928 25.769-16.928 25.335\" 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=\"var(--tk-soft-warn)\" stroke=\"none\" d=\"M122.384 26.176v-85.358h73.978v85.358Zm73.978-85.358\"\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(179.975 -90.252)\">\u003Cpath d=\"M-59.440 24.693Q-59.440 24.351-59.305 24.052Q-59.170 23.753-58.930 23.529Q-58.691 23.305-58.373 23.180Q-58.055 23.055-57.724 23.055Q-57.279 23.055-56.880 23.271Q-56.480 23.486-56.245 23.864Q-56.011 24.241-56.011 24.693Q-56.011 25.034-56.153 25.318Q-56.295 25.602-56.539 25.809Q-56.784 26.015-57.093 26.130Q-57.402 26.244-57.724 26.244Q-58.154 26.244-58.556 26.043Q-58.958 25.841-59.199 25.489Q-59.440 25.137-59.440 24.693M-57.724 25.995Q-57.122 25.995-56.898 25.617Q-56.674 25.239-56.674 24.607Q-56.674 23.995-56.909 23.636Q-57.143 23.278-57.724 23.278Q-58.776 23.278-58.776 24.607Q-58.776 25.239-58.551 25.617Q-58.325 25.995-57.724 25.995M-53.619 26.176L-55.352 26.176L-55.352 25.896Q-55.126 25.896-54.977 25.862Q-54.829 25.827-54.829 25.687L-54.829 23.438L-55.417 23.438L-55.417 23.158L-54.829 23.158L-54.829 22.341Q-54.829 22.023-54.651 21.775Q-54.473 21.528-54.183 21.387Q-53.892 21.247-53.581 21.247Q-53.325 21.247-53.121 21.389Q-52.918 21.531-52.918 21.774Q-52.918 21.910-53.017 22.009Q-53.116 22.109-53.253 22.109Q-53.390 22.109-53.489 22.009Q-53.588 21.910-53.588 21.774Q-53.588 21.593-53.448 21.500Q-53.526 21.473-53.626 21.473Q-53.834 21.473-53.988 21.606Q-54.142 21.739-54.222 21.943Q-54.302 22.146-54.302 22.355L-54.302 23.158L-53.414 23.158L-53.414 23.438L-54.275 23.438L-54.275 25.687Q-54.275 25.896-53.619 25.896\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M-50.577 26.176L-52.310 26.176L-52.310 25.896Q-52.084 25.896-51.935 25.862Q-51.787 25.827-51.787 25.687L-51.787 23.438L-52.375 23.438L-52.375 23.158L-51.787 23.158L-51.787 22.341Q-51.787 22.023-51.609 21.775Q-51.431 21.528-51.141 21.387Q-50.850 21.247-50.539 21.247Q-50.283 21.247-50.079 21.389Q-49.876 21.531-49.876 21.774Q-49.876 21.910-49.975 22.009Q-50.074 22.109-50.211 22.109Q-50.348 22.109-50.447 22.009Q-50.546 21.910-50.546 21.774Q-50.546 21.593-50.406 21.500Q-50.484 21.473-50.584 21.473Q-50.792 21.473-50.946 21.606Q-51.100 21.739-51.180 21.943Q-51.260 22.146-51.260 22.355L-51.260 23.158L-50.372 23.158L-50.372 23.438L-51.233 23.438L-51.233 25.687Q-51.233 25.896-50.577 25.896L-50.577 26.176M-48.013 24.922L-50.071 24.922L-50.071 24.419L-48.013 24.419L-48.013 24.922M-47.210 26.169L-47.210 25.106Q-47.210 25.082-47.183 25.055Q-47.155 25.028-47.131 25.028L-47.022 25.028Q-46.957 25.028-46.943 25.086Q-46.848 25.520-46.602 25.771Q-46.356 26.022-45.942 26.022Q-45.600 26.022-45.347 25.889Q-45.094 25.756-45.094 25.448Q-45.094 25.291-45.188 25.176Q-45.282 25.062-45.421 24.993Q-45.559 24.925-45.727 24.887L-46.308 24.788Q-46.663 24.720-46.937 24.499Q-47.210 24.279-47.210 23.937Q-47.210 23.688-47.099 23.513Q-46.988 23.339-46.802 23.240Q-46.615 23.141-46.400 23.098Q-46.185 23.055-45.942 23.055Q-45.528 23.055-45.248 23.237L-45.033 23.062Q-45.022 23.059-45.016 23.057Q-45.009 23.055-44.999 23.055L-44.947 23.055Q-44.920 23.055-44.896 23.079Q-44.872 23.103-44.872 23.131L-44.872 23.978Q-44.872 23.999-44.896 24.026Q-44.920 24.053-44.947 24.053L-45.060 24.053Q-45.087 24.053-45.113 24.028Q-45.139 24.002-45.139 23.978Q-45.139 23.742-45.245 23.578Q-45.351 23.414-45.533 23.332Q-45.716 23.250-45.949 23.250Q-46.277 23.250-46.533 23.353Q-46.790 23.455-46.790 23.732Q-46.790 23.927-46.607 24.036Q-46.424 24.146-46.195 24.187L-45.621 24.293Q-45.375 24.341-45.161 24.469Q-44.947 24.597-44.811 24.800Q-44.674 25.004-44.674 25.253Q-44.674 25.766-45.040 26.005Q-45.405 26.244-45.942 26.244Q-46.438 26.244-46.769 25.950L-47.036 26.224Q-47.056 26.244-47.084 26.244L-47.131 26.244Q-47.155 26.244-47.183 26.217Q-47.210 26.190-47.210 26.169M-43.471 25.342L-43.471 23.838Q-43.471 23.568-43.578 23.507Q-43.686 23.445-43.997 23.445L-43.997 23.165L-42.890 23.090L-42.890 25.322L-42.890 25.342Q-42.890 25.622-42.838 25.766Q-42.787 25.909-42.645 25.966Q-42.503 26.022-42.216 26.022Q-41.963 26.022-41.758 25.882Q-41.553 25.742-41.437 25.516Q-41.321 25.291-41.321 25.041L-41.321 23.838Q-41.321 23.568-41.429 23.507Q-41.536 23.445-41.847 23.445L-41.847 23.165L-40.740 23.090L-40.740 25.503Q-40.740 25.694-40.687 25.776Q-40.634 25.858-40.533 25.877Q-40.432 25.896-40.217 25.896L-40.217 26.176L-41.293 26.244L-41.293 25.680Q-41.403 25.862-41.548 25.985Q-41.693 26.108-41.880 26.176Q-42.066 26.244-42.268 26.244Q-43.471 26.244-43.471 25.342M-37.985 27.533L-39.615 27.533L-39.615 27.253Q-39.386 27.253-39.238 27.218Q-39.089 27.184-39.089 27.044L-39.089 23.698Q-39.089 23.527-39.226 23.486Q-39.362 23.445-39.615 23.445L-39.615 23.165L-38.535 23.090L-38.535 23.496Q-38.313 23.295-38.026 23.192Q-37.739 23.090-37.431 23.090Q-37.004 23.090-36.640 23.303Q-36.276 23.517-36.062 23.881Q-35.849 24.245-35.849 24.665Q-35.849 25.110-36.088 25.474Q-36.327 25.838-36.720 26.041Q-37.113 26.244-37.558 26.244Q-37.824 26.244-38.072 26.144Q-38.320 26.043-38.508 25.862L-38.508 27.044Q-38.508 27.181-38.359 27.217Q-38.210 27.253-37.985 27.253L-37.985 27.533M-38.508 23.845L-38.508 25.455Q-38.375 25.708-38.132 25.865Q-37.889 26.022-37.612 26.022Q-37.284 26.022-37.031 25.821Q-36.778 25.619-36.645 25.301Q-36.512 24.983-36.512 24.665Q-36.512 24.436-36.577 24.207Q-36.642 23.978-36.770 23.780Q-36.898 23.582-37.093 23.462Q-37.288 23.343-37.520 23.343Q-37.814 23.343-38.082 23.472Q-38.351 23.602-38.508 23.845M-33.569 27.533L-35.199 27.533L-35.199 27.253Q-34.970 27.253-34.822 27.218Q-34.673 27.184-34.673 27.044L-34.673 23.698Q-34.673 23.527-34.810 23.486Q-34.946 23.445-35.199 23.445L-35.199 23.165L-34.119 23.090L-34.119 23.496Q-33.897 23.295-33.610 23.192Q-33.323 23.090-33.015 23.090Q-32.588 23.090-32.224 23.303Q-31.860 23.517-31.646 23.881Q-31.433 24.245-31.433 24.665Q-31.433 25.110-31.672 25.474Q-31.911 25.838-32.304 26.041Q-32.697 26.244-33.142 26.244Q-33.408 26.244-33.656 26.144Q-33.904 26.043-34.092 25.862L-34.092 27.044Q-34.092 27.181-33.943 27.217Q-33.794 27.253-33.569 27.253L-33.569 27.533M-34.092 23.845L-34.092 25.455Q-33.959 25.708-33.716 25.865Q-33.473 26.022-33.196 26.022Q-32.868 26.022-32.615 25.821Q-32.362 25.619-32.229 25.301Q-32.096 24.983-32.096 24.665Q-32.096 24.436-32.161 24.207Q-32.226 23.978-32.354 23.780Q-32.482 23.582-32.677 23.462Q-32.872 23.343-33.104 23.343Q-33.398 23.343-33.666 23.472Q-33.935 23.602-34.092 23.845\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M-30.608 24.693Q-30.608 24.351-30.473 24.052Q-30.338 23.753-30.098 23.529Q-29.859 23.305-29.541 23.180Q-29.223 23.055-28.892 23.055Q-28.447 23.055-28.048 23.271Q-27.648 23.486-27.413 23.864Q-27.179 24.241-27.179 24.693Q-27.179 25.034-27.321 25.318Q-27.463 25.602-27.707 25.809Q-27.952 26.015-28.261 26.130Q-28.570 26.244-28.892 26.244Q-29.322 26.244-29.724 26.043Q-30.126 25.841-30.367 25.489Q-30.608 25.137-30.608 24.693M-28.892 25.995Q-28.290 25.995-28.066 25.617Q-27.842 25.239-27.842 24.607Q-27.842 23.995-28.077 23.636Q-28.311 23.278-28.892 23.278Q-29.944 23.278-29.944 24.607Q-29.944 25.239-29.719 25.617Q-29.493 25.995-28.892 25.995M-24.835 26.176L-26.571 26.176L-26.571 25.896Q-26.342 25.896-26.193 25.862Q-26.045 25.827-26.045 25.687L-26.045 23.838Q-26.045 23.568-26.152 23.507Q-26.260 23.445-26.571 23.445L-26.571 23.165L-25.542 23.090L-25.542 23.797Q-25.412 23.489-25.170 23.290Q-24.927 23.090-24.609 23.090Q-24.390 23.090-24.219 23.214Q-24.048 23.339-24.048 23.551Q-24.048 23.688-24.148 23.787Q-24.247 23.886-24.380 23.886Q-24.517 23.886-24.616 23.787Q-24.715 23.688-24.715 23.551Q-24.715 23.411-24.616 23.312Q-24.906 23.312-25.106 23.508Q-25.306 23.705-25.399 23.999Q-25.491 24.293-25.491 24.573L-25.491 25.687Q-25.491 25.896-24.835 25.896L-24.835 26.176M-22.938 25.335L-22.938 23.438L-23.577 23.438L-23.577 23.216Q-23.259 23.216-23.042 23.006Q-22.825 22.796-22.724 22.486Q-22.623 22.177-22.623 21.869L-22.357 21.869L-22.357 23.158L-21.280 23.158L-21.280 23.438L-22.357 23.438L-22.357 25.322Q-22.357 25.598-22.252 25.797Q-22.148 25.995-21.888 25.995Q-21.731 25.995-21.625 25.891Q-21.519 25.786-21.470 25.633Q-21.420 25.479-21.420 25.322L-21.420 24.908L-21.153 24.908L-21.153 25.335Q-21.153 25.561-21.253 25.771Q-21.352 25.981-21.536 26.113Q-21.721 26.244-21.950 26.244Q-22.387 26.244-22.662 26.007Q-22.938 25.769-22.938 25.335M-19.944 25.756Q-19.944 25.588-19.820 25.465Q-19.697 25.342-19.523 25.342Q-19.356 25.342-19.233 25.465Q-19.110 25.588-19.110 25.756Q-19.110 25.930-19.233 26.053Q-19.356 26.176-19.523 26.176Q-19.697 26.176-19.820 26.053Q-19.944 25.930-19.944 25.756M-19.944 23.572Q-19.944 23.404-19.820 23.281Q-19.697 23.158-19.523 23.158Q-19.356 23.158-19.233 23.281Q-19.110 23.404-19.110 23.572Q-19.110 23.746-19.233 23.869Q-19.356 23.992-19.523 23.992Q-19.697 23.992-19.820 23.869Q-19.944 23.746-19.944 23.572\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M-14.553 24.693Q-14.553 24.351-14.418 24.052Q-14.283 23.753-14.043 23.529Q-13.804 23.305-13.486 23.180Q-13.168 23.055-12.837 23.055Q-12.392 23.055-11.993 23.271Q-11.593 23.486-11.358 23.864Q-11.124 24.241-11.124 24.693Q-11.124 25.034-11.266 25.318Q-11.408 25.602-11.652 25.809Q-11.897 26.015-12.206 26.130Q-12.515 26.244-12.837 26.244Q-13.267 26.244-13.669 26.043Q-14.071 25.841-14.312 25.489Q-14.553 25.137-14.553 24.693M-12.837 25.995Q-12.235 25.995-12.011 25.617Q-11.787 25.239-11.787 24.607Q-11.787 23.995-12.022 23.636Q-12.256 23.278-12.837 23.278Q-13.889 23.278-13.889 24.607Q-13.889 25.239-13.664 25.617Q-13.438 25.995-12.837 25.995\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M-9.151 26.149L-10.279 23.650Q-10.351 23.503-10.481 23.471Q-10.611 23.438-10.840 23.438L-10.840 23.158L-9.326 23.158L-9.326 23.438Q-9.678 23.438-9.678 23.585Q-9.678 23.630-9.667 23.650L-8.803 25.568L-8.023 23.838Q-7.989 23.770-7.989 23.691Q-7.989 23.578-8.073 23.508Q-8.157 23.438-8.276 23.438L-8.276 23.158L-7.080 23.158L-7.080 23.438Q-7.299 23.438-7.470 23.541Q-7.640 23.643-7.729 23.838L-8.765 26.149Q-8.813 26.244-8.919 26.244L-8.997 26.244Q-9.103 26.244-9.151 26.149\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M-6.796 24.641Q-6.796 24.320-6.671 24.031Q-6.546 23.742-6.320 23.519Q-6.095 23.295-5.799 23.175Q-5.504 23.055-5.186 23.055Q-4.858 23.055-4.596 23.155Q-4.335 23.254-4.159 23.436Q-3.983 23.619-3.889 23.877Q-3.795 24.135-3.795 24.467Q-3.795 24.559-3.877 24.580L-6.132 24.580L-6.132 24.641Q-6.132 25.229-5.849 25.612Q-5.565 25.995-4.998 25.995Q-4.676 25.995-4.408 25.802Q-4.140 25.609-4.051 25.294Q-4.044 25.253-3.969 25.239L-3.877 25.239Q-3.795 25.263-3.795 25.335Q-3.795 25.342-3.801 25.369Q-3.914 25.766-4.285 26.005Q-4.656 26.244-5.080 26.244Q-5.517 26.244-5.917 26.036Q-6.317 25.827-6.556 25.460Q-6.796 25.093-6.796 24.641M-6.126 24.371L-4.311 24.371Q-4.311 24.094-4.408 23.842Q-4.506 23.589-4.704 23.433Q-4.902 23.278-5.186 23.278Q-5.463 23.278-5.676 23.436Q-5.890 23.595-6.008 23.850Q-6.126 24.105-6.126 24.371M-1.457 26.176L-3.193 26.176L-3.193 25.896Q-2.964 25.896-2.815 25.862Q-2.667 25.827-2.667 25.687L-2.667 23.838Q-2.667 23.568-2.774 23.507Q-2.882 23.445-3.193 23.445L-3.193 23.165L-2.164 23.090L-2.164 23.797Q-2.034 23.489-1.792 23.290Q-1.549 23.090-1.231 23.090Q-1.012 23.090-0.841 23.214Q-0.671 23.339-0.671 23.551Q-0.671 23.688-0.770 23.787Q-0.869 23.886-1.002 23.886Q-1.139 23.886-1.238 23.787Q-1.337 23.688-1.337 23.551Q-1.337 23.411-1.238 23.312Q-1.528 23.312-1.728 23.508Q-1.928 23.705-2.021 23.999Q-2.113 24.293-2.113 24.573L-2.113 25.687Q-2.113 25.896-1.457 25.896L-1.457 26.176M-0.086 26.169L-0.086 25.106Q-0.086 25.082-0.059 25.055Q-0.031 25.028-0.007 25.028L0.102 25.028Q0.167 25.028 0.181 25.086Q0.276 25.520 0.522 25.771Q0.768 26.022 1.182 26.022Q1.524 26.022 1.777 25.889Q2.030 25.756 2.030 25.448Q2.030 25.291 1.936 25.176Q1.842 25.062 1.703 24.993Q1.565 24.925 1.397 24.887L0.816 24.788Q0.461 24.720 0.187 24.499Q-0.086 24.279-0.086 23.937Q-0.086 23.688 0.025 23.513Q0.136 23.339 0.322 23.240Q0.509 23.141 0.724 23.098Q0.939 23.055 1.182 23.055Q1.596 23.055 1.876 23.237L2.091 23.062Q2.101 23.059 2.108 23.057Q2.115 23.055 2.125 23.055L2.177 23.055Q2.204 23.055 2.228 23.079Q2.252 23.103 2.252 23.131L2.252 23.978Q2.252 23.999 2.228 24.026Q2.204 24.053 2.177 24.053L2.064 24.053Q2.036 24.053 2.011 24.028Q1.985 24.002 1.985 23.978Q1.985 23.742 1.879 23.578Q1.773 23.414 1.590 23.332Q1.408 23.250 1.175 23.250Q0.847 23.250 0.591 23.353Q0.334 23.455 0.334 23.732Q0.334 23.927 0.517 24.036Q0.700 24.146 0.929 24.187L1.503 24.293Q1.749 24.341 1.963 24.469Q2.177 24.597 2.313 24.800Q2.450 25.004 2.450 25.253Q2.450 25.766 2.084 26.005Q1.719 26.244 1.182 26.244Q0.686 26.244 0.355 25.950L0.088 26.224Q0.068 26.244 0.040 26.244L-0.007 26.244Q-0.031 26.244-0.059 26.217Q-0.086 26.190-0.086 26.169M4.761 26.176L3.127 26.176L3.127 25.896Q3.356 25.896 3.504 25.862Q3.653 25.827 3.653 25.687L3.653 22.068Q3.653 21.798 3.546 21.736Q3.438 21.675 3.127 21.675L3.127 21.394L4.207 21.319L4.207 23.705Q4.313 23.520 4.491 23.378Q4.668 23.237 4.877 23.163Q5.085 23.090 5.311 23.090Q5.817 23.090 6.100 23.313Q6.384 23.537 6.384 24.033L6.384 25.687Q6.384 25.824 6.533 25.860Q6.681 25.896 6.907 25.896L6.907 26.176L5.277 26.176L5.277 25.896Q5.506 25.896 5.654 25.862Q5.803 25.827 5.803 25.687L5.803 24.047Q5.803 23.712 5.683 23.512Q5.564 23.312 5.249 23.312Q4.979 23.312 4.745 23.448Q4.511 23.585 4.373 23.819Q4.234 24.053 4.234 24.327L4.234 25.687Q4.234 25.824 4.385 25.860Q4.535 25.896 4.761 25.896L4.761 26.176M7.454 24.693Q7.454 24.351 7.589 24.052Q7.724 23.753 7.963 23.529Q8.202 23.305 8.520 23.180Q8.838 23.055 9.170 23.055Q9.614 23.055 10.014 23.271Q10.414 23.486 10.648 23.864Q10.882 24.241 10.882 24.693Q10.882 25.034 10.740 25.318Q10.598 25.602 10.354 25.809Q10.110 26.015 9.800 26.130Q9.491 26.244 9.170 26.244Q8.739 26.244 8.337 26.043Q7.936 25.841 7.695 25.489Q7.454 25.137 7.454 24.693M9.170 25.995Q9.771 25.995 9.995 25.617Q10.219 25.239 10.219 24.607Q10.219 23.995 9.985 23.636Q9.751 23.278 9.170 23.278Q8.117 23.278 8.117 24.607Q8.117 25.239 8.343 25.617Q8.568 25.995 9.170 25.995\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(179.975 -90.252)\">\u003Cpath d=\"M11.671 24.693Q11.671 24.351 11.806 24.052Q11.941 23.753 12.181 23.529Q12.420 23.305 12.738 23.180Q13.056 23.055 13.387 23.055Q13.832 23.055 14.231 23.271Q14.631 23.486 14.866 23.864Q15.100 24.241 15.100 24.693Q15.100 25.034 14.958 25.318Q14.816 25.602 14.572 25.809Q14.327 26.015 14.018 26.130Q13.709 26.244 13.387 26.244Q12.957 26.244 12.555 26.043Q12.153 25.841 11.912 25.489Q11.671 25.137 11.671 24.693M13.387 25.995Q13.989 25.995 14.213 25.617Q14.437 25.239 14.437 24.607Q14.437 23.995 14.202 23.636Q13.968 23.278 13.387 23.278Q12.335 23.278 12.335 24.607Q12.335 25.239 12.560 25.617Q12.786 25.995 13.387 25.995M16.221 25.335L16.221 23.438L15.582 23.438L15.582 23.216Q15.899 23.216 16.117 23.006Q16.334 22.796 16.434 22.486Q16.535 22.177 16.535 21.869L16.802 21.869L16.802 23.158L17.878 23.158L17.878 23.438L16.802 23.438L16.802 25.322Q16.802 25.598 16.906 25.797Q17.010 25.995 17.270 25.995Q17.427 25.995 17.533 25.891Q17.639 25.786 17.689 25.633Q17.738 