[{"data":1,"prerenderedAt":9355},["ShallowReactive",2],{"lesson:\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":3,"course-wordcounts":3574,"ref-card-index":4486,"nav:reinforcement-learning":9168,"tikz:960dcf7c0aacecbee9d7caa1c4ee88be608e4e15eb6997737aea7038db4910d7":9349,"tikz:ffa923082a8c2fd0c921d8064789d6bd895585717b007522819983387eefca3e":9350,"tikz:31b246d8bff6f4c6a2e5d8b1ba4646f44cb4932fdb1e09ed396ef84dc91b4157":9351,"tikz:4162e9fd73b83bd8a8dcabcf1f9a5f66cbad95dc05fbdeead8728cc4429ac24d":9352,"tikz:5824043938b9007e3251ac72ca9cc22149ea17d2fa4a2d7c006f0dae0bff5696":9353,"tikz:383a2c46bb3a6c32a02d773fb37a9b9005f336bf0019b002279515cc8cfe2b18":9354},{"id":4,"title":5,"blurb":6,"body":7,"brief":3546,"category":3547,"description":3548,"draft":3549,"extension":3550,"meta":3551,"module":3554,"navigation":3555,"path":3556,"practice":3557,"rawbody":3558,"readingTime":3559,"seo":3564,"sources":3565,"status":3568,"stem":3569,"summary":3570,"topics":3571,"__hash__":3573},"course\u002F07.reinforcement-learning\u002F06.minds-and-brains\u002F06.cognitive-maps-and-planning.md","Cognitive Maps and Model-Based Learning","",{"type":8,"value":9,"toc":3534},"minimark",[10,25,30,61,80,85,96,100,119,152,162,166,439,450,453,728,731,735,912,923,926,1272,2118,2498,3074,3077,3085,3089,3092,3114,3125,3128,3156,3174,3178,3181,3189],[11,12,13,14,19,20,24],"p",{},"This builds on ",[15,16,18],"a",{"href":17},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition","Animal Learning and Cognition",",\nwhich treated three ",[21,22,23],"em",{},"associative"," phenomena — blocking, higher-order conditioning,\nand delayed reinforcement — as reinforcement-learning mechanisms seen in behavior.\nNone of them needs the animal to hold a model of its world. This part takes up the\nphenomenon that does: the cognitive map, learned without reward and queried by\nplanning, which is model-based reinforcement learning measured in a rat.",[26,27,29],"h2",{"id":28},"cognitive-maps","Cognitive maps",[11,31,32,33,37,38,42,43,46,47,50,51],{},"Model-based reinforcement-learning algorithms use environment models that have\nelements in common with what psychologists call ",[34,35,36],"strong",{},"cognitive maps",". Recall from\n",[15,39,41],{"href":40},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning","planning and learning","\nthat by an environment model we mean anything an agent can use to predict how its\nenvironment will respond to its actions — in terms of state transitions and rewards\n— and by planning we mean any process that computes a policy from such a model. An\nenvironment model has two parts: a ",[34,44,45],{},"state-transition"," part, encoding the effect\nof actions on state changes, and a ",[34,48,49],{},"reward-model"," part, encoding the reward\nsignals expected for each state or state–action pair.",[52,53,54],"sup",{},[15,55,60],{"href":56,"ariaDescribedBy":57,"dataFootnoteRef":6,"id":59},"#user-content-fn-cog-map-sb",[58],"footnote-label","user-content-fnref-cog-map-sb","1",[11,62,63,64,67,68,71,72],{},"Whether animals use environment models, and if so what the models are like and how\nthey are learned, has driven a long argument in animal-learning research. Some\nresearchers challenged the then-prevailing ",[34,65,66],{},"stimulus–response (S–R)"," view of\nlearning — which corresponds to the simplest model-free way of learning policies —\nby demonstrating ",[34,69,70],{},"latent learning",".",[52,73,74],{},[15,75,79],{"href":76,"ariaDescribedBy":77,"dataFootnoteRef":6,"id":78},"#user-content-fn-latent-sb",[58],"user-content-fnref-latent-sb","2",[81,82,84],"h3",{"id":83},"latent-learning","Latent learning",[11,86,87,88],{},"In the earliest latent-learning experiment, two groups of rats were run in a maze.\nFor the experimental group there was no reward during the first stage of the\nexperiment, but food was suddenly introduced into the goal box at the start of the\nsecond stage. For the control group, food was in the goal box throughout both\nstages. The question was whether the experimental rats would have learned anything\nduring the first, unrewarded stage.",[52,89,90],{},[15,91,95],{"href":92,"ariaDescribedBy":93,"dataFootnoteRef":6,"id":94},"#user-content-fn-blodgett",[58],"user-content-fnref-blodgett","3",[97,98],"tikz-figure",{"hash":99},"960dcf7c0aacecbee9d7caa1c4ee88be608e4e15eb6997737aea7038db4910d7",[11,101,102,103,106,107,109,110,118],{},"Although the experimental rats did not ",[21,104,105],{},"appear"," to learn much during the first,\nunrewarded stage, as soon as they discovered the food in the second stage they\nrapidly caught up with the control rats. The conclusion was that during the\nnon-reward period the experimental rats were developing a ",[34,108,70],{}," of the\nmaze, which they were able to use as soon as reward was introduced.",[52,111,112],{},[15,113,117],{"href":114,"ariaDescribedBy":115,"dataFootnoteRef":6,"id":116},"#user-content-fn-blodgett-quote",[58],"user-content-fnref-blodgett-quote","4","\nThe rats had learned the maze's layout while it earned them nothing; the reward only\nrevealed a knowledge that was already there.",[11,120,121,122,125,126,134,135,139,140,143,144],{},"Latent learning is most closely associated with Edward Tolman, who interpreted\nresults like these as showing that animals learn a ",[34,123,124],{},"cognitive map"," of the\nenvironment in the absence of rewards or penalties, and can use the map later when\nmotivated to reach a goal.",[52,127,128],{},[15,129,133],{"href":130,"ariaDescribedBy":131,"dataFootnoteRef":6,"id":132},"#user-content-fn-tolman",[58],"user-content-fnref-tolman","5"," A cognitive map could let a rat plan a route to\nthe goal that is different from the route it used in its initial exploration — the\nsignature of a model you can query for paths you never walked. In modern terms,\ncognitive maps are not restricted to spatial layouts but are more generally\nenvironment models, models of an animal's ",[136,137,138],"q",{},"task space."," That a rat's spatial map\nhas a physical substrate is now familiar: the hippocampus contains place cells that\nfire selectively when the animal is at a particular location, a neural correlate of\nthe map Tolman inferred from behavior alone. The cognitive-map explanation of latent\nlearning is the claim that ",[34,141,142],{},"animals use model-based algorithms",", and that\nenvironment models can be learned even without explicit rewards or penalties, then\nused for planning once reward appears.",[52,145,146],{},[15,147,151],{"href":148,"ariaDescribedBy":149,"dataFootnoteRef":6,"id":150},"#user-content-fn-cog-model-based",[58],"user-content-fnref-cog-model-based","6",[153,154,156],"callout",{"type":155},"definition",[11,157,158,161],{},[34,159,160],{},"Definition (Cognitive map \u002F latent learning)."," A cognitive map is an internal\nmodel of the environment — its transitions and, where relevant, its rewards —\nthat an animal can build without reward and later use to plan. Latent learning is\nthe acquisition of such a model in the absence of reinforcement, invisible in\nbehavior until reward makes it worth acting on.",[81,163,165],{"id":164},"how-the-map-is-learned-system-identification","How the map is learned: system identification",[11,167,168,169,172,173,176,177,180,181,189,190,193,194,296,297,316,317,364,365,412,413,71,431],{},"Tolman's account of ",[21,170,171],{},"how"," animals learn cognitive maps is that they learn\n",[34,174,175],{},"stimulus–stimulus (S–S) associations"," by experiencing successions of stimuli as\nthey explore. In psychology this is called ",[34,178,179],{},"expectancy theory",": given the S–S\nassociations, the occurrence of one stimulus generates an expectation about the\nstimulus to come next.",[52,182,183],{},[15,184,188],{"href":185,"ariaDescribedBy":186,"dataFootnoteRef":6,"id":187},"#user-content-fn-expectancy",[58],"user-content-fnref-expectancy","7"," This is much like what control engineers call\n",[34,191,192],{},"system identification"," — learning a model of a system with unknown dynamics from\nlabeled training examples. In the simplest discrete-time version the training\nexamples are ",[195,196,199],"span",{"className":197},[198],"katex",[195,200,204,242],{"className":201,"ariaHidden":203},[202],"katex-html","true",[195,205,208,213,222,227,231,236,239],{"className":206},[207],"base",[195,209],{"className":210,"style":212},[211],"strut","height:0.7667em;vertical-align:-0.0833em;",[195,214,218],{"className":215},[216,217],"mord","text",[195,219,221],{"className":220},[216],"S",[195,223],{"className":224,"style":226},[225],"mspace","margin-right:-0.1667em;",[195,228],{"className":229,"style":230},[225],"margin-right:0.2222em;",[195,232,235],{"className":233},[234],"mbin","−",[195,237],{"className":238,"style":226},[225],[195,240],{"className":241,"style":230},[225],[195,243,245,249],{"className":244},[207],[195,246],{"className":247,"style":248},[211],"height:0.8251em;",[195,250,252,258],{"className":251},[216],[195,253,255],{"className":254},[216,217],[195,256,221],{"className":257},[216],[195,259,262],{"className":260},[261],"msupsub",[195,263,266],{"className":264},[265],"vlist-t",[195,267,270],{"className":268},[269],"vlist-r",[195,271,274],{"className":272,"style":248},[273],"vlist",[195,275,277,282],{"style":276},"top:-3.1362em;margin-right:0.05em;",[195,278],{"className":279,"style":281},[280],"pstrut","height:2.7em;",[195,283,289],{"className":284},[285,286,287,288],"sizing","reset-size6","size3","mtight",[195,290,292],{"className":291},[216,288],[195,293,295],{"className":294},[216,288],"′"," pairs, where ",[195,298,300],{"className":299},[198],[195,301,303],{"className":302,"ariaHidden":203},[202],[195,304,306,310],{"className":305},[207],[195,307],{"className":308,"style":309},[211],"height:0.6833em;",[195,311,313],{"className":312},[216,217],[195,314,221],{"className":315},[216]," is a state and\n",[195,318,320],{"className":319},[198],[195,321,323],{"className":322,"ariaHidden":203},[202],[195,324,326,329],{"className":325},[207],[195,327],{"className":328,"style":248},[211],[195,330,332,338],{"className":331},[216],[195,333,335],{"className":334},[216,217],[195,336,221],{"className":337},[216],[195,339,341],{"className":340},[261],[195,342,344],{"className":343},[265],[195,345,347],{"className":346},[269],[195,348,350],{"className":349,"style":248},[273],[195,351,352,355],{"style":276},[195,353],{"className":354,"style":281},[280],[195,356,358],{"className":357},[285,286,287,288],[195,359,361],{"className":360},[216,288],[195,362,295],{"className":363},[216,288]," the subsequent state, and the model learns to expect ",[195,366,368],{"className":367},[198],[195,369,371],{"className":370,"ariaHidden":203},[202],[195,372,374,377],{"className":373},[207],[195,375],{"className":376,"style":248},[211],[195,378,380,386],{"className":379},[216],[195,381,383],{"className":382},[216,217],[195,384,221],{"className":385},[216],[195,387,389],{"className":388},[261],[195,390,392],{"className":391},[265],[195,393,395],{"className":394},[269],[195,396,398],{"className":397,"style":248},[273],[195,399,400,403],{"style":276},[195,401],{"className":402,"style":281},[280],[195,404,406],{"className":405},[285,286,287,288],[195,407,409],{"className":408},[216,288],[195,410,295],{"className":411},[216,288]," after\n",[195,414,416],{"className":415},[198],[195,417,419],{"className":418,"ariaHidden":203},[202],[195,420,422,425],{"className":421},[207],[195,423],{"className":424,"style":309},[211],[195,426,428],{"className":427},[216,217],[195,429,221],{"className":430},[216],[52,432,433],{},[15,434,438],{"href":435,"ariaDescribedBy":436,"dataFootnoteRef":6,"id":437},"#user-content-fn-sysid",[58],"user-content-fnref-sysid","8",[11,440,441,442,445,446,449],{},"The S–S account is the exact contrast to the stimulus–response view it was raised\nagainst, and the contrast is what latent learning turns on. An ",[34,443,444],{},"S–R"," learner caches,\nfor each state, the action that has paid off there: a lookup table from situation to\nresponse, built only when reward stamps a response in. Take away reward and there is\nnothing to stamp in, so an S–R learner should acquire nothing during an unrewarded\nstage. An ",[34,447,448],{},"S–S"," learner instead records what follows what — state to next state — a\nmodel of the world that reward never enters. It fills in during exploration whether or\nnot anything is paid, and stands ready to be queried once a goal is supplied. The\nexperimental rats behaved like S–S learners: silent during the unrewarded stage, then\nimmediately competent when food appeared, because the map was already built.",[97,451],{"hash":452},"ffa923082a8c2fd0c921d8064789d6bd895585717b007522819983387eefca3e",[11,454,455,456,531,532,579,580,599,600,618,619,664,665,707,708,723,724,727],{},"Models useful for planning involve actions too, so examples become\n",[195,457,459],{"className":458},[198],[195,460,462,490],{"className":461,"ariaHidden":203},[202],[195,463,465,468,475,478,481,484,487],{"className":464},[207],[195,466],{"className":467,"style":212},[211],[195,469,471],{"className":470},[216,217],[195,472,474],{"className":473},[216],"SA",[195,476],{"className":477,"style":226},[225],[195,479],{"className":480,"style":230},[225],[195,482,235],{"className":483},[234],[195,485],{"className":486,"style":226},[225],[195,488],{"className":489,"style":230},[225],[195,491,493,496],{"className":492},[207],[195,494],{"className":495,"style":248},[211],[195,497,499,505],{"className":498},[216],[195,500,502],{"className":501},[216,217],[195,503,221],{"className":504},[216],[195,506,508],{"className":507},[261],[195,509,511],{"className":510},[265],[195,512,514],{"className":513},[269],[195,515,517],{"className":516,"style":248},[273],[195,518,519,522],{"style":276},[195,520],{"className":521,"style":281},[280],[195,523,525],{"className":524},[285,286,287,288],[195,526,528],{"className":527},[216,288],[195,529,295],{"className":530},[216,288],": state ",[195,533,535],{"className":534},[198],[195,536,538],{"className":537,"ariaHidden":203},[202],[195,539,541,544],{"className":540},[207],[195,542],{"className":543,"style":248},[211],[195,545,547,553],{"className":546},[216],[195,548,550],{"className":549},[216,217],[195,551,221],{"className":552},[216],[195,554,556],{"className":555},[261],[195,557,559],{"className":558},[265],[195,560,562],{"className":561},[269],[195,563,565],{"className":564,"style":248},[273],[195,566,567,570],{"style":276},[195,568],{"className":569,"style":281},[280],[195,571,573],{"className":572},[285,286,287,288],[195,574,576],{"className":575},[216,288],[195,577,295],{"className":578},[216,288]," is expected when action ",[195,581,583],{"className":582},[198],[195,584,586],{"className":585,"ariaHidden":203},[202],[195,587,589,592],{"className":588},[207],[195,590],{"className":591,"style":309},[211],[195,593,595],{"className":594},[216,217],[195,596,598],{"className":597},[216],"A"," is\ntaken in state ",[195,601,603],{"className":602},[198],[195,604,606],{"className":605,"ariaHidden":203},[202],[195,607,609,612],{"className":608},[207],[195,610],{"className":611,"style":309},[211],[195,613,615],{"className":614},[216,217],[195,616,221],{"className":617},[216],". And to learn where the rewards are, examples take the form\n",[195,620,622],{"className":621},[198],[195,623,625,652],{"className":624,"ariaHidden":203},[202],[195,626,628,631,637,640,643,646,649],{"className":627},[207],[195,629],{"className":630,"style":212},[211],[195,632,634],{"className":633},[216,217],[195,635,221],{"className":636},[216],[195,638],{"className":639,"style":226},[225],[195,641],{"className":642,"style":230},[225],[195,644,235],{"className":645},[234],[195,647],{"className":648,"style":226},[225],[195,650],{"className":651,"style":230},[225],[195,653,655,658],{"className":654},[207],[195,656],{"className":657,"style":309},[211],[195,659,663],{"className":660,"style":662},[216,661],"mathnormal","margin-right:0.0077em;","R"," or ",[195,666,668],{"className":667},[198],[195,669,671,698],{"className":670,"ariaHidden":203},[202],[195,672,674,677,683,686,689,692,695],{"className":673},[207],[195,675],{"className":676,"style":212},[211],[195,678,680],{"className":679},[216,217],[195,681,474],{"className":682},[216],[195,684],{"className":685,"style":226},[225],[195,687],{"className":688,"style":230},[225],[195,690,235],{"className":691},[234],[195,693],{"className":694,"style":226},[225],[195,696],{"className":697,"style":230},[225],[195,699,701,704],{"className":700},[207],[195,702],{"className":703,"style":309},[211],[195,705,663],{"className":706,"style":662},[216,661],", with ",[195,709,711],{"className":710},[198],[195,712,714],{"className":713,"ariaHidden":203},[202],[195,715,717,720],{"className":716},[207],[195,718],{"className":719,"style":309},[211],[195,721,663],{"className":722,"style":662},[216,661]," the reward signal associated with the\nstate or the state–action pair. These are all forms of ",[34,725,726],{},"supervised learning",", by\nwhich an agent can acquire cognitive-like maps whether or not it receives any\nnon-zero reward signal while exploring, and that is how the experimental rats\ncould build their map through a stage that paid them nothing.",[97,729],{"hash":730},"31b246d8bff6f4c6a2e5d8b1ba4646f44cb4932fdb1e09ed396ef84dc91b4157",[81,732,734],{"id":733},"planning-over-the-map-a-worked-maze","Planning over the map: a worked maze",[11,736,737,738,798,799,664,851,903,904],{},"The payoff of holding a model rather than cached values is what happens when the\nworld changes. Consider a rat in a maze with two turns and four distinctive goal\nboxes, each delivering a fixed reward. From the start state ",[195,739,741],{"className":740},[198],[195,742,744],{"className":743,"ariaHidden":203},[202],[195,745,747,751],{"className":746},[207],[195,748],{"className":749,"style":750},[211],"height:0.8333em;vertical-align:-0.15em;",[195,752,754,758],{"className":753},[216],[195,755,221],{"className":756,"style":757},[216,661],"margin-right:0.0576em;",[195,759,761],{"className":760},[261],[195,762,765,789],{"className":763},[265,764],"vlist-t2",[195,766,768,784],{"className":767},[269],[195,769,772],{"className":770,"style":771},[273],"height:0.3011em;",[195,773,775,778],{"style":774},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[195,776],{"className":777,"style":281},[280],[195,779,781],{"className":780},[285,286,287,288],[195,782,60],{"className":783},[216,288],[195,785,788],{"className":786},[787],"vlist-s","​",[195,790,792],{"className":791},[269],[195,793,796],{"className":794,"style":795},[273],"height:0.15em;",[195,797],{}," the rat chooses\nleft or right to reach ",[195,800,802],{"className":801},[198],[195,803,805],{"className":804,"ariaHidden":203},[202],[195,806,808,811],{"className":807},[207],[195,809],{"className":810,"style":750},[211],[195,812,814,817],{"className":813},[216],[195,815,221],{"className":816,"style":757},[216,661],[195,818,820],{"className":819},[261],[195,821,823,843],{"className":822},[265,764],[195,824,826,840],{"className":825},[269],[195,827,829],{"className":828,"style":771},[273],[195,830,831,834],{"style":774},[195,832],{"className":833,"style":281},[280],[195,835,837],{"className":836},[285,286,287,288],[195,838,79],{"className":839},[216,288],[195,841,788],{"className":842},[787],[195,844,846],{"className":845},[269],[195,847,849],{"className":848,"style":795},[273],[195,850],{},[195,852,854],{"className":853},[198],[195,855,857],{"className":856,"ariaHidden":203},[202],[195,858,860,863],{"className":859},[207],[195,861],{"className":862,"style":750},[211],[195,864,866,869],{"className":865},[216],[195,867,221],{"className":868,"style":757},[216,661],[195,870,872],{"className":871},[261],[195,873,875,895],{"className":874},[265,764],[195,876,878,892],{"className":877},[269],[195,879,881],{"className":880,"style":771},[273],[195,882,883,886],{"style":774},[195,884],{"className":885,"style":281},[280],[195,887,889],{"className":888},[285,286,287,288],[195,890,95],{"className":891},[216,288],[195,893,788],{"className":894},[787],[195,896,898],{"className":897},[269],[195,899,901],{"className":900,"style":795},[273],[195,902],{},", then chooses left or right again to land in\none of the four goal boxes.",[52,905,906],{},[15,907,911],{"href":908,"ariaDescribedBy":909,"dataFootnoteRef":6,"id":910},"#user-content-fn-maze-fig",[58],"user-content-fnref-maze-fig","9",[11,913,914,915,918,919,922],{},"A ",[34,916,917],{},"model-free"," rat solves this with stored action values, one number per\nstate–action pair, learned over many runs of the maze. Once those estimates are\ngood enough, it just picks the largest-valued action at each state. A ",[34,920,921],{},"model-based","\nrat instead learns a transition model (the branching structure of the maze) and a\nreward model (which goal box pays what), and decides by simulating action sequences\nto find the path with the highest return.",[97,924],{"hash":925},"4162e9fd73b83bd8a8dcabcf1f9a5f66cbad95dc05fbdeead8728cc4429ac24d",[11,927,928,929,976,977,1029,1030,1046,1047,1099,1100,1115,1116,1131,1132,1184,1185,1237,1238,1253,1254,1257,1258,1266,1267,1271],{},"With rewards ",[195,930,932],{"className":931},[198],[195,933,935],{"className":934,"ariaHidden":203},[202],[195,936,938,942,946,951,955,958,961,964,967,970,973],{"className":937},[207],[195,939],{"className":940,"style":941},[211],"height:0.8389em;vertical-align:-0.1944em;",[195,943,945],{"className":944},[216],"0",[195,947,950],{"className":948},[949],"mpunct",",",[195,952],{"className":953,"style":954},[225],"margin-right:0.1667em;",[195,956,117],{"className":957},[216],[195,959,950],{"className":960},[949],[195,962],{"className":963,"style":954},[225],[195,965,79],{"className":966},[216],[195,968,950],{"className":969},[949],[195,971],{"className":972,"style":954},[225],[195,974,95],{"className":975},[216]," across the four goal boxes, both strategies converge on\nthe same choice: go left to ",[195,978,980],{"className":979},[198],[195,981,983],{"className":982,"ariaHidden":203},[202],[195,984,986,989],{"className":985},[207],[195,987],{"className":988,"style":750},[211],[195,990,992,995],{"className":991},[216],[195,993,221],{"className":994,"style":757},[216,661],[195,996,998],{"className":997},[261],[195,999,1001,1021],{"className":1000},[265,764],[195,1002,1004,1018],{"className":1003},[269],[195,1005,1007],{"className":1006,"style":771},[273],[195,1008,1009,1012],{"style":774},[195,1010],{"className":1011,"style":281},[280],[195,1013,1015],{"className":1014},[285,286,287,288],[195,1016,79],{"className":1017},[216,288],[195,1019,788],{"className":1020},[787],[195,1022,1024],{"className":1023},[269],[195,1025,1027],{"className":1026,"style":795},[273],[195,1028],{},", then right, for a return of ",[195,1031,1033],{"className":1032},[198],[195,1034,1036],{"className":1035,"ariaHidden":203},[202],[195,1037,1039,1043],{"className":1038},[207],[195,1040],{"className":1041,"style":1042},[211],"height:0.6444em;",[195,1044,117],{"className":1045},[216],". The difference\nshows only under change. Suppose the rat is placed directly in the goal box right of\n",[195,1048,1050],{"className":1049},[198],[195,1051,1053],{"className":1052,"ariaHidden":203},[202],[195,1054,1056,1059],{"className":1055},[207],[195,1057],{"className":1058,"style":750},[211],[195,1060,1062,1065],{"className":1061},[216],[195,1063,221],{"className":1064,"style":757},[216,661],[195,1066,1068],{"className":1067},[261],[195,1069,1071,1091],{"className":1070},[265,764],[195,1072,1074,1088],{"className":1073},[269],[195,1075,1077],{"className":1076,"style":771},[273],[195,1078,1079,1082],{"style":774},[195,1080],{"className":1081,"style":281},[280],[195,1083,1085],{"className":1084},[285,286,287,288],[195,1086,79],{"className":1087},[216,288],[195,1089,788],{"className":1090},[787],[195,1092,1094],{"className":1093},[269],[195,1095,1097],{"className":1096,"style":795},[273],[195,1098],{}," and finds the reward there is now ",[195,1101,1103],{"className":1102},[198],[195,1104,1106],{"className":1105,"ariaHidden":203},[202],[195,1107,1109,1112],{"className":1108},[207],[195,1110],{"className":1111,"style":1042},[211],[195,1113,60],{"className":1114},[216]," instead of ",[195,1117,1119],{"className":1118},[198],[195,1120,1122],{"className":1121,"ariaHidden":203},[202],[195,1123,1125,1128],{"className":1124},[207],[195,1126],{"className":1127,"style":1042},[211],[195,1129,117],{"className":1130},[216],", without ever running the\nmaze. A model-based rat updates its reward model from that single experience;\nplanning then brings the new value to bear on maze-running with no further\nexperience in the maze, changing the policy to right turns at both ",[195,1133,1135],{"className":1134},[198],[195,1136,1138],{"className":1137,"ariaHidden":203},[202],[195,1139,1141,1144],{"className":1140},[207],[195,1142],{"className":1143,"style":750},[211],[195,1145,1147,1150],{"className":1146},[216],[195,1148,221],{"className":1149,"style":757},[216,661],[195,1151,1153],{"className":1152},[261],[195,1154,1156,1176],{"className":1155},[265,764],[195,1157,1159,1173],{"className":1158},[269],[195,1160,1162],{"className":1161,"style":771},[273],[195,1163,1164,1167],{"style":774},[195,1165],{"className":1166,"style":281},[280],[195,1168,1170],{"className":1169},[285,286,287,288],[195,1171,60],{"className":1172},[216,288],[195,1174,788],{"className":1175},[787],[195,1177,1179],{"className":1178},[269],[195,1180,1182],{"className":1181,"style":795},[273],[195,1183],{}," and ",[195,1186,1188],{"className":1187},[198],[195,1189,1191],{"className":1190,"ariaHidden":203},[202],[195,1192,1194,1197],{"className":1193},[207],[195,1195],{"className":1196,"style":750},[211],[195,1198,1200,1203],{"className":1199},[216],[195,1201,221],{"className":1202,"style":757},[216,661],[195,1204,1206],{"className":1205},[261],[195,1207,1209,1229],{"className":1208},[265,764],[195,1210,1212,1226],{"className":1211},[269],[195,1213,1215],{"className":1214,"style":771},[273],[195,1216,1217,1220],{"style":774},[195,1218],{"className":1219,"style":281},[280],[195,1221,1223],{"className":1222},[285,286,287,288],[195,1224,95],{"className":1225},[216,288],[195,1227,788],{"className":1228},[787],[195,1230,1232],{"className":1231},[269],[195,1233,1235],{"className":1234,"style":795},[273],[195,1236],{},"\nfor a return of ",[195,1239,1241],{"className":1240},[198],[195,1242,1244],{"className":1243,"ariaHidden":203},[202],[195,1245,1247,1250],{"className":1246},[207],[195,1248],{"className":1249,"style":1042},[211],[195,1251,95],{"className":1252},[216],". A model-free rat cannot do this. To change the action its\npolicy specifies at a state, or the value it has cached for a state, it has to move\n",[21,1255,1256],{},"to"," that state, act from it, and experience the consequences — possibly many\ntimes.",[52,1259,1260],{},[15,1261,1265],{"href":1262,"ariaDescribedBy":1263,"dataFootnoteRef":6,"id":1264},"#user-content-fn-devaluation-tie",[58],"user-content-fnref-devaluation-tie","10"," This is the mechanism behind the outcome-devaluation\nresult from the ",[15,1268,1270],{"href":1269},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement","previous\nlesson",": the\ngoal-directed rat re-plans over its map the instant the goal's value changes, while\nthe habitual rat must relearn by acting.",[11,1273,1274,1275,1327,1328,1343,1344,1396,1397,1462,1463,1515,1516,1184,1531,1546,1547,1599,1600,1654,1655,1707,1708,1723,1724,1739,1740,1755,1756,1808,1809,1824,1825,1840,1841,1893,1894,1948,1949,2001,2002,2017,2018,2070,2071,2086,2087,2102,2103,71],{},"Trace the replan explicitly. Before the change, the model-based rat backs values up the\ntree by taking the max at each choice. The right goal box of ",[195,1276,1278],{"className":1277},[198],[195,1279,1281],{"className":1280,"ariaHidden":203},[202],[195,1282,1284,1287],{"className":1283},[207],[195,1285],{"className":1286,"style":750},[211],[195,1288,1290,1293],{"className":1289},[216],[195,1291,221],{"className":1292,"style":757},[216,661],[195,1294,1296],{"className":1295},[261],[195,1297,1299,1319],{"className":1298},[265,764],[195,1300,1302,1316],{"className":1301},[269],[195,1303,1305],{"className":1304,"style":771},[273],[195,1306,1307,1310],{"style":774},[195,1308],{"className":1309,"style":281},[280],[195,1311,1313],{"className":1312},[285,286,287,288],[195,1314,79],{"className":1315},[216,288],[195,1317,788],{"className":1318},[787],[195,1320,1322],{"className":1321},[269],[195,1323,1325],{"className":1324,"style":795},[273],[195,1326],{}," pays ",[195,1329,1331],{"className":1330},[198],[195,1332,1334],{"className":1333,"ariaHidden":203},[202],[195,1335,1337,1340],{"className":1336},[207],[195,1338],{"className":1339,"style":1042},[211],[195,1341,117],{"className":1342},[216],", so\n",[195,1345,1347],{"className":1346},[198],[195,1348,1350],{"className":1349,"ariaHidden":203},[202],[195,1351,1353,1356],{"className":1352},[207],[195,1354],{"className":1355,"style":750},[211],[195,1357,1359,1362],{"className":1358},[216],[195,1360,221],{"className":1361,"style":757},[216,661],[195,1363,1365],{"className":1364},[261],[195,1366,1368,1388],{"className":1367},[265,764],[195,1369,1371,1385],{"className":1370},[269],[195,1372,1374],{"className":1373,"style":771},[273],[195,1375,1376,1379],{"style":774},[195,1377],{"className":1378,"style":281},[280],[195,1380,1382],{"className":1381},[285,286,287,288],[195,1383,79],{"className":1384},[216,288],[195,1386,788],{"className":1387},[787],[195,1389,1391],{"className":1390},[269],[195,1392,1394],{"className":1393,"style":795},[273],[195,1395],{},"'s value is ",[195,1398,1400],{"className":1399},[198],[195,1401,1403,1453],{"className":1402,"ariaHidden":203},[202],[195,1404,1406,1410,1419,1424,1427,1430,1433,1436,1441,1445,1450],{"className":1405},[207],[195,1407],{"className":1408,"style":1409},[211],"height:1em;vertical-align:-0.25em;",[195,1411,1414],{"className":1412},[1413],"mop",[195,1415,1418],{"className":1416},[216,1417],"mathrm","max",[195,1420,1423],{"className":1421},[1422],"mopen","(",[195,1425,945],{"className":1426},[216],[195,1428,950],{"className":1429},[949],[195,1431],{"className":1432,"style":954},[225],[195,1434,117],{"className":1435},[216],[195,1437,1440],{"className":1438},[1439],"mclose",")",[195,1442],{"className":1443,"style":1444},[225],"margin-right:0.2778em;",[195,1446,1449],{"className":1447},[1448],"mrel","=",[195,1451],{"className":1452,"style":1444},[225],[195,1454,1456,1459],{"className":1455},[207],[195,1457],{"className":1458,"style":1042},[211],[195,1460,117],{"className":1461},[216],"; the goal boxes of ",[195,1464,1466],{"className":1465},[198],[195,1467,1469],{"className":1468,"ariaHidden":203},[202],[195,1470,1472,1475],{"className":1471},[207],[195,1473],{"className":1474,"style":750},[211],[195,1476,1478,1481],{"className":1477},[216],[195,1479,221],{"className":1480,"style":757},[216,661],[195,1482,1484],{"className":1483},[261],[195,1485,1487,1507],{"className":1486},[265,764],[195,1488,1490,1504],{"className":1489},[269],[195,1491,1493],{"className":1492,"style":771},[273],[195,1494,1495,1498],{"style":774},[195,1496],{"className":1497,"style":281},[280],[195,1499,1501],{"className":1500},[285,286,287,288],[195,1502,95],{"className":1503},[216,288],[195,1505,788],{"className":1506},[787],[195,1508,1510],{"className":1509},[269],[195,1511,1513],{"className":1512,"style":795},[273],[195,1514],{}," pay ",[195,1517,1519],{"className":1518},[198],[195,1520,1522],{"className":1521,"ariaHidden":203},[202],[195,1523,1525,1528],{"className":1524},[207],[195,1526],{"className":1527,"style":1042},[211],[195,1529,79],{"className":1530},[216],[195,1532,1534],{"className":1533},[198],[195,1535,1537],{"className":1536,"ariaHidden":203},[202],[195,1538,1540,1543],{"className":1539},[207],[195,1541],{"className":1542,"style":1042},[211],[195,1544,95],{"className":1545},[216],", so ",[195,1548,1550],{"className":1549},[198],[195,1551,1553],{"className":1552,"ariaHidden":203},[202],[195,1554,1556,1559],{"className":1555},[207],[195,1557],{"className":1558,"style":750},[211],[195,1560,1562,1565],{"className":1561},[216],[195,1563,221],{"className":1564,"style":757},[216,661],[195,1566,1568],{"className":1567},[261],[195,1569,1571,1591],{"className":1570},[265,764],[195,1572,1574,1588],{"className":1573},[269],[195,1575,1577],{"className":1576,"style":771},[273],[195,1578,1579,1582],{"style":774},[195,1580],{"className":1581,"style":281},[280],[195,1583,1585],{"className":1584},[285,286,287,288],[195,1586,95],{"className":1587},[216,288],[195,1589,788],{"className":1590},[787],[195,1592,1594],{"className":1593},[269],[195,1595,1597],{"className":1596,"style":795},[273],[195,1598],{},"'s\nvalue is ",[195,1601,1603],{"className":1602},[198],[195,1604,1606,1645],{"className":1605,"ariaHidden":203},[202],[195,1607,1609,1612,1618,1621,1624,1627,1630,1633,1636,1639,1642],{"className":1608},[207],[195,1610],{"className":1611,"style":1409},[211],[195,1613,1615],{"className":1614},[1413],[195,1616,1418],{"className":1617},[216,1417],[195,1619,1423],{"className":1620},[1422],[195,1622,79],{"className":1623},[216],[195,1625,950],{"className":1626},[949],[195,1628],{"className":1629,"style":954},[225],[195,1631,95],{"className":1632},[216],[195,1634,1440],{"className":1635},[1439],[195,1637],{"className":1638,"style":1444},[225],[195,1640,1449],{"className":1641},[1448],[195,1643],{"className":1644,"style":1444},[225],[195,1646,1648,1651],{"className":1647},[207],[195,1649],{"className":1650,"style":1042},[211],[195,1652,95],{"className":1653},[216],". From ",[195,1656,1658],{"className":1657},[198],[195,1659,1661],{"className":1660,"ariaHidden":203},[202],[195,1662,1664,1667],{"className":1663},[207],[195,1665],{"className":1666,"style":750},[211],[195,1668,1670,1673],{"className":1669},[216],[195,1671,221],{"className":1672,"style":757},[216,661],[195,1674,1676],{"className":1675},[261],[195,1677,1679,1699],{"className":1678},[265,764],[195,1680,1682,1696],{"className":1681},[269],[195,1683,1685],{"className":1684,"style":771},[273],[195,1686,1687,1690],{"style":774},[195,1688],{"className":1689,"style":281},[280],[195,1691,1693],{"className":1692},[285,286,287,288],[195,1694,60],{"className":1695},[216,288],[195,1697,788],{"className":1698},[787],[195,1700,1702],{"className":1701},[269],[195,1703,1705],{"className":1704,"style":795},[273],[195,1706],{}," the rat compares going left (",[195,1709,1711],{"className":1710},[198],[195,1712,1714],{"className":1713,"ariaHidden":203},[202],[195,1715,1717,1720],{"className":1716},[207],[195,1718],{"className":1719,"style":1042},[211],[195,1721,117],{"className":1722},[216],") with going right\n(",[195,1725,1727],{"className":1726},[198],[195,1728,1730],{"className":1729,"ariaHidden":203},[202],[195,1731,1733,1736],{"className":1732},[207],[195,1734],{"className":1735,"style":1042},[211],[195,1737,95],{"className":1738},[216],") and turns left, then right, for a return of ",[195,1741,1743],{"className":1742},[198],[195,1744,1746],{"className":1745,"ariaHidden":203},[202],[195,1747,1749,1752],{"className":1748},[207],[195,1750],{"className":1751,"style":1042},[211],[195,1753,117],{"className":1754},[216],". Now devalue: the rat is dropped\ninto the right box of ",[195,1757,1759],{"className":1758},[198],[195,1760,1762],{"className":1761,"ariaHidden":203},[202],[195,1763,1765,1768],{"className":1764},[207],[195,1766],{"className":1767,"style":750},[211],[195,1769,1771,1774],{"className":1770},[216],[195,1772,221],{"className":1773,"style":757},[216,661],[195,1775,1777],{"className":1776},[261],[195,1778,1780,1800],{"className":1779},[265,764],[195,1781,1783,1797],{"className":1782},[269],[195,1784,1786],{"className":1785,"style":771},[273],[195,1787,1788,1791],{"style":774},[195,1789],{"className":1790,"style":281},[280],[195,1792,1794],{"className":1793},[285,286,287,288],[195,1795,79],{"className":1796},[216,288],[195,1798,788],{"className":1799},[787],[195,1801,1803],{"className":1802},[269],[195,1804,1806],{"className":1805,"style":795},[273],[195,1807],{}," and finds it pays ",[195,1810,1812],{"className":1811},[198],[195,1813,1815],{"className":1814,"ariaHidden":203},[202],[195,1816,1818,1821],{"className":1817},[207],[195,1819],{"className":1820,"style":1042},[211],[195,1822,60],{"className":1823},[216],", not ",[195,1826,1828],{"className":1827},[198],[195,1829,1831],{"className":1830,"ariaHidden":203},[202],[195,1832,1834,1837],{"className":1833},[207],[195,1835],{"className":1836,"style":1042},[211],[195,1838,117],{"className":1839},[216],", and it updates that one\nreward-model entry. Nothing else in the maze was touched, but the backup changes\nwholesale. ",[195,1842,1844],{"className":1843},[198],[195,1845,1847],{"className":1846,"ariaHidden":203},[202],[195,1848,1850,1853],{"className":1849},[207],[195,1851],{"className":1852,"style":750},[211],[195,1854,1856,1859],{"className":1855},[216],[195,1857,221],{"className":1858,"style":757},[216,661],[195,1860,1862],{"className":1861},[261],[195,1863,1865,1885],{"className":1864},[265,764],[195,1866,1868,1882],{"className":1867},[269],[195,1869,1871],{"className":1870,"style":771},[273],[195,1872,1873,1876],{"style":774},[195,1874],{"className":1875,"style":281},[280],[195,1877,1879],{"className":1878},[285,286,287,288],[195,1880,79],{"className":1881},[216,288],[195,1883,788],{"className":1884},[787],[195,1886,1888],{"className":1887},[269],[195,1889,1891],{"className":1890,"style":795},[273],[195,1892],{}," is now worth ",[195,1895,1897],{"className":1896},[198],[195,1898,1900,1939],{"className":1899,"ariaHidden":203},[202],[195,1901,1903,1906,1912,1915,1918,1921,1924,1927,1930,1933,1936],{"className":1902},[207],[195,1904],{"className":1905,"style":1409},[211],[195,1907,1909],{"className":1908},[1413],[195,1910,1418],{"className":1911},[216,1417],[195,1913,1423],{"className":1914},[1422],[195,1916,945],{"className":1917},[216],[195,1919,950],{"className":1920},[949],[195,1922],{"className":1923,"style":954},[225],[195,1925,60],{"className":1926},[216],[195,1928,1440],{"className":1929},[1439],[195,1931],{"className":1932,"style":1444},[225],[195,1934,1449],{"className":1935},[1448],[195,1937],{"className":1938,"style":1444},[225],[195,1940,1942,1945],{"className":1941},[207],[195,1943],{"className":1944,"style":1042},[211],[195,1946,60],{"className":1947},[216],"; ",[195,1950,1952],{"className":1951},[198],[195,1953,1955],{"className":1954,"ariaHidden":203},[202],[195,1956,1958,1961],{"className":1957},[207],[195,1959],{"className":1960,"style":750},[211],[195,1962,1964,1967],{"className":1963},[216],[195,1965,221],{"className":1966,"style":757},[216,661],[195,1968,1970],{"className":1969},[261],[195,1971,1973,1993],{"className":1972},[265,764],[195,1974,1976,1990],{"className":1975},[269],[195,1977,1979],{"className":1978,"style":771},[273],[195,1980,1981,1984],{"style":774},[195,1982],{"className":1983,"style":281},[280],[195,1985,1987],{"className":1986},[285,286,287,288],[195,1988,95],{"className":1989},[216,288],[195,1991,788],{"className":1992},[787],[195,1994,1996],{"className":1995},[269],[195,1997,1999],{"className":1998,"style":795},[273],[195,2000],{}," is untouched at ",[195,2003,2005],{"className":2004},[198],[195,2006,2008],{"className":2007,"ariaHidden":203},[202],[195,2009,2011,2014],{"className":2010},[207],[195,2012],{"className":2013,"style":1042},[211],[195,2015,95],{"className":2016},[216],"; from ",[195,2019,2021],{"className":2020},[198],[195,2022,2024],{"className":2023,"ariaHidden":203},[202],[195,2025,2027,2030],{"className":2026},[207],[195,2028],{"className":2029,"style":750},[211],[195,2031,2033,2036],{"className":2032},[216],[195,2034,221],{"className":2035,"style":757},[216,661],[195,2037,2039],{"className":2038},[261],[195,2040,2042,2062],{"className":2041},[265,764],[195,2043,2045,2059],{"className":2044},[269],[195,2046,2048],{"className":2047,"style":771},[273],[195,2049,2050,2053],{"style":774},[195,2051],{"className":2052,"style":281},[280],[195,2054,2056],{"className":2055},[285,286,287,288],[195,2057,60],{"className":2058},[216,288],[195,2060,788],{"className":2061},[787],[195,2063,2065],{"className":2064},[269],[195,2066,2068],{"className":2067,"style":795},[273],[195,2069],{},"\nthe comparison is now left (",[195,2072,2074],{"className":2073},[198],[195,2075,2077],{"className":2076,"ariaHidden":203},[202],[195,2078,2080,2083],{"className":2079},[207],[195,2081],{"className":2082,"style":1042},[211],[195,2084,60],{"className":2085},[216],") versus right (",[195,2088,2090],{"className":2089},[198],[195,2091,2093],{"className":2092,"ariaHidden":203},[202],[195,2094,2096,2099],{"className":2095},[207],[195,2097],{"className":2098,"style":1042},[211],[195,2100,95],{"className":2101},[216],"), so the plan flips to right, then\nright, for a return of ",[195,2104,2106],{"className":2105},[198],[195,2107,2109],{"className":2108,"ariaHidden":203},[202],[195,2110,2112,2115],{"className":2111},[207],[195,2113],{"className":2114,"style":1042},[211],[195,2116,95],{"className":2117},[216],[2119,2120,2121,2139],"table",{},[2122,2123,2124],"thead",{},[2125,2126,2127,2132,2136],"tr",{},[2128,2129,2131],"th",{"align":2130},"left","Value",[2128,2133,2135],{"align":2134},"right","before",[2128,2137,2138],{"align":2134},"after devaluation",[2140,2141,2142,2205,2316,2424,2489],"tbody",{},[2125,2143,2144,2201,2203],{},[2145,2146,2147,2148,2200],"td",{"align":2130},"right box of ",[195,2149,2151],{"className":2150},[198],[195,2152,2154],{"className":2153,"ariaHidden":203},[202],[195,2155,2157,2160],{"className":2156},[207],[195,2158],{"className":2159,"style":750},[211],[195,2161,2163,2166],{"className":2162},[216],[195,2164,221],{"className":2165,"style":757},[216,661],[195,2167,2169],{"className":2168},[261],[195,2170,2172,2192],{"className":2171},[265,764],[195,2173,2175,2189],{"className":2174},[269],[195,2176,2178],{"className":2177,"style":771},[273],[195,2179,2180,2183],{"style":774},[195,2181],{"className":2182,"style":281},[280],[195,2184,2186],{"className":2185},[285,286,287,288],[195,2187,79],{"className":2188},[216,288],[195,2190,788],{"className":2191},[787],[195,2193,2195],{"className":2194},[269],[195,2196,2198],{"className":2197,"style":795},[273],[195,2199],{}," (reward model)",[2145,2202,117],{"align":2134},[2145,2204,60],{"align":2134},[2125,2206,2207,2312,2314],{},[2145,2208,2209],{"align":2130},[195,2210,2212],{"className":2211},[198],[195,2213,2215,2281],{"className":2214,"ariaHidden":203},[202],[195,2216,2218,2221,2226,2229,2269,2272,2275,2278],{"className":2217},[207],[195,2219],{"className":2220,"style":1409},[211],[195,2222,2225],{"className":2223,"style":2224},[216,661],"margin-right:0.0359em;","v",[195,2227,1423],{"className":2228},[1422],[195,2230,2232,2235],{"className":2231},[216],[195,2233,221],{"className":2234,"style":757},[216,661],[195,2236,2238],{"className":2237},[261],[195,2239,2241,2261],{"className":2240},[265,764],[195,2242,2244,2258],{"className":2243},[269],[195,2245,2247],{"className":2246,"style":771},[273],[195,2248,2249,2252],{"style":774},[195,2250],{"className":2251,"style":281},[280],[195,2253,2255],{"className":2254},[285,286,287,288],[195,2256,79],{"className":2257},[216,288],[195,2259,788],{"className":2260},[787],[195,2262,2264],{"className":2263},[269],[195,2265,2267],{"className":2266,"style":795},[273],[195,2268],{},[195,2270,1440],{"className":2271},[1439],[195,2273],{"className":2274,"style":1444},[225],[195,2276,1449],{"className":2277},[1448],[195,2279],{"className":2280,"style":1444},[225],[195,2282,2284,2287,2293,2296,2299,2302,2305,2309],{"className":2283},[207],[195,2285],{"className":2286,"style":1409},[211],[195,2288,2290],{"className":2289},[1413],[195,2291,1418],{"className":2292},[216,1417],[195,2294,1423],{"className":2295},[1422],[195,2297,945],{"className":2298},[216],[195,2300,950],{"className":2301},[949],[195,2303],{"className":2304,"style":954},[225],[195,2306,2308],{"className":2307},[216],"⋅",[195,2310,1440],{"className":2311},[1439],[2145,2313,117],{"align":2134},[2145,2315,60],{"align":2134},[2125,2317,2318,2420,2422],{},[2145,2319,2320],{"align":2130},[195,2321,2323],{"className":2322},[198],[195,2324,2326,2390],{"className":2325,"ariaHidden":203},[202],[195,2327,2329,2332,2335,2338,2378,2381,2384,2387],{"className":2328},[207],[195,2330],{"className":2331,"style":1409},[211],[195,2333,2225],{"className":2334,"style":2224},[216,661],[195,2336,1423],{"className":2337},[1422],[195,2339,2341,2344],{"className":2340},[216],[195,2342,221],{"className":2343,"style":757},[216,661],[195,2345,2347],{"className":2346},[261],[195,2348,2350,2370],{"className":2349},[265,764],[195,2351,2353,2367],{"className":2352},[269],[195,2354,2356],{"className":2355,"style":771},[273],[195,2357,2358,2361],{"style":774},[195,2359],{"className":2360,"style":281},[280],[195,2362,2364],{"className":2363},[285,286,287,288],[195,2365,95],{"className":2366},[216,288],[195,2368,788],{"className":2369},[787],[195,2371,2373],{"className":2372},[269],[195,2374,2376],{"className":2375,"style":795},[273],[195,2377],{},[195,2379,1440],{"className":2380},[1439],[195,2382],{"className":2383,"style":1444},[225],[195,2385,1449],{"className":2386},[1448],[195,2388],{"className":2389,"style":1444},[225],[195,2391,2393,2396,2402,2405,2408,2411,2414,2417],{"className":2392},[207],[195,2394],{"className":2395,"style":1409},[211],[195,2397,2399],{"className":2398},[1413],[195,2400,1418],{"className":2401},[216,1417],[195,2403,1423],{"className":2404},[1422],[195,2406,79],{"className":2407},[216],[195,2409,950],{"className":2410},[949],[195,2412],{"className":2413,"style":954},[225],[195,2415,95],{"className":2416},[216],[195,2418,1440],{"className":2419},[1439],[2145,2421,95],{"align":2134},[2145,2423,95],{"align":2134},[2125,2425,2426,2481,2484],{},[2145,2427,2428,2429],{"align":2130},"choice at ",[195,2430,2432],{"className":2431},[198],[195,2433,2435],{"className":2434,"ariaHidden":203},[202],[195,2436,2438,2441],{"className":2437},[207],[195,2439],{"className":2440,"style":750},[211],[195,2442,2444,2447],{"className":2443},[216],[195,2445,221],{"className":2446,"style":757},[216,661],[195,2448,2450],{"className":2449},[261],[195,2451,2453,2473],{"className":2452},[265,764],[195,2454,2456,2470],{"className":2455},[269],[195,2457,2459],{"className":2458,"style":771},[273],[195,2460,2461,2464],{"style":774},[195,2462],{"className":2463,"style":281},[280],[195,2465,2467],{"className":2466},[285,286,287,288],[195,2468,60],{"className":2469},[216,288],[195,2471,788],{"className":2472},[787],[195,2474,2476],{"className":2475},[269],[195,2477,2479],{"className":2478,"style":795},[273],[195,2480],{},[2145,2482,2483],{"align":2134},"L (4 vs 3)",[2145,2485,2486,2488],{"align":2134},[34,2487,663],{}," (1 vs 3)",[2125,2490,2491,2494,2496],{},[2145,2492,2493],{"align":2130},"return",[2145,2495,117],{"align":2134},[2145,2497,95],{"align":2134},[11,2499,2500,2501,2589,2590,2675,2676,2743,2744,2759,2760,2763,2764,2816,2817,2884,2885,2937,2938,3005,3006,3073],{},"The model-based rat did no maze-running to produce the new plan; it re-solved the tree\nfrom one changed number. The model-free rat, holding only the cached action values\n",[195,2502,2504],{"className":2503},[198],[195,2505,2507,2580],{"className":2506,"ariaHidden":203},[202],[195,2508,2510,2513,2517,2557,2560,2563,2567,2571,2574,2577],{"className":2509},[207],[195,2511],{"className":2512,"style":1409},[211],[195,2514,2516],{"className":2515},[1422],"⟨",[195,2518,2520,2523],{"className":2519},[216],[195,2521,221],{"className":2522,"style":757},[216,661],[195,2524,2526],{"className":2525},[261],[195,2527,2529,2549],{"className":2528},[265,764],[195,2530,2532,2546],{"className":2531},[269],[195,2533,2535],{"className":2534,"style":771},[273],[195,2536,2537,2540],{"style":774},[195,2538],{"className":2539,"style":281},[280],[195,2541,2543],{"className":2542},[285,286,287,288],[195,2544,60],{"className":2545},[216,288],[195,2547,788],{"className":2548},[787],[195,2550,2552],{"className":2551},[269],[195,2553,2555],{"className":2554,"style":795},[273],[195,2556],{},[195,2558,950],{"className":2559},[949],[195,2561],{"className":2562,"style":954},[225],[195,2564,2566],{"className":2565},[216,661],"L",[195,2568,2570],{"className":2569},[1439],"⟩",[195,2572],{"className":2573,"style":1444},[225],[195,2575,1449],{"className":2576},[1448],[195,2578],{"className":2579,"style":1444},[225],[195,2581,2583,2586],{"className":2582},[207],[195,2584],{"className":2585,"style":1042},[211],[195,2587,117],{"className":2588},[216],", ",[195,2591,2593],{"className":2592},[198],[195,2594,2596,2666],{"className":2595,"ariaHidden":203},[202],[195,2597,2599,2602,2605,2645,2648,2651,2654,2657,2660,2663],{"className":2598},[207],[195,2600],{"className":2601,"style":1409},[211],[195,2603,2516],{"className":2604},[1422],[195,2606,2608,2611],{"className":2607},[216],[195,2609,221],{"className":2610,"style":757},[216,661],[195,2612,2614],{"className":2613},[261],[195,2615,2617,2637],{"className":2616},[265,764],[195,2618,2620,2634],{"className":2619},[269],[195,2621,2623],{"className":2622,"style":771},[273],[195,2624,2625,2628],{"style":774},[195,2626],{"className":2627,"style":281},[280],[195,2629,2631],{"className":2630},[285,286,287,288],[195,2632,60],{"className":2633},[216,288],[195,2635,788],{"className":2636},[787],[195,2638,2640],{"className":2639},[269],[195,2641,2643],{"className":2642,"style":795},[273],[195,2644],{},[195,2646,950],{"className":2647},[949],[195,2649],{"className":2650,"style":954},[225],[195,2652,663],{"className":2653,"style":662},[216,661],[195,2655,2570],{"className":2656},[1439],[195,2658],{"className":2659,"style":1444},[225],[195,2661,1449],{"className":2662},[1448],[195,2664],{"className":2665,"style":1444},[225],[195,2667,2669,2672],{"className":2668},[207],[195,2670],{"className":2671,"style":1042},[211],[195,2673,95],{"className":2674},[216],", and so on, has no such move. The\nvalue it cached for ",[195,2677,2679],{"className":2678},[198],[195,2680,2682],{"className":2681,"ariaHidden":203},[202],[195,2683,2685,2688,2691,2731,2734,2737,2740],{"className":2684},[207],[195,2686],{"className":2687,"style":1409},[211],[195,2689,2516],{"className":2690},[1422],[195,2692,2694,2697],{"className":2693},[216],[195,2695,221],{"className":2696,"style":757},[216,661],[195,2698,2700],{"className":2699},[261],[195,2701,2703,2723],{"className":2702},[265,764],[195,2704,2706,2720],{"className":2705},[269],[195,2707,2709],{"className":2708,"style":771},[273],[195,2710,2711,2714],{"style":774},[195,2712],{"className":2713,"style":281},[280],[195,2715,2717],{"className":2716},[285,286,287,288],[195,2718,60],{"className":2719},[216,288],[195,2721,788],{"className":2722},[787],[195,2724,2726],{"className":2725},[269],[195,2727,2729],{"className":2728,"style":795},[273],[195,2730],{},[195,2732,950],{"className":2733},[949],[195,2735],{"className":2736,"style":954},[225],[195,2738,2566],{"className":2739},[216,661],[195,2741,2570],{"className":2742},[1439]," was learned from full runs ending at the box\nthat used to pay ",[195,2745,2747],{"className":2746},[198],[195,2748,2750],{"className":2749,"ariaHidden":203},[202],[195,2751,2753,2756],{"className":2752},[207],[195,2754],{"className":2755,"style":1042},[211],[195,2757,117],{"className":2758},[216],"; a single visit to that box while ",[21,2761,2762],{},"not"," running the maze from ",[195,2765,2767],{"className":2766},[198],[195,2768,2770],{"className":2769,"ariaHidden":203},[202],[195,2771,2773,2776],{"className":2772},[207],[195,2774],{"className":2775,"style":750},[211],[195,2777,2779,2782],{"className":2778},[216],[195,2780,221],{"className":2781,"style":757},[216,661],[195,2783,2785],{"className":2784},[261],[195,2786,2788,2808],{"className":2787},[265,764],[195,2789,2791,2805],{"className":2790},[269],[195,2792,2794],{"className":2793,"style":771},[273],[195,2795,2796,2799],{"style":774},[195,2797],{"className":2798,"style":281},[280],[195,2800,2802],{"className":2801},[285,286,287,288],[195,2803,60],{"className":2804},[216,288],[195,2806,788],{"className":2807},[787],[195,2809,2811],{"className":2810},[269],[195,2812,2814],{"className":2813,"style":795},[273],[195,2815],{},"\ngives it no update to ",[195,2818,2820],{"className":2819},[198],[195,2821,2823],{"className":2822,"ariaHidden":203},[202],[195,2824,2826,2829,2832,2872,2875,2878,2881],{"className":2825},[207],[195,2827],{"className":2828,"style":1409},[211],[195,2830,2516],{"className":2831},[1422],[195,2833,2835,2838],{"className":2834},[216],[195,2836,221],{"className":2837,"style":757},[216,661],[195,2839,2841],{"className":2840},[261],[195,2842,2844,2864],{"className":2843},[265,764],[195,2845,2847,2861],{"className":2846},[269],[195,2848,2850],{"className":2849,"style":771},[273],[195,2851,2852,2855],{"style":774},[195,2853],{"className":2854,"style":281},[280],[195,2856,2858],{"className":2857},[285,286,287,288],[195,2859,60],{"className":2860},[216,288],[195,2862,788],{"className":2863},[787],[195,2865,2867],{"className":2866},[269],[195,2868,2870],{"className":2869,"style":795},[273],[195,2871],{},[195,2873,950],{"className":2874},[949],[195,2876],{"className":2877,"style":954},[225],[195,2879,2566],{"className":2880},[216,661],[195,2882,2570],{"className":2883},[1439]," at all. To fix the stale cache it must\nphysically go left from ",[195,2886,2888],{"className":2887},[198],[195,2889,2891],{"className":2890,"ariaHidden":203},[202],[195,2892,2894,2897],{"className":2893},[207],[195,2895],{"className":2896,"style":750},[211],[195,2898,2900,2903],{"className":2899},[216],[195,2901,221],{"className":2902,"style":757},[216,661],[195,2904,2906],{"className":2905},[261],[195,2907,2909,2929],{"className":2908},[265,764],[195,2910,2912,2926],{"className":2911},[269],[195,2913,2915],{"className":2914,"style":771},[273],[195,2916,2917,2920],{"style":774},[195,2918],{"className":2919,"style":281},[280],[195,2921,2923],{"className":2922},[285,286,287,288],[195,2924,60],{"className":2925},[216,288],[195,2927,788],{"className":2928},[787],[195,2930,2932],{"className":2931},[269],[195,2933,2935],{"className":2934,"style":795},[273],[195,2936],{},", walk to the now-devalued box, take the disappointing\nreward, and let the error propagate back — and repeat until ",[195,2939,2941],{"className":2940},[198],[195,2942,2944],{"className":2943,"ariaHidden":203},[202],[195,2945,2947,2950,2953,2993,2996,2999,3002],{"className":2946},[207],[195,2948],{"className":2949,"style":1409},[211],[195,2951,2516],{"className":2952},[1422],[195,2954,2956,2959],{"className":2955},[216],[195,2957,221],{"className":2958,"style":757},[216,661],[195,2960,2962],{"className":2961},[261],[195,2963,2965,2985],{"className":2964},[265,764],[195,2966,2968,2982],{"className":2967},[269],[195,2969,2971],{"className":2970,"style":771},[273],[195,2972,2973,2976],{"style":774},[195,2974],{"className":2975,"style":281},[280],[195,2977,2979],{"className":2978},[285,286,287,288],[195,2980,60],{"className":2981},[216,288],[195,2983,788],{"className":2984},[787],[195,2986,2988],{"className":2987},[269],[195,2989,2991],{"className":2990,"style":795},[273],[195,2992],{},[195,2994,950],{"className":2995},[949],[195,2997],{"className":2998,"style":954},[225],[195,3000,2566],{"className":3001},[216,661],[195,3003,2570],{"className":3004},[1439]," falls\nbelow ",[195,3007,3009],{"className":3008},[198],[195,3010,3012],{"className":3011,"ariaHidden":203},[202],[195,3013,3015,3018,3021,3061,3064,3067,3070],{"className":3014},[207],[195,3016],{"className":3017,"style":1409},[211],[195,3019,2516],{"className":3020},[1422],[195,3022,3024,3027],{"className":3023},[216],[195,3025,221],{"className":3026,"style":757},[216,661],[195,3028,3030],{"className":3029},[261],[195,3031,3033,3053],{"className":3032},[265,764],[195,3034,3036,3050],{"className":3035},[269],[195,3037,3039],{"className":3038,"style":771},[273],[195,3040,3041,3044],{"style":774},[195,3042],{"className":3043,"style":281},[280],[195,3045,3047],{"className":3046},[285,286,287,288],[195,3048,60],{"className":3049},[216,288],[195,3051,788],{"className":3052},[787],[195,3054,3056],{"className":3055},[269],[195,3057,3059],{"className":3058,"style":795},[273],[195,3060],{},[195,3062,950],{"className":3063},[949],[195,3065],{"className":3066,"style":954},[225],[195,3068,663],{"className":3069,"style":662},[216,661],[195,3071,2570],{"className":3072},[1439],". Same maze, same single fact learned; the map-holder acts on\nit at once, the cache-holder cannot act on it until it re-runs.",[97,3075],{"hash":3076},"5824043938b9007e3251ac72ca9cc22149ea17d2fa4a2d7c006f0dae0bff5696",[11,3078,3079,3080,3084],{},"The connection runs forward as well as back. The same model-based machinery — a\nlearned transition model, a learned reward model, and planning by simulated\nrollouts — is what\n",[15,3081,3083],{"href":3082},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl","modern model-based reinforcement learning","\nscales up with function approximation, learning the model with neural networks and\nplanning with search or imagined trajectories. Tolman's rat, silently mapping a maze\nit was not paid to explore, was doing in tissue what a model-based agent does in\nsilicon.",[26,3086,3088],{"id":3087},"the-successor-representation-and-evidence-for-maps","The successor representation and evidence for maps",[11,3090,3091],{},"Sutton and Barto leave the model-free\u002Fmodel-based split as a clean dichotomy — cache versus\ntree, habit versus plan. Work since has filled in the middle and pinned the split to\nbrain and behavior. Three lines are worth stating carefully, each for exactly what its\nauthors showed.",[11,3093,3094,3097,3098,3101,3102,3105,3106,3109,3110,3113],{},[34,3095,3096],{},"A predictive map between the two systems."," Dayan (1993, ",[21,3099,3100],{},"Neural Computation",")\nintroduced the ",[34,3103,3104],{},"successor representation",": instead of caching a scalar value per\nstate, or holding a full one-step transition model to plan over, cache for each state a\nvector of ",[21,3107,3108],{},"expected future occupancy"," — how often, discounted, each other state will be\nvisited if you follow the current policy from here. Value is then a dot product of this\npredictive map with the reward vector. The successor representation sits between\nmodel-free and model-based control. Like a cached value it is fast to read, needing no\nsearch; like a model it factors the world's dynamics apart from reward, so a change in\nreward re-values states without re-running the policy — though a change in the\n",[21,3111,3112],{},"transitions"," still requires relearning the map, which a full model would not. It is a\npartial cognitive map: predictive structure without a plannable model.",[11,3115,3116,3117,3120,3121,3124],{},"Stachenfeld, Botvinick and Gershman (2017, ",[21,3118,3119],{},"Nature Neuroscience",") argued that the\nhippocampus encodes a map of this kind. They showed that place-cell firing fields, and\nthe periodic fields of entorhinal grid cells, are well described as components of a\nsuccessor representation of the animal's environment — place fields skew and stretch\nalong frequently-taken routes, and grid patterns emerge as a low-dimensional\ndecomposition of the predictive map. This gives Tolman's cognitive map, inferred from\nbehavior in the 1940s, a concrete computational form and a candidate neural substrate:\nnot a static layout but a ",[21,3122,3123],{},"predictive"," map of where the animal is headed.",[97,3126],{"hash":3127},"383a2c46bb3a6c32a02d773fb37a9b9005f336bf0019b002279515cc8cfe2b18",[11,3129,3130,3133,3134,3137,3138,3141,3142,3144,3145,3148,3149,3151,3152,3155],{},[34,3131,3132],{},"Both systems in one human choice."," Daw, Gershman, Seymour, Dayan and Dolan (2011,\n",[21,3135,3136],{},"Neuron",") built a ",[34,3139,3140],{},"two-step Markov task"," that separates the two controllers in a\nsingle stream of choices. A first-stage choice leads probabilistically to one of two\nsecond-stage states, and reward there drifts over time. A pure model-free learner\nrepeats whatever first-stage action preceded reward, ignoring ",[21,3143,171],{}," the transition went;\na model-based learner uses its knowledge of the transition structure, so after a rare\ntransition it does the opposite. Human choices showed a ",[21,3146,3147],{},"mixture"," of the two signatures,\nand the balance shifted with circumstance — tilting model-free under cognitive load or\ntime pressure. This is the behavioral fingerprint of the habit\u002Fgoal-directed distinction\nthis lesson maps onto model-free and model-based control, measured in people rather than\ninferred. Gläscher, Daw, Dayan and O'Doherty (2010, ",[21,3150,3136],{},") added the learning-signal\nhalf: alongside the reward-prediction error that trains values, they found a distinct\n",[34,3153,3154],{},"state-prediction error"," — the surprise when a transition differs from what the model\nexpected — with its own neural correlate, the error signal that would train the model\nitself rather than its cached values.",[11,3157,3158,3161,3162,3165,3166,3169,3170,3173],{},[34,3159,3160],{},"Scaled-up maps in machines."," The engineering forward-pointer is real systems that\nlearn a world model and plan inside it. Ha and Schmidhuber (2018, ",[136,3163,3164],{},"World Models",") trained\nan agent to compress its environment into a learned latent dynamics model and then\nlearn a policy largely by ",[136,3167,3168],{},"dreaming"," — rolling the model forward instead of acting. Schrittwieser\net al. (2020, ",[21,3171,3172],{},"Nature","), in MuZero, learned a model whose only job is to support\nvalue-accurate planning, and combined it with Monte-Carlo tree search to reach top play\nin Go, chess, shogi, and Atari without being told the rules. These are Tolman's map plus\nplan built at scale: a learned model of consequences, queried by simulated rollouts, to\nchoose actions in situations never directly experienced. The rat mapping a maze it was\nnot paid to explore and the agent planning over a learned latent model are running the\nsame idea at very different sizes.",[26,3175,3177],{"id":3176},"the-one-claim","The one claim",[11,3179,3180],{},"Three phenomena, one point. Blocking is prediction error: an animal learns about a\nstimulus only to the extent that its outcome is surprising, which is the error term\nin every update in this course. Higher-order conditioning and conditioned\nreinforcement are bootstrapping: a learned prediction becomes a reward in its own\nright, which is what a value function is and what an actor–critic critic supplies.\nDelayed reinforcement is credit assignment, and the stimulus traces and lengthened\ngoal gradients that Pavlov and Hull proposed for it are eligibility traces and\nTD-learned values. Cognitive maps are environment models, learned by system\nidentification with or without reward, and queried by planning — model-based\nreinforcement learning, measured in a rat.",[11,3182,3183,3184,3188],{},"None of this asserts that the brain runs these exact equations. It asserts something\nnarrower and more durable: the structure of the learning problem is the same whether\nthe learner is tissue or silicon, so a theory built to solve it computationally keeps\nlanding on mechanisms that animal-learning researchers had already isolated in\nbehavior. The ",[15,3185,3187],{"href":3186},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error","next module thread","\nfollows the contact point one level deeper, into the neuron, where the dopamine\nsignal turns out to behave like the TD error that all of this has been circling.",[3190,3191,3194,3199],"section",{"className":3192,"dataFootnotes":6},[3193],"footnotes",[26,3195,3198],{"className":3196,"id":58},[3197],"sr-only","Footnotes",[3200,3201,3202,3217,3228,3239,3250,3264,3279,3290,3301,3470],"ol",{},[3203,3204,3206,3209,3210],"li",{"id":3205},"user-content-fn-cog-map-sb",[34,3207,3208],{},"Sutton & Barto",", §14.5 — Cognitive Maps: model-based reinforcement learning algorithms use environment models with elements in common with psychologists' cognitive maps; an environment model has a state-transition part (effect of actions on state changes) and a reward-model part (reward signals expected for each state or state–action pair), and planning computes a policy by simulating imagined sequences of decisions. ",[15,3211,3216],{"href":3212,"ariaLabel":3213,"className":3214,"dataFootnoteBackref":6},"#user-content-fnref-cog-map-sb","Back to reference 1",[3215],"data-footnote-backref","↩",[3203,3218,3220,3222,3223],{"id":3219},"user-content-fn-latent-sb",[34,3221,3208],{},", §14.5: some researchers challenged the prevailing stimulus–response (S–R) view — corresponding to the simplest model-free way of learning policies — by demonstrating latent learning. ",[15,3224,3216],{"href":3225,"ariaLabel":3226,"className":3227,"dataFootnoteBackref":6},"#user-content-fnref-latent-sb","Back to reference 2",[3215],[3203,3229,3231,3233,3234],{"id":3230},"user-content-fn-blodgett",[34,3232,3208],{},", §14.5: in the earliest latent-learning experiment, two groups of rats ran a maze; the experimental group had no reward in stage 1 with food suddenly introduced into the goal box at the start of stage 2, while the control group had food throughout; the question was whether the experimental group learned anything during the unrewarded first stage. ",[15,3235,3216],{"href":3236,"ariaLabel":3237,"className":3238,"dataFootnoteBackref":6},"#user-content-fnref-blodgett","Back to reference 3",[3215],[3203,3240,3242,3244,3245],{"id":3241},"user-content-fn-blodgett-quote",[34,3243,3208],{},", §14.5: although the experimental rats did not appear to learn much during the unrewarded first stage, once they discovered the stage-2 food they rapidly caught up to the control rats; Blodgett (1929) concluded that during the non-reward period the rats were developing a latent learning of the maze they could utilize as soon as reward was introduced. ",[15,3246,3216],{"href":3247,"ariaLabel":3248,"className":3249,"dataFootnoteBackref":6},"#user-content-fnref-blodgett-quote","Back to reference 4",[3215],[3203,3251,3253,3255,3256,3258,3259],{"id":3252},"user-content-fn-tolman",[34,3254,3208],{},", §14.5: latent learning is most closely associated with Edward Tolman (1948), who interpreted such results as animals learning a ",[136,3257,124],{}," of the environment in the absence of rewards or penalties, usable later when motivated to reach a goal; a cognitive map could let a rat plan a route different from the one used in initial exploration. ",[15,3260,3216],{"href":3261,"ariaLabel":3262,"className":3263,"dataFootnoteBackref":6},"#user-content-fnref-tolman","Back to reference 5",[3215],[3203,3265,3267,3269,3270,3273,3274],{"id":3266},"user-content-fn-cog-model-based",[34,3268,3208],{},", §14.5: in modern terms cognitive maps are not restricted to spatial layouts but are more generally environment models, models of an animal's ",[136,3271,3272],{},"task space"," (e.g. Wilson, Takahashi, Schoenbaum, and Niv, 2014); the cognitive-map explanation of latent learning is the claim that animals use model-based algorithms and that environment models can be learned even without explicit rewards or penalties, then used for planning when reward appears. ",[15,3275,3216],{"href":3276,"ariaLabel":3277,"className":3278,"dataFootnoteBackref":6},"#user-content-fnref-cog-model-based","Back to reference 6",[3215],[3203,3280,3282,3284,3285],{"id":3281},"user-content-fn-expectancy",[34,3283,3208],{},", §14.5: Tolman's account is that animals learn stimulus–stimulus (S–S) associations by experiencing successions of stimuli while exploring; in psychology this is expectancy theory — given S–S associations, the occurrence of a stimulus generates an expectation about the stimulus to come next. ",[15,3286,3216],{"href":3287,"ariaLabel":3288,"className":3289,"dataFootnoteBackref":6},"#user-content-fnref-expectancy","Back to reference 7",[3215],[3203,3291,3293,3295,3296],{"id":3292},"user-content-fn-sysid",[34,3294,3208],{},", §14.5: learning S–S associations is much like what control engineers call system identification — learning a model of a system with unknown dynamics from labeled examples; the simplest discrete-time examples are S–S′ pairs, action-inclusive examples are SA–S′, and reward examples are S–R or SA–R, all forms of supervised learning by which an agent can acquire cognitive-like maps whether or not it receives non-zero reward while exploring. ",[15,3297,3216],{"href":3298,"ariaLabel":3299,"className":3300,"dataFootnoteBackref":6},"#user-content-fnref-sysid","Back to reference 8",[3215],[3203,3302,3304,3306,3307,3359,3360,664,3412,3464,3465],{"id":3303},"user-content-fn-maze-fig",[34,3305,3208],{},", §14.6 \u002F Figure 14.5: a rat navigates a maze with distinctive goal boxes each delivering an associated reward; from ",[195,3308,3310],{"className":3309},[198],[195,3311,3313],{"className":3312,"ariaHidden":203},[202],[195,3314,3316,3319],{"className":3315},[207],[195,3317],{"className":3318,"style":750},[211],[195,3320,3322,3325],{"className":3321},[216],[195,3323,221],{"className":3324,"style":757},[216,661],[195,3326,3328],{"className":3327},[261],[195,3329,3331,3351],{"className":3330},[265,764],[195,3332,3334,3348],{"className":3333},[269],[195,3335,3337],{"className":3336,"style":771},[273],[195,3338,3339,3342],{"style":774},[195,3340],{"className":3341,"style":281},[280],[195,3343,3345],{"className":3344},[285,286,287,288],[195,3346,60],{"className":3347},[216,288],[195,3349,788],{"className":3350},[787],[195,3352,3354],{"className":3353},[269],[195,3355,3357],{"className":3356,"style":795},[273],[195,3358],{}," it selects L or R to reach ",[195,3361,3363],{"className":3362},[198],[195,3364,3366],{"className":3365,"ariaHidden":203},[202],[195,3367,3369,3372],{"className":3368},[207],[195,3370],{"className":3371,"style":750},[211],[195,3373,3375,3378],{"className":3374},[216],[195,3376,221],{"className":3377,"style":757},[216,661],[195,3379,3381],{"className":3380},[261],[195,3382,3384,3404],{"className":3383},[265,764],[195,3385,3387,3401],{"className":3386},[269],[195,3388,3390],{"className":3389,"style":771},[273],[195,3391,3392,3395],{"style":774},[195,3393],{"className":3394,"style":281},[280],[195,3396,3398],{"className":3397},[285,286,287,288],[195,3399,79],{"className":3400},[216,288],[195,3402,788],{"className":3403},[787],[195,3405,3407],{"className":3406},[269],[195,3408,3410],{"className":3409,"style":795},[273],[195,3411],{},[195,3413,3415],{"className":3414},[198],[195,3416,3418],{"className":3417,"ariaHidden":203},[202],[195,3419,3421,3424],{"className":3420},[207],[195,3422],{"className":3423,"style":750},[211],[195,3425,3427,3430],{"className":3426},[216],[195,3428,221],{"className":3429,"style":757},[216,661],[195,3431,3433],{"className":3432},[261],[195,3434,3436,3456],{"className":3435},[265,764],[195,3437,3439,3453],{"className":3438},[269],[195,3440,3442],{"className":3441,"style":771},[273],[195,3443,3444,3447],{"style":774},[195,3445],{"className":3446,"style":281},[280],[195,3448,3450],{"className":3449},[285,286,287,288],[195,3451,95],{"className":3452},[216,288],[195,3454,788],{"className":3455},[787],[195,3457,3459],{"className":3458},[269],[195,3460,3462],{"className":3461,"style":795},[273],[195,3463],{},", then L or R again to reach a goal box. A model-free strategy relies on stored action values for state–action pairs; a model-based strategy learns a state-transition model (a decision tree) and a reward model associating goal-box features with rewards, deciding by simulating action sequences to find the highest-return path. ",[15,3466,3216],{"href":3467,"ariaLabel":3468,"className":3469,"dataFootnoteBackref":6},"#user-content-fnref-maze-fig","Back to reference 9",[3215],[3203,3471,3473,3475,3476,3528,3529],{"id":3472},"user-content-fn-devaluation-tie",[34,3474,3208],{},", §14.6: with rewards 0, 4, 2, 3 both strategies select L then R from ",[195,3477,3479],{"className":3478},[198],[195,3480,3482],{"className":3481,"ariaHidden":203},[202],[195,3483,3485,3488],{"className":3484},[207],[195,3486],{"className":3487,"style":750},[211],[195,3489,3491,3494],{"className":3490},[216],[195,3492,221],{"className":3493,"style":757},[216,661],[195,3495,3497],{"className":3496},[261],[195,3498,3500,3520],{"className":3499},[265,764],[195,3501,3503,3517],{"className":3502},[269],[195,3504,3506],{"className":3505,"style":771},[273],[195,3507,3508,3511],{"style":774},[195,3509],{"className":3510,"style":281},[280],[195,3512,3514],{"className":3513},[285,286,287,288],[195,3515,60],{"className":3516},[216,288],[195,3518,788],{"className":3519},[787],[195,3521,3523],{"className":3522},[269],[195,3524,3526],{"className":3525,"style":795},[273],[195,3527],{}," for return 4; a model-based agent placed directly in a goal box whose reward changed (e.g. 4 to 1) updates its reward model and re-plans without maze experience, whereas a model-free agent must move to that state and act, possibly many times, to update — the logic underlying outcome-devaluation experiments. ",[15,3530,3216],{"href":3531,"ariaLabel":3532,"className":3533,"dataFootnoteBackref":6},"#user-content-fnref-devaluation-tie","Back to reference 10",[3215],{"title":6,"searchDepth":3535,"depth":3535,"links":3536},2,[3537,3543,3544,3545],{"id":28,"depth":3535,"text":29,"children":3538},[3539,3541,3542],{"id":83,"depth":3540,"text":84},3,{"id":164,"depth":3540,"text":165},{"id":733,"depth":3540,"text":734},{"id":3087,"depth":3535,"text":3088},{"id":3176,"depth":3535,"text":3177},{"id":58,"depth":3535,"text":3198},[],"computer-science","This builds on Animal Learning and Cognition,\nwhich treated three associative phenomena — blocking, higher-order conditioning,\nand delayed reinforcement — as reinforcement-learning mechanisms seen in behavior.\nNone of them needs the animal to hold a model of its world. This part takes up the\nphenomenon that does: the cognitive map, learned without reward and queried by\nplanning, which is model-based reinforcement learning measured in a rat.",false,"md",{"moduleNumber":3552,"lessonNumber":3552,"order":3553},6,606,"Reinforcement Learning in Minds and Brains",true,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning",[],"---\ntitle: \"Cognitive Maps and Model-Based Learning\"\nmodule: Reinforcement Learning in Minds and Brains\nmoduleNumber: 6\nlessonNumber: 6\norder: 606\nsummary: >\n  Tolman's rats learned the layout of a maze with no reward, then used it the moment\n  food appeared — latent learning, a cognitive map, and the behavioral face of\n  model-based reinforcement learning. The map is learned by system identification\n  (stimulus–stimulus associations), which fills in whether or not reward is present,\n  and queried by planning, which re-solves a route from a single changed reward. The\n  successor representation sits between cache and model, and hippocampal predictive\n  maps and scaled-up world models carry the same idea into brain and machine.\ntopics: [Minds and Brains]\nsources:\n  - book: Sutton & Barto\n    ref: \"Ch. 14 — Psychology; §14.5 Cognitive Maps; §14.6 Habitual and Goal-directed Behavior\"\n---\n\nThis builds on [Animal Learning and Cognition](\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition),\nwhich treated three _associative_ phenomena — blocking, higher-order conditioning,\nand delayed reinforcement — as reinforcement-learning mechanisms seen in behavior.\nNone of them needs the animal to hold a model of its world. This part takes up the\nphenomenon that does: the cognitive map, learned without reward and queried by\nplanning, which is model-based reinforcement learning measured in a rat.\n\n## Cognitive maps\n\nModel-based reinforcement-learning algorithms use environment models that have\nelements in common with what psychologists call **cognitive maps**. Recall from\n[planning and learning](\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning)\nthat by an environment model we mean anything an agent can use to predict how its\nenvironment will respond to its actions — in terms of state transitions and rewards\n— and by planning we mean any process that computes a policy from such a model. An\nenvironment model has two parts: a **state-transition** part, encoding the effect\nof actions on state changes, and a **reward-model** part, encoding the reward\nsignals expected for each state or state–action pair.[^cog-map-sb]\n\nWhether animals use environment models, and if so what the models are like and how\nthey are learned, has driven a long argument in animal-learning research. Some\nresearchers challenged the then-prevailing **stimulus–response (S–R)** view of\nlearning — which corresponds to the simplest model-free way of learning policies —\nby demonstrating **latent learning**.[^latent-sb]\n\n### Latent learning\n\nIn the earliest latent-learning experiment, two groups of rats were run in a maze.\nFor the experimental group there was no reward during the first stage of the\nexperiment, but food was suddenly introduced into the goal box at the start of the\nsecond stage. For the control group, food was in the goal box throughout both\nstages. The question was whether the experimental rats would have learned anything\nduring the first, unrewarded stage.[^blodgett]\n\n$$\n% caption: Latent learning. The control group is rewarded throughout and its\n% error count falls steadily. The experimental group is unrewarded until stage 2,\n% shows no improvement while unrewarded, then drops sharply to the control level\n% once food appears — revealing a map learned silently during the reward-free\n% stage.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % axes\n  \\draw[->, black] (0,0) -- (7.4,0) node[anchor=west, black, font=\\scriptsize] {days};\n  \\draw[->, black] (0,0) -- (0,3.4) node[anchor=south, black, font=\\scriptsize] {errors in maze};\n  % reward-introduction line for experimental group\n  \\draw[black, dotted] (4.4,0) -- (4.4,3.2);\n  \\node[anchor=south, font=\\scriptsize, text=black] at (4.4,3.15) {food introduced};\n  % control curve: steady decline\n  \\draw[acc, very thick] (0.2,3.0) .. controls (2.0,1.9) and (4.0,1.0) .. (7.0,0.55);\n  \\node[acc, anchor=west, font=\\scriptsize] at (7.05,0.55) {control};\n  % experimental curve: flat high, then sharp drop\n  \\draw[red, very thick] (0.2,3.05) .. controls (2.2,2.75) and (3.6,2.6) .. (4.4,2.55)\n    .. controls (5.0,1.4) and (5.6,0.85) .. (7.0,0.6);\n  \\node[red, anchor=west, font=\\scriptsize] at (7.05,0.75) {experimental};\n\\end{tikzpicture}\n$$\n\nAlthough the experimental rats did not _appear_ to learn much during the first,\nunrewarded stage, as soon as they discovered the food in the second stage they\nrapidly caught up with the control rats. The conclusion was that during the\nnon-reward period the experimental rats were developing a **latent learning** of the\nmaze, which they were able to use as soon as reward was introduced.[^blodgett-quote]\nThe rats had learned the maze's layout while it earned them nothing; the reward only\nrevealed a knowledge that was already there.\n\nLatent learning is most closely associated with Edward Tolman, who interpreted\nresults like these as showing that animals learn a **cognitive map** of the\nenvironment in the absence of rewards or penalties, and can use the map later when\nmotivated to reach a goal.[^tolman] A cognitive map could let a rat plan a route to\nthe goal that is different from the route it used in its initial exploration — the\nsignature of a model you can query for paths you never walked. In modern terms,\ncognitive maps are not restricted to spatial layouts but are more generally\nenvironment models, models of an animal's \"task space.\" That a rat's spatial map\nhas a physical substrate is now familiar: the hippocampus contains place cells that\nfire selectively when the animal is at a particular location, a neural correlate of\nthe map Tolman inferred from behavior alone. The cognitive-map explanation of latent\nlearning is the claim that **animals use model-based algorithms**, and that\nenvironment models can be learned even without explicit rewards or penalties, then\nused for planning once reward appears.[^cog-model-based]\n\n> **Definition (Cognitive map \u002F latent learning).** A cognitive map is an internal\n> model of the environment — its transitions and, where relevant, its rewards —\n> that an animal can build without reward and later use to plan. Latent learning is\n> the acquisition of such a model in the absence of reinforcement, invisible in\n> behavior until reward makes it worth acting on.\n\n### How the map is learned: system identification\n\nTolman's account of _how_ animals learn cognitive maps is that they learn\n**stimulus–stimulus (S–S) associations** by experiencing successions of stimuli as\nthey explore. In psychology this is called **expectancy theory**: given the S–S\nassociations, the occurrence of one stimulus generates an expectation about the\nstimulus to come next.[^expectancy] This is much like what control engineers call\n**system identification** — learning a model of a system with unknown dynamics from\nlabeled training examples. In the simplest discrete-time version the training\nexamples are $\\text{S}\\!-\\!\\text{S}'$ pairs, where $\\text{S}$ is a state and\n$\\text{S}'$ the subsequent state, and the model learns to expect $\\text{S}'$ after\n$\\text{S}$.[^sysid]\n\nThe S–S account is the exact contrast to the stimulus–response view it was raised\nagainst, and the contrast is what latent learning turns on. An **S–R** learner caches,\nfor each state, the action that has paid off there: a lookup table from situation to\nresponse, built only when reward stamps a response in. Take away reward and there is\nnothing to stamp in, so an S–R learner should acquire nothing during an unrewarded\nstage. An **S–S** learner instead records what follows what — state to next state — a\nmodel of the world that reward never enters. It fills in during exploration whether or\nnot anything is paid, and stands ready to be queried once a goal is supplied. The\nexperimental rats behaved like S–S learners: silent during the unrewarded stage, then\nimmediately competent when food appeared, because the map was already built.\n\n$$\n% caption: Stimulus-response versus stimulus-stimulus learning. S-R caches, per state,\n% the rewarded action - a policy table that only forms under reward. S-S records what\n% state follows what - a transition model that forms during exploration with no reward.\n% Latent learning is S-S: the map is built unrewarded, then queried once a goal appears.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  nd\u002F.style={draw, minimum width=9mm, minimum height=7mm, font=\\scriptsize},\n  cap2\u002F.style={font=\\scriptsize, text=black, align=center}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % ==== S-R (left) ====\n  \\node[font=\\small, anchor=south] at (1.5,2.5) {S-R (policy cache)};\n  \\node[nd, draw=red, text=red] (sr_s) at (0,1.4) {state};\n  \\node[nd] (sr_a) at (3.0,1.4) {action};\n  \\draw[->, red, thick] (sr_s) -- (sr_a) node[midway, above, font=\\scriptsize] {cached};\n  \\node[cap2] at (1.5,0.5) {needs reward to form;\\\\nothing learned unrewarded};\n  % ==== S-S (right) ====\n  \\begin{scope}[xshift=6.0cm]\n    \\node[font=\\small, anchor=south] at (1.5,2.5) {S-S (world model)};\n    \\node[nd, draw=acc, text=acc] (ss_s) at (0,1.4) {state};\n    \\node[nd, draw=acc, text=acc] (ss_s2) at (3.0,1.4) {next state};\n    \\draw[->, acc, thick] (ss_s) -- (ss_s2) node[midway, above, font=\\scriptsize] {predicts};\n    \\node[cap2] at (1.5,0.5) {forms during exploration;\\\\built with no reward};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nModels useful for planning involve actions too, so examples become\n$\\text{SA}\\!-\\!\\text{S}'$: state $\\text{S}'$ is expected when action $\\text{A}$ is\ntaken in state $\\text{S}$. And to learn where the rewards are, examples take the form\n$\\text{S}\\!-\\!R$ or $\\text{SA}\\!-\\!R$, with $R$ the reward signal associated with the\nstate or the state–action pair. These are all forms of **supervised learning**, by\nwhich an agent can acquire cognitive-like maps whether or not it receives any\nnon-zero reward signal while exploring, and that is how the experimental rats\ncould build their map through a stage that paid them nothing.\n\n$$\n% caption: A cognitive map is learned by system identification: supervised\n% examples of what follows what. State-transition examples S-S' or SA-S' give the\n% dynamics; reward examples S-R or SA-R give the reward model. None of this\n% requires reward to be present while learning, which is why latent learning works.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  ex\u002F.style={draw, minimum width=20mm, minimum height=8mm, font=\\scriptsize, align=center}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[ex, draw=acc, text=acc] (t1) at (0,1.0) {S, A  $\\Rightarrow$  S'};\n  \\node[ex, draw=acc, text=acc] (t2) at (0,-0.2) {SA  $\\Rightarrow$  R};\n  \\node[ex] (model) at (5.2,0.4) {environment\\\\model};\n  \\node[ex] (plan) at (5.2,-1.6) {planning\\\\(a policy)};\n  \\draw[->, acc, thick] (t1) -- (model.west) node[midway, above, font=\\scriptsize, text=black] {transition};\n  \\draw[->, acc, thick] (t2) -- (model.west);\n  \\draw[->, thick] (model) -- (plan) node[midway, right, font=\\scriptsize] {simulate paths};\n  \\node[anchor=north, font=\\scriptsize, text=black, align=center] at (0,-1.0)\n    {supervised examples\\\\(system identif\\\u002Fication)};\n\\end{tikzpicture}\n$$\n\n### Planning over the map: a worked maze\n\nThe payoff of holding a model rather than cached values is what happens when the\nworld changes. Consider a rat in a maze with two turns and four distinctive goal\nboxes, each delivering a fixed reward. From the start state $S_1$ the rat chooses\nleft or right to reach $S_2$ or $S_3$, then chooses left or right again to land in\none of the four goal boxes.