25.479 17.738 25.322L17.738 24.908L18.005 24.908L18.005 25.335Q18.005 25.561 17.906 25.771Q17.807 25.981 17.622 26.113Q17.438 26.244 17.209 26.244Q16.771 26.244 16.496 26.007Q16.221 25.769 16.221 25.335\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M16.264 11.95a2 2 0 1 0-4 0 2 2 0 0 0 4 0M36.181 11.95a2 2 0 1 0-4 0 2 2 0 0 0 4 0M53.253 11.95a2 2 0 1 0-4 0 2 2 0 0 0 4 0M73.17 11.95a2 2 0 1 0-4 0 2 2 0 0 0 4 0M87.396 11.95a2 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)\">\u003Cpath fill=\"none\" d=\"M14.264 11.95h5.967\"\u002F>\u003Cpath stroke=\"none\" d=\"m22.231 11.95-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M34.181 11.95h5.967\"\u002F>\u003Cpath stroke=\"none\" d=\"m42.148 11.95-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M51.253 11.95h5.966\"\u002F>\u003Cpath stroke=\"none\" d=\"m59.22 11.95-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M71.17 11.95h5.967\"\u002F>\u003Cpath stroke=\"none\" d=\"m79.137 11.95-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M85.396 11.95h5.967\"\u002F>\u003Cpath stroke=\"none\" d=\"m93.363 11.95-3.2-1.6 1.2 1.6-1.2 1.6\"\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(54.197 -1.565)\">\u003Cpath d=\"M-59.399 24.665Q-59.399 24.337-59.264 24.036Q-59.129 23.736-58.893 23.515Q-58.657 23.295-58.353 23.175Q-58.048 23.055-57.724 23.055Q-57.218 23.055-56.869 23.158Q-56.521 23.260-56.521 23.636Q-56.521 23.783-56.618 23.884Q-56.715 23.985-56.862 23.985Q-57.016 23.985-57.115 23.886Q-57.214 23.787-57.214 23.636Q-57.214 23.448-57.074 23.356Q-57.276 23.305-57.717 23.305Q-58.072 23.305-58.301 23.501Q-58.530 23.698-58.631 24.007Q-58.732 24.317-58.732 24.665Q-58.732 25.014-58.606 25.320Q-58.479 25.626-58.224 25.810Q-57.970 25.995-57.614 25.995Q-57.392 25.995-57.208 25.911Q-57.023 25.827-56.888 25.672Q-56.753 25.516-56.695 25.308Q-56.681 25.253-56.627 25.253L-56.514 25.253Q-56.483 25.253-56.461 25.277Q-56.439 25.301-56.439 25.335L-56.439 25.356Q-56.524 25.643-56.712 25.841Q-56.900 26.039-57.165 26.142Q-57.430 26.244-57.724 26.244Q-58.154 26.244-58.542 26.038Q-58.930 25.831-59.164 25.468Q-59.399 25.106-59.399 24.665M-55.793 25.448Q-55.793 25.116-55.569 24.889Q-55.345 24.662-55.001 24.534Q-54.658 24.405-54.285 24.353Q-53.913 24.300-53.609 24.300L-53.609 24.047Q-53.609 23.842-53.716 23.662Q-53.824 23.483-54.005 23.380Q-54.186 23.278-54.395 23.278Q-54.801 23.278-55.037 23.370Q-54.948 23.407-54.902 23.491Q-54.856 23.575-54.856 23.677Q-54.856 23.773-54.902 23.852Q-54.948 23.930-55.029 23.975Q-55.109 24.019-55.198 24.019Q-55.348 24.019-55.449 23.922Q-55.550 23.824-55.550 23.677Q-55.550 23.055-54.395 23.055Q-54.183 23.055-53.933 23.119Q-53.684 23.182-53.482 23.301Q-53.280 23.421-53.154 23.606Q-53.027 23.790-53.027 24.033L-53.027 25.609Q-53.027 25.725-52.966 25.821Q-52.904 25.916-52.792 25.916Q-52.682 25.916-52.617 25.822Q-52.552 25.728-52.552 25.609L-52.552 25.161L-52.286 25.161L-52.286 25.609Q-52.286 25.879-52.513 26.044Q-52.740 26.210-53.021 26.210Q-53.229 26.210-53.366 26.056Q-53.503 25.903-53.526 25.687Q-53.673 25.954-53.955 26.099Q-54.237 26.244-54.562 26.244Q-54.839 26.244-55.123 26.169Q-55.406 26.094-55.599 25.915Q-55.793 25.735-55.793 25.448M-55.177 25.448Q-55.177 25.622-55.077 25.752Q-54.976 25.882-54.820 25.952Q-54.665 26.022-54.501 26.022Q-54.282 26.022-54.073 25.925Q-53.865 25.827-53.737 25.646Q-53.609 25.465-53.609 25.239L-53.609 24.511Q-53.933 24.511-54.299 24.602Q-54.665 24.693-54.921 24.905Q-55.177 25.116-55.177 25.448M-50.187 26.176L-51.821 26.176L-51.821 25.896Q-51.592 25.896-51.443 25.862Q-51.295 25.827-51.295 25.687L-51.295 23.838Q-51.295 23.568-51.402 23.507Q-51.510 23.445-51.821 23.445L-51.821 23.165L-50.761 23.090L-50.761 23.739Q-50.590 23.431-50.286 23.260Q-49.982 23.090-49.637 23.090Q-49.131 23.090-48.847 23.313Q-48.564 23.537-48.564 24.033L-48.564 25.687Q-48.564 25.824-48.415 25.860Q-48.266 25.896-48.041 25.896L-48.041 26.176L-49.671 26.176L-49.671 25.896Q-49.442 25.896-49.293 25.862Q-49.145 25.827-49.145 25.687L-49.145 24.047Q-49.145 23.712-49.264 23.512Q-49.384 23.312-49.698 23.312Q-49.968 23.312-50.203 23.448Q-50.437 23.585-50.575 23.819Q-50.713 24.053-50.713 24.327L-50.713 25.687Q-50.713 25.824-50.563 25.860Q-50.413 25.896-50.187 25.896L-50.187 26.176M-47.453 24.665Q-47.453 24.327-47.313 24.036Q-47.172 23.746-46.928 23.532Q-46.684 23.319-46.380 23.204Q-46.075 23.090-45.751 23.090Q-45.481 23.090-45.217 23.189Q-44.954 23.288-44.763 23.466L-44.763 22.068Q-44.763 21.798-44.870 21.736Q-44.978 21.675-45.289 21.675L-45.289 21.394L-44.213 21.319L-44.213 25.503Q-44.213 25.691-44.158 25.774Q-44.103 25.858-44.002 25.877Q-43.901 25.896-43.686 25.896L-43.686 26.176L-44.794 26.244L-44.794 25.827Q-45.211 26.244-45.836 26.244Q-46.267 26.244-46.639 26.032Q-47.012 25.821-47.232 25.460Q-47.453 25.099-47.453 24.665M-45.778 26.022Q-45.569 26.022-45.383 25.950Q-45.197 25.879-45.043 25.742Q-44.889 25.605-44.794 25.427L-44.794 23.818Q-44.879 23.671-45.024 23.551Q-45.170 23.431-45.339 23.372Q-45.508 23.312-45.689 23.312Q-46.250 23.312-46.518 23.701Q-46.786 24.091-46.786 24.672Q-46.786 25.243-46.552 25.633Q-46.318 26.022-45.778 26.022M-41.420 26.176L-42.972 26.176L-42.972 25.896Q-42.746 25.896-42.598 25.862Q-42.449 25.827-42.449 25.687L-42.449 23.838Q-42.449 23.650-42.497 23.566Q-42.545 23.483-42.642 23.464Q-42.739 23.445-42.951 23.445L-42.951 23.165L-41.895 23.090L-41.895 25.687Q-41.895 25.827-41.764 25.862Q-41.632 25.896-41.420 25.896L-41.420 26.176M-42.692 21.869Q-42.692 21.698-42.568 21.579Q-42.445 21.459-42.275 21.459Q-42.107 21.459-41.984 21.579Q-41.861 21.698-41.861 21.869Q-41.861 22.044-41.984 22.167Q-42.107 22.290-42.275 22.290Q-42.445 22.290-42.568 22.167Q-42.692 22.044-42.692 21.869M-40.774 24.665Q-40.774 24.327-40.634 24.036Q-40.494 23.746-40.249 23.532Q-40.005 23.319-39.701 23.204Q-39.397 23.090-39.072 23.090Q-38.802 23.090-38.539 23.189Q-38.276 23.288-38.084 23.466L-38.084 22.068Q-38.084 21.798-38.192 21.736Q-38.299 21.675-38.610 21.675L-38.610 21.394L-37.534 21.319L-37.534 25.503Q-37.534 25.691-37.479 25.774Q-37.424 25.858-37.324 25.877Q-37.223 25.896-37.007 25.896L-37.007 26.176L-38.115 26.244L-38.115 25.827Q-38.532 26.244-39.157 26.244Q-39.588 26.244-39.961 26.032Q-40.333 25.821-40.554 25.460Q-40.774 25.099-40.774 24.665M-39.099 26.022Q-38.891 26.022-38.704 25.950Q-38.518 25.879-38.364 25.742Q-38.211 25.605-38.115 25.427L-38.115 23.818Q-38.200 23.671-38.346 23.551Q-38.491 23.431-38.660 23.372Q-38.829 23.312-39.010 23.312Q-39.571 23.312-39.839 23.701Q-40.108 24.091-40.108 24.672Q-40.108 25.243-39.873 25.633Q-39.639 26.022-39.099 26.022M-36.300 25.448Q-36.300 25.116-36.076 24.889Q-35.852 24.662-35.509 24.534Q-35.165 24.405-34.793 24.353Q-34.420 24.300-34.116 24.300L-34.116 24.047Q-34.116 23.842-34.223 23.662Q-34.331 23.483-34.512 23.380Q-34.693 23.278-34.902 23.278Q-35.309 23.278-35.545 23.370Q-35.456 23.407-35.410 23.491Q-35.363 23.575-35.363 23.677Q-35.363 23.773-35.410 23.852Q-35.456 23.930-35.536 23.975Q-35.616 24.019-35.705 24.019Q-35.856 24.019-35.956 23.922Q-36.057 23.824-36.057 23.677Q-36.057 23.055-34.902 23.055Q-34.690 23.055-34.441 23.119Q-34.191 23.182-33.989 23.301Q-33.788 23.421-33.661 23.606Q-33.535 23.790-33.535 24.033L-33.535 25.609Q-33.535 25.725-33.473 25.821Q-33.412 25.916-33.299 25.916Q-33.190 25.916-33.125 25.822Q-33.060 25.728-33.060 25.609L-33.060 25.161L-32.793 25.161L-32.793 25.609Q-32.793 25.879-33.020 26.044Q-33.248 26.210-33.528 26.210Q-33.736 26.210-33.873 26.056Q-34.010 25.903-34.034 25.687Q-34.181 25.954-34.463 26.099Q-34.745 26.244-35.069 26.244Q-35.346 26.244-35.630 26.169Q-35.914 26.094-36.107 25.915Q-36.300 25.735-36.300 25.448M-35.685 25.448Q-35.685 25.622-35.584 25.752Q-35.483 25.882-35.328 25.952Q-35.172 26.022-35.008 26.022Q-34.789 26.022-34.581 25.925Q-34.372 25.827-34.244 25.646Q-34.116 25.465-34.116 25.239L-34.116 24.511Q-34.441 24.511-34.806 24.602Q-35.172 24.693-35.428 24.905Q-35.685 25.116-35.685 25.448M-31.850 25.335L-31.850 23.438L-32.489 23.438L-32.489 23.216Q-32.171 23.216-31.954 23.006Q-31.737 22.796-31.636 22.486Q-31.535 22.177-31.535 21.869L-31.269 21.869L-31.269 23.158L-30.192 23.158L-30.192 23.438L-31.269 23.438L-31.269 25.322Q-31.269 25.598-31.164 25.797Q-31.060 25.995-30.800 25.995Q-30.643 25.995-30.537 25.891Q-30.431 25.786-30.382 25.633Q-30.332 25.479-30.332 25.322L-30.332 24.908L-30.066 24.908L-30.066 25.335Q-30.066 25.561-30.165 25.771Q-30.264 25.981-30.448 26.113Q-30.633 26.244-30.862 26.244Q-31.299 26.244-31.575 26.007Q-31.850 25.769-31.850 25.335M-29.296 24.641Q-29.296 24.320-29.172 24.031Q-29.047 23.742-28.821 23.519Q-28.596 23.295-28.300 23.175Q-28.005 23.055-27.687 23.055Q-27.359 23.055-27.097 23.155Q-26.836 23.254-26.660 23.436Q-26.484 23.619-26.390 23.877Q-26.296 24.135-26.296 24.467Q-26.296 24.559-26.378 24.580L-28.633 24.580L-28.633 24.641Q-28.633 25.229-28.350 25.612Q-28.066 25.995-27.499 25.995Q-27.177 25.995-26.909 25.802Q-26.641 25.609-26.552 25.294Q-26.545 25.253-26.470 25.239L-26.378 25.239Q-26.296 25.263-26.296 25.335Q-26.296 25.342-26.302 25.369Q-26.415 25.766-26.786 26.005Q-27.157 26.244-27.581 26.244Q-28.018 26.244-28.418 26.036Q-28.818 25.827-29.057 25.460Q-29.296 25.093-29.296 24.641M-28.627 24.371L-26.812 24.371Q-26.812 24.094-26.909 23.842Q-27.006 23.589-27.205 23.433Q-27.403 23.278-27.687 23.278Q-27.963 23.278-28.177 23.436Q-28.391 23.595-28.509 23.850Q-28.627 24.105-28.627 24.371M-25.708 26.169L-25.708 25.106Q-25.708 25.082-25.680 25.055Q-25.653 25.028-25.629 25.028L-25.520 25.028Q-25.455 25.028-25.441 25.086Q-25.345 25.520-25.099 25.771Q-24.853 26.022-24.440 26.022Q-24.098 26.022-23.845 25.889Q-23.592 25.756-23.592 25.448Q-23.592 25.291-23.686 25.176Q-23.780 25.062-23.918 24.993Q-24.057 24.925-24.224 24.887L-24.805 24.788Q-25.161 24.720-25.434 24.499Q-25.708 24.279-25.708 23.937Q-25.708 23.688-25.597 23.513Q-25.485 23.339-25.299 23.240Q-25.113 23.141-24.898 23.098Q-24.682 23.055-24.440 23.055Q-24.026 23.055-23.746 23.237L-23.530 23.062Q-23.520 23.059-23.513 23.057Q-23.506 23.055-23.496 23.055L-23.445 23.055Q-23.418 23.055-23.394 23.079Q-23.370 23.103-23.370 23.131L-23.370 23.978Q-23.370 23.999-23.394 24.026Q-23.418 24.053-23.445 24.053L-23.558 24.053Q-23.585 24.053-23.611 24.028Q-23.636 24.002-23.636 23.978Q-23.636 23.742-23.742 23.578Q-23.848 23.414-24.031 23.332Q-24.214 23.250-24.446 23.250Q-24.775 23.250-25.031 23.353Q-25.287 23.455-25.287 23.732Q-25.287 23.927-25.104 24.036Q-24.921 24.146-24.692 24.187L-24.118 24.293Q-23.872 24.341-23.659 24.469Q-23.445 24.597-23.308 24.800Q-23.171 25.004-23.171 25.253Q-23.171 25.766-23.537 26.005Q-23.903 26.244-24.440 26.244Q-24.935 26.244-25.267 25.950L-25.533 26.224Q-25.554 26.244-25.581 26.244L-25.629 26.244Q-25.653 26.244-25.680 26.217Q-25.708 26.190-25.708 26.169\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.197 -1.565)\">\u003Cpath d=\"M-17.230 26.856L-17.230 24.600L-19.479 24.600Q-19.547 24.590-19.593 24.544Q-19.639 24.498-19.639 24.426Q-19.639 24.282-19.479 24.259L-17.230 24.259L-17.230 22.003Q-17.219 21.934-17.173 21.888Q-17.127 21.842-17.055 21.842Q-16.912 21.842-16.888 22.003L-16.888 24.259L-14.646 24.259Q-14.485 24.282-14.485 24.426Q-14.485 24.498-14.531 24.544Q-14.577 24.590-14.646 24.600L-16.888 24.600L-16.888 26.856Q-16.912 27.017-17.055 27.017Q-17.127 27.017-17.173 26.971Q-17.219 26.925-17.230 26.856\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.197 -1.565)\">\u003Cpath d=\"M-10.981 26.169L-10.981 25.106Q-10.981 25.082-10.953 25.055Q-10.926 25.028-10.902 25.028L-10.793 25.028Q-10.728 25.028-10.714 25.086Q-10.618 25.520-10.372 25.771Q-10.126 26.022-9.712 26.022Q-9.371 26.022-9.118 25.889Q-8.865 25.756-8.865 25.448Q-8.865 25.291-8.959 25.176Q-9.053 25.062-9.191 24.993Q-9.330 24.925-9.497 24.887L-10.078 24.788Q-10.434 24.720-10.707 24.499Q-10.981 24.279-10.981 23.937Q-10.981 23.688-10.869 23.513Q-10.758 23.339-10.572 23.240Q-10.386 23.141-10.170 23.098Q-9.955 23.055-9.712 23.055Q-9.299 23.055-9.019 23.237L-8.803 23.062Q-8.793 23.059-8.786 23.057Q-8.779 23.055-8.769 23.055L-8.718 23.055Q-8.691 23.055-8.667 23.079Q-8.643 23.103-8.643 23.131L-8.643 23.978Q-8.643 23.999-8.667 24.026Q-8.691 24.053-8.718 24.053L-8.831 24.053Q-8.858 24.053-8.884 24.028Q-8.909 24.002-8.909 23.978Q-8.909 23.742-9.015 23.578Q-9.121 23.414-9.304 23.332Q-9.487 23.250-9.719 23.250Q-10.047 23.250-10.304 23.353Q-10.560 23.455-10.560 23.732Q-10.560 23.927-10.377 24.036Q-10.194 24.146-9.965 24.187L-9.391 24.293Q-9.145 24.341-8.931 24.469Q-8.718 24.597-8.581 24.800Q-8.444 25.004-8.444 25.253Q-8.444 25.766-8.810 26.005Q-9.176 26.244-9.712 26.244Q-10.208 26.244-10.540 25.950L-10.806 26.224Q-10.827 26.244-10.854 26.244L-10.902 26.244Q-10.926 26.244-10.953 26.217Q-10.981 26.190-10.981 26.169M-6.134 26.176L-7.768 26.176L-7.768 25.896Q-7.539 25.896-7.390 25.862Q-7.241 25.827-7.241 25.687L-7.241 23.838Q-7.241 23.568-7.349 23.507Q-7.457 23.445-7.768 23.445L-7.768 23.165L-6.708 23.090L-6.708 23.739Q-6.537 23.431-6.233 23.260Q-5.929 23.090-5.584 23.090Q-5.184 23.090-4.907 23.230Q-4.630 23.370-4.545 23.718Q-4.377 23.425-4.078 23.257Q-3.779 23.090-3.434 23.090Q-2.928 23.090-2.644 23.313Q-2.360 23.537-2.360 24.033L-2.360 25.687Q-2.360 25.824-2.212 25.860Q-2.063 25.896-1.837 25.896L-1.837 26.176L-3.468 26.176L-3.468 25.896Q-3.242 25.896-3.092 25.860Q-2.941 25.824-2.941 25.687L-2.941 24.047Q-2.941 23.712-3.061 23.512Q-3.181 23.312-3.495 23.312Q-3.765 23.312-3.999 23.448Q-4.233 23.585-4.372 23.819Q-4.510 24.053-4.510 24.327L-4.510 25.687Q-4.510 25.824-4.362 25.860Q-4.213 25.896-3.987 25.896L-3.987 26.176L-5.618 26.176L-5.618 25.896Q-5.389 25.896-5.240 25.862Q-5.091 25.827-5.091 25.687L-5.091 24.047Q-5.091 23.712-5.211 23.512Q-5.331 23.312-5.645 23.312Q-5.915 23.312-6.149 23.448Q-6.383 23.585-6.522 23.819Q-6.660 24.053-6.660 24.327L-6.660 25.687Q-6.660 25.824-6.510 25.860Q-6.359 25.896-6.134 25.896L-6.134 26.176M-1.191 25.448Q-1.191 25.116-0.968 24.889Q-0.744 24.662-0.400 24.534Q-0.057 24.405 0.316 24.353Q0.688 24.300 0.993 24.300L0.993 24.047Q0.993 23.842 0.885 23.662Q0.777 23.483 0.596 23.380Q0.415 23.278 0.206 23.278Q-0.200 23.278-0.436 23.370Q-0.347 23.407-0.301 23.491Q-0.255 23.575-0.255 23.677Q-0.255 23.773-0.301 23.852Q-0.347 23.930-0.428 23.975Q-0.508 24.019-0.597 24.019Q-0.747 24.019-0.848 23.922Q-0.949 23.824-0.949 23.677Q-0.949 23.055 0.206 23.055Q0.418 23.055 0.668 23.119Q0.917 23.182 1.119 23.301Q1.321 23.421 1.447 23.606Q1.574 23.790 1.574 24.033L1.574 25.609Q1.574 25.725 1.635 25.821Q1.697 25.916 1.809 25.916Q1.919 25.916 1.984 25.822Q2.049 25.728 2.049 25.609L2.049 25.161L2.315 25.161L2.315 25.609Q2.315 25.879 2.088 26.044Q1.861 26.210 1.580 26.210Q1.372 26.210 1.235 26.056Q1.099 25.903 1.075 25.687Q0.928 25.954 0.646 26.099Q0.364 26.244 0.039 26.244Q-0.238 26.244-0.522 26.169Q-0.805 26.094-0.998 25.915Q-1.191 25.735-1.191 25.448M-0.576 25.448Q-0.576 25.622-0.475 25.752Q-0.375 25.882-0.219 25.952Q-0.064 26.022 0.101 26.022Q0.319 26.022 0.528 25.925Q0.736 25.827 0.864 25.646Q0.993 25.465 0.993 25.239L0.993 24.511Q0.668 24.511 0.302 24.602Q-0.064 24.693-0.320 24.905Q-0.576 25.116-0.576 