[^maze-fig]\n\nA **model-free** rat solves this with stored action values, one number per\nstate–action pair, learned over many runs of the maze. Once those estimates are\ngood enough, it just picks the largest-valued action at each state. A **model-based**\nrat instead learns a transition model (the branching structure of the maze) and a\nreward model (which goal box pays what), and decides by simulating action sequences\nto find the path with the highest return.\n\n$$\n% caption: A model-free rat caches an action value per state-action pair (left);\n% a model-based rat holds a transition tree plus a reward model and simulates\n% paths to the best goal box (right). With rewards 0, 4, 2, 3, both pick L then R\n% from S1 for a return of 4.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  st\u002F.style={circle, draw, minimum size=7mm, inner sep=0pt, font=\\scriptsize},\n  gb\u002F.style={draw, minimum width=8mm, minimum height=6mm, font=\\scriptsize},\n  vc\u002F.style={draw, minimum width=8mm, minimum height=6mm, font=\\scriptsize},\n  kc\u002F.style={draw, minimum width=14mm, minimum height=6mm, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % ===== left: model-free cached values =====\n  \\node[font=\\small, anchor=south] at (1.5,3.15) {model-free};\n  \\foreach \\row\u002F\\lab\u002F\\val in {0\u002F{S1,L}\u002F4, 1\u002F{S1,R}\u002F3, 2\u002F{S2,L}\u002F0, 3\u002F{S2,R}\u002F4, 4\u002F{S3,L}\u002F2, 5\u002F{S3,R}\u002F3}{\n    \\node[kc, anchor=north west] (k\\row) at (0, 2.7-\\row*0.62) {\\lab};\n    \\node[vc, draw=acc, text=acc, anchor=north west] at (1.45, 2.7-\\row*0.62) {\\val};\n  }\n  \\node[anchor=north, font=\\scriptsize, text=black, align=center] at (1.5,-1.15)\n    {cached action values,\\\\pick the largest};\n  % ===== right: model-based tree =====\n  \\begin{scope}[xshift=5.4cm]\n    \\node[font=\\small, anchor=south] at (2.6,3.15) {model-based};\n    \\node[st] (s1) at (0,1.2) {S1};\n    \\node[st] (s2) at (2.2,2.4) {S2};\n    \\node[st] (s3) at (2.2,0.0) {S3};\n    \\node[gb] (g1) at (4.6,3.0) {=0};\n    \\node[gb] (g2) at (4.6,1.9) {=4};\n    \\node[gb] (g3) at (4.6,0.6) {=2};\n    \\node[gb] (g4) at (4.6,-0.5) {=3};\n    \\draw[->, acc] (s1) -- (s2) node[midway, above, font=\\scriptsize] {L};\n    \\draw[->, acc] (s1) -- (s3) node[midway, below, font=\\scriptsize] {R};\n    \\draw[->, red] (s2) -- (g1) node[near end, above, font=\\scriptsize] {L};\n    \\draw[->, red] (s2) -- (g2) node[near end, below, font=\\scriptsize] {R};\n    \\draw[->, red] (s3) -- (g3) node[near end, above, font=\\scriptsize] {L};\n    \\draw[->, red] (s3) -- (g4) node[near end, below, font=\\scriptsize] {R};\n    \\node[anchor=north, font=\\scriptsize, text=black, align=center] at (2.3,-1.15)\n      {transitions + reward model,\\\\simulate to best goal};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nWith rewards $0, 4, 2, 3$ across the four goal boxes, both strategies converge on\nthe same choice: go left to $S_2$, then right, for a return of $4$. The difference\nshows only under change. Suppose the rat is placed directly in the goal box right of\n$S_2$ and finds the reward there is now $1$ instead of $4$, without ever running the\nmaze. A model-based rat updates its reward model from that single experience;\nplanning then brings the new value to bear on maze-running with no further\nexperience in the maze, changing the policy to right turns at both $S_1$ and $S_3$\nfor a return of $3$. A model-free rat cannot do this. To change the action its\npolicy specifies at a state, or the value it has cached for a state, it has to move\n_to_ that state, act from it, and experience the consequences — possibly many\ntimes.[^devaluation-tie] This is the mechanism behind the outcome-devaluation\nresult from the [previous\nlesson](\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement): the\ngoal-directed rat re-plans over its map the instant the goal's value changes, while\nthe habitual rat must relearn by acting.\n\nTrace the replan explicitly. Before the change, the model-based rat backs values up the\ntree by taking the max at each choice. The right goal box of $S_2$ pays $4$, so\n$S_2$'s value is $\\max(0, 4) = 4$; the goal boxes of $S_3$ pay $2$ and $3$, so $S_3$'s\nvalue is $\\max(2, 3) = 3$. From $S_1$ the rat compares going left ($4$) with going right\n($3$) and turns left, then right, for a return of $4$. Now devalue: the rat is dropped\ninto the right box of $S_2$ and finds it pays $1$, not $4$, and it updates that one\nreward-model entry. Nothing else in the maze was touched, but the backup changes\nwholesale. $S_2$ is now worth $\\max(0, 1) = 1$; $S_3$ is untouched at $3$; from $S_1$\nthe comparison is now left ($1$) versus right ($3$), so the plan flips to right, then\nright, for a return of $3$.\n\n| Value | before | after devaluation |\n| :-- | ---: | ---: |\n| right box of $S_2$ (reward model) | 4 | 1 |\n| $v(S_2) = \\max(0, \\cdot)$ | 4 | 1 |\n| $v(S_3) = \\max(2, 3)$ | 3 | 3 |\n| choice at $S_1$ | L (4 vs 3) | **R** (1 vs 3) |\n| return | 4 | 3 |\n\nThe model-based rat did no maze-running to produce the new plan; it re-solved the tree\nfrom one changed number. The model-free rat, holding only the cached action values\n$\\langle S_1,L\\rangle = 4$, $\\langle S_1,R\\rangle = 3$, and so on, has no such move. The\nvalue it cached for $\\langle S_1,L\\rangle$ was learned from full runs ending at the box\nthat used to pay $4$; a single visit to that box while _not_ running the maze from $S_1$\ngives it no update to $\\langle S_1, L\\rangle$ at all. To fix the stale cache it must\nphysically go left from $S_1$, walk to the now-devalued box, take the disappointing\nreward, and let the error propagate back — and repeat until $\\langle S_1,L\\rangle$ falls\nbelow $\\langle S_1,R\\rangle$. Same maze, same single fact learned; the map-holder acts on\nit at once, the cache-holder cannot act on it until it re-runs.\n\n$$\n% caption: The devaluation replan. Left: before, both agents pick L then R for return 4.\n% Right: the model-based rat learns the S2-right box now pays 1 (not 4) from one direct\n% visit, re-backs the tree - S2 becomes max(0,1)=1, S3 stays 3 - and flips to R then R\n% for return 3. The changed numbers and the new best path are highlighted in red.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  st\u002F.style={circle, draw, minimum size=7mm, inner sep=0pt, font=\\scriptsize},\n  gb\u002F.style={draw, minimum width=8mm, minimum height=6mm, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[font=\\small, anchor=south] at (2.3,3.3) {after devaluation: re-plan from one changed reward};\n  \\node[st] (s1) at (0,1.1) {S1};\n  \\node[st] (s2) at (2.4,2.2) {S2};\n  \\node[st] (s3) at (2.4,0.0) {S3};\n  \\node[gb] (g1) at (4.9,3.0) {=0};\n  \\node[gb, draw=red, text=red] (g2) at (4.9,1.7) {=1};\n  \\node[gb] (g3) at (4.9,0.6) {=2};\n  \\node[gb, draw=red, text=red] (g4) at (4.9,-0.6) {=3};\n  % chosen path highlighted\n  \\draw[->, red, very thick] (s1) -- (s3) node[midway, below, font=\\scriptsize] {R};\n  \\draw[->, black] (s1) -- (s2) node[midway, above, font=\\scriptsize] {L};\n  \\draw[->, black] (s2) -- (g1) node[near end, above, font=\\scriptsize] {L};\n  \\draw[->, red, thick] (s2) -- (g2) node[near end, below, font=\\scriptsize] {R};\n  \\draw[->, black] (s3) -- (g3) node[near end, above, font=\\scriptsize] {L};\n  \\draw[->, red, very thick] (s3) -- (g4) node[near end, below, font=\\scriptsize] {R};\n  % value labels\n  \\node[anchor=east, font=\\scriptsize, text=black] at (2.15,2.55) {v=1};\n  \\node[anchor=east, font=\\scriptsize, text=black] at (2.15,-0.35) {v=3};\n  \\node[anchor=north, font=\\scriptsize, text=black, align=center] at (2.3,-1.1)\n    {new plan: R then R, return 3};\n\\end{tikzpicture}\n$$\n\nThe connection runs forward as well as back. The same model-based machinery — a\nlearned transition model, a learned reward model, and planning by simulated\nrollouts — is what\n[modern model-based reinforcement learning](\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl)\nscales up with function approximation, learning the model with neural networks and\nplanning with search or imagined trajectories. Tolman's rat, silently mapping a maze\nit was not paid to explore, was doing in tissue what a model-based agent does in\nsilicon.\n\n## The successor representation and evidence for maps\n\nSutton and Barto leave the model-free\u002Fmodel-based split as a clean dichotomy — cache versus\ntree, habit versus plan. Work since has filled in the middle and pinned the split to\nbrain and behavior. Three lines are worth stating carefully, each for exactly what its\nauthors showed.\n\n**A predictive map between the two systems.** Dayan (1993, _Neural Computation_)\nintroduced the **successor representation**: instead of caching a scalar value per\nstate, or holding a full one-step transition model to plan over, cache for each state a\nvector of _expected future occupancy_ — how often, discounted, each other state will be\nvisited if you follow the current policy from here. Value is then a dot product of this\npredictive map with the reward vector. The successor representation sits between\nmodel-free and model-based control. Like a cached value it is fast to read, needing no\nsearch; like a model it factors the world's dynamics apart from reward, so a change in\nreward re-values states without re-running the policy — though a change in the\n_transitions_ still requires relearning the map, which a full model would not. It is a\npartial cognitive map: predictive structure without a plannable model.\n\nStachenfeld, Botvinick and Gershman (2017, _Nature Neuroscience_) argued that the\nhippocampus encodes a map of this kind. They showed that place-cell firing fields, and\nthe periodic fields of entorhinal grid cells, are well described as components of a\nsuccessor representation of the animal's environment — place fields skew and stretch\nalong frequently-taken routes, and grid patterns emerge as a low-dimensional\ndecomposition of the predictive map. This gives Tolman's cognitive map, inferred from\nbehavior in the 1940s, a concrete computational form and a candidate neural substrate:\nnot a static layout but a _predictive_ map of where the animal is headed.\n\n$$\n% caption: The successor representation as a middle ground. Model-free caches a scalar\n% value per state (fast, but reward and dynamics are fused). The SR caches expected\n% future occupancy (a predictive map, re-valued instantly when reward changes).\n% Model-based holds the transition model and plans (flexible, but must search).\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  bx\u002F.style={draw, minimum width=26mm, minimum height=13mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[bx] (mf) at (0,0) {model-free\\\\value per state};\n  \\node[bx, draw=acc, text=acc, thick] (sr) at (4.4,0) {successor rep.\\\\future occupancy};\n  \\node[bx] (mb) at (8.8,0) {model-based\\\\transition model};\n  \\draw[->, black] (mf.east) -- (sr.west);\n  \\draw[->, black] (sr.east) -- (mb.west);\n  \\node[font=\\scriptsize, text=black, anchor=north] at (0,-0.95) {fast, in\\\u002Ff\\\u002Flexible};\n  \\node[font=\\scriptsize, text=acc, anchor=north] at (4.4,-0.95) {predictive, re-values on reward change};\n  \\node[font=\\scriptsize, text=black, anchor=north] at (8.8,-0.95) {f\\\u002Flexible, must plan};\n\\end{tikzpicture}\n$$\n\n**Both systems in one human choice.** Daw, Gershman, Seymour, Dayan and Dolan (2011,\n_Neuron_) built a **two-step Markov task** that separates the two controllers in a\nsingle stream of choices. A first-stage choice leads probabilistically to one of two\nsecond-stage states, and reward there drifts over time. A pure model-free learner\nrepeats whatever first-stage action preceded reward, ignoring _how_ the transition went;\na model-based learner uses its knowledge of the transition structure, so after a rare\ntransition it does the opposite. Human choices showed a _mixture_ of the two signatures,\nand the balance shifted with circumstance — tilting model-free under cognitive load or\ntime pressure. This is the behavioral fingerprint of the habit\u002Fgoal-directed distinction\nthis lesson maps onto model-free and model-based control, measured in people rather than\ninferred. Gläscher, Daw, Dayan and O'Doherty (2010, _Neuron_) added the learning-signal\nhalf: alongside the reward-prediction error that trains values, they found a distinct\n**state-prediction error** — the surprise when a transition differs from what the model\nexpected — with its own neural correlate, the error signal that would train the model\nitself rather than its cached values.\n\n**Scaled-up maps in machines.** The engineering forward-pointer is real systems that\nlearn a world model and plan inside it. Ha and Schmidhuber (2018, \"World Models\") trained\nan agent to compress its environment into a learned latent dynamics model and then\nlearn a policy largely by \"dreaming\" — rolling the model forward instead of acting. Schrittwieser\net al. (2020, _Nature_), in MuZero, learned a model whose only job is to support\nvalue-accurate planning, and combined it with Monte-Carlo tree search to reach top play\nin Go, chess, shogi, and Atari without being told the rules. These are Tolman's map plus\nplan built at scale: a learned model of consequences, queried by simulated rollouts, to\nchoose actions in situations never directly experienced. The rat mapping a maze it was\nnot paid to explore and the agent planning over a learned latent model are running the\nsame idea at very different sizes.\n\n## The one claim\n\nThree phenomena, one point. Blocking is prediction error: an animal learns about a\nstimulus only to the extent that its outcome is surprising, which is the error term\nin every update in this course. Higher-order conditioning and conditioned\nreinforcement are bootstrapping: a learned prediction becomes a reward in its own\nright, which is what a value function is and what an actor–critic critic supplies.\nDelayed reinforcement is credit assignment, and the stimulus traces and lengthened\ngoal gradients that Pavlov and Hull proposed for it are eligibility traces and\nTD-learned values. Cognitive maps are environment models, learned by system\nidentification with or without reward, and queried by planning — model-based\nreinforcement learning, measured in a rat.\n\nNone of this asserts that the brain runs these exact equations. It asserts something\nnarrower and more durable: the structure of the learning problem is the same whether\nthe learner is tissue or silicon, so a theory built to solve it computationally keeps\nlanding on mechanisms that animal-learning researchers had already isolated in\nbehavior. The [next module thread](\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error)\nfollows the contact point one level deeper, into the neuron, where the dopamine\nsignal turns out to behave like the TD error that all of this has been circling.\n\n[^cog-map-sb]: **Sutton & Barto**, §14.5 — Cognitive Maps: model-based reinforcement learning algorithms use environment models with elements in common with psychologists' cognitive maps; an environment model has a state-transition part (effect of actions on state changes) and a reward-model part (reward signals expected for each state or state–action pair), and planning computes a policy by simulating imagined sequences of decisions.\n[^latent-sb]: **Sutton & Barto**, §14.5: some researchers challenged the prevailing stimulus–response (S–R) view — corresponding to the simplest model-free way of learning policies — by demonstrating latent learning.\n[^blodgett]: **Sutton & Barto**, §14.5: in the earliest latent-learning experiment, two groups of rats ran a maze; the experimental group had no reward in stage 1 with food suddenly introduced into the goal box at the start of stage 2, while the control group had food throughout; the question was whether the experimental group learned anything during the unrewarded first stage.\n[^blodgett-quote]: **Sutton & Barto**, §14.5: although the experimental rats did not appear to learn much during the unrewarded first stage, once they discovered the stage-2 food they rapidly caught up to the control rats; Blodgett (1929) concluded that during the non-reward period the rats were developing a latent learning of the maze they could utilize as soon as reward was introduced.\n[^tolman]: **Sutton & Barto**, §14.5: latent learning is most closely associated with Edward Tolman (1948), who interpreted such results as animals learning a \"cognitive map\" of the environment in the absence of rewards or penalties, usable later when motivated to reach a goal; a cognitive map could let a rat plan a route different from the one used in initial exploration.\n[^cog-model-based]: **Sutton & Barto**, §14.5: in modern terms cognitive maps are not restricted to spatial layouts but are more generally environment models, models of an animal's \"task space\" (e.g. Wilson, Takahashi, Schoenbaum, and Niv, 2014); the cognitive-map explanation of latent learning is the claim that animals use model-based algorithms and that environment models can be learned even without explicit rewards or penalties, then used for planning when reward appears.\n[^expectancy]: **Sutton & Barto**, §14.5: Tolman's account is that animals learn stimulus–stimulus (S–S) associations by experiencing successions of stimuli while exploring; in psychology this is expectancy theory — given S–S associations, the occurrence of a stimulus generates an expectation about the stimulus to come next.\n[^sysid]: **Sutton & Barto**, §14.5: learning S–S associations is much like what control engineers call system identification — learning a model of a system with unknown dynamics from labeled examples; the simplest discrete-time examples are S–S′ pairs, action-inclusive examples are SA–S′, and reward examples are S–R or SA–R, all forms of supervised learning by which an agent can acquire cognitive-like maps whether or not it receives non-zero reward while exploring.\n[^maze-fig]: **Sutton & Barto**, §14.6 \u002F Figure 14.5: a rat navigates a maze with distinctive goal boxes each delivering an associated reward; from $S_1$ it selects L or R to reach $S_2$ or $S_3$, then L or R again to reach a goal box. A model-free strategy relies on stored action values for state–action pairs; a model-based strategy learns a state-transition model (a decision tree) and a reward model associating goal-box features with rewards, deciding by simulating action sequences to find the highest-return path.\n[^devaluation-tie]: **Sutton & Barto**, §14.6: with rewards 0, 4, 2, 3 both strategies select L then R from $S_1$ for return 4; a model-based agent placed directly in a goal box whose reward changed (e.g. 4 to 1) updates its reward model and re-plans without maze experience, whereas a model-free agent must move to that state and act, possibly many times, to update — the logic underlying outcome-devaluation experiments.\n",{"text":3560,"minutes":3561,"time":3562,"words":3563},"16 min read",15.15,909000,3030,{"title":5,"description":3548},[3566],{"book":3208,"ref":3567},"Ch. 14 — Psychology; §14.5 Cognitive Maps; §14.6 Habitual and Goal-directed Behavior","available","07.reinforcement-learning\u002F06.minds-and-brains\u002F06.cognitive-maps-and-planning","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",[3572],"Minds and Brains","Hx8URVp00unbBc094dUKe9MX3-zCciL3_VIqLV-6LTU",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":3575,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":3576,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":3577,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":3578,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":3579,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":3580,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":3581,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":3582,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":3583,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":3584,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":3585,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":3586,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":3587,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":3588,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":3589,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":3590,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":3591,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":3592,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":3593,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":3594,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":3595,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":3596,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":3597,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":3598,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":3599,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":3600,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":3601,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":3602,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":3603,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":3604,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":3605,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":3606,"\u002Falgorithms\u002Fsequences\u002Ftries":3607,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":3608,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":3609,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":3610,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":3611,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":3612,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":3613,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":3614,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":3615,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":3616,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":3617,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":3618,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":3619,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":3620,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":3621,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":3622,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":3623,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":3624,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":3625,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":3626,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":3627,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":3628,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":3629,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":3630,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":3631,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":3632,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":3633,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":3634,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":3635,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":3636,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":3637,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":3638,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":3639,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":3640,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":3641,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":3642,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":3643,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":3644,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":3645,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":3646,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":3647,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":3648,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":3649,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":3650,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":3651,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":3652,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":3653,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":3654,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":3655,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":3656,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":3657,"\u002Falgorithms":3658,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":3659,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":3660,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":3661,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":3662,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":3663,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":3664,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":3665,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":3666,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":3667,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":3668,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":3669,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":3670,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":3671,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":3672,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":3673,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":3674,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":3675,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":3676,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":3677,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":3678,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":3679,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":3680,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":3681,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":3682,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":3683,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":3684,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":3685,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":3686,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":3687,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":3688,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":3689,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":3690,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":3691,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":3692,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":3673,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":3693,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":3694,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":3695,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":3663,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":3696,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":3697,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":3698,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":3699,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":3700,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":3701,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":3702,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":3703,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":3704,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":3705,"\u002Fcalculus":3706,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":3707,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":3708,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":3709,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":3710,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":3711,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":3712,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":3713,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":3714,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":3715,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":3716,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":3717,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":3718,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":3719,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":3720,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":3721,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":3722,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":3723,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":3724,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":3725,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":3726,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":3727,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":3728,"\u002Fmechanics\u002Frotation\u002Frolling-motion":3729,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":3730,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":3731,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":3732,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":3733,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":3734,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":3735,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":3736,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":3737,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":3738,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":3739,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":3740,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":3741,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":3742,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":3743,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":3744,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":3745,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":3746,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":3747,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":3748,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":3749,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":3750,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":3751,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":3752,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":3753,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":3754,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":3755,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":3756,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":3757,"\u002Fmechanics":3758,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":3759,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":3760,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":3761,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":3762,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":3763,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":3764,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":3765,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":3766,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":3767,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":3768,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":3769,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":3770,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":3747,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":3771,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":3772,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":3773,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":3743,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":3608,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":3774,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":3734,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":3775,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":3776,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":3777,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":3778,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":3779,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":3780,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":3781,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":3782,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":3783,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":3708,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":3784,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":3785,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":3786,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":3787,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":3788,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":3789,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":3790,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":3791,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":3726,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":3725,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":3792,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":3793,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":3794,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":3795,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":3796,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":3797,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":3798,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":3799,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":3800,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":3752,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":3750,"\u002Felectricity-and-magnetism":3801,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":3802,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":3803,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":3804,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":3805,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":3806,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":3807,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":3808,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":3809,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":3810,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":3660,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":3811,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":3812,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":3664,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":3813,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":3814,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":3815,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":3816,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":3817,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":3818,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":3819,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":3820,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":3821,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":3822,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":3823,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":3824,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":3825,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":3826,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":3827,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":3828,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":3829,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":3830,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":3831,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":3832,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":3699,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":3833,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":3834,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":3835,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":3836,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":3837,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":3838,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":3839,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":3840,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":3841,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":3842,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":3843,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":3844,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":3845,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":3846,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":3847,"\u002Flinear-algebra":3848,"\u002Ftheory-of-computation":3849,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":3850,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":3851,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":3852,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":3853,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":3854,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":3855,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":3856,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":3857,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":3858,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":3859,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":3860,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":3861,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":3862,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":3863,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":3864,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":3865,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":3866,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":3867,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":3868,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":3869,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":3870,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":3871,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":3872,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":3873,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":3874,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":3875,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":3876,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":3877,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":3878,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":3879,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":3880,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":3881,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":3882,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":3883,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":3884,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":3885,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":3886,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":3887,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":3888,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":3889,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":3890,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":3891,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":3892,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":3893,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":3894,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":3895,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":3896,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":3897,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":3898,"\u002Fcomputer-architecture":3849,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":3899,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":3900,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":3901,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":3664,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":3902,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":3663,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":3670,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":3903,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":3904,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":3704,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":3905,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":3906,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":3907,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":3908,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":3909,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":3910,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":3911,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":3912,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":3913,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":3914,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":3915,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":3916,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":3911,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":3917,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":3918,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":3919,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":3920,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":3921,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":3922,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":3923,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":3924,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":3925,"\u002Fdifferential-equations":3926,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":3927,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":3928,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":3929,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":3930,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":3809,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":3931,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":3932,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":3933,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":3934,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":3935,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":3936,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":3679,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":3937,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":3938,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":3939,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":3940,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":3941,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":3942,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":3943,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":3944,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":3945,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":3946,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":3901,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":3947,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":3948,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":3949,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":3950,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":3951,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":3830,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":3952,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":3953,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":3840,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":3954,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":3955,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":3956,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":3957,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":3958,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":3959,"\u002Frelativity":3960,"\u002Fphysical-computing":3849,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":3961,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":3940,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":3962,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":3963,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":3964,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":3965,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":3966,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":3967,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":3968,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":3916,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":3840,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":3969,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":3970,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":3971,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":3972,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":3968,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":3973,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":3948,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":3701,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":3974,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":3975,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":3681,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":3976,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":3977,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":3978,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":3979,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":3980,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":3981,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":3982,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":3983,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":3984,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":3940,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":3985,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":3986,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":3987,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":3974,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":3662,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":3988,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":3989,"\u002Fquantum-mechanics":3990,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":3924,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":3991,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":3992,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":3813,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":3993,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":3699,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":3994,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":3995,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":3836,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":3946,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":3996,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":3997,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":3998,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":3999,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":4000,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":3973,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":4001,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":3806,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":4002,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":4003,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":3660,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":4004,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":4005,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":3962,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":3689,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":3840,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":4006,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":4007,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":3825,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":3950,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":4008,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":4009,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":4010,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":3845,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":4011,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":4012,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":4013,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":4013,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":4014,"\u002Freal-analysis":4015,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":4016,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":4017,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":4018,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":4019,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":4020,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":4021,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":4022,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":4023,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":4024,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":4025,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":3989,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":4026,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":4018,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":3935,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":4027,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":4028,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":4029,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":4030,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":4031,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":4032,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":4033,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":4034,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":4028,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":4035,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":4004,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":4036,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":4037,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":4038,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":4039,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":4040,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":4041,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":4042,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":4043,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":4044,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":4045,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":3678,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":4046,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":4047,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":3920,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":4048,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":4049,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":3977,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":4049,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":4050,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":4051,"\u002Fabstract-algebra":4052,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":4053,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":4054,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":4055,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":4056,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":4057,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":3969,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":4058,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":4059,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":4060,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":4061,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":4062,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":4063,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":4064,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":3803,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":3932,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":3678,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":4065,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":3662,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":4066,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":4067,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":3700,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":3994,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":4068,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":4069,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":4070,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":4071,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":4072,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":4073,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":4074,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":4075,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":4076,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":4077,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":4078,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":4079,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":4080,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":4081,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":4025,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":4082,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":4083,"\u002Fatomic-physics":4084,"\u002Fdatabases":3849,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":4085,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":4086,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":4087,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":4016,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":4088,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":4089,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":4090,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":4091,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":4092,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":4093,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":4094,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":4095,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":4096,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":4097,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":4098,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":4099,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":4100,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":4101,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":4102,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":4103,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":4104,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":4105,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":4106,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":4107,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":4098,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":4108,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":4109,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":4110,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":4111,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":4112,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":4057,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":4113,"\u002Fcategory-theory":4114,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":4115,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":4116,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":4117,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":4118,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":4119,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":4120,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":4079,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":4121,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":4122,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":4123,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":4124,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":4125,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":4126,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":4127,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":4128,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":4129,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":4130,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":4131,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":4132,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":4133,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":4134,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":4135,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":4136,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":4137,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":4138,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":4139,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":4140,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":4141,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":4142,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":4143,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":4144,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":4145,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":4146,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":4147,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":4091,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":4148,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":4149,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":4150,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":4151,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":4152,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":4153,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":4154,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":3643,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":4155,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":4156,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":3879,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":4157,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":4158,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":4159,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":4160,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":4161,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":4162,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":4163,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":4164,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":4165,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":4166,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":4167,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":4168,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":3852,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":4169,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":4170,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":4171,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":4172,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":3889,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":4173,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":4174,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":4175,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":4176,"\u002Fdeep-learning":3849,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":4177,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":3974,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":4178,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":4179,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":4180,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":4181,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":4182,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":4183,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":4184,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":4185,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":4021,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":4186,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":4187,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":4188,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":3908,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":3701,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":4189,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":4190,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":4191,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":3835,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":4192,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":4193,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":3913,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":3840,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":4194,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":3809,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":3679,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":3695,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":4195,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":4196,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":4197,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":3827,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":4198,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":3689,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":4199,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":4200,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":4201,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":4202,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":4023,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":4203,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":4204,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":4205,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":4206,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":3969,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":4207,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":3947,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":4208,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":3808,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":4209,"\u002Fstatistical-mechanics":4210,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":4211,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":3674,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":3934,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":4212,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":4213,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":4214,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":4215,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":4216,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":4217,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":3807,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":4218,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":4219,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":4220,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":4221,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":3950,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":4222,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":4223,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":4224,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":4225,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":4226,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":4005,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":3949,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":4227,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":4228,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":4229,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":4230,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":3940,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":4231,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":4232,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":4177,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":4077,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":3804,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":4233,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":3670,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":4234,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":4235,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":3815,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":4236,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":4070,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":4237,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":3662,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":4238,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":3673,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":4239,"\u002Fcondensed-matter":3990,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":4240,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":4241,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":4242,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":4243,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":3687,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":4244,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":4245,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":3687,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":4246,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":4100,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":4247,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":4248,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":4249,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":4247,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":4250,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":4251,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":4252,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":4253,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":4254,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":4255,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":4256,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":4257,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":4178,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":4258,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":4251,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":4259,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":4260,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":3893,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":4261,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":4078,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":4262,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":4263,"\u002Flogic":4264,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":4265,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":4266,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":3879,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":4267,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":4268,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":4269,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":4270,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":4260,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":4271,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":4272,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":4273,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":4171,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":4274,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":4275,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":4276,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":4277,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":4278,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":3896,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":4279,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":4280,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":4281,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":4282,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":4087,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":4283,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":4259,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":4284,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":4285,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":4286,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":3907,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":4287,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":4037,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":4288,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":4289,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":3869,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":4290,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":4291,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":4292,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":4293,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":4294,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":4147,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":4295,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":4296,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":4297,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":4182,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":4298,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":4299,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":4300,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":3896,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":3963,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":4301,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":4302,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":4303,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":4304,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":4305,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":4306,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":4307,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":4308,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":4309,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":4310,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":4311,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":4312,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":4313,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":4314,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":4315,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":4316,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":4317,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":4318,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":3563,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":4319,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":4320,"\u002Freinforcement-learning":3849,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":4321,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":4322,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":4323,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":4324,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":4325,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":4326,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":4327,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":4328,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":4329,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":4330,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":4331,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":4332,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":4333,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":4176,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":4031,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":4334,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":4335,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":4336,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":4337,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":4338,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":4339,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":4158,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":4340,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":4341,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":4342,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":4343,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":4344,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":4345,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":4346,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":4347,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":4348,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":4349,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":4350,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":4351,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":4090,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":4341,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":4352,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":3630,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":4353,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":4354,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":4355,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":4356,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":4357,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":4139,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":4358,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":4359,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":4360,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":4361,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":4362,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":4363,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":4364,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":4365,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":4366,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":4367,"\u002Fartificial-intelligence":3849,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":4106,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":4368,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":3666,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":3664,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":4200,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":4369,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":3692,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":4002,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":4370,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":4371,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":4372,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":4062,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":4373,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":4374,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":4375,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":4261,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":4376,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":4377,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":3676,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":3905,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":4378,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":4006,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":4379,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":4380,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":3901,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":3989,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":4381,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":4382,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":4383,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":3671,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":4046,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":3908,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":4384,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":3956,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":4020,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":4385,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":4386,"\u002Fnuclear-physics":4387,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":4388,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":4389,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":4027,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":4390,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":4391,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":4392,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":3875,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":4393,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":4394,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":4172,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":4395,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":4340,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":4396,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":4397,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":4398,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":4399,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":4400,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":4401,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":4402,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":3853,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":4403,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":4404,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":4346,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":4405,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":4406,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":4407,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":4408,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":4409,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":4410,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":4411,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":4412,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":4271,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":4413,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":4414,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":4415,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":4416,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":4417,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":4418,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":4419,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":4420,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":4421,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":4422,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":4423,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":4424,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":4125,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":4292,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":3881,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":4425,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":4426,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":4427,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":4428,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":4139,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":4429,"\u002Fnatural-language-processing":3849,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":4430,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":4047,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":4431,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":4192,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":4432,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":4433,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":4381,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":4434,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":4435,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":3997,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":3699,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":4436,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":3951,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":4437,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":3984,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":4438,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":4238,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":4439,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":4440,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":4207,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":4441,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":3670,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":4442,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":4443,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":4444,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":3988,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":4204,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":4445,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":4446,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":4063,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":4447,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":4108,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":4448,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":4449,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":3997,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":4030,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":4450,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":4451,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":4452,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":4086,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":4453,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":3827,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":4454,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":3930,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":4455,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":4456,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":4254,"\u002Fparticle-physics":4457,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":4048,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":4202,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":4458,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":3987,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":4459,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":4047,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":3959,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":4460,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":4461,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":4462,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":4463,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":4464,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":4253,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":4184,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":4465,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":4466,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":4467,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":4468,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":4380,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":4469,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":3943,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":4006,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":3988,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":4470,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":4072,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":3688,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":4471,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":4472,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":4203,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":4473,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":4474,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":4475,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":4476,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":3669,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":4477,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":4478,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":3930,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":4479,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":4480,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":4196,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":3851,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":4481,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":4482,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":4108,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":4094,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":4106,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":4203,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":4483,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":3668,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":3860,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":4484,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":3947,"\u002Fastrophysics-cosmology":4052,"\u002Fcolophon":4485,"\u002F":3849},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":4487,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":4492,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":4496,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":4500,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":4504,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":4508,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":4512,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":4517,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":4521,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":4525,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":4529,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":4534,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":4538,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":4542,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":4546,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":4551,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":4555,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":4559,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":4563,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":4567,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":4571,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":4575,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":4579,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":4583,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":4587,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":4591,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":4595,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":4600,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":4604,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":4608,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":4612,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":4616,"\u002Falgorithms\u002Fsequences\u002Ftries":4620,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":4624,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":4628,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":4633,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":4637,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":4641,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":4645,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":4649,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":4653,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":4657,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":4661,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":4665,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":4669,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":4673,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":4677,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":4681,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":4685,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":4690,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":4694,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":4698,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":4702,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":4706,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":4711,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":4715,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":4719,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":4723,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":4727,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":4731,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":4735,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":4739,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":4743,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":4747,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":4751,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":4756,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":4760,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":4764,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":4768,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":4773,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":4777,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":4781,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":4785,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":4789,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":4793,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":4797,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":4802,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":4806,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":4810,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":4814,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":4819,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":4823,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":4827,"\u002Falgorithms":4831,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":4834,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":4839,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":4843,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":4847,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":4851,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":4856,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":4860,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":4864,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":4868,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":4873,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":4877,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":4881,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":4885,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":4890,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":4894,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":4898,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":4903,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":4907,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":4911,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":4916,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":4920,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":4924,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":4929,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":4933,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":4937,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":4941,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":4946,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":4950,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":4954,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":4959,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":4963,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":4967,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":4971,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":4975,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":4980,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":4984,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":4988,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":4992,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":4996,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":5001,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":5004,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":5008,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":5012,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":5016,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":5021,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":5025,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":5029,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":5033,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":5037,"\u002Fcalculus":5041,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":5044,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":5048,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":5052,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":5057,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":5061,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":5065,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":5069,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":5073,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":5078,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":5082,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":5086,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":5090,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":5094,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":5099,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":5103,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":5107,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":5111,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":5115,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":5120,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":5124,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":5128,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":5133,"\u002Fmechanics\u002Frotation\u002Frolling-motion":5137,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":5141,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":5145,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":5149,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":5153,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":5158,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":5162,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":5166,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":5170,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":5174,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":5178,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":5182,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":5187,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":5191,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":5195,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":5199,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":5203,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":5207,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":5211,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":5215,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":5219,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":5223,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":5227,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":5231,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":5236,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":5240,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":5244,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":5248,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":5252,"\u002Fmechanics":5256,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":5259,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":5264,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":5268,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":5272,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":5276,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":5280,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":5285,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":5289,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":5294,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":5298,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":5302,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":5306,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":5310,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":5315,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":5319,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":5323,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":5327,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":5332,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":5336,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":5340,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":5345,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":5349,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":5353,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":5357,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":5361,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":5366,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":5370,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":5374,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":5378,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":5382,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":5386,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":5391,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":5395,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":5399,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":5403,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":5407,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":5411,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":5415,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":5419,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":5424,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":5428,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":5432,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":5436,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":5440,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":5445,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":5449,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":5453,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":5457,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":5461,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":5466,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":5470,"\u002Felectricity-and-magnetism":5474,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":5477,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":5482,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":5486,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":5490,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":5494,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":5498,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":5503,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":5507,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":5511,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":5515,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":5519,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":5524,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":5528,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":5532,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":5537,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":5541,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":5545,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":5549,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":5553,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":5557,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":5561,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":5566,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":5570,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":5574,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":5578,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":5582,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":5586,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":5590,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":5595,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":5599,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":5603,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":5607,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":5611,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":5615,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":5620,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":5624,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":5628,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":5632,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":5636,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":5641,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":5645,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":5649,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":5653,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":5657,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":5661,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":5666,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":5670,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":5674,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":5678,"\u002Flinear-algebra":5682,"\u002Ftheory-of-computation":5685,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":5688,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":5692,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":5696,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":5700,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":5704,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":5708,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":5713,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":5717,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":5721,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":5725,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":5729,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":5733,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":5737,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":5742,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":5746,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":5750,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":5754,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":5758,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":5763,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":5767,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":5771,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":5775,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":5779,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":5784,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":5788,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":5792,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":5796,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":5800,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":5805,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":5809,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":5813,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":5817,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":5821,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":5826,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":5830,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":5834,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":5838,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":5842,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":5847,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":5851,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":5855,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":5860,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":5864,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":5869,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":5873,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":5877,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":5881,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":5885,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":5890,"\u002Fcomputer-architecture":5894,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":5897,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":5901,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":5905,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":5910,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":5914,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":5918,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":5922,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":5926,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":5930,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":5935,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":5939,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":5943,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":5947,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":5951,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":5955,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":5960,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":5964,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":5968,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":5973,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":5977,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":5982,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":5986,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":5990,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":5995,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":5999,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":6004,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":6008,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":6012,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":6017,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":6021,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":6025,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":6030,"\u002Fdifferential-equations":6034,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":6037,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":6042,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":6046,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":6050,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":6054,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":6058,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":6063,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":6067,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":6071,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":6075,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":6080,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":6084,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":6088,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":6092,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":6097,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":6101,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":6105,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":6109,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":6114,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":6118,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":6122,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":6126,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":6130,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":6134,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":6139,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":6143,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":6147,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":6152,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":6156,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":6160,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":6164,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":6169,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":6173,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":6177,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":6182,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":6186,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":6190,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":6195,"\u002Frelativity":6199,"\u002Fphysical-computing":6202,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":6205,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":6210,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":6214,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":6218,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":6222,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":6227,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":6231,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":6235,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":6240,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":6244,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":6248,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":6252,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":6256,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":6260,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":6265,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":6269,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":6273,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":6277,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":6281,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":6285,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":6290,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":6294,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":6298,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":6302,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":6306,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":6310,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":6314,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":6319,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":6323,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":6327,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":6332,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":6336,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":6340,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":6345,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":6349,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":6354,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":6358,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":6362,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":6366,"\u002Fquantum-mechanics":6370,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":6373,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":6378,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":6382,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":6386,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":6390,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":6395,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":6399,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":6403,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":6407,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":6411,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":6415,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":6420,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":6424,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":6428,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":6432,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":6436,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":6440,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":6444,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":6448,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":6452,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":6456,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":6460,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":6465,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":6469,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":6473,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":6477,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":6482,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":6486,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":6490,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":6493,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":6497,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":6502,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":6506,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":6510,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":6514,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":6519,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":6523,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":6527,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":6531,"\u002Freal-analysis":6535,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":6538,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":6542,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":6546,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":6551,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":6555,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":6559,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":6563,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":6568,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":6572,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":6576,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":6580,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":6584,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":6588,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":6593,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":6597,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":6601,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":6605,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":6610,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":6614,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":6618,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":6622,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":6627,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":6631,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":6635,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":6640,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":6644,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":6648,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":6652,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":6657,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":6661,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":6665,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":6669,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":6674,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":6678,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":6682,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":6687,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":6691,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":6695,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":6699,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":6704,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":6708,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":6712,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":6716,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":6720,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":6725,"\u002Fabstract-algebra":6729,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":6732,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":6737,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":6741,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":6745,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":6749,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":6753,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":6758,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":6762,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":6766,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":6770,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":6774,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":6778,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":6782,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":6787,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":6791,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":6795,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":6799,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":6804,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":6808,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":6812,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":6817,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":6821,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":6825,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":6829,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":6833,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":6837,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":6842,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":6846,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":6850,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":6855,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":6859,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":6863,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":6867,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":6872,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":6876,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":6880,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":6885,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":6889,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":6893,"\u002Fatomic-physics":6897,"\u002Fdatabases":6900,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":6903,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":6907,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":6911,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":6915,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":6919,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":6923,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":6927,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":6932,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":6936,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":6940,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":6945,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":6949,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":6953,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":6958,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":6962,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":6966,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":6970,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":6974,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":6979,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":6983,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":6987,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":6991,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":6996,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":7000,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":7004,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":7008,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":7013,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":7017,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":7021,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":7025,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":7030,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":7034,"\u002Fcategory-theory":7038,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":7041,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":7045,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":7049,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":7053,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":7056,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":7060,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":7064,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":7068,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":7073,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":7077,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":7081,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":7085,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":7089,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":7094,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":7098,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":7102,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":7106,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":7110,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":7115,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":7119,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":7123,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":7127,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":7132,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":7136,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":7140,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":7144,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":7148,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":7152,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":7156,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":7160,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":7164,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":7169,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":7173,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":7177,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":7181,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":7185,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":7190,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":7194,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":7198,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":7202,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":7206,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":7210,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":7214,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":7219,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":7223,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":7227,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":7232,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":7236,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":7240,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":7244,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":7248,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":7252,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":7256,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":7261,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":7265,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":7269,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":7273,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":7277,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":7281,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":7285,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":7289,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":7293,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":7297,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":7301,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":7306,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":7310,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":7314,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":7318,"\u002Fdeep-learning":7322,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":7325,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":7329,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":7333,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":7337,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":7341,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":7345,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":7350,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":7354,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":7358,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":7362,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":7367,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":7371,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":7375,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":7379,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":7384,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":7388,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":7392,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":7396,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":7400,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":7405,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":7409,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":7413,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":7418,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":7422,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":7426,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":7431,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":7435,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":7439,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":7443,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":7448,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":7452,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":7456,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":7460,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":7464,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":7468,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":7473,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":7477,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":7481,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":7485,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":7490,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":7494,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":7498,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":7503,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":7507,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":7511,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":7515,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":7519,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":7524,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":7528,"\u002Fstatistical-mechanics":7532,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":7535,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":7540,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":7544,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":7548,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":7552,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":7557,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":7561,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":7565,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":7569,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":7574,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":7578,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":7582,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":7586,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":7591,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":7595,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":7599,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":7603,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":7608,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":7612,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":7616,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":7620,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":7625,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":7629,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":7633,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":7637,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":7642,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":7646,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":7650,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":7654,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":7658,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":7663,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":7667,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":7672,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":7676,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":7680,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":7684,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":7689,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":7693,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":7697,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":7701,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":7705,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":7710,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":7714,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":7718,"\u002Fcondensed-matter":7722,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":7725,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":7729,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":7734,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":7738,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":7742,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":7746,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":7750,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":7754,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":7758,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":7763,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":7767,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":7771,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":7775,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":7780,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":7784,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":7788,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":7792,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":7797,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":7801,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":7805,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":7809,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":7814,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":7818,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":7822,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":7826,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":7831,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":7835,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":7839,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":7844,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":7848,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":7853,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":7857,"\u002Flogic":7861,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":7864,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":7868,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":7872,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":7876,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":7880,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":7884,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":7888,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":7892,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":7896,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":7900,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":7904,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":7908,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":7912,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":7916,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":7920,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":7923,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":7927,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":7931,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":7935,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":7940,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":7944,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":7948,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":7952,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":7956,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":7960,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":7964,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":7968,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":7972,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":7976,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":7980,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":7984,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":7988,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":7992,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":7996,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":8000,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":8004,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":8008,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":8012,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":8016,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":8020,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":8024,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":8029,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":8033,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":8037,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":8041,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":8044,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":8048,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":8052,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":8056,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":8060,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":8064,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":8068,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":8072,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":8076,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":8080,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":8084,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":8088,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":8092,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":8096,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":8100,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":8104,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":8108,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":8112,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":8115,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":8119,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":8122,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":8126,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":8128,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":8129,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":8133,"\u002Freinforcement-learning":8137,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":8139,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":8143,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":8147,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":8151,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":8155,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":8160,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":8164,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":8168,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":8172,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":8176,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":8180,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":8184,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":8188,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":8192,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":8196,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":8200,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":8204,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":8209,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":8213,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":8217,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":8221,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":8225,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":8229,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":8233,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":8237,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":8241,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":8245,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":8249,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":8253,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":8258,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":8262,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":8266,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":8270,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":8274,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":8278,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":8282,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":8285,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":8289,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":8293,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":8298,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":8302,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":8306,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":8310,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":8313,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":8317,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":8321,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":8325,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":8330,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":8334,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":8338,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":8342,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":8346,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":8350,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":8354,"\u002Fartificial-intelligence":8358,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":8361,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":8366,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":8370,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":8374,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":8378,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":8382,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":8387,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":8391,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":8395,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":8399,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":8404,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":8408,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":8412,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":8416,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":8421,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":8425,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":8430,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":8434,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":8439,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":8443,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":8447,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":8451,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":8456,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":8460,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":8464,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":8469,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":8473,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":8477,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":8482,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":8486,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":8491,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":8495,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":8499,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":8504,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":8508,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":8512,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":8516,"\u002Fnuclear-physics":8520,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":8523,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":8527,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":8531,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":8535,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":8539,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":8543,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":8548,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":8552,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":8556,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":8560,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":8565,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":8569,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":8573,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":8577,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":8581,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":8585,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":8589,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":8592,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":8595,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":8599,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":8603,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":8607,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":8612,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":8616,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":8620,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":8624,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":8628,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":8632,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":8636,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":8640,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":8644,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":8648,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":8652,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":8656,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":8660,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":8664,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":8668,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":8672,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":8676,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":8680,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":8684,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":8688,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":8692,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":8696,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":8700,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":8704,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":8708,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":8712,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":8716,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":8720,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":8725,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":8729,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":8733,"\u002Fnatural-language-processing":8737,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":8740,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":8744,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":8748,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":8752,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":8757,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":8761,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":8765,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":8769,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":8774,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":8778,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":8782,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":8786,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":8791,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":8795,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":8799,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":8803,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":8808,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":8812,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":8816,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":8821,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":8825,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":8829,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":8833,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":8838,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":8842,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":8846,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":8850,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":8855,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":8859,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":8863,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":8867,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":8872,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":8876,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":8880,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":8884,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":8888,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":8893,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":8897,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":8901,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":8906,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":8910,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":8914,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":8918,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":8922,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":8926,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":8930,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":8934,"\u002Fparticle-physics":8938,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":8941,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":8946,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":8950,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":8954,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":8959,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":8963,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":8967,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":8971,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":8976,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":8980,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":8984,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":8988,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":8993,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":8997,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":9001,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":9005,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":9010,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":9014,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":9018,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":9022,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":9027,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":9031,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":9035,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":9040,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":9044,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":9048,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":9052,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":9056,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":9060,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":9064,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":9068,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":9072,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":9077,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":9081,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":9085,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":9089,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":9094,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":9098,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":9102,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":9106,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":9110,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":9115,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":9119,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":9122,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":9126,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":9130,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":9135,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":9139,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":9143,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":9147,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":9151,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":9155,"\u002Fastrophysics-cosmology":9159,"\u002Fcolophon":9162,"\u002F":9165},{"path":4488,"title":4489,"module":4490,"summary":4491},"\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":4493,"title":4494,"module":4490,"summary":4495},"\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":4497,"title":4498,"module":4490,"summary":4499},"\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":4501,"title":4502,"module":4490,"summary":4503},"\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":4505,"title":4506,"module":4490,"summary":4507},"\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":4509,"title":4510,"module":4490,"summary":4511},"\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":4513,"title":4514,"module":4515,"summary":4516},"\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":4518,"title":4519,"module":4515,"summary":4520},"\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":4522,"title":4523,"module":4515,"summary":4524},"\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":4526,"title":4527,"module":4515,"summary":4528},"\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":4530,"title":4531,"module":4532,"summary":4533},"\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":4535,"title":4536,"module":4532,"summary":4537},"\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":4539,"title":4540,"module":4532,"summary":4541},"\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":4543,"title":4544,"module":4532,"summary":4545},"\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":4547,"title":4548,"module":4549,"summary":4550},"\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":4552,"title":4553,"module":4549,"summary":4554},"\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":4556,"title":4557,"module":4549,"summary":4558},"\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":4560,"title":4561,"module":4549,"summary":4562},"\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":4564,"title":4565,"module":4549,"summary":4566},"\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":4568,"title":4569,"module":4549,"summary":4570},"\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":4572,"title":4573,"module":4549,"summary":4574},"\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":4576,"title":4577,"module":4549,"summary":4578},"\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":4580,"title":4581,"module":4549,"summary":4582},"\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":4584,"title":4585,"module":4549,"summary":4586},"\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":4588,"title":4589,"module":4549,"summary":4590},"\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":4592,"title":4593,"module":4549,"summary":4594},"\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":4596,"title":4597,"module":4598,"summary":4599},"\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":4601,"title":4602,"module":4598,"summary":4603},"\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":4605,"title":4606,"module":4598,"summary":4607},"\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":4609,"title":4610,"module":4598,"summary":4611},"\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":4613,"title":4614,"module":4598,"summary":4615},"\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":4617,"title":4618,"module":4598,"summary":4619},"\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":4621,"title":4622,"module":4598,"summary":4623},"\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":4625,"title":4626,"module":4598,"summary":4627},"\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":4629,"title":4630,"module":4631,"summary":4632},"\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":4634,"title":4635,"module":4631,"summary":4636},"\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":4638,"title":4639,"module":4631,"summary":4640},"\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":4642,"title":4643,"module":4631,"summary":4644},"\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":4646,"title":4647,"module":4631,"summary":4648},"\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":4650,"title":4651,"module":4631,"summary":4652},"\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":4654,"title":4655,"module":4631,"summary":4656},"\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":4658,"title":4659,"module":4631,"summary":4660},"\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":4662,"title":4663,"module":4631,"summary":4664},"\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":4666,"title":4667,"module":4631,"summary":4668},"\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":4670,"title":4671,"module":4631,"summary":4672},"\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":4674,"title":4675,"module":4631,"summary":4676},"\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":4678,"title":4679,"module":4631,"summary":4680},"\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":4682,"title":4683,"module":4631,"summary":4684},"\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":4686,"title":4687,"module":4688,"summary":4689},"\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":4691,"title":4692,"module":4688,"summary":4693},"\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":4695,"title":4696,"module":4688,"summary":4697},"\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":4699,"title":4700,"module":4688,"summary":4701},"\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":4703,"title":4704,"module":4688,"summary":4705},"\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":4707,"title":4708,"module":4709,"summary":4710},"\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":4712,"title":4713,"module":4709,"summary":4714},"\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":4716,"title":4717,"module":4709,"summary":4718},"\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":4720,"title":4721,"module":4709,"summary":4722},"\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":4724,"title":4725,"module":4709,"summary":4726},"\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":4728,"title":4729,"module":4709,"summary":4730},"\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":4732,"title":4733,"module":4709,"summary":4734},"\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":4736,"title":4737,"module":4709,"summary":4738},"\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":4740,"title":4741,"module":4709,"summary":4742},"\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":4744,"title":4745,"module":4709,"summary":4746},"\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":4748,"title":4749,"module":4709,"summary":4750},"\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":4752,"title":4753,"module":4754,"summary":4755},"\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":4757,"title":4758,"module":4754,"summary":4759},"\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":4761,"title":4762,"module":4754,"summary":4763},"\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":4765,"title":4766,"module":4754,"summary":4767},"\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":4769,"title":4770,"module":4771,"summary":4772},"\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":4774,"title":4775,"module":4771,"summary":4776},"\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":4778,"title":4779,"module":4771,"summary":4780},"\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":4782,"title":4783,"module":4771,"summary":4784},"\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":4786,"title":4787,"module":4771,"summary":4788},"\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":4790,"title":4791,"module":4771,"summary":4792},"\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":4794,"title":4795,"module":4771,"summary":4796},"\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":4798,"title":4799,"module":4800,"summary":4801},"\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":4803,"title":4804,"module":4800,"summary":4805},"\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":4807,"title":4808,"module":4800,"summary":4809},"\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":4811,"title":4812,"module":4800,"summary":4813},"\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":4815,"title":4816,"module":4817,"summary":4818},"\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":4820,"title":4821,"module":4817,"summary":4822},"\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":4824,"title":4825,"module":4817,"summary":4826},"\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":4828,"title":4829,"module":4817,"summary":4830},"\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":4832,"title":4833,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":4835,"title":4836,"module":4837,"summary":4838},"\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":4840,"title":4841,"module":4837,"summary":4842},"\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":4844,"title":4845,"module":4837,"summary":4846},"\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":4848,"title":4849,"module":4837,"summary":4850},"\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":4852,"title":4853,"module":4854,"summary":4855},"\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":4857,"title":4858,"module":4854,"summary":4859},"\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":4861,"title":4862,"module":4854,"summary":4863},"\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":4865,"title":4866,"module":4854,"summary":4867},"\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":4869,"title":4870,"module":4871,"summary":4872},"\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":4874,"title":4875,"module":4871,"summary":4876},"\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":4878,"title":4879,"module":4871,"summary":4880},"\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":4882,"title":4883,"module":4871,"summary":4884},"\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":4886,"title":4887,"module":4888,"summary":4889},"\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":4891,"title":4892,"module":4888,"summary":4893},"\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":4895,"title":4896,"module":4888,"summary":4897},"\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":4899,"title":4900,"module":4901,"summary":4902},"\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":4904,"title":4905,"module":4901,"summary":4906},"\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":4908,"title":4909,"module":4901,"summary":4910},"\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":4912,"title":4913,"module":4914,"summary":4915},"\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":4917,"title":4918,"module":4914,"summary":4919},"\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":4921,"title":4922,"module":4914,"summary":4923},"\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":4925,"title":4926,"module":4927,"summary":4928},"\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":4930,"title":4931,"module":4927,"summary":4932},"\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":4934,"title":4935,"module":4927,"summary":4936},"\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":4938,"title":4939,"module":4927,"summary":4940},"\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":4942,"title":4943,"module":4944,"summary":4945},"\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":4947,"title":4948,"module":4944,"summary":4949},"\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":4951,"title":4952,"module":4944,"summary":4953},"\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":4955,"title":4956,"module":4957,"summary":4958},"\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":4960,"title":4961,"module":4957,"summary":4962},"\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":4964,"title":4965,"module":4957,"summary":4966},"\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":4968,"title":4969,"module":4957,"summary":4970},"\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":4972,"title":4973,"module":4957,"summary":4974},"\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":4976,"title":4977,"module":4978,"summary":4979},"\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":4981,"title":4982,"module":4978,"summary":4983},"\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":4985,"title":4986,"module":4978,"summary":4987},"\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":4989,"title":4990,"module":4978,"summary":4991},"\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":4993,"title":4994,"module":4978,"summary":4995},"\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":4997,"title":4998,"module":4999,"summary":5000},"\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":5002,"title":4999,"module":4999,"summary":5003},"\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":5005,"title":5006,"module":4999,"summary":5007},"\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":5009,"title":5010,"module":4999,"summary":5011},"\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":5013,"title":5014,"module":4999,"summary":5015},"\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":5017,"title":5018,"module":5019,"summary":5020},"\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":5022,"title":5023,"module":5019,"summary":5024},"\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":5026,"title":5027,"module":5019,"summary":5028},"\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":5030,"title":5031,"module":5019,"summary":5032},"\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":5034,"title":5035,"module":5019,"summary":5036},"\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":5038,"title":5039,"module":5019,"summary":5040},"\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":5042,"title":5043,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":5045,"title":5046,"module":4490,"summary":5047},"\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":5049,"title":5050,"module":4490,"summary":5051},"\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":5053,"title":5054,"module":5055,"summary":5056},"\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":5058,"title":5059,"module":5055,"summary":5060},"\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":5062,"title":5063,"module":5055,"summary":5064},"\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":5066,"title":5067,"module":5055,"summary":5068},"\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":5070,"title":5071,"module":5055,"summary":5072},"\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":5074,"title":5075,"module":5076,"summary":5077},"\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":5079,"title":5080,"module":5076,"summary":5081},"\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":5083,"title":5084,"module":5076,"summary":5085},"\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":5087,"title":5088,"module":5076,"summary":5089},"\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":5091,"title":5092,"module":5076,"summary":5093},"\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":5095,"title":5096,"module":5097,"summary":5098},"\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":5100,"title":5101,"module":5097,"summary":5102},"\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":5104,"title":5105,"module":5097,"summary":5106},"\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":5108,"title":5109,"module":5097,"summary":5110},"\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":5112,"title":5113,"module":5097,"summary":5114},"\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":5116,"title":5117,"module":5118,"summary":5119},"\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":5121,"title":5122,"module":5118,"summary":5123},"\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":5125,"title":5126,"module":5118,"summary":5127},"\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":5129,"title":5130,"module":5131,"summary":5132},"\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":5134,"title":5135,"module":5131,"summary":5136},"\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":5138,"title":5139,"module":5131,"summary":5140},"\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":5142,"title":5143,"module":5131,"summary":5144},"\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":5146,"title":5147,"module":5131,"summary":5148},"\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":5150,"title":5151,"module":5131,"summary":5152},"\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":5154,"title":5155,"module":5156,"summary":5157},"\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":5159,"title":5160,"module":5156,"summary":5161},"\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":5163,"title":5164,"module":5156,"summary":5165},"\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":5167,"title":5168,"module":5156,"summary":5169},"\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":5171,"title":5172,"module":5156,"summary":5173},"\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":5175,"title":5176,"module":5156,"summary":5177},"\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":5179,"title":5180,"module":5156,"summary":5181},"\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":5183,"title":5184,"module":5185,"summary":5186},"\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":5188,"title":5189,"module":5185,"summary":5190},"\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":5192,"title":5193,"module":5185,"summary":5194},"\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":5196,"title":5197,"module":5185,"summary":5198},"\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