25.448M4.400 26.176L2.797 26.176L2.797 25.896Q3.023 25.896 3.172 25.862Q3.320 25.827 3.320 25.687L3.320 22.068Q3.320 21.798 3.213 21.736Q3.105 21.675 2.797 21.675L2.797 21.394L3.874 21.319L3.874 25.687Q3.874 25.824 4.024 25.860Q4.175 25.896 4.400 25.896L4.400 26.176M6.663 26.176L5.060 26.176L5.060 25.896Q5.286 25.896 5.434 25.862Q5.583 25.827 5.583 25.687L5.583 22.068Q5.583 21.798 5.475 21.736Q5.368 21.675 5.060 21.675L5.060 21.394L6.137 21.319L6.137 25.687Q6.137 25.824 6.287 25.860Q6.437 25.896 6.663 25.896\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.197 -1.565)\">\u003Cpath d=\"M11.609 27.533L9.979 27.533L9.979 27.253Q10.208 27.253 10.357 27.218Q10.505 27.184 10.505 27.044L10.505 23.698Q10.505 23.527 10.369 23.486Q10.232 23.445 9.979 23.445L9.979 23.165L11.059 23.090L11.059 23.496Q11.281 23.295 11.568 23.192Q11.856 23.090 12.163 23.090Q12.590 23.090 12.954 23.303Q13.318 23.517 13.532 23.881Q13.746 24.245 13.746 24.665Q13.746 25.110 13.506 25.474Q13.267 25.838 12.874 26.041Q12.481 26.244 12.037 26.244Q11.770 26.244 11.522 26.144Q11.275 26.043 11.087 25.862L11.087 27.044Q11.087 27.181 11.235 27.217Q11.384 27.253 11.609 27.253L11.609 27.533M11.087 23.845L11.087 25.455Q11.220 25.708 11.463 25.865Q11.705 26.022 11.982 26.022Q12.310 26.022 12.563 25.821Q12.816 25.619 12.949 25.301Q13.083 24.983 13.083 24.665Q13.083 24.436 13.018 24.207Q12.953 23.978 12.825 23.780Q12.696 23.582 12.502 23.462Q12.307 23.343 12.074 23.343Q11.780 23.343 11.512 23.472Q11.244 23.602 11.087 23.845\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.197 -1.565)\">\u003Cpath d=\"M14.556 24.641Q14.556 24.320 14.681 24.031Q14.806 23.742 15.032 23.519Q15.257 23.295 15.553 23.175Q15.848 23.055 16.166 23.055Q16.494 23.055 16.756 23.155Q17.017 23.254 17.193 23.436Q17.369 23.619 17.463 23.877Q17.557 24.135 17.557 24.467Q17.557 24.559 17.475 24.580L15.220 24.580L15.220 24.641Q15.220 25.229 15.503 25.612Q15.787 25.995 16.354 25.995Q16.676 25.995 16.944 25.802Q17.212 25.609 17.301 25.294Q17.308 25.253 17.383 25.239L17.475 25.239Q17.557 25.263 17.557 25.335Q17.557 25.342 17.551 25.369Q17.438 25.766 17.067 26.005Q16.696 26.244 16.272 26.244Q15.835 26.244 15.435 26.036Q15.035 25.827 14.796 25.460Q14.556 25.093 14.556 24.641M15.226 24.371L17.041 24.371Q17.041 24.094 16.944 23.842Q16.846 23.589 16.648 23.433Q16.450 23.278 16.166 23.278Q15.889 23.278 15.676 23.436Q15.462 23.595 15.344 23.850Q15.226 24.105 15.226 24.371M19.895 26.176L18.159 26.176L18.159 25.896Q18.388 25.896 18.537 25.862Q18.685 25.827 18.685 25.687L18.685 23.838Q18.685 23.568 18.578 23.507Q18.470 23.445 18.159 23.445L18.159 23.165L19.188 23.090L19.188 23.797Q19.318 23.489 19.560 23.290Q19.803 23.090 20.121 23.090Q20.340 23.090 20.511 23.214Q20.681 23.339 20.681 23.551Q20.681 23.688 20.582 23.787Q20.483 23.886 20.350 23.886Q20.213 23.886 20.114 23.787Q20.015 23.688 20.015 23.551Q20.015 23.411 20.114 23.312Q19.824 23.312 19.624 23.508Q19.424 23.705 19.331 23.999Q19.239 24.293 19.239 24.573L19.239 25.687Q19.239 25.896 19.895 25.896L19.895 26.176M21.792 25.335L21.792 23.438L21.153 23.438L21.153 23.216Q21.471 23.216 21.688 23.006Q21.905 22.796 22.006 22.486Q22.107 22.177 22.107 21.869L22.373 21.869L22.373 23.158L23.450 23.158L23.450 23.438L22.373 23.438L22.373 25.322Q22.373 25.598 22.478 25.797Q22.582 25.995 22.842 25.995Q22.999 25.995 23.105 25.891Q23.211 25.786 23.260 25.633Q23.310 25.479 23.310 25.322L23.310 24.908L23.576 24.908L23.576 25.335Q23.576 25.561 23.477 25.771Q23.378 25.981 23.194 26.113Q23.009 26.244 22.780 26.244Q22.343 26.244 22.067 26.007Q21.792 25.769 21.792 25.335M24.961 25.342L24.961 23.838Q24.961 23.568 24.853 23.507Q24.745 23.445 24.434 23.445L24.434 23.165L25.542 23.090L25.542 25.322L25.542 25.342Q25.542 25.622 25.593 25.766Q25.644 25.909 25.786 25.966Q25.928 26.022 26.215 26.022Q26.468 26.022 26.673 25.882Q26.878 25.742 26.994 25.516Q27.111 25.291 27.111 25.041L27.111 23.838Q27.111 23.568 27.003 23.507Q26.895 23.445 26.584 23.445L26.584 23.165L27.692 23.090L27.692 25.503Q27.692 25.694 27.745 25.776Q27.798 25.858 27.898 25.877Q27.999 25.896 28.215 25.896L28.215 26.176L27.138 26.244L27.138 25.680Q27.029 25.862 26.883 25.985Q26.738 26.108 26.552 26.176Q26.366 26.244 26.164 26.244Q24.961 26.244 24.961 25.342M30.553 26.176L28.816 26.176L28.816 25.896Q29.045 25.896 29.194 25.862Q29.343 25.827 29.343 25.687L29.343 23.838Q29.343 23.568 29.235 23.507Q29.127 23.445 28.816 23.445L28.816 23.165L29.845 23.090L29.845 23.797Q29.975 23.489 30.218 23.290Q30.460 23.090 30.778 23.090Q30.997 23.090 31.168 23.214Q31.339 23.339 31.339 23.551Q31.339 23.688 31.240 23.787Q31.140 23.886 31.007 23.886Q30.870 23.886 30.771 23.787Q30.672 23.688 30.672 23.551Q30.672 23.411 30.771 23.312Q30.481 23.312 30.281 23.508Q30.081 23.705 29.989 23.999Q29.896 24.293 29.896 24.573L29.896 25.687Q29.896 25.896 30.553 25.896L30.553 26.176M32.730 26.176L32.463 26.176L32.463 22.068Q32.463 21.798 32.356 21.736Q32.248 21.675 31.937 21.675L31.937 21.394L33.017 21.319L33.017 23.489Q33.225 23.298 33.511 23.194Q33.796 23.090 34.094 23.090Q34.411 23.090 34.709 23.211Q35.006 23.332 35.228 23.548Q35.450 23.763 35.577 24.048Q35.703 24.334 35.703 24.665Q35.703 25.110 35.464 25.474Q35.225 25.838 34.832 26.041Q34.439 26.244 33.994 26.244Q33.800 26.244 33.610 26.188Q33.420 26.132 33.260 26.027Q33.099 25.923 32.959 25.762L32.730 26.176M33.044 23.831L33.044 25.448Q33.181 25.708 33.422 25.865Q33.663 26.022 33.940 26.022Q34.234 26.022 34.446 25.915Q34.658 25.807 34.791 25.615Q34.924 25.424 34.982 25.185Q35.040 24.946 35.040 24.665Q35.040 24.306 34.946 24.002Q34.852 23.698 34.625 23.505Q34.398 23.312 34.032 23.312Q33.731 23.312 33.465 23.448Q33.198 23.585 33.044 23.831M36.397 25.448Q36.397 25.116 36.621 24.889Q36.845 24.662 37.189 24.534Q37.532 24.405 37.905 24.353Q38.277 24.300 38.581 24.300L38.581 24.047Q38.581 23.842 38.474 23.662Q38.366 23.483 38.185 23.380Q38.004 23.278 37.795 23.278Q37.388 23.278 37.153 23.370Q37.241 23.407 37.288 23.491Q37.334 23.575 37.334 23.677Q37.334 23.773 37.288 23.852Q37.241 23.930 37.161 23.975Q37.081 24.019 36.992 24.019Q36.842 24.019 36.741 23.922Q36.640 23.824 36.640 23.677Q36.640 23.055 37.795 23.055Q38.007 23.055 38.257 23.119Q38.506 23.182 38.708 23.301Q38.909 23.421 39.036 23.606Q39.162 23.790 39.162 24.033L39.162 25.609Q39.162 25.725 39.224 25.821Q39.285 25.916 39.398 25.916Q39.508 25.916 39.573 25.822Q39.637 25.728 39.637 25.609L39.637 25.161L39.904 25.161L39.904 25.609Q39.904 25.879 39.677 26.044Q39.450 26.210 39.169 26.210Q38.961 26.210 38.824 26.056Q38.687 25.903 38.663 25.687Q38.516 25.954 38.234 26.099Q37.952 26.244 37.628 26.244Q37.351 26.244 37.067 26.169Q36.783 26.094 36.590 25.915Q36.397 25.735 36.397 25.448M37.012 25.448Q37.012 25.622 37.113 25.752Q37.214 25.882 37.370 25.952Q37.525 26.022 37.689 26.022Q37.908 26.022 38.116 25.925Q38.325 25.827 38.453 25.646Q38.581 25.465 38.581 25.239L38.581 24.511Q38.257 24.511 37.891 24.602Q37.525 24.693 37.269 24.905Q37.012 25.116 37.012 25.448M40.847 25.335L40.847 23.438L40.208 23.438L40.208 23.216Q40.526 23.216 40.743 23.006Q40.960 22.796 41.061 22.486Q41.162 22.177 41.162 21.869L41.429 21.869L41.429 23.158L42.505 23.158L42.505 23.438L41.429 23.438L41.429 25.322Q41.429 25.598 41.533 25.797Q41.637 25.995 41.897 25.995Q42.054 25.995 42.160 25.891Q42.266 25.786 42.315 25.633Q42.365 25.479 42.365 25.322L42.365 24.908L42.632 24.908L42.632 25.335Q42.632 25.561 42.533 25.771Q42.433 25.981 42.249 26.113Q42.064 26.244 41.835 26.244Q41.398 26.244 41.123 26.007Q40.847 25.769 40.847 25.335M45.058 26.176L43.507 26.176L43.507 25.896Q43.732 25.896 43.881 25.862Q44.030 25.827 44.030 25.687L44.030 23.838Q44.030 23.650 43.982 23.566Q43.934 23.483 43.836 23.464Q43.739 23.445 43.527 23.445L43.527 23.165L44.583 23.090L44.583 25.687Q44.583 25.827 44.715 25.862Q44.846 25.896 45.058 25.896L45.058 26.176M43.787 21.869Q43.787 21.698 43.910 21.579Q44.033 21.459 44.204 21.459Q44.371 21.459 44.494 21.579Q44.617 21.698 44.617 21.869Q44.617 22.044 44.494 22.167Q44.371 22.290 44.204 22.290Q44.033 22.290 43.910 22.167Q43.787 22.044 43.787 21.869M45.663 24.693Q45.663 24.351 45.798 24.052Q45.933 23.753 46.173 23.529Q46.412 23.305 46.730 23.180Q47.048 23.055 47.379 23.055Q47.824 23.055 48.223 23.271Q48.623 23.486 48.857 23.864Q49.092 24.241 49.092 24.693Q49.092 25.034 48.950 25.318Q48.808 25.602 48.564 25.809Q48.319 26.015 48.010 26.130Q47.700 26.244 47.379 26.244Q46.949 26.244 46.547 26.043Q46.145 25.841 45.904 25.489Q45.663 25.137 45.663 24.693M47.379 25.995Q47.981 25.995 48.205 25.617Q48.429 25.239 48.429 24.607Q48.429 23.995 48.194 23.636Q47.960 23.278 47.379 23.278Q46.326 23.278 46.326 24.607Q46.326 25.239 46.552 25.617Q46.778 25.995 47.379 25.995M51.368 26.176L49.734 26.176L49.734 25.896Q49.963 25.896 50.112 25.862Q50.261 25.827 50.261 25.687L50.261 23.838Q50.261 23.568 50.153 23.507Q50.045 23.445 49.734 23.445L49.734 23.165L50.794 23.090L50.794 23.739Q50.965 23.431 51.269 23.260Q51.573 23.090 51.918 23.090Q52.424 23.090 52.708 23.313Q52.991 23.537 52.991 24.033L52.991 25.687Q52.991 25.824 53.140 25.860Q53.289 25.896 53.514 25.896L53.514 26.176L51.884 26.176L51.884 25.896Q52.113 25.896 52.262 25.862Q52.410 25.827 52.410 25.687L52.410 24.047Q52.410 23.712 52.291 23.512Q52.171 23.312 51.857 23.312Q51.587 23.312 51.353 23.448Q51.118 23.585 50.980 23.819Q50.842 24.053 50.842 24.327L50.842 25.687Q50.842 25.824 50.992 25.860Q51.142 25.896 51.368 25.896\" 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=\"var(--tk-accent)\" stroke=\"none\" d=\"M74.17 11.95a3 3 0 1 0-6 0 3 3 0 0 0 6 0m-3 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M71.17-19.348v25.03\"\u002F>\u003Cpath stroke=\"none\" d=\"m71.17 7.682 1.6-3.2-1.6 1.2-1.6-1.2\"\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(94.5 -50.808)\">\u003Cpath d=\"M-59.340 25.448Q-59.340 25.116-59.117 24.889Q-58.893 24.662-58.549 24.534Q-58.206 24.405-57.833 24.353Q-57.461 24.300-57.156 24.300L-57.156 24.047Q-57.156 23.842-57.264 23.662Q-57.372 23.483-57.553 23.380Q-57.734 23.278-57.942 23.278Q-58.349 23.278-58.585 23.370Q-58.496 23.407-58.450 23.491Q-58.404 23.575-58.404 23.677Q-58.404 23.773-58.450 23.852Q-58.496 23.930-58.577 23.975Q-58.657 24.019-58.746 24.019Q-58.896 24.019-58.997 23.922Q-59.098 23.824-59.098 23.677Q-59.098 23.055-57.942 23.055Q-57.731 23.055-57.481 23.119Q-57.232 23.182-57.030 23.301Q-56.828 23.421-56.702 23.606Q-56.575 23.790-56.575 24.033L-56.575 25.609Q-56.575 25.725-56.514 25.821Q-56.452 25.916-56.339 25.916Q-56.230 25.916-56.165 25.822Q-56.100 25.728-56.100 25.609L-56.100 25.161L-55.834 25.161L-55.834 25.609Q-55.834 25.879-56.061 26.044Q-56.288 26.210-56.568 26.210Q-56.777 26.210-56.914 26.056Q-57.050 25.903-57.074 25.687Q-57.221 25.954-57.503 26.099Q-57.785 26.244-58.110 26.244Q-58.387 26.244-58.671 26.169Q-58.954 26.094-59.147 25.915Q-59.340 25.735-59.340 25.448M-58.725 25.448Q-58.725 25.622-58.624 25.752Q-58.524 25.882-58.368 25.952Q-58.213 26.022-58.048 26.022Q-57.830 26.022-57.621 25.925Q-57.413 25.827-57.285 25.646Q-57.156 25.465-57.156 25.239L-57.156 24.511Q-57.481 24.511-57.847 24.602Q-58.213 24.693-58.469 24.905Q-58.725 25.116-58.725 25.448M-53.667 26.176L-55.403 26.176L-55.403 25.896Q-55.174 25.896-55.025 25.862Q-54.877 25.827-54.877 25.687L-54.877 23.838Q-54.877 23.568-54.984 23.507Q-55.092 23.445-55.403 23.445L-55.403 23.165L-54.374 23.090L-54.374 23.797Q-54.244 23.489-54.002 23.290Q-53.759 23.090-53.441 23.090Q-53.222 23.090-53.051 23.214Q-52.880 23.339-52.880 23.551Q-52.880 23.688-52.980 23.787Q-53.079 23.886-53.212 23.886Q-53.349 23.886-53.448 23.787Q-53.547 23.688-53.547 23.551Q-53.547 23.411-53.448 23.312Q-53.738 23.312-53.938 23.508Q-54.138 23.705-54.231 23.999Q-54.323 24.293-54.323 24.573L-54.323 25.687Q-54.323 25.896-53.667 25.896L-53.667 26.176M-52.337 26.709Q-52.337 26.463-52.140 26.279Q-51.944 26.094-51.688 26.015Q-51.824 25.903-51.896 25.742Q-51.968 25.581-51.968 25.400Q-51.968 25.079-51.756 24.833Q-52.091 24.535-52.091 24.125Q-52.091 23.664-51.701 23.377Q-51.312 23.090-50.833 23.090Q-50.361 23.090-50.026 23.336Q-49.852 23.182-49.642 23.100Q-49.432 23.018-49.203 23.018Q-49.039 23.018-48.917 23.125Q-48.796 23.233-48.796 23.397Q-48.796 23.493-48.868 23.565Q-48.940 23.636-49.032 23.636Q-49.131 23.636-49.201 23.563Q-49.271 23.489-49.271 23.390Q-49.271 23.336-49.257 23.305L-49.251 23.291Q-49.244 23.271-49.235 23.260Q-49.227 23.250-49.223 23.243Q-49.579 23.243-49.866 23.466Q-49.579 23.759-49.579 24.125Q-49.579 24.440-49.763 24.672Q-49.948 24.905-50.237 25.033Q-50.526 25.161-50.833 25.161Q-51.035 25.161-51.226 25.111Q-51.418 25.062-51.595 24.952Q-51.688 25.079-51.688 25.222Q-51.688 25.404-51.559 25.539Q-51.431 25.674-51.247 25.674L-50.614 25.674Q-50.167 25.674-49.797 25.745Q-49.428 25.817-49.169 26.046Q-48.909 26.275-48.909 26.709Q-48.909 27.030-49.204 27.232Q-49.500 27.434-49.903 27.523Q-50.307 27.612-50.621 27.612Q-50.939 27.612-51.342 27.523Q-51.746 27.434-52.041 27.232Q-52.337 27.030-52.337 26.709M-51.882 26.709Q-51.882 26.938-51.664 27.087Q-51.445 27.236-51.153 27.304Q-50.860 27.372-50.621 27.372Q-50.457 27.372-50.249 27.336Q-50.040 27.301-49.833 27.220Q-49.627 27.140-49.495 27.012Q-49.363 26.884-49.363 26.709Q-49.363 26.357-49.744 26.263Q-50.126 26.169-50.628 26.169L-51.247 26.169Q-51.486 26.169-51.684 26.320Q-51.882 26.470-51.882 26.709M-50.833 24.922Q-50.167 24.922-50.167 24.125Q-50.167 23.325-50.833 23.325Q-51.503 23.325-51.503 24.125Q-51.503 24.922-50.833 24.922M-46.632 26.176L-48.266 26.176L-48.266 25.896Q-48.037 25.896-47.889 25.862Q-47.740 25.827-47.740 25.687L-47.740 23.838Q-47.740 23.568-47.848 23.507Q-47.955 23.445-48.266 23.445L-48.266 23.165L-47.207 23.090L-47.207 23.739Q-47.036 23.431-46.732 23.260Q-46.427 23.090-46.082 23.090Q-45.682 23.090-45.405 23.230Q-45.129 23.370-45.043 23.718Q-44.876 23.425-44.577 23.257Q-44.277 23.090-43.932 23.090Q-43.426 23.090-43.143 23.313Q-42.859 23.537-42.859 24.033L-42.859 25.687Q-42.859 25.824-42.710 25.860Q-42.562 25.896-42.336 25.896L-42.336 26.176L-43.966 26.176L-43.966 25.896Q-43.741 25.896-43.590 25.860Q-43.440 25.824-43.440 25.687L-43.440 24.047Q-43.440 23.712-43.560 23.512Q-43.679 23.312-43.994 23.312Q-44.264 23.312-44.498 23.448Q-44.732 23.585-44.870 23.819Q-45.009 24.053-45.009 24.327L-45.009 25.687Q-45.009 25.824-44.860 25.860Q-44.712 25.896-44.486 25.896L-44.486 26.176L-46.116 26.176L-46.116 25.896Q-45.887 25.896-45.739 25.862Q-45.590 25.827-45.590 25.687L-45.590 24.047Q-45.590 23.712-45.710 23.512Q-45.829 23.312-46.144 23.312Q-46.414 23.312-46.648 23.448Q-46.882 23.585-47.020 23.819Q-47.159 24.053-47.159 24.327L-47.159 25.687Q-47.159 25.824-47.008 25.860Q-46.858 25.896-46.632 25.896L-46.632 26.176M-41.690 25.448Q-41.690 25.116-41.466 24.889Q-41.242 24.662-40.899 24.534Q-40.555 24.405-40.183 24.353Q-39.810 24.300-39.506 24.300L-39.506 24.047Q-39.506 23.842-39.614 23.662Q-39.721 23.483-39.902 23.380Q-40.084 23.278-40.292 23.278Q-40.699 23.278-40.935 23.370Q-40.846 23.407-40.800 23.491Q-40.754 23.575-40.754 23.677Q-40.754 23.773-40.800 23.852Q-40.846 23.930-40.926 23.975Q-41.006 