":5200,"title":5201,"module":5185,"summary":5202},"\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":5204,"title":5205,"module":5185,"summary":5206},"\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":5208,"title":5209,"module":5185,"summary":5210},"\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":5212,"title":5213,"module":5185,"summary":5214},"\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":5216,"title":5217,"module":5185,"summary":5218},"\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":5220,"title":5221,"module":5185,"summary":5222},"\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":5224,"title":5225,"module":5185,"summary":5226},"\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":5228,"title":5229,"module":5185,"summary":5230},"\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":5232,"title":5233,"module":5234,"summary":5235},"\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":5237,"title":5238,"module":5234,"summary":5239},"\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":5241,"title":5242,"module":5234,"summary":5243},"\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":5245,"title":5246,"module":5234,"summary":5247},"\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":5249,"title":5250,"module":5234,"summary":5251},"\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":5253,"title":5254,"module":5234,"summary":5255},"\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":5257,"title":5258,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":5260,"title":5261,"module":5262,"summary":5263},"\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":5265,"title":5266,"module":5262,"summary":5267},"\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":5269,"title":5270,"module":5262,"summary":5271},"\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":5273,"title":5274,"module":5262,"summary":5275},"\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":5277,"title":5278,"module":5262,"summary":5279},"\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":5281,"title":5282,"module":5283,"summary":5284},"\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":5286,"title":5287,"module":5283,"summary":5288},"\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":5290,"title":5291,"module":5292,"summary":5293},"\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":5295,"title":5296,"module":5292,"summary":5297},"\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":5299,"title":5300,"module":5292,"summary":5301},"\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":5303,"title":5304,"module":5292,"summary":5305},"\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":5307,"title":5308,"module":5292,"summary":5309},"\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":5311,"title":5312,"module":5313,"summary":5314},"\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":5316,"title":5317,"module":5313,"summary":5318},"\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":5320,"title":5321,"module":5313,"summary":5322},"\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":5324,"title":5325,"module":5313,"summary":5326},"\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":5328,"title":5329,"module":5330,"summary":5331},"\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":5333,"title":5334,"module":5330,"summary":5335},"\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":5337,"title":5338,"module":5330,"summary":5339},"\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":5341,"title":5342,"module":5343,"summary":5344},"\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":5346,"title":5347,"module":5343,"summary":5348},"\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":5350,"title":5351,"module":5343,"summary":5352},"\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":5354,"title":5355,"module":5343,"summary":5356},"\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":5358,"title":5359,"module":5343,"summary":5360},"\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":5362,"title":5363,"module":5364,"summary":5365},"\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":5367,"title":5368,"module":5364,"summary":5369},"\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":5371,"title":5372,"module":5364,"summary":5373},"\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":5375,"title":5376,"module":5364,"summary":5377},"\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":5379,"title":5380,"module":5364,"summary":5381},"\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":5383,"title":5384,"module":5364,"summary":5385},"\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":5387,"title":5388,"module":5389,"summary":5390},"\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":5392,"title":5393,"module":5389,"summary":5394},"\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":5396,"title":5397,"module":5389,"summary":5398},"\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":5400,"title":5401,"module":5389,"summary":5402},"\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":5404,"title":5405,"module":5389,"summary":5406},"\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":5408,"title":5409,"module":5389,"summary":5410},"\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":5412,"title":5413,"module":5389,"summary":5414},"\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":5416,"title":5417,"module":5389,"summary":5418},"\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":5420,"title":5421,"module":5422,"summary":5423},"\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":5425,"title":5426,"module":5422,"summary":5427},"\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":5429,"title":5430,"module":5422,"summary":5431},"\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":5433,"title":5434,"module":5422,"summary":5435},"\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":5437,"title":5438,"module":5422,"summary":5439},"\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":5441,"title":5442,"module":5443,"summary":5444},"\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":5446,"title":5447,"module":5443,"summary":5448},"\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":5450,"title":5451,"module":5443,"summary":5452},"\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":5454,"title":5455,"module":5443,"summary":5456},"\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":5458,"title":5459,"module":5443,"summary":5460},"\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":5462,"title":5463,"module":5464,"summary":5465},"\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":5467,"title":5468,"module":5464,"summary":5469},"\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":5471,"title":5472,"module":5464,"summary":5473},"\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":5475,"title":5476,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":5478,"title":5479,"module":5480,"summary":5481},"\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":5483,"title":5484,"module":5480,"summary":5485},"\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":5487,"title":5488,"module":5480,"summary":5489},"\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":5491,"title":5492,"module":5480,"summary":5493},"\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":5495,"title":5496,"module":5480,"summary":5497},"\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":5499,"title":5500,"module":5501,"summary":5502},"\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":5504,"title":5505,"module":5501,"summary":5506},"\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":5508,"title":5509,"module":5501,"summary":5510},"\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":5512,"title":5513,"module":5501,"summary":5514},"\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":5516,"title":5517,"module":5501,"summary":5518},"\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":5520,"title":5521,"module":5522,"summary":5523},"\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":5525,"title":5526,"module":5522,"summary":5527},"\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":5529,"title":5530,"module":5522,"summary":5531},"\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":5533,"title":5534,"module":5535,"summary":5536},"\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":5538,"title":5539,"module":5535,"summary":5540},"\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":5542,"title":5543,"module":5535,"summary":5544},"\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":5546,"title":5547,"module":5535,"summary":5548},"\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":5550,"title":5551,"module":5535,"summary":5552},"\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":5554,"title":5555,"module":5535,"summary":5556},"\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":5558,"title":5559,"module":5535,"summary":5560},"\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":5562,"title":5563,"module":5564,"summary":5565},"\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":5567,"title":5568,"module":5564,"summary":5569},"\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":5571,"title":5572,"module":5564,"summary":5573},"\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":5575,"title":5576,"module":5564,"summary":5577},"\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":5579,"title":5580,"module":5564,"summary":5581},"\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":5583,"title":5584,"module":5564,"summary":5585},"\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":5587,"title":5588,"module":5564,"summary":5589},"\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":5591,"title":5592,"module":5593,"summary":5594},"\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":5596,"title":5597,"module":5593,"summary":5598},"\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":5600,"title":5601,"module":5593,"summary":5602},"\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":5604,"title":5605,"module":5593,"summary":5606},"\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":5608,"title":5609,"module":5593,"summary":5610},"\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":5612,"title":5613,"module":5593,"summary":5614},"\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":5616,"title":5617,"module":5618,"summary":5619},"\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":5621,"title":5622,"module":5618,"summary":5623},"\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":5625,"title":5626,"module":5618,"summary":5627},"\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":5629,"title":5630,"module":5618,"summary":5631},"\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":5633,"title":5634,"module":5618,"summary":5635},"\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":5637,"title":5638,"module":5639,"summary":5640},"\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":5642,"title":5643,"module":5639,"summary":5644},"\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":5646,"title":5647,"module":5639,"summary":5648},"\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":5650,"title":5651,"module":5639,"summary":5652},"\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":5654,"title":5655,"module":5639,"summary":5656},"\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":5658,"title":5659,"module":5639,"summary":5660},"\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":5662,"title":5663,"module":5664,"summary":5665},"\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":5667,"title":5668,"module":5664,"summary":5669},"\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":5671,"title":5672,"module":5664,"summary":5673},"\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":5675,"title":5676,"module":5664,"summary":5677},"\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":5679,"title":5680,"module":5664,"summary":5681},"\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":5683,"title":5684,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":5686,"title":5687,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":5689,"title":5690,"module":4490,"summary":5691},"\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":5693,"title":5694,"module":4490,"summary":5695},"\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":5697,"title":5698,"module":4490,"summary":5699},"\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":5701,"title":5702,"module":4490,"summary":5703},"\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":5705,"title":5706,"module":4490,"summary":5707},"\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":5709,"title":5710,"module":5711,"summary":5712},"\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":5714,"title":5715,"module":5711,"summary":5716},"\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":5718,"title":5719,"module":5711,"summary":5720},"\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":5722,"title":5723,"module":5711,"summary":5724},"\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":5726,"title":5727,"module":5711,"summary":5728},"\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":5730,"title":5731,"module":5711,"summary":5732},"\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":5734,"title":5735,"module":5711,"summary":5736},"\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":5738,"title":5739,"module":5740,"summary":5741},"\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":5743,"title":5744,"module":5740,"summary":5745},"\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":5747,"title":5748,"module":5740,"summary":5749},"\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":5751,"title":5752,"module":5740,"summary":5753},"\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":5755,"title":5756,"module":5740,"summary":5757},"\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":5759,"title":5760,"module":5761,"summary":5762},"\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":5764,"title":5765,"module":5761,"summary":5766},"\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":5768,"title":5769,"module":5761,"summary":5770},"\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":5772,"title":5773,"module":5761,"summary":5774},"\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":5776,"title":5777,"module":5761,"summary":5778},"\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":5780,"title":5781,"module":5782,"summary":5783},"\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":5785,"title":5786,"module":5782,"summary":5787},"\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":5789,"title":5790,"module":5782,"summary":5791},"\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":5793,"title":5794,"module":5782,"summary":5795},"\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":5797,"title":5798,"module":5782,"summary":5799},"\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":5801,"title":5802,"module":5803,"summary":5804},"\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":5806,"title":5807,"module":5803,"summary":5808},"\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":5810,"title":5811,"module":5803,"summary":5812},"\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":5814,"title":5815,"module":5803,"summary":5816},"\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":5818,"title":5819,"module":5803,"summary":5820},"\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":5822,"title":5823,"module":5824,"summary":5825},"\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":5827,"title":5828,"module":5824,"summary":5829},"\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":5831,"title":5832,"module":5824,"summary":5833},"\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":5835,"title":5836,"module":5824,"summary":5837},"\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":5839,"title":5840,"module":5824,"summary":5841},"\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":5843,"title":5844,"module":5845,"summary":5846},"\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":5848,"title":5849,"module":5845,"summary":5850},"\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":5852,"title":5853,"module":5845,"summary":5854},"\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":5856,"title":5857,"module":5858,"summary":5859},"\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":5861,"title":5862,"module":5858,"summary":5863},"\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":5865,"title":5866,"module":5867,"summary":5868},"\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":5870,"title":5871,"module":5867,"summary":5872},"\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":5874,"title":5875,"module":5867,"summary":5876},"\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":5878,"title":5879,"module":5867,"summary":5880},"\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":5882,"title":5883,"module":5867,"summary":5884},"\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":5886,"title":5887,"module":5888,"summary":5889},"\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":5891,"title":5892,"module":5888,"summary":5893},"\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":5895,"title":5896,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":5898,"title":5899,"module":4490,"summary":5900},"\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":5902,"title":5903,"module":4490,"summary":5904},"\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":5906,"title":5907,"module":5908,"summary":5909},"\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":5911,"title":5912,"module":5908,"summary":5913},"\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":5915,"title":5916,"module":5908,"summary":5917},"\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":5919,"title":5920,"module":5908,"summary":5921},"\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":5923,"title":5924,"module":5908,"summary":5925},"\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":5927,"title":5928,"module":5908,"summary":5929},"\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":5931,"title":5932,"module":5933,"summary":5934},"\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":5936,"title":5937,"module":5933,"summary":5938},"\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":5940,"title":5941,"module":5933,"summary":5942},"\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":5944,"title":5945,"module":5933,"summary":5946},"\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":5948,"title":5949,"module":5933,"summary":5950},"\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":5952,"title":5953,"module":5933,"summary":5954},"\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":5956,"title":5957,"module":5958,"summary":5959},"\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":5961,"title":5962,"module":5958,"summary":5963},"\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":5965,"title":5966,"module":5958,"summary":5967},"\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":5969,"title":5970,"module":5971,"summary":5972},"\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":5974,"title":5975,"module":5971,"summary":5976},"\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":5978,"title":5979,"module":5980,"summary":5981},"\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":5983,"title":5984,"module":5980,"summary":5985},"\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":5987,"title":5988,"module":5980,"summary":5989},"\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":5991,"title":5992,"module":5993,"summary":5994},"\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":5996,"title":5997,"module":5993,"summary":5998},"\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":6000,"title":6001,"module":6002,"summary":6003},"\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":6005,"title":6006,"module":6002,"summary":6007},"\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":6009,"title":6010,"module":6002,"summary":6011},"\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":6013,"title":6014,"module":6015,"summary":6016},"\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":6018,"title":6019,"module":6015,"summary":6020},"\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":6022,"title":6023,"module":6015,"summary":6024},"\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":6026,"title":6027,"module":6028,"summary":6029},"\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":6031,"title":6032,"module":6028,"summary":6033},"\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":6035,"title":6036,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":6038,"title":6039,"module":6040,"summary":6041},"\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":6043,"title":6044,"module":6040,"summary":6045},"\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":6047,"title":6048,"module":6040,"summary":6049},"\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":6051,"title":6052,"module":6040,"summary":6053},"\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":6055,"title":6056,"module":6040,"summary":6057},"\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":6059,"title":6060,"module":6061,"summary":6062},"\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":6064,"title":6065,"module":6061,"summary":6066},"\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":6068,"title":6069,"module":6061,"summary":6070},"\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":6072,"title":6073,"module":6061,"summary":6074},"\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":6076,"title":6077,"module":6078,"summary":6079},"\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":6081,"title":6082,"module":6078,"summary":6083},"\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":6085,"title":6086,"module":6078,"summary":6087},"\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":6089,"title":6090,"module":6078,"summary":6091},"\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":6093,"title":6094,"module":6095,"summary":6096},"\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":6098,"title":6099,"module":6095,"summary":6100},"\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":6102,"title":6103,"module":6095,"summary":6104},"\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":6106,"title":6107,"module":6095,"summary":6108},"\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":6110,"title":6111,"module":6112,"summary":6113},"\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":6115,"title":6116,"module":6112,"summary":6117},"\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":6119,"title":6120,"module":6112,"summary":6121},"\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":6123,"title":6124,"module":6112,"summary":6125},"\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":6127,"title":6128,"module":6112,"summary":6129},"\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":6131,"title":6132,"module":6112,"summary":6133},"\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":6135,"title":6136,"module":6137,"summary":6138},"\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":6140,"title":6141,"module":6137,"summary":6142},"\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":6144,"title":6145,"module":6137,"summary":6146},"\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":6148,"title":6149,"module":6150,"summary":6151},"\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":6153,"title":6154,"module":6150,"summary":6155},"\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":6157,"title":6158,"module":6150,"summary":6159},"\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":6161,"title":6162,"module":6150,"summary":6163},"\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":6165,"title":6166,"module":6167,"summary":6168},"\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":6170,"title":6171,"module":6167,"summary":6172},"\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":6174,"title":6175,"module":6167,"summary":6176},"\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":6178,"title":6179,"module":6180,"summary":6181},"\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":6183,"title":6184,"module":6180,"summary":6185},"\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":6187,"title":6188,"module":6180,"summary":6189},"\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":6191,"title":6192,"module":6193,"summary":6194},"\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":6196,"title":6197,"module":6193,"summary":6198},"\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":6200,"title":6201,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":6203,"title":6204,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":6206,"title":6207,"module":6208,"summary":6209},"\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":6211,"title":6212,"module":6208,"summary":6213},"\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":6215,"title":6216,"module":6208,"summary":6217},"\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":6219,"title":6220,"module":6208,"summary":6221},"\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":6223,"title":6224,"module":6225,"summary":6226},"\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":6228,"title":6229,"module":6225,"summary":6230},"\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":6232,"title":6233,"module":6225,"summary":6234},"\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":6236,"title":6237,"module":6238,"summary":6239},"\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":6241,"title":6242,"module":6238,"summary":6243},"\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":6245,"title":6246,"module":6238,"summary":6247},"\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":6249,"title":6250,"module":6238,"summary":6251},"\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":6253,"title":6254,"module":6238,"summary":6255},"\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":6257,"title":6258,"module":6238,"summary":6259},"\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":6261,"title":6262,"module":6263,"summary":6264},"\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":6266,"title":6267,"module":6263,"summary":6268},"\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":6270,"title":6271,"module":6263,"summary":6272},"\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":6274,"title":6275,"module":6263,"summary":6276},"\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":6278,"title":6279,"module":6263,"summary":6280},"\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":6282,"title":6283,"module":6263,"summary":6284},"\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":6286,"title":6287,"module":6288,"summary":6289},"\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":6291,"title":6292,"module":6288,"summary":6293},"\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":6295,"title":6296,"module":6288,"summary":6297},"\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":6299,"title":6300,"module":6288,"summary":6301},"\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":6303,"title":6304,"module":5143,"summary":6305},"\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":6307,"title":6308,"module":5143,"summary":6309},"\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":6311,"title":6312,"module":5143,"summary":6313},"\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":6315,"title":6316,"module":6317,"summary":6318},"\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":6320,"title":6321,"module":6317,"summary":6322},"\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":6324,"title":6325,"module":6317,"summary":6326},"\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":6328,"title":6329,"module":6330,"summary":6331},"\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":6333,"title":6334,"module":6330,"summary":6335},"\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":6337,"title":6338,"module":6330,"summary":6339},"\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":6341,"title":6342,"module":6343,"summary":6344},"\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":6346,"title":6347,"module":6343,"summary":6348},"\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":6350,"title":6351,"module":6352,"summary":6353},"\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":6355,"title":6356,"module":6352,"summary":6357},"\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":6359,"title":6360,"module":6352,"summary":6361},"\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":6363,"title":6364,"module":6352,"summary":6365},"\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":6367,"title":6368,"module":6352,"summary":6369},"\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":6371,"title":6372,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":6374,"title":6375,"module":6376,"summary":6377},"\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":6379,"title":6380,"module":6376,"summary":6381},"\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":6383,"title":6384,"module":6376,"summary":6385},"\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":6387,"title":6388,"module":6376,"summary":6389},"\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":6391,"title":6392,"module":6393,"summary":6394},"\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":6396,"title":6397,"module":6393,"summary":6398},"\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":6400,"title":6401,"module":6393,"summary":6402},"\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":6404,"title":6405,"module":6393,"summary":6406},"\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":6408,"title":6409,"module":6393,"summary":6410},"\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":6412,"title":6413,"module":6393,"summary":6414},"\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":6416,"title":6417,"module":6418,"summary":6419},"\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":6421,"title":6422,"module":6418,"summary":6423},"\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":6425,"title":6426,"module":6418,"summary":6427},"\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":6429,"title":6430,"module":6418,"summary":6431},"\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":6433,"title":6434,"module":6418,"summary":6435},"\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":6437,"title":6438,"module":4837,"summary":6439},"\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":6441,"title":6442,"module":4837,"summary":6443},"\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":6445,"title":6446,"module":4837,"summary":6447},"\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":6449,"title":6450,"module":4837,"summary":6451},"\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":6453,"title":6454,"module":4837,"summary":6455},"\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":6457,"title":6458,"module":4837,"summary":6459},"\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":6461,"title":6462,"module":6463,"summary":6464},"\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":6466,"title":6467,"module":6463,"summary":6468},"\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":6470,"title":6471,"module":6463,"summary":6472},"\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":6474,"title":6475,"module":6463,"summary":6476},"\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":6478,"title":6479,"module":6480,"summary":6481},"\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":6483,"title":6484,"module":6480,"summary":6485},"\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":6487,"title":6488,"module":6480,"summary":6489},"\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":6491,"title":4892,"module":6480,"summary":6492},"\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":6494,"title":6495,"module":6480,"summary":6496},"\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":6498,"title":6499,"module":6500,"summary":6501},"\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":6503,"title":6504,"module":6500,"summary":6505},"\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":6507,"title":6508,"module":6500,"summary":6509},"\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":6511,"title":6512,"module":6500,"summary":6513},"\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":6515,"title":6516,"module":6517,"summary":6518},"\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":6520,"title":6521,"module":6517,"summary":6522},"\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":6524,"title":6525,"module":6517,"summary":6526},"\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":6528,"title":6529,"module":6517,"summary":6530},"\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":6532,"title":6533,"module":6517,"summary":6534},"\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":6536,"title":6537,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":6539,"title":6540,"module":4490,"summary":6541},"\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":6543,"title":6544,"module":4490,"summary":6545},"\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":6547,"title":6548,"module":6549,"summary":6550},"\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":6552,"title":6553,"module":6549,"summary":6554},"\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":6556,"title":6557,"module":6549,"summary":6558},"\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":6560,"title":6561,"module":6549,"summary":6562},"\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":6564,"title":6565,"module":6566,"summary":6567},"\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":6569,"title":6570,"module":6566,"summary":6571},"\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":6573,"title":6574,"module":6566,"summary":6575},"\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":6577,"title":6578,"module":6566,"summary":6579},"\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":6581,"title":6582,"module":6566,"summary":6583},"\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":6585,"title":6586,"module":6566,"summary":6587},"\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":6589,"title":6590,"module":6591,"summary":6592},"\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":6594,"title":6595,"module":6591,"summary":6596},"\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":6598,"title":6599,"module":6591,"summary":6600},"\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":6602,"title":6603,"module":6591,"summary":6604},"\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":6606,"title":6607,"module":6608,"summary":6609},"\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":6611,"title":6612,"module":6608,"summary":6613},"\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":6615,"title":6616,"module":6608,"summary":6617},"\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":6619,"title":6620,"module":6608,"summary":6621},"\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":6623,"title":6624,"module":6625,"summary":6626},"\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":6628,"title":6629,"module":6625,"summary":6630},"\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":6632,"title":6633,"module":6625,"summary":6634},"\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":6636,"title":6637,"module":6638,"summary":6639},"\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":6641,"title":6642,"module":6638,"summary":6643},"\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":6645,"title":6646,"module":6638,"summary":6647},"\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":6649,"title":6650,"module":6638,"summary":6651},"\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":6653,"title":6654,"module":6655,"summary":6656},"\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":6658,"title":6659,"module":6655,"summary":6660},"\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":6662,"title":6663,"module":6655,"summary":6664},"\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":6666,"title":6667,"module":6655,"summary":6668},"\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":6670,"title":6671,"module":6672,"summary":6673},"\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":6675,"title":6676,"module":6672,"summary":6677},"\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":6679,"title":6680,"module":6672,"summary":6681},"\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":6683,"title":6684,"module":6685,"summary":6686},"\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":6688,"title":6689,"module":6685,"summary":6690},"\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":6692,"title":6693,"module":6685,"summary":6694},"\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":6696,"title":6697,"module":6685,"summary":6698},"\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":6700,"title":6701,"module":6702,"summary":6703},"\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":6705,"title":6706,"module":6702,"summary":6707},"\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":6709,"title":6710,"module":6702,"summary":6711},"\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":6713,"title":6714,"module":6702,"summary":6715},"\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":6717,"title":6718,"module":6702,"summary":6719},"\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":6721,"title":6722,"module":6723,"summary":6724},"\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":6726,"title":6727,"module":6723,"summary":6728},"\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":6730,"title":6731,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":6733,"title":6734,"module":6735,"summary":6736},"\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":6738,"title":6739,"module":6735,"summary":6740},"\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":6742,"title":6743,"module":6735,"summary":6744},"\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":6746,"title":6747,"module":6735,"summary":6748},"\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":6750,"title":6751,"module":6735,"summary":6752},"\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":6754,"title":6755,"module":6756,"summary":6757},"\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":6759,"title":6760,"module":6756,"summary":6761},"\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":6763,"title":6764,"module":6756,"summary":6765},"\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":6767,"title":6768,"module":6756,"summary":6769},"\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":6771,"title":6772,"module":6756,"summary":6773},"\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":6775,"title":6776,"module":6756,"summary":6777},"\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":6779,"title":6780,"module":6756,"summary":6781},"\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":6783,"title":6784,"module":6785,"summary":6786},"\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":6788,"title":6789,"module":6785,"summary":6790},"\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":6792,"title":6793,"module":6785,"summary":6794},"\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":6796,"title":6797,"module":6785,"summary":6798},"\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":6800,"title":6801,"module":6802,"summary":6803},"\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":6805,"title":6806,"module":6802,"summary":6807},"\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":6809,"title":6810,"module":6802,"summary":6811},"\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":6813,"title":6814,"module":6815,"summary":6816},"\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":6818,"title":6819,"module":6815,"summary":6820},"\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":6822,"title":6823,"module":6815,"summary":6824},"\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":6826,"title":6827,"module":6815,"summary":6828},"\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":6830,"title":6831,"module":6815,"summary":6832},"\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":6834,"title":6835,"module":6815,"summary":6836},"\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":6838,"title":6839,"module":6840,"summary":6841},"\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":6843,"title":6844,"module":6840,"summary":6845},"\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":6847,"title":6848,"module":6840,"summary":6849},"\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":6851,"title":6852,"module":6853,"summary":6854},"\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":6856,"title":6857,"module":6853,"summary":6858},"\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":6860,"title":6861,"module":6853,"summary":6862},"\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":6864,"title":6865,"module":6853,"summary":6866},"\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":6868,"title":6869,"module":6870,"summary":6871},"\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":6873,"title":6874,"module":6870,"summary":6875},"\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":6877,"title":6878,"module":6870,"summary":6879},"\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":6881,"title":6882,"module":6883,"summary":6884},"\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":6886,"title":6887,"module":6883,"summary":6888},"\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":6890,"title":6891,"module":6883,"summary":6892},"\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":6894,"title":6895,"module":6883,"summary":6896},"\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":6898,"title":6899,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":6901,"title":6902,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":6904,"title":6905,"module":4490,"summary":6906},"\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":6908,"title":6909,"module":4490,"summary":6910},"\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":6912,"title":6913,"module":4490,"summary":6914},"\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":6916,"title":6917,"module":4490,"summary":6918},"\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":6920,"title":6921,"module":4490,"summary":6922},"\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":6924,"title":6925,"module":4490,"summary":6926},"\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":6928,"title":6929,"module":6930,"summary":6931},"\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":6933,"title":6934,"module":6930,"summary":6935},"\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":6937,"title":6938,"module":6930,"summary":6939},"\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":6941,"title":6942,"module":6943,"summary":6944},"\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":6946,"title":6947,"module":6943,"summary":6948},"\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":6950,"title":6951,"module":6943,"summary":6952},"\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":6954,"title":6955,"module":6956,"summary":6957},"\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":6959,"title":6960,"module":6956,"summary":6961},"\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":6963,"title":6964,"module":6956,"summary":6965},"\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":6967,"title":6968,"module":6956,"summary":6969},"\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":6971,"title":6972,"module":6956,"summary":6973},"\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":6975,"title":6976,"module":6977,"summary":6978},"\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":6980,"title":6981,"module":6977,"summary":6982},"\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":6984,"title":6985,"module":6977,"summary":6986},"\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":6988,"title":6989,"module":6977,"summary":6990},"\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":6992,"title":6993,"module":6994,"summary":6995},"\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":6997,"title":6998,"module":6994,"summary":6999},"\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":7001,"title":7002,"module":6994,"summary":7003},"\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":7005,"title":7006,"module":6994,"summary":7007},"\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":7009,"title":7010,"module":7011,"summary":7012},"\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":7014,"title":7015,"module":7011,"summary":7016},"\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":7018,"title":7019,"module":7011,"summary":7020},"\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":7022,"title":7023,"module":7011,"summary":7024},"\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":7026,"title":7027,"module":7028,"summary":7029},"\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":7031,"title":7032,"module":7028,"summary":7033},"\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":7035,"title":7036,"module":7028,"summary":7037},"\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":7039,"title":7040,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":7042,"title":5684,"module":7043,"summary":7044},"\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":7046,"title":7047,"module":7043,"summary":7048},"\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":7050,"title":7051,"module":7043,"summary":7052},"\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":7054,"title":5043,"module":7043,"summary":7055},"\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":7057,"title":7058,"module":4490,"summary":7059},"\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":7061,"title":7062,"module":4490,"summary":7063},"\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":7065,"title":7066,"module":4490,"summary":7067},"\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":7069,"title":7070,"module":7071,"summary":7072},"\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":7074,"title":7075,"module":7071,"summary":7076},"\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":7078,"title":7079,"module":7071,"summary":7080},"\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":7082,"title":7083,"module":7071,"summary":7084},"\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":7086,"title":7087,"module":7071,"summary":7088},"\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":7090,"title":7091,"module":7092,"summary":7093},"\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":7095,"title":7096,"module":7092,"summary":7097},"\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":7099,"title":7100,"module":7092,"summary":7101},"\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":7103,"title":7104,"module":7092,"summary":7105},"\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":7107,"title":7108,"module":7092,"summary":7109},"\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":7111,"title":7112,"module":7113,"summary":7114},"\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":7116,"title":7117,"module":7113,"summary":7118},"\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":7120,"title":7121,"module":7113,"summary":7122},"\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":7124,"title":7125,"module":7113,"summary":7126},"\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":7128,"title":7129,"module":7130,"summary":7131},"\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":7133,"title":7134,"module":7130,"summary":7135},"\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":7137,"title":7138,"module":7130,"summary":7139},"\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":7141,"title":7142,"module":7130,"summary":7143},"\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":7145,"title":7146,"module":7130,"summary":7147},"\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":7149,"title":7150,"module":7130,"summary":7151},"\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":7153,"title":7154,"module":7130,"summary":7155},"\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":7157,"title":7158,"module":7130,"summary":7159},"\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":7161,"title":7162,"module":7130,"summary":7163},"\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":7165,"title":7166,"module":7167,"summary":7168},"\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":7170,"title":7171,"module":7167,"summary":7172},"\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":7174,"title":7175,"module":7167,"summary":7176},"\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":7178,"title":7179,"module":7167,"summary":7180},"\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":7182,"title":7183,"module":7167,"summary":7184},"\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":7186,"title":7187,"module":7188,"summary":7189},"\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":7191,"title":7192,"module":7188,"summary":7193},"\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":7195,"title":7196,"module":7188,"summary":7197},"\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":7199,"title":7200,"module":7188,"summary":7201},"\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":7203,"title":7204,"module":7188,"summary":7205},"\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":7207,"title":7208,"module":7188,"summary":7209},"\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":7211,"title":7212,"module":7188,"summary":7213},"\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":7215,"title":7216,"module":7217,"summary":7218},"\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":7220,"title":7221,"module":7217,"summary":7222},"\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":7224,"title":7225,"module":7217,"summary":7226},"\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":7228,"title":7229,"module":7230,"summary":7231},"\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":7233,"title":7234,"module":7230,"summary":7235},"\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":7237,"title":7238,"module":7230,"summary":7239},"\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":7241,"title":7242,"module":7230,"summary":7243},"\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":7245,"title":7246,"module":7230,"summary":7247},"\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":7249,"title":7250,"module":7230,"summary":7251},"\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":7253,"title":7254,"module":7230,"summary":7255},"\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":7257,"title":7258,"module":7259,"summary":7260},"\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":7262,"title":7263,"module":7259,"summary":7264},"\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":7266,"title":7267,"module":7259,"summary":7268},"\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":7270,"title":7271,"module":7259,"summary":7272},"\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":7274,"title":7275,"module":7259,"summary":7276},"\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":7278,"title":7279,"module":7259,"summary":7280},"\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":7282,"title":7283,"module":7259,"summary":7284},"\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":7286,"title":7287,"module":7259,"summary":7288},"\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":7290,"title":7291,"module":7259,"summary":7292},"\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":7294,"title":7295,"module":7259,"summary":7296},"\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":7298,"title":7299,"module":7259,"summary":7300},"\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":7302,"title":7303,"module":7304,"summary":7305},"\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":7307,"title":7308,"module":7304,"summary":7309},"\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":7311,"title":7312,"module":7304,"summary":7313},"\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":7315,"title":7316,"module":7304,"summary":7317},"\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":7319,"title":7320,"module":7304,"summary":7321},"\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":7323,"title":7324,"module":6,"summary":6},"\u002Fdeep-learning","Deep Learning",{"path":7326,"title":7327,"module":5234,"summary":7328},"\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":7330,"title":7331,"module":5234,"summary":7332},"\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":7334,"title":7335,"module":5234,"summary":7336},"\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":7338,"title":7339,"module":5234,"summary":7340},"\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":7342,"title":7343,"module":5234,"summary":7344},"\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":7346,"title":7347,"module":7348,"summary":7349},"\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":7351,"title":7352,"module":7348,"summary":7353},"\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":7355,"title":7356,"module":7348,"summary":7357},"\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":7359,"title":7360,"module":7348,"summary":7361},"\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":7363,"title":7364,"module":7365,"summary":7366},"\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":7368,"title":7369,"module":7365,"summary":7370},"\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":7372,"title":7373,"module":7365,"summary":7374},"\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":7376,"title":7377,"module":7365,"summary":7378},"\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":7380,"title":7381,"module":7382,"summary":7383},"\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":7385,"title":7386,"module":7382,"summary":7387},"\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":7389,"title":7390,"module":7382,"summary":7391},"\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":7393,"title":7394,"module":7382,"summary":7395},"\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":7397,"title":7398,"module":7382,"summary":7399},"\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":7401,"title":7402,"module":7403,"summary":7404},"\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":7406,"title":7407,"module":7403,"summary":7408},"\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":7410,"title":7411,"module":7403,"summary":7412},"\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":7414,"title":7415,"module":7416,"summary":7417},"\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":7419,"title":7420,"module":7416,"summary":7421},"\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":7423,"title":7424,"module":7416,"summary":7425},"\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":7427,"title":7428,"module":7429,"summary":7430},"\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":7432,"title":7433,"module":7429,"summary":7434},"\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":7436,"title":7437,"module":7429,"summary":7438},"\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":7440,"title":7441,"module":7429,"summary":7442},"\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":7444,"title":7445,"module":7446,"summary":7447},"\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":7449,"title":7450,"module":7446,"summary":7451},"\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":7453,"title":7454,"module":7446,"summary":7455},"\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":7457,"title":7458,"module":7446,"summary":7459},"\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":7461,"title":7462,"module":7446,"summary":7463},"\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":7465,"title":7466,"module":7446,"summary":7467},"\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":7469,"title":7470,"module":7471,"summary":7472},"\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":7474,"title":7475,"module":7471,"summary":7476},"\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":7478,"title":7479,"module":7471,"summary":7480},"\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":7482,"title":7483,"module":7471,"summary":7484},"\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":7486,"title":7487,"module":7488,"summary":7489},"\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":7491,"title":7492,"module":7488,"summary":7493},"\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":7495,"title":7496,"module":7488,"summary":7497},"\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":7499,"title":7500,"module":7501,"summary":7502},"\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":7504,"title":7505,"module":7501,"summary":7506},"\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":7508,"title":7509,"module":7501,"summary":7510},"\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":7512,"title":7513,"module":7501,"summary":7514},"\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":7516,"title":7517,"module":7501,"summary":7518},"\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":7520,"title":7521,"module":7522,"summary":7523},"\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":7525,"title":7526,"module":7522,"summary":7527},"\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":7529,"title":7530,"module":7522,"summary":7531},"\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":7533,"title":7534,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":7536,"title":7537,"module":7538,"summary":7539},"\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":7541,"title":7542,"module":7538,"summary":7543},"\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":7545,"title":7546,"module":7538,"summary":7547},"\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":7549,"title":7550,"module":7538,"summary":7551},"\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":7553,"title":7554,"module":7555,"summary":7556},"\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":7558,"title":7559,"module":7555,"summary":7560},"\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":7562,"title":7563,"module":7555,"summary":7564},"\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":7566,"title":7567,"module":7555,"summary":7568},"\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":7570,"title":7571,"module":7572,"summary":7573},"\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":7575,"title":7576,"module":7572,"summary":7577},"\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":7579,"title":7580,"module":7572,"summary":7581},"\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":7583,"title":7584,"module":7572,"summary":7585},"\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":7587,"title":7588,"module":7589,"summary":7590},"\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":7592,"title":7593,"module":7589,"summary":7594},"\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":7596,"title":7597,"module":7589,"summary":7598},"\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":7600,"title":7601,"module":7589,"summary":7602},"\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":7604,"title":7605,"module":7606,"summary":7607},"\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":7609,"title":7610,"module":7606,"summary":7611},"\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":7613,"title":7614,"module":7606,"summary":7615},"\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":7617,"title":7618,"module":7606,"summary":7619},"\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":7621,"title":7622,"module":7623,"summary":7624},"\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":7626,"title":7627,"module":7623,"summary":7628},"\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":7630,"title":7631,"module":7623,"summary":7632},"\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":7634,"title":7635,"module":7623,"summary":7636},"\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":7638,"title":7639,"module":7640,"summary":7641},"\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":7643,"title":7644,"module":7640,"summary":7645},"\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":7647,"title":7648,"module":7640,"summary":7649},"\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":7651,"title":7652,"module":7640,"summary":7653},"\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":7655,"title":7656,"module":7640,"summary":7657},"\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":7659,"title":7660,"module":7661,"summary":7662},"\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":7664,"title":7665,"module":7661,"summary":7666},"\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":7668,"title":7669,"module":7670,"summary":7671},"\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":7673,"title":7674,"module":7670,"summary":7675},"\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":7677,"title":7678,"module":7670,"summary":7679},"\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":7681,"title":7682,"module":7670,"summary":7683},"\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":7685,"title":7686,"module":7687,"summary":7688},"\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":7690,"title":7691,"module":7687,"summary":7692},"\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":7694,"title":7695,"module":7687,"summary":7696},"\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":7698,"title":7699,"module":7687,"summary":7700},"\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":7702,"title":7703,"module":7687,"summary":7704},"\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":7706,"title":7707,"module":7708,"summary":7709},"\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":7711,"title":7712,"module":7708,"summary":7713},"\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":7715,"title":7716,"module":7708,"summary":7717},"\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":7719,"title":7720,"module":7708,"summary":7721},"\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":7723,"title":7724,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":7726,"title":7727,"module":4490,"summary":7728},"\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":7730,"title":7731,"module":7732,"summary":7733},"\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":7735,"title":7736,"module":7732,"summary":7737},"\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":7739,"title":7740,"module":7732,"summary":7741},"\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":7743,"title":7744,"module":7732,"summary":7745},"\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":7747,"title":7748,"module":7732,"summary":7749},"\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":7751,"title":7752,"module":7732,"summary":7753},"\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":7755,"title":7756,"module":7732,"summary":7757},"\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":7759,"title":7760,"module":7761,"summary":7762},"\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":7764,"title":7765,"module":7761,"summary":7766},"\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":7768,"title":7769,"module":7761,"summary":7770},"\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":7772,"title":7773,"module":7761,"summary":7774},"\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":7776,"title":7777,"module":7778,"summary":7779},"\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":7781,"title":7782,"module":7778,"summary":7783},"\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":7785,"title":7786,"module":7778,"summary":7787},"\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":7789,"title":7790,"module":7778,"summary":7791},"\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":7793,"title":7794,"module":7795,"summary":7796},"\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":7798,"title":7799,"module":7795,"summary":7800},"\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":7802,"title":7803,"module":7795,"summary":7804},"\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":7806,"title":7807,"module":7795,"summary":7808},"\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":7810,"title":7811,"module":7812,"summary":7813},"\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":7815,"title":7816,"module":7812,"summary":7817},"\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":7819,"title":7820,"module":7812,"summary":7821},"\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":7823,"title":7824,"module":7812,"summary":7825},"\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":7827,"title":7828,"module":7829,"summary":7830},"\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":7832,"title":7833,"module":7829,"summary":7834},"\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":7836,"title":7837,"module":7829,"summary":7838},"\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":7840,"title":7841,"module":7842,"summary":7843},"\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":7845,"title":7846,"module":7842,"summary":7847},"\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":7849,"title":7850,"module":7851,"summary":7852},"\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":7854,"title":7855,"module":7851,"summary":7856},"\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":7858,"title":7859,"module":7851,"summary":7860},"\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":7862,"title":7863,"module":6,"summary":6},"\u002Flogic","Logic",{"path":7865,"title":7866,"module":4490,"summary":7867},"\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":7869,"title":7870,"module":4490,"summary":7871},"\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":7873,"title":7874,"module":4490,"summary":7875},"\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":7877,"title":7878,"module":4490,"summary":7879},"\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":7881,"title":7882,"module":4490,"summary":7883},"\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":7885,"title":7886,"module":4490,"summary":7887},"\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":7889,"title":4709,"module":7890,"summary":7891},"\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":7893,"title":7894,"module":7890,"summary":7895},"\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":7897,"title":7898,"module":7890,"summary":7899},"\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":7901,"title":7902,"module":7890,"summary":7903},"\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":7905,"title":7906,"module":7890,"summary":7907},"\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":7909,"title":7910,"module":7890,"summary":7911},"\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":7913,"title":7914,"module":7890,"summary":7915},"\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":7917,"title":7918,"module":7890,"summary":7919},"\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":40,"title":7921,"module":7890,"summary":7922},"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":7924,"title":7925,"module":7890,"summary":7926},"\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":7928,"title":7929,"module":7890,"summary":7930},"\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":7932,"title":7933,"module":7890,"summary":7934},"\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":7936,"title":7937,"module":7938,"summary":7939},"\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":7941,"title":7942,"module":7938,"summary":7943},"\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":7945,"title":7946,"module":7938,"summary":7947},"\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":7949,"title":7950,"module":7938,"summary":7951},"\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":7953,"title":7954,"module":7938,"summary":7955},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","Off-Policy Methods and the Deadly Triad","Off-policy learning with function approximation is where the convergence guarantees of reinforcement learning fail. We extend the tabular off-policy updates to semi-gradient form with per-step importance sampling, show Baird's counterexample driving the weights to infinity, and identify the cause: the deadly triad of function approximation, bootstrapping, and off-policy training — any two are safe, all three can diverge. The divergence is not caused by sampling noise: a fully synchronous dynamic-programming update blows up just the same, which is what makes the triad a structural hazard rather than a fluke.\n",{"path":7957,"title":7958,"module":7938,"summary":7959},"\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":7961,"title":7962,"module":7938,"summary":7963},"\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":7965,"title":7966,"module":7938,"summary":7967},"\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":7969,"title":7970,"module":7938,"summary":7971},"\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":7973,"title":7974,"module":7938,"summary":7975},"\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":7977,"title":7978,"module":7938,"summary":7979},"\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":7981,"title":7982,"module":7938,"summary":7983},"\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":7985,"title":7986,"module":7938,"summary":7987},"\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":7989,"title":7990,"module":7938,"summary":7991},"\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":7993,"title":7312,"module":7994,"summary":7995},"\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":7997,"title":7998,"module":7994,"summary":7999},"\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":8001,"title":8002,"module":7994,"summary":8003},"\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":8005,"title":8006,"module":7994,"summary":8007},"\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":8009,"title":8010,"module":7994,"summary":8011},"\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":8013,"title":8014,"module":7994,"summary":8015},"\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":8017,"title":8018,"module":7994,"summary":8019},"\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":8021,"title":8022,"module":7994,"summary":8023},"\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":8025,"title":8026,"module":8027,"summary":8028},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow","Sharpening DQN: Improvements and the Distributional Idea","Modern Deep Reinforcement Learning","In the years after the 2015 DQN paper, a stream of focused improvements each fixed one weakness of the baseline without disturbing its frame. This lesson recaps five that keep the scalar $Q$-value — Double DQN, multi-step returns, dueling networks, prioritized replay, and NoisyNets, each changing a different slot of the same Q-learning loop — then develops the sixth, distributional RL, which changes the objective itself: learn the whole return distribution $Z(s,a)$. We build the distributional Bellman equation and the C51 categorical algorithm, projection step and all, worked end to end on real numbers. A companion lesson takes up QR-DQN, Rainbow, and the modern distributional line.\n",{"path":8030,"title":8031,"module":8027,"summary":8032},"\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":8034,"title":8035,"module":8027,"summary":8036},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control","Continuous Control: DDPG and TD3","When actions are real-valued, the $\\arg\\max_a Q(s,a)$ in Q-learning becomes an optimization problem on every step. This lesson builds the off-policy actor-critic family that sidesteps it: the deterministic policy gradient and DDPG, which replaces the max with a learned actor, and the three fixes of TD3 that counter the value overestimation DDPG inherits. A companion lesson takes up SAC's maximum-entropy objective and the methods built on this template.\n",{"path":8038,"title":8039,"module":8027,"summary":8040},"\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":3082,"title":8042,"module":8027,"summary":8043},"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":8045,"title":8046,"module":8027,"summary":8047},"\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":8049,"title":8050,"module":8027,"summary":8051},"\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":8053,"title":8054,"module":8027,"summary":8055},"\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":8057,"title":8058,"module":8027,"summary":8059},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl","Offline RL: The Problem and Value-Based Fixes","Offline reinforcement learning learns a policy from a fixed logged dataset with no further environment interaction — off-policy learning pushed to the extreme, and it breaks for the extreme version of the same reason. Bootstrapping queries the value function at out-of-distribution actions the data never covers, those errors are optimistic, and with no online feedback to correct them they compound through the Bellman backup. This lesson sets up the failure and off-policy evaluation, then builds the first two families of pessimistic fixes: policy constraint (BCQ) and conservative value estimation (CQL). A companion lesson takes up implicit methods, model-based offline RL, and Decision Transformer.\n",{"path":8061,"title":8062,"module":8027,"summary":8063},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2","Offline RL: Implicit Methods, Sequence Models, and Beyond","A companion to the offline-RL problem lesson. Policy constraint and conservative value estimation both still query a learned value function; implicit methods (IQL) avoid querying it off the data at all, using an in-sample expectile backup. We then build pessimism into a learned model (MOPO, COMBO) and drop bootstrapping entirely with Decision Transformer's return-conditioned sequence modeling, closing with offline-to-online fine-tuning, diffusion planners, and the offline view of RLHF. The one rule throughout: without online correction, be pessimistic about what you cannot verify.