24.019-41.095 24.019Q-41.246 24.019-41.347 23.922Q-41.447 23.824-41.447 23.677Q-41.447 23.055-40.292 23.055Q-40.080 23.055-39.831 23.119Q-39.581 23.182-39.380 23.301Q-39.178 23.421-39.051 23.606Q-38.925 23.790-38.925 24.033L-38.925 25.609Q-38.925 25.725-38.863 25.821Q-38.802 25.916-38.689 25.916Q-38.580 25.916-38.515 25.822Q-38.450 25.728-38.450 25.609L-38.450 25.161L-38.183 25.161L-38.183 25.609Q-38.183 25.879-38.411 26.044Q-38.638 26.210-38.918 26.210Q-39.127 26.210-39.263 26.056Q-39.400 25.903-39.424 25.687Q-39.571 25.954-39.853 26.099Q-40.135 26.244-40.460 26.244Q-40.736 26.244-41.020 26.169Q-41.304 26.094-41.497 25.915Q-41.690 25.735-41.690 25.448M-41.075 25.448Q-41.075 25.622-40.974 25.752Q-40.873 25.882-40.718 25.952Q-40.562 26.022-40.398 26.022Q-40.179 26.022-39.971 25.925Q-39.762 25.827-39.634 25.646Q-39.506 25.465-39.506 25.239L-39.506 24.511Q-39.831 24.511-40.196 24.602Q-40.562 24.693-40.818 24.905Q-41.075 25.116-41.075 25.448M-36.584 26.176L-37.906 26.176L-37.906 25.896Q-37.346 25.896-36.966 25.496L-36.252 24.699L-37.165 23.650Q-37.301 23.503-37.450 23.471Q-37.599 23.438-37.865 23.438L-37.865 23.158L-36.365 23.158L-36.365 23.438Q-36.556 23.438-36.556 23.572Q-36.556 23.602-36.526 23.650L-35.931 24.334L-35.490 23.838Q-35.377 23.708-35.377 23.592Q-35.377 23.530-35.415 23.484Q-35.452 23.438-35.510 23.438L-35.510 23.158L-34.194 23.158L-34.194 23.438Q-34.755 23.438-35.134 23.838L-35.756 24.539L-34.762 25.687Q-34.663 25.786-34.562 25.831Q-34.461 25.875-34.350 25.885Q-34.239 25.896-34.061 25.896L-34.061 26.176L-35.555 26.176L-35.555 25.896Q-35.490 25.896-35.430 25.862Q-35.370 25.827-35.370 25.762Q-35.370 25.715-35.401 25.687L-36.078 24.901L-36.611 25.496Q-36.724 25.626-36.724 25.742Q-36.724 25.807-36.683 25.851Q-36.642 25.896-36.584 25.896\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(94.5 -50.808)\">\u003Cpath d=\"M-28.720 27.926Q-29.270 27.526-29.641 26.971Q-30.012 26.415-30.193 25.769Q-30.374 25.123-30.374 24.426Q-30.374 23.913-30.274 23.418Q-30.173 22.922-29.968 22.471Q-29.763 22.020-29.450 21.628Q-29.137 21.237-28.720 20.933Q-28.710 20.929-28.703 20.928Q-28.696 20.926-28.686 20.926L-28.618 20.926Q-28.583 20.926-28.561 20.950Q-28.539 20.974-28.539 21.011Q-28.539 21.056-28.566 21.073Q-28.915 21.374-29.168 21.758Q-29.421 22.143-29.573 22.584Q-29.725 23.025-29.797 23.481Q-29.869 23.937-29.869 24.426Q-29.869 25.427-29.559 26.314Q-29.250 27.201-28.566 27.786Q-28.539 27.803-28.539 27.847Q-28.539 27.885-28.561 27.909Q-28.583 27.933-28.618 27.933L-28.686 27.933Q-28.693 27.929-28.701 27.928Q-28.710 27.926-28.720 27.926M-27.770 24.693Q-27.770 24.351-27.635 24.052Q-27.500 23.753-27.261 23.529Q-27.021 23.305-26.704 23.180Q-26.386 23.055-26.054 23.055Q-25.610 23.055-25.210 23.271Q-24.810 23.486-24.576 23.864Q-24.342 24.241-24.342 24.693Q-24.342 25.034-24.484 25.318Q-24.625 25.602-24.870 25.809Q-25.114 26.015-25.424 26.130Q-25.733 26.244-26.054 26.244Q-26.485 26.244-26.886 26.043Q-27.288 25.841-27.529 25.489Q-27.770 25.137-27.770 24.693M-26.054 25.995Q-25.453 25.995-25.229 25.617Q-25.005 25.239-25.005 24.607Q-25.005 23.995-25.239 23.636Q-25.473 23.278-26.054 23.278Q-27.107 23.278-27.107 24.607Q-27.107 25.239-26.881 25.617Q-26.656 25.995-26.054 25.995M-22.065 26.176L-23.699 26.176L-23.699 25.896Q-23.470 25.896-23.321 25.862Q-23.173 25.827-23.173 25.687L-23.173 23.838Q-23.173 23.568-23.280 23.507Q-23.388 23.445-23.699 23.445L-23.699 23.165L-22.640 23.090L-22.640 23.739Q-22.469 23.431-22.164 23.260Q-21.860 23.090-21.515 23.090Q-21.009 23.090-20.726 23.313Q-20.442 23.537-20.442 24.033L-20.442 25.687Q-20.442 25.824-20.293 25.860Q-20.144 25.896-19.919 25.896L-19.919 26.176L-21.549 26.176L-21.549 25.896Q-21.320 25.896-21.172 25.862Q-21.023 25.827-21.023 25.687L-21.023 24.047Q-21.023 23.712-21.143 23.512Q-21.262 23.312-21.577 23.312Q-21.847 23.312-22.081 23.448Q-22.315 23.585-22.453 23.819Q-22.592 24.053-22.592 24.327L-22.592 25.687Q-22.592 25.824-22.441 25.860Q-22.291 25.896-22.065 25.896\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(94.5 -50.808)\">\u003Cpath d=\"M-16.628 26.169L-16.628 25.106Q-16.628 25.082-16.600 25.055Q-16.573 25.028-16.549 25.028L-16.440 25.028Q-16.375 25.028-16.361 25.086Q-16.265 25.520-16.019 25.771Q-15.773 26.022-15.359 26.022Q-15.018 26.022-14.765 25.889Q-14.512 25.756-14.512 25.448Q-14.512 25.291-14.606 25.176Q-14.700 25.062-14.838 24.993Q-14.977 24.925-15.144 24.887L-15.725 24.788Q-16.081 24.720-16.354 24.499Q-16.628 24.279-16.628 23.937Q-16.628 23.688-16.516 23.513Q-16.405 23.339-16.219 23.240Q-16.033 23.141-15.817 23.098Q-15.602 23.055-15.359 23.055Q-14.946 23.055-14.666 23.237L-14.450 23.062Q-14.440 23.059-14.433 23.057Q-14.426 23.055-14.416 23.055L-14.365 23.055Q-14.338 23.055-14.314 23.079Q-14.290 23.103-14.290 23.131L-14.290 23.978Q-14.290 23.999-14.314 24.026Q-14.338 24.053-14.365 24.053L-14.478 24.053Q-14.505 24.053-14.531 24.028Q-14.556 24.002-14.556 23.978Q-14.556 23.742-14.662 23.578Q-14.768 23.414-14.951 23.332Q-15.134 23.250-15.366 23.250Q-15.694 23.250-15.951 23.353Q-16.207 23.455-16.207 23.732Q-16.207 23.927-16.024 24.036Q-15.841 24.146-15.612 24.187L-15.038 24.293Q-14.792 24.341-14.578 24.469Q-14.365 24.597-14.228 24.800Q-14.091 25.004-14.091 25.253Q-14.091 25.766-14.457 26.005Q-14.823 26.244-15.359 26.244Q-15.855 26.244-16.187 25.950L-16.453 26.224Q-16.474 26.244-16.501 26.244L-16.549 26.244Q-16.573 26.244-16.600 26.217Q-16.628 26.190-16.628 26.169M-12.888 25.342L-12.888 23.838Q-12.888 23.568-12.996 23.507Q-13.104 23.445-13.415 23.445L-13.415 23.165L-12.307 23.090L-12.307 25.322L-12.307 25.342Q-12.307 25.622-12.256 25.766Q-12.205 25.909-12.063 25.966Q-11.921 26.022-11.634 26.022Q-11.381 26.022-11.176 25.882Q-10.971 25.742-10.855 25.516Q-10.738 25.291-10.738 25.041L-10.738 23.838Q-10.738 23.568-10.846 23.507Q-10.954 23.445-11.265 23.445L-11.265 23.165L-10.157 23.090L-10.157 25.503Q-10.157 25.694-10.104 25.776Q-10.051 25.858-9.951 25.877Q-9.850 25.896-9.634 25.896L-9.634 26.176L-10.711 26.244L-10.711 25.680Q-10.820 25.862-10.966 25.985Q-11.111 26.108-11.297 26.176Q-11.484 26.244-11.685 26.244Q-12.888 26.244-12.888 25.342M-7.402 27.533L-9.033 27.533L-9.033 27.253Q-8.804 27.253-8.655 27.218Q-8.506 27.184-8.506 27.044L-8.506 23.698Q-8.506 23.527-8.643 23.486Q-8.780 23.445-9.033 23.445L-9.033 23.165L-7.953 23.090L-7.953 23.496Q-7.731 23.295-7.443 23.192Q-7.156 23.090-6.849 23.090Q-6.421 23.090-6.057 23.303Q-5.693 23.517-5.480 23.881Q-5.266 24.245-5.266 24.665Q-5.266 25.110-5.505 25.474Q-5.745 25.838-6.138 26.041Q-6.531 26.244-6.975 26.244Q-7.242 26.244-7.490 26.144Q-7.737 26.043-7.925 25.862L-7.925 27.044Q-7.925 27.181-7.777 27.217Q-7.628 27.253-7.402 27.253L-7.402 27.533M-7.925 23.845L-7.925 25.455Q-7.792 25.708-7.549 25.865Q-7.307 26.022-7.030 26.022Q-6.702 26.022-6.449 25.821Q-6.196 25.619-6.063 25.301Q-5.929 24.983-5.929 24.665Q-5.929 24.436-5.994 24.207Q-6.059 23.978-6.187 23.780Q-6.316 23.582-6.510 23.462Q-6.705 23.343-6.938 23.343Q-7.232 23.343-7.500 23.472Q-7.768 23.602-7.925 23.845M-2.986 27.533L-4.617 27.533L-4.617 27.253Q-4.388 27.253-4.239 27.218Q-4.090 27.184-4.090 27.044L-4.090 23.698Q-4.090 23.527-4.227 23.486Q-4.364 23.445-4.617 23.445L-4.617 23.165L-3.537 23.090L-3.537 23.496Q-3.315 23.295-3.027 23.192Q-2.740 23.090-2.433 23.090Q-2.005 23.090-1.641 23.303Q-1.277 23.517-1.064 23.881Q-0.850 24.245-0.850 24.665Q-0.850 25.110-1.089 25.474Q-1.329 25.838-1.722 26.041Q-2.115 26.244-2.559 26.244Q-2.826 26.244-3.074 26.144Q-3.321 26.043-3.509 25.862L-3.509 27.044Q-3.509 27.181-3.361 27.217Q-3.212 27.253-2.986 27.253L-2.986 27.533M-3.509 23.845L-3.509 25.455Q-3.376 25.708-3.133 25.865Q-2.891 26.022-2.614 26.022Q-2.286 26.022-2.033 25.821Q-1.780 25.619-1.647 25.301Q-1.513 24.983-1.513 24.665Q-1.513 24.436-1.578 24.207Q-1.643 23.978-1.771 23.780Q-1.900 23.582-2.094 23.462Q-2.289 23.343-2.522 23.343Q-2.816 23.343-3.084 23.472Q-3.352 23.602-3.509 23.845\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(94.5 -50.808)\">\u003Cpath d=\"M-0.035 24.693Q-0.035 24.351 0.100 24.052Q0.235 23.753 0.475 23.529Q0.714 23.305 1.032 23.180Q1.350 23.055 1.681 23.055Q2.126 23.055 2.525 23.271Q2.925 23.486 3.160 23.864Q3.394 24.241 3.394 24.693Q3.394 25.034 3.252 25.318Q3.110 25.602 2.866 25.809Q2.621 26.015 2.312 26.130Q2.003 26.244 1.681 26.244Q1.251 26.244 0.849 26.043Q0.447 25.841 0.206 25.489Q-0.035 25.137-0.035 24.693M1.681 25.995Q2.283 25.995 2.507 25.617Q2.731 25.239 2.731 24.607Q2.731 23.995 2.496 23.636Q2.262 23.278 1.681 23.278Q0.629 23.278 0.629 24.607Q0.629 25.239 0.854 25.617Q1.080 25.995 1.681 25.995M5.738 26.176L4.002 26.176L4.002 25.896Q4.231 25.896 4.380 25.862Q4.528 25.827 4.528 25.687L4.528 23.838Q4.528 23.568 4.421 23.507Q4.313 23.445 4.002 23.445L4.002 23.165L5.031 23.090L5.031 23.797Q5.161 23.489 5.403 23.290Q5.646 23.090 5.964 23.090Q6.183 23.090 6.354 23.214Q6.525 23.339 6.525 23.551Q6.525 23.688 6.425 23.787Q6.326 23.886 6.193 23.886Q6.056 23.886 5.957 23.787Q5.858 23.688 5.858 23.551Q5.858 23.411 5.957 23.312Q5.667 23.312 5.467 23.508Q5.267 23.705 5.174 23.999Q5.082 24.293 5.082 24.573L5.082 25.687Q5.082 25.896 5.738 25.896L5.738 26.176M7.635 25.335L7.635 23.438L6.996 23.438L6.996 23.216Q7.314 23.216 7.531 23.006Q7.748 22.796 7.849 22.486Q7.950 22.177 7.950 21.869L8.216 21.869L8.216 23.158L9.293 23.158L9.293 23.438L8.216 23.438L8.216 25.322Q8.216 25.598 8.321 25.797Q8.425 25.995 8.685 25.995Q8.842 25.995 8.948 25.891Q9.054 25.786 9.103 25.633Q9.153 25.479 9.153 25.322L9.153 24.908L9.420 24.908L9.420 25.335Q9.420 25.561 9.320 25.771Q9.221 25.981 9.037 26.113Q8.852 26.244 8.623 26.244Q8.186 26.244 7.911 26.007Q7.635 25.769 7.635 25.335M10.551 27.933L10.483 27.933Q10.448 27.933 10.426 27.907Q10.404 27.882 10.404 27.847Q10.404 27.803 10.435 27.786Q10.790 27.482 11.040 27.092Q11.289 26.702 11.441 26.270Q11.593 25.838 11.663 25.369Q11.734 24.901 11.734 24.426Q11.734 23.947 11.663 23.481Q11.593 23.014 11.440 22.579Q11.286 22.143 11.035 21.755Q10.783 21.367 10.435 21.073Q10.404 21.056 10.404 21.011Q10.404 20.977 10.426 20.952Q10.448 20.926 10.483 20.926L10.551 20.926Q10.561 20.926 10.570 20.928Q10.578 20.929 10.588 20.933Q11.132 21.333 11.505 21.886Q11.877 22.440 12.058 23.086Q12.239 23.732 12.239 24.426Q12.239 25.127 12.058 25.774Q11.877 26.422 11.503 26.976Q11.129 27.530 10.588 27.926Q10.578 27.926 10.570 27.928Q10.561 27.929 10.551 27.933\" 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\">BCQ's constrained argmax. A generative model of the behavior policy proposes candidate actions clustered on the data support (blue); each is allowed a small bounded perturbation (arrows); the argmax picks the best perturbed candidate. Because every candidate stays near the data, the value function is never queried at the far-off-support actions where it overshoots (red region).\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:442.083px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 331.562 151.710\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-45.303 61.208h265.456\"\u002F>\u003Cpath stroke=\"none\" d=\"m222.153 61.208-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-39.276 60.376Q-39.276 59.892-38.874 59.597Q-38.471 59.302-37.921 59.183Q-37.370 59.063-36.878 59.063L-36.878 58.774Q-36.878 58.548-36.993 58.341Q-37.108 58.134-37.305 58.015Q-37.503 57.895-37.733 57.895Q-38.159 57.895-38.444 58.001Q-38.374 58.028-38.327 58.083Q-38.280 58.138-38.255 58.208Q-38.229 58.278-38.229 58.353Q-38.229 58.458-38.280 58.550Q-38.331 58.642-38.423 58.692Q-38.514 58.743-38.620 58.743Q-38.725 58.743-38.817 58.692Q-38.909 58.642-38.960 58.550Q-39.010 58.458-39.010 58.353Q-39.010 57.935-38.622 57.788Q-38.233 57.642-37.733 57.642Q-37.401 57.642-37.048 57.772Q-36.694 57.903-36.466 58.157Q-36.237 58.411-36.237 58.759L-36.237 60.560Q-36.237 60.692-36.165 60.802Q-36.092 60.911-35.964 60.911Q-35.839 60.911-35.770 60.806Q-35.702 60.700-35.702 60.560L-35.702 60.048L-35.421 60.048L-35.421 60.560Q-35.421 60.763-35.538 60.921Q-35.655 61.079-35.837 61.163Q-36.018 61.247-36.221 61.247Q-36.452 61.247-36.604 61.075Q-36.757 60.903-36.788 60.673Q-36.948 60.954-37.257 61.120Q-37.565 61.286-37.917 61.286Q-38.428 61.286-38.852 61.063Q-39.276 60.841-39.276 60.376M-38.589 60.376Q-38.589 60.661-38.362 60.847Q-38.135 61.032-37.842 61.032Q-37.596 61.032-37.372 60.915Q-37.147 60.798-37.012 60.595Q-36.878 60.392-36.878 60.138L-36.878 59.306Q-37.143 59.306-37.428 59.360Q-37.714 59.415-37.985 59.544Q-38.257 59.673-38.423 59.880Q-38.589 60.087-38.589 60.376M-35.085 59.481Q-35.085 58.985-34.835 58.560Q-34.585 58.134-34.165 57.888Q-33.745 57.642-33.245 57.642Q-32.706 57.642-32.315 57.767Q-31.925 57.892-31.925 58.306Q-31.925 58.411-31.975 58.503Q-32.026 58.595-32.118 58.645Q-32.210 58.696-32.319 58.696Q-32.425 58.696-32.516 58.645Q-32.608 58.595-32.659 58.503Q-32.710 58.411-32.710 58.306Q-32.710 58.083-32.542 57.978Q-32.764 57.919-33.237 57.919Q-33.534 57.919-33.749 58.058Q-33.964 58.196-34.094 58.427Q-34.225 58.657-34.284 58.927Q-34.342 59.196-34.342 59.481Q-34.342 59.876-34.210 60.226Q-34.077 60.575-33.805 60.792Q-33.534 61.009-33.135 61.009Q-32.760 61.009-32.485 60.792Q-32.210 60.575-32.108 60.216Q-32.092 60.153-32.030 60.153L-31.925 60.153Q-31.889 60.153-31.864 60.181Q-31.839 60.208-31.839 60.247L-31.839 60.270Q-31.971 60.751-32.356 61.019Q-32.741 61.286-33.245 61.286Q-33.608 61.286-33.942 61.149Q-34.276 61.013-34.536 60.763Q-34.796 60.513-34.940 60.177Q-35.085 59.841-35.085 59.481M-30.725 60.247L-30.725 58.056L-31.428 58.056L-31.428 57.802Q-31.073 57.802-30.831 57.569Q-30.589 57.337-30.477 56.989Q-30.366 56.642-30.366 56.286L-30.085 56.286L-30.085 57.759L-28.909 57.759L-28.909 58.056L-30.085 58.056L-30.085 60.231Q-30.085 60.552-29.966 60.780Q-29.846 61.009-29.565 61.009Q-29.385 61.009-29.268 60.886Q-29.151 60.763-29.098 60.583Q-29.046 60.403-29.046 60.231L-29.046 59.759L-28.764 59.759L-28.764 60.247Q-28.764 60.501-28.870 60.741Q-28.975 60.981-29.173 61.134Q-29.370 61.286-29.628 61.286Q-29.944 61.286-30.196 61.163Q-30.448 61.040-30.587 60.806Q-30.725 60.571-30.725 60.247M-26.186 61.208L-27.964 61.208L-27.964 60.911Q-27.690 60.911-27.522 60.864Q-27.354 60.817-27.354 60.649L-27.354 58.513Q-27.354 58.298-27.411 58.202Q-27.467 58.106-27.581 58.085Q-27.694 58.063-27.940 58.063L-27.940 57.767L-26.741 57.681L-26.741 60.649Q-26.741 60.817-26.594 60.864Q-26.448 60.911-26.186 60.911L-26.186 61.208M-27.628 56.286Q-27.628 56.095-27.493 55.964Q-27.358 55.833-27.163 55.833Q-27.042 55.833-26.938 55.895Q-26.835 55.958-26.772 56.062Q-26.710 56.165-26.710 56.286Q-26.710 56.481-26.841 56.616Q-26.971 56.751-27.163 56.751Q-27.362 56.751-27.495 56.618Q-27.628 56.485-27.628 56.286M-25.686 59.513Q-25.686 59.009-25.430 58.577Q-25.175 58.145-24.739 57.894Q-24.303 57.642-23.803 57.642Q-23.417 57.642-23.075 57.786Q-22.733 57.931-22.471 58.192Q-22.210 58.454-22.067 58.790Q-21.925 59.126-21.925 59.513Q-21.925 60.005-22.188 60.415Q-22.452 60.825-22.882 61.056Q-23.311 61.286-23.803 61.286Q-24.296 61.286-24.729 61.054Q-25.163 60.821-25.425 60.413Q-25.686 60.005-25.686 59.513M-23.803 61.009Q-23.346 61.009-23.094 60.786Q-22.842 60.563-22.755 60.212Q-22.667 59.860-22.667 59.415Q-22.667 58.985-22.760 58.647Q-22.854 58.310-23.108 58.103Q-23.362 57.895-23.803 57.895Q-24.452 57.895-24.696 58.312Q-24.940 58.728-24.940 59.415Q-24.940 59.860-24.852 60.212Q-24.764 60.563-24.512 60.786Q-24.260 61.009-23.803 61.009M-19.510 61.208L-21.366 61.208L-21.366 60.911Q-21.092 60.911-20.925 60.864Q-20.757 60.817-20.757 60.649L-20.757 58.513Q-20.757 58.298-20.819 58.202Q-20.882 58.106-21.001 58.085Q-21.120 58.063-21.366 58.063L-21.366 57.767L-20.175 57.681L-20.175 58.415Q-20.061 58.200-19.868 58.032Q-19.675 57.864-19.436 57.772Q-19.198 57.681-18.944 57.681Q-17.776 57.681-17.776 58.759L-17.776 60.649Q-17.776 60.817-17.606 60.864Q-17.436 60.911-17.167 60.911L-17.167 61.208L-19.022 61.208L-19.022 60.911Q-18.749 60.911-18.581 60.864Q-18.413 60.817-18.413 60.649L-18.413 58.774Q-18.413 58.392-18.534 58.163Q-18.655 57.935-19.007 57.935Q-19.319 57.935-19.573 58.097Q-19.827 58.259-19.973 58.528Q-20.120 58.798-20.120 59.095L-20.120 60.649Q-20.120 60.817-19.950 60.864Q-19.780 60.911-19.510 60.911\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(246.758 8.898)\">\u003Cpath d=\"M-12.608 61.286Q-12.964 61.286-13.233 61.106Q-13.503 60.927-13.643 60.632Q-13.784 60.337-13.784 59.978Q-13.784 59.591-13.622 59.177Q-13.460 58.763-13.176 58.425Q-12.893 58.087-12.524 57.884Q-12.155 57.681-11.753 57.681Q-11.260 57.681-10.983 58.130Q-10.952 57.997-10.848 57.915Q-10.745 57.833-10.616 57.833Q-10.503 57.833-10.423 57.903Q-10.342 57.974-10.342 58.087Q-10.342 58.114-10.358 58.177L-10.913 60.376Q-10.952 60.622-10.952 60.673Q-10.952 61.032-10.706 61.032Q-10.561 61.032-10.460 60.925Q-10.358 60.817-10.294 60.663Q-10.229 60.509-10.180 60.319Q-10.132 60.130-10.112 60.032Q-10.085 59.962-10.022 59.962L-9.921 59.962Q-9.882 59.962-9.856 59.995Q-9.831 60.028-9.831 60.056Q-9.831 60.071-9.839 60.087Q-9.952 60.579-10.151 60.933Q-10.350 61.286-10.721 61.286Q-11.003 61.286-11.229 61.144Q-11.456 61.001-11.538 60.743Q-11.760 60.985-12.034 61.136Q-12.307 61.286-12.608 61.286M-12.592 61.032Q-12.382 61.032-12.173 60.919Q-11.964 60.806-11.794 60.632Q-11.624 60.458-11.495 60.263Q-11.507 60.278-11.516 60.292Q-11.526 60.306-11.538 60.321L-11.104 58.599Q-11.143 58.419-11.229 58.269Q-11.315 58.118-11.452 58.026Q-11.589 57.935-11.768 57.935Q-12.104 57.935-12.376 58.216Q-12.647 58.497-12.807 58.872Q-12.932 59.192-13.030 59.612Q-13.128 60.032-13.128 60.313Q-13.128 60.603-12.995 60.817Q-12.862 61.032-12.592 61.032\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-39.612 66.899V-61.983\"\u002F>\u003Cpath stroke=\"none\" d=\"m-39.612-63.983-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-22.658 -122.414)\">\u003Cpath d=\"M-37.573 61.177L-38.796 58.321Q-38.878 58.145-39.022 58.101Q-39.167 58.056-39.436 58.056L-39.436 57.759L-37.725 57.759L-37.725 58.056Q-38.147 58.056-38.147 58.239Q-38.147 58.274-38.132 58.321L-37.186 60.513L-36.346 58.536Q-36.307 58.458-36.307 58.368Q-36.307 58.228-36.413 58.142Q-36.518 58.056-36.659 58.056L-36.659 57.759L-35.307 57.759L-35.307 58.056Q-35.831 58.056-36.046 58.536L-37.171 61.177Q-37.233 61.286-37.339 61.286L-37.405 61.286Q-37.518 61.286-37.573 61.177\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-22.658 -122.414)\">\u003Cpath d=\"M-35.262 60.376Q-35.262 59.892-34.860 59.597Q-34.457 59.302-33.907 59.183Q-33.356 59.063-32.864 59.063L-32.864 58.774Q-32.864 58.548-32.979 58.341Q-33.094 58.134-33.291 58.015Q-33.489 57.895-33.719 57.895Q-34.145 57.895-34.430 58.001Q-34.360 58.028-34.313 58.083Q-34.266 58.138-34.241 58.208Q-34.215 58.278-34.215 58.353Q-34.215 58.458-34.266 58.550Q-34.317 58.642-34.409 58.692Q-34.500 58.743-34.606 58.743Q-34.711 58.743-34.803 58.692Q-34.895 58.642-34.946 58.550Q-34.996 58.458-34.996 58.353Q-34.996 57.935-34.608 57.788Q-34.219 57.642-33.719 57.642Q-33.387 57.642-33.034 57.772Q-32.680 57.903-32.452 58.157Q-32.223 58.411-32.223 58.759L-32.223 60.560Q-32.223 60.692-32.151 60.802Q-32.078 60.911-31.950 60.911Q-31.825 60.911-31.756 60.806Q-31.688 60.700-31.688 60.560L-31.688 60.048L-31.407 60.048L-31.407 60.560Q-31.407 60.763-31.524 60.921Q-31.641 61.079-31.823 61.163Q-32.004 61.247-32.207 61.247Q-32.438 61.247-32.590 61.075Q-32.743 60.903-32.774 60.673Q-32.934 60.954-33.243 61.120Q-33.551 61.286-33.903 61.286Q-34.414 61.286-34.838 61.063Q-35.262 60.841-35.262 60.376M-34.575 60.376Q-34.575 60.661-34.348 60.847Q-34.121 61.032-33.828 61.032Q-33.582 61.032-33.358 60.915Q-33.133 60.798-32.998 60.595Q-32.864 60.392-32.864 60.138L-32.864 59.306Q-33.129 59.306-33.414 59.360Q-33.700 59.415-33.971 59.544Q-34.243 59.673-34.409 59.880Q-34.575 60.087-34.575 60.376M-29.200 61.208L-31.032 61.208L-31.032 60.911Q-30.758 60.911-30.590 60.864Q-30.422 60.817-30.422 60.649L-30.422 56.489Q-30.422 56.274-30.485 56.179Q-30.547 56.083-30.666 56.062Q-30.785 56.040-31.032 56.040L-31.032 55.743L-29.809 55.657L-29.809 60.649Q-29.809 60.817-29.641 60.864Q-29.473 60.911-29.200 60.911L-29.200 61.208M-28.071 60.255L-28.071 58.513Q-28.071 58.298-28.133 58.202Q-28.196 58.106-28.315 58.085Q-28.434 58.063-28.680 58.063L-28.680 57.767L-27.434 57.681L-27.434 60.231L-27.434 60.255Q-27.434 60.567-27.379 60.729Q-27.325 60.892-27.174 60.962Q-27.024 61.032-26.703 61.032Q-26.274 61.032-26 60.694Q-25.727 60.356-25.727 59.911L-25.727 58.513Q-25.727 58.298-25.789 58.202Q-25.852 58.106-25.971 58.085Q-26.090 58.063-26.336 58.063L-26.336 57.767L-25.090 57.681L-25.090 60.466Q-25.090 60.677-25.028 60.772Q-24.965 60.868-24.846 60.890Q-24.727 60.911-24.481 60.911L-24.481 61.208L-25.703 61.286L-25.703 60.665Q-25.871 60.954-26.153 61.120Q-26.434 61.286-26.754 61.286Q-28.071 61.286-28.071 60.255M-24.035 59.454Q-24.035 58.974-23.803 58.558Q-23.571 58.142-23.160 57.892Q-22.750 57.642-22.274 57.642Q-21.543 57.642-21.145 58.083Q-20.746 58.524-20.746 59.255Q-20.746 59.360-20.840 59.384L-23.289 59.384L-23.289 59.454Q-23.289 59.864-23.168 60.220Q-23.047 60.575-22.776 60.792Q-22.504 61.009-22.075 61.009Q-21.711 61.009-21.414 60.780Q-21.118 60.552-21.016 60.200Q-21.008 60.153-20.922 60.138L-20.840 60.138Q-20.746 60.165-20.746 60.247Q-20.746 60.255-20.754 60.286Q-20.817 60.513-20.955 60.696Q-21.094 60.880-21.285 61.013Q-21.477 61.145-21.696 61.216Q-21.914 61.286-22.153 61.286Q-22.524 61.286-22.862 61.149Q-23.200 61.013-23.467 60.761Q-23.735 60.509-23.885 60.169Q-24.035 59.829-24.035 59.454M-23.282 59.145L-21.321 59.145Q-21.321 58.841-21.422 58.550Q-21.524 58.259-21.741 58.077Q-21.957 57.895-22.274 57.895Q-22.575 57.895-22.805 58.083Q-23.035 58.270-23.159 58.562Q-23.282 58.853-23.282 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"none\" d=\"M34.365 61.208v-119.5h79.668v119.5Zm79.668-119.5\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-37.557 61.286Q-38.038 61.286-38.446 61.042Q-38.854 60.798-39.092 60.384Q-39.331 59.970-39.331 59.481Q-39.331 58.989-39.073 58.573Q-38.815 58.157-38.383 57.919Q-37.952 57.681-37.460 57.681Q-36.839 57.681-36.389 58.118L-36.389 56.489Q-36.389 56.274-36.452 56.179Q-36.514 56.083-36.632 56.062Q-36.749 56.040-36.995 56.040L-36.995 55.743L-35.772 55.657L-35.772 60.466Q-35.772 60.677-35.710 60.772Q-35.647 60.868-35.530 60.890Q-35.413 60.911-35.163 60.911L-35.163 61.208L-36.413 61.286L-36.413 60.802Q-36.878 61.286-37.557 61.286M-37.491 61.032Q-37.151 61.032-36.858 60.841Q-36.565 60.649-36.413 60.353L-36.413 58.520Q-36.561 58.247-36.823 58.091Q-37.085 57.935-37.397 57.935Q-38.022 57.935-38.305 58.382Q-38.589 58.829-38.589 59.489Q-38.589 60.134-38.337 60.583Q-38.085 61.032-37.491 61.032M-34.557 60.376Q-34.557 59.892-34.155 59.597Q-33.753 59.302-33.202 59.183Q-32.651 59.063-32.159 59.063L-32.159 58.774Q-32.159 58.548-32.274 58.341Q-32.389 58.134-32.587 58.015Q-32.784 57.895-33.014 57.895Q-33.440 57.895-33.725 58.001Q-33.655 58.028-33.608 58.083Q-33.561 58.138-33.536 58.208Q-33.510 58.278-33.510 58.353Q-33.510 58.458-33.561 58.550Q-33.612 58.642-33.704 58.692Q-33.796 58.743-33.901 58.743Q-34.007 58.743-34.098 58.692Q-34.190 58.642-34.241 58.550Q-34.292 58.458-34.292 58.353Q-34.292 57.935-33.903 57.788Q-33.514 57.642-33.014 57.642Q-32.682 57.642-32.329 57.772Q-31.975 57.903-31.747 58.157Q-31.518 58.411-31.518 58.759L-31.518 60.560Q-31.518 60.692-31.446 60.802Q-31.374 60.911-31.245 60.911Q-31.120 60.911-31.051 60.806Q-30.983 60.700-30.983 60.560L-30.983 60.048L-30.702 60.048L-30.702 60.560Q-30.702 60.763-30.819 60.921Q-30.936 61.079-31.118 61.163Q-31.300 61.247-31.503 61.247Q-31.733 61.247-31.885 61.075Q-32.038 60.903-32.069 60.673Q-32.229 60.954-32.538 61.120Q-32.846 61.286-33.198 61.286Q-33.710 61.286-34.133 61.063Q-34.557 60.841-34.557 60.376M-33.870 60.376Q-33.870 60.661-33.643 60.847Q-33.417 61.032-33.124 61.032Q-32.878 61.032-32.653 60.915Q-32.428 60.798-32.294 60.595Q-32.159 60.392-32.159 60.138L-32.159 59.306Q-32.425 59.306-32.710 59.360Q-32.995 59.415-33.266 59.544Q-33.538 59.673-33.704 59.880Q-33.870 60.087-33.870 60.376M-29.784 60.247L-29.784 58.056L-30.487 58.056L-30.487 57.802Q-30.132 57.802-29.889 57.569Q-29.647 57.337-29.536 56.989Q-29.425 56.642-29.425 56.286L-29.143 56.286L-29.143 57.759L-27.967 57.759L-27.967 58.056L-29.143 58.056L-29.143 60.231Q-29.143 60.552-29.024 60.780Q-28.905 61.009-28.624 61.009Q-28.444 61.009-28.327 60.886Q-28.210 60.763-28.157 60.583Q-28.104 60.403-28.104 60.231L-28.104 59.759L-27.823 59.759L-27.823 60.247Q-27.823 60.501-27.928 60.741Q-28.034 60.981-28.231 61.134Q-28.428 61.286-28.686 61.286Q-29.003 61.286-29.255 61.163Q-29.507 61.040-29.645 60.806Q-29.784 60.571-29.784 60.247M-27.007 60.376Q-27.007 59.892-26.604 59.597Q-26.202 59.302-25.651 59.183Q-25.100 59.063-24.608 59.063L-24.608 58.774Q-24.608 58.548-24.723 58.341Q-24.839 58.134-25.036 58.015Q-25.233 57.895-25.464 57.895Q-25.889 57.895-26.175 58.001Q-26.104 58.028-26.057 58.083Q-26.010 58.138-25.985 58.208Q-25.960 58.278-25.960 58.353Q-25.960 58.458-26.010 58.550Q-26.061 58.642-26.153 58.692Q-26.245 58.743-26.350 58.743Q-26.456 58.743-26.548 58.692Q-26.639 58.642-26.690 58.550Q-26.741 58.458-26.741 58.353Q-26.741 57.935-26.352 57.788Q-25.964 57.642-25.464 57.642Q-25.132 57.642-24.778 57.772Q-24.425 57.903-24.196 58.157Q-23.967 58.411-23.967 58.759L-23.967 60.560Q-23.967 60.692-23.895 60.802Q-23.823 60.911-23.694 60.911Q-23.569 60.911-23.501 60.806Q-23.432 60.700-23.432 60.560L-23.432 60.048L-23.151 60.048L-23.151 60.560Q-23.151 60.763-23.268 60.921Q-23.385 61.079-23.567 61.163Q-23.749 61.247-23.952 61.247Q-24.182 61.247-24.335 61.075Q-24.487 60.903-24.518 60.673Q-24.678 60.954-24.987 61.120Q-25.296 61.286-25.647 61.286Q-26.159 61.286-26.583 61.063Q-27.007 60.841-27.007 60.376M-26.319 60.376Q-26.319 60.661-26.092 60.847Q-25.866 61.032-25.573 61.032Q-25.327 61.032-25.102 60.915Q-24.878 60.798-24.743 60.595Q-24.608 60.392-24.608 60.138L-24.608 59.306Q-24.874 59.306-25.159 59.360Q-25.444 59.415-25.716 59.544Q-25.987 59.673-26.153 59.880Q-26.319 60.087-26.319 60.376\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-19.970 61.200L-19.970 59.978Q-19.970 59.950-19.939 59.919Q-19.907 59.888-19.884 59.888L-19.778 59.888Q-19.708 59.888-19.692 59.950Q-19.630 60.270-19.491 60.511Q-19.353 60.751-19.120 60.892Q-18.888 61.032-18.579 61.032Q-18.341 61.032-18.132 60.972Q-17.923 60.911-17.786 60.763Q-17.649 60.614-17.649 60.368Q-17.649 60.114-17.860 59.948Q-18.071 59.782-18.341 59.728L-18.962 59.614Q-19.368 59.536-19.669 59.280Q-19.970 59.024-19.970 58.649Q-19.970 58.282-19.769 58.060Q-19.567 57.837-19.243 57.739Q-18.919 57.642-18.579 57.642Q-18.114 57.642-17.817 57.849L-17.595 57.665Q-17.571 57.642-17.540 57.642L-17.489 57.642Q-17.458 57.642-17.431 57.669Q-17.403 57.696-17.403 57.728L-17.403 58.712Q-17.403 58.743-17.429 58.772Q-17.454 58.802-17.489 58.802L-17.595 58.802Q-17.630 58.802-17.657 58.774Q-17.685 58.747-17.685 58.712Q-17.685 58.313-17.937 58.093Q-18.189 57.872-18.587 57.872Q-18.942 57.872-19.226 57.995Q-19.509 58.118-19.509 58.423Q-19.509 58.642-19.308 58.774Q-19.106 58.907-18.860 58.950L-18.235 59.063Q-17.806 59.153-17.497 59.450Q-17.189 59.747-17.189 60.161Q-17.189 60.731-17.587 61.009Q-17.985 61.286-18.579 61.286Q-19.130 61.286-19.481 60.950L-19.778 61.263Q-19.802 61.286-19.837 61.286L-19.884 61.286Q-19.907 61.286-19.939 61.255Q-19.970 61.224-19.970 61.200M-15.978 60.255L-15.978 58.513Q-15.978 58.298-16.040 58.202Q-16.103 58.106-16.222 58.085Q-16.341 58.063-16.587 58.063L-16.587 57.767L-15.341 57.681L-15.341 60.231L-15.341 60.255Q-15.341 60.567-15.286 60.729Q-15.231 60.892-15.081 60.962Q-14.931 61.032-14.610 61.032Q-14.181 61.032-13.907 60.694Q-13.634 60.356-13.634 59.911L-13.634 58.513Q-13.634 58.298-13.696 58.202Q-13.759 58.106-13.878 58.085Q-13.997 58.063-14.243 58.063L-14.243 57.767L-12.997 57.681L-12.997 60.466Q-12.997 60.677-12.935 60.772Q-12.872 60.868-12.753 60.890Q-12.634 60.911-12.388 60.911L-12.388 61.208L-13.610 61.286L-13.610 60.665Q-13.778 60.954-14.060 61.120Q-14.341 61.286-14.661 61.286Q-15.978 61.286-15.978 60.255M-10.060 62.759L-11.915 62.759L-11.915 62.466Q-11.646 62.466-11.478 62.421Q-11.310 62.376-11.310 62.200L-11.310 58.376Q-11.310 58.169-11.466 58.116Q-11.622 58.063-11.915 58.063L-11.915 57.767L-10.692 57.681L-10.692 58.145Q-10.462 57.923-10.147 57.802Q-9.833 57.681-9.493 57.681Q-9.021 57.681-8.616 57.927Q-8.212 58.173-7.980 58.589Q-7.747 59.005-7.747 59.481Q-7.747 59.856-7.896 60.185Q-8.044 60.513-8.314 60.765Q-8.583 61.017-8.927 61.151Q-9.271 61.286-9.630 61.286Q-9.919 61.286-10.190 61.165Q-10.462 61.044-10.669 60.833L-10.669 62.200Q-10.669 62.376-10.501 62.421Q-10.333 62.466-10.060 62.466L-10.060 62.759M-10.669 58.544L-10.669 60.384Q-10.517 60.673-10.255 60.853Q-9.993 61.032-9.685 61.032Q-9.399 61.032-9.177 60.894Q-8.954 60.755-8.802 60.524Q-8.649 60.294-8.571 60.022Q-8.493 59.751-8.493 59.481Q-8.493 59.149-8.618 58.792Q-8.743 58.435-8.991 58.198Q-9.239 57.962-9.587 57.962Q-9.911 57.962-10.206 58.118Q-10.501 58.274-10.669 58.544M-5.341 62.759L-7.196 62.759L-7.196 62.466Q-6.927 62.466-6.759 62.421Q-6.591 62.376-6.591 62.200L-6.591 58.376Q-6.591 58.169-6.747 58.116Q-6.903 58.063-7.196 58.063L-7.196 57.767L-5.974 57.681L-5.974 58.145Q-5.743 57.923-5.429 57.802Q-5.114 57.681-4.774 57.681Q-4.302 57.681-3.897 57.927Q-3.493 58.173-3.261 58.589Q-3.028 59.005-3.028 59.481Q-3.028 59.856-3.177 60.185Q-3.325 60.513-3.595 60.765Q-3.864 61.017-4.208 61.151Q-4.552 61.286-4.911 61.286Q-5.200 61.286-5.472 61.165Q-5.743 61.044-5.950 60.833L-5.950 62.200Q-5.950 62.376-5.782 62.421Q-5.614 62.466-5.341 62.466L-5.341 62.759M-5.950 58.544L-5.950 60.384Q-5.798 60.673-5.536 60.853Q-5.274 61.032-4.966 61.032Q-4.681 61.032-4.458 60.894Q-4.235 60.755-4.083 60.524Q-3.931 60.294-3.853 60.022Q-3.774 59.751-3.774 59.481Q-3.774 59.149-3.899 58.792Q-4.024 58.435-4.272 58.198Q-4.521 57.962-4.868 57.962Q-5.192 57.962-5.487 58.118Q-5.782 58.274-5.950 58.544\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(89.818 -124.59)\">\u003Cpath d=\"M-2.257 59.513Q-2.257 59.009-2.001 58.577Q-1.745 58.145-1.309 57.894Q-0.874 57.642-0.374 57.642Q0.013 57.642 0.355 57.786Q0.696 57.931 0.958 58.192Q1.220 58.454 1.362 58.790Q1.505 59.126 1.505 59.513Q1.505 