\n",{"path":8065,"title":8066,"module":8027,"summary":8067},"\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":8069,"title":8070,"module":8027,"summary":8071},"\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":8073,"title":8074,"module":8027,"summary":8075},"\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":8077,"title":8078,"module":8027,"summary":8079},"\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":8081,"title":8082,"module":8027,"summary":8083},"\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":8085,"title":8086,"module":8027,"summary":8087},"\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":8089,"title":8090,"module":8027,"summary":8091},"\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":8093,"title":8094,"module":8027,"summary":8095},"\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":8097,"title":8098,"module":8027,"summary":8099},"\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":8101,"title":8102,"module":8027,"summary":8103},"\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":8105,"title":8106,"module":8027,"summary":8107},"\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":8109,"title":8110,"module":8027,"summary":8111},"\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":1269,"title":8113,"module":3554,"summary":8114},"The Psychology of Reinforcement","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":8116,"title":8117,"module":3554,"summary":8118},"\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":3186,"title":8120,"module":3554,"summary":8121},"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":8123,"title":8124,"module":3554,"summary":8125},"\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":17,"title":18,"module":3554,"summary":8127},"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":3556,"title":5,"module":3554,"summary":3570},{"path":8130,"title":8131,"module":3554,"summary":8132},"\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":8134,"title":8135,"module":3554,"summary":8136},"\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":8138,"title":7304,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":8140,"title":8141,"module":4490,"summary":8142},"\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":8144,"title":8145,"module":4490,"summary":8146},"\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":8148,"title":8149,"module":4490,"summary":8150},"\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":8152,"title":8153,"module":4490,"summary":8154},"\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":8156,"title":8157,"module":8158,"summary":8159},"\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":8161,"title":8162,"module":8158,"summary":8163},"\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":8165,"title":8166,"module":8158,"summary":8167},"\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":8169,"title":8170,"module":8158,"summary":8171},"\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":8173,"title":8174,"module":8158,"summary":8175},"\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":8177,"title":8178,"module":8158,"summary":8179},"\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":8181,"title":8182,"module":8158,"summary":8183},"\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":8185,"title":8186,"module":8158,"summary":8187},"\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":8189,"title":8190,"module":8158,"summary":8191},"\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":8193,"title":8194,"module":8158,"summary":8195},"\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":8197,"title":8198,"module":8158,"summary":8199},"\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":8201,"title":8202,"module":8158,"summary":8203},"\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":8205,"title":8206,"module":8207,"summary":8208},"\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":8210,"title":8211,"module":8207,"summary":8212},"\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":8214,"title":8215,"module":8207,"summary":8216},"\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":8218,"title":8219,"module":8207,"summary":8220},"\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":8222,"title":8223,"module":8207,"summary":8224},"\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":8226,"title":8227,"module":8207,"summary":8228},"\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":8230,"title":8231,"module":8207,"summary":8232},"\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":8234,"title":8235,"module":8207,"summary":8236},"\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":8238,"title":8239,"module":8207,"summary":8240},"\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":8242,"title":8243,"module":8207,"summary":8244},"\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":8246,"title":8247,"module":8207,"summary":8248},"\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":8250,"title":8251,"module":8207,"summary":8252},"\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":8254,"title":8255,"module":8256,"summary":8257},"\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":8259,"title":8260,"module":8256,"summary":8261},"\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":8263,"title":8264,"module":8256,"summary":8265},"\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":8267,"title":8268,"module":8256,"summary":8269},"\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":8271,"title":8272,"module":8256,"summary":8273},"\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":8275,"title":8276,"module":8256,"summary":8277},"\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":8279,"title":8280,"module":8256,"summary":8281},"\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":8283,"title":7882,"module":8256,"summary":8284},"\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":8286,"title":8287,"module":8256,"summary":8288},"\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":8290,"title":8291,"module":8256,"summary":8292},"\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":8294,"title":8295,"module":8296,"summary":8297},"\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":8299,"title":8300,"module":8296,"summary":8301},"\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":8303,"title":8304,"module":8296,"summary":8305},"\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":8307,"title":8308,"module":8296,"summary":8309},"\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":8311,"title":7304,"module":8296,"summary":8312},"\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":8314,"title":8315,"module":8296,"summary":8316},"\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":8318,"title":8319,"module":8296,"summary":8320},"\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":8322,"title":8323,"module":8296,"summary":8324},"\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":8326,"title":8327,"module":8328,"summary":8329},"\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":8331,"title":8332,"module":8328,"summary":8333},"\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":8335,"title":8336,"module":8328,"summary":8337},"\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":8339,"title":8340,"module":8328,"summary":8341},"\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":8343,"title":8344,"module":8328,"summary":8345},"\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":8347,"title":8348,"module":8328,"summary":8349},"\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":8351,"title":8352,"module":8328,"summary":8353},"\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":8355,"title":8356,"module":8328,"summary":8357},"\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":8359,"title":8360,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":8362,"title":8363,"module":8364,"summary":8365},"\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":8367,"title":8368,"module":8364,"summary":8369},"\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":8371,"title":8372,"module":8364,"summary":8373},"\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":8375,"title":8376,"module":8364,"summary":8377},"\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":8379,"title":8380,"module":8364,"summary":8381},"\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":8383,"title":8384,"module":8385,"summary":8386},"\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":8388,"title":8389,"module":8385,"summary":8390},"\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":8392,"title":8393,"module":8385,"summary":8394},"\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":8396,"title":8397,"module":8385,"summary":8398},"\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":8400,"title":8401,"module":8402,"summary":8403},"\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":8405,"title":8406,"module":8402,"summary":8407},"\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":8409,"title":8410,"module":8402,"summary":8411},"\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":8413,"title":8414,"module":8402,"summary":8415},"\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":8417,"title":8418,"module":8419,"summary":8420},"\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":8422,"title":8423,"module":8419,"summary":8424},"\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":8426,"title":8427,"module":8428,"summary":8429},"\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":8431,"title":8432,"module":8428,"summary":8433},"\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":8435,"title":8436,"module":8437,"summary":8438},"\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":8440,"title":8441,"module":8437,"summary":8442},"\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":8444,"title":8445,"module":8437,"summary":8446},"\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":8448,"title":8449,"module":8437,"summary":8450},"\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":8452,"title":8453,"module":8454,"summary":8455},"\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":8457,"title":8458,"module":8454,"summary":8459},"\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":8461,"title":8462,"module":8454,"summary":8463},"\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":8465,"title":8466,"module":8467,"summary":8468},"\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":8470,"title":8471,"module":8467,"summary":8472},"\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":8474,"title":8475,"module":8467,"summary":8476},"\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":8478,"title":8479,"module":8480,"summary":8481},"\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":8483,"title":8484,"module":8480,"summary":8485},"\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":8487,"title":8488,"module":8489,"summary":8490},"\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":8492,"title":8493,"module":8489,"summary":8494},"\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":8496,"title":8497,"module":8489,"summary":8498},"\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":8500,"title":8501,"module":8502,"summary":8503},"\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":8505,"title":8506,"module":8502,"summary":8507},"\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":8509,"title":8510,"module":8502,"summary":8511},"\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":8513,"title":8514,"module":8502,"summary":8515},"\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":8517,"title":8518,"module":8502,"summary":8519},"\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":8521,"title":8522,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":8524,"title":8525,"module":4490,"summary":8526},"\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":8528,"title":8529,"module":4490,"summary":8530},"\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":8532,"title":8533,"module":4490,"summary":8534},"\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":8536,"title":8537,"module":4490,"summary":8538},"\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":8540,"title":8541,"module":4490,"summary":8542},"\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":8544,"title":8545,"module":8546,"summary":8547},"\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":8549,"title":8550,"module":8546,"summary":8551},"\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":8553,"title":8554,"module":8546,"summary":8555},"\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":8557,"title":8558,"module":8546,"summary":8559},"\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":8561,"title":8562,"module":8563,"summary":8564},"\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":8566,"title":8567,"module":8563,"summary":8568},"\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":8570,"title":8571,"module":8563,"summary":8572},"\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":8574,"title":8575,"module":4956,"summary":8576},"\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":8578,"title":8579,"module":4956,"summary":8580},"\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":8582,"title":8583,"module":4956,"summary":8584},"\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":8586,"title":8587,"module":5438,"summary":8588},"\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":8590,"title":7150,"module":5438,"summary":8591},"\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":8593,"title":7258,"module":5438,"summary":8594},"\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":8596,"title":8597,"module":5438,"summary":8598},"\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":8600,"title":8601,"module":5438,"summary":8602},"\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":8604,"title":8605,"module":5438,"summary":8606},"\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":8608,"title":8609,"module":8610,"summary":8611},"\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":8613,"title":8614,"module":8610,"summary":8615},"\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":8617,"title":8618,"module":8610,"summary":8619},"\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":8621,"title":8622,"module":8610,"summary":8623},"\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":8625,"title":8626,"module":8610,"summary":8627},"\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":8629,"title":8630,"module":8610,"summary":8631},"\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":8633,"title":8634,"module":8610,"summary":8635},"\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":8637,"title":8638,"module":8610,"summary":8639},"\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":8641,"title":8642,"module":8610,"summary":8643},"\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":8645,"title":8646,"module":8610,"summary":8647},"\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":8649,"title":8650,"module":8610,"summary":8651},"\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":8653,"title":8654,"module":8610,"summary":8655},"\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":8657,"title":8658,"module":8610,"summary":8659},"\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":8661,"title":8662,"module":8610,"summary":8663},"\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":8665,"title":8666,"module":8610,"summary":8667},"\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":8669,"title":8670,"module":8610,"summary":8671},"\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":8673,"title":8674,"module":8610,"summary":8675},"\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":8677,"title":8678,"module":8610,"summary":8679},"\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":8681,"title":8682,"module":8610,"summary":8683},"\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":8685,"title":8686,"module":8610,"summary":8687},"\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":8689,"title":8690,"module":7246,"summary":8691},"\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":8693,"title":8694,"module":7246,"summary":8695},"\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":8697,"title":8698,"module":7246,"summary":8699},"\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":8701,"title":8702,"module":7246,"summary":8703},"\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":8705,"title":8706,"module":7246,"summary":8707},"\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":8709,"title":8710,"module":7246,"summary":8711},"\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":8713,"title":8714,"module":7246,"summary":8715},"\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":8717,"title":8718,"module":7246,"summary":8719},"\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":8721,"title":8722,"module":8723,"summary":8724},"\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":8726,"title":8727,"module":8723,"summary":8728},"\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":8730,"title":8731,"module":8723,"summary":8732},"\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":8734,"title":8735,"module":8723,"summary":8736},"\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":8738,"title":8739,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":8741,"title":8742,"module":4490,"summary":8743},"\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":8745,"title":8746,"module":4490,"summary":8747},"\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":8749,"title":8750,"module":4490,"summary":8751},"\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":8753,"title":8754,"module":8755,"summary":8756},"\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":8758,"title":8759,"module":8755,"summary":8760},"\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":8762,"title":8763,"module":8755,"summary":8764},"\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":8766,"title":8767,"module":8755,"summary":8768},"\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":8770,"title":8771,"module":8772,"summary":8773},"\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":8775,"title":8776,"module":8772,"summary":8777},"\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":8779,"title":8780,"module":8772,"summary":8781},"\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":8783,"title":8784,"module":8772,"summary":8785},"\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":8787,"title":8788,"module":8789,"summary":8790},"\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":8792,"title":8793,"module":8789,"summary":8794},"\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":8796,"title":8797,"module":8789,"summary":8798},"\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":8800,"title":8801,"module":8789,"summary":8802},"\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":8804,"title":8805,"module":8806,"summary":8807},"\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":8809,"title":8810,"module":8806,"summary":8811},"\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":8813,"title":8814,"module":8806,"summary":8815},"\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":8817,"title":8818,"module":8819,"summary":8820},"\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":8822,"title":8823,"module":8819,"summary":8824},"\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":8826,"title":8827,"module":8819,"summary":8828},"\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":8830,"title":8831,"module":8819,"summary":8832},"\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":8834,"title":8835,"module":8836,"summary":8837},"\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":8839,"title":8840,"module":8836,"summary":8841},"\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":8843,"title":8844,"module":8836,"summary":8845},"\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":8847,"title":8848,"module":8836,"summary":8849},"\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":8851,"title":8852,"module":8853,"summary":8854},"\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":8856,"title":8857,"module":8853,"summary":8858},"\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":8860,"title":8861,"module":8853,"summary":8862},"\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":8864,"title":8865,"module":8853,"summary":8866},"\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":8868,"title":8869,"module":8870,"summary":8871},"\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":8873,"title":8874,"module":8870,"summary":8875},"\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":8877,"title":8878,"module":8870,"summary":8879},"\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":8881,"title":8882,"module":8870,"summary":8883},"\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":8885,"title":8886,"module":8870,"summary":8887},"\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":8889,"title":8890,"module":8891,"summary":8892},"\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":8894,"title":8895,"module":8891,"summary":8896},"\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":8898,"title":8899,"module":8891,"summary":8900},"\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":8902,"title":8903,"module":8904,"summary":8905},"\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":8907,"title":8908,"module":8904,"summary":8909},"\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":8911,"title":8912,"module":8904,"summary":8913},"\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":8915,"title":8916,"module":8916,"summary":8917},"\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":8919,"title":8920,"module":8916,"summary":8921},"\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":8923,"title":8924,"module":8916,"summary":8925},"\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":8927,"title":8928,"module":8916,"summary":8929},"\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":8931,"title":8932,"module":8916,"summary":8933},"\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":8935,"title":8936,"module":8916,"summary":8937},"\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":8939,"title":8940,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":8942,"title":8943,"module":8944,"summary":8945},"\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":8947,"title":8948,"module":8944,"summary":8949},"\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":8951,"title":8952,"module":8944,"summary":8953},"\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":8955,"title":8956,"module":8957,"summary":8958},"\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":8960,"title":8961,"module":8957,"summary":8962},"\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":8964,"title":8965,"module":8957,"summary":8966},"\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":8968,"title":8969,"module":8957,"summary":8970},"\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":8972,"title":8973,"module":8974,"summary":8975},"\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":8977,"title":8978,"module":8974,"summary":8979},"\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":8981,"title":8982,"module":8974,"summary":8983},"\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":8985,"title":8986,"module":8974,"summary":8987},"\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":8989,"title":8990,"module":8991,"summary":8992},"\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":8994,"title":8995,"module":8991,"summary":8996},"\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":8998,"title":8999,"module":8991,"summary":9000},"\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":9002,"title":9003,"module":8991,"summary":9004},"\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":9006,"title":9007,"module":9008,"summary":9009},"\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":9011,"title":9012,"module":9008,"summary":9013},"\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":9015,"title":9016,"module":9008,"summary":9017},"\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":9019,"title":9020,"module":9008,"summary":9021},"\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":9023,"title":9024,"module":9025,"summary":9026},"\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":9028,"title":9029,"module":9025,"summary":9030},"\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":9032,"title":9033,"module":9025,"summary":9034},"\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":9036,"title":9037,"module":9038,"summary":9039},"\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":9041,"title":9042,"module":9038,"summary":9043},"\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":9045,"title":9046,"module":9038,"summary":9047},"\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":9049,"title":9050,"module":9038,"summary":9051},"\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":9053,"title":7479,"module":9054,"summary":9055},"\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":9057,"title":9058,"module":9054,"summary":9059},"\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":9061,"title":9062,"module":9054,"summary":9063},"\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":9065,"title":9066,"module":9054,"summary":9067},"\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":9069,"title":9070,"module":9054,"summary":9071},"\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":9073,"title":9074,"module":9075,"summary":9076},"\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":9078,"title":9079,"module":9075,"summary":9080},"\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":9082,"title":9083,"module":9075,"summary":9084},"\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":9086,"title":9087,"module":9075,"summary":9088},"\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":9090,"title":9091,"module":9092,"summary":9093},"\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":9095,"title":9096,"module":9092,"summary":9097},"\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":9099,"title":9100,"module":9092,"summary":9101},"\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":9103,"title":9104,"module":9092,"summary":9105},"\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":9107,"title":9108,"module":9092,"summary":9109},"\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":9111,"title":9112,"module":9113,"summary":9114},"\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":9116,"title":9117,"module":9113,"summary":9118},"\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":9120,"title":6197,"module":9113,"summary":9121},"\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":9123,"title":9124,"module":9113,"summary":9125},"\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":9127,"title":9128,"module":9113,"summary":9129},"\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":9131,"title":9132,"module":9133,"summary":9134},"\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":9136,"title":9137,"module":9133,"summary":9138},"\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":9140,"title":9141,"module":9133,"summary":9142},"\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":9144,"title":9145,"module":9133,"summary":9146},"\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":9148,"title":9149,"module":9133,"summary":9150},"\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":9152,"title":9153,"module":9133,"summary":9154},"\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":9156,"title":9157,"module":9133,"summary":9158},"\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":9160,"title":9161,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":9163,"title":9164,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":9166,"title":9167,"module":6,"summary":6},"\u002F","Study Notes",[9169,9187,9221,9255,9275,9330],{"module":4490,"moduleNumber":9170,"slug":9171,"lessons":9172},1,"foundations",[9173,9175,9177,9179,9182,9185],{"title":7866,"path":7865,"lessonNumber":9170,"topics":9174,"summary":7867},[4490],{"title":7870,"path":7869,"lessonNumber":3535,"topics":9176,"summary":7871},[4490],{"title":7874,"path":7873,"lessonNumber":3540,"topics":9178,"summary":7875},[4490],{"title":7878,"path":7877,"lessonNumber":9180,"topics":9181,"summary":7879},4,[4490],{"title":7882,"path":7881,"lessonNumber":9183,"topics":9184,"summary":7883},5,[4490],{"title":7886,"path":7885,"lessonNumber":3552,"topics":9186,"summary":7887},[4490],{"module":7890,"moduleNumber":3535,"slug":9188,"lessons":9189},"tabular-methods",[9190,9193,9195,9197,9199,9201,9203,9206,9209,9212,9215,9218],{"title":4709,"path":7889,"lessonNumber":9170,"topics":9191,"summary":7891},[9192],"Tabular Methods",{"title":7894,"path":7893,"lessonNumber":3535,"topics":9194,"summary":7895},[9192],{"title":7898,"path":7897,"lessonNumber":3540,"topics":9196,"summary":7899},[9192],{"title":7902,"path":7901,"lessonNumber":9180,"topics":9198,"summary":7903},[9192],{"title":7906,"path":7905,"lessonNumber":9183,"topics":9200,"summary":7907},[9192],{"title":7910,"path":7909,"lessonNumber":3552,"topics":9202,"summary":7911},[9192],{"title":7914,"path":7913,"lessonNumber":9204,"topics":9205,"summary":7915},7,[9192],{"title":7918,"path":7917,"lessonNumber":9207,"topics":9208,"summary":7919},8,[9192],{"title":7921,"path":40,"lessonNumber":9210,"topics":9211,"summary":7922},9,[9192],{"title":7925,"path":7924,"lessonNumber":9213,"topics":9214,"summary":7926},10,[9192],{"title":7929,"path":7928,"lessonNumber":9216,"topics":9217,"summary":7930},11,[9192],{"title":7933,"path":7932,"lessonNumber":9219,"topics":9220,"summary":7934},12,[9192],{"module":7938,"moduleNumber":3540,"slug":9222,"lessons":9223},"approximation",[9224,9227,9229,9231,9233,9235,9237,9239,9241,9243,9245,9247,9249,9252],{"title":7937,"path":7936,"lessonNumber":9170,"topics":9225,"summary":7939},[9226],"Approximation",{"title":7942,"path":7941,"lessonNumber":3535,"topics":9228,"summary":7943},[9226],{"title":7946,"path":7945,"lessonNumber":3540,"topics":9230,"summary":7947},[9226],{"title":7950,"path":7949,"lessonNumber":9180,"topics":9232,"summary":7951},[9226],{"title":7954,"path":7953,"lessonNumber":9183,"topics":9234,"summary":7955},[9226],{"title":7958,"path":7957,"lessonNumber":3552,"topics":9236,"summary":7959},[9226],{"title":7962,"path":7961,"lessonNumber":9204,"topics":9238,"summary":7963},[9226],{"title":7966,"path":7965,"lessonNumber":9207,"topics":9240,"summary":7967},[9226],{"title":7970,"path":7969,"lessonNumber":9210,"topics":9242,"summary":7971},[9226],{"title":7974,"path":7973,"lessonNumber":9213,"topics":9244,"summary":7975},[9226],{"title":7978,"path":7977,"lessonNumber":9216,"topics":9246,"summary":7979},[9226],{"title":7982,"path":7981,"lessonNumber":9219,"topics":9248,"summary":7983},[9226],{"title":7986,"path":7985,"lessonNumber":9250,"topics":9251,"summary":7987},13,[9226],{"title":7990,"path":7989,"lessonNumber":9253,"topics":9254,"summary":7991},14,[9226],{"module":7994,"moduleNumber":9180,"slug":9256,"lessons":9257},"deep-rl",[9258,9261,9263,9265,9267,9269,9271,9273],{"title":7312,"path":7993,"lessonNumber":9170,"topics":9259,"summary":7995},[9260],"Deep RL",{"title":7998,"path":7997,"lessonNumber":3535,"topics":9262,"summary":7999},[9260],{"title":8002,"path":8001,"lessonNumber":3540,"topics":9264,"summary":8003},[9260],{"title":8006,"path":8005,"lessonNumber":9180,"topics":9266,"summary":8007},[9260],{"title":8010,"path":8009,"lessonNumber":9183,"topics":9268,"summary":8011},[9260],{"title":8014,"path":8013,"lessonNumber":3552,"topics":9270,"summary":8015},[9260],{"title":8018,"path":8017,"lessonNumber":9204,"topics":9272,"summary":8019},[9260],{"title":8022,"path":8021,"lessonNumber":9207,"topics":9274,"summary":8023},[9260],{"module":8027,"moduleNumber":9183,"slug":9276,"lessons":9277},"modern-deep-rl",[9278,9280,9282,9284,9286,9288,9290,9292,9294,9296,9298,9300,9302,9304,9306,9309,9312,9315,9318,9321,9324,9327],{"title":8026,"path":8025,"lessonNumber":9170,"topics":9279,"summary":8028},[9260],{"title":8031,"path":8030,"lessonNumber":3535,"topics":9281,"summary":8032},[9260],{"title":8035,"path":8034,"lessonNumber":3540,"topics":9283,"summary":8036},[9260],{"title":8039,"path":8038,"lessonNumber":9180,"topics":9285,"summary":8040},[9260],{"title":8042,"path":3082,"lessonNumber":9183,"topics":9287,"summary":8043},[9260],{"title":8046,"path":8045,"lessonNumber":3552,"topics":9289,"summary":8047},[9260],{"title":8050,"path":8049,"lessonNumber":9204,"topics":9291,"summary":8051},[9260],{"title":8054,"path":8053,"lessonNumber":9207,"topics":9293,"summary":8055},[9260],{"title":8058,"path":8057,"lessonNumber":9210,"topics":9295,"summary":8059},[9260],{"title":8062,"path":8061,"lessonNumber":9213,"topics":9297,"summary":8063},[9260],{"title":8066,"path":8065,"lessonNumber":9216,"topics":9299,"summary":8067},[9260],{"title":8070,"path":8069,"lessonNumber":9219,"topics":9301,"summary":8071},[9260],{"title":8074,"path":8073,"lessonNumber":9250,"topics":9303,"summary":8075},[9260],{"title":8078,"path":8077,"lessonNumber":9253,"topics":9305,"summary":8079},[9260],{"title":8082,"path":8081,"lessonNumber":9307,"topics":9308,"summary":8083},15,[9260],{"title":8086,"path":8085,"lessonNumber":9310,"topics":9311,"summary":8087},16,[9260],{"title":8090,"path":8089,"lessonNumber":9313,"topics":9314,"summary":8091},17,[9260],{"title":8094,"path":8093,"lessonNumber":9316,"topics":9317,"summary":8095},18,[9260],{"title":8098,"path":8097,"lessonNumber":9319,"topics":9320,"summary":8099},19,[9260],{"title":8102,"path":8101,"lessonNumber":9322,"topics":9323,"summary":8103},20,[9260],{"title":8106,"path":8105,"lessonNumber":9325,"topics":9326,"summary":8107},21,[9260],{"title":8110,"path":8109,"lessonNumber":9328,"topics":9329,"summary":8111},22,[9260],{"module":3554,"moduleNumber":3552,"slug":9331,"lessons":9332},"minds-and-brains",[9333,9335,9337,9339,9341,9343,9345,9347],{"title":8113,"path":1269,"lessonNumber":9170,"topics":9334,"summary":8114},[3572],{"title":8117,"path":8116,"lessonNumber":3535,"topics":9336,"summary":8118},[3572],{"title":8120,"path":3186,"lessonNumber":3540,"topics":9338,"summary":8121},[3572],{"title":8124,"path":8123,"lessonNumber":9180,"topics":9340,"summary":8125},[3572],{"title":18,"path":17,"lessonNumber":9183,"topics":9342,"summary":8127},[3572],{"title":5,"path":3556,"lessonNumber":3552,"topics":9344,"summary":3570},[3572],{"title":8131,"path":8130,"lessonNumber":9204,"topics":9346,"summary":8132},[3572],{"title":8135,"path":8134,"lessonNumber":9207,"topics":9348,"summary":8136},[3572],"\u003Csvg style=\"width:100%;max-width:391.204px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 293.403 120.758\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-37.36 36.043h208.55\"\u002F>\u003Cpath stroke=\"none\" d=\"m173.19 36.043-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(214.083 1.75)\">\u003Cpath d=\"M-37.046 34.532Q-37.046 34.194-36.905 33.903Q-36.765 33.613-36.521 33.399Q-36.277 33.186-35.972 33.071Q-35.668 32.957-35.343 32.957Q-35.073 32.957-34.810 33.056Q-34.547 33.155-34.356 33.333L-34.356 31.935Q-34.356 31.665-34.463 31.603Q-34.571 31.542-34.882 31.542L-34.882 31.261L-33.805 31.186L-33.805 35.370Q-33.805 35.558-33.751 35.641Q-33.696 35.725-33.595 35.744Q-33.494 35.763-33.279 35.763L-33.279 36.043L-34.386 36.111L-34.386 35.694Q-34.803 36.111-35.429 36.111Q-35.860 36.111-36.232 35.899Q-36.605 35.688-36.825 35.327Q-37.046 34.966-37.046 34.532M-35.371 35.889Q-35.162 35.889-34.976 35.817Q-34.790 35.746-34.636 35.609Q-34.482 35.472-34.386 35.294L-34.386 33.685Q-34.472 33.538-34.617 33.418Q-34.762 33.298-34.932 33.239Q-35.101 33.179-35.282 33.179Q-35.842 33.179-36.111 33.568Q-36.379 33.958-36.379 34.539Q-36.379 35.110-36.145 35.500Q-35.911 35.889-35.371 35.889M-32.571 35.315Q-32.571 34.983-32.348 34.756Q-32.124 34.529-31.780 34.401Q-31.437 34.272-31.064 34.220Q-30.692 34.167-30.387 34.167L-30.387 33.914Q-30.387 33.709-30.495 33.529Q-30.603 33.350-30.784 33.247Q-30.965 33.145-31.173 33.145Q-31.580 33.145-31.816 33.237Q-31.727 33.274-31.681 33.358Q-31.635 33.442-31.635 33.544Q-31.635 33.640-31.681 33.719Q-31.727 33.797-31.808 33.842Q-31.888 33.886-31.977 33.886Q-32.127 33.886-32.228 33.789Q-32.329 33.691-32.329 33.544Q-32.329 32.922-31.173 32.922Q-30.962 32.922-30.712 32.986Q-30.463 33.049-30.261 33.168Q-30.059 33.288-29.933 33.473Q-29.806 33.657-29.806 33.900L-29.806 35.476Q-29.806 35.592-29.745 35.688Q-29.683 35.783-29.570 35.783Q-29.461 35.783-29.396 35.689Q-29.331 35.595-29.331 35.476L-29.331 35.028L-29.065 35.028L-29.065 35.476Q-29.065 35.746-29.292 35.911Q-29.519 36.077-29.799 36.077Q-30.008 36.077-30.145 35.923Q-30.281 35.770-30.305 35.554Q-30.452 35.821-30.734 35.966Q-31.016 36.111-31.341 36.111Q-31.618 36.111-31.902 36.036Q-32.185 35.961-32.378 35.782Q-32.571 35.602-32.571 35.315M-31.956 35.315Q-31.956 35.489-31.855 35.619Q-31.755 35.749-31.599 35.819Q-31.443 35.889-31.279 35.889Q-31.061 35.889-30.852 35.792Q-30.644 35.694-30.516 35.513Q-30.387 35.332-30.387 35.106L-30.387 34.378Q-30.712 34.378-31.078 34.469Q-31.443 34.560-31.700 34.772Q-31.956 34.983-31.956 35.315\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(214.083 1.75)\">\u003Cpath d=\"M-28.523 37.178Q-28.393 37.246-28.256 37.246Q-28.085 37.246-27.935 37.157Q-27.784 37.068-27.673 36.923Q-27.562 36.778-27.484 36.610L-27.220 36.043L-28.389 33.517Q-28.464 33.370-28.594 33.338Q-28.724 33.305-28.957 33.305L-28.957 33.025L-27.436 33.025L-27.436 33.305Q-27.784 33.305-27.784 33.452Q-27.781 33.473-27.779 33.490Q-27.777 33.507-27.777 33.517L-26.920 35.376L-26.147 33.705Q-26.113 33.637-26.113 33.558Q-26.113 33.445-26.197 33.375Q-26.280 33.305-26.393 33.305L-26.393 33.025L-25.197 33.025L-25.197 33.305Q-25.416 33.305-25.588 33.409Q-25.761 33.514-25.853 33.705L-27.190 36.610Q-27.360 36.980-27.630 37.226Q-27.901 37.472-28.256 37.472Q-28.526 37.472-28.745 37.306Q-28.964 37.140-28.964 36.877Q-28.964 36.740-28.871 36.651Q-28.779 36.563-28.639 36.563Q-28.502 36.563-28.413 36.651Q-28.324 36.740-28.324 36.877Q-28.324 36.980-28.377 37.058Q-28.430 37.137-28.523 37.178M-24.657 36.036L-24.657 34.973Q-24.657 34.949-24.630 34.922Q-24.602 34.895-24.578 34.895L-24.469 34.895Q-24.404 34.895-24.390 34.953Q-24.295 35.387-24.048 35.638Q-23.802 35.889-23.389 35.889Q-23.047 35.889-22.794 35.756Q-22.541 35.623-22.541 35.315Q-22.541 35.158-22.635 35.043Q-22.729 34.929-22.868 34.860Q-23.006 34.792-23.173 34.754L-23.755 34.655Q-24.110 34.587-24.383 34.366Q-24.657 34.146-24.657 33.804Q-24.657 33.555-24.546 33.380Q-24.435 33.206-24.248 33.107Q-24.062 33.008-23.847 32.965Q-23.631 32.922-23.389 32.922Q-22.975 32.922-22.695 33.104L-22.480 32.929Q-22.469 32.926-22.463 32.924Q-22.456 32.922-22.445 32.922L-22.394 32.922Q-22.367 32.922-22.343 32.946Q-22.319 32.970-22.319 32.998L-22.319 33.845Q-22.319 33.866-22.343 33.893Q-22.367 33.920-22.394 33.920L-22.507 33.920Q-22.534 33.920-22.560 33.895Q-22.586 33.869-22.586 33.845Q-22.586 33.609-22.692 33.445Q-22.797 33.281-22.980 33.199Q-23.163 33.117-23.396 33.117Q-23.724 33.117-23.980 33.220Q-24.236 33.322-24.236 33.599Q-24.236 33.794-24.054 33.903Q-23.871 34.013-23.642 34.054L-23.068 34.160Q-22.821 34.208-22.608 34.336Q-22.394 34.464-22.257 34.667Q-22.121 34.871-22.121 35.120Q-22.121 35.633-22.486 35.872Q-22.852 36.111-23.389 36.111Q-23.884 36.111-24.216 35.817L-24.483 36.091Q-24.503 36.111-24.530 36.111L-24.578 36.111Q-24.602 36.111-24.630 36.084Q-24.657 36.057-24.657 36.036\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-37.36 36.043v-94.739\"\u002F>\u003Cpath stroke=\"none\" d=\"m-37.36-60.696-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-24.91 -100.272)\">\u003Cpath d=\"M-37.087 34.508Q-37.087 34.187-36.962 33.898Q-36.837 33.609-36.611 33.386Q-36.386 33.162-36.090 33.042Q-35.795 32.922-35.477 32.922Q-35.149 32.922-34.887 33.022Q-34.626 33.121-34.450 33.303Q-34.274 33.486-34.180 33.744Q-34.086 34.002-34.086 34.334Q-34.086 34.426-34.168 34.447L-36.423 34.447L-36.423 34.508Q-36.423 35.096-36.140 35.479Q-35.856 35.862-35.289 35.862Q-34.967 35.862-34.699 35.669Q-34.431 35.476-34.342 35.161Q-34.335 35.120-34.260 35.106L-34.168 35.106Q-34.086 35.130-34.086 35.202Q-34.086 35.209-34.092 35.236Q-34.205 35.633-34.576 35.872Q-34.947 36.111-35.371 36.111Q-35.808 36.111-36.208 35.903Q-36.608 35.694-36.847 35.327Q-37.087 34.960-37.087 34.508M-36.417 34.238L-34.602 34.238Q-34.602 33.961-34.699 33.709Q-34.797 33.456-34.995 33.300Q-35.193 33.145-35.477 33.145Q-35.754 33.145-35.967 33.303Q-36.181 33.462-36.299 33.717Q-36.417 33.972-36.417 34.238M-31.748 36.043L-33.484 36.043L-33.484 35.763Q-33.255 35.763-33.106 35.729Q-32.958 35.694-32.958 35.554L-32.958 33.705Q-32.958 33.435-33.065 33.374Q-33.173 33.312-33.484 33.312L-33.484 33.032L-32.455 32.957L-32.455 33.664Q-32.325 33.356-32.083 33.157Q-31.840 32.957-31.522 32.957Q-31.303 32.957-31.132 33.081Q-30.962 33.206-30.962 33.418Q-30.962 33.555-31.061 33.654Q-31.160 33.753-31.293 33.753Q-31.430 33.753-31.529 33.654Q-31.628 33.555-31.628 33.418Q-31.628 33.278-31.529 33.179Q-31.819 33.179-32.019 33.375Q-32.219 33.572-32.312 33.866Q-32.404 34.160-32.404 34.440L-32.404 35.554Q-32.404 35.763-31.748 35.763L-31.748 36.043M-28.627 36.043L-30.363 36.043L-30.363 35.763Q-30.134 35.763-29.986 35.729Q-29.837 35.694-29.837 35.554L-29.837 33.705Q-29.837 33.435-29.945 33.374Q-30.052 33.312-30.363 33.312L-30.363 33.032L-29.335 32.957L-29.335 33.664Q-29.205 33.356-28.962 33.157Q-28.719 32.957-28.402 32.957Q-28.183 32.957-28.012 33.081Q-27.841 33.206-27.841 33.418Q-27.841 33.555-27.940 33.654Q-28.039 33.753-28.172 33.753Q-28.309 33.753-28.408 33.654Q-28.507 33.555-28.507 33.418Q-28.507 33.278-28.408 33.179Q-28.699 33.179-28.899 33.375Q-29.099 33.572-29.191 33.866Q-29.283 34.160-29.283 34.440L-29.283 35.554Q-29.283 35.763-28.627 35.763L-28.627 36.043M-27.297 34.560Q-27.297 34.218-27.162 33.919Q-27.027 33.620-26.788 33.396Q-26.549 33.172-26.231 33.047Q-25.913 32.922-25.582 32.922Q-25.137 32.922-24.737 33.138Q-24.338 33.353-24.103 33.731Q-23.869 34.108-23.869 34.560Q-23.869 34.901-24.011 35.185Q-24.153 35.469-24.397 35.676Q-24.642 35.882-24.951 35.997Q-25.260 36.111-25.582 36.111Q-26.012 36.111-26.414 35.910Q-26.816 35.708-27.057 35.356Q-27.297 35.004-27.297 34.560M-25.582 35.862Q-24.980 35.862-24.756 35.484Q-24.532 35.106-24.532 34.474Q-24.532 33.862-24.766 33.503Q-25.001 33.145-25.582 33.145Q-26.634 33.145-26.634 34.474Q-26.634 35.106-26.409 35.484Q-26.183 35.862-25.582 35.862M-21.525 36.043L-23.261 36.043L-23.261 35.763Q-23.032 35.763-22.883 35.729Q-22.735 35.694-22.735 35.554L-22.735 33.705Q-22.735 33.435-22.842 33.374Q-22.950 33.312-23.261 33.312L-23.261 33.032L-22.232 32.957L-22.232 33.664Q-22.102 33.356-21.860 33.157Q-21.617 32.957-21.299 32.957Q-21.080 32.957-20.909 33.081Q-20.738 33.206-20.738 33.418Q-20.738 33.555-20.838 33.654Q-20.937 33.753-21.070 33.753Q-21.207 33.753-21.306 33.654Q-21.405 33.555-21.405 33.418Q-21.405 33.278-21.306 33.179Q-21.596 33.179-21.796 33.375Q-21.996 33.572-22.089 33.866Q-22.181 34.160-22.181 34.440L-22.181 35.554Q-22.181 35.763-21.525 35.763L-21.525 36.043M-20.154 36.036L-20.154 34.973Q-20.154 34.949-20.127 34.922Q-20.099 34.895-20.075 34.895L-19.966 34.895Q-19.901 34.895-19.887 34.953Q-19.792 35.387-19.546 35.638Q-19.299 35.889-18.886 35.889Q-18.544 35.889-18.291 35.756Q-18.038 35.623-18.038 35.315Q-18.038 35.158-18.132 35.043Q-18.226 34.929-18.365 34.860Q-18.503 34.792-18.671 34.754L-19.252 34.655Q-19.607 34.587-19.881 34.366Q-20.154 34.146-20.154 33.804Q-20.154 33.555-20.043 33.380Q-19.932 33.206-19.745 33.107Q-19.559 33.008-19.344 32.965Q-19.129 32.922-18.886 32.922Q-18.472 32.922-18.192 33.104L-17.977 32.929Q-17.966 32.926-17.960 32.924Q-17.953 32.922-17.943 32.922L-17.891 32.922Q-17.864 32.922-17.840 32.946Q-17.816 32.970-17.816 32.998L-17.816 33.845Q-17.816 33.866-17.840 33.893Q-17.864 33.920-17.891 33.920L-18.004 33.920Q-18.031 33.920-18.057 33.895Q-18.083 33.869-18.083 33.845Q-18.083 33.609-18.189 33.445Q-18.295 33.281-18.477 33.199Q-18.660 33.117-18.893 33.117Q-19.221 33.117-19.477 33.220Q-19.734 33.322-19.734 33.599Q-19.734 33.794-19.551 33.903Q-19.368 34.013-19.139 34.054L-18.565 34.160Q-18.318 34.208-18.105 34.336Q-17.891 34.464-17.755 34.667Q-17.618 34.871-17.618 35.120Q-17.618 35.633-17.984 35.872Q-18.349 36.111-18.886 36.111Q-19.381 36.111-19.713 35.817L-19.980 36.091Q-20 36.111-20.027 36.111L-20.075 36.111Q-20.099 36.111-20.127 36.084Q-20.154 36.057-20.154 36.036\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-24.91 -100.272)\">\u003Cpath d=\"M-12.650 36.043L-14.202 36.043L-14.202 35.763Q-13.976 35.763-13.827 35.729Q-13.679 35.694-13.679 35.554L-13.679 33.705Q-13.679 33.517-13.727 33.433Q-13.774 33.350-13.872 33.331Q-13.969 33.312-14.181 33.312L-14.181 33.032L-13.125 32.957L-13.125 35.554Q-13.125 35.694-12.993 35.729Q-12.862 35.763-12.650 35.763L-12.650 36.043M-13.921 31.736Q-13.921 31.565-13.798 31.446Q-13.675 31.326-13.504 31.326Q-13.337 31.326-13.214 31.446Q-13.091 31.565-13.091 31.736Q-13.091 31.911-13.214 32.034Q-13.337 32.157-13.504 32.157Q-13.675 32.157-13.798 32.034Q-13.921 31.911-13.921 31.736M-10.322 36.043L-11.956 36.043L-11.956 35.763Q-11.727 35.763-11.578 35.729Q-11.430 35.694-11.430 35.554L-11.430 33.705Q-11.430 33.435-11.537 33.374Q-11.645 33.312-11.956 33.312L-11.956 33.032L-10.896 32.957L-10.896 33.606Q-10.726 33.298-10.421 33.127Q-10.117 32.957-9.772 32.957Q-9.266 32.957-8.982 33.180Q-8.699 33.404-8.699 33.900L-8.699 35.554Q-8.699 35.691-8.550 35.727Q-8.401 35.763-8.176 35.763L-8.176 36.043L-9.806 36.043L-9.806 35.763Q-9.577 35.763-9.428 35.729Q-9.280 35.694-9.280 35.554L-9.280 33.914Q-9.280 33.579-9.399 33.379Q-9.519 33.179-9.833 33.179Q-10.103 33.179-10.338 33.315Q-10.572 33.452-10.710 33.686Q-10.849 33.920-10.849 34.194L-10.849 35.554Q-10.849 35.691-10.698 35.727Q-10.548 35.763-10.322 35.763\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-24.91 -100.272)\">\u003Cpath d=\"M-3.209 36.043L-4.843 36.043L-4.843 35.763Q-4.614 35.763-4.465 35.729Q-4.316 35.694-4.316 35.554L-4.316 33.705Q-4.316 33.435-4.424 33.374Q-4.532 33.312-4.843 33.312L-4.843 33.032L-3.783 32.957L-3.783 33.606Q-3.612 33.298-3.308 33.127Q-3.004 32.957-2.659 32.957Q-2.259 32.957-1.982 33.097Q-1.705 33.237-1.620 33.585Q-1.452 33.292-1.153 33.124Q-0.854 32.957-0.509 32.957Q-0.003 32.957 0.281 33.180Q0.565 33.404 0.565 33.900L0.565 35.554Q0.565 35.691 0.713 35.727Q0.862 35.763 1.087 35.763L1.087 36.043L-0.543 36.043L-0.543 35.763Q-0.317 35.763-0.167 35.727Q-0.017 35.691-0.017 35.554L-0.017 33.914Q-0.017 33.579-0.136 33.379Q-0.256 33.179-0.570 33.179Q-0.840 33.179-1.074 33.315Q-1.309 33.452-1.447 33.686Q-1.585 33.920-1.585 34.194L-1.585 35.554Q-1.585 35.691-1.437 35.727Q-1.288 35.763-1.062 35.763L-1.062 36.043L-2.693 36.043L-2.693 35.763Q-2.464 35.763-2.315 35.729Q-2.166 35.694-2.166 35.554L-2.166 33.914Q-2.166 33.579-2.286 33.379Q-2.406 33.179-2.720 33.179Q-2.990 33.179-3.224 33.315Q-3.458 33.452-3.597 33.686Q-3.735 33.920-3.735 34.194L-3.735 35.554Q-3.735 35.691-3.585 35.727Q-3.434 35.763-3.209 35.763L-3.209 36.043M1.733 35.315Q1.733 34.983 1.957 34.756Q2.181 34.529 2.525 34.401Q2.868 34.272 3.241 34.220Q3.613 34.167 3.918 34.167L3.918 33.914Q3.918 33.709 3.810 33.529Q3.702 33.350 3.521 33.247Q3.340 33.145 3.131 33.145Q2.725 33.145 2.489 33.237Q2.578 33.274 2.624 33.358Q2.670 33.442 2.670 33.544Q2.670 33.640 2.624 33.719Q2.578 33.797 2.497 33.842Q2.417 33.886 2.328 33.886Q2.178 33.886 2.077 33.789Q1.976 33.691 1.976 33.544Q1.976 32.922 3.131 32.922Q3.343 32.922 3.593 32.986Q3.842 33.049 4.044 33.168Q4.246 33.288 4.372 33.473Q4.499 33.657 4.499 33.900L4.499 35.476Q4.499 35.592 4.560 35.688Q4.622 35.783 4.734 35.783Q4.844 35.783 4.909 35.689Q4.974 35.595 4.974 35.476L4.974 35.028L5.240 35.028L5.240 35.476Q5.240 35.746 5.013 35.911Q4.786 36.077 4.505 36.077Q4.297 36.077 4.160 35.923Q4.024 35.770 4 35.554Q3.853 35.821 3.571 35.966Q3.289 36.111 2.964 36.111Q2.687 36.111 2.403 36.036Q2.120 35.961 1.927 35.782Q1.733 35.602 1.733 35.315M2.349 35.315Q2.349 35.489 2.450 35.619Q2.550 35.749 2.706 35.819Q2.861 35.889 3.025 35.889Q3.244 35.889 3.453 35.792Q3.661 35.694 3.789 35.513Q3.918 35.332 3.918 35.106L3.918 34.378Q3.593 34.378 3.227 34.469Q2.861 34.560 2.605 34.772Q2.349 34.983 2.349 35.315M8.388 36.043L5.715 36.043Q5.671 36.043 5.644 36.016Q5.616 35.988 5.616 35.944L5.616 35.876Q5.616 35.835 5.644 35.804L7.753 33.251L7.113 33.251Q6.690 33.251 6.457 33.317Q6.225 33.384 6.105 33.594Q5.985 33.804 5.985 34.225L5.722 34.225L5.804 33.025L8.395 33.025Q8.436 33.025 8.465 33.052Q8.494 33.080 8.494 33.124L8.494 33.172Q8.494 33.216 8.470 33.244L6.365 35.790L7.045 35.790Q7.373 35.790 7.590 35.754Q7.807 35.718 7.975 35.568Q8.115 35.431 8.168 35.207Q8.221 34.983 8.248 34.655L8.515 34.655L8.388 36.043M9.164 34.508Q9.164 34.187 9.289 33.898Q9.414 33.609 9.639 33.386Q9.865 33.162 10.160 33.042Q10.456 32.922 10.774 32.922Q11.102 32.922 11.364 33.022Q11.625 33.121 11.801 33.303Q11.977 33.486 12.071 33.744Q12.165 34.002 12.165 34.334Q12.165 34.426 12.083 34.447L9.827 34.447L9.827 34.508Q9.827 35.096 10.111 35.479Q10.395 35.862 10.962 35.862Q11.283 35.862 11.552 35.669Q11.820 35.476 11.909 35.161Q11.916 35.120 11.991 35.106L12.083 35.106Q12.165 35.130 12.165 35.202Q12.165 35.209 12.158 35.236Q12.045 35.633 11.675 35.872Q11.304 36.111 10.880 36.111Q10.442 36.111 10.043 35.903Q9.643 35.694 9.403 35.327Q9.164 34.960 9.164 34.508M9.834 34.238L11.649 34.238Q11.649 33.961 11.552 33.709Q11.454 33.456 11.256 33.300Q11.058 33.145 10.774 33.145Q10.497 33.145 10.284 33.303Q10.070 33.462 9.952 33.717Q9.834 33.972 9.834 34.238\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M87.832 36.043v-91.048\" style=\"stroke-dasharray:0.4,2.0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-35.248 36.043L-36.981 36.043L-36.981 35.763Q-36.755 35.763-36.606 35.729Q-36.458 35.694-36.458 35.554L-36.458 33.305L-37.046 33.305L-37.046 33.025L-36.458 33.025L-36.458 32.208Q-36.458 31.890-36.280 31.642Q-36.102 31.395-35.812 31.254Q-35.521 31.114-35.210 31.114Q-34.954 31.114-34.750 31.256Q-34.547 31.398-34.547 31.641Q-34.547 31.777-34.646 31.876Q-34.745 31.976-34.882 31.976Q-35.019 31.976-35.118 31.876Q-35.217 31.777-35.217 31.641Q-35.217 31.460-35.077 31.367Q-35.155 31.340-35.255 31.340Q-35.463 31.340-35.617 31.473Q-35.771 31.606-35.851 31.810Q-35.931 32.013-35.931 32.222L-35.931 33.025L-35.043 33.025L-35.043 33.305L-35.904 33.305L-35.904 35.554Q-35.904 35.763-35.248 35.763L-35.248 36.043M-34.609 34.560Q-34.609 34.218-34.474 33.919Q-34.339 33.620-34.099 33.396Q-33.860 33.172-33.542 33.047Q-33.224 32.922-32.893 32.922Q-32.448 32.922-32.048 33.138Q-31.649 33.353-31.414 33.731Q-31.180 34.108-31.180 34.560Q-31.180 34.901-31.322 35.185Q-31.464 35.469-31.708 35.676Q-31.953 35.882-32.262 35.997Q-32.571 36.111-32.893 36.111Q-33.323 36.111-33.725 35.910Q-34.127 35.708-34.368 35.356Q-34.609 35.004-34.609 34.560M-32.893 35.862Q-32.291 35.862-32.067 35.484Q-31.843 35.106-31.843 34.474Q-31.843 33.862-32.078 33.503Q-32.312 33.145-32.893 33.145Q-33.945 33.145-33.945 34.474Q-33.945 35.106-33.720 35.484Q-33.494 35.862-32.893 35.862\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-30.406 34.560Q-30.406 34.218-30.271 33.919Q-30.136 33.620-29.896 33.396Q-29.657 33.172-29.339 33.047Q-29.021 32.922-28.690 32.922Q-28.245 32.922-27.846 33.138Q-27.446 33.353-27.211 33.731Q-26.977 34.108-26.977 34.560Q-26.977 34.901-27.119 35.185Q-27.261 35.469-27.505 35.676Q-27.750 35.882-28.059 35.997Q-28.368 36.111-28.690 36.111Q-29.120 36.111-29.522 35.910Q-29.924 35.708-30.165 35.356Q-30.406 35.004-30.406 34.560M-28.690 35.862Q-28.088 35.862-27.864 35.484Q-27.640 35.106-27.640 34.474Q-27.640 33.862-27.875 33.503Q-28.109 33.145-28.690 33.145Q-29.742 33.145-29.742 34.474Q-29.742 35.106-29.517 35.484Q-29.291 35.862-28.690 35.862\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-26.164 34.532Q-26.164 34.194-26.023 33.903Q-25.883 33.613-25.639 33.399Q-25.395 33.186-25.090 33.071Q-24.786 32.957-24.461 32.957Q-24.191 32.957-23.928 33.056Q-23.665 33.155-23.474 33.333L-23.474 31.935Q-23.474 31.665-23.581 31.603Q-23.689 31.542-24 31.542L-24 31.261L-22.923 31.186L-22.923 35.370Q-22.923 35.558-22.869 35.641Q-22.814 35.725-22.713 35.744Q-22.612 35.763-22.397 35.763L-22.397 36.043L-23.504 36.111L-23.504 35.694Q-23.921 36.111-24.547 36.111Q-24.978 36.111-25.350 35.899Q-25.723 35.688-25.943 35.327Q-26.164 34.966-26.164 34.532M-24.489 35.889Q-24.280 35.889-24.094 35.817Q-23.908 35.746-23.754 35.609Q-23.600 35.472-23.504 35.294L-23.504 33.685Q-23.590 33.538-23.735 33.418Q-23.880 33.298-24.050 33.239Q-24.219 33.179-24.400 33.179Q-24.960 33.179-25.229 33.568Q-25.497 33.958-25.497 34.539Q-25.497 35.110-25.263 35.500Q-25.029 35.889-24.489 35.889\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-17.436 36.043L-18.988 36.043L-18.988 35.763Q-18.762 35.763-18.613 35.729Q-18.465 35.694-18.465 35.554L-18.465 33.705Q-18.465 33.517-18.513 33.433Q-18.560 33.350-18.658 33.331Q-18.755 33.312-18.967 33.312L-18.967 33.032L-17.911 32.957L-17.911 35.554Q-17.911 35.694-17.779 35.729Q-17.648 35.763-17.436 35.763L-17.436 36.043M-18.707 31.736Q-18.707 31.565-18.584 31.446Q-18.461 31.326-18.290 31.326Q-18.123 31.326-18 31.446Q-17.877 31.565-17.877 31.736Q-17.877 31.911-18 32.034Q-18.123 32.157-18.290 32.157Q-18.461 32.157-18.584 32.034Q-18.707 31.911-18.707 31.736M-15.108 36.043L-16.742 36.043L-16.742 35.763Q-16.513 35.763-16.364 35.729Q-16.216 35.694-16.216 35.554L-16.216 33.705Q-16.216 33.435-16.323 33.374Q-16.431 33.312-16.742 33.312L-16.742 33.032L-15.682 32.957L-15.682 33.606Q-15.512 33.298-15.207 33.127Q-14.903 32.957-14.558 32.957Q-14.052 32.957-13.768 33.180Q-13.485 33.404-13.485 33.900L-13.485 35.554Q-13.485 35.691-13.336 35.727Q-13.187 35.763-12.962 35.763L-12.962 36.043L-14.592 36.043L-14.592 35.763Q-14.363 35.763-14.214 35.729Q-14.066 35.694-14.066 35.554L-14.066 33.914Q-14.066 33.579-14.185 33.379Q-14.305 33.179-14.619 33.179Q-14.889 33.179-15.124 33.315Q-15.358 33.452-15.496 33.686Q-15.635 33.920-15.635 34.194L-15.635 35.554Q-15.635 35.691-15.484 35.727Q-15.334 35.763-15.108 35.763\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-12.060 35.202L-12.060 33.305L-12.699 33.305L-12.699 33.083Q-12.381 33.083-12.164 32.873Q-11.947 32.663-11.847 32.353Q-11.746 32.044-11.746 31.736L-11.479 31.736L-11.479 33.025L-10.402 33.025L-10.402 33.305L-11.479 33.305L-11.479 35.189Q-11.479 35.465-11.375 35.664Q-11.271 35.862-11.011 35.862Q-10.854 35.862-10.748 35.758Q-10.642 35.653-10.592 35.500Q-10.543 35.346-10.543 35.189L-10.543 34.775L-10.276 34.775L-10.276 35.202Q-10.276 35.428-10.375 35.638Q-10.474 35.848-10.659 35.980Q-10.843 36.111-11.072 36.111Q-11.510 36.111-11.785 35.874Q-12.060 35.636-12.060 35.202M-7.716 36.043L-9.452 36.043L-9.452 35.763Q-9.223 35.763-9.075 35.729Q-8.926 35.694-8.926 35.554L-8.926 33.705Q-8.926 33.435-9.034 33.374Q-9.141 33.312-9.452 33.312L-9.452 33.032L-8.423 32.957L-8.423 33.664Q-8.294 33.356-8.051 33.157Q-7.808 32.957-7.490 32.957Q-7.272 32.957-7.101 33.081Q-6.930 33.206-6.930 33.418Q-6.930 33.555-7.029 33.654Q-7.128 33.753-7.261 33.753Q-7.398 33.753-7.497 33.654Q-7.596 33.555-7.596 33.418Q-7.596 33.278-7.497 33.179Q-7.788 33.179-7.988 33.375Q-8.188 33.572-8.280 33.866Q-8.372 34.160-8.372 34.440L-8.372 35.554Q-8.372 35.763-7.716 35.763L-7.716 36.043M-6.386 34.560Q-6.386 34.218-6.251 33.919Q-6.116 33.620-5.877 33.396Q-5.638 33.172-5.320 33.047Q-5.002 32.922-4.671 32.922Q-4.226 32.922-3.826 33.138Q-3.426 33.353-3.192 33.731Q-2.958 34.108-2.958 34.560Q-2.958 34.901-3.100 35.185Q-3.242 35.469-3.486 35.676Q-3.731 35.882-4.040 35.997Q-4.349 36.111-4.671 36.111Q-5.101 36.111-5.503 35.910Q-5.904 35.708-6.145 35.356Q-6.386 35.004-6.386 34.560M-4.671 35.862Q-4.069 35.862-3.845 35.484Q-3.621 35.106-3.621 34.474Q-3.621 33.862-3.855 33.503Q-4.089 33.145-4.671 33.145Q-5.723 33.145-5.723 34.474Q-5.723 35.106-5.498 35.484Q-5.272 35.862-4.671 35.862\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(97.556 -93.159)\">\u003Cpath d=\"M-2.136 34.532Q-2.136 34.194-1.995 33.903Q-1.855 33.613-1.611 33.399Q-1.367 33.186-1.062 33.071Q-0.758 32.957-0.433 32.957Q-0.163 32.957 0.100 33.056Q0.363 33.155 0.554 33.333L0.554 31.935Q0.554 31.665 0.447 31.603Q0.339 31.542 0.028 31.542L0.028 31.261L1.105 31.186L1.105 35.370Q1.105 35.558 1.159 35.641Q1.214 35.725 1.315 35.744Q1.416 35.763 1.631 35.763L1.631 36.043L0.524 36.111L0.524 35.694Q0.107 36.111-0.519 36.111Q-0.950 36.111-1.322 35.899Q-1.695 35.688-1.915 35.327Q-2.136 34.966-2.136 34.532M-0.461 35.889Q-0.252 35.889-0.066 35.817Q0.120 35.746 0.274 35.609Q0.428 35.472 0.524 35.294L0.524 33.685Q0.438 33.538 0.293 33.418Q0.148 33.298-0.022 33.239Q-0.191 33.179-0.372 33.179Q-0.932 33.179-1.201 33.568Q-1.469 33.958-1.469 34.539Q-1.469 35.110-1.235 35.500Q-1.001 35.889-0.461 35.889M2.855 35.209L2.855 33.705Q2.855 33.435 2.747 33.374Q2.639 33.312 2.328 33.312L2.328 33.032L3.436 32.957L3.436 35.189L3.436 35.209Q3.436 35.489 3.487 35.633Q3.538 35.776 3.680 35.833Q3.822 35.889 4.109 35.889Q4.362 35.889 4.567 35.749Q4.772 35.609 4.888 35.383Q5.005 35.158 5.005 34.908L5.005 33.705Q5.005 33.435 4.897 33.374Q4.789 33.312 4.478 33.312L4.478 33.032L5.586 32.957L5.586 35.370Q5.586 35.561 5.639 35.643Q5.692 35.725 5.792 35.744Q5.893 35.763 6.109 35.763L6.109 36.043L5.032 36.111L5.032 35.547Q4.923 35.729 4.777 35.852Q4.632 35.975 4.446 36.043Q4.259 36.111 4.058 36.111Q2.855 36.111 2.855 35.209M6.696 34.532Q6.696 34.204 6.831 33.903Q6.967 33.603 7.202 33.382Q7.438 33.162 7.742 33.042Q8.047 32.922 8.371 32.922Q8.877 32.922 9.226 33.025Q9.574 33.127 9.574 33.503Q9.574 33.650 9.477 33.751Q9.380 33.852 9.233 33.852Q9.079 33.852 8.980 33.753Q8.881 33.654 8.881 33.503Q8.881 33.315 9.021 33.223Q8.819 33.172 8.378 33.172Q8.023 33.172 7.794 33.368Q7.565 33.565 7.464 33.874Q7.363 34.184 7.363 34.532Q7.363 34.881 7.489 35.187Q7.616 35.493 7.871 35.677Q8.125 35.862 8.481 35.862Q8.703 35.862 8.887 35.778Q9.072 35.694 9.207 35.539Q9.342 35.383 9.400 35.175Q9.414 35.120 9.468 35.120L9.581 35.120Q9.612 35.120 9.634 35.144Q9.656 35.168 9.656 35.202L9.656 35.223Q9.571 35.510 9.383 35.708Q9.195 35.906 8.930 36.009Q8.665 36.111 8.371 36.111Q7.941 36.111 7.553 35.905Q7.165 35.698 6.931 35.335Q6.696 34.973 6.696 34.532M10.203 34.508Q10.203 34.187 10.328 33.898Q10.453 33.609 10.678 33.386Q10.904 33.162 11.200 33.042Q11.495 32.922 11.813 32.922Q12.141 32.922 12.403 33.022Q12.664 33.121 12.840 33.303Q13.016 33.486 13.110 33.744Q13.204 34.002 13.204 34.334Q13.204 34.426 13.122 34.447L10.866 34.447L10.866 34.508Q10.866 35.096 11.150 35.479Q11.434 35.862 12.001 35.862Q12.322 35.862 12.591 35.669Q12.859 35.476 12.948 35.161Q12.955 35.120 13.030 35.106L13.122 35.106Q13.204 35.130 13.204 35.202Q13.204 35.209 13.197 35.236Q13.085 35.633 12.714 35.872Q12.343 36.111 11.919 36.111Q11.482 36.111 11.082 35.903Q10.682 35.694 10.443 35.327Q10.203 34.960 10.203 34.508M10.873 34.238L12.688 34.238Q12.688 33.961 12.591 33.709Q12.493 33.456 12.295 33.300Q12.097 33.145 11.813 33.145Q11.536 33.145 11.323 33.303Q11.109 33.462 10.991 33.717Q10.873 33.972 10.873 34.238M13.792 34.532Q13.792 34.194 13.932 33.903Q14.072 33.613 14.317 33.399Q14.561 33.186 14.865 33.071Q15.170 32.957 15.494 32.957Q15.764 32.957 16.028 33.056Q16.291 33.155 16.482 33.333L16.482 31.935Q16.482 31.665 16.374 31.603Q16.267 31.542 15.956 31.542L15.956 31.261L17.032 31.186L17.032 35.370Q17.032 35.558 17.087 35.641Q17.142 35.725 17.243 35.744Q17.343 35.763 17.559 35.763L17.559 36.043L16.451 36.111L16.451 35.694Q16.034 36.111 15.409 36.111Q14.978 36.111 14.606 35.899Q14.233 35.688 14.013 35.327Q13.792 34.966 13.792 34.532M15.467 35.889Q15.675 35.889 15.862 35.817Q16.048 35.746 16.202 35.609Q16.356 35.472 16.451 35.294L16.451 33.685Q16.366 33.538 16.221 33.418Q16.075 33.298 15.906 33.239Q15.737 33.179 15.556 33.179Q14.995 33.179 14.727 33.568Q14.459 33.958 14.459 34.539Q14.459 35.110 14.693 35.500Q14.927 35.889 15.467 35.889\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M-31.67-49.315C19.547-18.017 76.452 7.591 161.81 20.395\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(204.125 -13.219)\">\u003Cpath d=\"M-37.046 34.532Q-37.046 34.204-36.911 33.903Q-36.776 33.603-36.540 33.382Q-36.304 33.162-36 33.042Q-35.695 32.922-35.371 32.922Q-34.865 32.922-34.516 33.025Q-34.168 33.127-34.168 33.503Q-34.168 33.650-34.265 33.751Q-34.362 33.852-34.509 33.852Q-34.663 33.852-34.762 33.753Q-34.861 33.654-34.861 33.503Q-34.861 33.315-34.721 33.223Q-34.923 33.172-35.364 33.172Q-35.719 33.172-35.948 33.368Q-36.177 33.565-36.278 33.874Q-36.379 34.184-36.379 34.532Q-36.379 34.881-36.253 35.187Q-36.126 35.493-35.871 35.677Q-35.617 35.862-35.261 35.862Q-35.039 35.862-34.855 35.778Q-34.670 35.694-34.535 35.539Q-34.400 35.383-34.342 35.175Q-34.328 35.120-34.274 35.120L-34.161 35.120Q-34.130 35.120-34.108 35.144Q-34.086 35.168-34.086 35.202L-34.086 35.223Q-34.171 35.510-34.359 35.708Q-34.547 35.906-34.812 36.009Q-35.077 36.111-35.371 36.111Q-35.801 36.111-36.189 35.905Q-36.577 35.698-36.811 35.335Q-37.046 34.973-37.046 34.532M-33.539 34.560Q-33.539 34.218-33.404 33.919Q-33.269 33.620-33.029 33.396Q-32.790 33.172-32.472 33.047Q-32.154 32.922-31.823 32.922Q-31.379 32.922-30.979 33.138Q-30.579 33.353-30.345 33.731Q-30.110 34.108-30.110 34.560Q-30.110 34.901-30.252 35.185Q-30.394 35.469-30.639 35.676Q-30.883 35.882-31.192 35.997Q-31.502 36.111-31.823 36.111Q-32.254 36.111-32.655 35.910Q-33.057 35.708-33.298 35.356Q-33.539 35.004-33.539 34.560M-31.823 35.862Q-31.221 35.862-30.997 35.484Q-30.774 35.106-30.774 34.474Q-30.774 33.862-31.008 33.503Q-31.242 33.145-31.823 33.145Q-32.876 33.145-32.876 34.474Q-32.876 35.106-32.650 35.484Q-32.424 35.862-31.823 35.862M-27.834 36.043L-29.468 36.043L-29.468 35.763Q-29.239 35.763-29.090 35.729Q-28.942 35.694-28.942 35.554L-28.942 33.705Q-28.942 33.435-29.049 33.374Q-29.157 33.312-29.468 33.312L-29.468 33.032L-28.408 32.957L-28.408 33.606Q-28.237 33.298-27.933 33.127Q-27.629 32.957-27.284 32.957Q-26.778 32.957-26.494 33.180Q-26.211 33.404-26.211 33.900L-26.211 35.554Q-26.211 35.691-26.062 35.727Q-25.913 35.763-25.688 35.763L-25.688 36.043L-27.318 36.043L-27.318 35.763Q-27.089 35.763-26.940 35.729Q-26.792 35.694-26.792 35.554L-26.792 33.914Q-26.792 33.579-26.911 33.379Q-27.031 33.179-27.345 33.179Q-27.615 33.179-27.850 33.315Q-28.084 33.452-28.222 33.686Q-28.360 33.920-28.360 34.194L-28.360 35.554Q-28.360 35.691-28.210 35.727Q-28.060 35.763-27.834 35.763\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(204.125 -13.219)\">\u003Cpath d=\"M-24.776 35.202L-24.776 33.305L-25.415 33.305L-25.415 33.083Q-25.097 33.083-24.880 32.873Q-24.663 32.663-24.563 32.353Q-24.462 32.044-24.462 31.736L-24.195 31.736L-24.195 33.025L-23.118 33.025L-23.118 33.305L-24.195 33.305L-24.195 35.189Q-24.195 35.465-24.091 35.664Q-23.987 35.862-23.727 35.862Q-23.570 35.862-23.464 35.758Q-23.358 35.653-23.308 35.500Q-23.259 35.346-23.259 35.189L-23.259 34.775L-22.992 34.775L-22.992 35.202Q-22.992 35.428-23.091 35.638Q-23.190 35.848-23.375 35.980Q-23.559 36.111-23.788 36.111Q-24.226 36.111-24.501 35.874Q-24.776 35.636-24.776 35.202M-20.432 36.043L-22.168 36.043L-22.168 35.763Q-21.939 35.763-21.791 35.729Q-21.642 35.694-21.642 35.554L-21.642 33.705Q-21.642 33.435-21.750 33.374Q-21.857 33.312-22.168 33.312L-22.168 33.032L-21.139 32.957L-21.139 33.664Q-21.010 33.356-20.767 33.157Q-20.524 32.957-20.206 32.957Q-19.988 32.957-19.817 33.081Q-19.646 33.206-19.646 33.418Q-19.646 33.555-19.745 33.654Q-19.844 33.753-19.977 33.753Q-20.114 33.753-20.213 33.654Q-20.312 33.555-20.312 33.418Q-20.312 33.278-20.213 33.179Q-20.504 33.179-20.704 33.375Q-20.904 33.572-20.996 33.866Q-21.088 34.160-21.088 34.440L-21.088 35.554Q-21.088 35.763-20.432 35.763L-20.432 36.043M-19.102 34.560Q-19.102 34.218-18.967 33.919Q-18.832 33.620-18.593 33.396Q-18.354 33.172-18.036 33.047Q-17.718 32.922-17.387 32.922Q-16.942 32.922-16.542 33.138Q-16.142 33.353-15.908 33.731Q-15.674 34.108-15.674 34.560Q-15.674 34.901-15.816 35.185Q-15.958 35.469-16.202 35.676Q-16.447 35.882-16.756 35.997Q-17.065 36.111-17.387 36.111Q-17.817 36.111-18.219 35.910Q-18.620 35.708-18.861 35.356Q-19.102 35.004-19.102 34.560M-17.387 35.862Q-16.785 35.862-16.561 35.484Q-16.337 35.106-16.337 34.474Q-16.337 33.862-16.571 33.503Q-16.805 33.145-17.387 33.145Q-18.439 33.145-18.439 34.474Q-18.439 35.106-18.214 35.484Q-17.988 35.862-17.387 35.862M-13.411 36.043L-15.014 36.043L-15.014 35.763Q-14.789 35.763-14.640 35.729Q-14.492 35.694-14.492 35.554L-14.492 31.935Q-14.492 31.665-14.599 31.603Q-14.707 31.542-15.014 31.542L-15.014 31.261L-13.938 31.186L-13.938 35.554Q-13.938 35.691-13.787 35.727Q-13.637 35.763-13.411 35.763\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M-31.67-50.737c56.906 8.535 96.74 12.803 119.502 14.226 17.072 32.72 34.144 48.37 73.977 55.483\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(204.125 -19.59)\">\u003Cpath d=\"M-37.087 34.508Q-37.087 34.187-36.962 33.898Q-36.837 33.609-36.611 33.386Q-36.386 33.162-36.090 33.042Q-35.795 32.922-35.477 32.922Q-35.149 32.922-34.887 33.022Q-34.626 33.121-34.450 33.303Q-34.274 33.486-34.180 33.744Q-34.086 34.002-34.086 34.334Q-34.086 34.426-34.168 34.447L-36.423 34.447L-36.423 34.508Q-36.423 35.096-36.140 35.479Q-35.856 35.862-35.289 35.862Q-34.967 35.862-34.699 35.669Q-34.431 35.476-34.342 35.161Q-34.335 35.120-34.260 35.106L-34.168 35.106Q-34.086 35.130-34.086 35.202Q-34.086 35.209-34.092 35.236Q-34.205 35.633-34.576 35.872Q-34.947 36.111-35.371 36.111Q-35.808 36.111-36.208 35.903Q-36.608 35.694-36.847 35.327Q-37.087 34.960-37.087 34.508M-36.417 34.238L-34.602 34.238Q-34.602 33.961-34.699 33.709Q-34.797 33.456-34.995 33.300Q-35.193 33.145-35.477 33.145Q-35.754 33.145-35.967 33.303Q-36.181 33.462-36.299 33.717Q-36.417 33.972-36.417 34.238M-32.315 36.043L-33.638 36.043L-33.638 35.763Q-33.077 35.763-32.698 35.363L-31.984 34.566L-32.896 33.517Q-33.033 33.370-33.182 33.338Q-33.330 33.305-33.597 33.305L-33.597 33.025L-32.096 33.025L-32.096 33.305Q-32.288 33.305-32.288 33.439Q-32.288 33.469-32.257 33.517L-31.662 34.201L-31.221 33.705Q-31.109 33.575-31.109 33.459Q-31.109 33.397-31.146 33.351Q-31.184 33.305-31.242 33.305L-31.242 33.025L-29.926 33.025L-29.926 33.305Q-30.486 33.305-30.866 33.705L-31.488 34.406L-30.493 35.554Q-30.394 35.653-30.293 35.698Q-30.193 35.742-30.081 35.752Q-29.970 35.763-29.793 35.763L-29.793 36.043L-31.286 36.043L-31.286 35.763Q-31.221 35.763-31.162 35.729Q-31.102 35.694-31.102 35.629Q-31.102 35.582-31.132 35.554L-31.809 34.768L-32.342 35.363Q-32.455 35.493-32.455 35.609Q-32.455 35.674-32.414 35.718Q-32.373 35.763-32.315 35.763L-32.315 36.043M-27.653 37.400L-29.283 37.400L-29.283 37.120Q-29.054 37.120-28.906 37.085Q-28.757 37.051-28.757 36.911L-28.757 33.565Q-28.757 33.394-28.894 33.353Q-29.030 33.312-29.283 33.312L-29.283 33.032L-28.203 32.957L-28.203 33.363Q-27.981 33.162-27.694 33.059Q-27.407 32.957-27.099 32.957Q-26.672 32.957-26.308 33.170Q-25.944 33.384-25.730 33.748Q-25.517 34.112-25.517 34.532Q-25.517 34.977-25.756 35.341Q-25.995 35.705-26.388 35.908Q-26.781 36.111-27.226 36.111Q-27.492 36.111-27.740 36.011Q-27.988 35.910-28.176 35.729L-28.176 36.911Q-28.176 37.048-28.027 37.084Q-27.879 37.120-27.653 37.120L-27.653 37.400M-28.176 33.712L-28.176 35.322Q-28.043 35.575-27.800 35.732Q-27.557 35.889-27.280 35.889Q-26.952 35.889-26.699 35.688Q-26.446 35.486-26.313 35.168Q-26.180 34.850-26.180 34.532Q-26.180 34.303-26.245 34.074Q-26.310 33.845-26.438 33.647Q-26.566 33.449-26.761 33.329Q-26.956 33.210-27.188 33.210Q-27.482 33.210-27.750 33.339Q-28.019 33.469-28.176 33.712\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(204.125 -19.59)\">\u003Cpath d=\"M-24.698 34.508Q-24.698 34.187-24.573 33.898Q-24.448 33.609-24.222 33.386Q-23.997 33.162-23.701 33.042Q-23.406 32.922-23.088 32.922Q-22.760 32.922-22.498 33.022Q-22.237 33.121-22.061 33.303Q-21.885 33.486-21.791 33.744Q-21.697 34.002-21.697 34.334Q-21.697 34.426-21.779 34.447L-24.034 34.447L-24.034 34.508Q-24.034 35.096-23.751 35.479Q-23.467 35.862-22.900 35.862Q-22.578 35.862-22.310 35.669Q-22.042 35.476-21.953 35.161Q-21.946 35.120-21.871 35.106L-21.779 35.106Q-21.697 35.130-21.697 35.202Q-21.697 35.209-21.703 35.236Q-21.816 35.633-22.187 35.872Q-22.558 36.111-22.982 36.111Q-23.419 36.111-23.819 35.903Q-24.219 35.694-24.458 35.327Q-24.698 34.960-24.698 34.508M-24.028 34.238L-22.213 34.238Q-22.213 33.961-22.310 33.709Q-22.408 33.456-22.606 33.300Q-22.804 33.145-23.088 33.145Q-23.365 33.145-23.578 33.303Q-23.792 33.462-23.910 33.717Q-24.028 33.972-24.028 34.238M-19.359 36.043L-21.095 36.043L-21.095 35.763Q-20.866 35.763-20.717 35.729Q-20.569 35.694-20.569 35.554L-20.569 33.705Q-20.569 33.435-20.676 33.374Q-20.784 33.312-21.095 33.312L-21.095 33.032L-20.066 32.957L-20.066 33.664Q-19.936 33.356-19.694 33.157Q-19.451 32.957-19.133 32.957Q-18.914 32.957-18.743 33.081Q-18.573 33.206-18.573 33.418Q-18.573 33.555-18.672 33.654Q-18.771 33.753-18.904 33.753Q-19.041 33.753-19.140 33.654Q-19.239 33.555-19.239 33.418Q-19.239 33.278-19.140 33.179Q-19.430 33.179-19.630 33.375Q-19.830 33.572-19.923 33.866Q-20.015 34.160-20.015 34.440L-20.015 35.554Q-20.015 35.763-19.359 35.763L-19.359 36.043M-16.371 36.043L-17.923 36.043L-17.923 35.763Q-17.698 35.763-17.549 35.729Q-17.400 35.694-17.400 35.554L-17.400 33.705Q-17.400 33.517-17.448 33.433Q-17.496 33.350-17.593 33.331Q-17.691 33.312-17.903 33.312L-17.903 33.032L-16.846 32.957L-16.846 35.554Q-16.846 35.694-16.715 35.729Q-16.583 35.763-16.371 35.763L-16.371 36.043M-17.643 31.736Q-17.643 31.565-17.520 31.446Q-17.397 31.326-17.226 31.326Q-17.058 31.326-16.935 31.446Q-16.812 31.565-16.812 31.736Q-16.812 31.911-16.935 32.034Q-17.058 32.157-17.226 32.157Q-17.397 32.157-17.520 32.034Q-17.643 31.911-17.643 31.736M-14.044 36.043L-15.678 36.043L-15.678 35.763Q-15.449 35.763-15.300 35.729Q-15.151 35.694-15.151 35.554L-15.151 33.705Q-15.151 33.435-15.259 33.374Q-15.367 33.312-15.678 33.312L-15.678 33.032L-14.618 32.957L-14.618 33.606Q-14.447 33.298-14.143 33.127Q-13.839 32.957-13.493 32.957Q-13.094 32.957-12.817 33.097Q-12.540 33.237-12.454 33.585Q-12.287 33.292-11.988 33.124Q-11.689 32.957-11.344 32.957Q-10.838 32.957-10.554 33.180Q-10.270 33.404-10.270 33.900L-10.270 35.554Q-10.270 35.691-10.122 35.727Q-9.973 35.763-9.747 35.763L-9.747 36.043L-11.378 36.043L-11.378 35.763Q-11.152 35.763-11.002 35.727Q-10.851 35.691-10.851 35.554L-10.851 33.914Q-10.851 33.579-10.971 33.379Q-11.091 33.179-11.405 33.179Q-11.675 33.179-11.909 33.315Q-12.143 33.452-12.282 33.686Q-12.420 33.920-12.420 34.194L-12.420 35.554Q-12.420 35.691-12.272 35.727Q-12.123 35.763-11.897 35.763L-11.897 36.043L-13.528 36.043L-13.528 35.763Q-13.299 35.763-13.150 35.729Q-13.001 35.694-13.001 35.554L-13.001 33.914Q-13.001 33.579-13.121 33.379Q-13.241 33.179-13.555 33.179Q-13.825 33.179-14.059 33.315Q-14.293 33.452-14.432 33.686Q-14.570 33.920-14.570 34.194L-14.570 35.554Q-14.570 35.691-14.420 35.727Q-14.269 35.763-14.044 35.763L-14.044 36.043M-9.200 34.508Q-9.200 34.187-9.076 33.898Q-8.951 33.609-8.725 33.386Q-8.500 33.162-8.204 33.042Q-7.909 32.922-7.591 32.922Q-7.263 32.922-7.001 33.022Q-6.740 33.121-6.564 33.303Q-6.388 33.486-6.294 33.744Q-6.200 34.002-6.200 34.334Q-6.200 34.426-6.282 34.447L-8.537 34.447L-8.537 34.508Q-8.537 35.096-8.254 35.479Q-7.970 35.862-7.403 35.862Q-7.081 35.862-6.813 35.669Q-6.545 35.476-6.456 35.161Q-6.449 35.120-6.374 35.106L-6.282 35.106Q-6.200 35.130-6.200 35.202Q-6.200 35.209-6.206 35.236Q-6.319 35.633-6.690 35.872Q-7.061 36.111-7.485 36.111Q-7.922 36.111-8.322 35.903Q-8.722 35.694-8.961 35.327Q-9.200 34.960-9.200 34.508M-8.531 34.238L-6.716 34.238Q-6.716 33.961-6.813 33.709Q-6.910 33.456-7.109 33.300Q-7.307 33.145-7.591 33.145Q-7.867 33.145-8.081 33.303Q-8.295 33.462-8.413 33.717Q-8.531 33.972-8.531 34.238M-3.930 36.043L-5.564 36.043L-5.564 35.763Q-5.335 35.763-5.186 35.729Q-5.037 35.694-5.037 35.554L-5.037 33.705Q-5.037 33.435-5.145 33.374Q-5.253 33.312-5.564 33.312L-5.564 33.032L-4.504 32.957L-4.504 33.606Q-4.333 33.298-4.029 33.127Q-3.725 32.957-3.380 32.957Q-2.874 32.957-2.590 33.180Q-2.306 33.404-2.306 33.900L-2.306 35.554Q-2.306 35.691-2.158 35.727Q-2.009 35.763-1.784 35.763L-1.784 36.043L-3.414 36.043L-3.414 35.763Q-3.185 35.763-3.036 35.729Q-2.888 35.694-2.888 35.554L-2.888 33.914Q-2.888 33.579-3.007 33.379Q-3.127 33.179-3.441 33.179Q-3.711 33.179-3.945 33.315Q-4.179 33.452-4.318 33.686Q-4.456 33.920-4.456 34.194L-4.456 35.554Q-4.456 35.691-4.306 35.727Q-4.156 35.763-3.930 35.763\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(204.125 -19.59)\">\u003Cpath d=\"M-0.859 35.202L-0.859 33.305L-1.498 33.305L-1.498 33.083Q-1.180 33.083-0.963 32.873Q-0.746 32.663-0.646 32.353Q-0.545 32.044-0.545 31.736L-0.278 31.736L-0.278 33.025L0.799 33.025L0.799 33.305L-0.278 33.305L-0.278 35.189Q-0.278 35.465-0.174 35.664Q-0.070 35.862 0.190 35.862Q0.347 35.862 0.453 35.758Q0.559 35.653 0.609 35.500Q0.658 35.346 0.658 35.189L0.658 34.775L0.925 34.775L0.925 35.202Q0.925 35.428 0.826 35.638Q0.727 35.848 0.542 35.980Q0.358 36.111 0.129 36.111Q-0.309 36.111-0.584 35.874Q-0.859 35.636-0.859 35.202M1.793 35.315Q1.793 34.983 2.017 34.756Q2.241 34.529 2.584 34.401Q2.928 34.272 3.300 34.220Q3.673 34.167 3.977 34.167L3.977 33.914Q3.977 33.709 3.870 33.529Q3.762 33.350 3.581 33.247Q3.400 33.145 3.191 33.145Q2.784 33.145 2.549 33.237Q2.637 33.274 2.684 33.358Q2.730 33.442 2.730 33.544Q2.730 33.640 2.684 33.719Q2.637 33.797 2.557 33.842Q2.477 33.886 2.388 33.886Q2.237 33.886 2.137 33.789Q2.036 33.691 2.036 33.544Q2.036 32.922 3.191 32.922Q3.403 32.922 3.653 32.986Q3.902 33.049 4.104 33.168Q4.305 33.288 4.432 33.473Q4.558 33.657 4.558 33.900L4.558 35.476Q4.558 35.592 4.620 35.688Q4.681 35.783 4.794 35.783Q4.904 35.783 4.968 35.689Q5.033 35.595 5.033 35.476L5.033 35.028L5.300 35.028L5.300 35.476Q5.300 35.746 5.073 35.911Q4.845 36.077 4.565 36.077Q4.357 36.077 4.220 35.923Q4.083 35.770 4.059 35.554Q3.912 35.821 3.630 35.966Q3.348 36.111 3.024 36.111Q2.747 36.111 2.463 36.036Q2.179 35.961 1.986 35.782Q1.793 35.602 1.793 35.315M2.408 35.315Q2.408 35.489 2.509 35.619Q2.610 35.749 2.766 35.819Q2.921 35.889 3.085 35.889Q3.304 35.889 3.512 35.792Q3.721 35.694 3.849 35.513Q3.977 35.332 3.977 35.106L3.977 34.378Q3.653 34.378 3.287 34.469Q2.921 34.560 2.665 34.772Q2.408 34.983 2.408 35.315M7.385 36.043L5.782 36.043L5.782 35.763Q6.008 35.763 6.156 35.729Q6.305 35.694 6.305 35.554L6.305 31.935Q6.305 31.665 6.197 31.603Q6.090 31.542 5.782 31.542L5.782 31.261L6.859 31.186L6.859 35.554Q6.859 35.691 7.009 35.727Q7.159 35.763 7.385 35.763\" 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\">Latent learning. The control group is rewarded throughout and its error count falls steadily. The experimental group is unrewarded until stage 2, shows no improvement while unrewarded, then drops sharply to the control level once food appears — revealing a map learned silently during the reward-free stage.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:403.673px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 302.755 89.216\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(5.485 -76.915)\">\u003Cpath d=\"M-52.086 14.847L-52.086 12.768Q-52.086 12.742-52.057 12.713Q-52.029 12.685-51.998 12.685L-51.888 12.685Q-51.857 12.685-51.829 12.713Q-51.800 12.742-51.800 12.768Q-51.800 13.370-51.543 13.781Q-51.286 14.192-50.825 14.401Q-50.363 14.609-49.783 14.609Q-49.449 14.609-49.177 14.434Q-48.904 14.258-48.750 13.966Q-48.597 13.673-48.597 13.353Q-48.597 13.137-48.665 12.935Q-48.733 12.733-48.851 12.568Q-48.970 12.403-49.150 12.282Q-49.330 12.162-49.533 12.118L-50.728 11.832Q-51.110 11.740-51.420 11.491Q-51.730 11.243-51.908 10.881Q-52.086 10.518-52.086 10.127Q-52.086 9.657-51.842 9.252Q-51.598 8.848-51.185 8.615Q-50.772 8.382-50.297 8.382Q-49.858 8.382-49.478 8.538Q-49.098 8.694-48.821 9.002L-48.452 8.417Q-48.434 8.382-48.390 8.382L-48.333 8.382Q-48.302 8.382-48.274 8.411Q-48.245 8.439-48.245 8.466L-48.245 10.544Q-48.245 10.566-48.278 10.595Q-48.311 10.624-48.333 10.624L-48.443 10.624Q-48.465 10.624-48.498 10.595Q-48.531 10.566-48.531 10.544Q-48.531 10.360-48.572 10.171Q-48.614 9.982-48.698 9.782Q-48.781 9.582-48.884 9.408Q-48.988 9.235-49.106 9.121Q-49.348 8.883-49.645 8.778Q-49.941 8.672-50.288 8.672Q-50.592 8.672-50.869 8.824Q-51.145 8.976-51.310 9.239Q-51.475 9.503-51.475 9.824Q-51.475 10.079-51.356 10.316Q-51.238 10.553-51.036 10.711Q-50.833 10.870-50.565 10.940L-49.370 11.226Q-48.961 11.322-48.645 11.597Q-48.329 11.872-48.155 12.256Q-47.981 12.641-47.981 13.054Q-47.981 13.537-48.219 13.972Q-48.456 14.407-48.869 14.666Q-49.282 14.926-49.774 14.926Q-50.271 14.926-50.730 14.772Q-51.189 14.618-51.501 14.306L-51.879 14.891Q-51.897 14.926-51.941 14.926L-51.998 14.926Q-52.020 14.926-52.053 14.897Q-52.086 14.869-52.086 14.847M-44.896 13.080L-47.362 13.080L-47.362 12.487L-44.896 12.487L-44.896 13.080M-41.390 14.728L-44.044 14.728L-44.044 14.412Q-43.125 14.412-43.125 14.117L-43.125 9.191Q-43.125 8.896-44.044 8.896L-44.044 8.580L-41.183 8.580Q-40.809 8.580-40.385 8.679Q-39.961 8.778-39.588 8.978Q-39.214 9.178-38.979 9.488Q-38.744 9.797-38.744 10.210Q-38.744 10.580-38.970 10.881Q-39.197 11.182-39.548 11.379Q-39.900 11.577-40.265 11.669Q-39.346 11.995-39.192 12.830L-39.087 13.621Q-39.043 13.955-38.999 14.148Q-38.955 14.341-38.825 14.504Q-38.696 14.666-38.463 14.666Q-38.190 14.666-38.048 14.416Q-37.905 14.165-37.905 13.862Q-37.905 13.823-37.872 13.794Q-37.839 13.766-37.799 13.766L-37.716 13.766Q-37.667 13.766-37.643 13.799Q-37.619 13.832-37.619 13.880Q-37.619 14.117-37.720 14.365Q-37.821 14.614-38.017 14.770Q-38.212 14.926-38.463 14.926Q-39.122 14.926-39.581 14.601Q-40.040 14.275-40.040 13.638L-40.040 12.847Q-40.040 12.548-40.203 12.302Q-40.366 12.056-40.629 11.920Q-40.893 11.784-41.192 11.784L-42.308 11.784L-42.308 14.117Q-42.308 14.412-41.390 14.412L-41.390 14.728M-42.308 9.191L-42.308 11.524L-41.302 11.524Q-40.550 11.524-40.124 11.226Q-39.698 10.927-39.698 10.210Q-39.698 9.477-40.117 9.186Q-40.537 8.896-41.302 8.896L-41.847 8.896Q-42.097 8.896-42.203 8.945Q-42.308 8.993-42.308 9.191\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.485 -76.915)\">\u003Cpath d=\"M-31.634 16.969Q-32.139 16.582-32.508 16.077Q-32.878 15.572-33.121 14.972Q-33.365 14.372-33.480 13.755Q-33.594 13.137-33.594 12.478Q-33.594 11.819-33.480 11.204Q-33.365 10.588-33.126 9.995Q-32.886 9.402-32.513 8.892Q-32.139 8.382-31.634 7.996Q-31.599 7.978-31.577 7.978L-31.498 7.978Q-31.410 7.978-31.410 8.079Q-31.410 8.114-31.445 8.149Q-32.007 8.672-32.352 9.378Q-32.697 10.083-32.845 10.865Q-32.992 11.647-32.992 12.478Q-32.992 13.102-32.913 13.686Q-32.834 14.271-32.656 14.838Q-32.478 15.405-32.179 15.906Q-31.880 16.407-31.445 16.815Q-31.410 16.851-31.410 16.890Q-31.410 16.987-31.498 16.987L-31.577 16.987Q-31.599 16.987-31.634 16.969M-28.558 16.473L-30.645 16.473L-30.645 16.161Q-30.333 16.161-30.142 16.110Q-29.951 16.060-29.951 15.862L-29.951 11.524Q-29.951 11.283-30.124 11.223Q-30.298 11.164-30.645 11.164L-30.645 10.848L-29.274 10.751L-29.274 11.291Q-29.015 11.023-28.681 10.887Q-28.347 10.751-27.978 10.751Q-27.578 10.751-27.222 10.918Q-26.866 11.085-26.611 11.366Q-26.356 11.647-26.213 12.012Q-26.070 12.377-26.070 12.786Q-26.070 13.348-26.347 13.814Q-26.624 14.280-27.099 14.554Q-27.573 14.829-28.123 14.829Q-28.435 14.829-28.725 14.695Q-29.015 14.561-29.248 14.315L-29.248 15.862Q-29.248 16.060-29.057 16.110Q-28.865 16.161-28.558 16.161L-28.558 16.473M-29.248 11.705L-29.248 13.836Q-29.090 14.161-28.806 14.363Q-28.523 14.565-28.180 14.565Q-27.767 14.565-27.472 14.289Q-27.178 14.012-27.031 13.594Q-26.883 13.177-26.883 12.786Q-26.883 12.408-27.017 11.999Q-27.152 11.590-27.426 11.313Q-27.701 11.037-28.079 11.037Q-28.320 11.037-28.536 11.113Q-28.751 11.190-28.942 11.351Q-29.133 11.511-29.248 11.705\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.485 -76.915)\">\u003Cpath d=\"M-25.234 12.821Q-25.234 12.254-24.961 11.766Q-24.689 11.278-24.219 10.986Q-23.748 10.694-23.181 10.694Q-22.760 10.694-22.384 10.863Q-22.008 11.032-21.731 11.324Q-21.454 11.617-21.296 12.012Q-21.138 12.408-21.138 12.821Q-21.138 13.370-21.417 13.832Q-21.696 14.293-22.164 14.561Q-22.632 14.829-23.181 14.829Q-23.735 14.829-24.205 14.561Q-24.676 14.293-24.955 13.832Q-25.234 13.370-25.234 12.821M-23.181 14.539Q-22.685 14.539-22.408 14.278Q-22.131 14.016-22.039 13.612Q-21.947 13.207-21.947 12.711Q-21.947 12.236-22.045 11.847Q-22.144 11.458-22.417 11.208Q-22.689 10.957-23.181 10.957Q-23.893 10.957-24.157 11.452Q-24.421 11.946-24.421 12.711Q-24.421 13.511-24.166 14.025Q-23.911 14.539-23.181 14.539M-18.506 14.728L-20.567 14.728L-20.567 14.412Q-20.259 14.412-20.068 14.359Q-19.877 14.306-19.877 14.117L-19.877 9.402Q-19.877 9.160-19.947 9.052Q-20.017 8.945-20.151 8.921Q-20.285 8.896-20.567 8.896L-20.567 8.580L-19.200 8.483L-19.200 14.117Q-19.200 14.306-19.007 14.359Q-18.813 14.412-18.506 14.412L-18.506 14.728M-16.014 14.728L-18 14.728L-18 14.412Q-17.693 14.412-17.502 14.359Q-17.310 14.306-17.310 14.117L-17.310 11.669Q-17.310 11.423-17.376 11.318Q-17.442 11.212-17.567 11.188Q-17.693 11.164-17.965 11.164L-17.965 10.848L-16.634 10.751L-16.634 14.117Q-16.634 14.311-16.469 14.361Q-16.304 14.412-16.014 14.412L-16.014 14.728M-17.614 9.204Q-17.614 8.998-17.464 8.848Q-17.315 8.699-17.113 8.699Q-16.981 8.699-16.864 8.769Q-16.748 8.839-16.678 8.956Q-16.607 9.072-16.607 9.204Q-16.607 9.406-16.757 9.556Q-16.906 9.705-17.113 9.705Q-17.315 9.705-17.464 9.556Q-17.614 9.406-17.614 9.204M-13.430 14.829Q-13.979 14.829-14.439 14.550Q-14.898 14.271-15.166 13.801Q-15.434 13.331-15.434 12.786Q-15.434 12.373-15.285 11.995Q-15.135 11.617-14.860 11.322Q-14.586 11.028-14.217 10.861Q-13.847 10.694-13.430 10.694Q-13.096 10.694-12.775 10.760Q-12.454 10.826-12.226 11.012Q-11.997 11.199-11.997 11.524Q-11.997 11.700-12.125 11.828Q-12.252 11.955-12.428 11.955Q-12.613 11.955-12.742 11.830Q-12.872 11.705-12.872 11.524Q-12.872 11.388-12.797 11.276Q-12.722 11.164-12.591 11.111Q-12.898 10.984-13.430 10.984Q-13.865 10.984-14.133 11.261Q-14.401 11.538-14.513 11.953Q-14.625 12.368-14.625 12.786Q-14.625 13.216-14.487 13.618Q-14.348 14.020-14.054 14.280Q-13.760 14.539-13.320 14.539Q-12.898 14.539-12.595 14.289Q-12.292 14.038-12.186 13.629Q-12.178 13.599-12.153 13.574Q-12.129 13.550-12.098 13.550L-11.989 13.550Q-11.901 13.550-11.901 13.665Q-12.046 14.201-12.459 14.515Q-12.872 14.829-13.430 14.829M-10.960 16.205Q-10.793 16.310-10.613 16.310Q-10.288 16.310-10.051 16.066Q-9.813 15.822-9.664 15.475L-9.361 14.728L-10.727 11.463Q-10.820 11.265-10.982 11.215Q-11.145 11.164-11.448 11.164L-11.448 10.848L-9.523 10.848L-9.523 11.164Q-10.015 11.164-10.015 11.379Q-10.015 11.401-9.998 11.463L-8.992 13.853L-8.091 11.713Q-8.055 11.630-8.055 11.524Q-8.055 11.362-8.176 11.263Q-8.297 11.164-8.460 11.164L-8.460 10.848L-6.957 10.848L-6.957 11.164Q-7.243 11.164-7.456 11.307Q-7.669 11.450-7.774 11.713L-9.361 15.475Q-9.550 15.928-9.864 16.251Q-10.178 16.574-10.613 16.574Q-10.938 16.574-11.198 16.354Q-11.457 16.134-11.457 15.809Q-11.457 15.642-11.338 15.528Q-11.220 15.414-11.053 15.414Q-10.938 15.414-10.848 15.464Q-10.758 15.515-10.708 15.603Q-10.657 15.690-10.657 15.809Q-10.657 15.954-10.738 16.064Q-10.820 16.174-10.960 16.205\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.485 -76.915)\">\u003Cpath d=\"M-1.341 14.829Q-1.891 14.829-2.350 14.550Q-2.809 14.271-3.077 13.801Q-3.345 13.331-3.345 12.786Q-3.345 12.373-3.196 11.995Q-3.047 11.617-2.772 11.322Q-2.497 11.028-2.128 10.861Q-1.759 10.694-1.341 10.694Q-1.007 10.694-0.687 10.760Q-0.366 10.826-0.137 11.012Q0.091 11.199 0.091 11.524Q0.091 11.700-0.036 11.828Q-0.164 11.955-0.340 11.955Q-0.524 11.955-0.654 11.830Q-0.783 11.705-0.783 11.524Q-0.783 11.388-0.709 11.276Q-0.634 11.164-0.502 11.111Q-0.810 10.984-1.341 10.984Q-1.777 10.984-2.045 11.261Q-2.313 11.538-2.425 11.953Q-2.537 12.368-2.537 12.786Q-2.537 13.216-2.398 13.618Q-2.260 14.020-1.966 14.280Q-1.671 14.539-1.232 14.539Q-0.810 14.539-0.507 14.289Q-0.203 14.038-0.098 13.629Q-0.089 13.599-0.065 13.574Q-0.041 13.550-0.010 13.550L0.100 13.550Q0.188 13.550 0.188 13.665Q0.043 14.201-0.370 14.515Q-0.783 14.829-1.341 14.829M0.821 13.818Q0.821 13.278 1.253 12.944Q1.686 12.610 2.293 12.471Q2.899 12.333 3.431 12.333L3.431 11.999Q3.431 11.740 3.312 11.496Q3.194 11.252 2.985 11.105Q2.776 10.957 2.504 10.957Q1.941 10.957 1.629 11.155Q1.779 11.182 1.871 11.300Q1.963 11.419 1.963 11.577Q1.963 11.753 1.838 11.883Q1.713 12.012 1.533 12.012Q1.344 12.012 1.216 11.885Q1.089 11.757 1.089 11.577Q1.089 11.107 1.528 10.900Q1.968 10.694 2.504 10.694Q2.794 10.694 3.082 10.782Q3.369 10.870 3.602 11.030Q3.835 11.190 3.985 11.430Q4.134 11.669 4.134 11.964L4.134 13.999Q4.134 14.152 4.209 14.291Q4.284 14.429 4.429 14.429Q4.582 14.429 4.655 14.293Q4.727 14.157 4.727 13.999L4.727 13.423L5.013 13.423L5.013 13.999Q5.013 14.332 4.765 14.557Q4.516 14.781 4.187 14.781Q3.928 14.781 3.745 14.587Q3.563 14.394 3.519 14.117Q3.352 14.438 3.018 14.634Q2.684 14.829 2.315 14.829Q1.761 14.829 1.291 14.581Q0.821 14.332 0.821 13.818M1.576 13.818Q1.576 14.130 1.818 14.348Q2.060 14.565 2.376 14.565Q2.811 14.565 3.121 14.256Q3.431 13.946 3.431 13.515L3.431 12.588Q3.014 12.588 2.587 12.715Q2.161 12.843 1.869 13.120Q1.576 13.396 1.576 13.818M7.382 14.829Q6.832 14.829 6.373 14.550Q5.914 14.271 5.646 13.801Q5.378 13.331 5.378 12.786Q5.378 12.373 5.527 11.995Q5.677 11.617 5.951 11.322Q6.226 11.028 6.595 10.861Q6.964 10.694 7.382 10.694Q7.716 10.694 8.036 10.760Q8.357 10.826 8.586 11.012Q8.814 11.199 8.814 11.524Q8.814 11.700 8.687 11.828Q8.559 11.955 8.384 11.955Q8.199 11.955 8.069 11.830Q7.940 11.705 7.940 11.524Q7.940 11.388 8.014 11.276Q8.089 11.164 8.221 11.111Q7.913 10.984 7.382 10.984Q6.947 10.984 6.679 11.261Q6.410 11.538 6.298 11.953Q6.186 12.368 6.186 12.786Q6.186 13.216 6.325 13.618Q6.463 14.020 6.758 14.280Q7.052 14.539 7.492 14.539Q7.913 14.539 8.217 14.289Q8.520 14.038 8.625 13.629Q8.634 13.599 8.658 13.574Q8.682 13.550 8.713 13.550L8.823 13.550Q8.911 13.550 8.911 13.665Q8.766 14.201 8.353 14.515Q7.940 14.829 7.382 14.829\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.485 -76.915)\">\u003Cpath d=\"M11.323 14.728L9.236 14.728L9.236 14.412Q9.543 14.412 9.735 14.359Q9.926 14.306 9.926 14.117L9.926 9.402Q9.926 9.160 9.855 9.052Q9.785 8.945 9.651 8.921Q9.517 8.896 9.236 8.896L9.236 8.580L10.603 8.483L10.603 11.533Q10.800 11.177 11.154 10.964Q11.508 10.751 11.908 10.751Q13.187 10.751 13.187 11.964L13.187 14.117Q13.187 14.306 13.378 14.359Q13.569 14.412 13.876 14.412L13.876 14.728L11.789 14.728L11.789 14.412Q12.101 14.412 12.292 14.359Q12.483 14.306 12.483 14.117L12.483 11.999Q12.483 11.740 12.439 11.518Q12.395 11.296 12.250 11.153Q12.105 11.010 11.846 11.010Q11.503 11.010 11.222 11.199Q10.941 11.388 10.785 11.700Q10.629 12.012 10.629 12.359L10.629 14.117Q10.629 14.306 10.822 14.359Q11.016 14.412 11.323 14.412L11.323 14.728M16.386 14.829Q15.828 14.829 15.355 14.546Q14.883 14.262 14.608 13.785Q14.333 13.309 14.333 12.755Q14.333 12.359 14.476 11.984Q14.619 11.608 14.876 11.320Q15.133 11.032 15.491 10.863Q15.850 10.694 16.254 10.694Q16.799 10.694 17.170 10.931Q17.541 11.168 17.728 11.586Q17.915 12.003 17.915 12.540Q17.915 12.592 17.891 12.630Q17.867 12.667 17.818 12.667L15.146 12.667L15.146 12.746Q15.146 13.493 15.458 14.016Q15.770 14.539 16.469 14.539Q16.874 14.539 17.194 14.282Q17.515 14.025 17.638 13.621Q17.656 13.541 17.739 13.541L17.818 13.541Q17.858 13.541 17.886 13.572Q17.915 13.603 17.915 13.647L17.915 13.682Q17.810 14.025 17.588 14.284Q17.366 14.543 17.052 14.686Q16.737 14.829 16.386 14.829M15.155 12.416L17.269 12.416Q17.269 12.148 17.216 11.902Q17.164 11.656 17.043 11.434Q16.922 11.212 16.724 11.085Q16.526 10.957 16.254 10.957Q15.911 10.957 15.658 11.182Q15.406 11.406 15.281 11.744Q15.155 12.082 15.155 12.416M18.873 16.987L18.790 16.987Q18.702 16.987 18.702 16.890Q18.702 16.851 18.737 16.815Q19.572 16.042 19.932 14.908Q20.292 13.774 20.292 12.478Q20.292 11.858 20.213 11.267Q20.134 10.676 19.954 10.118Q19.774 9.560 19.473 9.052Q19.172 8.545 18.737 8.149Q18.702 8.114 18.702 8.079Q18.702 7.978 18.790 7.978L18.873 7.978Q18.891 7.978 18.926 7.996Q19.431 8.378 19.805 8.892Q20.178 9.406 20.416 9.984Q20.653 10.562 20.769 11.190Q20.886 11.819 20.886 12.478Q20.886 13.137 20.769 13.768Q20.653 14.398 20.413 14.983Q20.174 15.567 19.802 16.077Q19.431 16.587 18.926 16.969Q18.891 16.987 18.873 16.987\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M-65.403-15.147h25.607v-19.917h-25.607Z\"\u002F>\u003Cg transform=\"translate(-8.48 -37.68)\">\u003Cpath d=\"M-52.286 14.721L-52.286 13.658Q-52.286 13.634-52.258 13.607Q-52.231 13.580-52.207 13.580L-52.098 13.580Q-52.033 13.580-52.019 13.638Q-51.923 14.072-51.677 14.323Q-51.431 14.574-51.017 14.574Q-50.676 14.574-50.423 14.441Q-50.170 14.308-50.170 14Q-50.170 13.843-50.264 13.728Q-50.358 13.614-50.496 13.545Q-50.635 13.477-50.802 13.439L-51.383 13.340Q-51.739 13.272-52.012 13.051Q-52.286 12.831-52.286 12.489Q-52.286 12.240-52.174 12.065Q-52.063 11.891-51.877 11.792Q-51.691 11.693-51.475 11.650Q-51.260 11.607-51.017 11.607Q-50.604 11.607-50.324 11.789L-50.108 11.614Q-50.098 11.611-50.091 11.609Q-50.084 11.607-50.074 11.607L-50.023 11.607Q-49.996 11.607-49.972 11.631Q-49.948 11.655-49.948 11.683L-49.948 12.530Q-49.948 12.551-49.972 12.578Q-49.996 12.605-50.023 12.605L-50.136 12.605Q-50.163 12.605-50.189 12.580Q-50.214 12.554-50.214 12.530Q-50.214 12.294-50.320 12.130Q-50.426 11.966-50.609 11.884Q-50.792 11.802-51.024 11.802Q-51.352 11.802-51.609 11.905Q-51.865 12.007-51.865 12.284Q-51.865 12.479-51.682 12.588Q-51.499 12.698-51.270 12.739L-50.696 12.845Q-50.450 12.893-50.236 13.021Q-50.023 13.149-49.886 13.352Q-49.749 13.556-49.749 13.805Q-49.749 14.318-50.115 14.557Q-50.481 14.796-51.017 14.796Q-51.513 14.796-51.845 14.502L-52.111 14.776Q-52.132 14.796-52.159 14.796L-52.207 14.796Q-52.231 14.796-52.258 14.769Q-52.286 14.742-52.286 14.721M-48.594 13.887L-48.594 11.990L-49.233 11.990L-49.233 11.768Q-48.915 11.768-48.698 11.558Q-48.481 11.348-48.381 11.038Q-48.280 10.729-48.280 10.421L-48.013 10.421L-48.013 11.710L-46.936 11.710L-46.936 11.990L-48.013 11.990L-48.013 13.874Q-48.013 14.150-47.909 14.349Q-47.805 14.547-47.545 14.547Q-47.388 14.547-47.282 14.443Q-47.176 14.338-47.126 14.185Q-47.077 14.031-47.077 13.874L-47.077 13.460L-46.810 13.460L-46.810 13.887Q-46.810 14.113-46.909 14.323Q-47.008 14.533-47.193 14.665Q-47.377 14.796-47.606 14.796Q-48.044 14.796-48.319 14.559Q-48.594 14.321-48.594 13.887M-45.942 14Q-45.942 13.668-45.718 13.441Q-45.494 13.214-45.151 13.086Q-44.807 12.957-44.434 12.905Q-44.062 12.852-43.758 12.852L-43.758 12.599Q-43.758 12.394-43.865 12.214Q-43.973 12.035-44.154 11.932Q-44.335 11.830-44.544 11.830Q-44.951 11.830-45.186 11.922Q-45.098 11.959-45.051 12.043Q-45.005 12.127-45.005 12.229Q-45.005 12.325-45.051 12.404Q-45.098 12.482-45.178 12.527Q-45.258 12.571-45.347 12.571Q-45.497 12.571-45.598 12.474Q-45.699 12.376-45.699 12.229Q-45.699 11.607-44.544 11.607Q-44.332 11.607-44.082 11.671Q-43.833 11.734-43.631 11.853Q-43.430 11.973-43.303 12.158Q-43.177 12.342-43.177 12.585L-43.177 14.161Q-43.177 14.277-43.115 14.373Q-43.054 14.468-42.941 14.468Q-42.831 14.468-42.767 14.374Q-42.702 14.280-42.702 14.161L-42.702 13.713L-42.435 13.713L-42.435 14.161Q-42.435 14.431-42.662 14.596Q-42.890 14.762-43.170 14.762Q-43.378 14.762-43.515 14.608Q-43.652 14.455-43.676 14.239Q-43.823 14.506-44.105 14.651Q-44.387 14.796-44.711 14.796Q-44.988 14.796-45.272 14.721Q-45.556 14.646-45.749 14.467Q-45.942 14.287-45.942 14M-45.327 14Q-45.327 14.174-45.226 14.304Q-45.125 14.434-44.969 14.504Q-44.814 14.574-44.650 14.574Q-44.431 14.574-44.223 14.477Q-44.014 14.379-43.886 14.198Q-43.758 14.017-43.758 13.791L-43.758 13.063Q-44.082 13.063-44.448 13.154Q-44.814 13.245-45.070 13.457Q-45.327 13.668-45.327 14M-41.492 13.887L-41.492 11.990L-42.131 11.990L-42.131 11.768Q-41.813 11.768-41.596 11.558Q-41.379 11.348-41.278 11.038Q-41.177 10.729-41.177 10.421L-40.911 10.421L-40.911 11.710L-39.834 11.710L-39.834 11.990L-40.911 11.990L-40.911 13.874Q-40.911 14.150-40.806 14.349Q-40.702 14.547-40.442 14.547Q-40.285 14.547-40.179 14.443Q-40.073 14.338-40.024 14.185Q-39.974 14.031-39.974 13.874L-39.974 13.460L-39.707 13.460L-39.707 13.887Q-39.707 14.113-39.807 14.323Q-39.906 14.533-40.090 14.665Q-40.275 14.796-40.504 14.796Q-40.941 14.796-41.216 14.559Q-41.492 14.321-41.492 13.887M-38.938 13.193Q-38.938 12.872-38.814 12.583Q-38.689 12.294-38.463 12.071Q-38.238 11.847-37.942 11.727Q-37.646 11.607-37.329 11.607Q-37 11.607-36.739 11.707Q-36.477 11.806-36.301 11.988Q-36.125 12.171-36.031 12.429Q-35.937 12.687-35.937 13.019Q-35.937 13.111-36.019 13.132L-38.275 13.132L-38.275 13.193Q-38.275 13.781-37.992 14.164Q-37.708 14.547-37.141 14.547Q-36.819 14.547-36.551 14.354Q-36.283 14.161-36.194 13.846Q-36.187 13.805-36.112 13.791L-36.019 13.791Q-35.937 13.815-35.937 13.887Q-35.937 13.894-35.944 13.921Q-36.057 14.318-36.428 14.557Q-36.799 14.796-37.223 14.796Q-37.660 14.796-38.060 14.588Q-38.460 14.379-38.699 14.012Q-38.938 13.645-38.938 13.193M-38.268 12.923L-36.454 12.923Q-36.454 12.646-36.551 12.394Q-36.648 12.141-36.847 11.985Q-37.045 11.830-37.329 11.830Q-37.605 11.830-37.819 11.988Q-38.033 12.147-38.151 12.402Q-38.268 12.657-38.268 12.923\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M18.759-15.147h28v-19.917h-28Z\"\u002F>\u003Cg transform=\"translate(74.692 -37.48)\">\u003Cpath d=\"M-52.227 14Q-52.227 13.668-52.004 13.441Q-51.780 13.214-51.436 13.086Q-51.093 12.957-50.720 12.905Q-50.348 12.852-50.043 12.852L-50.043 12.599Q-50.043 12.394-50.151 12.214Q-50.259 12.035-50.440 11.932Q-50.621 11.830-50.829 11.830Q-51.236 11.830-51.472 11.922Q-51.383 11.959-51.337 12.043Q-51.291 12.127-51.291 12.229Q-51.291 12.325-51.337 12.404Q-51.383 12.482-51.464 12.527Q-51.544 12.571-51.633 12.571Q-51.783 12.571-51.884 12.474Q-51.985 12.376-51.985 12.229Q-51.985 11.607-50.829 11.607Q-50.618 11.607-50.368 11.671Q-50.119 11.734-49.917 11.853Q-49.715 11.973-49.589 12.158Q-49.462 12.342-49.462 12.585L-49.462 14.161Q-49.462 14.277-49.401 14.373Q-49.339 14.468-49.226 14.468Q-49.117 14.468-49.052 14.374Q-48.987 14.280-48.987 14.161L-48.987 13.713L-48.721 13.713L-48.721 14.161Q-48.721 14.431-48.948 14.596Q-49.175 14.762-49.455 14.762Q-49.664 14.762-49.801 14.608Q-49.937 14.455-49.961 14.239Q-50.108 14.506-50.390 14.651Q-50.672 14.796-50.997 14.796Q-51.274 14.796-51.558 14.721Q-51.841 14.646-52.034 14.467Q-52.227 14.287-52.227 14M-51.612 14Q-51.612 14.174-51.511 14.304Q-51.411 14.434-51.255 14.504Q-51.100 14.574-50.935 14.574Q-50.717 14.574-50.508 14.477Q-50.300 14.379-50.172 14.198Q-50.043 14.017-50.043 13.791L-50.043 13.063Q-50.368 13.063-50.734 13.154Q-51.100 13.245-51.356 13.457Q-51.612 13.668-51.612 14M-48.304 13.217Q-48.304 12.889-48.169 12.588Q-48.034 12.288-47.798 12.067Q-47.562 11.847-47.258 11.727Q-46.954 11.607-46.629 11.607Q-46.123 11.607-45.774 11.710Q-45.426 11.812-45.426 12.188Q-45.426 12.335-45.523 12.436Q-45.621 12.537-45.767 12.537Q-45.921 12.537-46.020 12.438Q-46.120 12.339-46.120 12.188Q-46.120 12-45.979 11.908Q-46.181 11.857-46.622 11.857Q-46.977 11.857-47.206 12.053Q-47.435 12.250-47.536 12.559Q-47.637 12.869-47.637 13.217Q-47.637 13.566-47.511 13.872Q-47.384 14.178-47.130 14.362Q-46.875 14.547-46.519 14.547Q-46.297 14.547-46.113 14.463Q-45.928 14.379-45.793 14.224Q-45.658 14.068-45.600 13.860Q-45.586 13.805-45.532 13.805L-45.419 13.805Q-45.388 13.805-45.366 13.829Q-45.344 13.853-45.344 13.887L-45.344 13.908Q-45.429 14.195-45.617 14.393Q-45.805 14.591-46.070 14.694Q-46.335 14.796-46.629 14.796Q-47.059 14.796-47.447 14.590Q-47.835 14.383-48.069 14.020Q-48.304 13.658-48.304 13.217M-44.229 13.887L-44.229 11.990L-44.869 11.990L-44.869 11.768Q-44.551 11.768-44.334 11.558Q-44.117 11.348-44.016 11.038Q-43.915 10.729-43.915 10.421L-43.648 10.421L-43.648 11.710L-42.572 11.710L-42.572 11.990L-43.648 11.990L-43.648 13.874Q-43.648 14.150-43.544 14.349Q-43.440 14.547-43.180 14.547Q-43.023 14.547-42.917 14.443Q-42.811 14.338-42.761 14.185Q-42.712 14.031-42.712 13.874L-42.712 13.460L-42.445 13.460L-42.445 13.887Q-42.445 14.113-42.544 14.323Q-42.643 14.533-42.828 14.665Q-43.013 14.796-43.242 14.796Q-43.679 14.796-43.954 14.559Q-44.229 14.321-44.229 13.887M-40.018 14.728L-41.570 14.728L-41.570 14.448Q-41.345 14.448-41.196 14.414Q-41.047 14.379-41.047 14.239L-41.047 12.390Q-41.047 12.202-41.095 12.118Q-41.143 12.035-41.240 12.016Q-41.338 11.997-41.550 11.997L-41.550 11.717L-40.494 11.642L-40.494 14.239Q-40.494 14.379-40.362 14.414Q-40.230 14.448-40.018 14.448L-40.018 14.728M-41.290 10.421Q-41.290 10.250-41.167 10.131Q-41.044 10.011-40.873 10.011Q-40.705 10.011-40.582 10.131Q-40.459 10.250-40.459 10.421Q-40.459 10.596-40.582 10.719Q-40.705 10.842-40.873 10.842Q-41.044 10.842-41.167 