60.005 1.241 60.415Q0.978 60.825 0.548 61.056Q0.118 61.286-0.374 61.286Q-0.866 61.286-1.300 61.054Q-1.733 60.821-1.995 60.413Q-2.257 60.005-2.257 59.513M-0.374 61.009Q0.083 61.009 0.335 60.786Q0.587 60.563 0.675 60.212Q0.763 59.860 0.763 59.415Q0.763 58.985 0.669 58.647Q0.575 58.310 0.321 58.103Q0.067 57.895-0.374 57.895Q-1.022 57.895-1.266 58.312Q-1.511 58.728-1.511 59.415Q-1.511 59.860-1.423 60.212Q-1.335 60.563-1.083 60.786Q-0.831 61.009-0.374 61.009M3.997 61.208L2.017 61.208L2.017 60.911Q2.286 60.911 2.454 60.866Q2.622 60.821 2.622 60.649L2.622 58.513Q2.622 58.298 2.560 58.202Q2.497 58.106 2.380 58.085Q2.263 58.063 2.017 58.063L2.017 57.767L3.185 57.681L3.185 58.466Q3.263 58.255 3.415 58.069Q3.567 57.884 3.767 57.782Q3.966 57.681 4.192 57.681Q4.439 57.681 4.630 57.825Q4.821 57.970 4.821 58.200Q4.821 58.356 4.716 58.466Q4.610 58.575 4.454 58.575Q4.298 58.575 4.189 58.466Q4.079 58.356 4.079 58.200Q4.079 58.040 4.185 57.935Q3.860 57.935 3.646 58.163Q3.431 58.392 3.335 58.731Q3.239 59.071 3.239 59.376L3.239 60.649Q3.239 60.817 3.466 60.864Q3.692 60.911 3.997 60.911L3.997 61.208M5.927 60.247L5.927 58.056L5.224 58.056L5.224 57.802Q5.579 57.802 5.821 57.569Q6.064 57.337 6.175 56.989Q6.286 56.642 6.286 56.286L6.567 56.286L6.567 57.759L7.743 57.759L7.743 58.056L6.567 58.056L6.567 60.231Q6.567 60.552 6.687 60.780Q6.806 61.009 7.087 61.009Q7.267 61.009 7.384 60.886Q7.501 60.763 7.554 60.583Q7.607 60.403 7.607 60.231L7.607 59.759L7.888 59.759L7.888 60.247Q7.888 60.501 7.782 60.741Q7.677 60.981 7.480 61.134Q7.282 61.286 7.025 61.286Q6.708 61.286 6.456 61.163Q6.204 61.040 6.066 60.806Q5.927 60.571 5.927 60.247\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-28.231 29.91s25.458-5.377 34.143-8.535S27.26 10.899 34.365 7.148 51.6-3.484 57.127-5.656c5.527-2.17 12.334-3.042 17.072-2.845 4.737.198 11.545 1.9 17.072 4.268 5.526 2.369 14.866 9.448 22.762 12.804 7.895 3.355 24.668 8.815 34.143 11.381s25.853 5.732 34.143 7.113 25.608 2.845 25.608 2.845\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(252.494 -28.52)\">\u003Cpath d=\"M-38.749 60.247L-38.749 58.056L-39.452 58.056L-39.452 57.802Q-39.096 57.802-38.854 57.569Q-38.612 57.337-38.501 56.989Q-38.389 56.642-38.389 56.286L-38.108 56.286L-38.108 57.759L-36.932 57.759L-36.932 58.056L-38.108 58.056L-38.108 60.231Q-38.108 60.552-37.989 60.780Q-37.870 61.009-37.589 61.009Q-37.409 61.009-37.292 60.886Q-37.175 60.763-37.122 60.583Q-37.069 60.403-37.069 60.231L-37.069 59.759L-36.788 59.759L-36.788 60.247Q-36.788 60.501-36.893 60.741Q-36.999 60.981-37.196 61.134Q-37.393 61.286-37.651 61.286Q-37.967 61.286-38.219 61.163Q-38.471 61.040-38.610 60.806Q-38.749 60.571-38.749 60.247M-34.061 61.208L-36.042 61.208L-36.042 60.911Q-35.772 60.911-35.604 60.866Q-35.436 60.821-35.436 60.649L-35.436 58.513Q-35.436 58.298-35.499 58.202Q-35.561 58.106-35.678 58.085Q-35.796 58.063-36.042 58.063L-36.042 57.767L-34.874 57.681L-34.874 58.466Q-34.796 58.255-34.643 58.069Q-34.491 57.884-34.292 57.782Q-34.092 57.681-33.866 57.681Q-33.620 57.681-33.428 57.825Q-33.237 57.970-33.237 58.200Q-33.237 58.356-33.342 58.466Q-33.448 58.575-33.604 58.575Q-33.760 58.575-33.870 58.466Q-33.979 58.356-33.979 58.200Q-33.979 58.040-33.874 57.935Q-34.198 57.935-34.413 58.163Q-34.628 58.392-34.723 58.731Q-34.819 59.071-34.819 59.376L-34.819 60.649Q-34.819 60.817-34.592 60.864Q-34.366 60.911-34.061 60.911L-34.061 61.208M-32.073 60.255L-32.073 58.513Q-32.073 58.298-32.135 58.202Q-32.198 58.106-32.317 58.085Q-32.436 58.063-32.682 58.063L-32.682 57.767L-31.436 57.681L-31.436 60.231L-31.436 60.255Q-31.436 60.567-31.382 60.729Q-31.327 60.892-31.176 60.962Q-31.026 61.032-30.706 61.032Q-30.276 61.032-30.003 60.694Q-29.729 60.356-29.729 59.911L-29.729 58.513Q-29.729 58.298-29.792 58.202Q-29.854 58.106-29.973 58.085Q-30.092 58.063-30.339 58.063L-30.339 57.767L-29.092 57.681L-29.092 60.466Q-29.092 60.677-29.030 60.772Q-28.967 60.868-28.848 60.890Q-28.729 60.911-28.483 60.911L-28.483 61.208L-29.706 61.286L-29.706 60.665Q-29.874 60.954-30.155 61.120Q-30.436 61.286-30.757 61.286Q-32.073 61.286-32.073 60.255M-28.038 59.454Q-28.038 58.974-27.805 58.558Q-27.573 58.142-27.163 57.892Q-26.753 57.642-26.276 57.642Q-25.546 57.642-25.147 58.083Q-24.749 58.524-24.749 59.255Q-24.749 59.360-24.842 59.384L-27.292 59.384L-27.292 59.454Q-27.292 59.864-27.171 60.220Q-27.050 60.575-26.778 60.792Q-26.507 61.009-26.077 61.009Q-25.714 61.009-25.417 60.780Q-25.120 60.552-25.018 60.200Q-25.010 60.153-24.925 60.138L-24.842 60.138Q-24.749 60.165-24.749 60.247Q-24.749 60.255-24.757 60.286Q-24.819 60.513-24.958 60.696Q-25.096 60.880-25.288 61.013Q-25.479 61.145-25.698 61.216Q-25.917 61.286-26.155 61.286Q-26.526 61.286-26.864 61.149Q-27.202 61.013-27.469 60.761Q-27.737 60.509-27.887 60.169Q-28.038 59.829-28.038 59.454M-27.284 59.145L-25.323 59.145Q-25.323 58.841-25.425 58.550Q-25.526 58.259-25.743 58.077Q-25.960 57.895-26.276 57.895Q-26.577 57.895-26.807 58.083Q-27.038 58.270-27.161 58.562Q-27.284 58.853-27.284 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(252.494 -28.52)\">\u003Cpath d=\"M-19.621 61.177L-20.844 58.321Q-20.926 58.145-21.070 58.101Q-21.215 58.056-21.484 58.056L-21.484 57.759L-19.773 57.759L-19.773 58.056Q-20.195 58.056-20.195 58.239Q-20.195 58.274-20.180 58.321L-19.234 60.513L-18.394 58.536Q-18.355 58.458-18.355 58.368Q-18.355 58.228-18.461 58.142Q-18.566 58.056-18.707 58.056L-18.707 57.759L-17.355 57.759L-17.355 58.056Q-17.879 58.056-18.094 58.536L-19.219 61.177Q-19.281 61.286-19.387 61.286L-19.453 61.286Q-19.566 61.286-19.621 61.177\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(252.494 -28.52)\">\u003Cpath d=\"M-17.311 60.376Q-17.311 59.892-16.909 59.597Q-16.506 59.302-15.956 59.183Q-15.405 59.063-14.913 59.063L-14.913 58.774Q-14.913 58.548-15.028 58.341Q-15.143 58.134-15.340 58.015Q-15.538 57.895-15.768 57.895Q-16.194 57.895-16.479 58.001Q-16.409 58.028-16.362 58.083Q-16.315 58.138-16.290 58.208Q-16.264 58.278-16.264 58.353Q-16.264 58.458-16.315 58.550Q-16.366 58.642-16.458 58.692Q-16.549 58.743-16.655 58.743Q-16.760 58.743-16.852 58.692Q-16.944 58.642-16.995 58.550Q-17.045 58.458-17.045 58.353Q-17.045 57.935-16.657 57.788Q-16.268 57.642-15.768 57.642Q-15.436 57.642-15.083 57.772Q-14.729 57.903-14.501 58.157Q-14.272 58.411-14.272 58.759L-14.272 60.560Q-14.272 60.692-14.200 60.802Q-14.127 60.911-13.999 60.911Q-13.874 60.911-13.805 60.806Q-13.737 60.700-13.737 60.560L-13.737 60.048L-13.456 60.048L-13.456 60.560Q-13.456 60.763-13.573 60.921Q-13.690 61.079-13.872 61.163Q-14.053 61.247-14.256 61.247Q-14.487 61.247-14.639 61.075Q-14.792 60.903-14.823 60.673Q-14.983 60.954-15.292 61.120Q-15.600 61.286-15.952 61.286Q-16.463 61.286-16.887 61.063Q-17.311 60.841-17.311 60.376M-16.624 60.376Q-16.624 60.661-16.397 60.847Q-16.170 61.032-15.877 61.032Q-15.631 61.032-15.407 60.915Q-15.182 60.798-15.047 60.595Q-14.913 60.392-14.913 60.138L-14.913 59.306Q-15.178 59.306-15.463 59.360Q-15.749 59.415-16.020 59.544Q-16.292 59.673-16.458 59.880Q-16.624 60.087-16.624 60.376M-11.249 61.208L-13.081 61.208L-13.081 60.911Q-12.807 60.911-12.639 60.864Q-12.471 60.817-12.471 60.649L-12.471 56.489Q-12.471 56.274-12.534 56.179Q-12.596 56.083-12.715 56.062Q-12.834 56.040-13.081 56.040L-13.081 55.743L-11.858 55.657L-11.858 60.649Q-11.858 60.817-11.690 60.864Q-11.522 60.911-11.249 60.911L-11.249 61.208M-10.120 60.255L-10.120 58.513Q-10.120 58.298-10.182 58.202Q-10.245 58.106-10.364 58.085Q-10.483 58.063-10.729 58.063L-10.729 57.767L-9.483 57.681L-9.483 60.231L-9.483 60.255Q-9.483 60.567-9.428 60.729Q-9.374 60.892-9.223 60.962Q-9.073 61.032-8.752 61.032Q-8.323 61.032-8.049 60.694Q-7.776 60.356-7.776 59.911L-7.776 58.513Q-7.776 58.298-7.838 58.202Q-7.901 58.106-8.020 58.085Q-8.139 58.063-8.385 58.063L-8.385 57.767L-7.139 57.681L-7.139 60.466Q-7.139 60.677-7.077 60.772Q-7.014 60.868-6.895 60.890Q-6.776 60.911-6.530 60.911L-6.530 61.208L-7.752 61.286L-7.752 60.665Q-7.920 60.954-8.202 61.120Q-8.483 61.286-8.803 61.286Q-10.120 61.286-10.120 60.255M-6.084 59.454Q-6.084 58.974-5.852 58.558Q-5.620 58.142-5.209 57.892Q-4.799 57.642-4.323 57.642Q-3.592 57.642-3.194 58.083Q-2.795 58.524-2.795 59.255Q-2.795 59.360-2.889 59.384L-5.338 59.384L-5.338 59.454Q-5.338 59.864-5.217 60.220Q-5.096 60.575-4.825 60.792Q-4.553 61.009-4.124 61.009Q-3.760 61.009-3.463 60.780Q-3.167 60.552-3.065 60.200Q-3.057 60.153-2.971 60.138L-2.889 60.138Q-2.795 60.165-2.795 60.247Q-2.795 60.255-2.803 60.286Q-2.866 60.513-3.004 60.696Q-3.143 60.880-3.334 61.013Q-3.526 61.145-3.745 61.216Q-3.963 61.286-4.202 61.286Q-4.573 61.286-4.911 61.149Q-5.249 61.013-5.516 60.761Q-5.784 60.509-5.934 60.169Q-6.084 59.829-6.084 59.454M-5.331 59.145L-3.370 59.145Q-3.370 58.841-3.471 58.550Q-3.573 58.259-3.790 58.077Q-4.006 57.895-4.323 57.895Q-4.624 57.895-4.854 58.083Q-5.084 58.270-5.208 58.562Q-5.331 58.853-5.331 59.145\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M-28.231-12.769S-2.773-26.32 5.912-24.149C14.598-21.98 27.26.51 34.365 2.88 41.471 5.248 51.6-5.302 57.127-7.079s12.334-3.043 17.072-2.845S85.744-7.235 91.27-5.656s14.866 10.074 22.762 7.114c7.895-2.961 24.668-22.137 34.143-28.453 9.475-6.317 25.853-13.716 34.143-17.072s25.608-7.113 25.608-7.113\" style=\"stroke-dasharray:3.0,3.0;stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(192.728 -118.146)\">\u003Cpath d=\"M-37.444 61.208L-39.300 61.208L-39.300 60.911Q-39.026 60.911-38.858 60.864Q-38.690 60.817-38.690 60.649L-38.690 58.513Q-38.690 58.298-38.753 58.202Q-38.815 58.106-38.934 58.085Q-39.053 58.063-39.300 58.063L-39.300 57.767L-38.108 57.681L-38.108 58.415Q-37.995 58.200-37.801 58.032Q-37.608 57.864-37.370 57.772Q-37.132 57.681-36.878 57.681Q-35.710 57.681-35.710 58.759L-35.710 60.649Q-35.710 60.817-35.540 60.864Q-35.370 60.911-35.100 60.911L-35.100 61.208L-36.956 61.208L-36.956 60.911Q-36.682 60.911-36.514 60.864Q-36.346 60.817-36.346 60.649L-36.346 58.774Q-36.346 58.392-36.467 58.163Q-36.589 57.935-36.940 57.935Q-37.253 57.935-37.507 58.097Q-37.760 58.259-37.907 58.528Q-38.053 58.798-38.053 59.095L-38.053 60.649Q-38.053 60.817-37.883 60.864Q-37.714 60.911-37.444 60.911L-37.444 61.208M-34.557 60.376Q-34.557 59.892-34.155 59.597Q-33.753 59.302-33.202 59.183Q-32.651 59.063-32.159 59.063L-32.159 58.774Q-32.159 58.548-32.274 58.341Q-32.389 58.134-32.587 58.015Q-32.784 57.895-33.014 57.895Q-33.440 57.895-33.725 58.001Q-33.655 58.028-33.608 58.083Q-33.561 58.138-33.536 58.208Q-33.510 58.278-33.510 58.353Q-33.510 58.458-33.561 58.550Q-33.612 58.642-33.704 58.692Q-33.796 58.743-33.901 58.743Q-34.007 58.743-34.098 58.692Q-34.190 58.642-34.241 58.550Q-34.292 58.458-34.292 58.353Q-34.292 57.935-33.903 57.788Q-33.514 57.642-33.014 57.642Q-32.682 57.642-32.329 57.772Q-31.975 57.903-31.747 58.157Q-31.518 58.411-31.518 58.759L-31.518 60.560Q-31.518 60.692-31.446 60.802Q-31.374 60.911-31.245 60.911Q-31.120 60.911-31.051 60.806Q-30.983 60.700-30.983 60.560L-30.983 60.048L-30.702 60.048L-30.702 60.560Q-30.702 60.763-30.819 60.921Q-30.936 61.079-31.118 61.163Q-31.300 61.247-31.503 61.247Q-31.733 61.247-31.885 61.075Q-32.038 60.903-32.069 60.673Q-32.229 60.954-32.538 61.120Q-32.846 61.286-33.198 61.286Q-33.710 61.286-34.133 61.063Q-34.557 60.841-34.557 60.376M-33.870 60.376Q-33.870 60.661-33.643 60.847Q-33.417 61.032-33.124 61.032Q-32.878 61.032-32.653 60.915Q-32.428 60.798-32.294 60.595Q-32.159 60.392-32.159 60.138L-32.159 59.306Q-32.425 59.306-32.710 59.360Q-32.995 59.415-33.266 59.544Q-33.538 59.673-33.704 59.880Q-33.870 60.087-33.870 60.376M-28.550 61.208L-30.327 61.208L-30.327 60.911Q-30.053 60.911-29.885 60.864Q-29.717 60.817-29.717 60.649L-29.717 58.513Q-29.717 58.298-29.774 58.202Q-29.831 58.106-29.944 58.085Q-30.057 58.063-30.303 58.063L-30.303 57.767L-29.104 57.681L-29.104 60.649Q-29.104 60.817-28.958 60.864Q-28.811 60.911-28.550 60.911L-28.550 61.208M-29.991 56.286Q-29.991 56.095-29.856 55.964Q-29.721 55.833-29.526 55.833Q-29.405 55.833-29.301 55.895Q-29.198 55.958-29.135 56.062Q-29.073 56.165-29.073 56.286Q-29.073 56.481-29.204 56.616Q-29.335 56.751-29.526 56.751Q-29.725 56.751-29.858 56.618Q-29.991 56.485-29.991 56.286M-26.249 61.177L-27.471 58.321Q-27.553 58.145-27.698 58.101Q-27.842 58.056-28.112 58.056L-28.112 57.759L-26.401 57.759L-26.401 58.056Q-26.823 58.056-26.823 58.239Q-26.823 58.274-26.807 58.321L-25.862 60.513L-25.022 58.536Q-24.983 58.458-24.983 58.368Q-24.983 58.228-25.089 58.142Q-25.194 58.056-25.335 58.056L-25.335 57.759L-23.983 57.759L-23.983 58.056Q-24.507 58.056-24.721 58.536L-25.846 61.177Q-25.909 61.286-26.014 61.286L-26.081 61.286Q-26.194 61.286-26.249 61.177\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(192.728 -118.146)\">\u003Cpath d=\"M-23.791 59.454Q-23.791 58.974-23.558 58.558Q-23.326 58.142-22.916 57.892Q-22.506 57.642-22.029 57.642Q-21.299 57.642-20.900 58.083Q-20.502 58.524-20.502 59.255Q-20.502 59.360-20.595 59.384L-23.045 59.384L-23.045 59.454Q-23.045 59.864-22.924 60.220Q-22.802 60.575-22.531 60.792Q-22.259 61.009-21.830 61.009Q-21.466 61.009-21.170 60.780Q-20.873 60.552-20.771 60.200Q-20.763 60.153-20.677 60.138L-20.595 60.138Q-20.502 60.165-20.502 60.247Q-20.502 60.255-20.509 60.286Q-20.572 60.513-20.711 60.696Q-20.849 60.880-21.041 61.013Q-21.232 61.145-21.451 61.216Q-21.670 61.286-21.908 61.286Q-22.279 61.286-22.617 61.149Q-22.955 61.013-23.222 60.761Q-23.490 60.509-23.640 60.169Q-23.791 59.829-23.791 59.454M-23.037 59.145L-21.076 59.145Q-21.076 58.841-21.177 58.550Q-21.279 58.259-21.496 58.077Q-21.713 57.895-22.029 57.895Q-22.330 57.895-22.560 58.083Q-22.791 58.270-22.914 58.562Q-23.037 58.853-23.037 59.145\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(192.728 -118.146)\">\u003Cpath d=\"M-15.378 61.177L-16.601 58.321Q-16.683 58.145-16.827 58.101Q-16.972 58.056-17.241 58.056L-17.241 57.759L-15.530 57.759L-15.530 58.056Q-15.952 58.056-15.952 58.239Q-15.952 58.274-15.937 58.321L-14.991 60.513L-14.151 58.536Q-14.112 58.458-14.112 58.368Q-14.112 58.228-14.218 58.142Q-14.323 58.056-14.464 58.056L-14.464 57.759L-13.112 57.759L-13.112 58.056Q-13.636 58.056-13.851 58.536L-14.976 61.177Q-15.038 61.286-15.144 61.286L-15.210 61.286Q-15.323 61.286-15.378 61.177\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(192.728 -118.146)\">\u003Cpath d=\"M-13.067 60.376Q-13.067 59.892-12.665 59.597Q-12.262 59.302-11.712 59.183Q-11.161 59.063-10.669 59.063L-10.669 58.774Q-10.669 58.548-10.784 58.341Q-10.899 58.134-11.096 58.015Q-11.294 57.895-11.524 57.895Q-11.950 57.895-12.235 58.001Q-12.165 58.028-12.118 58.083Q-12.071 58.138-12.046 58.208Q-12.020 58.278-12.020 58.353Q-12.020 58.458-12.071 58.550Q-12.122 58.642-12.214 58.692Q-12.305 58.743-12.411 58.743Q-12.516 58.743-12.608 58.692Q-12.700 58.642-12.751 58.550Q-12.801 58.458-12.801 58.353Q-12.801 57.935-12.413 57.788Q-12.024 57.642-11.524 57.642Q-11.192 57.642-10.839 57.772Q-10.485 57.903-10.257 58.157Q-10.028 58.411-10.028 58.759L-10.028 60.560Q-10.028 60.692-9.956 60.802Q-9.883 60.911-9.755 60.911Q-9.630 60.911-9.561 60.806Q-9.493 60.700-9.493 60.560L-9.493 60.048L-9.212 60.048L-9.212 60.560Q-9.212 60.763-9.329 60.921Q-9.446 