10.719Q-41.290 10.596-41.290 10.421M-39.413 13.245Q-39.413 12.903-39.278 12.604Q-39.143 12.305-38.904 12.081Q-38.665 11.857-38.347 11.732Q-38.029 11.607-37.698 11.607Q-37.253 11.607-36.853 11.823Q-36.454 12.038-36.219 12.416Q-35.985 12.793-35.985 13.245Q-35.985 13.586-36.127 13.870Q-36.269 14.154-36.513 14.361Q-36.758 14.567-37.067 14.682Q-37.376 14.796-37.698 14.796Q-38.128 14.796-38.530 14.595Q-38.932 14.393-39.173 14.041Q-39.413 13.689-39.413 13.245M-37.698 14.547Q-37.096 14.547-36.872 14.169Q-36.648 13.791-36.648 13.159Q-36.648 12.547-36.882 12.188Q-37.117 11.830-37.698 11.830Q-38.750 11.830-38.750 13.159Q-38.750 13.791-38.525 14.169Q-38.299 14.547-37.698 14.547M-33.709 14.728L-35.343 14.728L-35.343 14.448Q-35.114 14.448-34.965 14.414Q-34.816 14.379-34.816 14.239L-34.816 12.390Q-34.816 12.120-34.924 12.059Q-35.032 11.997-35.343 11.997L-35.343 11.717L-34.283 11.642L-34.283 12.291Q-34.112 11.983-33.808 11.812Q-33.504 11.642-33.159 11.642Q-32.653 11.642-32.369 11.865Q-32.085 12.089-32.085 12.585L-32.085 14.239Q-32.085 14.376-31.937 14.412Q-31.788 14.448-31.562 14.448L-31.562 14.728L-33.193 14.728L-33.193 14.448Q-32.964 14.448-32.815 14.414Q-32.666 14.379-32.666 14.239L-32.666 12.599Q-32.666 12.264-32.786 12.064Q-32.906 11.864-33.220 11.864Q-33.490 11.864-33.724 12Q-33.958 12.137-34.097 12.371Q-34.235 12.605-34.235 12.879L-34.235 14.239Q-34.235 14.376-34.085 14.412Q-33.934 14.448-33.709 14.448\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-39.596-25.106H15.96\"\u002F>\u003Cpath stroke=\"none\" d=\"m18.559-25.106-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(30.446 -43.567)\">\u003Cpath d=\"M-52.286 13.217Q-52.286 12.889-52.151 12.588Q-52.016 12.288-51.780 12.067Q-51.544 11.847-51.240 11.727Q-50.935 11.607-50.611 11.607Q-50.105 11.607-49.756 11.710Q-49.408 11.812-49.408 12.188Q-49.408 12.335-49.505 12.436Q-49.602 12.537-49.749 12.537Q-49.903 12.537-50.002 12.438Q-50.101 12.339-50.101 12.188Q-50.101 12-49.961 11.908Q-50.163 11.857-50.604 11.857Q-50.959 11.857-51.188 12.053Q-51.417 12.250-51.518 12.559Q-51.619 12.869-51.619 13.217Q-51.619 13.566-51.493 13.872Q-51.366 14.178-51.111 14.362Q-50.857 14.547-50.501 14.547Q-50.279 14.547-50.095 14.463Q-49.910 14.379-49.775 14.224Q-49.640 14.068-49.582 13.860Q-49.568 13.805-49.514 13.805L-49.401 13.805Q-49.370 13.805-49.348 13.829Q-49.326 13.853-49.326 13.887L-49.326 13.908Q-49.411 14.195-49.599 14.393Q-49.787 14.591-50.052 14.694Q-50.317 14.796-50.611 14.796Q-51.041 14.796-51.429 14.590Q-51.817 14.383-52.051 14.020Q-52.286 13.658-52.286 13.217M-48.680 14Q-48.680 13.668-48.456 13.441Q-48.232 13.214-47.888 13.086Q-47.545 12.957-47.172 12.905Q-46.800 12.852-46.496 12.852L-46.496 12.599Q-46.496 12.394-46.603 12.214Q-46.711 12.035-46.892 11.932Q-47.073 11.830-47.282 11.830Q-47.688 11.830-47.924 11.922Q-47.835 11.959-47.789 12.043Q-47.743 12.127-47.743 12.229Q-47.743 12.325-47.789 12.404Q-47.835 12.482-47.916 12.527Q-47.996 12.571-48.085 12.571Q-48.235 12.571-48.336 12.474Q-48.437 12.376-48.437 12.229Q-48.437 11.607-47.282 11.607Q-47.070 11.607-46.820 11.671Q-46.571 11.734-46.369 11.853Q-46.167 11.973-46.041 12.158Q-45.914 12.342-45.914 12.585L-45.914 14.161Q-45.914 14.277-45.853 14.373Q-45.791 14.468-45.679 14.468Q-45.569 14.468-45.504 14.374Q-45.439 14.280-45.439 14.161L-45.439 13.713L-45.173 13.713L-45.173 14.161Q-45.173 14.431-45.400 14.596Q-45.627 14.762-45.908 14.762Q-46.116 14.762-46.253 14.608Q-46.390 14.455-46.413 14.239Q-46.560 14.506-46.842 14.651Q-47.124 14.796-47.449 14.796Q-47.726 14.796-48.010 14.721Q-48.293 14.646-48.486 14.467Q-48.680 14.287-48.680 14M-48.064 14Q-48.064 14.174-47.964 14.304Q-47.863 14.434-47.707 14.504Q-47.552 14.574-47.388 14.574Q-47.169 14.574-46.960 14.477Q-46.752 14.379-46.624 14.198Q-46.496 14.017-46.496 13.791L-46.496 13.063Q-46.820 13.063-47.186 13.154Q-47.552 13.245-47.808 13.457Q-48.064 13.668-48.064 14M-44.756 13.217Q-44.756 12.889-44.621 12.588Q-44.486 12.288-44.250 12.067Q-44.014 11.847-43.710 11.727Q-43.406 11.607-43.081 11.607Q-42.575 11.607-42.226 11.710Q-41.878 11.812-41.878 12.188Q-41.878 12.335-41.975 12.436Q-42.073 12.537-42.220 12.537Q-42.373 12.537-42.473 12.438Q-42.572 12.339-42.572 12.188Q-42.572 12-42.432 11.908Q-42.633 11.857-43.074 11.857Q-43.430 11.857-43.659 12.053Q-43.888 12.250-43.988 12.559Q-44.089 12.869-44.089 13.217Q-44.089 13.566-43.963 13.872Q-43.836 14.178-43.582 14.362Q-43.327 14.547-42.972 14.547Q-42.749 14.547-42.565 14.463Q-42.380 14.379-42.245 14.224Q-42.110 14.068-42.052 13.860Q-42.038 13.805-41.984 13.805L-41.871 13.805Q-41.840 13.805-41.818 13.829Q-41.796 13.853-41.796 13.887L-41.796 13.908Q-41.881 14.195-42.069 14.393Q-42.257 14.591-42.522 14.694Q-42.787 14.796-43.081 14.796Q-43.512 14.796-43.900 14.590Q-44.288 14.383-44.522 14.020Q-44.756 13.658-44.756 13.217\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(30.446 -43.567)\">\u003Cpath d=\"M-39.722 14.728L-41.356 14.728L-41.356 14.448Q-41.127 14.448-40.978 14.414Q-40.829 14.379-40.829 14.239L-40.829 10.620Q-40.829 10.350-40.937 10.288Q-41.045 10.227-41.356 10.227L-41.356 9.946L-40.276 9.871L-40.276 12.257Q-40.170 12.072-39.992 11.930Q-39.814 11.789-39.606 11.715Q-39.397 11.642-39.172 11.642Q-38.666 11.642-38.382 11.865Q-38.098 12.089-38.098 12.585L-38.098 14.239Q-38.098 14.376-37.950 14.412Q-37.801 14.448-37.575 14.448L-37.575 14.728L-39.206 14.728L-39.206 14.448Q-38.977 14.448-38.828 14.414Q-38.679 14.379-38.679 14.239L-38.679 12.599Q-38.679 12.264-38.799 12.064Q-38.919 11.864-39.233 11.864Q-39.503 11.864-39.737 12Q-39.971 12.137-40.110 12.371Q-40.248 12.605-40.248 12.879L-40.248 14.239Q-40.248 14.376-40.098 14.412Q-39.947 14.448-39.722 14.448L-39.722 14.728M-37.029 13.193Q-37.029 12.872-36.904 12.583Q-36.779 12.294-36.553 12.071Q-36.328 11.847-36.032 11.727Q-35.737 11.607-35.419 11.607Q-35.091 11.607-34.829 11.707Q-34.568 11.806-34.392 11.988Q-34.216 12.171-34.122 12.429Q-34.028 12.687-34.028 13.019Q-34.028 13.111-34.110 13.132L-36.365 13.132L-36.365 13.193Q-36.365 13.781-36.082 14.164Q-35.798 14.547-35.231 14.547Q-34.909 14.547-34.641 14.354Q-34.373 14.161-34.284 13.846Q-34.277 13.805-34.202 13.791L-34.110 13.791Q-34.028 13.815-34.028 13.887Q-34.028 13.894-34.034 13.921Q-34.147 14.318-34.518 14.557Q-34.889 14.796-35.313 14.796Q-35.750 14.796-36.150 14.588Q-36.550 14.379-36.789 14.012Q-37.029 13.645-37.029 13.193M-36.359 12.923L-34.544 12.923Q-34.544 12.646-34.641 12.394Q-34.739 12.141-34.937 11.985Q-35.135 11.830-35.419 11.830Q-35.696 11.830-35.909 11.988Q-36.123 12.147-36.241 12.402Q-36.359 12.657-36.359 12.923M-33.440 13.217Q-33.440 12.879-33.300 12.588Q-33.159 12.298-32.915 12.084Q-32.671 11.871-32.366 11.756Q-32.062 11.642-31.738 11.642Q-31.468 11.642-31.204 11.741Q-30.941 11.840-30.750 12.018L-30.750 10.620Q-30.750 10.350-30.857 10.288Q-30.965 10.227-31.276 10.227L-31.276 9.946L-30.199 9.871L-30.199 14.055Q-30.199 14.243-30.145 14.326Q-30.090 14.410-29.989 14.429Q-29.888 14.448-29.673 14.448L-29.673 14.728L-30.781 14.796L-30.781 14.379Q-31.197 14.796-31.823 14.796Q-32.254 14.796-32.626 14.584Q-32.999 14.373-33.219 14.012Q-33.440 13.651-33.440 13.217M-31.765 14.574Q-31.556 14.574-31.370 14.502Q-31.184 14.431-31.030 14.294Q-30.876 14.157-30.781 13.979L-30.781 12.370Q-30.866 12.223-31.011 12.103Q-31.156 11.983-31.326 11.924Q-31.495 11.864-31.676 11.864Q-32.237 11.864-32.505 12.253Q-32.773 12.643-32.773 13.224Q-32.773 13.795-32.539 14.185Q-32.305 14.574-31.765 14.574\" 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(-6.234 -8.476)\">\u003Cpath d=\"M-39.906 6.728L-41.540 6.728L-41.540 6.448Q-41.311 6.448-41.162 6.414Q-41.013 6.379-41.013 6.239L-41.013 4.390Q-41.013 4.120-41.121 4.059Q-41.229 3.997-41.540 3.997L-41.540 3.717L-40.480 3.642L-40.480 4.291Q-40.309 3.983-40.005 3.812Q-39.701 3.642-39.356 3.642Q-38.850 3.642-38.566 3.865Q-38.282 4.089-38.282 4.585L-38.282 6.239Q-38.282 6.376-38.134 6.412Q-37.985 6.448-37.759 6.448L-37.759 6.728L-39.390 6.728L-39.390 6.448Q-39.161 6.448-39.012 6.414Q-38.863 6.379-38.863 6.239L-38.863 4.599Q-38.863 4.264-38.983 4.064Q-39.103 3.864-39.417 3.864Q-39.687 3.864-39.921 4Q-40.155 4.137-40.294 4.371Q-40.432 4.605-40.432 4.879L-40.432 6.239Q-40.432 6.376-40.282 6.412Q-40.131 6.448-39.906 6.448L-39.906 6.728M-37.213 5.193Q-37.213 4.872-37.088 4.583Q-36.963 4.294-36.737 4.071Q-36.512 3.847-36.216 3.727Q-35.921 3.607-35.603 3.607Q-35.275 3.607-35.013 3.707Q-34.752 3.806-34.576 3.988Q-34.400 4.171-34.306 4.429Q-34.212 4.687-34.212 5.019Q-34.212 5.111-34.294 5.132L-36.549 5.132L-36.549 5.193Q-36.549 5.781-36.266 6.164Q-35.982 6.547-35.415 6.547Q-35.093 6.547-34.825 6.354Q-34.557 6.161-34.468 5.846Q-34.461 5.805-34.386 5.791L-34.294 5.791Q-34.212 5.815-34.212 5.887Q-34.212 5.894-34.218 5.921Q-34.331 6.318-34.702 6.557Q-35.073 6.796-35.497 6.796Q-35.934 6.796-36.334 6.588Q-36.734 6.379-36.973 6.012Q-37.213 5.645-37.213 5.193M-36.543 4.923L-34.728 4.923Q-34.728 4.646-34.825 4.394Q-34.923 4.141-35.121 3.985Q-35.319 3.830-35.603 3.830Q-35.880 3.830-36.093 3.988Q-36.307 4.147-36.425 4.402Q-36.543 4.657-36.543 4.923M-33.665 5.193Q-33.665 4.872-33.540 4.583Q-33.415 4.294-33.190 4.071Q-32.964 3.847-32.668 3.727Q-32.373 3.607-32.055 3.607Q-31.727 3.607-31.465 3.707Q-31.204 3.806-31.028 3.988Q-30.852 4.171-30.758 4.429Q-30.664 4.687-30.664 5.019Q-30.664 5.111-30.746 5.132L-33.002 5.132L-33.002 5.193Q-33.002 5.781-32.718 6.164Q-32.434 6.547-31.867 6.547Q-31.546 6.547-31.277 6.354Q-31.009 6.161-30.920 5.846Q-30.913 5.805-30.838 5.791L-30.746 5.791Q-30.664 5.815-30.664 5.887Q-30.664 5.894-30.671 5.921Q-30.783 6.318-31.154 6.557Q-31.525 6.796-31.949 6.796Q-32.386 6.796-32.786 6.588Q-33.186 6.379-33.425 6.012Q-33.665 5.645-33.665 5.193M-32.995 4.923L-31.180 4.923Q-31.180 4.646-31.277 4.394Q-31.375 4.141-31.573 3.985Q-31.771 3.830-32.055 3.830Q-32.332 3.830-32.545 3.988Q-32.759 4.147-32.877 4.402Q-32.995 4.657-32.995 4.923M-30.076 5.217Q-30.076 4.879-29.936 4.588Q-29.796 4.298-29.551 4.084Q-29.307 3.871-29.003 3.756Q-28.698 3.642-28.374 3.642Q-28.104 3.642-27.840 3.741Q-27.577 3.840-27.386 4.018L-27.386 2.620Q-27.386 2.350-27.494 2.288Q-27.601 2.227-27.912 2.227L-27.912 1.946L-26.836 1.871L-26.836 6.055Q-26.836 6.243-26.781 6.326Q-26.726 6.410-26.625 6.429Q-26.525 6.448-26.309 6.448L-26.309 6.728L-27.417 6.796L-27.417 6.379Q-27.834 6.796-28.459 6.796Q-28.890 6.796-29.262 6.584Q-29.635 6.373-29.855 6.012Q-30.076 5.651-30.076 5.217M-28.401 6.574Q-28.193 6.574-28.006 6.502Q-27.820 6.431-27.666 6.294Q-27.512 6.157-27.417 5.979L-27.417 4.370Q-27.502 4.223-27.647 4.103Q-27.793 3.983-27.962 3.924Q-28.131 3.864-28.312 3.864Q-28.873 3.864-29.141 4.253Q-29.409 4.643-29.409 5.224Q-29.409 5.795-29.175 6.185Q-28.941 6.574-28.401 6.574M-25.660 6.721L-25.660 5.658Q-25.660 5.634-25.632 5.607Q-25.605 5.580-25.581 5.580L-25.472 5.580Q-25.407 5.580-25.393 5.638Q-25.298 6.072-25.051 6.323Q-24.805 6.574-24.392 6.574Q-24.050 6.574-23.797 6.441Q-23.544 6.308-23.544 6Q-23.544 5.843-23.638 5.728Q-23.732 5.614-23.871 5.545Q-24.009 5.477-24.176 5.439L-24.757 5.340Q-25.113 5.272-25.386 5.051Q-25.660 4.831-25.660 4.489Q-25.660 4.240-25.549 4.065Q-25.438 3.891-25.251 3.792Q-25.065 3.693-24.850 3.650Q-24.634 3.607-24.392 3.607Q-23.978 3.607-23.698 3.789L-23.483 3.614Q-23.472 3.611-23.465 3.609Q-23.459 3.607-23.448 3.607L-23.397 3.607Q-23.370 3.607-23.346 3.631Q-23.322 3.655-23.322 3.683L-23.322 4.530Q-23.322 4.551-23.346 4.578Q-23.370 4.605-23.397 4.605L-23.510 4.605Q-23.537 4.605-23.563 4.580Q-23.589 4.554-23.589 4.530Q-23.589 4.294-23.694 4.130Q-23.800 3.966-23.983 3.884Q-24.166 3.802-24.399 3.802Q-24.727 3.802-24.983 3.905Q-25.239 4.007-25.239 4.284Q-25.239 4.479-25.057 4.588Q-24.874 4.698-24.645 4.739L-24.070 4.845Q-23.824 4.893-23.611 5.021Q-23.397 5.149-23.260 5.352Q-23.124 5.556-23.124 5.805Q-23.124 6.318-23.489 6.557Q-23.855 6.796-24.392 6.796Q-24.887 6.796-25.219 6.502L-25.485 6.776Q-25.506 6.796-25.533 6.796L-25.581 6.796Q-25.605 6.796-25.632 6.769Q-25.660 6.742-25.660 6.721\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M-18.031 6.728L-19.767 6.728L-19.767 6.448Q-19.538 6.448-19.389 6.414Q-19.241 6.379-19.241 6.239L-19.241 4.390Q-19.241 4.120-19.348 4.059Q-19.456 3.997-19.767 3.997L-19.767 3.717L-18.738 3.642L-18.738 4.349Q-18.608 4.041-18.366 3.842Q-18.123 3.642-17.805 3.642Q-17.586 3.642-17.415 3.766Q-17.244 3.891-17.244 4.103Q-17.244 4.240-17.344 4.339Q-17.443 4.438-17.576 4.438Q-17.713 4.438-17.812 4.339Q-17.911 4.240-17.911 4.103Q-17.911 3.963-17.812 3.864Q-18.102 3.864-18.302 4.060Q-18.502 4.257-18.595 4.551Q-18.687 4.845-18.687 5.125L-18.687 6.239Q-18.687 6.448-18.031 6.448L-18.031 6.728M-16.701 5.193Q-16.701 4.872-16.576 4.583Q-16.451 4.294-16.226 4.071Q-16 3.847-15.705 3.727Q-15.409 3.607-15.091 3.607Q-14.763 3.607-14.501 3.707Q-14.240 3.806-14.064 3.988Q-13.888 4.171-13.794 4.429Q-13.700 4.687-13.700 5.019Q-13.700 5.111-13.782 5.132L-16.038 5.132L-16.038 5.193Q-16.038 5.781-15.754 6.164Q-15.470 6.547-14.903 6.547Q-14.582 6.547-14.314 6.354Q-14.045 6.161-13.956 5.846Q-13.949 5.805-13.874 5.791L-13.782 5.791Q-13.700 5.815-13.700 5.887Q-13.700 5.894-13.707 5.921Q-13.820 6.318-14.190 6.557Q-14.561 6.796-14.985 6.796Q-15.423 6.796-15.823 6.588Q-16.222 6.379-16.462 6.012Q-16.701 5.645-16.701 5.193M-16.031 4.923L-14.216 4.923Q-14.216 4.646-14.314 4.394Q-14.411 4.141-14.609 3.985Q-14.807 3.830-15.091 3.830Q-15.368 3.830-15.582 3.988Q-15.795 4.147-15.913 4.402Q-16.031 4.657-16.031 4.923M-11.724 6.701L-12.705 4.202Q-12.767 4.059-12.885 4.024Q-13.003 3.990-13.218 3.990L-13.218 3.710L-11.738 3.710L-11.738 3.990Q-12.117 3.990-12.117 4.151Q-12.117 4.161-12.104 4.202L-11.389 6.034L-10.716 4.329Q-10.747 4.257-10.747 4.229Q-10.747 4.202-10.774 4.202Q-10.836 4.055-10.954 4.023Q-11.072 3.990-11.283 3.990L-11.283 3.710L-9.886 3.710L-9.886 3.990Q-10.262 3.990-10.262 4.151Q-10.262 4.182-10.255 4.202L-9.499 6.140L-8.812 4.390Q-8.792 4.339-8.792 4.284Q-8.792 4.144-8.905 4.067Q-9.017 3.990-9.157 3.990L-9.157 3.710L-7.937 3.710L-7.937 3.990Q-8.142 3.990-8.298 4.096Q-8.453 4.202-8.525 4.390L-9.431 6.701Q-9.465 6.796-9.578 6.796L-9.646 6.796Q-9.756 6.796-9.793 6.701L-10.576 4.698L-11.362 6.701Q-11.396 6.796-11.509 6.796L-11.577 6.796Q-11.687 6.796-11.724 6.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M-7.549 6Q-7.549 5.668-7.326 5.441Q-7.102 5.214-6.758 5.086Q-6.415 4.957-6.042 4.905Q-5.670 4.852-5.365 4.852L-5.365 4.599Q-5.365 4.394-5.473 4.214Q-5.581 4.035-5.762 3.932Q-5.943 3.830-6.151 3.830Q-6.558 3.830-6.794 3.922Q-6.705 3.959-6.659 4.043Q-6.613 4.127-6.613 4.229Q-6.613 4.325-6.659 4.404Q-6.705 4.482-6.786 4.527Q-6.866 4.571-6.955 4.571Q-7.105 4.571-7.206 4.474Q-7.307 4.376-7.307 4.229Q-7.307 3.607-6.151 3.607Q-5.940 3.607-5.690 3.671Q-5.441 3.734-5.239 3.853Q-5.037 3.973-4.911 4.158Q-4.784 4.342-4.784 4.585L-4.784 6.161Q-4.784 6.277-4.723 6.373Q-4.661 6.468-4.548 6.468Q-4.439 6.468-4.374 6.374Q-4.309 6.280-4.309 6.161L-4.309 5.713L-4.043 5.713L-4.043 6.161Q-4.043 6.431-4.270 6.596Q-4.497 6.762-4.777 6.762Q-4.986 6.762-5.123 6.608Q-5.259 6.455-5.283 6.239Q-5.430 6.506-5.712 6.651Q-5.994 6.796-6.319 6.796Q-6.596 6.796-6.880 6.721Q-7.163 6.646-7.356 6.467Q-7.549 6.287-7.549 6M-6.934 6Q-6.934 6.174-6.833 6.304Q-6.733 6.434-6.577 6.504Q-6.422 6.574-6.257 6.574Q-6.039 6.574-5.830 6.477Q-5.622 6.379-5.494 6.198Q-5.365 6.017-5.365 5.791L-5.365 5.063Q-5.690 5.063-6.056 5.154Q-6.422 5.245-6.678 5.457Q-6.934 5.668-6.934 6M-1.876 6.728L-3.612 6.728L-3.612 6.448Q-3.383 6.448-3.234 6.414Q-3.086 6.379-3.086 6.239L-3.086 4.390Q-3.086 4.120-3.193 4.059Q-3.301 3.997-3.612 3.997L-3.612 3.717L-2.583 3.642L-2.583 4.349Q-2.453 4.041-2.211 3.842Q-1.968 3.642-1.650 3.642Q-1.431 3.642-1.260 3.766Q-1.089 3.891-1.089 4.103Q-1.089 4.240-1.189 4.339Q-1.288 4.438-1.421 4.438Q-1.558 4.438-1.657 4.339Q-1.756 4.240-1.756 4.103Q-1.756 3.963-1.657 3.864Q-1.947 3.864-2.147 4.060Q-2.347 4.257-2.440 4.551Q-2.532 4.845-2.532 5.125L-2.532 6.239Q-2.532 6.448-1.876 6.448L-1.876 6.728M-0.505 5.217Q-0.505 4.879-0.365 4.588Q-0.225 4.298 0.020 4.084Q0.264 3.871 0.568 3.756Q0.872 3.642 1.197 3.642Q1.467 3.642 1.730 3.741Q1.994 3.840 2.185 4.018L2.185 2.620Q2.185 2.350 2.077 2.288Q1.970 2.227 1.659 2.227L1.659 1.946L2.735 1.871L2.735 6.055Q2.735 6.243 2.790 6.326Q2.845 6.410 2.945 6.429Q3.046 6.448 3.262 6.448L3.262 6.728L2.154 6.796L2.154 6.379Q1.737 6.796 1.112 6.796Q0.681 6.796 0.308 6.584Q-0.064 6.373-0.285 6.012Q-0.505 5.651-0.505 5.217M1.170 6.574Q1.378 6.574 1.565 6.502Q1.751 6.431 1.905 6.294Q2.058 6.157 2.154 5.979L2.154 4.370Q2.069 4.223 1.923 4.103Q1.778 3.983 1.609 3.924Q1.440 3.864 1.259 3.864Q0.698 3.864 0.430 4.253Q0.161 4.643 0.161 5.224Q0.161 5.795 0.396 6.185Q0.630 6.574 1.170 6.574\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M7.142 5.887L7.142 3.990L6.503 3.990L6.503 3.768Q6.821 3.768 7.038 3.558Q7.255 3.348 7.355 3.038Q7.456 2.729 7.456 2.421L7.723 2.421L7.723 3.710L8.800 3.710L8.800 3.990L7.723 3.990L7.723 5.874Q7.723 6.150 7.827 6.349Q7.931 6.547 8.191 6.547Q8.348 6.547 8.454 6.443Q8.560 6.338 8.610 6.185Q8.659 6.031 8.659 5.874L8.659 5.460L8.926 5.460L8.926 5.887Q8.926 6.113 8.827 6.323Q8.728 6.533 8.543 6.665Q8.359 6.796 8.130 6.796Q7.692 6.796 7.417 6.559Q7.142 6.321 7.142 5.887M9.695 5.245Q9.695 4.903 9.830 4.604Q9.965 4.305 10.204 4.081Q10.444 3.857 10.761 3.732Q11.079 3.607 11.411 3.607Q11.855 3.607 12.255 3.823Q12.655 4.038 12.889 4.416Q13.123 4.793 13.123 5.245Q13.123 5.586 12.981 5.870Q12.840 6.154 12.595 6.361Q12.351 6.567 12.041 6.682Q11.732 6.796 11.411 6.796Q10.980 6.796 10.579 6.595Q10.177 6.393 9.936 6.041Q9.695 5.689 9.695 5.245M11.411 6.547Q12.012 6.547 12.236 6.169Q12.460 5.791 12.460 5.159Q12.460 4.547 12.226 4.188Q11.992 3.830 11.411 3.830Q10.358 3.830 10.358 5.159Q10.358 5.791 10.584 6.169Q10.809 6.547 11.411 6.547\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M18.218 6.728L16.485 6.728L16.485 6.448Q16.711 6.448 16.860 6.414Q17.008 6.379 17.008 6.239L17.008 3.990L16.420 3.990L16.420 3.710L17.008 3.710L17.008 2.893Q17.008 2.575 17.186 2.327Q17.364 2.080 17.654 1.939Q17.945 1.799 18.256 1.799Q18.512 1.799 18.716 1.941Q18.919 2.083 18.919 2.326Q18.919 2.462 18.820 2.561Q18.721 2.661 18.584 2.661Q18.447 2.661 18.348 2.561Q18.249 2.462 18.249 2.326Q18.249 2.145 18.389 2.052Q18.311 2.025 18.211 2.025Q18.003 2.025 17.849 2.158Q17.695 2.291 17.615 2.495Q17.535 2.698 17.535 2.907L17.535 3.710L18.423 3.710L18.423 3.990L17.562 3.990L17.562 6.239Q17.562 6.448 18.218 6.448L18.218 6.728M18.857 5.245Q18.857 4.903 18.992 4.604Q19.127 4.305 19.367 4.081Q19.606 3.857 19.924 3.732Q20.242 3.607 20.573 3.607Q21.018 3.607 21.418 3.823Q21.817 4.038 22.052 4.416Q22.286 4.793 22.286 5.245Q22.286 5.586 22.144 5.870Q22.002 6.154 21.758 6.361Q21.513 6.567 21.204 6.682Q20.895 6.796 20.573 6.796Q20.143 6.796 19.741 6.595Q19.339 6.393 19.098 6.041Q18.857 5.689 18.857 5.245M20.573 6.547Q21.175 6.547 21.399 6.169Q21.623 5.791 21.623 5.159Q21.623 4.547 21.388 4.188Q21.154 3.830 20.573 3.830Q19.521 3.830 19.521 5.159Q19.521 5.791 19.746 6.169Q19.972 6.547 20.573 6.547M24.630 6.728L22.894 6.728L22.894 6.448Q23.123 6.448 23.272 6.414Q23.420 6.379 23.420 6.239L23.420 4.390Q23.420 4.120 23.313 4.059Q23.205 3.997 22.894 3.997L22.894 3.717L23.923 3.642L23.923 4.349Q24.053 4.041 24.295 3.842Q24.538 3.642 24.856 3.642Q25.075 3.642 25.246 3.766Q25.417 3.891 25.417 4.103Q25.417 4.240 25.317 4.339Q25.218 4.438 25.085 4.438Q24.948 4.438 24.849 4.339Q24.750 4.240 24.750 4.103Q24.750 3.963 24.849 3.864Q24.559 3.864 24.359 4.060Q24.159 4.257 24.066 4.551Q23.974 4.845 23.974 5.125L23.974 6.239Q23.974 6.448 24.630 6.448L24.630 6.728M27.683 6.728L26.049 6.728L26.049 6.448Q26.278 6.448 26.427 6.414Q26.575 6.379 26.575 6.239L26.575 4.390Q26.575 4.120 26.468 4.059Q26.360 3.997 26.049 3.997L26.049 3.717L27.108 3.642L27.108 4.291Q27.279 3.983 27.584 3.812Q27.888 3.642 28.233 3.642Q28.633 3.642 28.910 3.782Q29.187 3.922 29.272 4.270Q29.439 3.977 29.739 3.809Q30.038 3.642 30.383 3.642Q30.889 3.642 31.172 3.865Q31.456 4.089 31.456 4.585L31.456 6.239Q31.456 6.376 31.605 6.412Q31.753 6.448 31.979 6.448L31.979 6.728L30.349 6.728L30.349 6.448Q30.574 6.448 30.725 6.412Q30.875 6.376 30.875 6.239L30.875 4.599Q30.875 4.264 30.755 4.064Q30.636 3.864 30.321 3.864Q30.051 3.864 29.817 4Q29.583 4.137 29.445 4.371Q29.306 4.605 29.306 4.879L29.306 6.239Q29.306 6.376 29.455 6.412Q29.604 6.448 29.829 6.448L29.829 6.728L28.199 6.728L28.199 6.448Q28.428 6.448 28.576 6.414Q28.725 6.379 28.725 6.239L28.725 4.599Q28.725 4.264 28.606 4.064Q28.486 3.864 28.171 3.864Q27.901 3.864 27.667 4Q27.433 4.137 27.295 4.371Q27.156 4.605 27.156 4.879L27.156 6.239Q27.156 6.376 27.307 6.412Q27.457 6.448 27.683 6.448L27.683 6.728M33.066 7.958Q33.066 7.924 33.086 7.904Q33.326 7.665 33.461 7.338Q33.596 7.012 33.596 6.680Q33.510 6.728 33.387 6.728Q33.206 6.728 33.086 6.608Q32.967 6.489 32.967 6.308Q32.967 6.133 33.086 6.014Q33.206 5.894 33.387 5.894Q33.558 5.894 33.656 6.017Q33.753 6.140 33.787 6.316Q33.821 6.492 33.821 6.666Q33.821 7.049 33.674 7.413Q33.527 7.777 33.254 8.058Q33.227 8.085 33.189 8.085Q33.148 8.085 33.107 8.044Q33.066 8.003 33.066 7.958M32.967 4.124Q32.967 3.956 33.090 3.833Q33.213 3.710 33.387 3.710Q33.555 3.710 33.678 3.833Q33.801 3.956 33.801 4.124Q33.801 4.298 33.678 4.421Q33.555 4.544 33.387 4.544Q33.213 4.544 33.090 4.421Q32.967 4.298 32.967 4.124\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M-50.604 14.728L-52.238 14.728L-52.238 14.448Q-52.009 14.448-51.860 14.414Q-51.711 14.379-51.711 14.239L-51.711 12.390Q-51.711 12.120-51.819 12.059Q-51.927 11.997-52.238 11.997L-52.238 11.717L-51.178 11.642L-51.178 12.291Q-51.007 11.983-50.703 11.812Q-50.399 11.642-50.054 11.642Q-49.548 11.642-49.264 11.865Q-48.980 12.089-48.980 12.585L-48.980 14.239Q-48.980 14.376-48.832 14.412Q-48.683 14.448-48.457 14.448L-48.457 14.728L-50.088 14.728L-50.088 14.448Q-49.859 14.448-49.710 14.414Q-49.561 14.379-49.561 14.239L-49.561 12.599Q-49.561 12.264-49.681 12.064Q-49.801 11.864-50.115 11.864Q-50.385 11.864-50.619 12Q-50.853 12.137-50.992 12.371Q-51.130 12.605-51.130 12.879L-51.130 14.239Q-51.130 14.376-50.980 14.412Q-50.829 14.448-50.604 14.448L-50.604 14.728M-47.911 13.245Q-47.911 12.903-47.776 12.604Q-47.641 12.305-47.401 12.081Q-47.162 11.857-46.844 11.732Q-46.526 11.607-46.195 11.607Q-45.750 11.607-45.350 11.823Q-44.951 12.038-44.716 12.416Q-44.482 12.793-44.482 13.245Q-44.482 13.586-44.624 13.870Q-44.766 14.154-45.010 14.361Q-45.255 14.567-45.564 14.682Q-45.873 14.796-46.195 14.796Q-46.625 14.796-47.027 14.595Q-47.429 14.393-47.670 14.041Q-47.911 13.689-47.911 13.245M-46.195 14.547Q-45.593 14.547-45.369 14.169Q-45.145 13.791-45.145 13.159Q-45.145 12.547-45.380 12.188Q-45.614 11.830-46.195 11.830Q-47.247 11.830-47.247 13.159Q-47.247 13.791-47.022 14.169Q-46.796 14.547-46.195 14.547M-43.361 13.887L-43.361 11.990L-44 11.990L-44 11.768Q-43.683 11.768-43.465 11.558Q-43.248 11.348-43.148 11.038Q-43.047 10.729-43.047 10.421L-42.780 10.421L-42.780 11.710L-41.704 11.710L-41.704 11.990L-42.780 11.990L-42.780 13.874Q-42.780 14.150-42.676 14.349Q-42.572 14.547-42.312 14.547Q-42.155 14.547-42.049 14.443Q-41.943 14.338-41.893 14.185Q-41.844 14.031-41.844 13.874L-41.844 13.460L-41.577 13.460L-41.577 13.887Q-41.577 14.113-41.676 14.323Q-41.775 14.533-41.960 14.665Q-42.144 14.796-42.373 14.796Q-42.811 14.796-43.086 14.559Q-43.361 14.321-43.361 13.887M-39.085 14.728L-40.719 14.728L-40.719 14.448Q-40.490 14.448-40.341 14.414Q-40.193 14.379-40.193 14.239L-40.193 10.620Q-40.193 10.350-40.300 10.288Q-40.408 10.227-40.719 10.227L-40.719 9.946L-39.639 9.871L-39.639 12.257Q-39.533 12.072-39.355 11.930Q-39.178 11.789-38.969 11.715Q-38.761 11.642-38.535 11.642Q-38.029 11.642-37.746 11.865Q-37.462 12.089-37.462 12.585L-37.462 14.239Q-37.462 14.376-37.313 14.412Q-37.164 14.448-36.939 14.448L-36.939 14.728L-38.569 14.728L-38.569 14.448Q-38.340 14.448-38.192 14.414Q-38.043 14.379-38.043 14.239L-38.043 12.599Q-38.043 12.264-38.163 12.064Q-38.282 11.864-38.597 11.864Q-38.867 11.864-39.101 12Q-39.335 12.137-39.473 12.371Q-39.612 12.605-39.612 12.879L-39.612 14.239Q-39.612 14.376-39.461 14.412Q-39.311 14.448-39.085 14.448L-39.085 14.728M-34.734 14.728L-36.286 14.728L-36.286 14.448Q-36.060 14.448-35.912 14.414Q-35.763 14.379-35.763 14.239L-35.763 12.390Q-35.763 12.202-35.811 12.118Q-35.859 12.035-35.956 12.016Q-36.054 11.997-36.266 11.997L-36.266 11.717L-35.209 11.642L-35.209 14.239Q-35.209 14.379-35.078 14.414Q-34.946 14.448-34.734 14.448L-34.734 14.728M-36.006 10.421Q-36.006 10.250-35.883 10.131Q-35.760 10.011-35.589 10.011Q-35.421 10.011-35.298 10.131Q-35.175 10.250-35.175 10.421Q-35.175 10.596-35.298 10.719Q-35.421 10.842-35.589 10.842Q-35.760 10.842-35.883 10.719Q-36.006 10.596-36.006 10.421M-32.407 14.728L-34.040 14.728L-34.040 14.448Q-33.811 14.448-33.663 14.414Q-33.514 14.379-33.514 14.239L-33.514 12.390Q-33.514 12.120-33.622 12.059Q-33.729 11.997-34.040 11.997L-34.040 11.717L-32.981 11.642L-32.981 12.291Q-32.810 11.983-32.506 11.812Q-32.202 11.642-31.856 11.642Q-31.350 11.642-31.067 11.865Q-30.783 12.089-30.783 12.585L-30.783 14.239Q-30.783 14.376-30.634 14.412Q-30.486 14.448-30.260 14.448L-30.260 14.728L-31.891 14.728L-31.891 14.448Q-31.662 14.448-31.513 14.414Q-31.364 14.379-31.364 14.239L-31.364 12.599Q-31.364 12.264-31.484 12.064Q-31.603 11.864-31.918 11.864Q-32.188 11.864-32.422 12Q-32.656 12.137-32.795 12.371Q-32.933 12.605-32.933 12.879L-32.933 14.239Q-32.933 14.376-32.783 14.412Q-32.632 14.448-32.407 14.448L-32.407 14.728M-29.713 15.261Q-29.713 15.015-29.517 14.831Q-29.320 14.646-29.064 14.567Q-29.201 14.455-29.272 14.294Q-29.344 14.133-29.344 13.952Q-29.344 13.631-29.132 13.385Q-29.467 13.087-29.467 12.677Q-29.467 12.216-29.078 11.929Q-28.688 11.642-28.209 11.642Q-27.738 11.642-27.403 11.888Q-27.228 11.734-27.018 11.652Q-26.808 11.570-26.579 11.570Q-26.415 11.570-26.294 11.677Q-26.172 11.785-26.172 11.949Q-26.172 12.045-26.244 12.117Q-26.316 12.188-26.408 12.188Q-26.507 12.188-26.577 12.115Q-26.647 12.041-26.647 11.942Q-26.647 11.888-26.634 11.857L-26.627 11.843Q-26.620 11.823-26.611 11.812Q-26.603 11.802-26.600 11.795Q-26.955 11.795-27.242 12.018Q-26.955 12.311-26.955 12.677Q-26.955 12.992-27.140 13.224Q-27.324 13.457-27.613 13.585Q-27.902 13.713-28.209 13.713Q-28.411 13.713-28.602 13.663Q-28.794 13.614-28.972 13.504Q-29.064 13.631-29.064 13.774Q-29.064 13.956-28.936 14.091Q-28.808 14.226-28.623 14.226L-27.991 14.226Q-27.543 14.226-27.174 14.297Q-26.805 14.369-26.545 14.598Q-26.285 14.827-26.285 15.261Q-26.285 15.582-26.581 15.784Q-26.876 15.986-27.280 16.075Q-27.683 16.164-27.997 16.164Q-28.315 16.164-28.719 16.075Q-29.122 15.986-29.418 15.784Q-29.713 15.582-29.713 15.261M-29.259 15.261Q-29.259 15.490-29.040 15.639Q-28.821 15.788-28.529 15.856Q-28.237 15.924-27.997 15.924Q-27.833 15.924-27.625 15.888Q-27.416 15.853-27.210 15.772Q-27.003 15.692-26.871 15.564Q-26.740 15.436-26.740 15.261Q-26.740 14.909-27.121 14.815Q-27.502 14.721-28.004 14.721L-28.623 14.721Q-28.862 14.721-29.060 14.872Q-29.259 15.022-29.259 15.261M-28.209 13.474Q-27.543 13.474-27.543 12.677Q-27.543 11.877-28.209 11.877Q-28.879 11.877-28.879 12.677Q-28.879 13.474-28.209 13.474\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M-21.312 14.728L-22.915 14.728L-22.915 14.448Q-22.689 14.448-22.540 14.414Q-22.392 14.379-22.392 14.239L-22.392 10.620Q-22.392 10.350-22.499 10.288Q-22.607 10.227-22.915 10.227L-22.915 9.946L-21.838 9.871L-21.838 14.239Q-21.838 14.376-21.688 14.412Q-21.537 14.448-21.312 14.448L-21.312 14.728M-20.758 13.193Q-20.758 12.872-20.633 12.583Q-20.508 12.294-20.283 12.071Q-20.057 11.847-19.762 11.727Q-19.466 11.607-19.148 11.607Q-18.820 11.607-18.558 11.707Q-18.297 11.806-18.121 11.988Q-17.945 12.171-17.851 12.429Q-17.757 12.687-17.757 13.019Q-17.757 13.111-17.839 13.132L-20.095 13.132L-20.095 13.193Q-20.095 13.781-19.811 14.164Q-19.527 14.547-18.960 14.547Q-18.639 14.547-18.370 14.354Q-18.102 14.161-18.013 13.846Q-18.006 13.805-17.931 13.791L-17.839 13.791Q-17.757 13.815-17.757 13.887Q-17.757 13.894-17.764 13.921Q-17.877 14.318-18.247 14.557Q-18.618 14.796-19.042 14.796Q-19.480 14.796-19.879 14.588Q-20.279 14.379-20.519 14.012Q-20.758 13.645-20.758 13.193M-20.088 12.923L-18.273 12.923Q-18.273 12.646-18.370 12.394Q-18.468 12.141-18.666 11.985Q-18.864 11.830-19.148 11.830Q-19.425 11.830-19.638 11.988Q-19.852 12.147-19.970 12.402Q-20.088 12.657-20.088 12.923M-17.111 14Q-17.111 13.668-16.887 13.441Q-16.663 13.214-16.320 13.086Q-15.976 12.957-15.604 12.905Q-15.231 12.852-14.927 12.852L-14.927 12.599Q-14.927 12.394-15.034 12.214Q-15.142 12.035-15.323 11.932Q-15.504 11.830-15.713 11.830Q-16.120 11.830-16.356 11.922Q-16.267 11.959-16.221 12.043Q-16.174 12.127-16.174 12.229Q-16.174 12.325-16.221 12.404Q-16.267 12.482-16.347 12.527Q-16.427 12.571-16.516 12.571Q-16.667 12.571-16.767 12.474Q-16.868 12.376-16.868 12.229Q-16.868 11.607-15.713 11.607Q-15.501 11.607-15.252 11.671Q-15.002 11.734-14.800 11.853Q-14.599 11.973-14.472 12.158Q-14.346 12.342-14.346 12.585L-14.346 14.161Q-14.346 14.277-14.284 14.373Q-14.223 14.468-14.110 14.468Q-14.001 14.468-13.936 14.374Q-13.871 14.280-13.871 14.161L-13.871 13.713L-13.604 13.713L-13.604 14.161Q-13.604 14.431-13.831 14.596Q-14.059 14.762-14.339 14.762Q-14.547 14.762-14.684 14.608Q-14.821 14.455-14.845 14.239Q-14.992 14.506-15.274 14.651Q-15.556 14.796-15.880 14.796Q-16.157 14.796-16.441 14.721Q-16.725 14.646-16.918 14.467Q-17.111 14.287-17.111 14M-16.496 14Q-16.496 14.174-16.395 14.304Q-16.294 14.434-16.138 14.504Q-15.983 14.574-15.819 14.574Q-15.600 14.574-15.392 14.477Q-15.183 14.379-15.055 14.198Q-14.927 14.017-14.927 13.791L-14.927 13.063Q-15.252 13.063-15.617 13.154Q-15.983 13.245-16.239 13.457Q-16.496 13.668-16.496 14M-11.437 14.728L-13.173 14.728L-13.173 14.448Q-12.944 14.448-12.796 14.414Q-12.647 14.379-12.647 14.239L-12.647 12.390Q-12.647 12.120-12.755 12.059Q-12.862 11.997-13.173 11.997L-13.173 11.717L-12.145 11.642L-12.145 12.349Q-12.015 12.041-11.772 11.842Q-11.529 11.642-11.211 11.642Q-10.993 11.642-10.822 11.766Q-10.651 11.891-10.651 12.103Q-10.651 12.240-10.750 12.339Q-10.849 12.438-10.982 12.438Q-11.119 12.438-11.218 12.339Q-11.317 12.240-11.317 12.103Q-11.317 11.963-11.218 11.864Q-11.509 11.864-11.709 12.060Q-11.909 12.257-12.001 12.551Q-12.093 12.845-12.093 13.125L-12.093 14.239Q-12.093 14.448-11.437 14.448L-11.437 14.728M-8.385 14.728L-10.019 14.728L-10.019 14.448Q-9.790 14.448-9.641 14.414Q-9.492 14.379-9.492 14.239L-9.492 12.390Q-9.492 12.120-9.600 12.059Q-9.708 11.997-10.019 11.997L-10.019 11.717L-8.959 11.642L-8.959 12.291Q-8.788 11.983-8.484 11.812Q-8.180 11.642-7.835 11.642Q-7.329 11.642-7.045 11.865Q-6.761 12.089-6.761 12.585L-6.761 14.239Q-6.761 14.376-6.613 14.412Q-6.464 14.448-6.238 14.448L-6.238 14.728L-7.869 14.728L-7.869 14.448Q-7.640 14.448-7.491 14.414Q-7.342 14.379-7.342 14.239L-7.342 12.599Q-7.342 12.264-7.462 12.064Q-7.582 11.864-7.896 11.864Q-8.166 11.864-8.400 12Q-8.634 12.137-8.773 12.371Q-8.911 12.605-8.911 12.879L-8.911 14.239Q-8.911 14.376-8.761 14.412Q-8.610 14.448-8.385 14.448L-8.385 14.728M-5.691 13.193Q-5.691 12.872-5.567 12.583Q-5.442 12.294-5.216 12.071Q-4.991 11.847-4.695 11.727Q-4.399 11.607-4.082 11.607Q-3.753 11.607-3.492 11.707Q-3.231 11.806-3.054 11.988Q-2.878 12.171-2.784 12.429Q-2.690 12.687-2.690 13.019Q-2.690 13.111-2.773 13.132L-5.028 13.132L-5.028 13.193Q-5.028 13.781-4.745 14.164Q-4.461 14.547-3.894 14.547Q-3.572 14.547-3.304 14.354Q-3.036 14.161-2.947 13.846Q-2.940 13.805-2.865 13.791L-2.773 13.791Q-2.690 13.815-2.690 13.887Q-2.690 13.894-2.697 13.921Q-2.810 14.318-3.181 14.557Q-3.552 14.796-3.976 14.796Q-4.413 14.796-4.813 14.588Q-5.213 14.379-5.452 14.012Q-5.691 13.645-5.691 13.193M-5.022 12.923L-3.207 12.923Q-3.207 12.646-3.304 12.394Q-3.401 12.141-3.600 11.985Q-3.798 11.830-4.082 11.830Q-4.358 11.830-4.572 11.988Q-4.786 12.147-4.904 12.402Q-5.022 12.657-5.022 12.923M-2.103 13.217Q-2.103 12.879-1.962 12.588Q-1.822 12.298-1.578 12.084Q-1.334 11.871-1.029 11.756Q-0.725 11.642-0.400 11.642Q-0.130 11.642 0.133 11.741Q0.396 11.840 0.587 12.018L0.587 10.620Q0.587 10.350 0.480 10.288Q0.372 10.227 0.061 10.227L0.061 9.946L1.138 9.871L1.138 14.055Q1.138 14.243 1.192 14.326Q1.247 14.410 1.348 14.429Q1.449 14.448 1.664 14.448L1.664 14.728L0.557 14.796L0.557 14.379Q0.140 14.796-0.486 14.796Q-0.917 14.796-1.289 14.584Q-1.662 14.373-1.882 14.012Q-2.103 13.651-2.103 13.217M-0.428 14.574Q-0.219 14.574-0.033 14.502Q0.153 14.431 0.307 14.294Q0.461 14.157 0.557 13.979L0.557 12.370Q0.471 12.223 0.326 12.103Q0.181 11.983 0.011 11.924Q-0.158 11.864-0.339 11.864Q-0.899 11.864-1.168 12.253Q-1.436 12.643-1.436 13.224Q-1.436 13.795-1.202 14.185Q-0.968 14.574-0.428 14.574\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M5.609 13.894L5.609 12.390Q5.609 12.120 5.501 12.059Q5.393 11.997 5.082 11.997L5.082 11.717L6.190 11.642L6.190 13.874L6.190 13.894Q6.190 14.174 6.241 14.318Q6.292 14.461 6.434 14.518Q6.576 14.574 6.863 14.574Q7.116 14.574 7.321 14.434Q7.526 14.294 7.642 14.068Q7.759 13.843 7.759 13.593L7.759 12.390Q7.759 12.120 7.651 12.059Q7.543 11.997 7.232 11.997L7.232 11.717L8.340 11.642L8.340 14.055Q8.340 14.246 8.393 14.328Q8.446 14.410 8.546 14.429Q8.647 14.448 8.863 14.448L8.863 14.728L7.786 14.796L7.786 14.232Q7.677 14.414 7.531 14.537Q7.386 14.660 7.200 14.728Q7.013 14.796 6.812 14.796Q5.609 14.796 5.609 13.894M11.132 14.728L9.498 14.728L9.498 14.448Q9.727 14.448 9.876 14.414Q10.025 14.379 10.025 14.239L10.025 12.390Q10.025 12.120 9.917 12.059Q9.809 11.997 9.498 11.997L9.498 11.717L10.558 11.642L10.558 12.291Q10.729 11.983 11.033 11.812Q11.337 11.642 11.682 11.642Q12.188 11.642 12.472 11.865Q12.756 12.089 12.756 12.585L12.756 14.239Q12.756 14.376 12.904 14.412Q13.053 14.448 13.279 14.448L13.279 14.728L11.648 14.728L11.648 14.448Q11.877 14.448 12.026 14.414Q12.175 14.379 12.175 14.239L12.175 12.599Q12.175 12.264 12.055 12.064Q11.935 11.864 11.621 11.864Q11.351 11.864 11.117 12Q10.883 12.137 10.744 12.371Q10.606 12.605 10.606 12.879L10.606 14.239Q10.606 14.376 10.756 14.412Q10.907 14.448 11.132 14.448L11.132 14.728M15.616 14.728L13.880 14.728L13.880 14.448Q14.109 14.448 14.258 14.414Q14.407 14.379 14.407 14.239L14.407 12.390Q14.407 12.120 14.299 12.059Q14.191 11.997 13.880 11.997L13.880 11.717L14.909 11.642L14.909 12.349Q15.039 12.041 15.282 11.842Q15.524 11.642 15.842 11.642Q16.061 11.642 16.232 11.766Q16.403 11.891 16.403 12.103Q16.403 12.240 16.303 12.339Q16.204 12.438 16.071 12.438Q15.934 12.438 15.835 12.339Q15.736 12.240 15.736 12.103Q15.736 11.963 15.835 11.864Q15.545 11.864 15.345 12.060Q15.145 12.257 15.053 12.551Q14.960 12.845 14.960 13.125L14.960 14.239Q14.960 14.448 15.616 14.448L15.616 14.728M16.946 13.193Q16.946 12.872 17.071 12.583Q17.196 12.294 17.421 12.071Q17.647 11.847 17.942 11.727Q18.238 11.607 18.556 11.607Q18.884 11.607 19.146 11.707Q19.407 11.806 19.583 11.988Q19.759 12.171 19.853 12.429Q19.947 12.687 19.947 13.019Q19.947 13.111 19.865 13.132L17.609 13.132L17.609 13.193Q17.609 13.781 17.893 14.164Q18.177 14.547 18.744 14.547Q19.065 14.547 19.334 14.354Q19.602 14.161 19.691 13.846Q19.698 13.805 19.773 13.791L19.865 13.791Q19.947 13.815 19.947 13.887Q19.947 13.894 19.940 13.921Q19.827 14.318 19.457 14.557Q19.086 14.796 18.662 14.796Q18.224 14.796 17.824 14.588Q17.425 14.379 17.185 14.012Q16.946 13.645 16.946 13.193M17.616 12.923L19.431 12.923Q19.431 12.646 19.334 12.394Q19.236 12.141 19.038 11.985Q18.840 11.830 18.556 11.830Q18.279 11.830 18.065 11.988Q17.852 12.147 17.734 12.402Q17.616 12.657 17.616 12.923M21.923 14.701L20.942 12.202Q20.880 12.059 20.762 12.024Q20.644 11.990 20.429 11.990L20.429 11.710L21.909 11.710L21.909 11.990Q21.530 11.990 21.530 12.151Q21.530 12.161 21.543 12.202L22.258 14.034L22.931 12.329Q22.900 12.257 22.900 12.229Q22.900 12.202 22.873 12.202Q22.811 12.055 22.693 12.023Q22.575 11.990 22.364 11.990L22.364 11.710L23.762 11.710L23.762 11.990Q23.386 11.990 23.386 12.151Q23.386 12.182 23.392 12.202L24.148 14.140L24.835 12.390Q24.855 12.339 24.855 12.284Q24.855 12.144 24.742 12.067Q24.630 11.990 24.490 11.990L24.490 11.710L25.710 11.710L25.710 11.990Q25.505 11.990 25.349 12.096Q25.194 12.202 25.122 12.390L24.216 14.701Q24.182 14.796 24.069 14.796L24.001 14.796Q23.891 14.796 23.854 14.701L23.071 12.698L22.285 14.701Q22.251 14.796 22.138 14.796L22.070 14.796Q21.960 14.796 21.923 14.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-6.234 -8.476)\">\u003Cpath d=\"M26.100 14Q26.100 13.668 26.323 13.441Q26.547 13.214 26.891 13.086Q27.234 12.957 27.607 12.905Q27.979 12.852 28.284 12.852L28.284 12.599Q28.284 12.394 28.176 12.214Q28.068 12.035 27.887 11.932Q27.706 11.830 27.498 11.830Q27.091 11.830 26.855 11.922Q26.944 11.959 26.990 12.043Q27.036 12.127 27.036 12.229Q27.036 12.325 26.990 12.404Q26.944 12.482 26.863 12.527Q26.783 12.571 26.694 12.571Q26.544 12.571 26.443 12.474Q26.342 12.376 26.342 12.229Q26.342 11.607 27.498 11.607Q27.709 11.607 27.959 11.671Q28.208 11.734 28.410 11.853Q28.612 11.973 28.738 12.158Q28.865 12.342 28.865 12.585L28.865 14.161Q28.865 14.277 28.926 14.373Q28.988 14.468 29.101 14.468Q29.210 14.468 29.275 14.374Q29.340 14.280 29.340 14.161L29.340 13.713L29.606 13.713L29.606 14.161Q29.606 14.431 29.379 14.596Q29.152 14.762 28.872 14.762Q28.663 14.762 28.526 14.608Q28.390 14.455 28.366 14.239Q28.219 14.506 27.937 14.651Q27.655 14.796 27.330 14.796Q27.053 14.796 26.769 14.721Q26.486 14.646 26.293 14.467Q26.100 14.287 26.100 14M26.715 14Q26.715 14.174 26.816 14.304Q26.916 14.434 27.072 14.504Q27.227 14.574 27.392 14.574Q27.610 14.574 27.819 14.477Q28.027 14.379 28.155 14.198Q28.284 14.017 28.284 13.791L28.284 13.063Q27.959 13.063 27.593 13.154Q27.227 13.245 26.971 13.457Q26.715 13.668 26.715 14M31.773 14.728L30.037 14.728L30.037 14.448Q30.266 14.448 30.415 14.414Q30.563 14.379 30.563 14.239L30.563 12.390Q30.563 12.120 30.456 12.059Q30.348 11.997 30.037 11.997L30.037 11.717L31.066 11.642L31.066 12.349Q31.196 12.041 31.438 11.842Q31.681 11.642 31.999 11.642Q32.218 11.642 32.389 11.766Q32.560 11.891 32.560 12.103Q32.560 12.240 32.460 12.339Q32.361 12.438 32.228 12.438Q32.091 12.438 31.992 12.339Q31.893 12.240 31.893 12.103Q31.893 11.963 31.992 11.864Q31.702 11.864 31.502 12.060Q31.302 12.257 31.209 12.551Q31.117 12.845 31.117 13.125L31.117 14.239Q31.117 14.448 31.773 14.448L31.773 14.728M33.144 13.217Q33.144 12.879 33.284 12.588Q33.424 12.298 33.669 12.084Q33.913 11.871 34.217 11.756Q34.521 11.642 34.846 11.642Q35.116 11.642 35.379 11.741Q35.643 11.840 35.834 12.018L35.834 10.620Q35.834 10.350 35.726 10.288Q35.619 10.227 35.308 10.227L35.308 9.946L36.384 9.871L36.384 14.055Q36.384 14.243 36.439 14.326Q36.494 14.410 36.594 14.429Q36.695 14.448 36.911 14.448L36.911 14.728L35.803 14.796L35.803 14.379Q35.386 14.796 34.761 14.796Q34.330 14.796 33.957 14.584Q33.585 14.373 33.364 14.012Q33.144 13.651 33.144 13.217M34.819 14.574Q35.027 14.574 35.214 14.502Q35.400 14.431 35.554 14.294Q35.707 14.157 35.803 13.979L35.803 12.370Q35.718 12.223 35.572 12.103Q35.427 11.983 35.258 11.924Q35.089 11.864 34.908 11.864Q34.347 11.864 34.079 12.253Q33.810 12.643 33.810 13.224Q33.810 13.795 34.045 14.185Q34.279 14.574 34.819 14.574M37.519 13.193Q37.519 12.872 37.644 12.583Q37.769 12.294 37.994 12.071Q38.220 11.847 38.515 11.727Q38.811 11.607 39.129 11.607Q39.457 11.607 39.718 11.707Q39.980 11.806 40.156 11.988Q40.332 12.171 40.426 12.429Q40.520 12.687 40.520 13.019Q40.520 13.111 40.438 13.132L38.182 13.132L38.182 13.193Q38.182 13.781 38.466 14.164Q38.749 14.547 39.317 14.547Q39.638 14.547 39.906 14.354Q40.175 14.161 40.264 13.846Q40.270 13.805 40.346 13.791L40.438 13.791Q40.520 13.815 40.520 13.887Q40.520 13.894 40.513 13.921Q40.400 14.318 40.029 14.557Q39.659 14.796 39.235 14.796Q38.797 14.796 38.397 14.588Q37.998 14.379 37.758 14.012Q37.519 13.645 37.519 13.193M38.189 12.923L40.004 12.923Q40.004 12.646 39.906 12.394Q39.809 12.141 39.611 11.985Q39.413 11.830 39.129 11.830Q38.852 11.830 38.638 11.988Q38.425 12.147 38.307 12.402Q38.189 12.657 38.189 12.923M41.108 13.217Q41.108 12.879 41.248 12.588Q41.388 12.298 41.633 12.084Q41.877 11.871 42.181 11.756Q42.485 11.642 42.810 11.642Q43.080 11.642 43.343 11.741Q43.606 11.840 43.798 12.018L43.798 10.620Q43.798 10.350 43.690 10.288Q43.582 10.227 43.271 10.227L43.271 9.946L44.348 9.871L44.348 14.055Q44.348 14.243 44.403 14.326Q44.457 14.410 44.558 14.429Q44.659 14.448 44.874 14.448L44.874 14.728L43.767 14.796L43.767 14.379Q43.350 14.796 42.725 14.796Q42.294 14.796 41.921 14.584Q41.549 14.373 41.328 14.012Q41.108 13.651 41.108 13.217M42.783 14.574Q42.991 14.574 43.177 14.502Q43.364 14.431 43.518 14.294Q43.671 14.157 43.767 13.979L43.767 12.370Q43.682 12.223 43.536 12.103Q43.391 11.983 43.222 11.924Q43.053 11.864 42.872 11.864Q42.311 11.864 42.043 12.253Q41.774 12.643 41.774 13.224Q41.774 13.795 42.008 14.185Q42.243 14.574 42.783 14.574\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(176.64 -76.915)\">\u003Cpath d=\"M-52.086 14.847L-52.086 12.768Q-52.086 12.742-52.057 12.713Q-52.029 12.685-51.998 12.685L-51.888 12.685Q-51.857 12.685-51.829 12.713Q-51.800 12.742-51.800 12.768Q-51.800 13.370-51.543 13.781Q-51.286 14.192-50.825 14.401Q-50.363 14.609-49.783 14.609Q-49.449 14.609-49.177 14.434Q-48.904 14.258-48.750 13.966Q-48.597 13.673-48.597 13.353Q-48.597 13.137-48.665 12.935Q-48.733 12.733-48.851 12.568Q-48.970 12.403-49.150 12.282Q-49.330 12.162-49.533 12.118L-50.728 11.832Q-51.110 11.740-51.420 11.491Q-51.730 11.243-51.908 10.881Q-52.086 10.518-52.086 10.127Q-52.086 9.657-51.842 9.252Q-51.598 8.848-51.185 8.615Q-50.772 8.382-50.297 8.382Q-49.858 8.382-49.478 8.538Q-49.098 8.694-48.821 9.002L-48.452 8.417Q-48.434 8.382-48.390 8.382L-48.333 8.382Q-48.302 8.382-48.274 8.411Q-48.245 8.439-48.245 8.466L-48.245 10.544Q-48.245 10.566-48.278 10.595Q-48.311 10.624-48.333 10.624L-48.443 10.624Q-48.465 10.624-48.498 10.595Q-48.531 10.566-48.531 10.544Q-48.531 10.360-48.572 10.171Q-48.614 9.982-48.698 9.782Q-48.781 9.582-48.884 9.408Q-48.988 9.235-49.106 9.121Q-49.348 8.883-49.645 8.778Q-49.941 8.672-50.288 8.672Q-50.592 8.672-50.869 8.824Q-51.145 8.976-51.310 9.239Q-51.475 9.503-51.475 9.824Q-51.475 10.079-51.356 10.316Q-51.238 10.553-51.036 10.711Q-50.833 10.870-50.565 10.940L-49.370 11.226Q-48.961 11.322-48.645 11.597Q-48.329 11.872-48.155 12.256Q-47.981 12.641-47.981 13.054Q-47.981 13.537-48.219 13.972Q-48.456 14.407-48.869 14.666Q-49.282 14.926-49.774 14.926Q-50.271 14.926-50.730 14.772Q-51.189 14.618-51.501 14.306L-51.879 14.891Q-51.897 14.926-51.941 14.926L-51.998 14.926Q-52.020 14.926-52.053 14.897Q-52.086 14.869-52.086 14.847M-44.896 13.080L-47.362 13.080L-47.362 12.487L-44.896 12.487L-44.896 13.080M-43.872 14.847L-43.872 12.768Q-43.872 12.742-43.844 12.713Q-43.815 12.685-43.785 12.685L-43.675 12.685Q-43.644 12.685-43.615 12.713Q-43.587 12.742-43.587 12.768Q-43.587 13.370-43.330 13.781Q-43.073 14.192-42.611 14.401Q-42.150 14.609-41.570 14.609Q-41.236 14.609-40.963 14.434Q-40.691 14.258-40.537 13.966Q-40.383 13.673-40.383 13.353Q-40.383 13.137-40.451 12.935Q-40.519 12.733-40.638 12.568Q-40.757 12.403-40.937 12.282Q-41.117 12.162-41.319 12.118L-42.515 11.832Q-42.897 11.740-43.207 11.491Q-43.517 11.243-43.694 10.881Q-43.872 10.518-43.872 10.127Q-43.872 9.657-43.629 9.252Q-43.385 8.848-42.972 8.615Q-42.558 8.382-42.084 8.382Q-41.644 8.382-41.264 8.538Q-40.884 8.694-40.607 9.002L-40.238 8.417Q-40.221 8.382-40.177 8.382L-40.120 8.382Q-40.089 8.382-40.060 8.411Q-40.032 8.439-40.032 8.466L-40.032 10.544Q-40.032 10.566-40.065 10.595Q-40.098 10.624-40.120 10.624L-40.229 10.624Q-40.251 10.624-40.284 10.595Q-40.317 10.566-40.317 10.544Q-40.317 10.360-40.359 10.171Q-40.401 9.982-40.484 9.782Q-40.568 9.582-40.671 9.408Q-40.774 9.235-40.893 9.121Q-41.135 8.883-41.431 8.778Q-41.728 8.672-42.075 8.672Q-42.378 8.672-42.655 8.824Q-42.932 8.976-43.097 9.239Q-43.262 9.503-43.262 9.824Q-43.262 10.079-43.143 10.316Q-43.024 10.553-42.822 10.711Q-42.620 10.870-42.352 10.940L-41.157 11.226Q-40.748 11.322-40.432 11.597Q-40.115 11.872-39.942 12.256Q-39.768 12.641-39.768 13.054Q-39.768 13.537-40.005 13.972Q-40.243 14.407-40.656 14.666Q-41.069 14.926-41.561 14.926Q-42.058 14.926-42.517 14.772Q-42.976 14.618-43.288 14.306L-43.666 14.891Q-43.683 14.926-43.727 14.926L-43.785 14.926Q-43.807 14.926-43.840 14.897Q-43.872 14.869-43.872 14.847\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(176.64 -76.915)\">\u003Cpath d=\"M-33.303 16.969Q-33.808 16.582-34.177 16.077Q-34.547 15.572-34.790 14.972Q-35.034 14.372-35.149 13.755Q-35.263 13.137-35.263 12.478Q-35.263 11.819-35.149 11.204Q-35.034 10.588-34.795 9.995Q-34.555 9.402-34.182 8.892Q-33.808 8.382-33.303 7.996Q-33.268 7.978-33.246 7.978L-33.167 7.978Q-33.079 7.978-33.079 8.079Q-33.079 8.114-33.114 8.149Q-33.676 8.672-34.021 9.378Q-34.366 10.083-34.514 10.865Q-34.661 11.647-34.661 12.478Q-34.661 13.102-34.582 13.686Q-34.503 14.271-34.325 14.838Q-34.147 15.405-33.848 15.906Q-33.549 16.407-33.114 16.815Q-33.079 16.851-33.079 16.890Q-33.079 16.987-33.167 16.987L-33.246 16.987Q-33.268 16.987-33.303 16.969M-30.539 14.710L-31.730 11.463Q-31.804 11.265-31.949 11.215Q-32.094 11.164-32.384 11.164L-32.384 10.848L-30.504 10.848L-30.504 11.164Q-31.027 11.164-31.027 11.388Q-31.027 11.419-31.009 11.463L-30.143 13.845L-29.387 11.757L-29.497 11.463Q-29.572 11.265-29.719 11.215Q-29.866 11.164-30.152 11.164L-30.152 10.848L-28.363 10.848L-28.363 11.164Q-28.873 11.164-28.873 11.388Q-28.873 11.441-28.864 11.463L-27.946 13.963L-27.129 11.713Q-27.111 11.669-27.111 11.568Q-27.111 11.375-27.263 11.270Q-27.414 11.164-27.616 11.164L-27.616 10.848L-26.074 10.848L-26.074 11.164Q-26.342 11.164-26.538 11.313Q-26.733 11.463-26.821 11.713L-27.920 14.710Q-27.950 14.829-28.082 14.829L-28.144 14.829Q-28.262 14.829-28.306 14.710L-29.225 12.179L-30.152 14.710Q-30.196 14.829-30.315 14.829L-30.376 14.829Q-30.508 14.829-30.539 14.710\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(176.64 -76.915)\">\u003Cpath d=\"M-25.876 12.821Q-25.876 12.254-25.603 11.766Q-25.331 11.278-24.861 10.986Q-24.390 10.694-23.823 10.694Q-23.402 10.694-23.026 10.863Q-22.650 11.032-22.373 11.324Q-22.096 11.617-21.938 12.012Q-21.780 12.408-21.780 12.821Q-21.780 13.370-22.059 13.832Q-22.338 14.293-22.806 14.561Q-23.274 14.829-23.823 14.829Q-24.377 14.829-24.847 14.561Q-25.318 14.293-25.597 13.832Q-25.876 13.370-25.876 12.821M-23.823 14.539Q-23.327 14.539-23.050 14.278Q-22.773 14.016-22.681 13.612Q-22.589 13.207-22.589 12.711Q-22.589 12.236-22.687 11.847Q-22.786 11.458-23.059 11.208Q-23.331 10.957-23.823 10.957Q-24.535 10.957-24.799 11.452Q-25.063 11.946-25.063 12.711Q-25.063 13.511-24.808 14.025Q-24.553 14.539-23.823 14.539M-19.033 14.728L-21.266 14.728L-21.266 14.412Q-20.954 14.412-20.763 14.359Q-20.572 14.306-20.572 14.117L-20.572 11.669Q-20.572 11.428-20.642 11.320Q-20.712 11.212-20.846 11.188Q-20.980 11.164-21.266 11.164L-21.266 10.848L-19.952 10.751L-19.952 11.612Q-19.789 11.221-19.521 10.986Q-19.253 10.751-18.862 10.751Q-18.590 10.751-18.374 10.914Q-18.159 11.076-18.159 11.335Q-18.159 11.511-18.278 11.630Q-18.396 11.749-18.572 11.749Q-18.752 11.749-18.871 11.630Q-18.989 11.511-18.989 11.335Q-18.989 11.120-18.836 11.010L-18.853 11.010Q-19.231 11.010-19.464 11.272Q-19.697 11.533-19.796 11.920Q-19.895 12.307-19.895 12.667L-19.895 14.117Q-19.895 14.306-19.638 14.359Q-19.381 14.412-19.033 14.412L-19.033 14.728M-15.531 14.728L-17.592 14.728L-17.592 14.412Q-17.284 14.412-17.093 14.359Q-16.902 14.306-16.902 14.117L-16.902 9.402Q-16.902 9.160-16.972 9.052Q-17.043 8.945-17.177 8.921Q-17.311 8.896-17.592 8.896L-17.592 8.580L-16.225 8.483L-16.225 14.117Q-16.225 14.306-16.032 14.359Q-15.839 14.412-15.531 14.412L-15.531 14.728M-13.083 14.829Q-13.628 14.829-14.072 14.546Q-14.516 14.262-14.771 13.790Q-15.026 13.317-15.026 12.786Q-15.026 12.232-14.749 11.766Q-14.472 11.300-14 11.026Q-13.527 10.751-12.982 10.751Q-12.648 10.751-12.345 10.883Q-12.042 11.015-11.822 11.252L-11.822 9.402Q-11.822 9.160-11.892 9.052Q-11.963 8.945-12.097 8.921Q-12.231 8.896-12.516 8.896L-12.516 8.580L-11.150 8.483L-11.150 13.911Q-11.150 14.148-11.079 14.256Q-11.009 14.363-10.873 14.387Q-10.737 14.412-10.455 14.412L-10.455 14.728L-11.848 14.829L-11.848 14.280Q-12.099 14.543-12.417 14.686Q-12.736 14.829-13.083 14.829M-13.022 14.565Q-12.653 14.565-12.343 14.354Q-12.033 14.144-11.848 13.801L-11.848 11.669Q-12.020 11.366-12.301 11.188Q-12.582 11.010-12.921 11.010Q-13.611 11.010-13.914 11.531Q-14.217 12.052-14.217 12.794Q-14.217 13.515-13.947 14.040Q-13.677 14.565-13.022 14.565\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(176.64 -76.915)\">\u003Cpath d=\"M-4.714 14.728L-6.801 14.728L-6.801 14.412Q-6.494 14.412-6.302 14.359Q-6.111 14.306-6.111 14.117L-6.111 11.669Q-6.111 11.428-6.182 11.320Q-6.252 11.212-6.386 11.188Q-6.520 11.164-6.801 11.164L-6.801 10.848L-5.461 10.751L-5.461 11.586Q-5.263 11.208-4.903 10.979Q-4.542 10.751-4.121 10.751Q-3.075 10.751-2.890 11.560Q-2.688 11.190-2.330 10.971Q-1.972 10.751-1.554 10.751Q-0.930 10.751-0.605 11.045Q-0.280 11.340-0.280 11.964L-0.280 14.117Q-0.280 14.306-0.086 14.359Q0.107 14.412 0.415 14.412L0.415 14.728L-1.673 14.728L-1.673 14.412Q-1.365 14.412-1.172 14.359Q-0.978 14.306-0.978 14.117L-0.978 11.999Q-0.978 11.568-1.106 11.289Q-1.233 11.010-1.620 11.010Q-1.963 11.010-2.246 11.199Q-2.530 11.388-2.686 11.700Q-2.842 12.012-2.842 12.359L-2.842 14.117Q-2.842 14.306-2.651 14.359Q-2.459 14.412-2.152 14.412L-2.152 14.728L-4.239 14.728L-4.239 14.412Q-3.927 14.412-3.736 14.359Q-3.545 14.306-3.545 14.117L-3.545 11.999Q-3.545 11.740-3.589 11.518Q-3.633 11.296-3.778 11.153Q-3.923 11.010-4.182 11.010Q-4.718 11.010-5.063 11.417Q-5.408 11.823-5.408 12.359L-5.408 14.117Q-5.408 14.306-5.215 14.359Q-5.021 14.412-4.714 14.412L-4.714 14.728M0.863 12.821Q0.863 12.254 1.135 11.766Q1.408 11.278 1.878 10.986Q2.348 10.694 2.915 10.694Q3.337 10.694 3.713 10.863Q4.088 11.032 4.365 11.324Q4.642 11.617 4.800 12.012Q4.959 12.408 4.959 12.821Q4.959 13.370 4.680 13.832Q4.400 14.293 3.932 14.561Q3.464 14.829 2.915 14.829Q2.361 14.829 1.891 14.561Q1.421 14.293 1.142 13.832Q0.863 13.370 0.863 12.821M2.915 14.539Q3.412 14.539 3.689 14.278Q3.965 14.016 4.058 13.612Q4.150 13.207 4.150 12.711Q4.150 12.236 4.051 11.847Q3.952 11.458 3.680 11.208Q3.407 10.957 2.915 10.957Q2.203 10.957 1.940 11.452Q1.676 11.946 1.676 12.711Q1.676 13.511 1.931 14.025Q2.186 14.539 2.915 14.539\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(176.64 -76.915)\">\u003Cpath d=\"M7.740 14.829Q7.195 14.829 6.751 14.546Q6.307 14.262 6.053 13.790Q5.798 13.317 5.798 12.786Q5.798 12.232 6.074 11.766Q6.351 11.300 6.824 11.026Q7.296 10.751 7.841 10.751Q8.175 10.751 8.478 10.883Q8.782 11.015 9.001 11.252L9.001 9.402Q9.001 9.160 8.931 9.052Q8.861 8.945 8.727 8.921Q8.593 8.896 8.307 8.896L8.307 8.580L9.674 8.483L9.674 13.911Q9.674 14.148 9.744 14.256Q9.814 14.363 9.950 14.387Q10.087 14.412 10.368 14.412L10.368 14.728L8.975 14.829L8.975 14.280Q8.724 14.543 8.406 14.686Q8.087 14.829 7.740 14.829M7.802 14.565Q8.171 14.565 8.480 14.354Q8.790 14.144 8.975 13.801L8.975 11.669Q8.803 11.366 8.522 11.188Q8.241 11.010 7.903 11.010Q7.213 11.010 6.909 11.531Q6.606 12.052 6.606 12.794Q6.606 13.515 6.876 14.040Q7.147 14.565 7.802 14.565M12.939 14.829Q12.381 14.829 11.908 14.546Q11.436 14.262 11.161 13.785Q10.886 13.309 10.886 12.755Q10.886 12.359 11.029 11.984Q11.172 11.608 11.429 11.320Q11.686 11.032 12.044 10.863Q12.403 10.694 12.807 10.694Q13.352 10.694 13.723 10.931Q14.094 11.168 14.281 11.586Q14.468 12.003 14.468 12.540Q14.468 12.592 14.444 12.630Q14.420 12.667 14.371 12.667L11.699 12.667L11.699 12.746Q11.699 13.493 12.011 14.016Q12.323 14.539 13.022 14.539Q13.427 14.539 13.747 14.282Q14.068 14.025 14.191 13.621Q14.209 13.541 14.292 13.541L14.371 13.541Q14.411 13.541 14.439 13.572Q14.468 13.603 14.468 13.647L14.468 13.682Q14.363 14.025 14.141 14.284Q13.919 14.543 13.605 14.686Q13.290 14.829 12.939 14.829M11.708 12.416L13.822 12.416Q13.822 12.148 13.769 11.902Q13.717 11.656 13.596 11.434Q13.475 11.212 13.277 11.085Q13.079 10.957 12.807 10.957Q12.464 10.957 12.211 11.182Q11.959 11.406 11.834 11.744Q11.708 12.082 11.708 12.416M17.109 14.728L15.048 14.728L15.048 14.412Q15.356 14.412 15.547 14.359Q15.738 14.306 15.738 14.117L15.738 9.402Q15.738 9.160 15.668 9.052Q15.597 8.945 15.463 8.921Q15.329 8.896 15.048 8.896L15.048 8.580L16.415 8.483L16.415 14.117Q16.415 14.306 16.608 14.359Q16.802 14.412 17.109 14.412L17.109 14.728M17.992 16.987L17.909 16.987Q17.821 16.987 17.821 16.890Q17.821 16.851 17.856 16.815Q18.691 16.042 19.052 14.908Q19.412 13.774 19.412 12.478Q19.412 11.858 19.333 11.267Q19.254 10.676 19.073 10.118Q18.893 9.560 18.592 9.052Q18.291 8.545 17.856 8.149Q17.821 8.114 17.821 8.079Q17.821 7.978 17.909 7.978L17.992 7.978Q18.010 7.978 18.045 7.996Q18.551 8.378 18.924 8.892Q19.298 9.406 19.535 9.984Q19.772 10.562 19.889 11.190Q20.005 11.819 20.005 12.478Q20.005 13.137 19.889 13.768Q19.772 14.398 19.533 14.983Q19.293 15.567 18.922 16.077Q18.551 16.587 18.045 16.969Q18.010 16.987 17.992 16.987\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M105.313-15.147h25.608v-19.917h-25.608Z\"\u002F>\u003Cg transform=\"translate(162.237 -37.68)\">\u003Cpath d=\"M-52.286 14.721L-52.286 13.658Q-52.286 13.634-52.258 13.607Q-52.231 13.580-52.207 13.580L-52.098 13.580Q-52.033 13.580-52.019 13.638Q-51.923 14.072-51.677 14.323Q-51.431 14.574-51.017 14.574Q-50.676 14.574-50.423 14.441Q-50.170 14.308-50.170 14Q-50.170 13.843-50.264 13.728Q-50.358 13.614-50.496 13.545Q-50.635 13.477-50.802 13.439L-51.383 13.340Q-51.739 13.272-52.012 13.051Q-52.286 12.831-52.286 12.489Q-52.286 12.240-52.174 12.065Q-52.063 11.891-51.877 11.792Q-51.691 11.693-51.475 11.650Q-51.260 11.607-51.017 11.607Q-50.604 11.607-50.324 11.789L-50.108 11.614Q-50.098 11.611-50.091 11.609Q-50.084 11.607-50.074 11.607L-50.023 11.607Q-49.996 11.607-49.972 11.631Q-49.948 11.655-49.948 11.683L-49.948 12.530Q-49.948 12.551-49.972 12.578Q-49.996 12.605-50.023 12.605L-50.136 12.605Q-50.163 12.605-50.189 12.580Q-50.214 12.554-50.214 12.530Q-50.214 12.294-50.320 12.130Q-50.426 11.966-50.609 11.884Q-50.792 11.802-51.024 11.802Q-51.352 11.802-51.609 11.905Q-51.865 12.007-51.865 12.284Q-51.865 12.479-51.682 12.588Q-51.499 12.698-51.270 12.739L-50.696 12.845Q-50.450 12.893-50.236 13.021Q-50.023 13.149-49.886 13.352Q-49.749 13.556-49.749 13.805Q-49.749 14.318-50.115 14.557Q-50.481 14.796-51.017 14.796Q-51.513 14.796-51.845 14.502L-52.111 14.776Q-52.132 14.796-52.159 14.796L-52.207 14.796Q-52.231 14.796-52.258 14.769Q-52.286 14.742-52.286 14.721M-48.594 13.887L-48.594 11.990L-49.233 11.990L-49.233 11.768Q-48.915 11.768-48.698 11.558Q-48.481 11.348-48.381 11.038Q-48.280 10.729-48.280 10.421L-48.013 10.421L-48.013 11.710L-46.936 11.710L-46.936 11.990L-48.013 11.990L-48.013 13.874Q-48.013 14.150-47.909 14.349Q-47.805 14.547-47.545 14.547Q-47.388 14.547-47.282 14.443Q-47.176 14.338-47.126 14.185Q-47.077 14.031-47.077 13.874L-47.077 13.460L-46.810 13.460L-46.810 13.887Q-46.810 14.113-46.909 14.323Q-47.008 14.533-47.193 14.665Q-47.377 14.796-47.606 14.796Q-48.044 14.796-48.319 14.559Q-48.594 14.321-48.594 13.887M-45.942 14Q-45.942 13.668-45.718 13.441Q-45.494 13.214-45.151 13.086Q-44.807 12.957-44.434 12.905Q-44.062 12.852-43.758 12.852L-43.758 12.599Q-43.758 12.394-43.865 12.214Q-43.973 12.035-44.154 11.932Q-44.335 11.830-44.544 11.830Q-44.951 11.830-45.186 11.922Q-45.098 11.959-45.051 12.043Q-45.005 12.127-45.005 12.229Q-45.005 12.325-45.051 12.404Q-45.098 12.482-45.178 12.527Q-45.258 12.571-45.347 12.571Q-45.497 12.571-45.598 12.474Q-45.699 12.376-45.699 12.229Q-45.699 11.607-44.544 11.607Q-44.332 11.607-44.082 11.671Q-43.833 11.734-43.631 11.853Q-43.430 11.973-43.303 12.158Q-43.177 12.342-43.177 12.585L-43.177 14.161Q-43.177 14.277-43.115 14.373Q-43.054 14.468-42.941 14.468Q-42.831 14.468-42.767 14.374Q-42.702 14.280-42.702 14.161L-42.702 13.713L-42.435 13.713L-42.435 14.161Q-42.435 14.431-42.662 14.596Q-42.890 14.762-43.170 14.762Q-43.378 14.762-43.515 14.608Q-43.652 14.455-43.676 14.239Q-43.823 14.506-44.105 14.651Q-44.387 14.796-44.711 14.796Q-44.988 14.796-45.272 14.721Q-45.556 14.646-45.749 14.467Q-45.942 14.287-45.942 14M-45.327 14Q-45.327 14.174-45.226 14.304Q-45.125 14.434-44.969 14.504Q-44.814 14.574-44.650 14.574Q-44.431 14.574-44.223 14.477Q-44.014 14.379-43.886 14.198Q-43.758 14.017-43.758 13.791L-43.758 13.063Q-44.082 13.063-44.448 13.154Q-44.814 13.245-45.070 13.457Q-45.327 13.668-45.327 14M-41.492 13.887L-41.492 11.990L-42.131 11.990L-42.131 11.768Q-41.813 11.768-41.596 11.558Q-41.379 11.348-41.278 11.038Q-41.177 10.729-41.177 10.421L-40.911 10.421L-40.911 11.710L-39.834 11.710L-39.834 11.990L-40.911 11.990L-40.911 13.874Q-40.911 14.150-40.806 14.349Q-40.702 14.547-40.442 14.547Q-40.285 14.547-40.179 14.443Q-40.073 14.338-40.024 14.185Q-39.974 14.031-39.974 13.874L-39.974 13.460L-39.707 13.460L-39.707 13.887Q-39.707 14.113-39.807 14.323Q-39.906 14.533-40.090 14.665Q-40.275 14.796-40.504 14.796Q-40.941 14.796-41.216 14.559Q-41.492 14.321-41.492 13.887M-38.938 13.193Q-38.938 12.872-38.814 12.583Q-38.689 12.294-38.463 12.071Q-38.238 11.847-37.942 11.727Q-37.646 11.607-37.329 11.607Q-37 11.607-36.739 11.707Q-36.477 11.806-36.301 11.988Q-36.125 12.171-36.031 12.429Q-35.937 12.687-35.937 13.019Q-35.937 13.111-36.019 13.132L-38.275 13.132L-38.275 13.193Q-38.275 13.781-37.992 14.164Q-37.708 14.547-37.141 14.547Q-36.819 14.547-36.551 14.354Q-36.283 14.161-36.194 13.846Q-36.187 13.805-36.112 13.791L-36.019 13.791Q-35.937 13.815-35.937 13.887Q-35.937 13.894-35.944 13.921Q-36.057 14.318-36.428 14.557Q-36.799 14.796-37.223 14.796Q-37.660 14.796-38.060 14.588Q-38.460 14.379-38.699 14.012Q-38.938 13.645-38.938 13.193M-38.268 12.923L-36.454 12.923Q-36.454 12.646-36.551 12.394Q-36.648 12.141-36.847 11.985Q-37.045 11.830-37.329 11.830Q-37.605 11.830-37.819 11.988Q-38.033 12.147-38.151 12.402Q-38.268 12.657-38.268 12.923\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M182.666-15.147h41.619v-19.917h-41.62Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(238.598 -37.68)\">\u003Cpath d=\"M-50.604 14.728L-52.238 14.728L-52.238 14.448Q-52.009 14.448-51.860 14.414Q-51.711 14.379-51.711 14.239L-51.711 12.390Q-51.711 12.120-51.819 12.059Q-51.927 11.997-52.238 11.997L-52.238 11.717L-51.178 11.642L-51.178 12.291Q-51.007 11.983-50.703 11.812Q-50.399 11.642-50.054 11.642Q-49.548 11.642-49.264 11.865Q-48.980 12.089-48.980 12.585L-48.980 14.239Q-48.980 14.376-48.832 14.412Q-48.683 14.448-48.457 14.448L-48.457 14.728L-50.088 14.728L-50.088 14.448Q-49.859 14.448-49.710 14.414Q-49.561 14.379-49.561 14.239L-49.561 12.599Q-49.561 12.264-49.681 12.064Q-49.801 11.864-50.115 11.864Q-50.385 11.864-50.619 12Q-50.853 12.137-50.992 12.371Q-51.130 12.605-51.130 12.879L-51.130 14.239Q-51.130 14.376-50.980 14.412Q-50.829 14.448-50.604 14.448L-50.604 14.728M-47.911 13.193Q-47.911 12.872-47.786 12.583Q-47.661 12.294-47.435 12.071Q-47.210 11.847-46.914 11.727Q-46.619 11.607-46.301 11.607Q-45.973 11.607-45.711 11.707Q-45.450 11.806-45.274 11.988Q-45.098 12.171-45.004 12.429Q-44.910 12.687-44.910 13.019Q-44.910 13.111-44.992 13.132L-47.247 13.132L-47.247 13.193Q-47.247 13.781-46.964 14.164Q-46.680 14.547-46.113 14.547Q-45.791 14.547-45.523 14.354Q-45.255 14.161-45.166 13.846Q-45.159 13.805-45.084 13.791L-44.992 13.791Q-44.910 13.815-44.910 13.887Q-44.910 13.894-44.916 13.921Q-45.029 14.318-45.400 14.557Q-45.771 14.796-46.195 14.796Q-46.632 14.796-47.032 14.588Q-47.432 14.379-47.671 14.012Q-47.911 13.645-47.911 13.193M-47.241 12.923L-45.426 12.923Q-45.426 12.646-45.523 12.394Q-45.621 12.141-45.819 11.985Q-46.017 11.830-46.301 11.830Q-46.578 11.830-46.791 11.988Q-47.005 12.147-47.123 12.402Q-47.241 12.657-47.241 12.923M-43.139 14.728L-44.462 14.728L-44.462 14.448Q-43.901 14.448-43.522 14.048L-42.808 13.251L-43.720 12.202Q-43.857 12.055-44.006 12.023Q-44.154 11.990-44.421 11.990L-44.421 11.710L-42.920 11.710L-42.920 11.990Q-43.112 11.990-43.112 12.124Q-43.112 12.154-43.081 12.202L-42.486 12.886L-42.045 12.390Q-41.933 12.260-41.933 12.144Q-41.933 12.082-41.970 12.036Q-42.008 11.990-42.066 11.990L-42.066 11.710L-40.750 11.710L-40.750 11.990Q-41.310 11.990-41.690 12.390L-42.312 13.091L-41.317 14.239Q-41.218 14.338-41.117 14.383Q-41.017 14.427-40.905 14.437Q-40.794 14.448-40.617 14.448L-40.617 14.728L-42.110 14.728L-42.110 14.448Q-42.045 14.448-41.985 14.414Q-41.926 14.379-41.926 14.314Q-41.926 14.267-41.956 14.239L-42.633 13.453L-43.166 14.048Q-43.279 14.178-43.279 14.294Q-43.279 14.359-43.238 14.403Q-43.197 14.448-43.139 14.448L-43.139 14.728M-39.595 13.887L-39.595 11.990L-40.234 11.990L-40.234 11.768Q-39.916 11.768-39.699 11.558Q-39.482 11.348-39.381 11.038Q-39.280 10.729-39.280 10.421L-39.014 10.421L-39.014 11.710L-37.937 11.710L-37.937 11.990L-39.014 11.990L-39.014 13.874Q-39.014 14.150-38.909 14.349Q-38.805 14.547-38.545 14.547Q-38.388 14.547-38.282 14.443Q-38.176 14.338-38.127 14.185Q-38.077 14.031-38.077 13.874L-38.077 13.460L-37.810 13.460L-37.810 13.887Q-37.810 14.113-37.910 14.323Q-38.009 14.533-38.193 14.665Q-38.378 14.796-38.607 14.796Q-39.044 14.796-39.319 14.559Q-39.595 14.321-39.595 13.887\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(238.598 -37.68)\">\u003Cpath d=\"M-34.292 14.721L-34.292 13.658Q-34.292 13.634-34.264 13.607Q-34.237 13.580-34.213 13.580L-34.104 13.580Q-34.039 13.580-34.025 13.638Q-33.929 14.072-33.683 14.323Q-33.437 14.574-33.023 14.574Q-32.682 14.574-32.429 14.441Q-32.176 14.308-32.176 14Q-32.176 13.843-32.270 13.728Q-32.364 13.614-32.502 13.545Q-32.641 13.477-32.808 13.439L-33.389 13.340Q-33.745 13.272-34.018 13.051Q-34.292 12.831-34.292 12.489Q-34.292 12.240-34.180 12.065Q-34.069 11.891-33.883 11.792Q-33.697 11.693-33.481 11.650Q-33.266 11.607-33.023 11.607Q-32.610 11.607-32.330 11.789L-32.114 11.614Q-32.104 11.611-32.097 11.609Q-32.090 11.607-32.080 11.607L-32.029 11.607Q-32.002 11.607-31.978 11.631Q-31.954 11.655-31.954 11.683L-31.954 12.530Q-31.954 12.551-31.978 12.578Q-32.002 12.605-32.029 12.605L-32.142 12.605Q-32.169 12.605-32.195 12.580Q-32.220 12.554-32.220 12.530Q-32.220 12.294-32.326 12.130Q-32.432 11.966-32.615 11.884Q-32.798 11.802-33.030 11.802Q-33.358 11.802-33.615 11.905Q-33.871 12.007-33.871 12.284Q-33.871 12.479-33.688 12.588Q-33.505 12.698-33.276 12.739L-32.702 12.845Q-32.456 12.893-32.242 13.021Q-32.029 13.149-31.892 13.352Q-31.755 13.556-31.755 13.805Q-31.755 14.318-32.121 14.557Q-32.487 14.796-33.023 14.796Q-33.519 14.796-33.851 14.502L-34.117 14.776Q-34.138 14.796-34.165 14.796L-34.213 14.796Q-34.237 14.796-34.264 14.769Q-34.292 14.742-34.292 14.721M-30.600 13.887L-30.600 11.990L-31.239 11.990L-31.239 11.768Q-30.921 11.768-30.704 11.558Q-30.487 11.348-30.387 11.038Q-30.286 10.729-30.286 10.421L-30.019 10.421L-30.019 11.710L-28.942 11.710L-28.942 11.990L-30.019 11.990L-30.019 13.874Q-30.019 14.150-29.915 14.349Q-29.811 14.547-29.551 14.547Q-29.394 14.547-29.288 14.443Q-29.182 14.338-29.132 14.185Q-29.083 14.031-29.083 13.874L-29.083 13.460L-28.816 13.460L-28.816 13.887Q-28.816 14.113-28.915 14.323Q-29.014 14.533-29.199 14.665Q-29.383 14.796-29.612 14.796Q-30.050 14.796-30.325 14.559Q-30.600 14.321-30.600 13.887M-27.948 14Q-27.948 13.668-27.724 13.441Q-27.500 13.214-27.157 13.086Q-26.813 12.957-26.440 12.905Q-26.068 12.852-25.764 12.852L-25.764 12.599Q-25.764 12.394-25.871 12.214Q-25.979 12.035-26.160 11.932Q-26.341 11.830-26.550 11.830Q-26.957 11.830-27.192 11.922Q-27.104 11.959-27.057 12.043Q-27.011 12.127-27.011 12.229Q-27.011 12.325-27.057 12.404Q-27.104 12.482-27.184 12.527Q-27.264 12.571-27.353 12.571Q-27.503 12.571-27.604 12.474Q-27.705 12.376-27.705 12.229Q-27.705 11.607-26.550 11.607Q-26.338 11.607-26.088 11.671Q-25.839 11.734-25.637 11.853Q-25.436 11.973-25.309 12.158Q-25.183 12.342-25.183 12.585L-25.183 14.161Q-25.183 14.277-25.121 14.373Q-25.060 14.468-24.947 14.468Q-24.837 14.468-24.773 14.374Q-24.708 14.280-24.708 14.161L-24.708 13.713L-24.441 13.713L-24.441 14.161Q-24.441 14.431-24.668 14.596Q-24.896 14.762-25.176 14.762Q-25.384 14.762-25.521 14.608Q-25.658 14.455-25.682 14.239Q-25.829 14.506-26.111 14.651Q-26.393 14.796-26.717 14.796Q-26.994 14.796-27.278 14.721Q-27.562 14.646-27.755 14.467Q-27.948 14.287-27.948 14M-27.333 14Q-27.333 14.174-27.232 14.304Q-27.131 14.434-26.975 14.504Q-26.820 14.574-26.656 14.574Q-26.437 14.574-26.229 14.477Q-26.020 14.379-25.892 14.198Q-25.764 14.017-25.764 13.791L-25.764 13.063Q-26.088 13.063-26.454 13.154Q-26.820 13.245-27.076 13.457Q-27.333 13.668-27.333 14M-23.498 13.887L-23.498 11.990L-24.137 11.990L-24.137 11.768Q-23.819 11.768-23.602 11.558Q-23.385 11.348-23.284 11.038Q-23.183 10.729-23.183 10.421L-22.917 10.421L-22.917 11.710L-21.840 11.710L-21.840 11.990L-22.917 11.990L-22.917 13.874Q-22.917 14.150-22.812 14.349Q-22.708 14.547-22.448 14.547Q-22.291 14.547-22.185 14.443Q-22.079 14.338-22.030 14.185Q-21.980 14.031-21.980 13.874L-21.980 13.460L-21.713 13.460L-21.713 13.887Q-21.713 14.113-21.813 14.323Q-21.912 14.533-22.096 14.665Q-22.281 14.796-22.510 14.796Q-22.947 14.796-23.222 14.559Q-23.498 14.321-23.498 13.887M-20.944 13.193Q-20.944 12.872-20.820 12.583Q-20.695 12.294-20.469 12.071Q-20.244 11.847-19.948 11.727Q-19.652 11.607-19.335 11.607Q-19.006 11.607-18.745 11.707Q-18.483 11.806-18.307 11.988Q-18.131 12.171-18.037 12.429Q-17.943 12.687-17.943 13.019Q-17.943 13.111-18.025 13.132L-20.281 13.132L-20.281 13.193Q-20.281 13.781-19.998 14.164Q-19.714 14.547-19.147 14.547Q-18.825 14.547-18.557 14.354Q-18.289 14.161-18.200 13.846Q-18.193 13.805-18.118 13.791L-18.025 13.791Q-17.943 13.815-17.943 13.887Q-17.943 13.894-17.950 13.921Q-18.063 14.318-18.434 14.557Q-18.805 14.796-19.229 14.796Q-19.666 14.796-20.066 14.588Q-20.466 14.379-20.705 14.012Q-20.944 13.645-20.944 13.193M-20.274 12.923L-18.460 12.923Q-18.460 12.646-18.557 12.394Q-18.654 12.141-18.853 11.985Q-19.051 11.830-19.335 11.830Q-19.611 11.830-19.825 11.988Q-20.039 12.147-20.157 12.402Q-20.274 12.657-20.274 12.923\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M131.12-25.106h48.746\"\u002F>\u003Cpath stroke=\"none\" d=\"m182.466-25.106-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg transform=\"translate(195.58 -44.928)\">\u003Cpath d=\"M-50.642 16.085L-52.272 16.085L-52.272 15.805Q-52.043 15.805-51.894 15.770Q-51.746 15.736-51.746 15.596L-51.746 12.250Q-51.746 12.079-51.882 12.038Q-52.019 11.997-52.272 11.997L-52.272 11.717L-51.192 11.642L-51.192 12.048Q-50.970 11.847-50.683 11.744Q-50.395 11.642-50.088 11.642Q-49.661 11.642-49.297 11.855Q-48.933 12.069-48.719 12.433Q-48.505 12.797-48.505 13.217Q-48.505 13.662-48.745 14.026Q-48.984 14.390-49.377 14.593Q-49.770 14.796-50.214 14.796Q-50.481 14.796-50.729 14.696Q-50.976 14.595-51.164 14.414L-51.164 15.596Q-51.164 15.733-51.016 15.769Q-50.867 15.805-50.642 15.805L-50.642 16.085M-51.164 12.397L-51.164 14.007Q-51.031 14.260-50.788 14.417Q-50.546 14.574-50.269 14.574Q-49.941 14.574-49.688 14.373Q-49.435 14.171-49.302 13.853Q-49.168 13.535-49.168 13.217Q-49.168 12.988-49.233 12.759Q-49.298 12.530-49.426 12.332Q-49.555 12.134-49.749 12.014Q-49.944 11.895-50.177 11.895Q-50.471 11.895-50.739 12.024Q-51.007 12.154-51.164 12.397M-46.120 14.728L-47.856 14.728L-47.856 14.448Q-47.627 14.448-47.478 14.414Q-47.329 14.379-47.329 14.239L-47.329 12.390Q-47.329 12.120-47.437 12.059Q-47.545 11.997-47.856 11.997L-47.856 11.717L-46.827 11.642L-46.827 12.349Q-46.697 12.041-46.454 11.842Q-46.212 11.642-45.894 11.642Q-45.675 11.642-45.504 11.766Q-45.333 11.891-45.333 12.103Q-45.333 12.240-45.433 12.339Q-45.532 12.438-45.665 12.438Q-45.802 12.438-45.901 12.339Q-46 12.240-46 12.103Q-46 11.963-45.901 11.864Q-46.191 11.864-46.391 12.060Q-46.591 12.257-46.683 12.551Q-46.776 12.845-46.776 13.125L-46.776 14.239Q-46.776 14.448-46.120 14.448L-46.120 14.728M-44.790 13.193Q-44.790 12.872-44.665 12.583Q-44.540 12.294-44.315 12.071Q-44.089 11.847-43.794 11.727Q-43.498 11.607-43.180 11.607Q-42.852 11.607-42.590 11.707Q-42.329 11.806-42.153 11.988Q-41.977 12.171-41.883 12.429Q-41.789 12.687-41.789 13.019Q-41.789 13.111-41.871 13.132L-44.127 13.132L-44.127 13.193Q-44.127 13.781-43.843 14.164Q-43.559 14.547-42.992 14.547Q-42.671 14.547-42.402 14.354Q-42.134 14.161-42.045 13.846Q-42.038 13.805-41.963 13.791L-41.871 13.791Q-41.789 13.815-41.789 13.887Q-41.789 13.894-41.796 13.921Q-41.909 14.318-42.279 14.557Q-42.650 14.796-43.074 14.796Q-43.512 14.796-43.912 14.588Q-44.311 14.379-44.551 14.012Q-44.790 13.645-44.790 13.193M-44.120 12.923L-42.305 12.923Q-42.305 12.646-42.402 12.394Q-42.500 12.141-42.698 11.985Q-42.896 11.830-43.180 11.830Q-43.457 11.830-43.671 11.988Q-43.884 12.147-44.002 12.402Q-44.120 12.657-44.120 12.923M-41.201 13.217Q-41.201 12.879-41.061 12.588Q-40.921 12.298-40.676 12.084Q-40.432 11.871-40.128 11.756Q-39.824 11.642-39.499 11.642Q-39.229 11.642-38.966 11.741Q-38.703 11.840-38.511 12.018L-38.511 10.620Q-38.511 10.350-38.619 10.288Q-38.726 10.227-39.038 10.227L-39.038 9.946L-37.961 9.871L-37.961 14.055Q-37.961 14.243-37.906 14.326Q-37.851 14.410-37.751 14.429Q-37.650 14.448-37.434 14.448L-37.434 14.728L-38.542 14.796L-38.542 14.379Q-38.959 14.796-39.584 14.796Q-40.015 14.796-40.388 14.584Q-40.760 14.373-40.981 14.012Q-41.201 13.651-41.201 13.217M-39.526 14.574Q-39.318 14.574-39.131 14.502Q-38.945 14.431-38.791 14.294Q-38.638 14.157-38.542 13.979L-38.542 12.370Q-38.627 12.223-38.773 12.103Q-38.918 11.983-39.087 11.924Q-39.256 11.864-39.437 11.864Q-39.998 11.864-40.266 12.253Q-40.535 12.643-40.535 13.224Q-40.535 13.795-40.300 14.185Q-40.066 14.574-39.526 14.574M-35.168 14.728L-36.720 14.728L-36.720 14.448Q-36.495 14.448-36.346 14.414Q-36.197 14.379-36.197 14.239L-36.197 12.390Q-36.197 12.202-36.245 12.118Q-36.293 12.035-36.390 12.016Q-36.488 11.997-36.700 11.997L-36.700 11.717L-35.643 11.642L-35.643 14.239Q-35.643 14.379-35.512 14.414Q-35.380 14.448-35.168 14.448L-35.168 14.728M-36.440 10.421Q-36.440 10.250-36.317 10.131Q-36.194 10.011-36.023 10.011Q-35.855 10.011-35.732 10.131Q-35.609 10.250-35.609 10.421Q-35.609 10.596-35.732 10.719Q-35.855 10.842-36.023 10.842Q-36.194 10.842-36.317 10.719Q-36.440 10.596-36.440 10.421M-34.522 13.217Q-34.522 12.889-34.387 12.588Q-34.252 12.288-34.017 12.067Q-33.781 11.847-33.476 11.727Q-33.172 11.607-32.848 11.607Q-32.342 11.607-31.993 11.710Q-31.644 11.812-31.644 12.188Q-31.644 12.335-31.742 12.436Q-31.839 12.537-31.986 12.537Q-32.140 12.537-32.239 12.438Q-32.338 12.339-32.338 12.188Q-32.338 12-32.198 11.908Q-32.400 11.857-32.841 11.857Q-33.196 11.857-33.425 12.053Q-33.654 12.250-33.755 12.559Q-33.856 12.869-33.856 13.217Q-33.856 13.566-33.729 13.872Q-33.603 14.178-33.348 14.362Q-33.094 14.547-32.738 14.547Q-32.516 14.547-32.331 14.463Q-32.147 14.379-32.012 14.224Q-31.877 14.068-31.819 13.860Q-31.805 13.805-31.750 13.805L-31.638 13.805Q-31.607 13.805-31.585 13.829Q-31.562 13.853-31.562 13.887L-31.562 13.908Q-31.648 14.195-31.836 14.393Q-32.024 14.591-32.289 14.694Q-32.554 14.796-32.848 14.796Q-33.278 14.796-33.666 14.590Q-34.054 14.383-34.288 14.020Q-34.522 13.658-34.522 13.217M-30.448 13.887L-30.448 11.990L-31.087 11.990L-31.087 11.768Q-30.769 11.768-30.552 11.558Q-30.335 11.348-30.235 11.038Q-30.134 10.729-30.134 10.421L-29.867 10.421L-29.867 11.710L-28.790 11.710L-28.790 11.990L-29.867 11.990L-29.867 13.874Q-29.867 14.150-29.763 14.349Q-29.659 14.547-29.399 14.547Q-29.242 14.547-29.136 14.443Q-29.030 14.338-28.980 14.185Q-28.931 14.031-28.931 13.874L-28.931 13.460L-28.664 13.460L-28.664 13.887Q-28.664 14.113-28.763 14.323Q-28.862 14.533-29.047 14.665Q-29.231 14.796-29.460 14.796Q-29.898 14.796-30.173 14.559Q-30.448 14.321-30.448 13.887M-27.854 14.721L-27.854 13.658Q-27.854 13.634-27.827 13.607Q-27.799 13.580-27.775 13.580L-27.666 13.580Q-27.601 13.580-27.587 13.638Q-27.492 14.072-27.246 14.323Q-26.999 14.574-26.586 14.574Q-26.244 14.574-25.991 14.441Q-25.738 14.308-25.738 14Q-25.738 13.843-25.832 13.728Q-25.926 13.614-26.065 13.545Q-26.203 13.477-26.371 13.439L-26.952 13.340Q-27.307 13.272-27.580 13.051Q-27.854 12.831-27.854 12.489Q-27.854 12.240-27.743 12.065Q-27.632 11.891-27.445 11.792Q-27.259 11.693-27.044 11.650Q-26.829 11.607-26.586 11.607Q-26.172 11.607-25.892 11.789L-25.677 11.614Q-25.666 11.611-25.660 11.609Q-25.653 11.607-25.642 11.607L-25.591 11.607Q-25.564 11.607-25.540 11.631Q-25.516 11.655-25.516 11.683L-25.516 12.530Q-25.516 12.551-25.540 12.578Q-25.564 12.605-25.591 12.605L-25.704 12.605Q-25.731 12.605-25.757 12.580Q-25.783 12.554-25.783 12.530Q-25.783 12.294-25.889 12.130Q-25.995 11.966-26.177 11.884Q-26.360 11.802-26.593 11.802Q-26.921 11.802-27.177 11.905Q-27.433 12.007-27.433 12.284Q-27.433 12.479-27.251 12.588Q-27.068 12.698-26.839 12.739L-26.265 12.845Q-26.018 12.893-25.805 13.021Q-25.591 13.149-25.454 13.352Q-25.318 13.556-25.318 13.805Q-25.318 14.318-25.683 14.557Q-26.049 14.796-26.586 14.796Q-27.081 14.796-27.413 14.502L-27.680 14.776Q-27.700 14.796-27.727 14.796L-27.775 14.796Q-27.799 14.796-27.827 14.769Q-27.854 14.742-27.854 14.721\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-50.488 6.728L-52.221 6.728L-52.221 6.448Q-51.995 6.448-51.846 6.414Q-51.698 6.379-51.698 6.239L-51.698 3.990L-52.286 3.990L-52.286 3.710L-51.698 3.710L-51.698 2.893Q-51.698 2.575-51.520 2.327Q-51.342 2.080-51.052 1.939Q-50.761 1.799-50.450 1.799Q-50.194 1.799-49.990 1.941Q-49.787 2.083-49.787 2.326Q-49.787 2.462-49.886 2.561Q-49.985 2.661-50.122 2.661Q-50.259 2.661-50.358 2.561Q-50.457 2.462-50.457 2.326Q-50.457 2.145-50.317 2.052Q-50.395 2.025-50.495 2.025Q-50.703 2.025-50.857 2.158Q-51.011 2.291-51.091 2.495Q-51.171 2.698-51.171 2.907L-51.171 3.710L-50.283 3.710L-50.283 3.990L-51.144 3.990L-51.144 6.239Q-51.144 6.448-50.488 6.448L-50.488 6.728M-49.849 5.245Q-49.849 4.903-49.714 4.604Q-49.579 4.305-49.339 4.081Q-49.100 3.857-48.782 3.732Q-48.464 3.607-48.133 3.607Q-47.688 3.607-47.288 3.823Q-46.889 4.038-46.654 4.416Q-46.420 4.793-46.420 5.245Q-46.420 5.586-46.562 5.870Q-46.704 6.154-46.948 6.361Q-47.193 6.567-47.502 6.682Q-47.811 6.796-48.133 6.796Q-48.563 6.796-48.965 6.595Q-49.367 6.393-49.608 6.041Q-49.849 5.689-49.849 5.245M-48.133 6.547Q-47.531 6.547-47.307 6.169Q-47.083 5.791-47.083 5.159Q-47.083 4.547-47.318 4.188Q-47.552 3.830-48.133 3.830Q-49.185 3.830-49.185 5.159Q-49.185 5.791-48.960 6.169Q-48.734 6.547-48.133 6.547M-44.076 6.728L-45.812 6.728L-45.812 6.448Q-45.583 6.448-45.434 6.414Q-45.286 6.379-45.286 6.239L-45.286 4.390Q-45.286 4.120-45.393 4.059Q-45.501 3.997-45.812 3.997L-45.812 3.717L-44.783 3.642L-44.783 4.349Q-44.653 4.041-44.411 3.842Q-44.168 3.642-43.850 3.642Q-43.631 3.642-43.460 3.766Q-43.289 3.891-43.289 4.103Q-43.289 4.240-43.389 4.339Q-43.488 4.438-43.621 4.438Q-43.758 4.438-43.857 4.339Q-43.956 4.240-43.956 4.103Q-43.956 3.963-43.857 3.864Q-44.147 3.864-44.347 4.060Q-44.547 4.257-44.640 4.551Q-44.732 4.845-44.732 5.125L-44.732 6.239Q-44.732 6.448-44.076 6.448L-44.076 6.728M-41.023 6.728L-42.657 6.728L-42.657 6.448Q-42.428 6.448-42.279 6.414Q-42.131 6.379-42.131 6.239L-42.131 4.390Q-42.131 4.120-42.238 4.059Q-42.346 3.997-42.657 3.997L-42.657 3.717L-41.598 3.642L-41.598 4.291Q-41.427 3.983-41.122 3.812Q-40.818 3.642-40.473 3.642Q-40.073 3.642-39.796 3.782Q-39.519 3.922-39.434 4.270Q-39.267 3.977-38.967 3.809Q-38.668 3.642-38.323 3.642Q-37.817 3.642-37.534 3.865Q-37.250 4.089-37.250 4.585L-37.250 6.239Q-37.250 6.376-37.101 6.412Q-36.953 6.448-36.727 6.448L-36.727 6.728L-38.357 6.728L-38.357 6.448Q-38.132 6.448-37.981 6.412Q-37.831 6.376-37.831 6.239L-37.831 4.599Q-37.831 4.264-37.951 4.064Q-38.070 3.864-38.385 3.864Q-38.655 3.864-38.889 4Q-39.123 4.137-39.261 4.371Q-39.400 4.605-39.400 4.879L-39.400 6.239Q-39.400 6.376-39.251 6.412Q-39.102 6.448-38.877 6.448L-38.877 6.728L-40.507 6.728L-40.507 6.448Q-40.278 6.448-40.130 6.414Q-39.981 6.379-39.981 6.239L-39.981 4.599Q-39.981 4.264-40.100 4.064Q-40.220 3.864-40.535 3.864Q-40.805 3.864-41.039 4Q-41.273 4.137-41.411 4.371Q-41.550 4.605-41.550 4.879L-41.550 6.239Q-41.550 6.376-41.399 6.412Q-41.249 6.448-41.023 6.448L-41.023 6.728M-36.139 6.721L-36.139 5.658Q-36.139 5.634-36.112 5.607Q-36.084 5.580-36.060 5.580L-35.951 5.580Q-35.886 5.580-35.872 5.638Q-35.777 6.072-35.531 6.323Q-35.285 6.574-34.871 6.574Q-34.529 6.574-34.276 6.441Q-34.023 6.308-34.023 6Q-34.023 5.843-34.117 5.728Q-34.211 5.614-34.350 5.545Q-34.488 5.477-34.656 5.439L-35.237 5.340Q-35.592 5.272-35.866 5.051Q-36.139 4.831-36.139 4.489Q-36.139 4.240-36.028 4.065Q-35.917 3.891-35.731 3.792Q-35.544 3.693-35.329 3.650Q-35.114 3.607-34.871 3.607Q-34.457 3.607-34.177 3.789L-33.962 3.614Q-33.952 3.611-33.945 3.609Q-33.938 3.607-33.928 3.607L-33.876 3.607Q-33.849 3.607-33.825 3.631Q-33.801 3.655-33.801 3.683L-33.801 4.530Q-33.801 4.551-33.825 4.578Q-33.849 4.605-33.876 4.605L-33.989 4.605Q-34.017 4.605-34.042 4.580Q-34.068 4.554-34.068 4.530Q-34.068 4.294-34.174 4.130Q-34.280 3.966-34.463 3.884Q-34.645 3.802-34.878 3.802Q-35.206 3.802-35.462 3.905Q-35.719 4.007-35.719 4.284Q-35.719 4.479-35.536 4.588Q-35.353 4.698-35.124 4.739L-34.550 4.845Q-34.304 4.893-34.090 5.021Q-33.876 5.149-33.740 5.352Q-33.603 5.556-33.603 5.805Q-33.603 6.318-33.969 6.557Q-34.334 6.796-34.871 6.796Q-35.367 6.796-35.698 6.502L-35.965 6.776Q-35.985 6.796-36.013 6.796L-36.060 6.796Q-36.084 6.796-36.112 6.769Q-36.139 6.742-36.139 6.721\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-30.263 5.217Q-30.263 4.879-30.122 4.588Q-29.982 4.298-29.738 4.084Q-29.494 3.871-29.189 3.756Q-28.885 3.642-28.560 3.642Q-28.290 3.642-28.027 3.741Q-27.764 3.840-27.573 4.018L-27.573 2.620Q-27.573 2.350-27.680 2.288Q-27.788 2.227-28.099 2.227L-28.099 1.946L-27.022 1.871L-27.022 6.055Q-27.022 6.243-26.968 6.326Q-26.913 6.410-26.812 6.429Q-26.711 6.448-26.496 6.448L-26.496 6.728L-27.603 6.796L-27.603 6.379Q-28.020 6.796-28.646 6.796Q-29.077 6.796-29.449 6.584Q-29.822 6.373-30.042 6.012Q-30.263 5.651-30.263 5.217M-28.588 6.574Q-28.379 6.574-28.193 6.502Q-28.007 6.431-27.853 6.294Q-27.699 6.157-27.603 5.979L-27.603 4.370Q-27.689 4.223-27.834 4.103Q-27.979 3.983-28.149 3.924Q-28.318 3.864-28.499 3.864Q-29.059 3.864-29.328 4.253Q-29.596 4.643-29.596 5.224Q-29.596 5.795-29.362 6.185Q-29.128 6.574-28.588 6.574M-25.272 5.894L-25.272 4.390Q-25.272 4.120-25.380 4.059Q-25.488 3.997-25.799 3.997L-25.799 3.717L-24.691 3.642L-24.691 5.874L-24.691 5.894Q-24.691 6.174-24.640 6.318Q-24.589 6.461-24.447 6.518Q-24.305 6.574-24.018 6.574Q-23.765 6.574-23.560 6.434Q-23.355 6.294-23.239 6.068Q-23.122 5.843-23.122 5.593L-23.122 4.390Q-23.122 4.120-23.230 4.059Q-23.338 3.997-23.649 3.997L-23.649 3.717L-22.541 3.642L-22.541 6.055Q-22.541 6.246-22.488 6.328Q-22.435 6.410-22.335 6.429Q-22.234 6.448-22.018 6.448L-22.018 6.728L-23.095 6.796L-23.095 6.232Q-23.204 6.414-23.350 6.537Q-23.495 6.660-23.681 6.728Q-23.868 6.796-24.069 6.796Q-25.272 6.796-25.272 5.894M-19.681 6.728L-21.417 6.728L-21.417 6.448Q-21.188 6.448-21.039 6.414Q-20.890 6.379-20.890 6.239L-20.890 4.390Q-20.890 4.120-20.998 4.059Q-21.106 3.997-21.417 3.997L-21.417 3.717L-20.388 3.642L-20.388 4.349Q-20.258 4.041-20.015 3.842Q-19.773 3.642-19.455 3.642Q-19.236 3.642-19.065 3.766Q-18.894 3.891-18.894 4.103Q-18.894 4.240-18.994 4.339Q-19.093 4.438-19.226 4.438Q-19.363 4.438-19.462 4.339Q-19.561 4.240-19.561 4.103Q-19.561 3.963-19.462 3.864Q-19.752 3.864-19.952 4.060Q-20.152 4.257-20.244 4.551Q-20.337 4.845-20.337 5.125L-20.337 6.239Q-20.337 6.448-19.681 6.448L-19.681 6.728M-16.693 6.728L-18.245 6.728L-18.245 6.448Q-18.019 6.448-17.871 6.414Q-17.722 6.379-17.722 6.239L-17.722 4.390Q-17.722 4.202-17.770 4.118Q-17.818 4.035-17.915 4.016Q-18.013 3.997-18.224 3.997L-18.224 3.717L-17.168 3.642L-17.168 6.239Q-17.168 6.379-17.037 6.414Q-16.905 6.448-16.693 6.448L-16.693 6.728M-17.965 2.421Q-17.965 2.250-17.842 2.131Q-17.719 2.011-17.548 2.011Q-17.380 2.011-17.257 2.131Q-17.134 2.250-17.134 2.421Q-17.134 2.596-17.257 2.719Q-17.380 2.842-17.548 2.842Q-17.719 2.842-17.842 2.719Q-17.965 2.596-17.965 2.421M-14.366 6.728L-15.999 6.728L-15.999 6.448Q-15.770 6.448-15.622 6.414Q-15.473 6.379-15.473 6.239L-15.473 4.390Q-15.473 4.120-15.581 4.059Q-15.688 3.997-15.999 3.997L-15.999 3.717L-14.940 3.642L-14.940 4.291Q-14.769 3.983-14.465 3.812Q-14.160 3.642-13.815 3.642Q-13.309 3.642-13.026 3.865Q-12.742 4.089-12.742 4.585L-12.742 6.239Q-12.742 6.376-12.593 6.412Q-12.445 6.448-12.219 6.448L-12.219 6.728L-13.849 6.728L-13.849 6.448Q-13.620 6.448-13.472 6.414Q-13.323 6.379-13.323 6.239L-13.323 4.599Q-13.323 4.264-13.443 4.064Q-13.562 3.864-13.877 3.864Q-14.147 3.864-14.381 4Q-14.615 4.137-14.754 4.371Q-14.892 4.605-14.892 4.879L-14.892 6.239Q-14.892 6.376-14.742 6.412Q-14.591 6.448-14.366 6.448L-14.366 6.728M-11.672 7.261Q-11.672 7.015-11.476 6.831Q-11.279 6.646-11.023 6.567Q-11.160 6.455-11.231 6.294Q-11.303 6.133-11.303 5.952Q-11.303 5.631-11.091 5.385Q-11.426 5.087-11.426 4.677Q-11.426 4.216-11.036 3.929Q-10.647 3.642-10.168 3.642Q-9.697 3.642-9.362 3.888Q-9.187 3.734-8.977 3.652Q-8.767 3.570-8.538 3.570Q-8.374 3.570-8.253 3.677Q-8.131 3.785-8.131 3.949Q-8.131 4.045-8.203 4.117Q-8.275 4.188-8.367 4.188Q-8.466 4.188-8.536 4.115Q-8.606 4.041-8.606 3.942Q-8.606 3.888-8.593 3.857L-8.586 3.843Q-8.579 3.823-8.570 3.812Q-8.562 3.802-8.558 3.795Q-8.914 3.795-9.201 4.018Q-8.914 4.311-8.914 4.677Q-8.914 4.992-9.098 5.224Q-9.283 5.457-9.572 5.585Q-9.861 5.713-10.168 5.713Q-10.370 5.713-10.561 5.663Q-10.753 5.614-10.931 5.504Q-11.023 5.631-11.023 5.774Q-11.023 5.956-10.895 6.091Q-10.766 6.226-10.582 6.226L-9.950 6.226Q-9.502 6.226-9.133 6.297Q-8.764 6.369-8.504 6.598Q-8.244 6.827-8.244 7.261Q-8.244 7.582-8.540 7.784Q-8.835 7.986-9.239 8.075Q-9.642 8.164-9.956 8.164Q-10.274 8.164-10.678 8.075Q-11.081 7.986-11.377 7.784Q-11.672 7.582-11.672 7.261M-11.218 7.261Q-11.218 7.490-10.999 7.639Q-10.780 7.788-10.488 7.856Q-10.196 7.924-9.956 7.924Q-9.792 7.924-9.584 7.888Q-9.375 7.853-9.169 7.772Q-8.962 7.692-8.830 7.564Q-8.699 7.436-8.699 7.261Q-8.699 6.909-9.080 6.815Q-9.461 6.721-9.963 6.721L-10.582 6.721Q-10.821 6.721-11.019 6.872Q-11.218 7.022-11.218 7.261M-10.168 5.474Q-9.502 5.474-9.502 4.677Q-9.502 3.877-10.168 3.877Q-10.838 3.877-10.838 4.677Q-10.838 5.474-10.168 5.474\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-4.985 5.193Q-4.985 4.872-4.860 4.583Q-4.735 4.294-4.509 4.071Q-4.284 3.847-3.988 3.727Q-3.693 3.607-3.375 3.607Q-3.047 3.607-2.785 3.707Q-2.524 3.806-2.348 3.988Q-2.172 4.171-2.078 4.429Q-1.984 4.687-1.984 5.019Q-1.984 5.111-2.066 5.132L-4.321 5.132L-4.321 5.193Q-4.321 5.781-4.038 6.164Q-3.754 6.547-3.187 6.547Q-2.865 6.547-2.597 6.354Q-2.329 6.161-2.240 5.846Q-2.233 5.805-2.158 5.791L-2.066 5.791Q-1.984 5.815-1.984 5.887Q-1.984 5.894-1.990 5.921Q-2.103 6.318-2.474 6.557Q-2.845 6.796-3.269 6.796Q-3.706 6.796-4.106 6.588Q-4.506 6.379-4.745 6.012Q-4.985 5.645-4.985 5.193M-4.315 4.923L-2.500 4.923Q-2.500 4.646-2.597 4.394Q-2.695 4.141-2.893 3.985Q-3.091 3.830-3.375 3.830Q-3.652 3.830-3.865 3.988Q-4.079 4.147-4.197 4.402Q-4.315 4.657-4.315 4.923M-0.213 6.728L-1.536 6.728L-1.536 6.448Q-0.975 6.448-0.596 6.048L0.118 5.251L-0.794 4.202Q-0.931 4.055-1.080 4.023Q-1.228 3.990-1.495 3.990L-1.495 3.710L0.006 3.710L0.006 3.990Q-0.186 3.990-0.186 4.124Q-0.186 4.154-0.155 4.202L0.440 4.886L0.881 4.390Q0.993 4.260 0.993 4.144Q0.993 4.082 0.956 4.036Q0.918 3.990 0.860 3.990L0.860 3.710L2.176 3.710L2.176 3.990Q1.616 3.990 1.236 4.390L0.614 5.091L1.609 6.239Q1.708 6.338 1.809 6.383Q1.909 6.427 2.021 6.437Q2.132 6.448 2.309 6.448L2.309 6.728L0.816 6.728L0.816 6.448Q0.881 6.448 0.940 6.414Q1 6.379 1 6.314Q1 6.267 0.970 6.239L0.293 5.453L-0.240 6.048Q-0.353 6.178-0.353 6.294Q-0.353 6.359-0.312 6.403Q-0.271 6.448-0.213 6.448L-0.213 6.728M4.449 8.085L2.819 8.085L2.819 7.805Q3.048 7.805 3.196 7.770Q3.345 7.736 3.345 7.596L3.345 4.250Q3.345 4.079 3.208 4.038Q3.072 3.997 2.819 3.997L2.819 3.717L3.899 3.642L3.899 4.048Q4.121 3.847 4.408 3.744Q4.695 3.642 5.003 3.642Q5.430 3.642 5.794 3.855Q6.158 4.069 6.372 4.433Q6.585 4.797 6.585 5.217Q6.585 5.662 6.346 6.026Q6.107 6.390 5.714 6.593Q5.321 6.796 4.876 6.796Q4.610 6.796 4.362 6.696Q4.114 6.595 3.926 6.414L3.926 7.596Q3.926 7.733 4.075 7.769Q4.223 7.805 4.449 7.805L4.449 8.085M3.926 4.397L3.926 6.007Q4.059 6.260 4.302 6.417Q4.545 6.574 4.822 6.574Q5.150 6.574 5.403 6.373Q5.656 6.171 5.789 5.853Q5.922 5.535 5.922 5.217Q5.922 4.988 5.857 4.759Q5.792 4.530 5.664 4.332Q5.536 4.134 5.341 4.014Q5.146 3.895 4.914 3.895Q4.620 3.895 4.352 4.024Q4.083 4.154 3.926 4.397M8.889 6.728L7.286 6.728L7.286 6.448Q7.512 6.448 7.660 6.414Q7.809 6.379 7.809 6.239L7.809 2.620Q7.809 2.350 7.701 2.288Q7.594 2.227 7.286 2.227L7.286 1.946L8.363 1.871L8.363 6.239Q8.363 6.376 8.513 6.412Q8.663 6.448 8.889 6.448L8.889 6.728M9.443 5.245Q9.443 4.903 9.578 4.604Q9.713 4.305 9.952 4.081Q10.191 3.857 10.509 3.732Q10.827 3.607 11.159 3.607Q11.603 3.607 12.003 3.823Q12.403 4.038 12.637 4.416Q12.871 4.793 12.871 5.245Q12.871 5.586 12.729 5.870Q12.587 6.154 12.343 6.361Q12.098 6.567 11.789 6.682Q11.480 6.796 11.159 6.796Q10.728 6.796 10.326 6.595Q9.925 6.393 9.684 6.041Q9.443 5.689 9.443 5.245M11.159 6.547Q11.760 6.547 11.984 6.169Q12.208 5.791 12.208 5.159Q12.208 4.547 11.974 4.188Q11.740 3.830 11.159 3.830Q10.106 3.830 10.106 5.159Q10.106 5.791 10.331 6.169Q10.557 6.547 11.159 6.547M15.216 6.728L13.479 6.728L13.479 6.448Q13.708 6.448 13.857 6.414Q14.006 6.379 14.006 6.239L14.006 4.390Q14.006 4.120 13.898 4.059Q13.790 3.997 13.479 3.997L13.479 3.717L14.508 3.642L14.508 4.349Q14.638 4.041 14.881 3.842Q15.123 3.642 15.441 3.642Q15.660 3.642 15.831 3.766Q16.002 3.891 16.002 4.103Q16.002 4.240 15.903 4.339Q15.804 4.438 15.670 4.438Q15.534 4.438 15.434 4.339Q15.335 4.240 15.335 4.103Q15.335 3.963 15.434 3.864Q15.144 3.864 14.944 4.060Q14.744 4.257 14.652 4.551Q14.559 4.845 14.559 5.125L14.559 6.239Q14.559 6.448 15.216 6.448L15.216 6.728M16.644 6Q16.644 5.668 16.868 5.441Q17.092 5.214 17.436 5.086Q17.779 4.957 18.152 4.905Q18.524 4.852 18.828 4.852L18.828 4.599Q18.828 4.394 18.721 4.214Q18.613 4.035 18.432 3.932Q18.251 3.830 18.042 3.830Q17.636 3.830 17.400 3.922Q17.489 3.959 17.535 4.043Q17.581 4.127 17.581 4.229Q17.581 4.325 17.535 4.404Q17.489 4.482 17.408 4.527Q17.328 4.571 17.239 4.571Q17.089 4.571 16.988 4.474Q16.887 4.376 16.887 4.229Q16.887 3.607 18.042 3.607Q18.254 3.607 18.504 3.671Q18.753 3.734 18.955 3.853Q19.157 3.973 19.283 4.158Q19.409 4.342 19.409 4.585L19.409 6.161Q19.409 6.277 19.471 6.373Q19.533 6.468 19.645 6.468Q19.755 6.468 19.820 6.374Q19.885 6.280 19.885 6.161L19.885 5.713L20.151 5.713L20.151 6.161Q20.151 6.431 19.924 6.596Q19.697 6.762 19.416 6.762Q19.208 6.762 19.071 6.608Q18.934 6.455 18.910 6.239Q18.763 6.506 18.482 6.651Q18.200 6.796 17.875 6.796Q17.598 6.796 17.314 6.721Q17.031 6.646 16.837 6.467Q16.644 6.287 16.644 6M17.260 6Q17.260 6.174 17.360 6.304Q17.461 6.434 17.617 6.504Q17.772 6.574 17.936 6.574Q18.155 6.574 18.364 6.477Q18.572 6.379 18.700 6.198Q18.828 6.017 18.828 5.791L18.828 5.063Q18.504 5.063 18.138 5.154Q17.772 5.245 17.516 5.457Q17.260 5.668 17.260 6M21.095 5.887L21.095 3.990L20.455 3.990L20.455 3.768Q20.773 3.768 20.990 3.558Q21.207 3.348 21.308 3.038Q21.409 2.729 21.409 2.421L21.676 2.421L21.676 3.710L22.752 3.710L22.752 3.990L21.676 3.990L21.676 5.874Q21.676 6.150 21.780 6.349Q21.884 6.547 22.144 6.547Q22.301 6.547 22.407 6.443Q22.513 6.338 22.563 6.185Q22.612 6.031 22.612 5.874L22.612 5.460L22.879 5.460L22.879 5.887Q22.879 6.113 22.780 6.323Q22.680 6.533 22.496 6.665Q22.311 6.796 22.082 6.796Q21.645 6.796 21.370 6.559Q21.095 6.321 21.095 5.887M25.305 6.728L23.754 6.728L23.754 6.448Q23.979 6.448 24.128 6.414Q24.277 6.379 24.277 6.239L24.277 4.390Q24.277 4.202 24.229 4.118Q24.181 4.035 24.084 4.016Q23.986 3.997 23.774 3.997L23.774 3.717L24.830 3.642L24.830 6.239Q24.830 6.379 24.962 6.414Q25.094 6.448 25.305 6.448L25.305 6.728M24.034 2.421Q24.034 2.250 24.157 2.131Q24.280 2.011 24.451 2.011Q24.618 2.011 24.742 2.131Q24.865 2.250 24.865 2.421Q24.865 2.596 24.742 2.719Q24.618 2.842 24.451 2.842Q24.280 2.842 24.157 2.719Q24.034 2.596 24.034 2.421M25.910 5.245Q25.910 4.903 26.045 4.604Q26.180 4.305 26.420 4.081Q26.659 3.857 26.977 3.732Q27.295 3.607 27.626 3.607Q28.071 3.607 28.471 3.823Q28.870 4.038 29.105 4.416Q29.339 4.793 29.339 5.245Q29.339 5.586 29.197 5.870Q29.055 6.154 28.811 6.361Q28.566 6.567 28.257 6.682Q27.948 6.796 27.626 6.796Q27.196 6.796 26.794 6.595Q26.392 6.393 26.151 6.041Q25.910 5.689 25.910 5.245M27.626 6.547Q28.228 6.547 28.452 6.169Q28.676 5.791 28.676 5.159Q28.676 4.547 28.441 4.188Q28.207 3.830 27.626 3.830Q26.574 3.830 26.574 5.159Q26.574 5.791 26.799 6.169Q27.025 6.547 27.626 6.547M31.615 6.728L29.981 6.728L29.981 6.448Q30.210 6.448 30.359 6.414Q30.508 6.379 30.508 6.239L30.508 4.390Q30.508 4.120 30.400 4.059Q30.292 3.997 29.981 3.997L29.981 3.717L31.041 3.642L31.041 4.291Q31.212 3.983 31.516 3.812Q31.820 3.642 32.165 3.642Q32.671 3.642 32.955 3.865Q33.239 4.089 33.239 4.585L33.239 6.239Q33.239 6.376 33.387 6.412Q33.536 6.448 33.762 6.448L33.762 6.728L32.131 6.728L32.131 6.448Q32.360 6.448 32.509 6.414Q32.658 6.379 32.658 6.239L32.658 4.599Q32.658 4.264 32.538 4.064Q32.418 3.864 32.104 3.864Q31.834 3.864 31.600 4Q31.366 4.137 31.227 4.371Q31.089 4.605 31.089 4.879L31.089 6.239Q31.089 6.376 31.239 6.412Q31.389 6.448 31.615 6.448L31.615 6.728M34.848 7.958Q34.848 7.924 34.869 7.904Q35.108 7.665 35.243 7.338Q35.378 7.012 35.378 6.680Q35.293 6.728 35.170 6.728Q34.989 6.728 34.869 6.608Q34.749 6.489 34.749 6.308Q34.749 6.133 34.869 6.014Q34.989 5.894 35.170 5.894Q35.341 5.894 35.438 6.017Q35.535 6.140 35.570 6.316Q35.604 6.492 35.604 6.666Q35.604 7.049 35.457 7.413Q35.310 7.777 35.036 8.058Q35.009 8.085 34.971 8.085Q34.930 8.085 34.889 8.044Q34.848 8.003 34.848 7.958M34.749 4.124Q34.749 3.956 34.872 3.833Q34.995 3.710 35.170 3.710Q35.337 3.710 35.460 3.833Q35.583 3.956 35.583 4.124Q35.583 4.298 35.460 4.421Q35.337 4.544 35.170 4.544Q34.995 4.544 34.872 4.421Q34.749 4.298 34.749 4.124\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-43.106 14.728L-43.373 14.728L-43.373 10.620Q-43.373 10.350-43.480 10.288Q-43.588 10.227-43.899 10.227L-43.899 9.946L-42.819 9.871L-42.819 12.041Q-42.610 11.850-42.325 11.746Q-42.039 11.642-41.742 11.642Q-41.424 11.642-41.127 11.763Q-40.830 11.884-40.607 12.100Q-40.385 12.315-40.259 12.600Q-40.132 12.886-40.132 13.217Q-40.132 13.662-40.372 14.026Q-40.611 14.390-41.004 14.593Q-41.397 14.796-41.841 14.796Q-42.036 14.796-42.226 14.740Q-42.415 14.684-42.576 14.579Q-42.737 14.475-42.877 14.314L-43.106 14.728M-42.791 12.383L-42.791 14Q-42.655 14.260-42.414 14.417Q-42.173 14.574-41.896 14.574Q-41.602 14.574-41.390 14.467Q-41.178 14.359-41.045 14.167Q-40.912 13.976-40.853 13.737Q-40.795 13.498-40.795 13.217Q-40.795 12.858-40.889 12.554Q-40.983 12.250-41.211 12.057Q-41.438 11.864-41.804 11.864Q-42.104 11.864-42.371 12Q-42.638 12.137-42.791 12.383M-38.922 13.894L-38.922 12.390Q-38.922 12.120-39.030 12.059Q-39.138 11.997-39.449 11.997L-39.449 11.717L-38.341 11.642L-38.341 13.874L-38.341 13.894Q-38.341 14.174-38.290 14.318Q-38.239 14.461-38.097 14.518Q-37.955 14.574-37.668 14.574Q-37.415 14.574-37.210 14.434Q-37.005 14.294-36.889 14.068Q-36.772 13.843-36.772 13.593L-36.772 12.390Q-36.772 12.120-36.880 12.059Q-36.988 11.997-37.299 11.997L-37.299 11.717L-36.191 11.642L-36.191 14.055Q-36.191 14.246-36.138 14.328Q-36.085 14.410-35.985 14.429Q-35.884 14.448-35.668 14.448L-35.668 14.728L-36.745 14.796L-36.745 14.232Q-36.854 14.414-37 14.537Q-37.145 14.660-37.331 14.728Q-37.518 14.796-37.719 14.796Q-38.922 14.796-38.922 13.894M-33.464 14.728L-35.016 14.728L-35.016 14.448Q-34.790 14.448-34.641 14.414Q-34.493 14.379-34.493 14.239L-34.493 12.390Q-34.493 12.202-34.540 12.118Q-34.588 12.035-34.686 12.016Q-34.783 11.997-34.995 11.997L-34.995 11.717L-33.939 11.642L-33.939 14.239Q-33.939 14.379-33.807 14.414Q-33.676 14.448-33.464 14.448L-33.464 14.728M-34.735 10.421Q-34.735 10.250-34.612 10.131Q-34.489 10.011-34.318 10.011Q-34.151 10.011-34.028 10.131Q-33.905 10.250-33.905 10.421Q-33.905 10.596-34.028 10.719Q-34.151 10.842-34.318 10.842Q-34.489 10.842-34.612 10.719Q-34.735 10.596-34.735 10.421M-31.150 14.728L-32.753 14.728L-32.753 14.448Q-32.527 14.448-32.379 14.414Q-32.230 14.379-32.230 14.239L-32.230 10.620Q-32.230 10.350-32.338 10.288Q-32.445 10.227-32.753 10.227L-32.753 9.946L-31.676 9.871L-31.676 14.239Q-31.676 14.376-31.526 14.412Q-31.375 14.448-31.150 14.448L-31.150 14.728M-30.029 13.887L-30.029 11.990L-30.668 11.990L-30.668 11.768Q-30.350 11.768-30.133 11.558Q-29.916 11.348-29.815 11.038Q-29.714 10.729-29.714 10.421L-29.448 10.421L-29.448 11.710L-28.371 11.710L-28.371 11.990L-29.448 11.990L-29.448 13.874Q-29.448 14.150-29.343 14.349Q-29.239 14.547-28.979 14.547Q-28.822 14.547-28.716 14.443Q-28.610 14.338-28.561 14.185Q-28.511 14.031-28.511 13.874L-28.511 13.460L-28.245 13.460L-28.245 13.887Q-28.245 14.113-28.344 14.323Q-28.443 14.533-28.627 14.665Q-28.812 14.796-29.041 14.796Q-29.478 14.796-29.754 14.559Q-30.029 14.321-30.029 13.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-23.345 14.701L-24.326 12.202Q-24.387 12.059-24.505 12.024Q-24.623 11.990-24.839 11.990L-24.839 11.710L-23.359 11.710L-23.359 11.990Q-23.738 11.990-23.738 12.151Q-23.738 12.161-23.724 12.202L-23.010 14.034L-22.337 12.329Q-22.367 12.257-22.367 12.229Q-22.367 12.202-22.395 12.202Q-22.456 12.055-22.574 12.023Q-22.692 11.990-22.904 11.990L-22.904 11.710L-21.506 11.710L-21.506 11.990Q-21.882 11.990-21.882 12.151Q-21.882 12.182-21.875 12.202L-21.120 14.140L-20.433 12.390Q-20.412 12.339-20.412 12.284Q-20.412 12.144-20.525 12.067Q-20.638 11.990-20.778 11.990L-20.778 11.710L-19.558 11.710L-19.558 11.990Q-19.763 11.990-19.918 12.096Q-20.074 12.202-20.146 12.390L-21.051 14.701Q-21.086 14.796-21.198 14.796L-21.267 14.796Q-21.376 14.796-21.414 14.701L-22.196 12.698L-22.983 14.701Q-23.017 14.796-23.130 14.796L-23.198 14.796Q-23.307 14.796-23.345 14.701M-17.411 14.728L-18.963 14.728L-18.963 14.448Q-18.737 14.448-18.589 14.414Q-18.440 14.379-18.440 14.239L-18.440 12.390Q-18.440 12.202-18.488 12.118Q-18.536 12.035-18.633 12.016Q-18.731 11.997-18.943 11.997L-18.943 11.717L-17.886 11.642L-17.886 14.239Q-17.886 14.379-17.755 14.414Q-17.623 14.448-17.411 14.448L-17.411 14.728M-18.683 10.421Q-18.683 10.250-18.560 10.131Q-18.437 10.011-18.266 10.011Q-18.098 10.011-17.975 10.131Q-17.852 10.250-17.852 10.421Q-17.852 10.596-17.975 10.719Q-18.098 10.842-18.266 10.842Q-18.437 10.842-18.560 10.719Q-18.683 10.596-18.683 10.421M-16.239 13.887L-16.239 11.990L-16.878 11.990L-16.878 11.768Q-16.560 11.768-16.343 11.558Q-16.126 11.348-16.025 11.038Q-15.924 10.729-15.924 10.421L-15.658 10.421L-15.658 11.710L-14.581 11.710L-14.581 11.990L-15.658 11.990L-15.658 13.874Q-15.658 14.150-15.554 14.349Q-15.449 14.547-15.190 14.547Q-15.032 14.547-14.926 14.443Q-14.820 14.338-14.771 14.185Q-14.721 14.031-14.721 13.874L-14.721 13.460L-14.455 13.460L-14.455 13.887Q-14.455 14.113-14.554 14.323Q-14.653 14.533-14.838 14.665Q-15.022 14.796-15.251 14.796Q-15.689 14.796-15.964 14.559Q-16.239 14.321-16.239 13.887M-11.963 14.728L-13.597 14.728L-13.597 14.448Q-13.368 14.448-13.219 14.414Q-13.070 14.379-13.070 14.239L-13.070 10.620Q-13.070 10.350-13.178 10.288Q-13.286 10.227-13.597 10.227L-13.597 9.946L-12.517 9.871L-12.517 12.257Q-12.411 12.072-12.233 11.930Q-12.055 11.789-11.847 11.715Q-11.638 11.642-11.413 11.642Q-10.907 11.642-10.623 11.865Q-10.339 12.089-10.339 12.585L-10.339 14.239Q-10.339 14.376-10.191 14.412Q-10.042 14.448-9.817 14.448L-9.817 14.728L-11.447 14.728L-11.447 14.448Q-11.218 14.448-11.069 14.414Q-10.921 14.379-10.921 14.239L-10.921 12.599Q-10.921 12.264-11.040 12.064Q-11.160 11.864-11.474 11.864Q-11.744 11.864-11.978 12Q-12.213 12.137-12.351 12.371Q-12.489 12.605-12.489 12.879L-12.489 14.239Q-12.489 14.376-12.339 14.412Q-12.189 14.448-11.963 14.448\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M-4.842 14.728L-6.476 14.728L-6.476 14.448Q-6.247 14.448-6.098 14.414Q-5.949 14.379-5.949 14.239L-5.949 12.390Q-5.949 12.120-6.057 12.059Q-6.165 11.997-6.476 11.997L-6.476 11.717L-5.416 11.642L-5.416 12.291Q-5.245 11.983-4.941 11.812Q-4.637 11.642-4.292 11.642Q-3.786 11.642-3.502 11.865Q-3.218 12.089-3.218 12.585L-3.218 14.239Q-3.218 14.376-3.070 14.412Q-2.921 14.448-2.695 14.448L-2.695 14.728L-4.326 14.728L-4.326 14.448Q-4.097 14.448-3.948 14.414Q-3.799 14.379-3.799 14.239L-3.799 12.599Q-3.799 12.264-3.919 12.064Q-4.039 11.864-4.353 11.864Q-4.623 11.864-4.857 12Q-5.091 12.137-5.230 12.371Q-5.368 12.605-5.368 12.879L-5.368 14.239Q-5.368 14.376-5.218 14.412Q-5.067 14.448-4.842 14.448L-4.842 14.728M-2.149 13.245Q-2.149 12.903-2.014 12.604Q-1.879 12.305-1.639 12.081Q-1.400 11.857-1.082 11.732Q-0.764 11.607-0.433 11.607Q0.012 11.607 0.412 11.823Q0.811 12.038 1.046 12.416Q1.280 12.793 1.280 13.245Q1.280 13.586 1.138 13.870Q0.996 14.154 0.752 14.361Q0.507 14.567 0.198 14.682Q-0.111 14.796-0.433 14.796Q-0.863 14.796-1.265 14.595Q-1.667 14.393-1.908 14.041Q-2.149 13.689-2.149 13.245M-0.433 14.547Q0.169 14.547 0.393 14.169Q0.617 13.791 0.617 13.159Q0.617 12.547 0.382 12.188Q0.148 11.830-0.433 11.830Q-1.485 11.830-1.485 13.159Q-1.485 13.791-1.260 14.169Q-1.034 14.547-0.433 14.547\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M6.323 14.728L4.587 14.728L4.587 14.448Q4.816 14.448 4.965 14.414Q5.113 14.379 5.113 14.239L5.113 12.390Q5.113 12.120 5.006 12.059Q4.898 11.997 4.587 11.997L4.587 11.717L5.616 11.642L5.616 12.349Q5.746 12.041 5.988 11.842Q6.231 11.642 6.549 11.642Q6.768 11.642 6.939 11.766Q7.110 11.891 7.110 12.103Q7.110 12.240 7.010 12.339Q6.911 12.438 6.778 12.438Q6.641 12.438 6.542 12.339Q6.443 12.240 6.443 12.103Q6.443 11.963 6.542 11.864Q6.252 11.864 6.052 12.060Q5.852 12.257 5.759 12.551Q5.667 12.845 5.667 13.125L5.667 14.239Q5.667 14.448 6.323 14.448L6.323 14.728M7.653 13.193Q7.653 12.872 7.778 12.583Q7.903 12.294 8.128 12.071Q8.354 11.847 8.649 11.727Q8.945 11.607 9.263 11.607Q9.591 11.607 9.853 11.707Q10.114 11.806 10.290 11.988Q10.466 12.171 10.560 12.429Q10.654 12.687 10.654 13.019Q10.654 13.111 10.572 13.132L8.316 13.132L8.316 13.193Q8.316 13.781 8.600 14.164Q8.884 14.547 9.451 14.547Q9.772 14.547 10.040 14.354Q10.309 14.161 10.398 13.846Q10.405 13.805 10.480 13.791L10.572 13.791Q10.654 13.815 10.654 13.887Q10.654 13.894 10.647 13.921Q10.534 14.318 10.164 14.557Q9.793 14.796 9.369 14.796Q8.931 14.796 8.531 14.588Q8.132 14.379 7.892 14.012Q7.653 13.645 7.653 13.193M8.323 12.923L10.138 12.923Q10.138 12.646 10.040 12.394Q9.943 12.141 9.745 11.985Q9.547 11.830 9.263 11.830Q8.986 11.830 8.772 11.988Q8.559 12.147 8.441 12.402Q8.323 12.657 8.323 12.923M12.630 14.701L11.649 12.202Q11.587 12.059 11.469 12.024Q11.351 11.990 11.136 11.990L11.136 11.710L12.616 11.710L12.616 11.990Q12.237 11.990 12.237 12.151Q12.237 12.161 12.250 12.202L12.965 14.034L13.638 12.329Q13.607 12.257 13.607 12.229Q13.607 12.202 13.580 12.202Q13.518 12.055 13.400 12.023Q13.282 11.990 13.071 11.990L13.071 11.710L14.468 11.710L14.468 11.990Q14.092 11.990 14.092 12.151Q14.092 12.182 14.099 12.202L14.855 14.140L15.542 12.390Q15.562 12.339 15.562 12.284Q15.562 12.144 15.449 12.067Q15.337 11.990 15.197 11.990L15.197 11.710L16.417 11.710L16.417 11.990Q16.212 11.990 16.056 12.096Q15.901 12.202 15.829 12.390L14.923 14.701Q14.889 14.796 14.776 14.796L14.708 14.796Q14.598 14.796 14.561 14.701L13.778 12.698L12.992 14.701Q12.958 14.796 12.845 14.796L12.777 14.796Q12.667 14.796 12.630 14.701\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(168.93 -7.796)\">\u003Cpath d=\"M16.806 14Q16.806 13.668 17.029 13.441Q17.253 13.214 17.597 13.086Q17.940 12.957 18.313 12.905Q18.685 12.852 18.990 12.852L18.990 12.599Q18.990 12.394 18.882 12.214Q18.774 12.035 18.593 11.932Q18.412 11.830 18.204 11.830Q17.797 11.830 17.561 11.922Q17.650 11.959 17.696 12.043Q17.742 12.127 17.742 12.229Q17.742 12.325 17.696 12.404Q17.650 12.482 17.569 12.527Q17.489 12.571 17.400 12.571Q17.250 12.571 17.149 12.474Q17.048 12.376 17.048 12.229Q17.048 11.607 18.204 11.607Q18.415 11.607 18.665 11.671Q18.914 11.734 19.116 11.853Q19.318 11.973 19.444 12.158Q19.571 12.342 19.571 12.585L19.571 14.161Q19.571 14.277 19.632 14.373Q19.694 14.468 19.807 14.468Q19.916 14.468 19.981 14.374Q20.046 14.280 20.046 14.161L20.046 13.713L20.312 13.713L20.312 14.161Q20.312 14.431 20.085 14.596Q19.858 14.762 19.578 14.762Q19.369 14.762 19.232 14.608Q19.096 14.455 19.072 14.239Q18.925 14.506 18.643 14.651Q18.361 14.796 18.036 14.796Q17.759 14.796 17.475 14.721Q17.192 14.646 16.999 14.467Q16.806 14.287 16.806 14M17.421 14Q17.421 14.174 17.522 14.304Q17.622 14.434 17.778 14.504Q17.933 14.574 18.098 14.574Q18.316 14.574 18.525 14.477Q18.733 14.379 18.861 14.198Q18.990 14.017 18.990 13.791L18.990 13.063Q18.665 13.063 18.299 13.154Q17.933 13.245 17.677 13.457Q17.421 13.668 17.421 14M22.479 14.728L20.743 14.728L20.743 14.448Q20.972 14.448 21.121 14.414Q21.269 14.379 21.269 14.239L21.269 12.390Q21.269 12.120 21.162 12.059Q21.054 11.997 20.743 11.997L20.743 11.717L21.772 11.642L21.772 12.349Q21.902 12.041 22.144 11.842Q22.387 11.642 22.705 11.642Q22.924 11.642 23.095 11.766Q23.266 11.891 23.266 12.103Q23.266 12.240 23.166 12.339Q23.067 12.438 22.934 12.438Q22.797 12.438 22.698 12.339Q22.599 12.240 22.599 12.103Q22.599 11.963 22.698 11.864Q22.408 11.864 22.208 12.060Q22.008 12.257 21.915 12.551Q21.823 12.845 21.823 13.125L21.823 14.239Q21.823 14.448 22.479 14.448L22.479 14.728M23.850 13.217Q23.850 12.879 23.990 12.588Q24.130 12.298 24.375 12.084Q24.619 11.871 24.923 11.756Q25.227 11.642 25.552 11.642Q25.822 11.642 26.085 11.741Q26.349 11.840 26.540 12.018L26.540 10.620Q26.540 10.350 26.432 10.288Q26.325 10.227 26.014 10.227L26.014 9.946L27.090 9.871L27.090 14.055Q27.090 14.243 27.145 14.326Q27.200 14.410 27.300 14.429Q27.401 14.448 27.617 14.448L27.617 14.728L26.509 14.796L26.509 14.379Q26.092 14.796 25.467 14.796Q25.036 14.796 24.663 14.584Q24.291 14.373 24.070 14.012Q23.850 13.651 23.850 13.217M25.525 14.574Q25.733 14.574 25.920 14.502Q26.106 14.431 26.260 14.294Q26.413 14.157 26.509 13.979L26.509 12.370Q26.424 12.223 26.278 12.103Q26.133 11.983 25.964 11.924Q25.795 11.864 25.614 11.864Q25.053 11.864 24.785 12.253Q24.516 12.643 24.516 13.224Q24.516 13.795 24.751 14.185Q24.985 14.574 25.525 14.574\" 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\">Stimulus-response versus stimulus-stimulus learning. S-R caches, per state, the rewarded action - a policy table that only forms under reward. S-S records what state follows what - a transition model that forms during exploration with no reward. Latent learning is S-S: the map is built unrewarded, then queried once a goal appears.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:344.299px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 258.224 103.139\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-54.46-49.308H2.447V-72.07h-56.905Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(-17.66 -26.703)\">\u003Cpath d=\"M-25.518-32.174L-25.518-33.747Q-25.518-33.774-25.493-33.800Q-25.467-33.825-25.440-33.825L-25.327-33.825Q-25.299-33.825-25.276-33.798Q-25.252-33.771-25.252-33.747Q-25.252-33.402-25.120-33.138Q-24.988-32.875-24.759-32.706Q-24.530-32.537-24.228-32.456Q-23.925-32.376-23.584-32.376Q-23.317-32.376-23.081-32.504Q-22.845-32.632-22.700-32.855Q-22.555-33.077-22.555-33.343Q-22.555-33.566-22.661-33.762Q-22.767-33.959-22.948-34.094Q-23.129-34.229-23.355-34.280L-24.383-34.512Q-24.695-34.584-24.954-34.770Q-25.214-34.957-25.366-35.228Q-25.518-35.500-25.518-35.815Q-25.518-36.201-25.305-36.508Q-25.091-36.816-24.744-36.987Q-24.397-37.158-24.018-37.158Q-23.789-37.158-23.560-37.105Q-23.331-37.052-23.132-36.944Q-22.934-36.837-22.780-36.673L-22.486-37.113Q-22.463-37.158-22.422-37.158L-22.374-37.158Q-22.343-37.158-22.321-37.132Q-22.299-37.107-22.299-37.079L-22.299-35.504Q-22.299-35.483-22.322-35.456Q-22.346-35.428-22.374-35.428L-22.486-35.428Q-22.548-35.428-22.562-35.504Q-22.603-35.917-22.784-36.237Q-22.965-36.556-23.276-36.731Q-23.587-36.905-24.018-36.905Q-24.267-36.905-24.507-36.794Q-24.746-36.683-24.896-36.485Q-25.047-36.286-25.047-36.023Q-25.047-35.811-24.939-35.630Q-24.831-35.449-24.655-35.329Q-24.479-35.210-24.271-35.169L-23.242-34.940Q-22.924-34.868-22.657-34.663Q-22.391-34.458-22.239-34.164Q-22.087-33.870-22.087-33.538Q-22.087-33.145-22.292-32.810Q-22.497-32.475-22.842-32.286Q-23.187-32.096-23.584-32.096Q-24.004-32.096-24.383-32.209Q-24.763-32.321-25.033-32.571L-25.327-32.137Q-25.354-32.096-25.392-32.096L-25.440-32.096Q-25.467-32.096-25.493-32.121Q-25.518-32.147-25.518-32.174M-20.778-31.006Q-20.778-31.040-20.750-31.067Q-20.480-31.296-20.331-31.619Q-20.183-31.942-20.183-32.298L-20.183-32.335Q-20.292-32.236-20.456-32.236Q-20.637-32.236-20.757-32.356Q-20.877-32.475-20.877-32.656Q-20.877-32.831-20.757-32.950Q-20.637-33.070-20.456-33.070Q-20.200-33.070-20.080-32.831Q-19.961-32.591-19.961-32.298Q-19.961-31.898-20.130-31.527Q-20.299-31.156-20.596-30.900Q-20.627-30.879-20.654-30.879Q-20.695-30.879-20.736-30.920Q-20.778-30.961-20.778-31.006\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-17.66 -26.703)\">\u003Cpath d=\"M-14.728-32.236L-16.318-32.236L-16.318-32.516Q-15.675-32.516-15.518-32.916L-13.874-37.131Q-13.840-37.226-13.727-37.226L-13.645-37.226Q-13.535-37.226-13.494-37.131L-11.775-32.725Q-11.707-32.585-11.517-32.550Q-11.327-32.516-11.054-32.516L-11.054-32.236L-13.053-32.236L-13.053-32.516Q-12.489-32.516-12.489-32.691Q-12.489-32.708-12.491-32.715Q-12.493-32.721-12.496-32.725L-12.917-33.791L-14.875-33.791L-15.217-32.916Q-15.231-32.916-15.231-32.838Q-15.231-32.677-15.068-32.597Q-14.906-32.516-14.728-32.516L-14.728-32.236M-13.894-36.310L-14.762-34.071L-13.019-34.071\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-17.66 -26.703)\">\u003Cpath d=\"M-2.784-33.043L-7.326-33.043Q-7.487-33.067-7.487-33.217Q-7.487-33.361-7.326-33.384L-2.411-33.384Q-1.971-33.743-1.413-33.986Q-1.960-34.212-2.411-34.581L-7.326-34.581Q-7.391-34.591-7.439-34.639Q-7.487-34.687-7.487-34.755Q-7.487-34.899-7.326-34.923L-2.784-34.923Q-2.975-35.131-3.223-35.471Q-3.471-35.811-3.471-35.917Q-3.471-35.989-3.379-36.016L-3.211-36.016Q-3.150-36.003-3.133-35.962Q-2.890-35.480-2.519-35.099Q-2.148-34.717-1.677-34.459Q-1.205-34.201-0.672-34.078Q-0.644-34.075-0.629-34.046Q-0.614-34.017-0.614-33.986Q-0.614-33.911-0.699-33.887Q-1.229-33.760-1.687-33.504Q-2.145-33.248-2.517-32.865Q-2.890-32.482-3.133-32.004Q-3.167-31.956-3.211-31.949L-3.379-31.949Q-3.471-31.976-3.471-32.048Q-3.471-32.161-3.208-32.518Q-2.945-32.875-2.784-33.043\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-17.66 -26.703)\">\u003Cpath d=\"M3.121-32.174L3.121-33.747Q3.121-33.774 3.146-33.800Q3.172-33.825 3.199-33.825L3.312-33.825Q3.340-33.825 3.363-33.798Q3.387-33.771 3.387-33.747Q3.387-33.402 3.519-33.138Q3.651-32.875 3.880-32.706Q4.109-32.537 4.411-32.456Q4.714-32.376 5.055-32.376Q5.322-32.376 5.558-32.504Q5.794-32.632 5.939-32.855Q6.084-33.077 6.084-33.343Q6.084-33.566 5.978-33.762Q5.872-33.959 5.691-34.094Q5.510-34.229 5.284-34.280L4.256-34.512Q3.945-34.584 3.685-34.770Q3.425-34.957 3.273-35.228Q3.121-35.500 3.121-35.815Q3.121-36.201 3.334-36.508Q3.548-36.816 3.895-36.987Q4.242-37.158 4.621-37.158Q4.850-37.158 5.079-37.105Q5.308-37.052 5.507-36.944Q5.705-36.837 5.859-36.673L6.153-37.113Q6.176-37.158 6.217-37.158L6.265-37.158Q6.296-37.158 6.318-37.132Q6.340-37.107 6.340-37.079L6.340-35.504Q6.340-35.483 6.317-35.456Q6.293-35.428 6.265-35.428L6.153-35.428Q6.091-35.428 6.077-35.504Q6.036-35.917 5.855-36.237Q5.674-36.556 5.363-36.731Q5.052-36.905 4.621-36.905Q4.372-36.905 4.132-36.794Q3.893-36.683 3.743-36.485Q3.592-36.286 3.592-36.023Q3.592-35.811 3.700-35.630Q3.808-35.449 3.984-35.329Q4.160-35.210 4.368-35.169L5.397-34.940Q5.715-34.868 5.982-34.663Q6.248-34.458 6.400-34.164Q6.552-33.870 6.552-33.538Q6.552-33.145 6.347-32.810Q6.142-32.475 5.797-32.286Q5.452-32.096 5.055-32.096Q4.635-32.096 4.256-32.209Q3.876-32.321 3.606-32.571L3.312-32.137Q3.285-32.096 3.247-32.096L3.199-32.096Q3.172-32.096 3.146-32.121Q3.121-32.147 3.121-32.174M7.861-35.028Q7.861-35.069 7.889-35.093Q8.073-35.240 8.207-35.439Q8.340-35.637 8.408-35.862Q8.477-36.088 8.477-36.324L8.477-36.372Q8.364-36.262 8.183-36.262Q8.002-36.262 7.882-36.382Q7.762-36.502 7.762-36.679Q7.762-36.854 7.882-36.973Q8.002-37.093 8.183-37.093Q8.364-37.093 8.480-36.973Q8.596-36.854 8.648-36.681Q8.699-36.508 8.699-36.324Q8.699-35.921 8.528-35.553Q8.357-35.186 8.043-34.923Q8.012-34.902 7.985-34.902Q7.940-34.902 7.901-34.945Q7.861-34.987 7.861-35.028\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-54.46-15.165H2.447v-22.762h-56.905Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(-14.743 8.082)\">\u003Cpath d=\"M-25.518-32.174L-25.518-33.747Q-25.518-33.774-25.493-33.800Q-25.467-33.825-25.440-33.825L-25.327-33.825Q-25.299-33.825-25.276-33.798Q-25.252-33.771-25.252-33.747Q-25.252-33.402-25.120-33.138Q-24.988-32.875-24.759-32.706Q-24.530-32.537-24.228-32.456Q-23.925-32.376-23.584-32.376Q-23.317-32.376-23.081-32.504Q-22.845-32.632-22.700-32.855Q-22.555-33.077-22.555-33.343Q-22.555-33.566-22.661-33.762Q-22.767-33.959-22.948-34.094Q-23.129-34.229-23.355-34.280L-24.383-34.512Q-24.695-34.584-24.954-34.770Q-25.214-34.957-25.366-35.228Q-25.518-35.500-25.518-35.815Q-25.518-36.201-25.305-36.508Q-25.091-36.816-24.744-36.987Q-24.397-37.158-24.018-37.158Q-23.789-37.158-23.560-37.105Q-23.331-37.052-23.132-36.944Q-22.934-36.837-22.780-36.673L-22.486-37.113Q-22.463-37.158-22.422-37.158L-22.374-37.158Q-22.343-37.158-22.321-37.132Q-22.299-37.107-22.299-37.079L-22.299-35.504Q-22.299-35.483-22.322-35.456Q-22.346-35.428-22.374-35.428L-22.486-35.428Q-22.548-35.428-22.562-35.504Q-22.603-35.917-22.784-36.237Q-22.965-36.556-23.276-36.731Q-23.587-36.905-24.018-36.905Q-24.267-36.905-24.507-36.794Q-24.746-36.683-24.896-36.485Q-25.047-36.286-25.047-36.023Q-25.047-35.811-24.939-35.630Q-24.831-35.449-24.655-35.329Q-24.479-35.210-24.271-35.169L-23.242-34.940Q-22.924-34.868-22.657-34.663Q-22.391-34.458-22.239-34.164Q-22.087-33.870-22.087-33.538Q-22.087-33.145-22.292-32.810Q-22.497-32.475-22.842-32.286Q-23.187-32.096-23.584-32.096Q-24.004-32.096-24.383-32.209Q-24.763-32.321-25.033-32.571L-25.327-32.137Q-25.354-32.096-25.392-32.096L-25.440-32.096Q-25.467-32.096-25.493-32.121Q-25.518-32.147-25.518-32.174M-19.687-32.236L-21.277-32.236L-21.277-32.516Q-20.634-32.516-20.477-32.916L-18.833-37.131Q-18.799-37.226-18.686-37.226L-18.604-37.226Q-18.494-37.226-18.453-37.131L-16.734-32.725Q-16.666-32.585-16.476-32.550Q-16.286-32.516-16.013-32.516L-16.013-32.236L-18.012-32.236L-18.012-32.516Q-17.448-32.516-17.448-32.691Q-17.448-32.708-17.450-32.715Q-17.452-32.721-17.455-32.725L-17.876-33.791L-19.834-33.791L-20.176-32.916Q-20.190-32.916-20.190-32.838Q-20.190-32.677-20.027-32.597Q-19.865-32.516-19.687-32.516L-19.687-32.236M-18.853-36.310L-19.721-34.071L-17.978-34.071\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-14.743 8.082)\">\u003Cpath d=\"M-7.743-33.043L-12.285-33.043Q-12.446-33.067-12.446-33.217Q-12.446-33.361-12.285-33.384L-7.370-33.384Q-6.930-33.743-6.372-33.986Q-6.919-34.212-7.370-34.581L-12.285-34.581Q-12.350-34.591-12.398-34.639Q-12.446-34.687-12.446-34.755Q-12.446-34.899-12.285-34.923L-7.743-34.923Q-7.934-35.131-8.182-35.471Q-8.430-35.811-8.430-35.917Q-8.430-35.989-8.338-36.016L-8.170-36.016Q-8.109-36.003-8.092-35.962Q-7.849-35.480-7.478-35.099Q-7.107-34.717-6.636-34.459Q-6.164-34.201-5.631-34.078Q-5.603-34.075-5.588-34.046Q-5.573-34.017-5.573-33.986Q-5.573-33.911-5.658-33.887Q-6.188-33.760-6.646-33.504Q-7.104-33.248-7.476-32.865Q-7.849-32.482-8.092-32.004Q-8.126-31.956-8.170-31.949L-8.338-31.949Q-8.430-31.976-8.430-32.048Q-8.430-32.161-8.167-32.518Q-7.904-32.875-7.743-33.043\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-14.743 8.082)\">\u003Cpath d=\"M0.173-32.236L-1.933-32.236L-1.933-32.516Q-1.212-32.516-1.212-32.725L-1.212-36.526Q-1.212-36.737-1.933-36.737L-1.933-37.018L0.405-37.018Q0.716-37.018 1.059-36.944Q1.403-36.871 1.724-36.715Q2.046-36.560 2.247-36.314Q2.449-36.068 2.449-35.743Q2.449-35.456 2.261-35.225Q2.073-34.994 1.796-34.846Q1.519-34.697 1.229-34.615Q1.564-34.512 1.798-34.280Q2.032-34.048 2.076-33.719L2.162-33.111Q2.189-32.899 2.235-32.730Q2.281-32.561 2.386-32.441Q2.490-32.321 2.678-32.321Q2.893-32.321 3.009-32.506Q3.126-32.691 3.126-32.923Q3.143-32.988 3.218-33.005L3.310-33.005Q3.392-32.985 3.392-32.903Q3.392-32.697 3.302-32.511Q3.211-32.325 3.045-32.210Q2.880-32.096 2.678-32.096Q2.340-32.096 2.046-32.197Q1.752-32.298 1.567-32.520Q1.382-32.742 1.382-33.090L1.382-33.699Q1.382-33.948 1.239-34.135Q1.095-34.321 0.872-34.420Q0.648-34.519 0.398-34.519L-0.549-34.519L-0.549-32.725Q-0.549-32.516 0.173-32.516L0.173-32.236M-0.549-36.526L-0.549-34.741L0.306-34.741Q0.593-34.741 0.832-34.784Q1.071-34.827 1.259-34.936Q1.447-35.046 1.559-35.244Q1.670-35.442 1.670-35.743Q1.670-36.320 1.302-36.529Q0.935-36.737 0.306-36.737L-0.183-36.737Q-0.371-36.737-0.460-36.703Q-0.549-36.669-0.549-36.526\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M93.495-32.236H150.4v-22.762H93.495Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(126.353 -5.027)\">\u003Cpath d=\"M-25.734-41.771Q-25.734-42.092-25.609-42.381Q-25.484-42.670-25.258-42.893Q-25.033-43.117-24.737-43.237Q-24.442-43.357-24.124-43.357Q-23.796-43.357-23.534-43.257Q-23.273-43.158-23.097-42.976Q-22.921-42.793-22.827-42.535Q-22.733-42.277-22.733-41.945Q-22.733-41.853-22.815-41.832L-25.070-41.832L-25.070-41.771Q-25.070-41.183-24.787-40.800Q-24.503-40.417-23.936-40.417Q-23.614-40.417-23.346-40.610Q-23.078-40.803-22.989-41.118Q-22.982-41.159-22.907-41.173L-22.815-41.173Q-22.733-41.149-22.733-41.077Q-22.733-41.070-22.739-41.043Q-22.852-40.646-23.223-40.407Q-23.594-40.168-24.018-40.168Q-24.455-40.168-24.855-40.376Q-25.255-40.585-25.494-40.952Q-25.734-41.319-25.734-41.771M-25.064-42.041L-23.249-42.041Q-23.249-42.318-23.346-42.570Q-23.444-42.823-23.642-42.979Q-23.840-43.134-24.124-43.134Q-24.401-43.134-24.614-42.976Q-24.828-42.817-24.946-42.562Q-25.064-42.307-25.064-42.041M-20.463-40.236L-22.097-40.236L-22.097-40.516Q-21.868-40.516-21.719-40.550Q-21.570-40.585-21.570-40.725L-21.570-42.574Q-21.570-42.844-21.678-42.905Q-21.786-42.967-22.097-42.967L-22.097-43.247L-21.037-43.322L-21.037-42.673Q-20.866-42.981-20.562-43.152Q-20.258-43.322-19.913-43.322Q-19.407-43.322-19.123-43.099Q-18.840-42.875-18.840-42.379L-18.840-40.725Q-18.840-40.588-18.691-40.552Q-18.542-40.516-18.317-40.516L-18.317-40.236L-19.947-40.236L-19.947-40.516Q-19.718-40.516-19.569-40.550Q-19.421-40.585-19.421-40.725L-19.421-42.365Q-19.421-42.700-19.540-42.900Q-19.660-43.100-19.974-43.100Q-20.244-43.100-20.478-42.964Q-20.713-42.827-20.851-42.593Q-20.989-42.359-20.989-42.085L-20.989-40.725Q-20.989-40.588-20.839-40.552Q-20.689-40.516-20.463-40.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(126.353 -5.027)\">\u003Cpath d=\"M-16.346-40.263L-17.474-42.762Q-17.546-42.909-17.676-42.941Q-17.806-42.974-18.035-42.974L-18.035-43.254L-16.521-43.254L-16.521-42.974Q-16.873-42.974-16.873-42.827Q-16.873-42.782-16.862-42.762L-15.998-40.844L-15.218-42.574Q-15.184-42.642-15.184-42.721Q-15.184-42.834-15.268-42.904Q-15.352-42.974-15.471-42.974L-15.471-43.254L-14.275-43.254L-14.275-42.974Q-14.494-42.974-14.665-42.871Q-14.835-42.769-14.924-42.574L-15.960-40.263Q-16.008-40.168-16.114-40.168L-16.192-40.168Q-16.298-40.168-16.346-40.263M-12.118-40.236L-13.670-40.236L-13.670-40.516Q-13.444-40.516-13.296-40.550Q-13.147-40.585-13.147-40.725L-13.147-42.574Q-13.147-42.762-13.195-42.846Q-13.243-42.929-13.340-42.948Q-13.437-42.967-13.649-42.967L-13.649-43.247L-12.593-43.322L-12.593-40.725Q-12.593-40.585-12.462-40.550Q-12.330-40.516-12.118-40.516L-12.118-40.236M-13.390-44.543Q-13.390-44.714-13.267-44.833Q-13.144-44.953-12.973-44.953Q-12.805-44.953-12.682-44.833Q-12.559-44.714-12.559-44.543Q-12.559-44.368-12.682-44.245Q-12.805-44.122-12.973-44.122Q-13.144-44.122-13.267-44.245Q-13.390-44.368-13.390-44.543M-9.722-40.236L-11.458-40.236L-11.458-40.516Q-11.229-40.516-11.081-40.550Q-10.932-40.585-10.932-40.725L-10.932-42.574Q-10.932-42.844-11.040-42.905Q-11.147-42.967-11.458-42.967L-11.458-43.247L-10.430-43.322L-10.430-42.615Q-10.300-42.923-10.057-43.122Q-9.814-43.322-9.497-43.322Q-9.278-43.322-9.107-43.198Q-8.936-43.073-8.936-42.861Q-8.936-42.724-9.035-42.625Q-9.134-42.526-9.268-42.526Q-9.404-42.526-9.503-42.625Q-9.603-42.724-9.603-42.861Q-9.603-43.001-9.503-43.100Q-9.794-43.100-9.994-42.904Q-10.194-42.707-10.286-42.413Q-10.378-42.119-10.378-41.839L-10.378-40.725Q-10.378-40.516-9.722-40.516L-9.722-40.236M-8.393-41.719Q-8.393-42.061-8.258-42.360Q-8.123-42.659-7.883-42.883Q-7.644-43.107-7.326-43.232Q-7.008-43.357-6.677-43.357Q-6.232-43.357-5.833-43.141Q-5.433-42.926-5.198-42.548Q-4.964-42.171-4.964-41.719Q-4.964-41.378-5.106-41.094Q-5.248-40.810-5.492-40.603Q-5.737-40.397-6.046-40.282Q-6.355-40.168-6.677-40.168Q-7.107-40.168-7.509-40.369Q-7.911-40.571-8.152-40.923Q-8.393-41.275-8.393-41.719M-6.677-40.417Q-6.075-40.417-5.851-40.795Q-5.627-41.173-5.627-41.805Q-5.627-42.417-5.862-42.776Q-6.096-43.134-6.677-43.134Q-7.729-43.134-7.729-41.805Q-7.729-41.173-7.504-40.795Q-7.278-40.417-6.677-40.417M-2.688-40.236L-4.322-40.236L-4.322-40.516Q-4.093-40.516-3.944-40.550Q-3.795-40.585-3.795-40.725L-3.795-42.574Q-3.795-42.844-3.903-42.905Q-4.011-42.967-4.322-42.967L-4.322-43.247L-3.262-43.322L-3.262-42.673Q-3.091-42.981-2.787-43.152Q-2.483-43.322-2.138-43.322Q-1.632-43.322-1.348-43.099Q-1.064-42.875-1.064-42.379L-1.064-40.725Q-1.064-40.588-0.916-40.552Q-0.767-40.516-0.542-40.516L-0.542-40.236L-2.172-40.236L-2.172-40.516Q-1.943-40.516-1.794-40.550Q-1.646-40.585-1.646-40.725L-1.646-42.365Q-1.646-42.700-1.765-42.900Q-1.885-43.100-2.199-43.100Q-2.469-43.100-2.703-42.964Q-2.937-42.827-3.076-42.593Q-3.214-42.359-3.214-42.085L-3.214-40.725Q-3.214-40.588-3.064-40.552Q-2.914-40.516-2.688-40.516L-2.688-40.236M1.728-40.236L0.094-40.236L0.094-40.516Q0.323-40.516 0.472-40.550Q0.621-40.585 0.621-40.725L0.621-42.574Q0.621-42.844 0.513-42.905Q0.405-42.967 0.094-42.967L0.094-43.247L1.154-43.322L1.154-42.673Q1.325-42.981 1.629-43.152Q1.933-43.322 2.278-43.322Q2.678-43.322 2.955-43.182Q3.232-43.042 3.317-42.694Q3.485-42.987 3.784-43.155Q4.083-43.322 4.428-43.322Q4.934-43.322 5.218-43.099Q5.501-42.875 5.501-42.379L5.501-40.725Q5.501-40.588 5.650-40.552Q5.799-40.516 6.024-40.516L6.024-40.236L4.394-40.236L4.394-40.516Q4.620-40.516 4.770-40.552Q4.920-40.588 4.920-40.725L4.920-42.365Q4.920-42.700 4.801-42.900Q4.681-43.100 4.367-43.100Q4.097-43.100 3.863-42.964Q3.628-42.827 3.490-42.593Q3.352-42.359 3.352-42.085L3.352-40.725Q3.352-40.588 3.500-40.552Q3.649-40.516 3.875-40.516L3.875-40.236L2.244-40.236L2.244-40.516Q2.473-40.516 2.622-40.550Q2.771-40.585 2.771-40.725L2.771-42.365Q2.771-42.700 2.651-42.900Q2.531-43.100 2.217-43.100Q1.947-43.100 1.713-42.964Q1.479-42.827 1.340-42.593Q1.202-42.359 1.202-42.085L1.202-40.725Q1.202-40.588 1.352-40.552Q1.502-40.516 1.728-40.516L1.728-40.236M6.571-41.771Q6.571-42.092 6.696-42.381Q6.821-42.670 7.046-42.893Q7.272-43.117 7.568-43.237Q7.863-43.357 8.181-43.357Q8.509-43.357 8.771-43.257Q9.032-43.158 9.208-42.976Q9.384-42.793 9.478-42.535Q9.572-42.277 9.572-41.945Q9.572-41.853 9.490-41.832L7.234-41.832L7.234-41.771Q7.234-41.183 7.518-40.800Q7.802-40.417 8.369-40.417Q8.690-40.417 8.959-40.610Q9.227-40.803 9.316-41.118Q9.323-41.159 9.398-41.173L9.490-41.173Q9.572-41.149 9.572-41.077Q9.572-41.070 9.565-41.043Q9.453-40.646 9.082-40.407Q8.711-40.168 8.287-40.168Q7.850-40.168 7.450-40.376Q7.050-40.585 6.811-40.952Q6.571-41.319 6.571-41.771M7.241-42.041L9.056-42.041Q9.056-42.318 8.959-42.570Q8.861-42.823 8.663-42.979Q8.465-43.134 8.181-43.134Q7.904-43.134 7.691-42.976Q7.477-42.817 7.359-42.562Q7.241-42.307 7.241-42.041M11.842-40.236L10.208-40.236L10.208-40.516Q10.437-40.516 10.586-40.550Q10.734-40.585 10.734-40.725L10.734-42.574Q10.734-42.844 10.627-42.905Q10.519-42.967 10.208-42.967L10.208-43.247L11.268-43.322L11.268-42.673Q11.438-42.981 11.743-43.152Q12.047-43.322 12.392-43.322Q12.898-43.322 13.182-43.099Q13.465-42.875 13.465-42.379L13.465-40.725Q13.465-40.588 13.614-40.552Q13.763-40.516 13.988-40.516L13.988-40.236L12.358-40.236L12.358-40.516Q12.587-40.516 12.736-40.550Q12.884-40.585 12.884-40.725L12.884-42.365Q12.884-42.700 12.765-42.900Q12.645-43.100 12.331-43.100Q12.061-43.100 11.826-42.964Q11.592-42.827 11.454-42.593Q11.315-42.359 11.315-42.085L11.315-40.725Q11.315-40.588 11.466-40.552Q11.616-40.516 11.842-40.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(126.353 -5.027)\">\u003Cpath d=\"M14.911-41.077L14.911-42.974L14.272-42.974L14.272-43.196Q14.590-43.196 14.807-43.406Q15.024-43.616 15.124-43.926Q15.225-44.235 15.225-44.543L15.492-44.543L15.492-43.254L16.569-43.254L16.569-42.974L15.492-42.974L15.492-41.090Q15.492-40.814 15.596-40.615Q15.700-40.417 15.960-40.417Q16.117-40.417 16.223-40.521Q16.329-40.626 16.379-40.779Q16.428-40.933 16.428-41.090L16.428-41.504L16.695-41.504L16.695-41.077Q16.695-40.851 16.596-40.641Q16.497-40.431 16.312-40.299Q16.128-40.168 15.899-40.168Q15.461-40.168 15.186-40.405Q14.911-40.643 14.911-41.077\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(126.353 -5.027)\">\u003Cpath d=\"M-12.913-32.236L-14.547-32.236L-14.547-32.516Q-14.318-32.516-14.169-32.550Q-14.020-32.585-14.020-32.725L-14.020-34.574Q-14.020-34.844-14.128-34.905Q-14.236-34.967-14.547-34.967L-14.547-35.247L-13.487-35.322L-13.487-34.673Q-13.316-34.981-13.012-35.152Q-12.708-35.322-12.363-35.322Q-11.963-35.322-11.686-35.182Q-11.409-35.042-11.324-34.694Q-11.156-34.987-10.857-35.155Q-10.558-35.322-10.213-35.322Q-9.707-35.322-9.423-35.099Q-9.139-34.875-9.139-34.379L-9.139-32.725Q-9.139-32.588-8.991-32.552Q-8.842-32.516-8.617-32.516L-8.617-32.236L-10.247-32.236L-10.247-32.516Q-10.021-32.516-9.871-32.552Q-9.721-32.588-9.721-32.725L-9.721-34.365Q-9.721-34.700-9.840-34.900Q-9.960-35.100-10.274-35.100Q-10.544-35.100-10.778-34.964Q-11.013-34.827-11.151-34.593Q-11.289-34.359-11.289-34.085L-11.289-32.725Q-11.289-32.588-11.141-32.552Q-10.992-32.516-10.766-32.516L-10.766-32.236L-12.397-32.236L-12.397-32.516Q-12.168-32.516-12.019-32.550Q-11.870-32.585-11.870-32.725L-11.870-34.365Q-11.870-34.700-11.990-34.900Q-12.110-35.100-12.424-35.100Q-12.694-35.100-12.928-34.964Q-13.162-34.827-13.301-34.593Q-13.439-34.359-13.439-34.085L-13.439-32.725Q-13.439-32.588-13.289-32.552Q-13.138-32.516-12.913-32.516L-12.913-32.236M-8.070-33.719Q-8.070-34.061-7.935-34.360Q-7.800-34.659-7.560-34.883Q-7.321-35.107-7.003-35.232Q-6.685-35.357-6.354-35.357Q-5.909-35.357-5.510-35.141Q-5.110-34.926-4.876-34.548Q-4.641-34.171-4.641-33.719Q-4.641-33.378-4.783-33.094Q-4.925-32.810-5.169-32.603Q-5.414-32.397-5.723-32.282Q-6.033-32.168-6.354-32.168Q-6.784-32.168-7.186-32.369Q-7.588-32.571-7.829-32.923Q-8.070-33.275-8.070-33.719M-6.354-32.417Q-5.752-32.417-5.528-32.795Q-5.305-33.173-5.305-33.805Q-5.305-34.417-5.539-34.776Q-5.773-35.134-6.354-35.134Q-7.407-35.134-7.407-33.805Q-7.407-33.173-7.181-32.795Q-6.955-32.417-6.354-32.417\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(126.353 -5.027)\">\u003Cpath d=\"M-3.824-33.747Q-3.824-34.085-3.683-34.376Q-3.543-34.666-3.299-34.880Q-3.055-35.093-2.750-35.208Q-2.446-35.322-2.121-35.322Q-1.851-35.322-1.588-35.223Q-1.325-35.124-1.134-34.946L-1.134-36.344Q-1.134-36.614-1.241-36.676Q-1.349-36.737-1.660-36.737L-1.660-37.018L-0.583-37.093L-0.583-32.909Q-0.583-32.721-0.529-32.638Q-0.474-32.554-0.373-32.535Q-0.272-32.516-0.057-32.516L-0.057-32.236L-1.164-32.168L-1.164-32.585Q-1.581-32.168-2.207-32.168Q-2.638-32.168-3.010-32.380Q-3.383-32.591-3.603-32.952Q-3.824-33.313-3.824-33.747M-2.149-32.390Q-1.940-32.390-1.754-32.462Q-1.568-32.533-1.414-32.670Q-1.260-32.807-1.164-32.985L-1.164-34.594Q-1.250-34.741-1.395-34.861Q-1.540-34.981-1.710-35.040Q-1.879-35.100-2.060-35.100Q-2.620-35.100-2.889-34.711Q-3.157-34.321-3.157-33.740Q-3.157-33.169-2.923-32.779Q-2.689-32.390-2.149-32.390M0.551-33.771Q0.551-34.092 0.676-34.381Q0.801-34.670 1.027-34.893Q1.252-35.117 1.548-35.237Q1.843-35.357 2.161-35.357Q2.489-35.357 2.751-35.257Q3.012-35.158 3.188-34.976Q3.364-34.793 3.458-34.535Q3.552-34.277 3.552-33.945Q3.552-33.853 3.470-33.832L1.215-33.832L1.215-33.771Q1.215-33.183 1.498-32.800Q1.782-32.417 2.349-32.417Q2.671-32.417 2.939-32.610Q3.207-32.803 3.296-33.118Q3.303-33.159 3.378-33.173L3.470-33.173Q3.552-33.149 3.552-33.077Q3.552-33.070 3.546-33.043Q3.433-32.646 3.062-32.407Q2.691-32.168 2.267-32.168Q1.830-32.168 1.430-32.376Q1.030-32.585 0.791-32.952Q0.551-33.319 0.551-33.771M1.221-34.041L3.036-34.041Q3.036-34.318 2.939-34.570Q2.841-34.823 2.643-34.979Q2.445-35.134 2.161-35.134Q1.884-35.134 1.671-34.976Q1.457-34.817 1.339-34.562Q1.221-34.307 1.221-34.041M5.808-32.236L4.205-32.236L4.205-32.516Q4.431-32.516 4.580-32.550Q4.728-32.585 4.728-32.725L4.728-36.344Q4.728-36.614 4.621-36.676Q4.513-36.737 4.205-36.737L4.205-37.018L5.282-37.093L5.282-32.725Q5.282-32.588 5.432-32.552Q5.583-32.516 5.808-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M93.495 24.67H150.4V1.906H93.495Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(131.037 51.08)\">\u003Cpath d=\"M-22.215-38.879L-23.845-38.879L-23.845-39.159Q-23.616-39.159-23.467-39.194Q-23.319-39.228-23.319-39.368L-23.319-42.714Q-23.319-42.885-23.455-42.926Q-23.592-42.967-23.845-42.967L-23.845-43.247L-22.765-43.322L-22.765-42.916Q-22.543-43.117-22.256-43.220Q-21.968-43.322-21.661-43.322Q-21.234-43.322-20.870-43.109Q-20.506-42.895-20.292-42.531Q-20.078-42.167-20.078-41.747Q-20.078-41.302-20.318-40.938Q-20.557-40.574-20.950-40.371Q-21.343-40.168-21.787-40.168Q-22.054-40.168-22.302-40.268Q-22.549-40.369-22.737-40.550L-22.737-39.368Q-22.737-39.231-22.589-39.195Q-22.440-39.159-22.215-39.159L-22.215-38.879M-22.737-42.567L-22.737-40.957Q-22.604-40.704-22.361-40.547Q-22.119-40.390-21.842-40.390Q-21.514-40.390-21.261-40.591Q-21.008-40.793-20.875-41.111Q-20.741-41.429-20.741-41.747Q-20.741-41.976-20.806-42.205Q-20.871-42.434-20.999-42.632Q-21.128-42.830-21.322-42.950Q-21.517-43.069-21.750-43.069Q-22.044-43.069-22.312-42.940Q-22.580-42.810-22.737-42.567M-17.775-40.236L-19.378-40.236L-19.378-40.516Q-19.152-40.516-19.003-40.550Q-18.855-40.585-18.855-40.725L-18.855-44.344Q-18.855-44.614-18.962-44.676Q-19.070-44.737-19.378-44.737L-19.378-45.018L-18.301-45.093L-18.301-40.725Q-18.301-40.588-18.151-40.552Q-18-40.516-17.775-40.516L-17.775-40.236M-17.122-40.964Q-17.122-41.296-16.898-41.523Q-16.674-41.750-16.330-41.878Q-15.987-42.007-15.614-42.059Q-15.242-42.112-14.938-42.112L-14.938-42.365Q-14.938-42.570-15.045-42.750Q-15.153-42.929-15.334-43.032Q-15.515-43.134-15.724-43.134Q-16.131-43.134-16.366-43.042Q-16.277-43.005-16.231-42.921Q-16.185-42.837-16.185-42.735Q-16.185-42.639-16.231-42.560Q-16.277-42.482-16.358-42.437Q-16.438-42.393-16.527-42.393Q-16.677-42.393-16.778-42.490Q-16.879-42.588-16.879-42.735Q-16.879-43.357-15.724-43.357Q-15.512-43.357-15.262-43.293Q-15.013-43.230-14.811-43.111Q-14.610-42.991-14.483-42.806Q-14.357-42.622-14.357-42.379L-14.357-40.803Q-14.357-40.687-14.295-40.591Q-14.234-40.496-14.121-40.496Q-14.011-40.496-13.946-40.590Q-13.881-40.684-13.881-40.803L-13.881-41.251L-13.615-41.251L-13.615-40.803Q-13.615-40.533-13.842-40.368Q-14.069-40.202-14.350-40.202Q-14.558-40.202-14.695-40.356Q-14.832-40.509-14.856-40.725Q-15.003-40.458-15.285-40.313Q-15.567-40.168-15.891-40.168Q-16.168-40.168-16.452-40.243Q-16.735-40.318-16.929-40.497Q-17.122-40.677-17.122-40.964M-16.506-40.964Q-16.506-40.790-16.406-40.660Q-16.305-40.530-16.149-40.460Q-15.994-40.390-15.830-40.390Q-15.611-40.390-15.402-40.487Q-15.194-40.585-15.066-40.766Q-14.938-40.947-14.938-41.173L-14.938-41.901Q-15.262-41.901-15.628-41.810Q-15.994-41.719-16.250-41.507Q-16.506-41.296-16.506-40.964M-11.516-40.236L-13.150-40.236L-13.150-40.516Q-12.921-40.516-12.772-40.550Q-12.624-40.585-12.624-40.725L-12.624-42.574Q-12.624-42.844-12.731-42.905Q-12.839-42.967-13.150-42.967L-13.150-43.247L-12.090-43.322L-12.090-42.673Q-11.920-42.981-11.615-43.152Q-11.311-43.322-10.966-43.322Q-10.460-43.322-10.176-43.099Q-9.893-42.875-9.893-42.379L-9.893-40.725Q-9.893-40.588-9.744-40.552Q-9.595-40.516-9.370-40.516L-9.370-40.236L-11-40.236L-11-40.516Q-10.771-40.516-10.622-40.550Q-10.474-40.585-10.474-40.725L-10.474-42.365Q-10.474-42.700-10.593-42.900Q-10.713-43.100-11.027-43.100Q-11.298-43.100-11.532-42.964Q-11.766-42.827-11.904-42.593Q-12.043-42.359-12.043-42.085L-12.043-40.725Q-12.043-40.588-11.892-40.552Q-11.742-40.516-11.516-40.516L-11.516-40.236M-7.100-40.236L-8.734-40.236L-8.734-40.516Q-8.505-40.516-8.356-40.550Q-8.208-40.585-8.208-40.725L-8.208-42.574Q-8.208-42.844-8.315-42.905Q-8.423-42.967-8.734-42.967L-8.734-43.247L-7.674-43.322L-7.674-42.673Q-7.504-42.981-7.199-43.152Q-6.895-43.322-6.550-43.322Q-6.044-43.322-5.760-43.099Q-5.477-42.875-5.477-42.379L-5.477-40.725Q-5.477-40.588-5.328-40.552Q-5.179-40.516-4.954-40.516L-4.954-40.236L-6.584-40.236L-6.584-40.516Q-6.355-40.516-6.206-40.550Q-6.058-40.585-6.058-40.725L-6.058-42.365Q-6.058-42.700-6.177-42.900Q-6.297-43.100-6.611-43.100Q-6.881-43.100-7.116-42.964Q-7.350-42.827-7.488-42.593Q-7.627-42.359-7.627-42.085L-7.627-40.725Q-7.627-40.588-7.476-40.552Q-7.326-40.516-7.100-40.516L-7.100-40.236M-2.749-40.236L-4.301-40.236L-4.301-40.516Q-4.075-40.516-3.927-40.550Q-3.778-40.585-3.778-40.725L-3.778-42.574Q-3.778-42.762-3.826-42.846Q-3.874-42.929-3.971-42.948Q-4.069-42.967-4.280-42.967L-4.280-43.247L-3.224-43.322L-3.224-40.725Q-3.224-40.585-3.093-40.550Q-2.961-40.516-2.749-40.516L-2.749-40.236M-4.021-44.543Q-4.021-44.714-3.898-44.833Q-3.775-44.953-3.604-44.953Q-3.436-44.953-3.313-44.833Q-3.190-44.714-3.190-44.543Q-3.190-44.368-3.313-44.245Q-3.436-44.122-3.604-44.122Q-3.775-44.122-3.898-44.245Q-4.021-44.368-4.021-44.543M-0.422-40.236L-2.055-40.236L-2.055-40.516Q-1.826-40.516-1.678-40.550Q-1.529-40.585-1.529-40.725L-1.529-42.574Q-1.529-42.844-1.637-42.905Q-1.744-42.967-2.055-42.967L-2.055-43.247L-0.996-43.322L-0.996-42.673Q-0.825-42.981-0.521-43.152Q-0.216-43.322 0.129-43.322Q0.635-43.322 0.918-43.099Q1.202-42.875 1.202-42.379L1.202-40.725Q1.202-40.588 1.351-40.552Q1.499-40.516 1.725-40.516L1.725-40.236L0.095-40.236L0.095-40.516Q0.324-40.516 0.472-40.550Q0.621-40.585 0.621-40.725L0.621-42.365Q0.621-42.700 0.501-42.900Q0.382-43.100 0.067-43.100Q-0.203-43.100-0.437-42.964Q-0.671-42.827-0.809-42.593Q-0.948-42.359-0.948-42.085L-0.948-40.725Q-0.948-40.588-0.798-40.552Q-0.647-40.516-0.422-40.516L-0.422-40.236M2.272-39.703Q2.272-39.949 2.468-40.133Q2.665-40.318 2.921-40.397Q2.785-40.509 2.713-40.670Q2.641-40.831 2.641-41.012Q2.641-41.333 2.853-41.579Q2.518-41.877 2.518-42.287Q2.518-42.748 2.908-43.035Q3.297-43.322 3.776-43.322Q4.247-43.322 4.582-43.076Q4.757-43.230 4.967-43.312Q5.177-43.394 5.406-43.394Q5.570-43.394 5.692-43.287Q5.813-43.179 5.813-43.015Q5.813-42.919 5.741-42.847Q5.669-42.776 5.577-42.776Q5.478-42.776 5.408-42.849Q5.338-42.923 5.338-43.022Q5.338-43.076 5.351-43.107L5.358-43.121Q5.365-43.141 5.374-43.152Q5.382-43.162 5.386-43.169Q5.030-43.169 4.743-42.946Q5.030-42.653 5.030-42.287Q5.030-41.972 4.846-41.740Q4.661-41.507 4.372-41.379Q4.083-41.251 3.776-41.251Q3.574-41.251 3.383-41.301Q3.191-41.350 3.014-41.460Q2.921-41.333 2.921-41.190Q2.921-41.008 3.049-40.873Q3.178-40.738 3.362-40.738L3.994-40.738Q4.442-40.738 4.811-40.667Q5.181-40.595 5.440-40.366Q5.700-40.137 5.700-39.703Q5.700-39.382 5.404-39.180Q5.109-38.978 4.705-38.889Q4.302-38.800 3.988-38.800Q3.670-38.800 3.266-38.889Q2.863-38.978 2.567-39.180Q2.272-39.382 2.272-39.703M2.726-39.703Q2.726-39.474 2.945-39.325Q3.164-39.176 3.456-39.108Q3.748-39.040 3.988-39.040Q4.152-39.040 4.360-39.076Q4.569-39.111 4.775-39.192Q4.982-39.272 5.114-39.400Q5.245-39.528 5.245-39.703Q5.245-40.055 4.864-40.149Q4.483-40.243 3.981-40.243L3.362-40.243Q3.123-40.243 2.925-40.092Q2.726-39.942 2.726-39.703M3.776-41.490Q4.442-41.490 4.442-42.287Q4.442-43.087 3.776-43.087Q3.106-43.087 3.106-42.287Q3.106-41.490 3.776-41.490\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(131.037 51.08)\">\u003Cpath d=\"M-23.563-30.486Q-24.113-30.886-24.484-31.441Q-24.855-31.997-25.036-32.643Q-25.217-33.289-25.217-33.986Q-25.217-34.499-25.117-34.994Q-25.016-35.490-24.811-35.941Q-24.606-36.392-24.293-36.784Q-23.980-37.175-23.563-37.479Q-23.553-37.483-23.546-37.484Q-23.539-37.486-23.529-37.486L-23.461-37.486Q-23.426-37.486-23.404-37.462Q-23.382-37.438-23.382-37.401Q-23.382-37.356-23.409-37.339Q-23.758-37.038-24.011-36.654Q-24.264-36.269-24.416-35.828Q-24.568-35.387-24.640-34.931Q-24.712-34.475-24.712-33.986Q-24.712-32.985-24.402-32.098Q-24.093-31.211-23.409-30.626Q-23.382-30.609-23.382-30.565Q-23.382-30.527-23.404-30.503Q-23.426-30.479-23.461-30.479L-23.529-30.479Q-23.536-30.483-23.544-30.484Q-23.553-30.486-23.563-30.486M-22.514-32.964Q-22.514-33.296-22.290-33.523Q-22.066-33.750-21.723-33.878Q-21.379-34.007-21.007-34.059Q-20.634-34.112-20.330-34.112L-20.330-34.365Q-20.330-34.570-20.437-34.750Q-20.545-34.929-20.726-35.032Q-20.907-35.134-21.116-35.134Q-21.523-35.134-21.758-35.042Q-21.670-35.005-21.623-34.921Q-21.577-34.837-21.577-34.735Q-21.577-34.639-21.623-34.560Q-21.670-34.482-21.750-34.437Q-21.830-34.393-21.919-34.393Q-22.070-34.393-22.170-34.490Q-22.271-34.588-22.271-34.735Q-22.271-35.357-21.116-35.357Q-20.904-35.357-20.654-35.293Q-20.405-35.230-20.203-35.111Q-20.002-34.991-19.875-34.806Q-19.749-34.622-19.749-34.379L-19.749-32.803Q-19.749-32.687-19.687-32.591Q-19.626-32.496-19.513-32.496Q-19.403-32.496-19.339-32.590Q-19.274-32.684-19.274-32.803L-19.274-33.251L-19.007-33.251L-19.007-32.803Q-19.007-32.533-19.234-32.368Q-19.462-32.202-19.742-32.202Q-19.950-32.202-20.087-32.356Q-20.224-32.509-20.248-32.725Q-20.395-32.458-20.677-32.313Q-20.959-32.168-21.283-32.168Q-21.560-32.168-21.844-32.243Q-22.128-32.318-22.321-32.497Q-22.514-32.677-22.514-32.964M-21.899-32.964Q-21.899-32.790-21.798-32.660Q-21.697-32.530-21.541-32.460Q-21.386-32.390-21.222-32.390Q-21.003-32.390-20.795-32.487Q-20.586-32.585-20.458-32.766Q-20.330-32.947-20.330-33.173L-20.330-33.901Q-20.654-33.901-21.020-33.810Q-21.386-33.719-21.642-33.507Q-21.899-33.296-21.899-32.964\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(131.037 51.08)\">\u003Cpath d=\"M-14.243-30.879L-15.873-30.879L-15.873-31.159Q-15.644-31.159-15.495-31.194Q-15.347-31.228-15.347-31.368L-15.347-34.714Q-15.347-34.885-15.483-34.926Q-15.620-34.967-15.873-34.967L-15.873-35.247L-14.793-35.322L-14.793-34.916Q-14.571-35.117-14.284-35.220Q-13.996-35.322-13.689-35.322Q-13.262-35.322-12.898-35.109Q-12.534-34.895-12.320-34.531Q-12.106-34.167-12.106-33.747Q-12.106-33.302-12.346-32.938Q-12.585-32.574-12.978-32.371Q-13.371-32.168-13.815-32.168Q-14.082-32.168-14.330-32.268Q-14.577-32.369-14.765-32.550L-14.765-31.368Q-14.765-31.231-14.617-31.195Q-14.468-31.159-14.243-31.159L-14.243-30.879M-14.765-34.567L-14.765-32.957Q-14.632-32.704-14.389-32.547Q-14.147-32.390-13.870-32.390Q-13.542-32.390-13.289-32.591Q-13.036-32.793-12.903-33.111Q-12.769-33.429-12.769-33.747Q-12.769-33.976-12.834-34.205Q-12.899-34.434-13.027-34.632Q-13.156-34.830-13.350-34.950Q-13.545-35.069-13.778-35.069Q-14.072-35.069-14.340-34.940Q-14.608-34.810-14.765-34.567\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(131.037 51.08)\">\u003Cpath d=\"M-11.296-33.719Q-11.296-34.061-11.161-34.360Q-11.026-34.659-10.786-34.883Q-10.547-35.107-10.229-35.232Q-9.911-35.357-9.580-35.357Q-9.135-35.357-8.736-35.141Q-8.336-34.926-8.101-34.548Q-7.867-34.171-7.867-33.719Q-7.867-33.378-8.009-33.094Q-8.151-32.810-8.395-32.603Q-8.640-32.397-8.949-32.282Q-9.258-32.168-9.580-32.168Q-10.010-32.168-10.412-32.369Q-10.814-32.571-11.055-32.923Q-11.296-33.275-11.296-33.719M-9.580-32.417Q-8.978-32.417-8.754-32.795Q-8.530-33.173-8.530-33.805Q-8.530-34.417-8.765-34.776Q-8.999-35.134-9.580-35.134Q-10.632-35.134-10.632-33.805Q-10.632-33.173-10.407-32.795Q-10.181-32.417-9.580-32.417M-5.605-32.236L-7.208-32.236L-7.208-32.516Q-6.982-32.516-6.833-32.550Q-6.685-32.585-6.685-32.725L-6.685-36.344Q-6.685-36.614-6.792-36.676Q-6.900-36.737-7.208-36.737L-7.208-37.018L-6.131-37.093L-6.131-32.725Q-6.131-32.588-5.981-32.552Q-5.830-32.516-5.605-32.516L-5.605-32.236M-3.393-32.236L-4.945-32.236L-4.945-32.516Q-4.719-32.516-4.571-32.550Q-4.422-32.585-4.422-32.725L-4.422-34.574Q-4.422-34.762-4.470-34.846Q-4.518-34.929-4.615-34.948Q-4.713-34.967-4.924-34.967L-4.924-35.247L-3.868-35.322L-3.868-32.725Q-3.868-32.585-3.737-32.550Q-3.605-32.516-3.393-32.516L-3.393-32.236M-4.665-36.543Q-4.665-36.714-4.542-36.833Q-4.419-36.953-4.248-36.953Q-4.080-36.953-3.957-36.833Q-3.834-36.714-3.834-36.543Q-3.834-36.368-3.957-36.245Q-4.080-36.122-4.248-36.122Q-4.419-36.122-4.542-36.245Q-4.665-36.368-4.665-36.543M-2.747-33.747Q-2.747-34.075-2.612-34.376Q-2.477-34.676-2.241-34.897Q-2.006-35.117-1.701-35.237Q-1.397-35.357-1.072-35.357Q-0.567-35.357-0.218-35.254Q0.131-35.152 0.131-34.776Q0.131-34.629 0.033-34.528Q-0.064-34.427-0.211-34.427Q-0.365-34.427-0.464-34.526Q-0.563-34.625-0.563-34.776Q-0.563-34.964-0.423-35.056Q-0.625-35.107-1.066-35.107Q-1.421-35.107-1.650-34.911Q-1.879-34.714-1.980-34.405Q-2.081-34.095-2.081-33.747Q-2.081-33.398-1.954-33.092Q-1.828-32.786-1.573-32.602Q-1.319-32.417-0.963-32.417Q-0.741-32.417-0.556-32.501Q-0.372-32.585-0.237-32.740Q-0.102-32.896-0.044-33.104Q-0.030-33.159 0.025-33.159L0.138-33.159Q0.168-33.159 0.191-33.135Q0.213-33.111 0.213-33.077L0.213-33.056Q0.127-32.769-0.061-32.571Q-0.249-32.373-0.514-32.270Q-0.778-32.168-1.072-32.168Q-1.503-32.168-1.891-32.374Q-2.279-32.581-2.513-32.944Q-2.747-33.306-2.747-33.747M1.136-31.101Q1.265-31.033 1.402-31.033Q1.573-31.033 1.723-31.122Q1.874-31.211 1.985-31.356Q2.096-31.501 2.175-31.669L2.438-32.236L1.269-34.762Q1.194-34.909 1.064-34.941Q0.934-34.974 0.702-34.974L0.702-35.254L2.223-35.254L2.223-34.974Q1.874-34.974 1.874-34.827Q1.877-34.806 1.879-34.789Q1.881-34.772 1.881-34.762L2.739-32.903L3.511-34.574Q3.545-34.642 3.545-34.721Q3.545-34.834 3.462-34.904Q3.378-34.974 3.265-34.974L3.265-35.254L4.461-35.254L4.461-34.974Q4.243-34.974 4.070-34.870Q3.897-34.765 3.805-34.574L2.469-31.669Q2.298-31.299 2.028-31.053Q1.758-30.807 1.402-30.807Q1.132-30.807 0.913-30.973Q0.695-31.139 0.695-31.402Q0.695-31.539 0.787-31.628Q0.879-31.716 1.019-31.716Q1.156-31.716 1.245-31.628Q1.334-31.539 1.334-31.402Q1.334-31.299 1.281-31.221Q1.228-31.142 1.136-31.101M5.323-30.479L5.254-30.479Q5.220-30.479 5.198-30.505Q5.176-30.530 5.176-30.565Q5.176-30.609 5.206-30.626Q5.562-30.930 5.811-31.320Q6.061-31.710 6.213-32.142Q6.365-32.574 6.435-33.043Q6.505-33.511 6.505-33.986Q6.505-34.465 6.435-34.931Q6.365-35.398 6.211-35.833Q6.057-36.269 5.806-36.657Q5.555-37.045 5.206-37.339Q5.176-37.356 5.176-37.401Q5.176-37.435 5.198-37.460Q5.220-37.486 5.254-37.486L5.323-37.486Q5.333-37.486 5.341-37.484Q5.350-37.483 5.360-37.479Q5.904-37.079 6.276-36.526Q6.649-35.972 6.830-35.326Q7.011-34.680 7.011-33.986Q7.011-33.285 6.830-32.638Q6.649-31.990 6.275-31.436Q5.900-30.882 5.360-30.486Q5.350-30.486 5.341-30.484Q5.333-30.483 5.323-30.479\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"m2.646-56.591 88.075 12.606\"\u002F>\u003Cpath stroke=\"none\" d=\"m93.295-43.617-3.823-2.648 1.25 2.28-1.84 1.837\"\u002F>\u003Cg transform=\"translate(57.039 -21.601)\">\u003Cpath d=\"M-25.166-33.077L-25.166-34.974L-25.805-34.974L-25.805-35.196Q-25.487-35.196-25.270-35.406Q-25.053-35.616-24.953-35.926Q-24.852-36.235-24.852-36.543L-24.585-36.543L-24.585-35.254L-23.508-35.254L-23.508-34.974L-24.585-34.974L-24.585-33.090Q-24.585-32.814-24.481-32.615Q-24.377-32.417-24.117-32.417Q-23.960-32.417-23.854-32.521Q-23.748-32.626-23.698-32.779Q-23.649-32.933-23.649-33.090L-23.649-33.504L-23.382-33.504L-23.382-33.077Q-23.382-32.851-23.481-32.641Q-23.580-32.431-23.765-32.299Q-23.949-32.168-24.178-32.168Q-24.616-32.168-24.891-32.405Q-25.166-32.643-25.166-33.077M-20.822-32.236L-22.558-32.236L-22.558-32.516Q-22.329-32.516-22.181-32.550Q-22.032-32.585-22.032-32.725L-22.032-34.574Q-22.032-34.844-22.140-34.905Q-22.247-34.967-22.558-34.967L-22.558-35.247L-21.529-35.322L-21.529-34.615Q-21.400-34.923-21.157-35.122Q-20.914-35.322-20.596-35.322Q-20.378-35.322-20.207-35.198Q-20.036-35.073-20.036-34.861Q-20.036-34.724-20.135-34.625Q-20.234-34.526-20.367-34.526Q-20.504-34.526-20.603-34.625Q-20.702-34.724-20.702-34.861Q-20.702-35.001-20.603-35.100Q-20.894-35.100-21.094-34.904Q-21.294-34.707-21.386-34.413Q-21.478-34.119-21.478-33.839L-21.478-32.725Q-21.478-32.516-20.822-32.516L-20.822-32.236M-19.393-32.964Q-19.393-33.296-19.169-33.523Q-18.945-33.750-18.602-33.878Q-18.258-34.007-17.886-34.059Q-17.513-34.112-17.209-34.112L-17.209-34.365Q-17.209-34.570-17.317-34.750Q-17.424-34.929-17.606-35.032Q-17.787-35.134-17.995-35.134Q-18.402-35.134-18.638-35.042Q-18.549-35.005-18.503-34.921Q-18.457-34.837-18.457-34.735Q-18.457-34.639-18.503-34.560Q-18.549-34.482-18.629-34.437Q-18.710-34.393-18.799-34.393Q-18.949-34.393-19.050-34.490Q-19.151-34.588-19.151-34.735Q-19.151-35.357-17.995-35.357Q-17.783-35.357-17.534-35.293Q-17.284-35.230-17.083-35.111Q-16.881-34.991-16.755-34.806Q-16.628-34.622-16.628-34.379L-16.628-32.803Q-16.628-32.687-16.567-32.591Q-16.505-32.496-16.392-32.496Q-16.283-32.496-16.218-32.590Q-16.153-32.684-16.153-32.803L-16.153-33.251L-15.886-33.251L-15.886-32.803Q-15.886-32.533-16.114-32.368Q-16.341-32.202-16.621-32.202Q-16.830-32.202-16.966-32.356Q-17.103-32.509-17.127-32.725Q-17.274-32.458-17.556-32.313Q-17.838-32.168-18.163-32.168Q-18.440-32.168-18.723-32.243Q-19.007-32.318-19.200-32.497Q-19.393-32.677-19.393-32.964M-18.778-32.964Q-18.778-32.790-18.677-32.660Q-18.576-32.530-18.421-32.460Q-18.265-32.390-18.101-32.390Q-17.882-32.390-17.674-32.487Q-17.465-32.585-17.337-32.766Q-17.209-32.947-17.209-33.173L-17.209-33.901Q-17.534-33.901-17.900-33.810Q-18.265-33.719-18.522-33.507Q-18.778-33.296-18.778-32.964M-13.788-32.236L-15.422-32.236L-15.422-32.516Q-15.193-32.516-15.044-32.550Q-14.895-32.585-14.895-32.725L-14.895-34.574Q-14.895-34.844-15.003-34.905Q-15.111-34.967-15.422-34.967L-15.422-35.247L-14.362-35.322L-14.362-34.673Q-14.191-34.981-13.887-35.152Q-13.583-35.322-13.237-35.322Q-12.732-35.322-12.448-35.099Q-12.164-34.875-12.164-34.379L-12.164-32.725Q-12.164-32.588-12.016-32.552Q-11.867-32.516-11.641-32.516L-11.641-32.236L-13.272-32.236L-13.272-32.516Q-13.043-32.516-12.894-32.550Q-12.745-32.585-12.745-32.725L-12.745-34.365Q-12.745-34.700-12.865-34.900Q-12.985-35.100-13.299-35.100Q-13.569-35.100-13.803-34.964Q-14.037-34.827-14.176-34.593Q-14.314-34.359-14.314-34.085L-14.314-32.725Q-14.314-32.588-14.164-32.552Q-14.013-32.516-13.788-32.516L-13.788-32.236M-11.053-32.243L-11.053-33.306Q-11.053-33.330-11.026-33.357Q-10.999-33.384-10.975-33.384L-10.865-33.384Q-10.800-33.384-10.787-33.326Q-10.691-32.892-10.445-32.641Q-10.199-32.390-9.785-32.390Q-9.444-32.390-9.191-32.523Q-8.938-32.656-8.938-32.964Q-8.938-33.121-9.032-33.236Q-9.126-33.350-9.264-33.419Q-9.403-33.487-9.570-33.525L-10.151-33.624Q-10.507-33.692-10.780-33.913Q-11.053-34.133-11.053-34.475Q-11.053-34.724-10.942-34.899Q-10.831-35.073-10.645-35.172Q-10.459-35.271-10.243-35.314Q-10.028-35.357-9.785-35.357Q-9.372-35.357-9.091-35.175L-8.876-35.350Q-8.866-35.353-8.859-35.355Q-8.852-35.357-8.842-35.357L-8.791-35.357Q-8.763-35.357-8.739-35.333Q-8.715-35.309-8.715-35.281L-8.715-34.434Q-8.715-34.413-8.739-34.386Q-8.763-34.359-8.791-34.359L-8.903-34.359Q-8.931-34.359-8.956-34.384Q-8.982-34.410-8.982-34.434Q-8.982-34.670-9.088-34.834Q-9.194-34.998-9.377-35.080Q-9.560-35.162-9.792-35.162Q-10.120-35.162-10.377-35.059Q-10.633-34.957-10.633-34.680Q-10.633-34.485-10.450-34.376Q-10.267-34.266-10.038-34.225L-9.464-34.119Q-9.218-34.071-9.004-33.943Q-8.791-33.815-8.654-33.612Q-8.517-33.408-8.517-33.159Q-8.517-32.646-8.883-32.407Q-9.249-32.168-9.785-32.168Q-10.281-32.168-10.612-32.462L-10.879-32.188Q-10.900-32.168-10.927-32.168L-10.975-32.168Q-10.999-32.168-11.026-32.195Q-11.053-32.222-11.053-32.243M-6.272-32.236L-7.823-32.236L-7.823-32.516Q-7.598-32.516-7.449-32.550Q-7.300-32.585-7.300-32.725L-7.300-34.574Q-7.300-34.762-7.348-34.846Q-7.396-34.929-7.494-34.948Q-7.591-34.967-7.803-34.967L-7.803-35.247L-6.747-35.322L-6.747-32.725Q-6.747-32.585-6.615-32.550Q-6.484-32.516-6.272-32.516L-6.272-32.236M-7.543-36.543Q-7.543-36.714-7.420-36.833Q-7.297-36.953-7.126-36.953Q-6.959-36.953-6.836-36.833Q-6.713-36.714-6.713-36.543Q-6.713-36.368-6.836-36.245Q-6.959-36.122-7.126-36.122Q-7.297-36.122-7.420-36.245Q-7.543-36.368-7.543-36.543M-5.099-33.077L-5.099-34.974L-5.738-34.974L-5.738-35.196Q-5.421-35.196-5.204-35.406Q-4.986-35.616-4.886-35.926Q-4.785-36.235-4.785-36.543L-4.518-36.543L-4.518-35.254L-3.442-35.254L-3.442-34.974L-4.518-34.974L-4.518-33.090Q-4.518-32.814-4.414-32.615Q-4.310-32.417-4.050-32.417Q-3.893-32.417-3.787-32.521Q-3.681-32.626-3.631-32.779Q-3.582-32.933-3.582-33.090L-3.582-33.504L-3.315-33.504L-3.315-33.077Q-3.315-32.851-3.414-32.641Q-3.513-32.431-3.698-32.299Q-3.882-32.168-4.111-32.168Q-4.549-32.168-4.824-32.405Q-5.099-32.643-5.099-33.077M-0.888-32.236L-2.440-32.236L-2.440-32.516Q-2.215-32.516-2.066-32.550Q-1.917-32.585-1.917-32.725L-1.917-34.574Q-1.917-34.762-1.965-34.846Q-2.013-34.929-2.110-34.948Q-2.208-34.967-2.420-34.967L-2.420-35.247L-1.363-35.322L-1.363-32.725Q-1.363-32.585-1.232-32.550Q-1.100-32.516-0.888-32.516L-0.888-32.236M-2.160-36.543Q-2.160-36.714-2.037-36.833Q-1.914-36.953-1.743-36.953Q-1.575-36.953-1.452-36.833Q-1.329-36.714-1.329-36.543Q-1.329-36.368-1.452-36.245Q-1.575-36.122-1.743-36.122Q-1.914-36.122-2.037-36.245Q-2.160-36.368-2.160-36.543M-0.283-33.719Q-0.283-34.061-0.148-34.360Q-0.013-34.659 0.226-34.883Q0.465-35.107 0.783-35.232Q1.101-35.357 1.432-35.357Q1.877-35.357 2.277-35.141Q2.677-34.926 2.911-34.548Q3.145-34.171 3.145-33.719Q3.145-33.378 3.003-33.094Q2.861-32.810 2.617-32.603Q2.372-32.397 2.063-32.282Q1.754-32.168 1.432-32.168Q1.002-32.168 0.600-32.369Q0.199-32.571-0.042-32.923Q-0.283-33.275-0.283-33.719M1.432-32.417Q2.034-32.417 2.258-32.795Q2.482-33.173 2.482-33.805Q2.482-34.417 2.248-34.776Q2.014-35.134 1.432-35.134Q0.380-35.134 0.380-33.805Q0.380-33.173 0.605-32.795Q0.831-32.417 1.432-32.417M5.421-32.236L3.787-32.236L3.787-32.516Q4.016-32.516 4.165-32.550Q4.314-32.585 4.314-32.725L4.314-34.574Q4.314-34.844 4.206-34.905Q4.098-34.967 3.787-34.967L3.787-35.247L4.847-35.322L4.847-34.673Q5.018-34.981 5.322-35.152Q5.626-35.322 5.972-35.322Q6.477-35.322 6.761-35.099Q7.045-34.875 7.045-34.379L7.045-32.725Q7.045-32.588 7.193-32.552Q7.342-32.516 7.568-32.516L7.568-32.236L5.937-32.236L5.937-32.516Q6.166-32.516 6.315-32.550Q6.464-32.585 6.464-32.725L6.464-34.365Q6.464-34.700 6.344-34.900Q6.224-35.100 5.910-35.100Q5.640-35.100 5.406-34.964Q5.172-34.827 5.033-34.593Q4.895-34.359 4.895-34.085L4.895-32.725Q4.895-32.588 5.045-32.552Q5.196-32.516 5.421-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M2.646-30.643 90.721-43.25\"\u002F>\u003Cpath stroke=\"none\" d=\"m93.295-43.617-4.413-1.47 1.84 1.838-1.25 2.28\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M121.948-32.036V-.893\"\u002F>\u003Cpath stroke=\"none\" d=\"m121.948 1.707 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(151.687 18.822)\">\u003Cpath d=\"M-25.693-32.243L-25.693-33.306Q-25.693-33.330-25.665-33.357Q-25.638-33.384-25.614-33.384L-25.505-33.384Q-25.440-33.384-25.426-33.326Q-25.330-32.892-25.084-32.641Q-24.838-32.390-24.424-32.390Q-24.083-32.390-23.830-32.523Q-23.577-32.656-23.577-32.964Q-23.577-33.121-23.671-33.236Q-23.765-33.350-23.903-33.419Q-24.042-33.487-24.209-33.525L-24.790-33.624Q-25.146-33.692-25.419-33.913Q-25.693-34.133-25.693-34.475Q-25.693-34.724-25.581-34.899Q-25.470-35.073-25.284-35.172Q-25.098-35.271-24.882-35.314Q-24.667-35.357-24.424-35.357Q-24.011-35.357-23.731-35.175L-23.515-35.350Q-23.505-35.353-23.498-35.355Q-23.491-35.357-23.481-35.357L-23.430-35.357Q-23.403-35.357-23.379-35.333Q-23.355-35.309-23.355-35.281L-23.355-34.434Q-23.355-34.413-23.379-34.386Q-23.403-34.359-23.430-34.359L-23.543-34.359Q-23.570-34.359-23.596-34.384Q-23.621-34.410-23.621-34.434Q-23.621-34.670-23.727-34.834Q-23.833-34.998-24.016-35.080Q-24.199-35.162-24.431-35.162Q-24.759-35.162-25.016-35.059Q-25.272-34.957-25.272-34.680Q-25.272-34.485-25.089-34.376Q-24.906-34.266-24.677-34.225L-24.103-34.119Q-23.857-34.071-23.643-33.943Q-23.430-33.815-23.293-33.612Q-23.156-33.408-23.156-33.159Q-23.156-32.646-23.522-32.407Q-23.888-32.168-24.424-32.168Q-24.920-32.168-25.252-32.462L-25.518-32.188Q-25.539-32.168-25.566-32.168L-25.614-32.168Q-25.638-32.168-25.665-32.195Q-25.693-32.222-25.693-32.243M-20.911-32.236L-22.463-32.236L-22.463-32.516Q-22.237-32.516-22.088-32.550Q-21.940-32.585-21.940-32.725L-21.940-34.574Q-21.940-34.762-21.987-34.846Q-22.035-34.929-22.133-34.948Q-22.230-34.967-22.442-34.967L-22.442-35.247L-21.386-35.322L-21.386-32.725Q-21.386-32.585-21.254-32.550Q-21.123-32.516-20.911-32.516L-20.911-32.236M-22.182-36.543Q-22.182-36.714-22.059-36.833Q-21.936-36.953-21.765-36.953Q-21.598-36.953-21.475-36.833Q-21.352-36.714-21.352-36.543Q-21.352-36.368-21.475-36.245Q-21.598-36.122-21.765-36.122Q-21.936-36.122-22.059-36.245Q-22.182-36.368-22.182-36.543M-18.583-32.236L-20.217-32.236L-20.217-32.516Q-19.988-32.516-19.839-32.550Q-19.691-32.585-19.691-32.725L-19.691-34.574Q-19.691-34.844-19.798-34.905Q-19.906-34.967-20.217-34.967L-20.217-35.247L-19.157-35.322L-19.157-34.673Q-18.986-34.981-18.682-35.152Q-18.378-35.322-18.033-35.322Q-17.633-35.322-17.356-35.182Q-17.079-35.042-16.994-34.694Q-16.826-34.987-16.527-35.155Q-16.228-35.322-15.883-35.322Q-15.377-35.322-15.093-35.099Q-14.810-34.875-14.810-34.379L-14.810-32.725Q-14.810-32.588-14.661-32.552Q-14.512-32.516-14.287-32.516L-14.287-32.236L-15.917-32.236L-15.917-32.516Q-15.692-32.516-15.541-32.552Q-15.391-32.588-15.391-32.725L-15.391-34.365Q-15.391-34.700-15.510-34.900Q-15.630-35.100-15.945-35.100Q-16.215-35.100-16.449-34.964Q-16.683-34.827-16.821-34.593Q-16.960-34.359-16.960-34.085L-16.960-32.725Q-16.960-32.588-16.811-32.552Q-16.662-32.516-16.437-32.516L-16.437-32.236L-18.067-32.236L-18.067-32.516Q-17.838-32.516-17.689-32.550Q-17.541-32.585-17.541-32.725L-17.541-34.365Q-17.541-34.700-17.660-34.900Q-17.780-35.100-18.094-35.100Q-18.364-35.100-18.599-34.964Q-18.833-34.827-18.971-34.593Q-19.110-34.359-19.110-34.085L-19.110-32.725Q-19.110-32.588-18.959-32.552Q-18.809-32.516-18.583-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(151.687 18.822)\">\u003Cpath d=\"M-13.331-33.070L-13.331-34.574Q-13.331-34.844-13.439-34.905Q-13.547-34.967-13.858-34.967L-13.858-35.247L-12.750-35.322L-12.750-33.090L-12.750-33.070Q-12.750-32.790-12.699-32.646Q-12.648-32.503-12.506-32.446Q-12.364-32.390-12.077-32.390Q-11.824-32.390-11.619-32.530Q-11.414-32.670-11.298-32.896Q-11.181-33.121-11.181-33.371L-11.181-34.574Q-11.181-34.844-11.289-34.905Q-11.397-34.967-11.708-34.967L-11.708-35.247L-10.600-35.322L-10.600-32.909Q-10.600-32.718-10.547-32.636Q-10.494-32.554-10.394-32.535Q-10.293-32.516-10.077-32.516L-10.077-32.236L-11.154-32.168L-11.154-32.732Q-11.263-32.550-11.409-32.427Q-11.554-32.304-11.740-32.236Q-11.927-32.168-12.128-32.168Q-13.331-32.168-13.331-33.070M-7.822-32.236L-9.425-32.236L-9.425-32.516Q-9.199-32.516-9.050-32.550Q-8.902-32.585-8.902-32.725L-8.902-36.344Q-8.902-36.614-9.009-36.676Q-9.117-36.737-9.425-36.737L-9.425-37.018L-8.348-37.093L-8.348-32.725Q-8.348-32.588-8.198-32.552Q-8.047-32.516-7.822-32.516L-7.822-32.236M-7.169-32.964Q-7.169-33.296-6.945-33.523Q-6.721-33.750-6.377-33.878Q-6.034-34.007-5.661-34.059Q-5.289-34.112-4.985-34.112L-4.985-34.365Q-4.985-34.570-5.092-34.750Q-5.200-34.929-5.381-35.032Q-5.562-35.134-5.771-35.134Q-6.178-35.134-6.413-35.042Q-6.324-35.005-6.278-34.921Q-6.232-34.837-6.232-34.735Q-6.232-34.639-6.278-34.560Q-6.324-34.482-6.405-34.437Q-6.485-34.393-6.574-34.393Q-6.724-34.393-6.825-34.490Q-6.926-34.588-6.926-34.735Q-6.926-35.357-5.771-35.357Q-5.559-35.357-5.309-35.293Q-5.060-35.230-4.858-35.111Q-4.657-34.991-4.530-34.806Q-4.404-34.622-4.404-34.379L-4.404-32.803Q-4.404-32.687-4.342-32.591Q-4.281-32.496-4.168-32.496Q-4.058-32.496-3.993-32.590Q-3.928-32.684-3.928-32.803L-3.928-33.251L-3.662-33.251L-3.662-32.803Q-3.662-32.533-3.889-32.368Q-4.116-32.202-4.397-32.202Q-4.605-32.202-4.742-32.356Q-4.879-32.509-4.903-32.725Q-5.050-32.458-5.332-32.313Q-5.614-32.168-5.938-32.168Q-6.215-32.168-6.499-32.243Q-6.783-32.318-6.976-32.497Q-7.169-32.677-7.169-32.964M-6.553-32.964Q-6.553-32.790-6.453-32.660Q-6.352-32.530-6.196-32.460Q-6.041-32.390-5.877-32.390Q-5.658-32.390-5.449-32.487Q-5.241-32.585-5.113-32.766Q-4.985-32.947-4.985-33.173L-4.985-33.901Q-5.309-33.901-5.675-33.810Q-6.041-33.719-6.297-33.507Q-6.553-33.296-6.553-32.964M-2.719-33.077L-2.719-34.974L-3.358-34.974L-3.358-35.196Q-3.040-35.196-2.823-35.406Q-2.606-35.616-2.505-35.926Q-2.404-36.235-2.404-36.543L-2.137-36.543L-2.137-35.254L-1.061-35.254L-1.061-34.974L-2.137-34.974L-2.137-33.090Q-2.137-32.814-2.033-32.615Q-1.929-32.417-1.669-32.417Q-1.512-32.417-1.406-32.521Q-1.300-32.626-1.251-32.779Q-1.201-32.933-1.201-33.090L-1.201-33.504L-0.934-33.504L-0.934-33.077Q-0.934-32.851-1.033-32.641Q-1.133-32.431-1.317-32.299Q-1.502-32.168-1.731-32.168Q-2.168-32.168-2.443-32.405Q-2.719-32.643-2.719-33.077M-0.165-33.771Q-0.165-34.092-0.041-34.381Q0.084-34.670 0.310-34.893Q0.535-35.117 0.831-35.237Q1.127-35.357 1.445-35.357Q1.773-35.357 2.034-35.257Q2.296-35.158 2.472-34.976Q2.648-34.793 2.742-34.535Q2.836-34.277 2.836-33.945Q2.836-33.853 2.754-33.832L0.498-33.832L0.498-33.771Q0.498-33.183 0.781-32.800Q1.065-32.417 1.633-32.417Q1.954-32.417 2.222-32.610Q2.490-32.803 2.579-33.118Q2.586-33.159 2.661-33.173L2.754-33.173Q2.836-33.149 2.836-33.077Q2.836-33.070 2.829-33.043Q2.716-32.646 2.345-32.407Q1.974-32.168 1.551-32.168Q1.113-32.168 0.713-32.376Q0.313-32.585 0.074-32.952Q-0.165-33.319-0.165-33.771M0.505-34.041L2.320-34.041Q2.320-34.318 2.222-34.570Q2.125-34.823 1.926-34.979Q1.728-35.134 1.445-35.134Q1.168-35.134 0.954-34.976Q0.740-34.817 0.623-34.562Q0.505-34.307 0.505-34.041\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(151.687 18.822)\">\u003Cpath d=\"M7.779-30.879L6.149-30.879L6.149-31.159Q6.378-31.159 6.527-31.194Q6.675-31.228 6.675-31.368L6.675-34.714Q6.675-34.885 6.539-34.926Q6.402-34.967 6.149-34.967L6.149-35.247L7.229-35.322L7.229-34.916Q7.451-35.117 7.738-35.220Q8.026-35.322 8.333-35.322Q8.760-35.322 9.124-35.109Q9.488-34.895 9.702-34.531Q9.916-34.167 9.916-33.747Q9.916-33.302 9.676-32.938Q9.437-32.574 9.044-32.371Q8.651-32.168 8.207-32.168Q7.940-32.168 7.692-32.268Q7.445-32.369 7.257-32.550L7.257-31.368Q7.257-31.231 7.405-31.195Q7.554-31.159 7.779-31.159L7.779-30.879M7.257-34.567L7.257-32.957Q7.390-32.704 7.633-32.547Q7.875-32.390 8.152-32.390Q8.480-32.390 8.733-32.591Q8.986-32.793 9.119-33.111Q9.253-33.429 9.253-33.747Q9.253-33.976 9.188-34.205Q9.123-34.434 8.995-34.632Q8.866-34.830 8.672-34.950Q8.477-35.069 8.244-35.069Q7.950-35.069 7.682-34.940Q7.414-34.810 7.257-34.567M10.610-32.964Q10.610-33.296 10.833-33.523Q11.057-33.750 11.401-33.878Q11.744-34.007 12.117-34.059Q12.489-34.112 12.794-34.112L12.794-34.365Q12.794-34.570 12.686-34.750Q12.578-34.929 12.397-35.032Q12.216-35.134 12.008-35.134Q11.601-35.134 11.365-35.042Q11.454-35.005 11.500-34.921Q11.546-34.837 11.546-34.735Q11.546-34.639 11.500-34.560Q11.454-34.482 11.373-34.437Q11.293-34.393 11.204-34.393Q11.054-34.393 10.953-34.490Q10.852-34.588 10.852-34.735Q10.852-35.357 12.008-35.357Q12.219-35.357 12.469-35.293Q12.718-35.230 12.920-35.111Q13.122-34.991 13.248-34.806Q13.375-34.622 13.375-34.379L13.375-32.803Q13.375-32.687 13.436-32.591Q13.498-32.496 13.611-32.496Q13.720-32.496 13.785-32.590Q13.850-32.684 13.850-32.803L13.850-33.251L14.116-33.251L14.116-32.803Q14.116-32.533 13.889-32.368Q13.662-32.202 13.382-32.202Q13.173-32.202 13.036-32.356Q12.900-32.509 12.876-32.725Q12.729-32.458 12.447-32.313Q12.165-32.168 11.840-32.168Q11.563-32.168 11.279-32.243Q10.996-32.318 10.803-32.497Q10.610-32.677 10.610-32.964M11.225-32.964Q11.225-32.790 11.326-32.660Q11.426-32.530 11.582-32.460Q11.738-32.390 11.902-32.390Q12.120-32.390 12.329-32.487Q12.537-32.585 12.665-32.766Q12.794-32.947 12.794-33.173L12.794-33.901Q12.469-33.901 12.103-33.810Q11.738-33.719 11.481-33.507Q11.225-33.296 11.225-32.964M15.060-33.077L15.060-34.974L14.421-34.974L14.421-35.196Q14.738-35.196 14.956-35.406Q15.173-35.616 15.273-35.926Q15.374-36.235 15.374-36.543L15.641-36.543L15.641-35.254L16.717-35.254L16.717-34.974L15.641-34.974L15.641-33.090Q15.641-32.814 15.745-32.615Q15.849-32.417 16.109-32.417Q16.266-32.417 16.372-32.521Q16.478-32.626 16.528-32.779Q16.577-32.933 16.577-33.090L16.577-33.504L16.844-33.504L16.844-33.077Q16.844-32.851 16.745-32.641Q16.646-32.431 16.461-32.299Q16.277-32.168 16.048-32.168Q15.610-32.168 15.335-32.405Q15.060-32.643 15.060-33.077M19.336-32.236L17.702-32.236L17.702-32.516Q17.931-32.516 18.080-32.550Q18.228-32.585 18.228-32.725L18.228-36.344Q18.228-36.614 18.121-36.676Q18.013-36.737 17.702-36.737L17.702-37.018L18.782-37.093L18.782-34.707Q18.888-34.892 19.066-35.034Q19.243-35.175 19.452-35.249Q19.660-35.322 19.886-35.322Q20.392-35.322 20.675-35.099Q20.959-34.875 20.959-34.379L20.959-32.725Q20.959-32.588 21.108-32.552Q21.257-32.516 21.482-32.516L21.482-32.236L19.852-32.236L19.852-32.516Q20.081-32.516 20.229-32.550Q20.378-32.585 20.378-32.725L20.378-34.365Q20.378-34.700 20.258-34.900Q20.139-35.100 19.824-35.100Q19.554-35.100 19.320-34.964Q19.086-34.827 18.948-34.593Q18.809-34.359 18.809-34.085L18.809-32.725Q18.809-32.588 18.960-32.552Q19.110-32.516 19.336-32.516L19.336-32.236M22.070-32.243L22.070-33.306Q22.070-33.330 22.097-33.357Q22.125-33.384 22.149-33.384L22.258-33.384Q22.323-33.384 22.337-33.326Q22.432-32.892 22.678-32.641Q22.925-32.390 23.338-32.390Q23.680-32.390 23.933-32.523Q24.186-32.656 24.186-32.964Q24.186-33.121 24.092-33.236Q23.998-33.350 23.859-33.419Q23.721-33.487 23.553-33.525L22.972-33.624Q22.617-33.692 22.343-33.913Q22.070-34.133 22.070-34.475Q22.070-34.724 22.181-34.899Q22.292-35.073 22.478-35.172Q22.665-35.271 22.880-35.314Q23.095-35.357 23.338-35.357Q23.752-35.357 24.032-35.175L24.247-35.350Q24.258-35.353 24.264-35.355Q24.271-35.357 24.281-35.357L24.333-35.357Q24.360-35.357 24.384-35.333Q24.408-35.309 24.408-35.281L24.408-34.434Q24.408-34.413 24.384-34.386Q24.360-34.359 24.333-34.359L24.220-34.359Q24.193-34.359 24.167-34.384Q24.141-34.410 24.141-34.434Q24.141-34.670 24.035-34.834Q23.929-34.998 23.747-35.080Q23.564-35.162 23.331-35.162Q23.003-35.162 22.747-35.059Q22.490-34.957 22.490-34.680Q22.490-34.485 22.673-34.376Q22.856-34.266 23.085-34.225L23.659-34.119Q23.905-34.071 24.119-33.943Q24.333-33.815 24.469-33.612Q24.606-33.408 24.606-33.159Q24.606-32.646 24.240-32.407Q23.875-32.168 23.338-32.168Q22.842-32.168 22.511-32.462L22.244-32.188Q22.224-32.168 22.196-32.168L22.149-32.168Q22.125-32.168 22.097-32.195Q22.070-32.222 22.070-32.243\" fill=\"currentColor\" stroke=\"currentColor\" 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(-39.597 44.847)\">\u003Cpath d=\"M-21.553-40.243L-21.553-41.306Q-21.553-41.330-21.525-41.357Q-21.498-41.384-21.474-41.384L-21.365-41.384Q-21.300-41.384-21.286-41.326Q-21.190-40.892-20.944-40.641Q-20.698-40.390-20.284-40.390Q-19.943-40.390-19.690-40.523Q-19.437-40.656-19.437-40.964Q-19.437-41.121-19.531-41.236Q-19.625-41.350-19.763-41.419Q-19.902-41.487-20.069-41.525L-20.650-41.624Q-21.006-41.692-21.279-41.913Q-21.553-42.133-21.553-42.475Q-21.553-42.724-21.441-42.899Q-21.330-43.073-21.144-43.172Q-20.958-43.271-20.742-43.314Q-20.527-43.357-20.284-43.357Q-19.871-43.357-19.591-43.175L-19.375-43.350Q-19.365-43.353-19.358-43.355Q-19.351-43.357-19.341-43.357L-19.290-43.357Q-19.263-43.357-19.239-43.333Q-19.215-43.309-19.215-43.281L-19.215-42.434Q-19.215-42.413-19.239-42.386Q-19.263-42.359-19.290-42.359L-19.403-42.359Q-19.430-42.359-19.456-42.384Q-19.481-42.410-19.481-42.434Q-19.481-42.670-19.587-42.834Q-19.693-42.998-19.876-43.080Q-20.059-43.162-20.291-43.162Q-20.619-43.162-20.876-43.059Q-21.132-42.957-21.132-42.680Q-21.132-42.485-20.949-42.376Q-20.766-42.266-20.537-42.225L-19.963-42.119Q-19.717-42.071-19.503-41.943Q-19.290-41.815-19.153-41.612Q-19.016-41.408-19.016-41.159Q-19.016-40.646-19.382-40.407Q-19.748-40.168-20.284-40.168Q-20.780-40.168-21.112-40.462L-21.378-40.188Q-21.399-40.168-21.426-40.168L-21.474-40.168Q-21.498-40.168-21.525-40.195Q-21.553-40.222-21.553-40.243M-17.813-41.070L-17.813-42.574Q-17.813-42.844-17.921-42.905Q-18.029-42.967-18.340-42.967L-18.340-43.247L-17.232-43.322L-17.232-41.090L-17.232-41.070Q-17.232-40.790-17.181-40.646Q-17.130-40.503-16.988-40.446Q-16.846-40.390-16.559-40.390Q-16.306-40.390-16.101-40.530Q-15.896-40.670-15.780-40.896Q-15.663-41.121-15.663-41.371L-15.663-42.574Q-15.663-42.844-15.771-42.905Q-15.879-42.967-16.190-42.967L-16.190-43.247L-15.082-43.322L-15.082-40.909Q-15.082-40.718-15.029-40.636Q-14.976-40.554-14.876-40.535Q-14.775-40.516-14.559-40.516L-14.559-40.236L-15.636-40.168L-15.636-40.732Q-15.745-40.550-15.891-40.427Q-16.036-40.304-16.222-40.236Q-16.409-40.168-16.610-40.168Q-17.813-40.168-17.813-41.070M-12.327-38.879L-13.958-38.879L-13.958-39.159Q-13.729-39.159-13.580-39.194Q-13.431-39.228-13.431-39.368L-13.431-42.714Q-13.431-42.885-13.568-42.926Q-13.705-42.967-13.958-42.967L-13.958-43.247L-12.878-43.322L-12.878-42.916Q-12.656-43.117-12.368-43.220Q-12.081-43.322-11.774-43.322Q-11.346-43.322-10.982-43.109Q-10.618-42.895-10.405-42.531Q-10.191-42.167-10.191-41.747Q-10.191-41.302-10.430-40.938Q-10.670-40.574-11.063-40.371Q-11.456-40.168-11.900-40.168Q-12.167-40.168-12.415-40.268Q-12.662-40.369-12.850-40.550L-12.850-39.368Q-12.850-39.231-12.702-39.195Q-12.553-39.159-12.327-39.159L-12.327-38.879M-12.850-42.567L-12.850-40.957Q-12.717-40.704-12.474-40.547Q-12.232-40.390-11.955-40.390Q-11.627-40.390-11.374-40.591Q-11.121-40.793-10.988-41.111Q-10.854-41.429-10.854-41.747Q-10.854-41.976-10.919-42.205Q-10.984-42.434-11.112-42.632Q-11.241-42.830-11.435-42.950Q-11.630-43.069-11.863-43.069Q-12.157-43.069-12.425-42.940Q-12.693-42.810-12.850-42.567\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M-9.377-41.771Q-9.377-42.092-9.252-42.381Q-9.127-42.670-8.901-42.893Q-8.676-43.117-8.380-43.237Q-8.085-43.357-7.767-43.357Q-7.439-43.357-7.177-43.257Q-6.916-43.158-6.740-42.976Q-6.564-42.793-6.470-42.535Q-6.376-42.277-6.376-41.945Q-6.376-41.853-6.458-41.832L-8.713-41.832L-8.713-41.771Q-8.713-41.183-8.430-40.800Q-8.146-40.417-7.579-40.417Q-7.257-40.417-6.989-40.610Q-6.721-40.803-6.632-41.118Q-6.625-41.159-6.550-41.173L-6.458-41.173Q-6.376-41.149-6.376-41.077Q-6.376-41.070-6.382-41.043Q-6.495-40.646-6.866-40.407Q-7.237-40.168-7.661-40.168Q-8.098-40.168-8.498-40.376Q-8.898-40.585-9.137-40.952Q-9.377-41.319-9.377-41.771M-8.707-42.041L-6.892-42.041Q-6.892-42.318-6.989-42.570Q-7.087-42.823-7.285-42.979Q-7.483-43.134-7.767-43.134Q-8.044-43.134-8.257-42.976Q-8.471-42.817-8.589-42.562Q-8.707-42.307-8.707-42.041M-4.038-40.236L-5.774-40.236L-5.774-40.516Q-5.545-40.516-5.396-40.550Q-5.248-40.585-5.248-40.725L-5.248-42.574Q-5.248-42.844-5.355-42.905Q-5.463-42.967-5.774-42.967L-5.774-43.247L-4.745-43.322L-4.745-42.615Q-4.615-42.923-4.373-43.122Q-4.130-43.322-3.812-43.322Q-3.593-43.322-3.422-43.198Q-3.252-43.073-3.252-42.861Q-3.252-42.724-3.351-42.625Q-3.450-42.526-3.583-42.526Q-3.720-42.526-3.819-42.625Q-3.918-42.724-3.918-42.861Q-3.918-43.001-3.819-43.100Q-4.109-43.100-4.309-42.904Q-4.509-42.707-4.602-42.413Q-4.694-42.119-4.694-41.839L-4.694-40.725Q-4.694-40.516-4.038-40.516L-4.038-40.236M-1.078-40.263L-2.206-42.762Q-2.277-42.909-2.407-42.941Q-2.537-42.974-2.766-42.974L-2.766-43.254L-1.252-43.254L-1.252-42.974Q-1.604-42.974-1.604-42.827Q-1.604-42.782-1.594-42.762L-0.729-40.844L0.050-42.574Q0.084-42.642 0.084-42.721Q0.084-42.834 0.001-42.904Q-0.083-42.974-0.203-42.974L-0.203-43.254L0.994-43.254L0.994-42.974Q0.775-42.974 0.604-42.871Q0.433-42.769 0.344-42.574L-0.692-40.263Q-0.739-40.168-0.845-40.168L-0.924-40.168Q-1.030-40.168-1.078-40.263M3.150-40.236L1.599-40.236L1.599-40.516Q1.824-40.516 1.973-40.550Q2.121-40.585 2.121-40.725L2.121-42.574Q2.121-42.762 2.074-42.846Q2.026-42.929 1.928-42.948Q1.831-42.967 1.619-42.967L1.619-43.247L2.675-43.322L2.675-40.725Q2.675-40.585 2.807-40.550Q2.938-40.516 3.150-40.516L3.150-40.236M1.879-44.543Q1.879-44.714 2.002-44.833Q2.125-44.953 2.296-44.953Q2.463-44.953 2.586-44.833Q2.709-44.714 2.709-44.543Q2.709-44.368 2.586-44.245Q2.463-44.122 2.296-44.122Q2.125-44.122 2.002-44.245Q1.879-44.368 1.879-44.543M3.796-40.243L3.796-41.306Q3.796-41.330 3.824-41.357Q3.851-41.384 3.875-41.384L3.984-41.384Q4.049-41.384 4.063-41.326Q4.159-40.892 4.405-40.641Q4.651-40.390 5.064-40.390Q5.406-40.390 5.659-40.523Q5.912-40.656 5.912-40.964Q5.912-41.121 5.818-41.236Q5.724-41.350 5.586-41.419Q5.447-41.487 5.280-41.525L4.699-41.624Q4.343-41.692 4.070-41.913Q3.796-42.133 3.796-42.475Q3.796-42.724 3.907-42.899Q4.018-43.073 4.205-43.172Q4.391-43.271 4.606-43.314Q4.822-43.357 5.064-43.357Q5.478-43.357 5.758-43.175L5.974-43.350Q5.984-43.353 5.991-43.355Q5.997-43.357 6.008-43.357L6.059-43.357Q6.086-43.357 6.110-43.333Q6.134-43.309 6.134-43.281L6.134-42.434Q6.134-42.413 6.110-42.386Q6.086-42.359 6.059-42.359L5.946-42.359Q5.919-42.359 5.893-42.384Q5.868-42.410 5.868-42.434Q5.868-42.670 5.762-42.834Q5.656-42.998 5.473-43.080Q5.290-43.162 5.058-43.162Q4.729-43.162 4.473-43.059Q4.217-42.957 4.217-42.680Q4.217-42.485 4.400-42.376Q4.582-42.266 4.811-42.225L5.386-42.119Q5.632-42.071 5.845-41.943Q6.059-41.815 6.196-41.612Q6.332-41.408 6.332-41.159Q6.332-40.646 5.967-40.407Q5.601-40.168 5.064-40.168Q4.569-40.168 4.237-40.462L3.971-40.188Q3.950-40.168 3.923-40.168L3.875-40.168Q3.851-40.168 3.824-40.195Q3.796-40.222 3.796-40.243M6.920-41.771Q6.920-42.092 7.045-42.381Q7.170-42.670 7.395-42.893Q7.621-43.117 7.917-43.237Q8.212-43.357 8.530-43.357Q8.858-43.357 9.120-43.257Q9.381-43.158 9.557-42.976Q9.733-42.793 9.827-42.535Q9.921-42.277 9.921-41.945Q9.921-41.853 9.839-41.832L7.583-41.832L7.583-41.771Q7.583-41.183 7.867-40.800Q8.151-40.417 8.718-40.417Q9.039-40.417 9.308-40.610Q9.576-40.803 9.665-41.118Q9.672-41.159 9.747-41.173L9.839-41.173Q9.921-41.149 9.921-41.077Q9.921-41.070 9.914-41.043Q9.802-40.646 9.431-40.407Q9.060-40.168 8.636-40.168Q8.199-40.168 7.799-40.376Q7.399-40.585 7.160-40.952Q6.920-41.319 6.920-41.771M7.590-42.041L9.405-42.041Q9.405-42.318 9.308-42.570Q9.210-42.823 9.012-42.979Q8.814-43.134 8.530-43.134Q8.253-43.134 8.040-42.976Q7.826-42.817 7.708-42.562Q7.590-42.307 7.590-42.041M10.509-41.747Q10.509-42.085 10.649-42.376Q10.789-42.666 11.034-42.880Q11.278-43.093 11.582-43.208Q11.887-43.322 12.211-43.322Q12.481-43.322 12.745-43.223Q13.008-43.124 13.199-42.946L13.199-44.344Q13.199-44.614 13.091-44.676Q12.984-44.737 12.673-44.737L12.673-45.018L13.749-45.093L13.749-40.909Q13.749-40.721 13.804-40.638Q13.859-40.554 13.960-40.535Q14.060-40.516 14.276-40.516L14.276-40.236L13.168-40.168L13.168-40.585Q12.751-40.168 12.126-40.168Q11.695-40.168 11.323-40.380Q10.950-40.591 10.730-40.952Q10.509-41.313 10.509-41.747M12.184-40.390Q12.392-40.390 12.579-40.462Q12.765-40.533 12.919-40.670Q13.073-40.807 13.168-40.985L13.168-42.594Q13.083-42.741 12.938-42.861Q12.792-42.981 12.623-43.040Q12.454-43.100 12.273-43.100Q11.712-43.100 11.444-42.711Q11.176-42.321 11.176-41.740Q11.176-41.169 11.410-40.779Q11.644-40.390 12.184-40.390\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M17.604-41.771Q17.604-42.092 17.729-42.381Q17.854-42.670 18.080-42.893Q18.305-43.117 18.601-43.237Q18.896-43.357 19.214-43.357Q19.542-43.357 19.804-43.257Q20.065-43.158 20.241-42.976Q20.417-42.793 20.511-42.535Q20.605-42.277 20.605-41.945Q20.605-41.853 20.523-41.832L18.268-41.832L18.268-41.771Q18.268-41.183 18.551-40.800Q18.835-40.417 19.402-40.417Q19.724-40.417 19.992-40.610Q20.260-40.803 20.349-41.118Q20.356-41.159 20.431-41.173L20.523-41.173Q20.605-41.149 20.605-41.077Q20.605-41.070 20.599-41.043Q20.486-40.646 20.115-40.407Q19.744-40.168 19.320-40.168Q18.883-40.168 18.483-40.376Q18.083-40.585 17.844-40.952Q17.604-41.319 17.604-41.771M18.274-42.041L20.089-42.041Q20.089-42.318 19.992-42.570Q19.894-42.823 19.696-42.979Q19.498-43.134 19.214-43.134Q18.937-43.134 18.724-42.976Q18.510-42.817 18.392-42.562Q18.274-42.307 18.274-42.041M22.376-40.236L21.053-40.236L21.053-40.516Q21.614-40.516 21.993-40.916L22.707-41.713L21.795-42.762Q21.658-42.909 21.509-42.941Q21.361-42.974 21.094-42.974L21.094-43.254L22.595-43.254L22.595-42.974Q22.403-42.974 22.403-42.840Q22.403-42.810 22.434-42.762L23.029-42.078L23.470-42.574Q23.582-42.704 23.582-42.820Q23.582-42.882 23.545-42.928Q23.507-42.974 23.449-42.974L23.449-43.254L24.765-43.254L24.765-42.974Q24.205-42.974 23.825-42.574L23.203-41.873L24.198-40.725Q24.297-40.626 24.398-40.581Q24.498-40.537 24.610-40.527Q24.721-40.516 24.898-40.516L24.898-40.236L23.405-40.236L23.405-40.516Q23.470-40.516 23.529-40.550Q23.589-40.585 23.589-40.650Q23.589-40.697 23.559-40.725L22.882-41.511L22.349-40.916Q22.236-40.786 22.236-40.670Q22.236-40.605 22.277-40.561Q22.318-40.516 22.376-40.516L22.376-40.236M25.452-40.964Q25.452-41.296 25.676-41.523Q25.900-41.750 26.243-41.878Q26.587-42.007 26.959-42.059Q27.332-42.112 27.636-42.112L27.636-42.365Q27.636-42.570 27.529-42.750Q27.421-42.929 27.240-43.032Q27.059-43.134 26.850-43.134Q26.443-43.134 26.207-43.042Q26.296-43.005 26.342-42.921Q26.389-42.837 26.389-42.735Q26.389-42.639 26.342-42.560Q26.296-42.482 26.216-42.437Q26.136-42.393 26.047-42.393Q25.896-42.393 25.796-42.490Q25.695-42.588 25.695-42.735Q25.695-43.357 26.850-43.357Q27.062-43.357 27.311-43.293Q27.561-43.230 27.763-43.111Q27.964-42.991 28.091-42.806Q28.217-42.622 28.217-42.379L28.217-40.803Q28.217-40.687 28.279-40.591Q28.340-40.496 28.453-40.496Q28.562-40.496 28.627-40.590Q28.692-40.684 28.692-40.803L28.692-41.251L28.959-41.251L28.959-40.803Q28.959-40.533 28.732-40.368Q28.504-40.202 28.224-40.202Q28.016-40.202 27.879-40.356Q27.742-40.509 27.718-40.725Q27.571-40.458 27.289-40.313Q27.007-40.168 26.683-40.168Q26.406-40.168 26.122-40.243Q25.838-40.318 25.645-40.497Q25.452-40.677 25.452-40.964M26.067-40.964Q26.067-40.790 26.168-40.660Q26.269-40.530 26.425-40.460Q26.580-40.390 26.744-40.390Q26.963-40.390 27.171-40.487Q27.380-40.585 27.508-40.766Q27.636-40.947 27.636-41.173L27.636-41.901Q27.311-41.901 26.946-41.810Q26.580-41.719 26.324-41.507Q26.067-41.296 26.067-40.964M31.058-40.236L29.424-40.236L29.424-40.516Q29.653-40.516 29.801-40.550Q29.950-40.585 29.950-40.725L29.950-42.574Q29.950-42.844 29.842-42.905Q29.735-42.967 29.424-42.967L29.424-43.247L30.483-43.322L30.483-42.673Q30.654-42.981 30.958-43.152Q31.263-43.322 31.608-43.322Q32.008-43.322 32.285-43.182Q32.561-43.042 32.647-42.694Q32.814-42.987 33.113-43.155Q33.413-43.322 33.758-43.322Q34.264-43.322 34.547-43.099Q34.831-42.875 34.831-42.379L34.831-40.725Q34.831-40.588 34.980-40.552Q35.128-40.516 35.354-40.516L35.354-40.236L33.724-40.236L33.724-40.516Q33.949-40.516 34.100-40.552Q34.250-40.588 34.250-40.725L34.250-42.365Q34.250-42.700 34.130-42.900Q34.011-43.100 33.696-43.100Q33.426-43.100 33.192-42.964Q32.958-42.827 32.820-42.593Q32.681-42.359 32.681-42.085L32.681-40.725Q32.681-40.588 32.830-40.552Q32.978-40.516 33.204-40.516L33.204-40.236L31.574-40.236L31.574-40.516Q31.803-40.516 31.951-40.550Q32.100-40.585 32.100-40.725L32.100-42.365Q32.100-42.700 31.980-42.900Q31.861-43.100 31.546-43.100Q31.276-43.100 31.042-42.964Q30.808-42.827 30.670-42.593Q30.531-42.359 30.531-42.085L30.531-40.725Q30.531-40.588 30.682-40.552Q30.832-40.516 31.058-40.516L31.058-40.236M37.586-38.879L35.956-38.879L35.956-39.159Q36.185-39.159 36.333-39.194Q36.482-39.228 36.482-39.368L36.482-42.714Q36.482-42.885 36.345-42.926Q36.208-42.967 35.956-42.967L35.956-43.247L37.036-43.322L37.036-42.916Q37.258-43.117 37.545-43.220Q37.832-43.322 38.140-43.322Q38.567-43.322 38.931-43.109Q39.295-42.895 39.508-42.531Q39.722-42.167 39.722-41.747Q39.722-41.302 39.483-40.938Q39.244-40.574 38.851-40.371Q38.457-40.168 38.013-40.168Q37.747-40.168 37.499-40.268Q37.251-40.369 37.063-40.550L37.063-39.368Q37.063-39.231 37.212-39.195Q37.360-39.159 37.586-39.159L37.586-38.879M37.063-42.567L37.063-40.957Q37.196-40.704 37.439-40.547Q37.682-40.390 37.958-40.390Q38.287-40.390 38.539-40.591Q38.792-40.793 38.926-41.111Q39.059-41.429 39.059-41.747Q39.059-41.976 38.994-42.205Q38.929-42.434 38.801-42.632Q38.673-42.830 38.478-42.950Q38.283-43.069 38.051-43.069Q37.757-43.069 37.488-42.940Q37.220-42.810 37.063-42.567M42.026-40.236L40.423-40.236L40.423-40.516Q40.648-40.516 40.797-40.550Q40.946-40.585 40.946-40.725L40.946-44.344Q40.946-44.614 40.838-44.676Q40.730-44.737 40.423-44.737L40.423-45.018L41.499-45.093L41.499-40.725Q41.499-40.588 41.650-40.552Q41.800-40.516 42.026-40.516L42.026-40.236M42.580-41.771Q42.580-42.092 42.704-42.381Q42.829-42.670 43.055-42.893Q43.280-43.117 43.576-43.237Q43.872-43.357 44.189-43.357Q44.518-43.357 44.779-43.257Q45.040-43.158 45.216-42.976Q45.393-42.793 45.487-42.535Q45.581-42.277 45.581-41.945Q45.581-41.853 45.498-41.832L43.243-41.832L43.243-41.771Q43.243-41.183 43.526-40.800Q43.810-40.417 44.377-40.417Q44.699-40.417 44.967-40.610Q45.235-40.803 45.324-41.118Q45.331-41.159 45.406-41.173L45.498-41.173Q45.581-41.149 45.581-41.077Q45.581-41.070 45.574-41.043Q45.461-40.646 45.090-40.407Q44.719-40.168 44.295-40.168Q43.858-40.168 43.458-40.376Q43.058-40.585 42.819-40.952Q42.580-41.319 42.580-41.771M43.249-42.041L45.064-42.041Q45.064-42.318 44.967-42.570Q44.870-42.823 44.671-42.979Q44.473-43.134 44.189-43.134Q43.913-43.134 