61.079-9.628 61.163Q-9.809 61.247-10.012 61.247Q-10.243 61.247-10.395 61.075Q-10.548 60.903-10.579 60.673Q-10.739 60.954-11.048 61.120Q-11.356 61.286-11.708 61.286Q-12.219 61.286-12.643 61.063Q-13.067 60.841-13.067 60.376M-12.380 60.376Q-12.380 60.661-12.153 60.847Q-11.926 61.032-11.633 61.032Q-11.387 61.032-11.163 60.915Q-10.938 60.798-10.803 60.595Q-10.669 60.392-10.669 60.138L-10.669 59.306Q-10.934 59.306-11.219 59.360Q-11.505 59.415-11.776 59.544Q-12.048 59.673-12.214 59.880Q-12.380 60.087-12.380 60.376M-7.005 61.208L-8.837 61.208L-8.837 60.911Q-8.563 60.911-8.395 60.864Q-8.227 60.817-8.227 60.649L-8.227 56.489Q-8.227 56.274-8.290 56.179Q-8.352 56.083-8.471 56.062Q-8.591 56.040-8.837 56.040L-8.837 55.743L-7.614 55.657L-7.614 60.649Q-7.614 60.817-7.446 60.864Q-7.278 60.911-7.005 60.911L-7.005 61.208M-5.876 60.255L-5.876 58.513Q-5.876 58.298-5.938 58.202Q-6.001 58.106-6.120 58.085Q-6.239 58.063-6.485 58.063L-6.485 57.767L-5.239 57.681L-5.239 60.231L-5.239 60.255Q-5.239 60.567-5.184 60.729Q-5.130 60.892-4.979 60.962Q-4.829 61.032-4.508 61.032Q-4.079 61.032-3.805 60.694Q-3.532 60.356-3.532 59.911L-3.532 58.513Q-3.532 58.298-3.594 58.202Q-3.657 58.106-3.776 58.085Q-3.895 58.063-4.141 58.063L-4.141 57.767L-2.895 57.681L-2.895 60.466Q-2.895 60.677-2.833 60.772Q-2.770 60.868-2.651 60.890Q-2.532 60.911-2.286 60.911L-2.286 61.208L-3.508 61.286L-3.508 60.665Q-3.676 60.954-3.958 61.120Q-4.239 61.286-4.559 61.286Q-5.876 61.286-5.876 60.255M-1.841 59.454Q-1.841 58.974-1.608 58.558Q-1.376 58.142-0.966 57.892Q-0.555 57.642-0.079 57.642Q0.652 57.642 1.050 58.083Q1.449 58.524 1.449 59.255Q1.449 59.360 1.355 59.384L-1.094 59.384L-1.094 59.454Q-1.094 59.864-0.973 60.220Q-0.852 60.575-0.581 60.792Q-0.309 61.009 0.120 61.009Q0.484 61.009 0.781 60.780Q1.077 60.552 1.179 60.200Q1.187 60.153 1.273 60.138L1.355 60.138Q1.449 60.165 1.449 60.247Q1.449 60.255 1.441 60.286Q1.378 60.513 1.240 60.696Q1.101 60.880 0.909 61.013Q0.718 61.145 0.499 61.216Q0.281 61.286 0.042 61.286Q-0.329 61.286-0.667 61.149Q-1.005 61.013-1.272 60.761Q-1.540 60.509-1.690 60.169Q-1.841 59.829-1.841 59.454M-1.087 59.145L0.874 59.145Q0.874 58.841 0.773 58.550Q0.671 58.259 0.454 58.077Q0.238 57.895-0.079 57.895Q-0.380 57.895-0.610 58.083Q-0.841 58.270-0.964 58.562Q-1.087 58.853-1.087 59.145\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M-28.231 46.982s25.458-2.178 34.143-7.113S27.26 17.14 34.365 11.416 51.6.784 57.127-1.387 69.461-4.43 74.199-4.233 85.744-2.333 91.27.035c5.526 2.369 14.866 7.672 22.762 12.804 7.895 5.132 24.668 19.645 34.143 24.184s25.853 7.155 34.143 8.536c8.29 1.382 25.608 1.423 25.608 1.423\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(162.869 -17.708)\">\u003Cpath d=\"M-39.331 59.481Q-39.331 58.985-39.081 58.560Q-38.831 58.134-38.411 57.888Q-37.991 57.642-37.491 57.642Q-36.952 57.642-36.561 57.767Q-36.171 57.892-36.171 58.306Q-36.171 58.411-36.221 58.503Q-36.272 58.595-36.364 58.645Q-36.456 58.696-36.565 58.696Q-36.671 58.696-36.762 58.645Q-36.854 58.595-36.905 58.503Q-36.956 58.411-36.956 58.306Q-36.956 58.083-36.788 57.978Q-37.010 57.919-37.483 57.919Q-37.780 57.919-37.995 58.058Q-38.210 58.196-38.341 58.427Q-38.471 58.657-38.530 58.927Q-38.589 59.196-38.589 59.481Q-38.589 59.876-38.456 60.226Q-38.323 60.575-38.051 60.792Q-37.780 61.009-37.382 61.009Q-37.007 61.009-36.731 60.792Q-36.456 60.575-36.354 60.216Q-36.339 60.153-36.276 60.153L-36.171 60.153Q-36.135 60.153-36.110 60.181Q-36.085 60.208-36.085 60.247L-36.085 60.270Q-36.217 60.751-36.602 61.019Q-36.987 61.286-37.491 61.286Q-37.854 61.286-38.188 61.149Q-38.522 61.013-38.782 60.763Q-39.042 60.513-39.186 60.177Q-39.331 59.841-39.331 59.481M-35.596 59.513Q-35.596 59.009-35.341 58.577Q-35.085 58.145-34.649 57.894Q-34.214 57.642-33.714 57.642Q-33.327 57.642-32.985 57.786Q-32.643 57.931-32.382 58.192Q-32.120 58.454-31.977 58.790Q-31.835 59.126-31.835 59.513Q-31.835 60.005-32.098 60.415Q-32.362 60.825-32.792 61.056Q-33.221 61.286-33.714 61.286Q-34.206 61.286-34.639 61.054Q-35.073 60.821-35.335 60.413Q-35.596 60.005-35.596 59.513M-33.714 61.009Q-33.257 61.009-33.005 60.786Q-32.753 60.563-32.665 60.212Q-32.577 59.860-32.577 59.415Q-32.577 58.985-32.671 58.647Q-32.764 58.310-33.018 58.103Q-33.272 57.895-33.714 57.895Q-34.362 57.895-34.606 58.312Q-34.850 58.728-34.850 59.415Q-34.850 59.860-34.762 60.212Q-34.675 60.563-34.423 60.786Q-34.171 61.009-33.714 61.009M-29.421 61.208L-31.276 61.208L-31.276 60.911Q-31.003 60.911-30.835 60.864Q-30.667 60.817-30.667 60.649L-30.667 58.513Q-30.667 58.298-30.729 58.202Q-30.792 58.106-30.911 58.085Q-31.030 58.063-31.276 58.063L-31.276 57.767L-30.085 57.681L-30.085 58.415Q-29.971 58.200-29.778 58.032Q-29.585 57.864-29.346 57.772Q-29.108 57.681-28.854 57.681Q-27.686 57.681-27.686 58.759L-27.686 60.649Q-27.686 60.817-27.516 60.864Q-27.346 60.911-27.077 60.911L-27.077 61.208L-28.932 61.208L-28.932 60.911Q-28.659 60.911-28.491 60.864Q-28.323 60.817-28.323 60.649L-28.323 58.774Q-28.323 58.392-28.444 58.163Q-28.565 57.935-28.917 57.935Q-29.229 57.935-29.483 58.097Q-29.737 58.259-29.883 58.528Q-30.030 58.798-30.030 59.095L-30.030 60.649Q-30.030 60.817-29.860 60.864Q-29.690 60.911-29.421 60.911L-29.421 61.208M-26.589 61.200L-26.589 59.978Q-26.589 59.950-26.557 59.919Q-26.526 59.888-26.503 59.888L-26.397 59.888Q-26.327 59.888-26.311 59.950Q-26.249 60.270-26.110 60.511Q-25.971 60.751-25.739 60.892Q-25.507 61.032-25.198 61.032Q-24.960 61.032-24.751 60.972Q-24.542 60.911-24.405 60.763Q-24.268 60.614-24.268 60.368Q-24.268 60.114-24.479 59.948Q-24.690 59.782-24.960 59.728L-25.581 59.614Q-25.987 59.536-26.288 59.280Q-26.589 59.024-26.589 58.649Q-26.589 58.282-26.387 58.060Q-26.186 57.837-25.862 57.739Q-25.538 57.642-25.198 57.642Q-24.733 57.642-24.436 57.849L-24.214 57.665Q-24.190 57.642-24.159 57.642L-24.108 57.642Q-24.077 57.642-24.050 57.669Q-24.022 57.696-24.022 57.728L-24.022 58.712Q-24.022 58.743-24.048 58.772Q-24.073 58.802-24.108 58.802L-24.214 58.802Q-24.249 58.802-24.276 58.774Q-24.303 58.747-24.303 58.712Q-24.303 58.313-24.555 58.093Q-24.807 57.872-25.206 57.872Q-25.561 57.872-25.844 57.995Q-26.128 58.118-26.128 58.423Q-26.128 58.642-25.926 58.774Q-25.725 58.907-25.479 58.950L-24.854 59.063Q-24.425 59.153-24.116 59.450Q-23.807 59.747-23.807 60.161Q-23.807 60.731-24.206 61.009Q-24.604 61.286-25.198 61.286Q-25.749 61.286-26.100 60.950L-26.397 61.263Q-26.421 61.286-26.456 61.286L-26.503 61.286Q-26.526 61.286-26.557 61.255Q-26.589 61.224-26.589 61.200M-23.280 59.454Q-23.280 58.974-23.048 58.558Q-22.815 58.142-22.405 57.892Q-21.995 57.642-21.518 57.642Q-20.788 57.642-20.389 58.083Q-19.991 58.524-19.991 59.255Q-19.991 59.360-20.085 59.384L-22.534 59.384L-22.534 59.454Q-22.534 59.864-22.413 60.220Q-22.292 60.575-22.020 60.792Q-21.749 61.009-21.319 61.009Q-20.956 61.009-20.659 60.780Q-20.362 60.552-20.260 60.200Q-20.253 60.153-20.167 60.138L-20.085 60.138Q-19.991 60.165-19.991 60.247Q-19.991 60.255-19.999 60.286Q-20.061 60.513-20.200 60.696Q-20.339 60.880-20.530 61.013Q-20.721 61.145-20.940 61.216Q-21.159 61.286-21.397 61.286Q-21.768 61.286-22.106 61.149Q-22.444 61.013-22.712 60.761Q-22.979 60.509-23.130 60.169Q-23.280 59.829-23.280 59.454M-22.526 59.145L-20.565 59.145Q-20.565 58.841-20.667 58.550Q-20.768 58.259-20.985 58.077Q-21.202 57.895-21.518 57.895Q-21.819 57.895-22.050 58.083Q-22.280 58.270-22.403 58.562Q-22.526 58.853-22.526 59.145M-17.495 61.208L-19.475 61.208L-19.475 60.911Q-19.206 60.911-19.038 60.866Q-18.870 60.821-18.870 60.649L-18.870 58.513Q-18.870 58.298-18.932 58.202Q-18.995 58.106-19.112 58.085Q-19.229 58.063-19.475 58.063L-19.475 57.767L-18.307 57.681L-18.307 58.466Q-18.229 58.255-18.077 58.069Q-17.925 57.884-17.725 57.782Q-17.526 57.681-17.300 57.681Q-17.053 57.681-16.862 57.825Q-16.671 57.970-16.671 58.200Q-16.671 58.356-16.776 58.466Q-16.882 58.575-17.038 58.575Q-17.194 58.575-17.303 58.466Q-17.413 58.356-17.413 58.200Q-17.413 58.040-17.307 57.935Q-17.632 57.935-17.846 58.163Q-18.061 58.392-18.157 58.731Q-18.253 59.071-18.253 59.376L-18.253 60.649Q-18.253 60.817-18.026 60.864Q-17.800 60.911-17.495 60.911L-17.495 61.208M-14.389 61.177L-15.612 58.321Q-15.694 58.145-15.839 58.101Q-15.983 58.056-16.253 58.056L-16.253 57.759L-14.542 57.759L-14.542 58.056Q-14.964 58.056-14.964 58.239Q-14.964 58.274-14.948 58.321L-14.003 60.513L-13.163 58.536Q-13.124 58.458-13.124 58.368Q-13.124 58.228-13.229 58.142Q-13.335 58.056-13.475 58.056L-13.475 57.759L-12.124 57.759L-12.124 58.056Q-12.647 58.056-12.862 58.536L-13.987 61.177Q-14.050 61.286-14.155 61.286L-14.221 61.286Q-14.335 61.286-14.389 61.177\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(162.869 -17.708)\">\u003Cpath d=\"M-12.069 60.376Q-12.069 59.892-11.667 59.597Q-11.264 59.302-10.714 59.183Q-10.163 59.063-9.671 59.063L-9.671 58.774Q-9.671 58.548-9.786 58.341Q-9.901 58.134-10.098 58.015Q-10.296 57.895-10.526 57.895Q-10.952 57.895-11.237 58.001Q-11.167 58.028-11.120 58.083Q-11.073 58.138-11.048 58.208Q-11.022 58.278-11.022 58.353Q-11.022 58.458-11.073 58.550Q-11.124 58.642-11.216 58.692Q-11.307 58.743-11.413 58.743Q-11.518 58.743-11.610 58.692Q-11.702 58.642-11.753 58.550Q-11.803 58.458-11.803 58.353Q-11.803 57.935-11.415 57.788Q-11.026 57.642-10.526 57.642Q-10.194 57.642-9.841 57.772Q-9.487 57.903-9.259 58.157Q-9.030 58.411-9.030 58.759L-9.030 60.560Q-9.030 60.692-8.958 60.802Q-8.885 60.911-8.757 60.911Q-8.632 60.911-8.563 60.806Q-8.495 60.700-8.495 60.560L-8.495 60.048L-8.214 60.048L-8.214 60.560Q-8.214 60.763-8.331 60.921Q-8.448 61.079-8.630 61.163Q-8.811 61.247-9.014 61.247Q-9.245 61.247-9.397 61.075Q-9.550 60.903-9.581 60.673Q-9.741 60.954-10.050 61.120Q-10.358 61.286-10.710 61.286Q-11.221 61.286-11.645 61.063Q-12.069 60.841-12.069 60.376M-11.382 60.376Q-11.382 60.661-11.155 60.847Q-10.928 61.032-10.635 61.032Q-10.389 61.032-10.165 60.915Q-9.940 60.798-9.805 60.595Q-9.671 60.392-9.671 60.138L-9.671 59.306Q-9.936 59.306-10.221 59.360Q-10.507 59.415-10.778 59.544Q-11.050 59.673-11.216 59.880Q-11.382 60.087-11.382 60.376M-7.296 60.247L-7.296 58.056L-7.999 58.056L-7.999 57.802Q-7.643 57.802-7.401 57.569Q-7.159 57.337-7.048 56.989Q-6.936 56.642-6.936 56.286L-6.655 56.286L-6.655 57.759L-5.479 57.759L-5.479 58.056L-6.655 58.056L-6.655 60.231Q-6.655 60.552-6.536 60.780Q-6.417 61.009-6.135 61.009Q-5.956 61.009-5.839 60.886Q-5.721 60.763-5.669 60.583Q-5.616 60.403-5.616 60.231L-5.616 59.759L-5.335 59.759L-5.335 60.247Q-5.335 60.501-5.440 60.741Q-5.546 60.981-5.743 61.134Q-5.940 61.286-6.198 61.286Q-6.514 61.286-6.766 61.163Q-7.018 61.040-7.157 60.806Q-7.296 60.571-7.296 60.247M-2.757 61.208L-4.534 61.208L-4.534 60.911Q-4.260 60.911-4.092 60.864Q-3.925 60.817-3.925 60.649L-3.925 58.513Q-3.925 58.298-3.981 58.202Q-4.038 58.106-4.151 58.085Q-4.264 58.063-4.510 58.063L-4.510 57.767L-3.311 57.681L-3.311 60.649Q-3.311 60.817-3.165 60.864Q-3.018 60.911-2.757 60.911L-2.757 61.208M-4.198 56.286Q-4.198 56.095-4.063 55.964Q-3.928 55.833-3.733 55.833Q-3.612 55.833-3.509 55.895Q-3.405 55.958-3.342 56.062Q-3.280 56.165-3.280 56.286Q-3.280 56.481-3.411 56.616Q-3.542 56.751-3.733 56.751Q-3.932 56.751-4.065 56.618Q-4.198 56.485-4.198 56.286M-0.456 61.177L-1.678 58.321Q-1.760 58.145-1.905 58.101Q-2.050 58.056-2.319 58.056L-2.319 57.759L-0.608 57.759L-0.608 58.056Q-1.030 58.056-1.030 58.239Q-1.030 58.274-1.014 58.321L-0.069 60.513L0.771 58.536Q0.810 58.458 0.810 58.368Q0.810 58.228 0.704 58.142Q0.599 58.056 0.458 58.056L0.458 57.759L1.810 57.759L1.810 58.056Q1.286 58.056 1.072 58.536L-0.053 61.177Q-0.116 61.286-0.221 61.286L-0.288 61.286Q-0.401 61.286-0.456 61.177\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(162.869 -17.708)\">\u003Cpath d=\"M2 59.454Q2 58.974 2.233 58.558Q2.465 58.142 2.875 57.892Q3.285 57.642 3.762 57.642Q4.492 57.642 4.891 58.083Q5.289 58.524 5.289 59.255Q5.289 59.360 5.196 59.384L2.746 59.384L2.746 59.454Q2.746 59.864 2.867 60.220Q2.989 60.575 3.260 60.792Q3.532 61.009 3.961 61.009Q4.325 61.009 4.621 60.780Q4.918 60.552 5.020 60.200Q5.028 60.153 5.114 60.138L5.196 60.138Q5.289 60.165 5.289 60.247Q5.289 60.255 5.282 60.286Q5.219 60.513 5.080 60.696Q4.942 60.880 4.750 61.013Q4.559 61.145 4.340 61.216Q4.121 61.286 3.883 61.286Q3.512 61.286 3.174 61.149Q2.836 61.013 2.569 60.761Q2.301 60.509 2.151 60.169Q2 59.829 2 59.454M2.754 59.145L4.715 59.145Q4.715 58.841 4.614 58.550Q4.512 58.259 4.295 58.077Q4.078 57.895 3.762 57.895Q3.461 57.895 3.231 58.083Q3 58.270 2.877 58.562Q2.754 58.853 2.754 59.145\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(162.869 -17.708)\">\u003Cpath d=\"M10.412 61.177L9.189 58.321Q9.107 58.145 8.963 58.101Q8.818 58.056 8.549 58.056L8.549 57.759L10.260 57.759L10.260 58.056Q9.838 58.056 9.838 58.239Q9.838 58.274 9.853 58.321L10.799 60.513L11.639 58.536Q11.678 58.458 11.678 58.368Q11.678 58.228 11.572 58.142Q11.467 58.056 11.326 58.056L11.326 57.759L12.678 57.759L12.678 58.056Q12.154 58.056 11.939 58.536L10.814 61.177Q10.752 61.286 10.646 61.286L10.580 61.286Q10.467 61.286 10.412 61.177\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(162.869 -17.708)\">\u003Cpath d=\"M12.723 60.376Q12.723 59.892 13.125 59.597Q13.528 59.302 14.078 59.183Q14.629 59.063 15.121 59.063L15.121 58.774Q15.121 58.548 15.006 58.341Q14.891 58.134 14.694 58.015Q14.496 57.895 14.266 57.895Q13.840 57.895 13.555 58.001Q13.625 58.028 13.672 58.083Q13.719 58.138 13.744 58.208Q13.770 58.278 13.770 58.353Q13.770 58.458 13.719 58.550Q13.668 58.642 13.576 58.692Q13.485 58.743 13.379 58.743Q13.274 58.743 13.182 58.692Q13.090 58.642 13.039 58.550Q12.989 58.458 12.989 58.353Q12.989 57.935 13.377 57.788Q13.766 57.642 14.266 57.642Q14.598 57.642 14.951 57.772Q15.305 57.903 15.533 58.157Q15.762 58.411 15.762 58.759L15.762 60.560Q15.762 60.692 15.834 60.802Q15.907 60.911 16.035 60.911Q16.160 60.911 16.229 60.806Q16.297 60.700 16.297 60.560L16.297 60.048L16.578 60.048L16.578 60.560Q16.578 60.763 16.461 60.921Q16.344 61.079 16.162 61.163Q15.981 61.247 15.778 61.247Q15.547 61.247 15.395 61.075Q15.242 60.903 15.211 60.673Q15.051 60.954 14.742 61.120Q14.434 61.286 14.082 61.286Q13.571 61.286 13.147 61.063Q12.723 60.841 12.723 60.376M13.410 60.376Q13.410 60.661 13.637 60.847Q13.864 61.032 14.157 61.032Q14.403 61.032 14.627 60.915Q14.852 60.798 14.987 60.595Q15.121 60.392 15.121 60.138L15.121 59.306Q14.856 59.306 14.571 59.360Q14.285 59.415 14.014 59.544Q13.742 59.673 13.576 59.880Q13.410 60.087 13.410 60.376M18.785 61.208L16.953 61.208L16.953 60.911Q17.227 60.911 17.395 60.864Q17.563 60.817 17.563 60.649L17.563 56.489Q17.563 56.274 17.500 56.179Q17.438 56.083 17.319 56.062Q17.200 56.040 16.953 56.040L16.953 55.743L18.176 55.657L18.176 60.649Q18.176 60.817 18.344 60.864Q18.512 60.911 18.785 60.911L18.785 61.208M19.914 60.255L19.914 58.513Q19.914 58.298 19.852 58.202Q19.789 58.106 19.670 58.085Q19.551 58.063 19.305 58.063L19.305 57.767L20.551 57.681L20.551 60.231L20.551 60.255Q20.551 60.567 20.606 60.729Q20.660 60.892 20.811 60.962Q20.961 61.032 21.282 61.032Q21.711 61.032 21.985 60.694Q22.258 60.356 22.258 59.911L22.258 58.513Q22.258 58.298 22.196 58.202Q22.133 58.106 22.014 58.085Q21.895 58.063 21.649 58.063L21.649 57.767L22.895 57.681L22.895 60.466Q22.895 60.677 22.957 60.772Q23.020 60.868 23.139 60.890Q23.258 60.911 23.504 60.911L23.504 61.208L22.282 61.286L22.282 60.665Q22.114 60.954 21.832 61.120Q21.551 61.286 21.231 61.286Q19.914 61.286 19.914 60.255M23.950 59.454Q23.950 58.974 24.182 58.558Q24.414 58.142 24.825 57.892Q25.235 57.642 25.711 57.642Q26.442 57.642 26.840 58.083Q27.239 58.524 27.239 59.255Q27.239 59.360 27.145 59.384L24.696 59.384L24.696 59.454Q24.696 59.864 24.817 60.220Q24.938 60.575 25.209 60.792Q25.481 61.009 25.910 61.009Q26.274 61.009 26.571 60.780Q26.867 60.552 26.969 60.200Q26.977 60.153 27.063 60.138L27.145 60.138Q27.239 60.165 27.239 60.247Q27.239 60.255 27.231 60.286Q27.168 60.513 27.030 60.696Q26.891 60.880 26.700 61.013Q26.508 61.145 26.289 61.216Q26.071 61.286 25.832 61.286Q25.461 61.286 25.123 61.149Q24.785 61.013 24.518 60.761Q24.250 60.509 24.100 60.169Q23.950 59.829 23.950 59.454M24.703 59.145L26.664 59.145Q26.664 58.841 26.563 58.550Q26.461 58.259 26.244 58.077Q26.028 57.895 25.711 57.895Q25.410 57.895 25.180 58.083Q24.950 58.270 24.826 58.562Q24.703 58.853 24.703 59.145\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M73.553-4.233a2.2 2.2 0 1 0-4.4 0 2.2 2.2 0 0 0 4.4 0m-2.2 0\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M207.282-50.326a2.2 2.2 0 1 0-4.4 0 2.2 2.2 0 0 0 4.4 0m-2.2 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M37.21-25.572C48.582-16.457 56.127-12.324 68.037-8.29\"\u002F>\u003Cpath stroke=\"none\" d=\"m69.931-7.647-2.517-2.542.623 1.9-1.65 1.13\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(38.887 -90.447)\">\u003Cpath d=\"M-39.276 60.376Q-39.276 59.892-38.874 59.597Q-38.471 59.302-37.921 59.183Q-37.370 59.063-36.878 59.063L-36.878 58.774Q-36.878 58.548-36.993 58.341Q-37.108 58.134-37.305 58.015Q-37.503 57.895-37.733 57.895Q-38.159 57.895-38.444 58.001Q-38.374 58.028-38.327 58.083Q-38.280 58.138-38.255 58.208Q-38.229 58.278-38.229 58.353Q-38.229 58.458-38.280 58.550Q-38.331 58.642-38.423 58.692Q-38.514 58.743-38.620 58.743Q-38.725 58.743-38.817 58.692Q-38.909 58.642-38.960 58.550Q-39.010 58.458-39.010 58.353Q-39.010 57.935-38.622 57.788Q-38.233 57.642-37.733 57.642Q-37.401 57.642-37.048 57.772Q-36.694 57.903-36.466 58.157Q-36.237 58.411-36.237 58.759L-36.237 60.560Q-36.237 60.692-36.165 60.802Q-36.092 60.911-35.964 60.911Q-35.839 60.911-35.770 60.806Q-35.702 60.700-35.702 60.560L-35.702 60.048L-35.421 60.048L-35.421 60.560Q-35.421 60.763-35.538 60.921Q-35.655 61.079-35.837 61.163Q-36.018 61.247-36.221 61.247Q-36.452 61.247-36.604 61.075Q-36.757 60.903-36.788 60.673Q-36.948 60.954-37.257 61.120Q-37.565 61.286-37.917 61.286Q-38.428 61.286-38.852 61.063Q-39.276 60.841-39.276 60.376M-38.589 60.376Q-38.589 60.661-38.362 60.847Q-38.135 61.032-37.842 61.032Q-37.596 61.032-37.372 60.915Q-37.147 60.798-37.012 60.595Q-36.878 60.392-36.878 60.138L-36.878 59.306Q-37.143 59.306-37.428 59.360Q-37.714 59.415-37.985 59.544Q-38.257 59.673-38.423 59.880Q-38.589 60.087-38.589 60.376M-33.120 61.208L-35.100 61.208L-35.100 60.911Q-34.831 60.911-34.663 60.866Q-34.495 60.821-34.495 60.649L-34.495 58.513Q-34.495 58.298-34.557 58.202Q-34.620 58.106-34.737 58.085Q-34.854 58.063-35.100 58.063L-35.100 57.767L-33.932 57.681L-33.932 58.466Q-33.854 58.255-33.702 58.069Q-33.550 57.884-33.350 57.782Q-33.151 57.681-32.925 57.681Q-32.678 57.681-32.487 57.825Q-32.296 57.970-32.296 58.200Q-32.296 58.356-32.401 58.466Q-32.507 58.575-32.663 58.575Q-32.819 58.575-32.928 58.466Q-33.038 58.356-33.038 58.200Q-33.038 58.040-32.932 57.935Q-33.257 57.935-33.471 58.163Q-33.686 58.392-33.782 58.731Q-33.878 59.071-33.878 59.376L-33.878 60.649Q-33.878 60.817-33.651 60.864Q-33.425 60.911-33.120 60.911L-33.120 61.208M-31.815 61.817Q-31.815 61.536-31.604 61.325Q-31.393 61.114-31.108 61.024Q-31.264 60.899-31.342 60.710Q-31.421 60.520-31.421 60.321Q-31.421 59.966-31.190 59.673Q-31.557 59.333-31.557 58.864Q-31.557 58.513-31.354 58.243Q-31.151 57.974-30.831 57.827Q-30.510 57.681-30.167 57.681Q-29.647 57.681-29.276 57.962Q-28.913 57.591-28.366 57.591Q-28.186 57.591-28.059 57.718Q-27.932 57.845-27.932 58.024Q-27.932 58.130-28.010 58.208Q-28.089 58.286-28.198 58.286Q-28.307 58.286-28.383 58.210Q-28.460 58.134-28.460 58.024Q-28.460 57.923-28.421 57.872Q-28.413 57.864-28.409 57.858Q-28.405 57.853-28.405 57.849Q-28.780 57.849-29.100 58.103Q-28.780 58.442-28.780 58.864Q-28.780 59.134-28.897 59.351Q-29.014 59.567-29.219 59.726Q-29.425 59.884-29.667 59.966Q-29.909 60.048-30.167 60.048Q-30.385 60.048-30.598 59.989Q-30.811 59.931-31.007 59.810Q-31.100 59.950-31.100 60.130Q-31.100 60.337-30.964 60.489Q-30.827 60.642-30.620 60.642L-29.925 60.642Q-29.436 60.642-29.024 60.726Q-28.612 60.810-28.333 61.067Q-28.053 61.325-28.053 61.817Q-28.053 62.181-28.374 62.413Q-28.694 62.645-29.135 62.747Q-29.577 62.849-29.932 62.849Q-30.288 62.849-30.731 62.747Q-31.175 62.645-31.495 62.413Q-31.815 62.181-31.815 61.817M-31.311 61.817Q-31.311 62.013-31.167 62.161Q-31.022 62.310-30.809 62.399Q-30.596 62.489-30.356 62.536Q-30.116 62.583-29.932 62.583Q-29.690 62.583-29.360 62.505Q-29.030 62.427-28.794 62.253Q-28.557 62.079-28.557 61.817Q-28.557 61.411-28.967 61.302Q-29.378 61.192-29.940 61.192L-30.620 61.192Q-30.889 61.192-31.100 61.370Q-31.311 61.548-31.311 61.817M-30.167 59.782Q-29.444 59.782-29.444 58.864Q-29.444 57.942-30.167 57.942Q-30.893 57.942-30.893 58.864Q-30.893 59.782-30.167 59.782M-25.639 61.208L-27.495 61.208L-27.495 60.911Q-27.221 60.911-27.053 60.864Q-26.885 60.817-26.885 60.649L-26.885 58.513Q-26.885 58.298-26.948 58.202Q-27.010 58.106-27.130 58.085Q-27.249 58.063-27.495 58.063L-27.495 57.767L-26.303 57.681L-26.303 58.415Q-26.190 58.200-25.997 58.032Q-25.803 57.864-25.565 57.772Q-25.327 57.681-25.073 57.681Q-24.112 57.681-23.936 58.392Q-23.753 58.063-23.425 57.872Q-23.096 57.681-22.717 57.681Q-21.542 57.681-21.542 58.759L-21.542 60.649Q-21.542 60.817-21.374 60.864Q-21.206 60.911-20.936 60.911L-20.936 61.208L-22.792 61.208L-22.792 60.911Q-22.518 60.911-22.350 60.866Q-22.182 60.821-22.182 60.649L-22.182 58.774Q-22.182 58.388-22.307 58.161Q-22.432 57.935-22.784 57.935Q-23.089 57.935-23.344 58.097Q-23.600 58.259-23.749 58.528Q-23.897 58.798-23.897 59.095L-23.897 60.649Q-23.897 60.817-23.727 60.864Q-23.557 60.911-23.288 60.911L-23.288 61.208L-25.143 61.208L-25.143 60.911Q-24.870 60.911-24.702 60.864Q-24.534 60.817-24.534 60.649L-24.534 58.774Q-24.534 58.388-24.659 58.161Q-24.784 57.935-25.135 57.935Q-25.440 57.935-25.696 58.097Q-25.952 58.259-26.100 58.528Q-26.249 58.798-26.249 59.095L-26.249 60.649Q-26.249 60.817-26.079 60.864Q-25.909 60.911-25.639 60.911L-25.639 61.208M-20.393 60.376Q-20.393 59.892-19.991 59.597Q-19.589 59.302-19.038 59.183Q-18.487 59.063-17.995 59.063L-17.995 58.774Q-17.995 58.548-18.110 58.341Q-18.225 58.134-18.423 58.015Q-18.620 57.895-18.850 57.895Q-19.276 57.895-19.561 58.001Q-19.491 58.028-19.444 58.083Q-19.397 58.138-19.372 58.208Q-19.346 58.278-19.346 58.353Q-19.346 58.458-19.397 58.550Q-19.448 58.642-19.540 58.692Q-19.632 58.743-19.737 58.743Q-19.842 58.743-19.934 58.692Q-20.026 58.642-20.077 58.550Q-20.128 58.458-20.128 58.353Q-20.128 57.935-19.739 57.788Q-19.350 57.642-18.850 57.642Q-18.518 57.642-18.165 57.772Q-17.811 57.903-17.583 58.157Q-17.354 58.411-17.354 58.759L-17.354 60.560Q-17.354 60.692-17.282 60.802Q-17.210 60.911-17.081 60.911Q-16.956 60.911-16.887 60.806Q-16.819 60.700-16.819 60.560L-16.819 60.048L-16.538 60.048L-16.538 60.560Q-16.538 60.763-16.655 60.921Q-16.772 61.079-16.954 61.163Q-17.135 61.247-17.339 61.247Q-17.569 61.247-17.721 61.075Q-17.874 60.903-17.905 60.673Q-18.065 60.954-18.374 61.120Q-18.682 61.286-19.034 61.286Q-19.546 61.286-19.969 61.063Q-20.393 60.841-20.393 60.376M-19.706 60.376Q-19.706 60.661-19.479 60.847Q-19.253 61.032-18.960 61.032Q-18.714 61.032-18.489 60.915Q-18.264 60.798-18.130 60.595Q-17.995 60.392-17.995 60.138L-17.995 59.306Q-18.260 59.306-18.546 59.360Q-18.831 59.415-19.102 59.544Q-19.374 59.673-19.540 59.880Q-19.706 60.087-19.706 60.376M-14.858 61.208L-16.354 61.208L-16.354 60.911Q-15.721 60.911-15.300 60.431L-14.530 59.520L-15.522 58.321Q-15.678 58.142-15.841 58.099Q-16.003 58.056-16.307 58.056L-16.307 57.759L-14.620 57.759L-14.620 58.056Q-14.714 58.056-14.790 58.099Q-14.866 58.142-14.866 58.231Q-14.866 58.274-14.835 58.321L-14.178 59.110L-13.698 58.536Q-13.581 58.399-13.581 58.263Q-13.581 58.173-13.632 58.114Q-13.682 58.056-13.764 58.056L-13.764 57.759L-12.276 57.759L-12.276 58.056Q-12.913 58.056-13.323 58.536L-14.003 59.337L-12.917 60.649Q-12.757 60.825-12.596 60.868Q-12.436 60.911-12.132 60.911L-12.132 61.208L-13.819 61.208L-13.819 60.911Q-13.729 60.911-13.651 60.868Q-13.573 60.825-13.573 60.735Q-13.573 60.712-13.604 60.649L-14.346 59.743L-14.932 60.431Q-15.050 60.567-15.050 60.704Q-15.050 60.790-14.999 60.851Q-14.948 60.911-14.858 60.911\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(38.887 -90.447)\">\u003Cpath d=\"M-8.908 59.513Q-8.908 59.009-8.652 58.577Q-8.396 58.145-7.960 57.894Q-7.525 57.642-7.025 57.642Q-6.638 57.642-6.296 57.786Q-5.955 57.931-5.693 58.192Q-5.431 58.454-5.289 58.790Q-5.146 59.126-5.146 59.513Q-5.146 60.005-5.410 60.415Q-5.673 60.825-6.103 61.056Q-6.533 61.286-7.025 61.286Q-7.517 61.286-7.951 61.054Q-8.384 60.821-8.646 60.413Q-8.908 60.005-8.908 59.513M-7.025 61.009Q-6.568 61.009-6.316 60.786Q-6.064 60.563-5.976 60.212Q-5.888 59.860-5.888 59.415Q-5.888 58.985-5.982 58.647Q-6.076 58.310-6.330 58.103Q-6.584 57.895-7.025 57.895Q-7.673 57.895-7.917 58.312Q-8.162 58.728-8.162 59.415Q-8.162 59.860-8.074 60.212Q-7.986 60.563-7.734 60.786Q-7.482 61.009-7.025 61.009M-2.732 61.208L-4.587 61.208L-4.587 60.911Q-4.314 60.911-4.146 60.864Q-3.978 60.817-3.978 60.649L-3.978 58.513Q-3.978 58.298-4.041 58.202Q-4.103 58.106-4.222 58.085Q-4.341 58.063-4.587 58.063L-4.587 57.767L-3.396 57.681L-3.396 58.415Q-3.283 58.200-3.089 58.032Q-2.896 57.864-2.658 57.772Q-2.419 57.681-2.166 57.681Q-0.998 57.681-0.998 58.759L-0.998 60.649Q-0.998 60.817-0.828 60.864Q-0.658 60.911-0.388 60.911L-0.388 61.208L-2.244 61.208L-2.244 60.911Q-1.970 60.911-1.802 60.864Q-1.634 60.817-1.634 60.649L-1.634 58.774Q-1.634 58.392-1.755 58.163Q-1.876 57.935-2.228 57.935Q-2.541 57.935-2.794 58.097Q-3.048 58.259-3.195 58.528Q-3.341 58.798-3.341 59.095L-3.341 60.649Q-3.341 60.817-3.171 60.864Q-3.001 60.911-2.732 60.911\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(38.887 -90.447)\">\u003Cpath d=\"M4.714 61.286Q4.233 61.286 3.825 61.042Q3.417 60.798 3.179 60.384Q2.940 59.970 2.940 59.481Q2.940 58.989 3.198 58.573Q3.456 58.157 3.888 57.919Q4.319 57.681 4.811 57.681Q5.432 57.681 5.882 58.118L5.882 56.489Q5.882 56.274 5.819 56.179Q5.757 56.083 5.639 56.062Q5.522 56.040 5.276 56.040L5.276 55.743L6.499 55.657L6.499 60.466Q6.499 60.677 6.561 60.772Q6.624 60.868 6.741 60.890Q6.858 60.911 7.108 60.911L7.108 61.208L5.858 61.286L5.858 60.802Q5.393 61.286 4.714 61.286M4.780 61.032Q5.120 61.032 5.413 60.841Q5.706 60.649 5.858 60.353L5.858 58.520Q5.710 58.247 5.448 58.091Q5.186 57.935 4.874 57.935Q4.249 57.935 3.966 58.382Q3.682 58.829 3.682 59.489Q3.682 60.134 3.934 60.583Q4.186 61.032 4.780 61.032M7.714 60.376Q7.714 59.892 8.116 59.597Q8.518 59.302 9.069 59.183Q9.620 59.063 10.112 59.063L10.112 58.774Q10.112 58.548 9.997 58.341Q9.882 58.134 9.684 58.015Q9.487 57.895 9.257 57.895Q8.831 57.895 8.546 58.001Q8.616 58.028 8.663 58.083Q8.710 58.138 8.735 58.208Q8.761 58.278 8.761 58.353Q8.761 58.458 8.710 58.550Q8.659 58.642 8.567 58.692Q8.475 58.743 8.370 58.743Q8.264 58.743 8.173 58.692Q8.081 58.642 8.030 58.550Q7.979 58.458 7.979 58.353Q7.979 57.935 8.368 57.788Q8.757 57.642 9.257 57.642Q9.589 57.642 9.942 57.772Q10.296 57.903 10.524 58.157Q10.753 58.411 10.753 58.759L10.753 60.560Q10.753 60.692 10.825 60.802Q10.897 60.911 11.026 60.911Q11.151 60.911 11.220 60.806Q11.288 60.700 11.288 60.560L11.288 60.048L11.569 60.048L11.569 60.560Q11.569 60.763 11.452 60.921Q11.335 61.079 11.153 61.163Q10.971 61.247 10.768 61.247Q10.538 61.247 10.386 61.075Q10.233 60.903 10.202 60.673Q10.042 60.954 9.733 61.120Q9.425 61.286 9.073 61.286Q8.561 61.286 8.138 61.063Q7.714 60.841 7.714 60.376M8.401 60.376Q8.401 60.661 8.628 60.847Q8.854 61.032 9.147 61.032Q9.393 61.032 9.618 60.915Q9.843 60.798 9.977 60.595Q10.112 60.392 10.112 60.138L10.112 59.306Q9.846 59.306 9.561 59.360Q9.276 59.415 9.005 59.544Q8.733 59.673 8.567 59.880Q8.401 60.087 8.401 60.376M12.487 60.247L12.487 58.056L11.784 58.056L11.784 57.802Q12.139 57.802 12.382 57.569Q12.624 57.337 12.735 56.989Q12.846 56.642 12.846 56.286L13.128 56.286L13.128 57.759L14.304 57.759L14.304 58.056L13.128 58.056L13.128 60.231Q13.128 60.552 13.247 60.780Q13.366 61.009 13.647 61.009Q13.827 61.009 13.944 60.886Q14.061 60.763 14.114 60.583Q14.167 60.403 14.167 60.231L14.167 59.759L14.448 59.759L14.448 60.247Q14.448 60.501 14.343 60.741Q14.237 60.981 14.040 61.134Q13.843 61.286 13.585 61.286Q13.268 61.286 13.016 61.163Q12.764 61.040 12.626 60.806Q12.487 60.571 12.487 60.247M15.264 60.376Q15.264 59.892 15.667 59.597Q16.069 59.302 16.620 59.183Q17.171 59.063 17.663 59.063L17.663 58.774Q17.663 58.548 17.548 58.341Q17.432 58.134 17.235 58.015Q17.038 57.895 16.807 57.895Q16.382 57.895 16.096 58.001Q16.167 58.028 16.214 58.083Q16.261 58.138 16.286 58.208Q16.311 58.278 16.311 58.353Q16.311 58.458 16.261 58.550Q16.210 58.642 16.118 58.692Q16.026 58.743 15.921 58.743Q15.815 58.743 15.723 58.692Q15.632 58.642 15.581 58.550Q15.530 58.458 15.530 58.353Q15.530 57.935 15.919 57.788Q16.307 57.642 16.807 57.642Q17.139 57.642 17.493 57.772Q17.846 57.903 18.075 58.157Q18.304 58.411 18.304 58.759L18.304 60.560Q18.304 60.692 18.376 60.802Q18.448 60.911 18.577 60.911Q18.702 60.911 18.770 60.806Q18.839 60.700 18.839 60.560L18.839 60.048L19.120 60.048L19.120 60.560Q19.120 60.763 19.003 60.921Q18.886 61.079 18.704 61.163Q18.522 61.247 18.319 61.247Q18.089 61.247 17.936 61.075Q17.784 60.903 17.753 60.673Q17.593 60.954 17.284 61.120Q16.975 61.286 16.624 61.286Q16.112 61.286 15.688 61.063Q15.264 60.841 15.264 60.376M15.952 60.376Q15.952 60.661 16.179 60.847Q16.405 61.032 16.698 61.032Q16.944 61.032 17.169 60.915Q17.393 60.798 17.528 60.595Q17.663 60.392 17.663 60.138L17.663 59.306Q17.397 59.306 17.112 59.360Q16.827 59.415 16.555 59.544Q16.284 59.673 16.118 59.880Q15.952 60.087 15.952 60.376\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Conservative versus naive value landscapes at a fixed state. The naive \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">Q\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:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> (dashed) fits the data in-support but extrapolates upward off it, handing the argmax a fictitious peak. CQL adds a penalty that pushes down out-of-distribution values, so the conservative \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">Q\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:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> (solid) lower-bounds the true value and its argmax stays on the data.\u003C\u002Ffigcaption>",1785117797561]