43.699-42.976Q43.485-42.817 43.367-42.562Q43.249-42.307 43.249-42.041M46.168-40.243L46.168-41.306Q46.168-41.330 46.196-41.357Q46.223-41.384 46.247-41.384L46.356-41.384Q46.421-41.384 46.435-41.326Q46.531-40.892 46.777-40.641Q47.023-40.390 47.436-40.390Q47.778-40.390 48.031-40.523Q48.284-40.656 48.284-40.964Q48.284-41.121 48.190-41.236Q48.096-41.350 47.958-41.419Q47.819-41.487 47.652-41.525L47.071-41.624Q46.715-41.692 46.442-41.913Q46.168-42.133 46.168-42.475Q46.168-42.724 46.279-42.899Q46.391-43.073 46.577-43.172Q46.763-43.271 46.978-43.314Q47.194-43.357 47.436-43.357Q47.850-43.357 48.130-43.175L48.346-43.350Q48.356-43.353 48.363-43.355Q48.370-43.357 48.380-43.357L48.431-43.357Q48.458-43.357 48.482-43.333Q48.506-43.309 48.506-43.281L48.506-42.434Q48.506-42.413 48.482-42.386Q48.458-42.359 48.431-42.359L48.318-42.359Q48.291-42.359 48.265-42.384Q48.240-42.410 48.240-42.434Q48.240-42.670 48.134-42.834Q48.028-42.998 47.845-43.080Q47.662-43.162 47.430-43.162Q47.102-43.162 46.845-43.059Q46.589-42.957 46.589-42.680Q46.589-42.485 46.772-42.376Q46.955-42.266 47.184-42.225L47.758-42.119Q48.004-42.071 48.217-41.943Q48.431-41.815 48.568-41.612Q48.705-41.408 48.705-41.159Q48.705-40.646 48.339-40.407Q47.973-40.168 47.436-40.168Q46.941-40.168 46.609-40.462L46.343-40.188Q46.322-40.168 46.295-40.168L46.247-40.168Q46.223-40.168 46.196-40.195Q46.168-40.222 46.168-40.243\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M-23.563-30.486Q-24.113-30.886-24.484-31.441Q-24.855-31.997-25.036-32.643Q-25.217-33.289-25.217-33.986Q-25.217-34.499-25.117-34.994Q-25.016-35.490-24.811-35.941Q-24.606-36.392-24.293-36.784Q-23.980-37.175-23.563-37.479Q-23.553-37.483-23.546-37.484Q-23.539-37.486-23.529-37.486L-23.461-37.486Q-23.426-37.486-23.404-37.462Q-23.382-37.438-23.382-37.401Q-23.382-37.356-23.409-37.339Q-23.758-37.038-24.011-36.654Q-24.264-36.269-24.416-35.828Q-24.568-35.387-24.640-34.931Q-24.712-34.475-24.712-33.986Q-24.712-32.985-24.402-32.098Q-24.093-31.211-23.409-30.626Q-23.382-30.609-23.382-30.565Q-23.382-30.527-23.404-30.503Q-23.426-30.479-23.461-30.479L-23.529-30.479Q-23.536-30.483-23.544-30.484Q-23.553-30.486-23.563-30.486M-22.572-32.243L-22.572-33.306Q-22.572-33.330-22.545-33.357Q-22.517-33.384-22.493-33.384L-22.384-33.384Q-22.319-33.384-22.305-33.326Q-22.210-32.892-21.964-32.641Q-21.717-32.390-21.304-32.390Q-20.962-32.390-20.709-32.523Q-20.456-32.656-20.456-32.964Q-20.456-33.121-20.550-33.236Q-20.644-33.350-20.783-33.419Q-20.921-33.487-21.089-33.525L-21.670-33.624Q-22.025-33.692-22.299-33.913Q-22.572-34.133-22.572-34.475Q-22.572-34.724-22.461-34.899Q-22.350-35.073-22.163-35.172Q-21.977-35.271-21.762-35.314Q-21.547-35.357-21.304-35.357Q-20.890-35.357-20.610-35.175L-20.395-35.350Q-20.384-35.353-20.378-35.355Q-20.371-35.357-20.361-35.357L-20.309-35.357Q-20.282-35.357-20.258-35.333Q-20.234-35.309-20.234-35.281L-20.234-34.434Q-20.234-34.413-20.258-34.386Q-20.282-34.359-20.309-34.359L-20.422-34.359Q-20.449-34.359-20.475-34.384Q-20.501-34.410-20.501-34.434Q-20.501-34.670-20.607-34.834Q-20.713-34.998-20.895-35.080Q-21.078-35.162-21.311-35.162Q-21.639-35.162-21.895-35.059Q-22.152-34.957-22.152-34.680Q-22.152-34.485-21.969-34.376Q-21.786-34.266-21.557-34.225L-20.983-34.119Q-20.736-34.071-20.523-33.943Q-20.309-33.815-20.173-33.612Q-20.036-33.408-20.036-33.159Q-20.036-32.646-20.402-32.407Q-20.767-32.168-21.304-32.168Q-21.799-32.168-22.131-32.462L-22.398-32.188Q-22.418-32.168-22.445-32.168L-22.493-32.168Q-22.517-32.168-22.545-32.195Q-22.572-32.222-22.572-32.243M-19.072-31.101Q-18.942-31.033-18.805-31.033Q-18.634-31.033-18.484-31.122Q-18.334-31.211-18.223-31.356Q-18.111-31.501-18.033-31.669L-17.770-32.236L-18.939-34.762Q-19.014-34.909-19.144-34.941Q-19.274-34.974-19.506-34.974L-19.506-35.254L-17.985-35.254L-17.985-34.974Q-18.334-34.974-18.334-34.827Q-18.330-34.806-18.329-34.789Q-18.327-34.772-18.327-34.762L-17.469-32.903L-16.696-34.574Q-16.662-34.642-16.662-34.721Q-16.662-34.834-16.746-34.904Q-16.830-34.974-16.943-34.974L-16.943-35.254L-15.746-35.254L-15.746-34.974Q-15.965-34.974-16.138-34.870Q-16.310-34.765-16.403-34.574L-17.739-31.669Q-17.910-31.299-18.180-31.053Q-18.450-30.807-18.805-30.807Q-19.075-30.807-19.294-30.973Q-19.513-31.139-19.513-31.402Q-19.513-31.539-19.421-31.628Q-19.328-31.716-19.188-31.716Q-19.051-31.716-18.963-31.628Q-18.874-31.539-18.874-31.402Q-18.874-31.299-18.927-31.221Q-18.980-31.142-19.072-31.101M-15.206-32.243L-15.206-33.306Q-15.206-33.330-15.179-33.357Q-15.152-33.384-15.128-33.384L-15.018-33.384Q-14.953-33.384-14.940-33.326Q-14.844-32.892-14.598-32.641Q-14.352-32.390-13.938-32.390Q-13.596-32.390-13.343-32.523Q-13.090-32.656-13.090-32.964Q-13.090-33.121-13.184-33.236Q-13.278-33.350-13.417-33.419Q-13.555-33.487-13.723-33.525L-14.304-33.624Q-14.659-33.692-14.933-33.913Q-15.206-34.133-15.206-34.475Q-15.206-34.724-15.095-34.899Q-14.984-35.073-14.798-35.172Q-14.611-35.271-14.396-35.314Q-14.181-35.357-13.938-35.357Q-13.525-35.357-13.244-35.175L-13.029-35.350Q-13.019-35.353-13.012-35.355Q-13.005-35.357-12.995-35.357L-12.944-35.357Q-12.916-35.357-12.892-35.333Q-12.868-35.309-12.868-35.281L-12.868-34.434Q-12.868-34.413-12.892-34.386Q-12.916-34.359-12.944-34.359L-13.056-34.359Q-13.084-34.359-13.109-34.384Q-13.135-34.410-13.135-34.434Q-13.135-34.670-13.241-34.834Q-13.347-34.998-13.530-35.080Q-13.713-35.162-13.945-35.162Q-14.273-35.162-14.529-35.059Q-14.786-34.957-14.786-34.680Q-14.786-34.485-14.603-34.376Q-14.420-34.266-14.191-34.225L-13.617-34.119Q-13.371-34.071-13.157-33.943Q-12.944-33.815-12.807-33.612Q-12.670-33.408-12.670-33.159Q-12.670-32.646-13.036-32.407Q-13.402-32.168-13.938-32.168Q-14.434-32.168-14.765-32.462L-15.032-32.188Q-15.052-32.168-15.080-32.168L-15.128-32.168Q-15.152-32.168-15.179-32.195Q-15.206-32.222-15.206-32.243M-11.515-33.077L-11.515-34.974L-12.154-34.974L-12.154-35.196Q-11.836-35.196-11.619-35.406Q-11.402-35.616-11.301-35.926Q-11.200-36.235-11.200-36.543L-10.934-36.543L-10.934-35.254L-9.857-35.254L-9.857-34.974L-10.934-34.974L-10.934-33.090Q-10.934-32.814-10.830-32.615Q-10.725-32.417-10.465-32.417Q-10.308-32.417-10.202-32.521Q-10.096-32.626-10.047-32.779Q-9.997-32.933-9.997-33.090L-9.997-33.504L-9.731-33.504L-9.731-33.077Q-9.731-32.851-9.830-32.641Q-9.929-32.431-10.113-32.299Q-10.298-32.168-10.527-32.168Q-10.965-32.168-11.240-32.405Q-11.515-32.643-11.515-33.077M-8.962-33.771Q-8.962-34.092-8.837-34.381Q-8.712-34.670-8.486-34.893Q-8.261-35.117-7.965-35.237Q-7.670-35.357-7.352-35.357Q-7.024-35.357-6.762-35.257Q-6.501-35.158-6.325-34.976Q-6.149-34.793-6.055-34.535Q-5.961-34.277-5.961-33.945Q-5.961-33.853-6.043-33.832L-8.299-33.832L-8.299-33.771Q-8.299-33.183-8.015-32.800Q-7.731-32.417-7.164-32.417Q-6.842-32.417-6.574-32.610Q-6.306-32.803-6.217-33.118Q-6.210-33.159-6.135-33.173L-6.043-33.173Q-5.961-33.149-5.961-33.077Q-5.961-33.070-5.967-33.043Q-6.080-32.646-6.451-32.407Q-6.822-32.168-7.246-32.168Q-7.683-32.168-8.083-32.376Q-8.483-32.585-8.722-32.952Q-8.962-33.319-8.962-33.771M-8.292-34.041L-6.477-34.041Q-6.477-34.318-6.574-34.570Q-6.672-34.823-6.870-34.979Q-7.068-35.134-7.352-35.134Q-7.629-35.134-7.842-34.976Q-8.056-34.817-8.174-34.562Q-8.292-34.307-8.292-34.041M-3.691-32.236L-5.325-32.236L-5.325-32.516Q-5.096-32.516-4.947-32.550Q-4.799-32.585-4.799-32.725L-4.799-34.574Q-4.799-34.844-4.906-34.905Q-5.014-34.967-5.325-34.967L-5.325-35.247L-4.265-35.322L-4.265-34.673Q-4.094-34.981-3.790-35.152Q-3.486-35.322-3.141-35.322Q-2.741-35.322-2.464-35.182Q-2.187-35.042-2.102-34.694Q-1.934-34.987-1.635-35.155Q-1.336-35.322-0.991-35.322Q-0.485-35.322-0.201-35.099Q0.082-34.875 0.082-34.379L0.082-32.725Q0.082-32.588 0.231-32.552Q0.380-32.516 0.605-32.516L0.605-32.236L-1.025-32.236L-1.025-32.516Q-0.799-32.516-0.649-32.552Q-0.499-32.588-0.499-32.725L-0.499-34.365Q-0.499-34.700-0.618-34.900Q-0.738-35.100-1.052-35.100Q-1.322-35.100-1.557-34.964Q-1.791-34.827-1.929-34.593Q-2.068-34.359-2.068-34.085L-2.068-32.725Q-2.068-32.588-1.919-32.552Q-1.770-32.516-1.545-32.516L-1.545-32.236L-3.175-32.236L-3.175-32.516Q-2.946-32.516-2.797-32.550Q-2.649-32.585-2.649-32.725L-2.649-34.365Q-2.649-34.700-2.768-34.900Q-2.888-35.100-3.202-35.100Q-3.472-35.100-3.706-34.964Q-3.941-34.827-4.079-34.593Q-4.217-34.359-4.217-34.085L-4.217-32.725Q-4.217-32.588-4.067-32.552Q-3.917-32.516-3.691-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M5.531-32.236L3.979-32.236L3.979-32.516Q4.205-32.516 4.354-32.550Q4.502-32.585 4.502-32.725L4.502-34.574Q4.502-34.762 4.454-34.846Q4.407-34.929 4.309-34.948Q4.212-34.967 4-34.967L4-35.247L5.056-35.322L5.056-32.725Q5.056-32.585 5.188-32.550Q5.319-32.516 5.531-32.516L5.531-32.236M4.260-36.543Q4.260-36.714 4.383-36.833Q4.506-36.953 4.677-36.953Q4.844-36.953 4.967-36.833Q5.090-36.714 5.090-36.543Q5.090-36.368 4.967-36.245Q4.844-36.122 4.677-36.122Q4.506-36.122 4.383-36.245Q4.260-36.368 4.260-36.543M6.177-33.747Q6.177-34.085 6.317-34.376Q6.457-34.666 6.702-34.880Q6.946-35.093 7.250-35.208Q7.555-35.322 7.879-35.322Q8.149-35.322 8.412-35.223Q8.676-35.124 8.867-34.946L8.867-36.344Q8.867-36.614 8.759-36.676Q8.652-36.737 8.341-36.737L8.341-37.018L9.417-37.093L9.417-32.909Q9.417-32.721 9.472-32.638Q9.527-32.554 9.628-32.535Q9.728-32.516 9.944-32.516L9.944-32.236L8.836-32.168L8.836-32.585Q8.419-32.168 7.794-32.168Q7.363-32.168 6.991-32.380Q6.618-32.591 6.398-32.952Q6.177-33.313 6.177-33.747M7.852-32.390Q8.060-32.390 8.247-32.462Q8.433-32.533 8.587-32.670Q8.741-32.807 8.836-32.985L8.836-34.594Q8.751-34.741 8.606-34.861Q8.460-34.981 8.291-35.040Q8.122-35.100 7.941-35.100Q7.380-35.100 7.112-34.711Q6.844-34.321 6.844-33.740Q6.844-33.169 7.078-32.779Q7.312-32.390 7.852-32.390M10.552-33.771Q10.552-34.092 10.677-34.381Q10.802-34.670 11.027-34.893Q11.253-35.117 11.548-35.237Q11.844-35.357 12.162-35.357Q12.490-35.357 12.752-35.257Q13.013-35.158 13.189-34.976Q13.365-34.793 13.459-34.535Q13.553-34.277 13.553-33.945Q13.553-33.853 13.471-33.832L11.215-33.832L11.215-33.771Q11.215-33.183 11.499-32.800Q11.783-32.417 12.350-32.417Q12.671-32.417 12.940-32.610Q13.208-32.803 13.297-33.118Q13.304-33.159 13.379-33.173L13.471-33.173Q13.553-33.149 13.553-33.077Q13.553-33.070 13.546-33.043Q13.433-32.646 13.063-32.407Q12.692-32.168 12.268-32.168Q11.830-32.168 11.431-32.376Q11.031-32.585 10.791-32.952Q10.552-33.319 10.552-33.771M11.222-34.041L13.037-34.041Q13.037-34.318 12.940-34.570Q12.842-34.823 12.644-34.979Q12.446-35.134 12.162-35.134Q11.885-35.134 11.672-34.976Q11.458-34.817 11.340-34.562Q11.222-34.307 11.222-34.041M15.823-32.236L14.189-32.236L14.189-32.516Q14.418-32.516 14.567-32.550Q14.715-32.585 14.715-32.725L14.715-34.574Q14.715-34.844 14.608-34.905Q14.500-34.967 14.189-34.967L14.189-35.247L15.248-35.322L15.248-34.673Q15.419-34.981 15.724-35.152Q16.028-35.322 16.373-35.322Q16.879-35.322 17.163-35.099Q17.446-34.875 17.446-34.379L17.446-32.725Q17.446-32.588 17.595-32.552Q17.744-32.516 17.969-32.516L17.969-32.236L16.339-32.236L16.339-32.516Q16.568-32.516 16.716-32.550Q16.865-32.585 16.865-32.725L16.865-34.365Q16.865-34.700 16.746-34.900Q16.626-35.100 16.311-35.100Q16.041-35.100 15.807-34.964Q15.573-34.827 15.435-34.593Q15.296-34.359 15.296-34.085L15.296-32.725Q15.296-32.588 15.447-32.552Q15.597-32.516 15.823-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M18.879-33.077L18.879-34.974L18.240-34.974L18.240-35.196Q18.558-35.196 18.775-35.406Q18.992-35.616 19.092-35.926Q19.193-36.235 19.193-36.543L19.460-36.543L19.460-35.254L20.537-35.254L20.537-34.974L19.460-34.974L19.460-33.090Q19.460-32.814 19.564-32.615Q19.668-32.417 19.928-32.417Q20.085-32.417 20.191-32.521Q20.297-32.626 20.347-32.779Q20.396-32.933 20.396-33.090L20.396-33.504L20.663-33.504L20.663-33.077Q20.663-32.851 20.564-32.641Q20.465-32.431 20.280-32.299Q20.096-32.168 19.867-32.168Q19.429-32.168 19.154-32.405Q18.879-32.643 18.879-33.077M23.090-32.236L21.538-32.236L21.538-32.516Q21.764-32.516 21.912-32.550Q22.061-32.585 22.061-32.725L22.061-34.574Q22.061-34.762 22.013-34.846Q21.965-34.929 21.868-34.948Q21.770-34.967 21.559-34.967L21.559-35.247L22.615-35.322L22.615-32.725Q22.615-32.585 22.746-32.550Q22.878-32.516 23.090-32.516L23.090-32.236M21.818-36.543Q21.818-36.714 21.941-36.833Q22.064-36.953 22.235-36.953Q22.403-36.953 22.526-36.833Q22.649-36.714 22.649-36.543Q22.649-36.368 22.526-36.245Q22.403-36.122 22.235-36.122Q22.064-36.122 21.941-36.245Q21.818-36.368 21.818-36.543M25.534-32.236L23.801-32.236L23.801-32.516Q24.026-32.516 24.175-32.550Q24.324-32.585 24.324-32.725L24.324-34.974L23.736-34.974L23.736-35.254L24.324-35.254L24.324-36.071Q24.324-36.389 24.501-36.637Q24.679-36.884 24.970-37.025Q25.260-37.165 25.571-37.165Q25.828-37.165 26.031-37.023Q26.234-36.881 26.234-36.638Q26.234-36.502 26.135-36.403Q26.036-36.303 25.899-36.303Q25.763-36.303 25.663-36.403Q25.564-36.502 25.564-36.638Q25.564-36.819 25.705-36.912Q25.626-36.939 25.527-36.939Q25.318-36.939 25.164-36.806Q25.011-36.673 24.930-36.469Q24.850-36.266 24.850-36.057L24.850-35.254L25.739-35.254L25.739-34.974L24.877-34.974L24.877-32.725Q24.877-32.516 25.534-32.516\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.597 44.847)\">\u003Cpath d=\"M28.396-32.236L26.844-32.236L26.844-32.516Q27.070-32.516 27.219-32.550Q27.367-32.585 27.367-32.725L27.367-34.574Q27.367-34.762 27.319-34.846Q27.272-34.929 27.174-34.948Q27.077-34.967 26.865-34.967L26.865-35.247L27.921-35.322L27.921-32.725Q27.921-32.585 28.053-32.550Q28.184-32.516 28.396-32.516L28.396-32.236M27.125-36.543Q27.125-36.714 27.248-36.833Q27.371-36.953 27.542-36.953Q27.709-36.953 27.832-36.833Q27.955-36.714 27.955-36.543Q27.955-36.368 27.832-36.245Q27.709-36.122 27.542-36.122Q27.371-36.122 27.248-36.245Q27.125-36.368 27.125-36.543M29.042-33.747Q29.042-34.075 29.177-34.376Q29.312-34.676 29.548-34.897Q29.784-35.117 30.088-35.237Q30.392-35.357 30.717-35.357Q31.223-35.357 31.571-35.254Q31.920-35.152 31.920-34.776Q31.920-34.629 31.823-34.528Q31.725-34.427 31.578-34.427Q31.424-34.427 31.325-34.526Q31.226-34.625 31.226-34.776Q31.226-34.964 31.366-35.056Q31.165-35.107 30.724-35.107Q30.368-35.107 30.139-34.911Q29.910-34.714 29.809-34.405Q29.709-34.095 29.709-33.747Q29.709-33.398 29.835-33.092Q29.962-32.786 30.216-32.602Q30.471-32.417 30.826-32.417Q31.048-32.417 31.233-32.501Q31.418-32.585 31.553-32.740Q31.688-32.896 31.746-33.104Q31.759-33.159 31.814-33.159L31.927-33.159Q31.958-33.159 31.980-33.135Q32.002-33.111 32.002-33.077L32.002-33.056Q31.917-32.769 31.729-32.571Q31.541-32.373 31.276-32.270Q31.011-32.168 30.717-32.168Q30.286-32.168 29.898-32.374Q29.510-32.581 29.276-32.944Q29.042-33.306 29.042-33.747M32.648-32.964Q32.648-33.296 32.872-33.523Q33.096-33.750 33.439-33.878Q33.783-34.007 34.155-34.059Q34.528-34.112 34.832-34.112L34.832-34.365Q34.832-34.570 34.725-34.750Q34.617-34.929 34.436-35.032Q34.255-35.134 34.046-35.134Q33.639-35.134 33.403-35.042Q33.492-35.005 33.538-34.921Q33.585-34.837 33.585-34.735Q33.585-34.639 33.538-34.560Q33.492-34.482 33.412-34.437Q33.332-34.393 33.243-34.393Q33.092-34.393 32.992-34.490Q32.891-34.588 32.891-34.735Q32.891-35.357 34.046-35.357Q34.258-35.357 34.507-35.293Q34.757-35.230 34.959-35.111Q35.160-34.991 35.287-34.806Q35.413-34.622 35.413-34.379L35.413-32.803Q35.413-32.687 35.475-32.591Q35.536-32.496 35.649-32.496Q35.758-32.496 35.823-32.590Q35.888-32.684 35.888-32.803L35.888-33.251L36.155-33.251L36.155-32.803Q36.155-32.533 35.928-32.368Q35.700-32.202 35.420-32.202Q35.212-32.202 35.075-32.356Q34.938-32.509 34.914-32.725Q34.767-32.458 34.485-32.313Q34.203-32.168 33.879-32.168Q33.602-32.168 33.318-32.243Q33.034-32.318 32.841-32.497Q32.648-32.677 32.648-32.964M33.263-32.964Q33.263-32.790 33.364-32.660Q33.465-32.530 33.621-32.460Q33.776-32.390 33.940-32.390Q34.159-32.390 34.367-32.487Q34.576-32.585 34.704-32.766Q34.832-32.947 34.832-33.173L34.832-33.901Q34.507-33.901 34.142-33.810Q33.776-33.719 33.520-33.507Q33.263-33.296 33.263-32.964M37.098-33.077L37.098-34.974L36.459-34.974L36.459-35.196Q36.777-35.196 36.994-35.406Q37.211-35.616 37.312-35.926Q37.413-36.235 37.413-36.543L37.679-36.543L37.679-35.254L38.756-35.254L38.756-34.974L37.679-34.974L37.679-33.090Q37.679-32.814 37.784-32.615Q37.888-32.417 38.148-32.417Q38.305-32.417 38.411-32.521Q38.517-32.626 38.566-32.779Q38.616-32.933 38.616-33.090L38.616-33.504L38.882-33.504L38.882-33.077Q38.882-32.851 38.783-32.641Q38.684-32.431 38.500-32.299Q38.315-32.168 38.086-32.168Q37.649-32.168 37.373-32.405Q37.098-32.643 37.098-33.077M41.309-32.236L39.757-32.236L39.757-32.516Q39.983-32.516 40.132-32.550Q40.280-32.585 40.280-32.725L40.280-34.574Q40.280-34.762 40.233-34.846Q40.185-34.929 40.087-34.948Q39.990-34.967 39.778-34.967L39.778-35.247L40.834-35.322L40.834-32.725Q40.834-32.585 40.966-32.550Q41.097-32.516 41.309-32.516L41.309-32.236M40.038-36.543Q40.038-36.714 40.161-36.833Q40.284-36.953 40.455-36.953Q40.622-36.953 40.745-36.833Q40.868-36.714 40.868-36.543Q40.868-36.368 40.745-36.245Q40.622-36.122 40.455-36.122Q40.284-36.122 40.161-36.245Q40.038-36.368 40.038-36.543M41.914-33.719Q41.914-34.061 42.049-34.360Q42.184-34.659 42.423-34.883Q42.663-35.107 42.981-35.232Q43.298-35.357 43.630-35.357Q44.074-35.357 44.474-35.141Q44.874-34.926 45.108-34.548Q45.342-34.171 45.342-33.719Q45.342-33.378 45.201-33.094Q45.059-32.810 44.814-32.603Q44.570-32.397 44.261-32.282Q43.951-32.168 43.630-32.168Q43.199-32.168 42.798-32.369Q42.396-32.571 42.155-32.923Q41.914-33.275 41.914-33.719M43.630-32.417Q44.232-32.417 44.455-32.795Q44.679-33.173 44.679-33.805Q44.679-34.417 44.445-34.776Q44.211-35.134 43.630-35.134Q42.577-35.134 42.577-33.805Q42.577-33.173 42.803-32.795Q43.028-32.417 43.630-32.417M47.619-32.236L45.985-32.236L45.985-32.516Q46.214-32.516 46.363-32.550Q46.511-32.585 46.511-32.725L46.511-34.574Q46.511-34.844 46.404-34.905Q46.296-34.967 45.985-34.967L45.985-35.247L47.045-35.322L47.045-34.673Q47.215-34.981 47.520-35.152Q47.824-35.322 48.169-35.322Q48.675-35.322 48.959-35.099Q49.242-34.875 49.242-34.379L49.242-32.725Q49.242-32.588 49.391-32.552Q49.540-32.516 49.765-32.516L49.765-32.236L48.135-32.236L48.135-32.516Q48.364-32.516 48.513-32.550Q48.661-32.585 48.661-32.725L48.661-34.365Q48.661-34.700 48.542-34.900Q48.422-35.100 48.108-35.100Q47.838-35.100 47.603-34.964Q47.369-34.827 47.231-34.593Q47.092-34.359 47.092-34.085L47.092-32.725Q47.092-32.588 47.243-32.552Q47.393-32.516 47.619-32.516L47.619-32.236M50.674-30.479L50.606-30.479Q50.572-30.479 50.550-30.505Q50.528-30.530 50.528-30.565Q50.528-30.609 50.558-30.626Q50.914-30.930 51.163-31.320Q51.413-31.710 51.565-32.142Q51.717-32.574 51.787-33.043Q51.857-33.511 51.857-33.986Q51.857-34.465 51.787-34.931Q51.717-35.398 51.563-35.833Q51.409-36.269 51.158-36.657Q50.907-37.045 50.558-37.339Q50.528-37.356 50.528-37.401Q50.528-37.435 50.550-37.460Q50.572-37.486 50.606-37.486L50.674-37.486Q50.685-37.486 50.693-37.484Q50.702-37.483 50.712-37.479Q51.256-37.079 51.628-36.526Q52.001-35.972 52.182-35.326Q52.363-34.680 52.363-33.986Q52.363-33.285 52.182-32.638Q52.001-31.990 51.626-31.436Q51.252-30.882 50.712-30.486Q50.702-30.486 50.693-30.484Q50.685-30.483 50.674-30.479\" 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\">A cognitive map is learned by system identification: supervised examples of what follows what. State-transition examples S-S' or SA-S' give the dynamics; reward examples S-R or SA-R give the reward model. None of this requires reward to be present while learning, which is why latent learning works.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:411.700px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 308.775 162.551\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(21.594 -93.159)\">\u003Cpath d=\"M-63.217 30.472L-65.304 30.472L-65.304 30.156Q-64.997 30.156-64.805 30.103Q-64.614 30.050-64.614 29.861L-64.614 27.413Q-64.614 27.172-64.685 27.064Q-64.755 26.956-64.889 26.932Q-65.023 26.908-65.304 26.908L-65.304 26.592L-63.964 26.495L-63.964 27.330Q-63.766 26.952-63.406 26.723Q-63.045 26.495-62.624 26.495Q-61.578 26.495-61.393 27.304Q-61.191 26.934-60.833 26.715Q-60.475 26.495-60.057 26.495Q-59.433 26.495-59.108 26.789Q-58.783 27.084-58.783 27.708L-58.783 29.861Q-58.783 30.050-58.589 30.103Q-58.396 30.156-58.088 30.156L-58.088 30.472L-60.176 30.472L-60.176 30.156Q-59.868 30.156-59.675 30.103Q-59.481 30.050-59.481 29.861L-59.481 27.743Q-59.481 27.312-59.609 27.033Q-59.736 26.754-60.123 26.754Q-60.466 26.754-60.749 26.943Q-61.033 27.132-61.189 27.444Q-61.345 27.756-61.345 28.103L-61.345 29.861Q-61.345 30.050-61.154 30.103Q-60.962 30.156-60.655 30.156L-60.655 30.472L-62.742 30.472L-62.742 30.156Q-62.430 30.156-62.239 30.103Q-62.048 30.050-62.048 29.861L-62.048 27.743Q-62.048 27.484-62.092 27.262Q-62.136 27.040-62.281 26.897Q-62.426 26.754-62.685 26.754Q-63.221 26.754-63.566 27.161Q-63.911 27.567-63.911 28.103L-63.911 29.861Q-63.911 30.050-63.718 30.103Q-63.524 30.156-63.217 30.156L-63.217 30.472M-57.640 28.565Q-57.640 27.998-57.368 27.510Q-57.095 27.022-56.625 26.730Q-56.155 26.438-55.588 26.438Q-55.166 26.438-54.790 26.607Q-54.415 26.776-54.138 27.068Q-53.861 27.361-53.703 27.756Q-53.544 28.152-53.544 28.565Q-53.544 29.114-53.823 29.576Q-54.103 30.037-54.571 30.305Q-55.039 30.573-55.588 30.573Q-56.142 30.573-56.612 30.305Q-57.082 30.037-57.361 29.576Q-57.640 29.114-57.640 28.565M-55.588 30.283Q-55.091 30.283-54.814 30.022Q-54.538 29.760-54.445 29.356Q-54.353 28.951-54.353 28.455Q-54.353 27.980-54.452 27.591Q-54.551 27.202-54.823 26.952Q-55.096 26.701-55.588 26.701Q-56.300 26.701-56.563 27.196Q-56.827 27.690-56.827 28.455Q-56.827 29.255-56.572 29.769Q-56.317 30.283-55.588 30.283\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(21.594 -93.159)\">\u003Cpath d=\"M-50.763 30.573Q-51.308 30.573-51.752 30.290Q-52.196 30.006-52.450 29.534Q-52.705 29.061-52.705 28.530Q-52.705 27.976-52.429 27.510Q-52.152 27.044-51.679 26.770Q-51.207 26.495-50.662 26.495Q-50.328 26.495-50.025 26.627Q-49.721 26.759-49.502 26.996L-49.502 25.146Q-49.502 24.904-49.572 24.796Q-49.642 24.689-49.776 24.665Q-49.910 24.640-50.196 24.640L-50.196 24.324L-48.829 24.227L-48.829 29.655Q-48.829 29.892-48.759 30Q-48.689 30.107-48.553 30.131Q-48.416 30.156-48.135 30.156L-48.135 30.472L-49.528 30.573L-49.528 30.024Q-49.779 30.287-50.097 30.430Q-50.416 30.573-50.763 30.573M-50.701 30.309Q-50.332 30.309-50.023 30.098Q-49.713 29.888-49.528 29.545L-49.528 27.413Q-49.700 27.110-49.981 26.932Q-50.262 26.754-50.600 26.754Q-51.290 26.754-51.594 27.275Q-51.897 27.796-51.897 28.538Q-51.897 29.259-51.627 29.784Q-51.356 30.309-50.701 30.309M-45.564 30.573Q-46.122 30.573-46.595 30.290Q-47.067 30.006-47.342 29.529Q-47.617 29.053-47.617 28.499Q-47.617 28.103-47.474 27.728Q-47.331 27.352-47.074 27.064Q-46.817 26.776-46.459 26.607Q-46.100 26.438-45.696 26.438Q-45.151 26.438-44.780 26.675Q-44.409 26.912-44.222 27.330Q-44.035 27.747-44.035 28.284Q-44.035 28.336-44.059 28.374Q-44.083 28.411-44.132 28.411L-46.804 28.411L-46.804 28.490Q-46.804 29.237-46.492 29.760Q-46.180 30.283-45.481 30.283Q-45.076 30.283-44.756 30.026Q-44.435 29.769-44.312 29.365Q-44.294 29.285-44.211 29.285L-44.132 29.285Q-44.092 29.285-44.064 29.316Q-44.035 29.347-44.035 29.391L-44.035 29.426Q-44.140 29.769-44.362 30.028Q-44.584 30.287-44.898 30.430Q-45.213 30.573-45.564 30.573M-46.795 28.160L-44.681 28.160Q-44.681 27.892-44.734 27.646Q-44.786 27.400-44.907 27.178Q-45.028 26.956-45.226 26.829Q-45.424 26.701-45.696 26.701Q-46.039 26.701-46.292 26.926Q-46.544 27.150-46.669 27.488Q-46.795 27.826-46.795 28.160M-41.394 30.472L-43.455 30.472L-43.455 30.156Q-43.147 30.156-42.956 30.103Q-42.765 30.050-42.765 29.861L-42.765 25.146Q-42.765 24.904-42.835 24.796Q-42.906 24.689-43.040 24.665Q-43.174 24.640-43.455 24.640L-43.455 24.324L-42.088 24.227L-42.088 29.861Q-42.088 30.050-41.895 30.103Q-41.701 30.156-41.394 30.156L-41.394 30.472M-38.630 28.824L-41.095 28.824L-41.095 28.231L-38.630 28.231L-38.630 28.824M-35.580 30.472L-37.812 30.472L-37.812 30.156Q-37.505 30.156-37.314 30.103Q-37.122 30.050-37.122 29.861L-37.122 26.908L-37.812 26.908L-37.812 26.592L-37.122 26.592L-37.122 25.524Q-37.122 25.141-36.909 24.818Q-36.696 24.495-36.342 24.311Q-35.989 24.126-35.611 24.126Q-35.290 24.126-35.039 24.304Q-34.789 24.482-34.789 24.794Q-34.789 24.970-34.908 25.089Q-35.026 25.207-35.202 25.207Q-35.382 25.207-35.505 25.089Q-35.628 24.970-35.628 24.794Q-35.628 24.553-35.413 24.425Q-35.518 24.390-35.655 24.390Q-35.923 24.390-36.105 24.572Q-36.287 24.755-36.380 25.025Q-36.472 25.295-36.472 25.559L-36.472 26.592L-35.422 26.592L-35.422 26.908L-36.446 26.908L-36.446 29.861Q-36.446 30.050-36.189 30.103Q-35.931 30.156-35.580 30.156L-35.580 30.472M-32.811 30.472L-35.044 30.472L-35.044 30.156Q-34.732 30.156-34.541 30.103Q-34.349 30.050-34.349 29.861L-34.349 27.413Q-34.349 27.172-34.420 27.064Q-34.490 26.956-34.624 26.932Q-34.758 26.908-35.044 26.908L-35.044 26.592L-33.730 26.495L-33.730 27.356Q-33.567 26.965-33.299 26.730Q-33.031 26.495-32.640 26.495Q-32.367 26.495-32.152 26.658Q-31.937 26.820-31.937 27.079Q-31.937 27.255-32.055 27.374Q-32.174 27.493-32.350 27.493Q-32.530 27.493-32.649 27.374Q-32.767 27.255-32.767 27.079Q-32.767 26.864-32.614 26.754L-32.631 26.754Q-33.009 26.754-33.242 27.016Q-33.475 27.277-33.574 27.664Q-33.673 28.051-33.673 28.411L-33.673 29.861Q-33.673 30.050-33.416 30.103Q-33.159 30.156-32.811 30.156L-32.811 30.472M-29.366 30.573Q-29.924 30.573-30.397 30.290Q-30.869 30.006-31.144 29.529Q-31.418 29.053-31.418 28.499Q-31.418 28.103-31.275 27.728Q-31.133 27.352-30.876 27.064Q-30.618 26.776-30.260 26.607Q-29.902 26.438-29.498 26.438Q-28.953 26.438-28.582 26.675Q-28.210 26.912-28.023 27.330Q-27.837 27.747-27.837 28.284Q-27.837 28.336-27.861 28.374Q-27.885 28.411-27.933 28.411L-30.605 28.411L-30.605 28.490Q-30.605 29.237-30.293 29.760Q-29.981 30.283-29.283 30.283Q-28.878 30.283-28.557 30.026Q-28.237 29.769-28.114 29.365Q-28.096 29.285-28.013 29.285L-27.933 29.285Q-27.894 29.285-27.865 29.316Q-27.837 29.347-27.837 29.391L-27.837 29.426Q-27.942 29.769-28.164 30.028Q-28.386 30.287-28.700 30.430Q-29.014 30.573-29.366 30.573M-30.596 28.160L-28.483 28.160Q-28.483 27.892-28.535 27.646Q-28.588 27.400-28.709 27.178Q-28.830 26.956-29.028 26.829Q-29.225 26.701-29.498 26.701Q-29.841 26.701-30.093 26.926Q-30.346 27.150-30.471 27.488Q-30.596 27.826-30.596 28.160M-25.253 30.573Q-25.811 30.573-26.283 30.290Q-26.756 30.006-27.030 29.529Q-27.305 29.053-27.305 28.499Q-27.305 28.103-27.162 27.728Q-27.019 27.352-26.762 27.064Q-26.505 26.776-26.147 26.607Q-25.789 26.438-25.385 26.438Q-24.840 26.438-24.468 26.675Q-24.097 26.912-23.910 27.330Q-23.723 27.747-23.723 28.284Q-23.723 28.336-23.748 28.374Q-23.772 28.411-23.820 28.411L-26.492 28.411L-26.492 28.490Q-26.492 29.237-26.180 29.760Q-25.868 30.283-25.169 30.283Q-24.765 30.283-24.444 30.026Q-24.123 29.769-24 29.365Q-23.983 29.285-23.899 29.285L-23.820 29.285Q-23.781 29.285-23.752 29.316Q-23.723 29.347-23.723 29.391L-23.723 29.426Q-23.829 29.769-24.051 30.028Q-24.273 30.287-24.587 30.430Q-24.901 30.573-25.253 30.573M-26.483 28.160L-24.369 28.160Q-24.369 27.892-24.422 27.646Q-24.475 27.400-24.596 27.178Q-24.717 26.956-24.914 26.829Q-25.112 26.701-25.385 26.701Q-25.727 26.701-25.980 26.926Q-26.233 27.150-26.358 27.488Q-26.483 27.826-26.483 28.160\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403-29.079h39.833V-46.15h-39.833Z\"\u002F>\u003Cg transform=\"translate(12.311 -66.375)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.391 30.472L-60.391 30.192Q-59.423 30.192-59.423 29.983L-59.423 26.364Q-59.816 26.552-60.438 26.552L-60.438 26.271Q-60.021 26.271-59.657 26.170Q-59.293 26.070-59.037 25.824L-58.911 25.824Q-58.846 25.841-58.829 25.909L-58.829 29.983Q-58.829 30.192-57.861 30.192L-57.861 30.472M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-50.595 30.472L-54.563 30.472L-54.563 30.192Q-53.842 30.192-53.842 29.983L-53.842 26.182Q-53.842 25.971-54.563 25.971L-54.563 25.690L-52.249 25.690L-52.249 25.971Q-52.567 25.971-52.859 26.006Q-53.151 26.042-53.151 26.182L-53.151 29.983Q-53.151 30.123-53.062 30.158Q-52.974 30.192-52.786 30.192L-52.164 30.192Q-51.750 30.192-51.471 30.089Q-51.193 29.987-51.027 29.785Q-50.861 29.583-50.779 29.296Q-50.697 29.009-50.653 28.602L-50.386 28.602\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147-29.079h22.762V-46.15h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 -65.831)\">\u003Cpath d=\"M-63.286 29.324L-65.330 29.324L-65.330 29.043L-62.999 25.871Q-62.964 25.824-62.899 25.824L-62.763 25.824Q-62.718 25.824-62.691 25.851Q-62.664 25.878-62.664 25.923L-62.664 29.043L-61.901 29.043L-61.901 29.324L-62.664 29.324L-62.664 29.983Q-62.664 30.192-61.908 30.192L-61.908 30.472L-64.041 30.472L-64.041 30.192Q-63.286 30.192-63.286 29.983L-63.286 29.324M-63.238 26.599L-65.029 29.043L-63.238 29.043\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403-11.438h39.833V-28.51h-39.833Z\"\u002F>\u003Cg transform=\"translate(11.88 -48.735)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.391 30.472L-60.391 30.192Q-59.423 30.192-59.423 29.983L-59.423 26.364Q-59.816 26.552-60.438 26.552L-60.438 26.271Q-60.021 26.271-59.657 26.170Q-59.293 26.070-59.037 25.824L-58.911 25.824Q-58.846 25.841-58.829 25.909L-58.829 29.983Q-58.829 30.192-57.861 30.192L-57.861 30.472M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-52.444 30.472L-54.549 30.472L-54.549 30.192Q-53.828 30.192-53.828 29.983L-53.828 26.182Q-53.828 25.971-54.549 25.971L-54.549 25.690L-52.211 25.690Q-51.900 25.690-51.557 25.764Q-51.213 25.837-50.892 25.993Q-50.571 26.148-50.369 26.394Q-50.167 26.640-50.167 26.965Q-50.167 27.252-50.355 27.483Q-50.543 27.714-50.820 27.862Q-51.097 28.011-51.388 28.093Q-51.053 28.196-50.819 28.428Q-50.584 28.660-50.540 28.989L-50.455 29.597Q-50.427 29.809-50.381 29.978Q-50.335 30.147-50.231 30.267Q-50.126 30.387-49.938 30.387Q-49.723 30.387-49.607 30.202Q-49.491 30.017-49.491 29.785Q-49.474 29.720-49.398 29.703L-49.306 29.703Q-49.224 29.723-49.224 29.805Q-49.224 30.011-49.315 30.197Q-49.405 30.383-49.571 30.498Q-49.737 30.612-49.938 30.612Q-50.277 30.612-50.571 30.511Q-50.865 30.410-51.049 30.188Q-51.234 29.966-51.234 29.618L-51.234 29.009Q-51.234 28.760-51.377 28.573Q-51.521 28.387-51.745 28.288Q-51.969 28.189-52.218 28.189L-53.165 28.189L-53.165 29.983Q-53.165 30.192-52.444 30.192L-52.444 30.472M-53.165 26.182L-53.165 27.967L-52.311 27.967Q-52.023 27.967-51.784 27.924Q-51.545 27.881-51.357 27.772Q-51.169 27.662-51.058 27.464Q-50.947 27.266-50.947 26.965Q-50.947 26.388-51.314 26.179Q-51.682 25.971-52.311 25.971L-52.799 25.971Q-52.987 25.971-53.076 26.005Q-53.165 26.039-53.165 26.182\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147-11.438h22.762V-28.51h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 -48.19)\">\u003Cpath d=\"M-64.807 29.925Q-64.687 30.082-64.496 30.181Q-64.304 30.281-64.089 30.320Q-63.874 30.359-63.651 30.359Q-63.354 30.359-63.159 30.204Q-62.964 30.048-62.874 29.794Q-62.783 29.539-62.783 29.255Q-62.783 28.961-62.875 28.710Q-62.968 28.459-63.166 28.303Q-63.364 28.148-63.658 28.148L-64.174 28.148Q-64.202 28.148-64.227 28.122Q-64.253 28.097-64.253 28.073L-64.253 28.001Q-64.253 27.970-64.227 27.948Q-64.202 27.926-64.174 27.926L-63.733 27.895Q-63.371 27.895-63.151 27.538Q-62.930 27.180-62.930 26.791Q-62.930 26.463-63.125 26.259Q-63.320 26.056-63.651 26.056Q-63.938 26.056-64.191 26.140Q-64.444 26.223-64.608 26.411Q-64.461 26.411-64.361 26.526Q-64.260 26.640-64.260 26.791Q-64.260 26.941-64.366 27.051Q-64.472 27.160-64.629 27.160Q-64.790 27.160-64.899 27.051Q-65.008 26.941-65.008 26.791Q-65.008 26.466-64.800 26.247Q-64.591 26.029-64.275 25.926Q-63.959 25.824-63.651 25.824Q-63.333 25.824-63.005 25.928Q-62.677 26.032-62.450 26.254Q-62.223 26.476-62.223 26.791Q-62.223 27.225-62.510 27.550Q-62.797 27.874-63.231 28.021Q-62.920 28.086-62.640 28.252Q-62.359 28.418-62.182 28.676Q-62.004 28.934-62.004 29.255Q-62.004 29.665-62.248 29.975Q-62.493 30.284-62.874 30.448Q-63.255 30.612-63.651 30.612Q-64.020 30.612-64.378 30.499Q-64.735 30.387-64.979 30.137Q-65.224 29.888-65.224 29.518Q-65.224 29.347-65.107 29.235Q-64.991 29.122-64.820 29.122Q-64.711 29.122-64.620 29.173Q-64.530 29.224-64.475 29.317Q-64.420 29.409-64.420 29.518Q-64.420 29.686-64.533 29.805Q-64.646 29.925-64.807 29.925\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403 6.202h39.833v-17.071h-39.833Z\"\u002F>\u003Cg transform=\"translate(12.311 -31.094)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.746 30.472L-60.746 30.270Q-60.746 30.240-60.719 30.212L-59.471 28.995Q-59.399 28.920-59.357 28.878Q-59.314 28.835-59.235 28.756Q-58.822 28.343-58.591 27.985Q-58.360 27.628-58.360 27.204Q-58.360 26.972-58.439 26.769Q-58.518 26.565-58.659 26.415Q-58.801 26.264-58.996 26.184Q-59.191 26.104-59.423 26.104Q-59.734 26.104-59.992 26.263Q-60.250 26.422-60.380 26.699L-60.360 26.699Q-60.192 26.699-60.085 26.810Q-59.977 26.921-59.977 27.085Q-59.977 27.242-60.086 27.355Q-60.196 27.468-60.360 27.468Q-60.520 27.468-60.633 27.355Q-60.746 27.242-60.746 27.085Q-60.746 26.709-60.538 26.422Q-60.329 26.135-59.994 25.979Q-59.659 25.824-59.304 25.824Q-58.880 25.824-58.500 25.982Q-58.121 26.141-57.887 26.458Q-57.653 26.774-57.653 27.204Q-57.653 27.515-57.793 27.784Q-57.933 28.052-58.138 28.257Q-58.343 28.462-58.706 28.744Q-59.068 29.026-59.177 29.122L-60.032 29.850L-59.389 29.850Q-59.126 29.850-58.837 29.848Q-58.548 29.847-58.330 29.838Q-58.111 29.829-58.094 29.812Q-58.032 29.747-57.995 29.580Q-57.957 29.412-57.919 29.170L-57.653 29.170L-57.861 30.472M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-50.595 30.472L-54.563 30.472L-54.563 30.192Q-53.842 30.192-53.842 29.983L-53.842 26.182Q-53.842 25.971-54.563 25.971L-54.563 25.690L-52.249 25.690L-52.249 25.971Q-52.567 25.971-52.859 26.006Q-53.151 26.042-53.151 26.182L-53.151 29.983Q-53.151 30.123-53.062 30.158Q-52.974 30.192-52.786 30.192L-52.164 30.192Q-51.750 30.192-51.471 30.089Q-51.193 29.987-51.027 29.785Q-50.861 29.583-50.779 29.296Q-50.697 29.009-50.653 28.602L-50.386 28.602\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147 6.202h22.762v-17.071h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 -30.55)\">\u003Cpath d=\"M-63.614 30.612Q-64.249 30.612-64.613 30.267Q-64.978 29.922-65.113 29.397Q-65.248 28.872-65.248 28.247Q-65.248 27.222-64.892 26.523Q-64.537 25.824-63.614 25.824Q-62.687 25.824-62.335 26.523Q-61.983 27.222-61.983 28.247Q-61.983 28.872-62.118 29.397Q-62.253 29.922-62.616 30.267Q-62.978 30.612-63.614 30.612M-63.614 30.387Q-63.176 30.387-62.963 30.012Q-62.749 29.638-62.699 29.171Q-62.650 28.705-62.650 28.127Q-62.650 27.574-62.699 27.146Q-62.749 26.719-62.961 26.384Q-63.173 26.049-63.614 26.049Q-63.956 26.049-64.159 26.256Q-64.362 26.463-64.449 26.775Q-64.537 27.088-64.559 27.404Q-64.581 27.721-64.581 28.127Q-64.581 28.544-64.559 28.886Q-64.537 29.228-64.448 29.576Q-64.359 29.925-64.154 30.156Q-63.949 30.387-63.614 30.387\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403 23.843h39.833V6.77h-39.833Z\"\u002F>\u003Cg transform=\"translate(11.88 -13.454)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.746 30.472L-60.746 30.270Q-60.746 30.240-60.719 30.212L-59.471 28.995Q-59.399 28.920-59.357 28.878Q-59.314 28.835-59.235 28.756Q-58.822 28.343-58.591 27.985Q-58.360 27.628-58.360 27.204Q-58.360 26.972-58.439 26.769Q-58.518 26.565-58.659 26.415Q-58.801 26.264-58.996 26.184Q-59.191 26.104-59.423 26.104Q-59.734 26.104-59.992 26.263Q-60.250 26.422-60.380 26.699L-60.360 26.699Q-60.192 26.699-60.085 26.810Q-59.977 26.921-59.977 27.085Q-59.977 27.242-60.086 27.355Q-60.196 27.468-60.360 27.468Q-60.520 27.468-60.633 27.355Q-60.746 27.242-60.746 27.085Q-60.746 26.709-60.538 26.422Q-60.329 26.135-59.994 25.979Q-59.659 25.824-59.304 25.824Q-58.880 25.824-58.500 25.982Q-58.121 26.141-57.887 26.458Q-57.653 26.774-57.653 27.204Q-57.653 27.515-57.793 27.784Q-57.933 28.052-58.138 28.257Q-58.343 28.462-58.706 28.744Q-59.068 29.026-59.177 29.122L-60.032 29.850L-59.389 29.850Q-59.126 29.850-58.837 29.848Q-58.548 29.847-58.330 29.838Q-58.111 29.829-58.094 29.812Q-58.032 29.747-57.995 29.580Q-57.957 29.412-57.919 29.170L-57.653 29.170L-57.861 30.472M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-52.444 30.472L-54.549 30.472L-54.549 30.192Q-53.828 30.192-53.828 29.983L-53.828 26.182Q-53.828 25.971-54.549 25.971L-54.549 25.690L-52.211 25.690Q-51.900 25.690-51.557 25.764Q-51.213 25.837-50.892 25.993Q-50.571 26.148-50.369 26.394Q-50.167 26.640-50.167 26.965Q-50.167 27.252-50.355 27.483Q-50.543 27.714-50.820 27.862Q-51.097 28.011-51.388 28.093Q-51.053 28.196-50.819 28.428Q-50.584 28.660-50.540 28.989L-50.455 29.597Q-50.427 29.809-50.381 29.978Q-50.335 30.147-50.231 30.267Q-50.126 30.387-49.938 30.387Q-49.723 30.387-49.607 30.202Q-49.491 30.017-49.491 29.785Q-49.474 29.720-49.398 29.703L-49.306 29.703Q-49.224 29.723-49.224 29.805Q-49.224 30.011-49.315 30.197Q-49.405 30.383-49.571 30.498Q-49.737 30.612-49.938 30.612Q-50.277 30.612-50.571 30.511Q-50.865 30.410-51.049 30.188Q-51.234 29.966-51.234 29.618L-51.234 29.009Q-51.234 28.760-51.377 28.573Q-51.521 28.387-51.745 28.288Q-51.969 28.189-52.218 28.189L-53.165 28.189L-53.165 29.983Q-53.165 30.192-52.444 30.192L-52.444 30.472M-53.165 26.182L-53.165 27.967L-52.311 27.967Q-52.023 27.967-51.784 27.924Q-51.545 27.881-51.357 27.772Q-51.169 27.662-51.058 27.464Q-50.947 27.266-50.947 26.965Q-50.947 26.388-51.314 26.179Q-51.682 25.971-52.311 25.971L-52.799 25.971Q-52.987 25.971-53.076 26.005Q-53.165 26.039-53.165 26.182\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147 23.843h22.762V6.77h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 -12.91)\">\u003Cpath d=\"M-63.286 29.324L-65.330 29.324L-65.330 29.043L-62.999 25.871Q-62.964 25.824-62.899 25.824L-62.763 25.824Q-62.718 25.824-62.691 25.851Q-62.664 25.878-62.664 25.923L-62.664 29.043L-61.901 29.043L-61.901 29.324L-62.664 29.324L-62.664 29.983Q-62.664 30.192-61.908 30.192L-61.908 30.472L-64.041 30.472L-64.041 30.192Q-63.286 30.192-63.286 29.983L-63.286 29.324M-63.238 26.599L-65.029 29.043L-63.238 29.043\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403 41.483h39.833V24.412h-39.833Z\"\u002F>\u003Cg transform=\"translate(12.311 4.187)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-60.391 29.925Q-60.271 30.082-60.080 30.181Q-59.888 30.281-59.673 30.320Q-59.457 30.359-59.235 30.359Q-58.938 30.359-58.743 30.204Q-58.548 30.048-58.458 29.794Q-58.367 29.539-58.367 29.255Q-58.367 28.961-58.459 28.710Q-58.552 28.459-58.750 28.303Q-58.948 28.148-59.242 28.148L-59.758 28.148Q-59.786 28.148-59.811 28.122Q-59.837 28.097-59.837 28.073L-59.837 28.001Q-59.837 27.970-59.811 27.948Q-59.786 27.926-59.758 27.926L-59.317 27.895Q-58.955 27.895-58.735 27.538Q-58.514 27.180-58.514 26.791Q-58.514 26.463-58.709 26.259Q-58.904 26.056-59.235 26.056Q-59.522 26.056-59.775 26.140Q-60.028 26.223-60.192 26.411Q-60.045 26.411-59.945 26.526Q-59.844 26.640-59.844 26.791Q-59.844 26.941-59.950 27.051Q-60.056 27.160-60.213 27.160Q-60.374 27.160-60.483 27.051Q-60.592 26.941-60.592 26.791Q-60.592 26.466-60.384 26.247Q-60.175 26.029-59.859 25.926Q-59.543 25.824-59.235 25.824Q-58.917 25.824-58.589 25.928Q-58.261 26.032-58.034 26.254Q-57.807 26.476-57.807 26.791Q-57.807 27.225-58.094 27.550Q-58.381 27.874-58.815 28.021Q-58.504 28.086-58.224 28.252Q-57.943 28.418-57.766 28.676Q-57.588 28.934-57.588 29.255Q-57.588 29.665-57.832 29.975Q-58.077 30.284-58.458 30.448Q-58.839 30.612-59.235 30.612Q-59.604 30.612-59.962 30.499Q-60.319 30.387-60.563 30.137Q-60.808 29.888-60.808 29.518Q-60.808 29.347-60.691 29.235Q-60.575 29.122-60.404 29.122Q-60.295 29.122-60.204 29.173Q-60.114 29.224-60.059 29.317Q-60.004 29.409-60.004 29.518Q-60.004 29.686-60.117 29.805Q-60.230 29.925-60.391 29.925M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-50.595 30.472L-54.563 30.472L-54.563 30.192Q-53.842 30.192-53.842 29.983L-53.842 26.182Q-53.842 25.971-54.563 25.971L-54.563 25.690L-52.249 25.690L-52.249 25.971Q-52.567 25.971-52.859 26.006Q-53.151 26.042-53.151 26.182L-53.151 29.983Q-53.151 30.123-53.062 30.158Q-52.974 30.192-52.786 30.192L-52.164 30.192Q-51.750 30.192-51.471 30.089Q-51.193 29.987-51.027 29.785Q-50.861 29.583-50.779 29.296Q-50.697 29.009-50.653 28.602L-50.386 28.602\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147 41.483h22.762V24.412h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 4.731)\">\u003Cpath d=\"M-62.277 30.472L-65.162 30.472L-65.162 30.270Q-65.162 30.240-65.135 30.212L-63.887 28.995Q-63.815 28.920-63.773 28.878Q-63.730 28.835-63.651 28.756Q-63.238 28.343-63.007 27.985Q-62.776 27.628-62.776 27.204Q-62.776 26.972-62.855 26.769Q-62.934 26.565-63.075 26.415Q-63.217 26.264-63.412 26.184Q-63.607 26.104-63.839 26.104Q-64.150 26.104-64.408 26.263Q-64.666 26.422-64.796 26.699L-64.776 26.699Q-64.608 26.699-64.501 26.810Q-64.393 26.921-64.393 27.085Q-64.393 27.242-64.502 27.355Q-64.612 27.468-64.776 27.468Q-64.936 27.468-65.049 27.355Q-65.162 27.242-65.162 27.085Q-65.162 26.709-64.954 26.422Q-64.745 26.135-64.410 25.979Q-64.075 25.824-63.720 25.824Q-63.296 25.824-62.916 25.982Q-62.537 26.141-62.303 26.458Q-62.069 26.774-62.069 27.204Q-62.069 27.515-62.209 27.784Q-62.349 28.052-62.554 28.257Q-62.759 28.462-63.122 28.744Q-63.484 29.026-63.593 29.122L-64.448 29.850L-63.805 29.850Q-63.542 29.850-63.253 29.848Q-62.964 29.847-62.746 29.838Q-62.527 29.829-62.510 29.812Q-62.448 29.747-62.411 29.580Q-62.373 29.412-62.335 29.170L-62.069 29.170\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-65.403 59.124h39.833V42.052h-39.833Z\"\u002F>\u003Cg transform=\"translate(11.88 21.827)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-60.391 29.925Q-60.271 30.082-60.080 30.181Q-59.888 30.281-59.673 30.320Q-59.457 30.359-59.235 30.359Q-58.938 30.359-58.743 30.204Q-58.548 30.048-58.458 29.794Q-58.367 29.539-58.367 29.255Q-58.367 28.961-58.459 28.710Q-58.552 28.459-58.750 28.303Q-58.948 28.148-59.242 28.148L-59.758 28.148Q-59.786 28.148-59.811 28.122Q-59.837 28.097-59.837 28.073L-59.837 28.001Q-59.837 27.970-59.811 27.948Q-59.786 27.926-59.758 27.926L-59.317 27.895Q-58.955 27.895-58.735 27.538Q-58.514 27.180-58.514 26.791Q-58.514 26.463-58.709 26.259Q-58.904 26.056-59.235 26.056Q-59.522 26.056-59.775 26.140Q-60.028 26.223-60.192 26.411Q-60.045 26.411-59.945 26.526Q-59.844 26.640-59.844 26.791Q-59.844 26.941-59.950 27.051Q-60.056 27.160-60.213 27.160Q-60.374 27.160-60.483 27.051Q-60.592 26.941-60.592 26.791Q-60.592 26.466-60.384 26.247Q-60.175 26.029-59.859 25.926Q-59.543 25.824-59.235 25.824Q-58.917 25.824-58.589 25.928Q-58.261 26.032-58.034 26.254Q-57.807 26.476-57.807 26.791Q-57.807 27.225-58.094 27.550Q-58.381 27.874-58.815 28.021Q-58.504 28.086-58.224 28.252Q-57.943 28.418-57.766 28.676Q-57.588 28.934-57.588 29.255Q-57.588 29.665-57.832 29.975Q-58.077 30.284-58.458 30.448Q-58.839 30.612-59.235 30.612Q-59.604 30.612-59.962 30.499Q-60.319 30.387-60.563 30.137Q-60.808 29.888-60.808 29.518Q-60.808 29.347-60.691 29.235Q-60.575 29.122-60.404 29.122Q-60.295 29.122-60.204 29.173Q-60.114 29.224-60.059 29.317Q-60.004 29.409-60.004 29.518Q-60.004 29.686-60.117 29.805Q-60.230 29.925-60.391 29.925M-56.392 31.702Q-56.392 31.668-56.364 31.641Q-56.094 31.412-55.946 31.089Q-55.797 30.766-55.797 30.410L-55.797 30.373Q-55.906 30.472-56.070 30.472Q-56.251 30.472-56.371 30.352Q-56.491 30.233-56.491 30.052Q-56.491 29.877-56.371 29.758Q-56.251 29.638-56.070 29.638Q-55.814 29.638-55.694 29.877Q-55.575 30.117-55.575 30.410Q-55.575 30.810-55.744 31.181Q-55.913 31.552-56.210 31.808Q-56.241 31.829-56.269 31.829Q-56.310 31.829-56.351 31.788Q-56.392 31.747-56.392 31.702M-52.444 30.472L-54.549 30.472L-54.549 30.192Q-53.828 30.192-53.828 29.983L-53.828 26.182Q-53.828 25.971-54.549 25.971L-54.549 25.690L-52.211 25.690Q-51.900 25.690-51.557 25.764Q-51.213 25.837-50.892 25.993Q-50.571 26.148-50.369 26.394Q-50.167 26.640-50.167 26.965Q-50.167 27.252-50.355 27.483Q-50.543 27.714-50.820 27.862Q-51.097 28.011-51.388 28.093Q-51.053 28.196-50.819 28.428Q-50.584 28.660-50.540 28.989L-50.455 29.597Q-50.427 29.809-50.381 29.978Q-50.335 30.147-50.231 30.267Q-50.126 30.387-49.938 30.387Q-49.723 30.387-49.607 30.202Q-49.491 30.017-49.491 29.785Q-49.474 29.720-49.398 29.703L-49.306 29.703Q-49.224 29.723-49.224 29.805Q-49.224 30.011-49.315 30.197Q-49.405 30.383-49.571 30.498Q-49.737 30.612-49.938 30.612Q-50.277 30.612-50.571 30.511Q-50.865 30.410-51.049 30.188Q-51.234 29.966-51.234 29.618L-51.234 29.009Q-51.234 28.760-51.377 28.573Q-51.521 28.387-51.745 28.288Q-51.969 28.189-52.218 28.189L-53.165 28.189L-53.165 29.983Q-53.165 30.192-52.444 30.192L-52.444 30.472M-53.165 26.182L-53.165 27.967L-52.311 27.967Q-52.023 27.967-51.784 27.924Q-51.545 27.881-51.357 27.772Q-51.169 27.662-51.058 27.464Q-50.947 27.266-50.947 26.965Q-50.947 26.388-51.314 26.179Q-51.682 25.971-52.311 25.971L-52.799 25.971Q-52.987 25.971-53.076 26.005Q-53.165 26.039-53.165 26.182\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-24.147 59.124h22.762V42.052h-22.762Z\"\u002F>\u003Cg transform=\"translate(50.844 22.372)\">\u003Cpath d=\"M-64.807 29.925Q-64.687 30.082-64.496 30.181Q-64.304 30.281-64.089 30.320Q-63.874 30.359-63.651 30.359Q-63.354 30.359-63.159 30.204Q-62.964 30.048-62.874 29.794Q-62.783 29.539-62.783 29.255Q-62.783 28.961-62.875 28.710Q-62.968 28.459-63.166 28.303Q-63.364 28.148-63.658 28.148L-64.174 28.148Q-64.202 28.148-64.227 28.122Q-64.253 28.097-64.253 28.073L-64.253 28.001Q-64.253 27.970-64.227 27.948Q-64.202 27.926-64.174 27.926L-63.733 27.895Q-63.371 27.895-63.151 27.538Q-62.930 27.180-62.930 26.791Q-62.930 26.463-63.125 26.259Q-63.320 26.056-63.651 26.056Q-63.938 26.056-64.191 26.140Q-64.444 26.223-64.608 26.411Q-64.461 26.411-64.361 26.526Q-64.260 26.640-64.260 26.791Q-64.260 26.941-64.366 27.051Q-64.472 27.160-64.629 27.160Q-64.790 27.160-64.899 27.051Q-65.008 26.941-65.008 26.791Q-65.008 26.466-64.800 26.247Q-64.591 26.029-64.275 25.926Q-63.959 25.824-63.651 25.824Q-63.333 25.824-63.005 25.928Q-62.677 26.032-62.450 26.254Q-62.223 26.476-62.223 26.791Q-62.223 27.225-62.510 27.550Q-62.797 27.874-63.231 28.021Q-62.920 28.086-62.640 28.252Q-62.359 28.418-62.182 28.676Q-62.004 28.934-62.004 29.255Q-62.004 29.665-62.248 29.975Q-62.493 30.284-62.874 30.448Q-63.255 30.612-63.651 30.612Q-64.020 30.612-64.378 30.499Q-64.735 30.387-64.979 30.137Q-65.224 29.888-65.224 29.518Q-65.224 29.347-65.107 29.235Q-64.991 29.122-64.820 29.122Q-64.711 29.122-64.620 29.173Q-64.530 29.224-64.475 29.317Q-64.420 29.409-64.420 29.518Q-64.420 29.686-64.533 29.805Q-64.646 29.925-64.807 29.925\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-65.289 20.961Q-65.289 20.633-65.154 20.332Q-65.019 20.032-64.783 19.811Q-64.547 19.591-64.243 19.471Q-63.938 19.351-63.614 19.351Q-63.108 19.351-62.759 19.454Q-62.411 19.556-62.411 19.932Q-62.411 20.079-62.508 20.180Q-62.605 20.281-62.752 20.281Q-62.906 20.281-63.005 20.182Q-63.104 20.083-63.104 19.932Q-63.104 19.744-62.964 19.652Q-63.166 19.601-63.607 19.601Q-63.962 19.601-64.191 19.797Q-64.420 19.994-64.521 20.303Q-64.622 20.613-64.622 20.961Q-64.622 21.310-64.496 21.616Q-64.369 21.922-64.114 22.106Q-63.860 22.291-63.504 22.291Q-63.282 22.291-63.098 22.207Q-62.913 22.123-62.778 21.968Q-62.643 21.812-62.585 21.604Q-62.571 21.549-62.517 21.549L-62.404 21.549Q-62.373 21.549-62.351 21.573Q-62.329 21.597-62.329 21.631L-62.329 21.652Q-62.414 21.939-62.602 22.137Q-62.790 22.335-63.055 22.438Q-63.320 22.540-63.614 22.540Q-64.044 22.540-64.432 22.334Q-64.820 22.127-65.054 21.764Q-65.289 21.402-65.289 20.961M-61.683 21.744Q-61.683 21.412-61.459 21.185Q-61.235 20.958-60.891 20.830Q-60.548 20.701-60.175 20.649Q-59.803 20.596-59.499 20.596L-59.499 20.343Q-59.499 20.138-59.606 19.958Q-59.714 19.779-59.895 19.676Q-60.076 19.574-60.285 19.574Q-60.691 19.574-60.927 19.666Q-60.838 19.703-60.792 19.787Q-60.746 19.871-60.746 19.973Q-60.746 20.069-60.792 20.148Q-60.838 20.226-60.919 20.271Q-60.999 20.315-61.088 20.315Q-61.238 20.315-61.339 20.218Q-61.440 20.120-61.440 19.973Q-61.440 19.351-60.285 19.351Q-60.073 19.351-59.823 19.415Q-59.574 19.478-59.372 19.597Q-59.170 19.717-59.044 19.902Q-58.917 20.086-58.917 20.329L-58.917 21.905Q-58.917 22.021-58.856 22.117Q-58.794 22.212-58.682 22.212Q-58.572 22.212-58.507 22.118Q-58.442 22.024-58.442 21.905L-58.442 21.457L-58.176 21.457L-58.176 21.905Q-58.176 22.175-58.403 22.340Q-58.630 22.506-58.911 22.506Q-59.119 22.506-59.256 22.352Q-59.393 22.199-59.416 21.983Q-59.563 22.250-59.845 22.395Q-60.127 22.540-60.452 22.540Q-60.729 22.540-61.013 22.465Q-61.296 22.390-61.489 22.211Q-61.683 22.031-61.683 21.744M-61.067 21.744Q-61.067 21.918-60.967 22.048Q-60.866 22.178-60.710 22.248Q-60.555 22.318-60.391 22.318Q-60.172 22.318-59.963 22.221Q-59.755 22.123-59.627 21.942Q-59.499 21.761-59.499 21.535L-59.499 20.807Q-59.823 20.807-60.189 20.898Q-60.555 20.989-60.811 21.201Q-61.067 21.412-61.067 21.744M-57.759 20.961Q-57.759 20.633-57.624 20.332Q-57.489 20.032-57.253 19.811Q-57.017 19.591-56.713 19.471Q-56.409 19.351-56.084 19.351Q-55.578 19.351-55.229 19.454Q-54.881 19.556-54.881 19.932Q-54.881 20.079-54.978 20.180Q-55.076 20.281-55.223 20.281Q-55.376 20.281-55.476 20.182Q-55.575 20.083-55.575 19.932Q-55.575 19.744-55.435 19.652Q-55.636 19.601-56.077 19.601Q-56.433 19.601-56.662 19.797Q-56.891 19.994-56.991 20.303Q-57.092 20.613-57.092 20.961Q-57.092 21.310-56.966 21.616Q-56.839 21.922-56.585 22.106Q-56.330 22.291-55.975 22.291Q-55.752 22.291-55.568 22.207Q-55.383 22.123-55.248 21.968Q-55.113 21.812-55.055 21.604Q-55.041 21.549-54.987 21.549L-54.874 21.549Q-54.843 21.549-54.821 21.573Q-54.799 21.597-54.799 21.631L-54.799 21.652Q-54.884 21.939-55.072 22.137Q-55.260 22.335-55.525 22.438Q-55.790 22.540-56.084 22.540Q-56.515 22.540-56.903 22.334Q-57.290 22.127-57.525 21.764Q-57.759 21.402-57.759 20.961\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-52.725 22.472L-54.359 22.472L-54.359 22.192Q-54.130 22.192-53.981 22.158Q-53.832 22.123-53.832 21.983L-53.832 18.364Q-53.832 18.094-53.940 18.032Q-54.048 17.971-54.359 17.971L-54.359 17.690L-53.279 17.615L-53.279 20.001Q-53.173 19.816-52.995 19.674Q-52.817 19.533-52.609 19.459Q-52.400 19.386-52.175 19.386Q-51.669 19.386-51.385 19.609Q-51.101 19.833-51.101 20.329L-51.101 21.983Q-51.101 22.120-50.953 22.156Q-50.804 22.192-50.578 22.192L-50.578 22.472L-52.209 22.472L-52.209 22.192Q-51.980 22.192-51.831 22.158Q-51.682 22.123-51.682 21.983L-51.682 20.343Q-51.682 20.008-51.802 19.808Q-51.922 19.608-52.236 19.608Q-52.506 19.608-52.740 19.744Q-52.974 19.881-53.113 20.115Q-53.251 20.349-53.251 20.623L-53.251 21.983Q-53.251 22.120-53.101 22.156Q-52.950 22.192-52.725 22.192L-52.725 22.472M-50.032 20.937Q-50.032 20.616-49.907 20.327Q-49.782 20.038-49.556 19.815Q-49.331 19.591-49.035 19.471Q-48.740 19.351-48.422 19.351Q-48.094 19.351-47.832 19.451Q-47.571 19.550-47.395 19.732Q-47.219 19.915-47.125 20.173Q-47.031 20.431-47.031 20.763Q-47.031 20.855-47.113 20.876L-49.368 20.876L-49.368 20.937Q-49.368 21.525-49.085 21.908Q-48.801 22.291-48.234 22.291Q-47.912 22.291-47.644 22.098Q-47.376 21.905-47.287 21.590Q-47.280 21.549-47.205 21.535L-47.113 21.535Q-47.031 21.559-47.031 21.631Q-47.031 21.638-47.037 21.665Q-47.150 22.062-47.521 22.301Q-47.892 22.540-48.316 22.540Q-48.753 22.540-49.153 22.332Q-49.553 22.123-49.792 21.756Q-50.032 21.389-50.032 20.937M-49.362 20.667L-47.547 20.667Q-47.547 20.390-47.644 20.138Q-47.742 19.885-47.940 19.729Q-48.138 19.574-48.422 19.574Q-48.699 19.574-48.912 19.732Q-49.126 19.891-49.244 20.146Q-49.362 20.401-49.362 20.667M-46.443 20.961Q-46.443 20.623-46.303 20.332Q-46.162 20.042-45.918 19.828Q-45.674 19.615-45.369 19.500Q-45.065 19.386-44.741 19.386Q-44.471 19.386-44.207 19.485Q-43.944 19.584-43.753 19.762L-43.753 18.364Q-43.753 18.094-43.860 18.032Q-43.968 17.971-44.279 17.971L-44.279 17.690L-43.202 17.615L-43.202 21.799Q-43.202 21.987-43.148 22.070Q-43.093 22.154-42.992 22.173Q-42.891 22.192-42.676 22.192L-42.676 22.472L-43.783 22.540L-43.783 22.123Q-44.200 22.540-44.826 22.540Q-45.257 22.540-45.629 22.328Q-46.002 22.117-46.222 21.756Q-46.443 21.395-46.443 20.961M-44.768 22.318Q-44.559 22.318-44.373 22.246Q-44.187 22.175-44.033 22.038Q-43.879 21.901-43.783 21.723L-43.783 20.114Q-43.869 19.967-44.014 19.847Q-44.159 19.727-44.329 19.668Q-44.498 19.608-44.679 19.608Q-45.240 19.608-45.508 19.997Q-45.776 20.387-45.776 20.968Q-45.776 21.539-45.542 21.929Q-45.308 22.318-44.768 22.318\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-39.265 21.744Q-39.265 21.412-39.042 21.185Q-38.818 20.958-38.474 20.830Q-38.131 20.701-37.758 20.649Q-37.386 20.596-37.081 20.596L-37.081 20.343Q-37.081 20.138-37.189 19.958Q-37.297 19.779-37.478 19.676Q-37.659 19.574-37.867 19.574Q-38.274 19.574-38.510 19.666Q-38.421 19.703-38.375 19.787Q-38.329 19.871-38.329 19.973Q-38.329 20.069-38.375 20.148Q-38.421 20.226-38.502 20.271Q-38.582 20.315-38.671 20.315Q-38.821 20.315-38.922 20.218Q-39.023 20.120-39.023 19.973Q-39.023 19.351-37.867 19.351Q-37.656 19.351-37.406 19.415Q-37.157 19.478-36.955 19.597Q-36.753 19.717-36.627 19.902Q-36.500 20.086-36.500 20.329L-36.500 21.905Q-36.500 22.021-36.439 22.117Q-36.377 22.212-36.264 22.212Q-36.155 22.212-36.090 22.118Q-36.025 22.024-36.025 21.905L-36.025 21.457L-35.759 21.457L-35.759 21.905Q-35.759 22.175-35.986 22.340Q-36.213 22.506-36.493 22.506Q-36.702 22.506-36.839 22.352Q-36.975 22.199-36.999 21.983Q-37.146 22.250-37.428 22.395Q-37.710 22.540-38.035 22.540Q-38.312 22.540-38.596 22.465Q-38.879 22.390-39.072 22.211Q-39.265 22.031-39.265 21.744M-38.650 21.744Q-38.650 21.918-38.549 22.048Q-38.449 22.178-38.293 22.248Q-38.138 22.318-37.973 22.318Q-37.755 22.318-37.546 22.221Q-37.338 22.123-37.210 21.942Q-37.081 21.761-37.081 21.535L-37.081 20.807Q-37.406 20.807-37.772 20.898Q-38.138 20.989-38.394 21.201Q-38.650 21.412-38.650 21.744M-35.342 20.961Q-35.342 20.633-35.207 20.332Q-35.072 20.032-34.836 19.811Q-34.600 19.591-34.296 19.471Q-33.992 19.351-33.667 19.351Q-33.161 19.351-32.812 19.454Q-32.464 19.556-32.464 19.932Q-32.464 20.079-32.561 20.180Q-32.659 20.281-32.805 20.281Q-32.959 20.281-33.058 20.182Q-33.158 20.083-33.158 19.932Q-33.158 19.744-33.017 19.652Q-33.219 19.601-33.660 19.601Q-34.015 19.601-34.244 19.797Q-34.473 19.994-34.574 20.303Q-34.675 20.613-34.675 20.961Q-34.675 21.310-34.549 21.616Q-34.422 21.922-34.168 22.106Q-33.913 22.291-33.557 22.291Q-33.335 22.291-33.151 22.207Q-32.966 22.123-32.831 21.968Q-32.696 21.812-32.638 21.604Q-32.624 21.549-32.570 21.549L-32.457 21.549Q-32.426 21.549-32.404 21.573Q-32.382 21.597-32.382 21.631L-32.382 21.652Q-32.467 21.939-32.655 22.137Q-32.843 22.335-33.108 22.438Q-33.373 22.540-33.667 22.540Q-34.097 22.540-34.485 22.334Q-34.873 22.127-35.107 21.764Q-35.342 21.402-35.342 20.961M-31.267 21.631L-31.267 19.734L-31.907 19.734L-31.907 19.512Q-31.589 19.512-31.372 19.302Q-31.155 19.092-31.054 18.782Q-30.953 18.473-30.953 18.165L-30.686 18.165L-30.686 19.454L-29.610 19.454L-29.610 19.734L-30.686 19.734L-30.686 21.618Q-30.686 21.894-30.582 22.093Q-30.478 22.291-30.218 22.291Q-30.061 22.291-29.955 22.187Q-29.849 22.082-29.799 21.929Q-29.750 21.775-29.750 21.618L-29.750 21.204L-29.483 21.204L-29.483 21.631Q-29.483 21.857-29.582 22.067Q-29.681 22.277-29.866 22.409Q-30.051 22.540-30.280 22.540Q-30.717 22.540-30.992 22.303Q-31.267 22.065-31.267 21.631M-27.056 22.472L-28.608 22.472L-28.608 22.192Q-28.383 22.192-28.234 22.158Q-28.085 22.123-28.085 21.983L-28.085 20.134Q-28.085 19.946-28.133 19.862Q-28.181 19.779-28.278 19.760Q-28.376 19.741-28.588 19.741L-28.588 19.461L-27.532 19.386L-27.532 21.983Q-27.532 22.123-27.400 22.158Q-27.268 22.192-27.056 22.192L-27.056 22.472M-28.328 18.165Q-28.328 17.994-28.205 17.875Q-28.082 17.755-27.911 17.755Q-27.743 17.755-27.620 17.875Q-27.497 17.994-27.497 18.165Q-27.497 18.340-27.620 18.463Q-27.743 18.586-27.911 18.586Q-28.082 18.586-28.205 18.463Q-28.328 18.340-28.328 18.165M-26.451 20.989Q-26.451 20.647-26.316 20.348Q-26.181 20.049-25.942 19.825Q-25.703 19.601-25.385 19.476Q-25.067 19.351-24.736 19.351Q-24.291 19.351-23.891 19.567Q-23.492 19.782-23.257 20.160Q-23.023 20.537-23.023 20.989Q-23.023 21.330-23.165 21.614Q-23.307 21.898-23.551 22.105Q-23.796 22.311-24.105 22.426Q-24.414 22.540-24.736 22.540Q-25.166 22.540-25.568 22.339Q-25.970 22.137-26.211 21.785Q-26.451 21.433-26.451 20.989M-24.736 22.291Q-24.134 22.291-23.910 21.913Q-23.686 21.535-23.686 20.903Q-23.686 20.291-23.920 19.932Q-24.155 19.574-24.736 19.574Q-25.788 19.574-25.788 20.903Q-25.788 21.535-25.563 21.913Q-25.337 22.291-24.736 22.291M-20.747 22.472L-22.381 22.472L-22.381 22.192Q-22.152 22.192-22.003 22.158Q-21.854 22.123-21.854 21.983L-21.854 20.134Q-21.854 19.864-21.962 19.803Q-22.070 19.741-22.381 19.741L-22.381 19.461L-21.321 19.386L-21.321 20.035Q-21.150 19.727-20.846 19.556Q-20.542 19.386-20.197 19.386Q-19.691 19.386-19.407 19.609Q-19.123 19.833-19.123 20.329L-19.123 21.983Q-19.123 22.120-18.975 22.156Q-18.826 22.192-18.600 22.192L-18.600 22.472L-20.231 22.472L-20.231 22.192Q-20.002 22.192-19.853 22.158Q-19.704 22.123-19.704 21.983L-19.704 20.343Q-19.704 20.008-19.824 19.808Q-19.944 19.608-20.258 19.608Q-20.528 19.608-20.762 19.744Q-20.996 19.881-21.135 20.115Q-21.273 20.349-21.273 20.623L-21.273 21.983Q-21.273 22.120-21.123 22.156Q-20.972 22.192-20.747 22.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-13.706 22.445L-14.834 19.946Q-14.906 19.799-15.036 19.767Q-15.166 19.734-15.395 19.734L-15.395 19.454L-13.881 19.454L-13.881 19.734Q-14.233 19.734-14.233 19.881Q-14.233 19.926-14.222 19.946L-13.358 21.864L-12.578 20.134Q-12.544 20.066-12.544 19.987Q-12.544 19.874-12.628 19.804Q-12.712 19.734-12.831 19.734L-12.831 19.454L-11.635 19.454L-11.635 19.734Q-11.854 19.734-12.025 19.837Q-12.195 19.939-12.284 20.134L-13.320 22.445Q-13.368 22.540-13.474 22.540L-13.552 22.540Q-13.658 22.540-13.706 22.445\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-11.466 21.744Q-11.466 21.412-11.243 21.185Q-11.019 20.958-10.675 20.830Q-10.332 20.701-9.959 20.649Q-9.587 20.596-9.282 20.596L-9.282 20.343Q-9.282 20.138-9.390 19.958Q-9.498 19.779-9.679 19.676Q-9.860 19.574-10.068 19.574Q-10.475 19.574-10.711 19.666Q-10.622 19.703-10.576 19.787Q-10.530 19.871-10.530 19.973Q-10.530 20.069-10.576 20.148Q-10.622 20.226-10.703 20.271Q-10.783 20.315-10.872 20.315Q-11.022 20.315-11.123 20.218Q-11.224 20.120-11.224 19.973Q-11.224 19.351-10.068 19.351Q-9.857 19.351-9.607 19.415Q-9.358 19.478-9.156 19.597Q-8.954 19.717-8.828 19.902Q-8.701 20.086-8.701 20.329L-8.701 21.905Q-8.701 22.021-8.640 22.117Q-8.578 22.212-8.465 22.212Q-8.356 22.212-8.291 22.118Q-8.226 22.024-8.226 21.905L-8.226 21.457L-7.960 21.457L-7.960 21.905Q-7.960 22.175-8.187 22.340Q-8.414 22.506-8.694 22.506Q-8.903 22.506-9.040 22.352Q-9.176 22.199-9.200 21.983Q-9.347 22.250-9.629 22.395Q-9.911 22.540-10.236 22.540Q-10.513 22.540-10.797 22.465Q-11.080 22.390-11.273 22.211Q-11.466 22.031-11.466 21.744M-10.851 21.744Q-10.851 21.918-10.750 22.048Q-10.650 22.178-10.494 22.248Q-10.339 22.318-10.174 22.318Q-9.956 22.318-9.747 22.221Q-9.539 22.123-9.411 21.942Q-9.282 21.761-9.282 21.535L-9.282 20.807Q-9.607 20.807-9.973 20.898Q-10.339 20.989-10.595 21.201Q-10.851 21.412-10.851 21.744M-5.875 22.472L-7.478 22.472L-7.478 22.192Q-7.252 22.192-7.103 22.158Q-6.955 22.123-6.955 21.983L-6.955 18.364Q-6.955 18.094-7.062 18.032Q-7.170 17.971-7.478 17.971L-7.478 17.690L-6.401 17.615L-6.401 21.983Q-6.401 22.120-6.251 22.156Q-6.100 22.192-5.875 22.192L-5.875 22.472M-4.706 21.638L-4.706 20.134Q-4.706 19.864-4.813 19.803Q-4.921 19.741-5.232 19.741L-5.232 19.461L-4.125 19.386L-4.125 21.618L-4.125 21.638Q-4.125 21.918-4.073 22.062Q-4.022 22.205-3.880 22.262Q-3.738 22.318-3.451 22.318Q-3.198 22.318-2.993 22.178Q-2.788 22.038-2.672 21.812Q-2.556 21.587-2.556 21.337L-2.556 20.134Q-2.556 19.864-2.663 19.803Q-2.771 19.741-3.082 19.741L-3.082 19.461L-1.975 19.386L-1.975 21.799Q-1.975 21.990-1.922 22.072Q-1.869 22.154-1.768 22.173Q-1.667 22.192-1.452 22.192L-1.452 22.472L-2.528 22.540L-2.528 21.976Q-2.638 22.158-2.783 22.281Q-2.928 22.404-3.115 22.472Q-3.301 22.540-3.503 22.540Q-4.706 22.540-4.706 21.638M-0.905 20.937Q-0.905 20.616-0.780 20.327Q-0.655 20.038-0.430 19.815Q-0.204 19.591 0.091 19.471Q0.387 19.351 0.705 19.351Q1.033 19.351 1.295 19.451Q1.556 19.550 1.732 19.732Q1.908 19.915 2.002 20.173Q2.096 20.431 2.096 20.763Q2.096 20.855 2.014 20.876L-0.242 20.876L-0.242 20.937Q-0.242 21.525 0.042 21.908Q0.326 22.291 0.893 22.291Q1.214 22.291 1.483 22.098Q1.751 21.905 1.840 21.590Q1.847 21.549 1.922 21.535L2.014 21.535Q2.096 21.559 2.096 21.631Q2.096 21.638 2.089 21.665Q1.976 22.062 1.606 22.301Q1.235 22.540 0.811 22.540Q0.373 22.540-0.027 22.332Q-0.426 22.123-0.666 21.756Q-0.905 21.389-0.905 20.937M-0.235 20.667L1.580 20.667Q1.580 20.390 1.483 20.138Q1.385 19.885 1.187 19.729Q0.989 19.574 0.705 19.574Q0.428 19.574 0.214 19.732Q0.001 19.891-0.117 20.146Q-0.235 20.401-0.235 20.667M2.684 22.465L2.684 21.402Q2.684 21.378 2.711 21.351Q2.739 21.324 2.763 21.324L2.872 21.324Q2.937 21.324 2.951 21.382Q3.046 21.816 3.292 22.067Q3.538 22.318 3.952 22.318Q4.294 22.318 4.547 22.185Q4.800 22.052 4.800 21.744Q4.800 21.587 4.706 21.472Q4.612 21.358 4.473 21.289Q4.335 21.221 4.167 21.183L3.586 21.084Q3.231 21.016 2.957 20.795Q2.684 20.575 2.684 20.233Q2.684 19.984 2.795 19.809Q2.906 19.635 3.092 19.536Q3.279 19.437 3.494 19.394Q3.709 19.351 3.952 19.351Q4.366 19.351 4.646 19.533L4.861 19.358Q4.871 19.355 4.878 19.353Q4.885 19.351 4.895 19.351L4.947 19.351Q4.974 19.351 4.998 19.375Q5.022 19.399 5.022 19.427L5.022 20.274Q5.022 20.295 4.998 20.322Q4.974 20.349 4.947 20.349L4.834 20.349Q4.807 20.349 4.781 20.324Q4.755 20.298 4.755 20.274Q4.755 20.038 4.649 19.874Q4.543 19.710 4.360 19.628Q4.178 19.546 3.945 19.546Q3.617 19.546 3.361 19.649Q3.104 19.751 3.104 20.028Q3.104 20.223 3.287 20.332Q3.470 20.442 3.699 20.483L4.273 20.589Q4.519 20.637 4.733 20.765Q4.947 20.893 5.083 21.096Q5.220 21.300 5.220 21.549Q5.220 22.062 4.854 22.301Q4.489 22.540 3.952 22.540Q3.456 22.540 3.125 22.246L2.858 22.520Q2.838 22.540 2.810 22.540L2.763 22.540Q2.739 22.540 2.711 22.513Q2.684 22.486 2.684 22.465M6.348 23.702Q6.348 23.668 6.375 23.641Q6.645 23.412 6.794 23.089Q6.943 22.766 6.943 22.410L6.943 22.373Q6.833 22.472 6.669 22.472Q6.488 22.472 6.369 22.352Q6.249 22.233 6.249 22.052Q6.249 21.877 6.369 21.758Q6.488 21.638 6.669 21.638Q6.926 21.638 7.045 21.877Q7.165 22.117 7.165 22.410Q7.165 22.810 6.996 23.181Q6.827 23.552 6.529 23.808Q6.498 23.829 6.471 23.829Q6.430 23.829 6.389 23.788Q6.348 23.747 6.348 23.702\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-53.895 31.829L-55.525 31.829L-55.525 31.549Q-55.296 31.549-55.147 31.514Q-54.999 31.480-54.999 31.340L-54.999 27.994Q-54.999 27.823-55.135 27.782Q-55.272 27.741-55.525 27.741L-55.525 27.461L-54.445 27.386L-54.445 27.792Q-54.223 27.591-53.936 27.488Q-53.648 27.386-53.341 27.386Q-52.914 27.386-52.550 27.599Q-52.186 27.813-51.972 28.177Q-51.758 28.541-51.758 28.961Q-51.758 29.406-51.998 29.770Q-52.237 30.134-52.630 30.337Q-53.023 30.540-53.467 30.540Q-53.734 30.540-53.982 30.440Q-54.229 30.339-54.417 30.158L-54.417 31.340Q-54.417 31.477-54.269 31.513Q-54.120 31.549-53.895 31.549L-53.895 31.829M-54.417 28.141L-54.417 29.751Q-54.284 30.004-54.041 30.161Q-53.799 30.318-53.522 30.318Q-53.194 30.318-52.941 30.117Q-52.688 29.915-52.555 29.597Q-52.421 29.279-52.421 28.961Q-52.421 28.732-52.486 28.503Q-52.551 28.274-52.679 28.076Q-52.808 27.878-53.002 27.758Q-53.197 27.639-53.430 27.639Q-53.724 27.639-53.992 27.768Q-54.260 27.898-54.417 28.141M-49.506 30.472L-51.058 30.472L-51.058 30.192Q-50.832 30.192-50.683 30.158Q-50.535 30.123-50.535 29.983L-50.535 28.134Q-50.535 27.946-50.582 27.862Q-50.630 27.779-50.728 27.760Q-50.825 27.741-51.037 27.741L-51.037 27.461L-49.981 27.386L-49.981 29.983Q-49.981 30.123-49.849 30.158Q-49.718 30.192-49.506 30.192L-49.506 30.472M-50.777 26.165Q-50.777 25.994-50.654 25.875Q-50.531 25.755-50.360 25.755Q-50.193 25.755-50.070 25.875Q-49.947 25.994-49.947 26.165Q-49.947 26.340-50.070 26.463Q-50.193 26.586-50.360 26.586Q-50.531 26.586-50.654 26.463Q-50.777 26.340-50.777 26.165M-48.860 28.961Q-48.860 28.633-48.725 28.332Q-48.590 28.032-48.354 27.811Q-48.118 27.591-47.814 27.471Q-47.510 27.351-47.185 27.351Q-46.679 27.351-46.331 27.454Q-45.982 27.556-45.982 27.932Q-45.982 28.079-46.079 28.180Q-46.177 28.281-46.324 28.281Q-46.478 28.281-46.577 28.182Q-46.676 28.083-46.676 27.932Q-46.676 27.744-46.536 27.652Q-46.737 27.601-47.178 27.601Q-47.534 27.601-47.763 27.797Q-47.992 27.994-48.093 28.303Q-48.193 28.613-48.193 28.961Q-48.193 29.310-48.067 29.616Q-47.940 29.922-47.686 30.106Q-47.431 30.291-47.076 30.291Q-46.853 30.291-46.669 30.207Q-46.484 30.123-46.349 29.968Q-46.214 29.812-46.156 29.604Q-46.143 29.549-46.088 29.549L-45.975 29.549Q-45.944 29.549-45.922 29.573Q-45.900 29.597-45.900 29.631L-45.900 29.652Q-45.985 29.939-46.173 30.137Q-46.361 30.335-46.626 30.438Q-46.891 30.540-47.185 30.540Q-47.616 30.540-48.004 30.334Q-48.392 30.127-48.626 29.764Q-48.860 29.402-48.860 28.961\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-43.921 30.472L-45.504 30.472L-45.504 30.192Q-45.275 30.192-45.126 30.158Q-44.978 30.123-44.978 29.983L-44.978 26.364Q-44.978 26.094-45.085 26.032Q-45.193 25.971-45.504 25.971L-45.504 25.690L-44.424 25.615L-44.424 28.903L-43.439 28.134Q-43.234 27.997-43.234 27.847Q-43.234 27.803-43.275 27.768Q-43.316 27.734-43.361 27.734L-43.361 27.454L-41.997 27.454L-41.997 27.734Q-42.486 27.734-43.005 28.134L-43.562 28.568L-42.585 29.792Q-42.383 30.038-42.250 30.115Q-42.117 30.192-41.830 30.192L-41.830 30.472L-43.262 30.472L-43.262 30.192Q-43.074 30.192-43.074 30.079Q-43.074 29.983-43.228 29.792L-43.962 28.883L-44.444 29.262L-44.444 29.983Q-44.444 30.120-44.296 30.156Q-44.147 30.192-43.921 30.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-38.096 29.631L-38.096 27.734L-38.735 27.734L-38.735 27.512Q-38.417 27.512-38.200 27.302Q-37.983 27.092-37.883 26.782Q-37.782 26.473-37.782 26.165L-37.515 26.165L-37.515 27.454L-36.438 27.454L-36.438 27.734L-37.515 27.734L-37.515 29.618Q-37.515 29.894-37.411 30.093Q-37.307 30.291-37.047 30.291Q-36.890 30.291-36.784 30.187Q-36.678 30.082-36.628 29.929Q-36.579 29.775-36.579 29.618L-36.579 29.204L-36.312 29.204L-36.312 29.631Q-36.312 29.857-36.411 30.067Q-36.510 30.277-36.695 30.409Q-36.879 30.540-37.108 30.540Q-37.546 30.540-37.821 30.303Q-38.096 30.065-38.096 29.631M-33.820 30.472L-35.454 30.472L-35.454 30.192Q-35.225 30.192-35.076 30.158Q-34.928 30.123-34.928 29.983L-34.928 26.364Q-34.928 26.094-35.035 26.032Q-35.143 25.971-35.454 25.971L-35.454 25.690L-34.374 25.615L-34.374 28.001Q-34.268 27.816-34.090 27.674Q-33.913 27.533-33.704 27.459Q-33.496 27.386-33.270 27.386Q-32.764 27.386-32.480 27.609Q-32.197 27.833-32.197 28.329L-32.197 29.983Q-32.197 30.120-32.048 30.156Q-31.899 30.192-31.674 30.192L-31.674 30.472L-33.304 30.472L-33.304 30.192Q-33.075 30.192-32.927 30.158Q-32.778 30.123-32.778 29.983L-32.778 28.343Q-32.778 28.008-32.897 27.808Q-33.017 27.608-33.332 27.608Q-33.602 27.608-33.836 27.744Q-34.070 27.881-34.208 28.115Q-34.347 28.349-34.347 28.623L-34.347 29.983Q-34.347 30.120-34.196 30.156Q-34.046 30.192-33.820 30.192L-33.820 30.472M-31.127 28.937Q-31.127 28.616-31.002 28.327Q-30.877 28.038-30.652 27.815Q-30.426 27.591-30.131 27.471Q-29.835 27.351-29.517 27.351Q-29.189 27.351-28.927 27.451Q-28.666 27.550-28.490 27.732Q-28.314 27.915-28.220 28.173Q-28.126 28.431-28.126 28.763Q-28.126 28.855-28.208 28.876L-30.464 28.876L-30.464 28.937Q-30.464 29.525-30.180 29.908Q-29.896 30.291-29.329 30.291Q-29.008 30.291-28.739 30.098Q-28.471 29.905-28.382 29.590Q-28.375 29.549-28.300 29.535L-28.208 29.535Q-28.126 29.559-28.126 29.631Q-28.126 29.638-28.133 29.665Q-28.246 30.062-28.616 30.301Q-28.987 30.540-29.411 30.540Q-29.849 30.540-30.249 30.332Q-30.648 30.123-30.888 29.756Q-31.127 29.389-31.127 28.937M-30.457 28.667L-28.642 28.667Q-28.642 28.390-28.739 28.138Q-28.837 27.885-29.035 27.729Q-29.233 27.574-29.517 27.574Q-29.794 27.574-30.008 27.732Q-30.221 27.891-30.339 28.146Q-30.457 28.401-30.457 28.667\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(5.97 49.115)\">\u003Cpath d=\"M-23.163 30.472L-24.766 30.472L-24.766 30.192Q-24.540 30.192-24.391 30.158Q-24.243 30.123-24.243 29.983L-24.243 26.364Q-24.243 26.094-24.350 26.032Q-24.458 25.971-24.766 25.971L-24.766 25.690L-23.689 25.615L-23.689 29.983Q-23.689 30.120-23.539 30.156Q-23.388 30.192-23.163 30.192L-23.163 30.472M-22.510 29.744Q-22.510 29.412-22.286 29.185Q-22.062 28.958-21.718 28.830Q-21.375 28.701-21.002 28.649Q-20.630 28.596-20.326 28.596L-20.326 28.343Q-20.326 28.138-20.433 27.958Q-20.541 27.779-20.722 27.676Q-20.903 27.574-21.112 27.574Q-21.519 27.574-21.754 27.666Q-21.666 27.703-21.619 27.787Q-21.573 27.871-21.573 27.973Q-21.573 28.069-21.619 28.148Q-21.666 28.226-21.746 28.271Q-21.826 28.315-21.915 28.315Q-22.065 28.315-22.166 28.218Q-22.267 28.120-22.267 27.973Q-22.267 27.351-21.112 27.351Q-20.900 27.351-20.650 27.415Q-20.401 27.478-20.199 27.597Q-19.998 27.717-19.871 27.902Q-19.745 28.086-19.745 28.329L-19.745 29.905Q-19.745 30.021-19.683 30.117Q-19.622 30.212-19.509 30.212Q-19.399 30.212-19.334 30.118Q-19.270 30.024-19.270 29.905L-19.270 29.457L-19.003 29.457L-19.003 29.905Q-19.003 30.175-19.230 30.340Q-19.457 30.506-19.738 30.506Q-19.946 30.506-20.083 30.352Q-20.220 30.199-20.244 29.983Q-20.391 30.250-20.673 30.395Q-20.955 30.540-21.279 30.540Q-21.556 30.540-21.840 30.465Q-22.124 30.390-22.317 30.211Q-22.510 30.031-22.510 29.744M-21.895 29.744Q-21.895 29.918-21.794 30.048Q-21.693 30.178-21.537 30.248Q-21.382 30.318-21.218 30.318Q-20.999 30.318-20.791 30.221Q-20.582 30.123-20.454 29.942Q-20.326 29.761-20.326 29.535L-20.326 28.807Q-20.650 28.807-21.016 28.898Q-21.382 28.989-21.638 29.201Q-21.895 29.412-21.895 29.744M-16.836 30.472L-18.572 30.472L-18.572 30.192Q-18.343 30.192-18.195 30.158Q-18.046 30.123-18.046 29.983L-18.046 28.134Q-18.046 27.864-18.154 27.803Q-18.261 27.741-18.572 27.741L-18.572 27.461L-17.543 27.386L-17.543 28.093Q-17.414 27.785-17.171 27.586Q-16.928 27.386-16.610 27.386Q-16.392 27.386-16.221 27.510Q-16.050 27.635-16.050 27.847Q-16.050 27.984-16.149 28.083Q-16.248 28.182-16.381 28.182Q-16.518 28.182-16.617 28.083Q-16.716 27.984-16.716 27.847Q-16.716 27.707-16.617 27.608Q-16.908 27.608-17.108 27.804Q-17.308 28.001-17.400 28.295Q-17.492 28.589-17.492 28.869L-17.492 29.983Q-17.492 30.192-16.836 30.192L-16.836 30.472M-15.506 31.005Q-15.506 30.759-15.310 30.575Q-15.113 30.390-14.857 30.311Q-14.994 30.199-15.065 30.038Q-15.137 29.877-15.137 29.696Q-15.137 29.375-14.925 29.129Q-15.260 28.831-15.260 28.421Q-15.260 27.960-14.871 27.673Q-14.481 27.386-14.002 27.386Q-13.531 27.386-13.196 27.632Q-13.021 27.478-12.811 27.396Q-12.601 27.314-12.372 27.314Q-12.208 27.314-12.087 27.421Q-11.965 27.529-11.965 27.693Q-11.965 27.789-12.037 27.861Q-12.109 27.932-12.201 27.932Q-12.300 27.932-12.370 27.859Q-12.440 27.785-12.440 27.686Q-12.440 27.632-12.427 27.601L-12.420 27.587Q-12.413 27.567-12.405 27.556Q-12.396 27.546-12.393 27.539Q-12.748 27.539-13.035 27.762Q-12.748 28.055-12.748 28.421Q-12.748 28.736-12.933 28.968Q-13.117 29.201-13.406 29.329Q-13.695 29.457-14.002 29.457Q-14.204 29.457-14.395 29.407Q-14.587 29.358-14.765 29.248Q-14.857 29.375-14.857 29.518Q-14.857 29.700-14.729 29.835Q-14.601 29.970-14.416 29.970L-13.784 29.970Q-13.336 29.970-12.967 30.041Q-12.598 30.113-12.338 30.342Q-12.078 30.571-12.078 31.005Q-12.078 31.326-12.374 31.528Q-12.669 31.730-13.073 31.819Q-13.476 31.908-13.791 31.908Q-14.108 31.908-14.512 31.819Q-14.915 31.730-15.211 31.528Q-15.506 31.326-15.506 31.005M-15.052 31.005Q-15.052 31.234-14.833 31.383Q-14.614 31.532-14.322 31.600Q-14.030 31.668-13.791 31.668Q-13.626 31.668-13.418 31.632Q-13.209 31.597-13.003 31.516Q-12.796 31.436-12.664 31.308Q-12.533 31.180-12.533 31.005Q-12.533 30.653-12.914 30.559Q-13.295 30.465-13.797 30.465L-14.416 30.465Q-14.655 30.465-14.853 30.616Q-15.052 30.766-15.052 31.005M-14.002 29.218Q-13.336 29.218-13.336 28.421Q-13.336 27.621-14.002 27.621Q-14.672 27.621-14.672 28.421Q-14.672 29.218-14.002 29.218M-11.524 28.937Q-11.524 28.616-11.400 28.327Q-11.275 28.038-11.049 27.815Q-10.824 27.591-10.528 27.471Q-10.232 27.351-9.915 27.351Q-9.586 27.351-9.325 27.451Q-9.063 27.550-8.887 27.732Q-8.711 27.915-8.617 28.173Q-8.523 28.431-8.523 28.763Q-8.523 28.855-8.605 28.876L-10.861 28.876L-10.861 28.937Q-10.861 29.525-10.578 29.908Q-10.294 30.291-9.727 30.291Q-9.405 30.291-9.137 30.098Q-8.869 29.905-8.780 29.590Q-8.773 29.549-8.698 29.535L-8.605 29.535Q-8.523 29.559-8.523 29.631Q-8.523 29.638-8.530 29.665Q-8.643 30.062-9.014 30.301Q-9.385 30.540-9.809 30.540Q-10.246 30.540-10.646 30.332Q-11.046 30.123-11.285 29.756Q-11.524 29.389-11.524 28.937M-10.854 28.667L-9.040 28.667Q-9.040 28.390-9.137 28.138Q-9.234 27.885-9.433 27.729Q-9.631 27.574-9.915 27.574Q-10.191 27.574-10.405 27.732Q-10.619 27.891-10.737 28.146Q-10.854 28.401-10.854 28.667M-7.936 30.465L-7.936 29.402Q-7.936 29.378-7.908 29.351Q-7.881 29.324-7.857 29.324L-7.748 29.324Q-7.683 29.324-7.669 29.382Q-7.573 29.816-7.327 30.067Q-7.081 30.318-6.667 30.318Q-6.326 30.318-6.073 30.185Q-5.820 30.052-5.820 29.744Q-5.820 29.587-5.914 29.472Q-6.008 29.358-6.146 29.289Q-6.285 29.221-6.452 29.183L-7.033 29.084Q-7.389 29.016-7.662 28.795Q-7.936 28.575-7.936 28.233Q-7.936 27.984-7.824 27.809Q-7.713 27.635-7.527 27.536Q-7.341 27.437-7.125 27.394Q-6.910 27.351-6.667 27.351Q-6.254 27.351-5.974 27.533L-5.758 27.358Q-5.748 27.355-5.741 27.353Q-5.734 27.351-5.724 27.351L-5.673 27.351Q-5.645 27.351-5.622 27.375Q-5.598 27.399-5.598 27.427L-5.598 28.274Q-5.598 28.295-5.622 28.322Q-5.645 28.349-5.673 28.349L-5.786 28.349Q-5.813 28.349-5.839 28.324Q-5.864 28.298-5.864 28.274Q-5.864 28.038-5.970 27.874Q-6.076 27.710-6.259 27.628Q-6.442 27.546-6.674 27.546Q-7.002 27.546-7.259 27.649Q-7.515 27.751-7.515 28.028Q-7.515 28.223-7.332 28.332Q-7.149 28.442-6.920 28.483L-6.346 28.589Q-6.100 28.637-5.886 28.765Q-5.673 28.893-5.536 29.096Q-5.399 29.300-5.399 29.549Q-5.399 30.062-5.765 30.301Q-6.131 30.540-6.667 30.540Q-7.163 30.540-7.495 30.246L-7.761 30.520Q-7.782 30.540-7.809 30.540L-7.857 30.540Q-7.881 30.540-7.908 30.513Q-7.936 30.486-7.936 30.465M-4.244 29.631L-4.244 27.734L-4.883 27.734L-4.883 27.512Q-4.565 27.512-4.348 27.302Q-4.131 27.092-4.030 26.782Q-3.930 26.473-3.930 26.165L-3.663 26.165L-3.663 27.454L-2.586 27.454L-2.586 27.734L-3.663 27.734L-3.663 29.618Q-3.663 29.894-3.559 30.093Q-3.455 30.291-3.195 30.291Q-3.038 30.291-2.932 30.187Q-2.826 30.082-2.776 29.929Q-2.727 29.775-2.727 29.618L-2.727 29.204L-2.460 29.204L-2.460 29.631Q-2.460 29.857-2.559 30.067Q-2.658 30.277-2.843 30.409Q-3.027 30.540-3.256 30.540Q-3.694 30.540-3.969 30.303Q-4.244 30.065-4.244 29.631\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(202.54 -93.159)\">\u003Cpath d=\"M-63.217 30.472L-65.304 30.472L-65.304 30.156Q-64.997 30.156-64.805 30.103Q-64.614 30.050-64.614 29.861L-64.614 27.413Q-64.614 27.172-64.685 27.064Q-64.755 26.956-64.889 26.932Q-65.023 26.908-65.304 26.908L-65.304 26.592L-63.964 26.495L-63.964 27.330Q-63.766 26.952-63.406 26.723Q-63.045 26.495-62.624 26.495Q-61.578 26.495-61.393 27.304Q-61.191 26.934-60.833 26.715Q-60.475 26.495-60.057 26.495Q-59.433 26.495-59.108 26.789Q-58.783 27.084-58.783 27.708L-58.783 29.861Q-58.783 30.050-58.589 30.103Q-58.396 30.156-58.088 30.156L-58.088 30.472L-60.176 30.472L-60.176 30.156Q-59.868 30.156-59.675 30.103Q-59.481 30.050-59.481 29.861L-59.481 27.743Q-59.481 27.312-59.609 27.033Q-59.736 26.754-60.123 26.754Q-60.466 26.754-60.749 26.943Q-61.033 27.132-61.189 27.444Q-61.345 27.756-61.345 28.103L-61.345 29.861Q-61.345 30.050-61.154 30.103Q-60.962 30.156-60.655 30.156L-60.655 30.472L-62.742 30.472L-62.742 30.156Q-62.430 30.156-62.239 30.103Q-62.048 30.050-62.048 29.861L-62.048 27.743Q-62.048 27.484-62.092 27.262Q-62.136 27.040-62.281 26.897Q-62.426 26.754-62.685 26.754Q-63.221 26.754-63.566 27.161Q-63.911 27.567-63.911 28.103L-63.911 29.861Q-63.911 30.050-63.718 30.103Q-63.524 30.156-63.217 30.156L-63.217 30.472M-57.640 28.565Q-57.640 27.998-57.368 27.510Q-57.095 27.022-56.625 26.730Q-56.155 26.438-55.588 26.438Q-55.166 26.438-54.790 26.607Q-54.415 26.776-54.138 27.068Q-53.861 27.361-53.703 27.756Q-53.544 28.152-53.544 28.565Q-53.544 29.114-53.823 29.576Q-54.103 30.037-54.571 30.305Q-55.039 30.573-55.588 30.573Q-56.142 30.573-56.612 30.305Q-57.082 30.037-57.361 29.576Q-57.640 29.114-57.640 28.565M-55.588 30.283Q-55.091 30.283-54.814 30.022Q-54.538 29.760-54.445 29.356Q-54.353 28.951-54.353 28.455Q-54.353 27.980-54.452 27.591Q-54.551 27.202-54.823 26.952Q-55.096 26.701-55.588 26.701Q-56.300 26.701-56.563 27.196Q-56.827 27.690-56.827 28.455Q-56.827 29.255-56.572 29.769Q-56.317 30.283-55.588 30.283\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.54 -93.159)\">\u003Cpath d=\"M-50.763 30.573Q-51.308 30.573-51.752 30.290Q-52.196 30.006-52.450 29.534Q-52.705 29.061-52.705 28.530Q-52.705 27.976-52.429 27.510Q-52.152 27.044-51.679 26.770Q-51.207 26.495-50.662 26.495Q-50.328 26.495-50.025 26.627Q-49.721 26.759-49.502 26.996L-49.502 25.146Q-49.502 24.904-49.572 24.796Q-49.642 24.689-49.776 24.665Q-49.910 24.640-50.196 24.640L-50.196 24.324L-48.829 24.227L-48.829 29.655Q-48.829 29.892-48.759 30Q-48.689 30.107-48.553 30.131Q-48.416 30.156-48.135 30.156L-48.135 30.472L-49.528 30.573L-49.528 30.024Q-49.779 30.287-50.097 30.430Q-50.416 30.573-50.763 30.573M-50.701 30.309Q-50.332 30.309-50.023 30.098Q-49.713 29.888-49.528 29.545L-49.528 27.413Q-49.700 27.110-49.981 26.932Q-50.262 26.754-50.600 26.754Q-51.290 26.754-51.594 27.275Q-51.897 27.796-51.897 28.538Q-51.897 29.259-51.627 29.784Q-51.356 30.309-50.701 30.309M-45.564 30.573Q-46.122 30.573-46.595 30.290Q-47.067 30.006-47.342 29.529Q-47.617 29.053-47.617 28.499Q-47.617 28.103-47.474 27.728Q-47.331 27.352-47.074 27.064Q-46.817 26.776-46.459 26.607Q-46.100 26.438-45.696 26.438Q-45.151 26.438-44.780 26.675Q-44.409 26.912-44.222 27.330Q-44.035 27.747-44.035 28.284Q-44.035 28.336-44.059 28.374Q-44.083 28.411-44.132 28.411L-46.804 28.411L-46.804 28.490Q-46.804 29.237-46.492 29.760Q-46.180 30.283-45.481 30.283Q-45.076 30.283-44.756 30.026Q-44.435 29.769-44.312 29.365Q-44.294 29.285-44.211 29.285L-44.132 29.285Q-44.092 29.285-44.064 29.316Q-44.035 29.347-44.035 29.391L-44.035 29.426Q-44.140 29.769-44.362 30.028Q-44.584 30.287-44.898 30.430Q-45.213 30.573-45.564 30.573M-46.795 28.160L-44.681 28.160Q-44.681 27.892-44.734 27.646Q-44.786 27.400-44.907 27.178Q-45.028 26.956-45.226 26.829Q-45.424 26.701-45.696 26.701Q-46.039 26.701-46.292 26.926Q-46.544 27.150-46.669 27.488Q-46.795 27.826-46.795 28.160M-41.394 30.472L-43.455 30.472L-43.455 30.156Q-43.147 30.156-42.956 30.103Q-42.765 30.050-42.765 29.861L-42.765 25.146Q-42.765 24.904-42.835 24.796Q-42.906 24.689-43.040 24.665Q-43.174 24.640-43.455 24.640L-43.455 24.324L-42.088 24.227L-42.088 29.861Q-42.088 30.050-41.895 30.103Q-41.701 30.156-41.394 30.156L-41.394 30.472M-38.630 28.824L-41.095 28.824L-41.095 28.231L-38.630 28.231L-38.630 28.824M-36.885 30.472L-37.175 30.472L-37.175 25.146Q-37.175 24.904-37.245 24.796Q-37.316 24.689-37.450 24.665Q-37.584 24.640-37.869 24.640L-37.869 24.324L-36.498 24.227L-36.498 27.027Q-36.252 26.776-35.918 26.636Q-35.584 26.495-35.233 26.495Q-34.833 26.495-34.470 26.658Q-34.108 26.820-33.851 27.099Q-33.594 27.378-33.444 27.756Q-33.295 28.134-33.295 28.530Q-33.295 29.092-33.572 29.558Q-33.848 30.024-34.323 30.298Q-34.798 30.573-35.347 30.573Q-35.712 30.573-36.033 30.404Q-36.353 30.235-36.573 29.940L-36.885 30.472M-36.472 27.440L-36.472 29.571Q-36.314 29.905-36.030 30.107Q-35.747 30.309-35.404 30.309Q-34.714 30.309-34.411 29.789Q-34.108 29.268-34.108 28.530Q-34.108 27.805-34.374 27.279Q-34.639 26.754-35.303 26.754Q-35.663 26.754-35.978 26.939Q-36.292 27.123-36.472 27.440M-32.614 29.562Q-32.614 29.022-32.181 28.688Q-31.748 28.354-31.141 28.215Q-30.535 28.077-30.003 28.077L-30.003 27.743Q-30.003 27.484-30.122 27.240Q-30.241 26.996-30.449 26.849Q-30.658 26.701-30.930 26.701Q-31.493 26.701-31.805 26.899Q-31.656 26.926-31.563 27.044Q-31.471 27.163-31.471 27.321Q-31.471 27.497-31.596 27.627Q-31.721 27.756-31.902 27.756Q-32.091 27.756-32.218 27.629Q-32.346 27.501-32.346 27.321Q-32.346 26.851-31.906 26.644Q-31.467 26.438-30.930 26.438Q-30.640 26.438-30.353 26.526Q-30.065 26.614-29.832 26.774Q-29.599 26.934-29.450 27.174Q-29.300 27.413-29.300 27.708L-29.300 29.743Q-29.300 29.896-29.225 30.035Q-29.151 30.173-29.006 30.173Q-28.852 30.173-28.779 30.037Q-28.707 29.901-28.707 29.743L-28.707 29.167L-28.421 29.167L-28.421 29.743Q-28.421 30.076-28.669 30.301Q-28.918 30.525-29.247 30.525Q-29.507 30.525-29.689 30.331Q-29.871 30.138-29.915 29.861Q-30.082 30.182-30.416 30.378Q-30.750 30.573-31.119 30.573Q-31.673 30.573-32.143 30.325Q-32.614 30.076-32.614 29.562M-31.858 29.562Q-31.858 29.874-31.616 30.092Q-31.374 30.309-31.058 30.309Q-30.623 30.309-30.313 30Q-30.003 29.690-30.003 29.259L-30.003 28.332Q-30.421 28.332-30.847 28.459Q-31.273 28.587-31.565 28.864Q-31.858 29.140-31.858 29.562M-28.056 30.490L-28.056 29.048Q-28.056 29.017-28.028 28.993Q-27.999 28.969-27.969 28.969L-27.859 28.969Q-27.824 28.969-27.802 28.991Q-27.780 29.013-27.771 29.048Q-27.512 30.309-26.545 30.309Q-26.118 30.309-25.824 30.125Q-25.530 29.940-25.530 29.536Q-25.530 29.242-25.760 29.046Q-25.991 28.850-26.303 28.789L-26.905 28.670Q-27.371 28.582-27.714 28.299Q-28.056 28.015-28.056 27.576Q-28.056 26.983-27.619 26.710Q-27.182 26.438-26.545 26.438Q-26.066 26.438-25.719 26.684L-25.468 26.460Q-25.411 26.438-25.411 26.438L-25.358 26.438Q-25.332 26.438-25.299 26.464Q-25.266 26.491-25.266 26.521L-25.266 27.681Q-25.266 27.712-25.301 27.739Q-25.336 27.765-25.358 27.765L-25.468 27.765Q-25.490 27.765-25.523 27.736Q-25.556 27.708-25.556 27.681Q-25.556 27.216-25.822 26.945Q-26.088 26.675-26.554 26.675Q-26.958 26.675-27.261 26.820Q-27.564 26.965-27.564 27.321Q-27.564 27.567-27.347 27.721Q-27.129 27.875-26.852 27.932L-26.224 28.059Q-25.908 28.121-25.637 28.288Q-25.367 28.455-25.200 28.721Q-25.033 28.987-25.033 29.303Q-25.033 29.945-25.459 30.259Q-25.886 30.573-26.545 30.573Q-26.817 30.573-27.083 30.479Q-27.349 30.384-27.529 30.195L-27.850 30.534Q-27.867 30.573-27.916 30.573L-27.969 30.573Q-27.991 30.573-28.023 30.545Q-28.056 30.516-28.056 30.490M-22.409 30.573Q-22.968 30.573-23.440 30.290Q-23.912 30.006-24.187 29.529Q-24.462 29.053-24.462 28.499Q-24.462 28.103-24.319 27.728Q-24.176 27.352-23.919 27.064Q-23.662 26.776-23.304 26.607Q-22.946 26.438-22.541 26.438Q-21.996 26.438-21.625 26.675Q-21.254 26.912-21.067 27.330Q-20.880 27.747-20.880 28.284Q-20.880 28.336-20.904 28.374Q-20.929 28.411-20.977 28.411L-23.649 28.411L-23.649 28.490Q-23.649 29.237-23.337 29.760Q-23.025 30.283-22.326 30.283Q-21.922 30.283-21.601 30.026Q-21.280 29.769-21.157 29.365Q-21.139 29.285-21.056 29.285L-20.977 29.285Q-20.937 29.285-20.909 29.316Q-20.880 29.347-20.880 29.391L-20.880 29.426Q-20.986 29.769-21.208 30.028Q-21.430 30.287-21.744 30.430Q-22.058 30.573-22.409 30.573M-23.640 28.160L-21.526 28.160Q-21.526 27.892-21.579 27.646Q-21.632 27.400-21.753 27.178Q-21.873 26.956-22.071 26.829Q-22.269 26.701-22.541 26.701Q-22.884 26.701-23.137 26.926Q-23.389 27.150-23.515 27.488Q-23.640 27.826-23.640 28.160M-18.358 30.573Q-18.903 30.573-19.346 30.290Q-19.790 30.006-20.045 29.534Q-20.300 29.061-20.300 28.530Q-20.300 27.976-20.023 27.510Q-19.746 27.044-19.274 26.770Q-18.802 26.495-18.257 26.495Q-17.923 26.495-17.619 26.627Q-17.316 26.759-17.096 26.996L-17.096 25.146Q-17.096 24.904-17.167 24.796Q-17.237 24.689-17.371 24.665Q-17.505 24.640-17.791 24.640L-17.791 24.324L-16.424 24.227L-16.424 29.655Q-16.424 29.892-16.354 30Q-16.284 30.107-16.147 30.131Q-16.011 30.156-15.730 30.156L-15.730 30.472L-17.123 30.573L-17.123 30.024Q-17.373 30.287-17.692 30.430Q-18.011 30.573-18.358 30.573M-18.296 30.309Q-17.927 30.309-17.617 30.098Q-17.307 29.888-17.123 29.545L-17.123 27.413Q-17.294 27.110-17.575 26.932Q-17.857 26.754-18.195 26.754Q-18.885 26.754-19.188 27.275Q-19.492 27.796-19.492 28.538Q-19.492 29.259-19.221 29.784Q-18.951 30.309-18.296 30.309\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M98-3.671c0-5.5-4.459-9.959-9.959-9.959S78.083-9.17 78.083-3.67s4.458 9.958 9.958 9.958S98 1.83 98-3.67Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(149.443 -31.752)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.391 30.472L-60.391 30.192Q-59.423 30.192-59.423 29.983L-59.423 26.364Q-59.816 26.552-60.438 26.552L-60.438 26.271Q-60.021 26.271-59.657 26.170Q-59.293 26.070-59.037 25.824L-58.911 25.824Q-58.846 25.841-58.829 25.909L-58.829 29.983Q-58.829 30.192-57.861 30.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M160.596-37.814c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.459 9.959-9.958Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(212.04 -65.895)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-57.861 30.472L-60.746 30.472L-60.746 30.270Q-60.746 30.240-60.719 30.212L-59.471 28.995Q-59.399 28.920-59.357 28.878Q-59.314 28.835-59.235 28.756Q-58.822 28.343-58.591 27.985Q-58.360 27.628-58.360 27.204Q-58.360 26.972-58.439 26.769Q-58.518 26.565-58.659 26.415Q-58.801 26.264-58.996 26.184Q-59.191 26.104-59.423 26.104Q-59.734 26.104-59.992 26.263Q-60.250 26.422-60.380 26.699L-60.360 26.699Q-60.192 26.699-60.085 26.810Q-59.977 26.921-59.977 27.085Q-59.977 27.242-60.086 27.355Q-60.196 27.468-60.360 27.468Q-60.520 27.468-60.633 27.355Q-60.746 27.242-60.746 27.085Q-60.746 26.709-60.538 26.422Q-60.329 26.135-59.994 25.979Q-59.659 25.824-59.304 25.824Q-58.880 25.824-58.500 25.982Q-58.121 26.141-57.887 26.458Q-57.653 26.774-57.653 27.204Q-57.653 27.515-57.793 27.784Q-57.933 28.052-58.138 28.257Q-58.343 28.462-58.706 28.744Q-59.068 29.026-59.177 29.122L-60.032 29.850L-59.389 29.850Q-59.126 29.850-58.837 29.848Q-58.548 29.847-58.330 29.838Q-58.111 29.829-58.094 29.812Q-58.032 29.747-57.995 29.580Q-57.957 29.412-57.919 29.170L-57.653 29.170\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M160.596 30.472c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(212.04 2.392)\">\u003Cpath d=\"M-65.114 30.534L-65.114 28.961Q-65.114 28.934-65.089 28.908Q-65.063 28.883-65.036 28.883L-64.923 28.883Q-64.895 28.883-64.872 28.910Q-64.848 28.937-64.848 28.961Q-64.848 29.306-64.716 29.570Q-64.584 29.833-64.355 30.002Q-64.126 30.171-63.824 30.252Q-63.521 30.332-63.180 30.332Q-62.913 30.332-62.677 30.204Q-62.441 30.076-62.296 29.853Q-62.151 29.631-62.151 29.365Q-62.151 29.142-62.257 28.946Q-62.363 28.749-62.544 28.614Q-62.725 28.479-62.951 28.428L-63.979 28.196Q-64.290 28.124-64.550 27.938Q-64.810 27.751-64.962 27.480Q-65.114 27.208-65.114 26.893Q-65.114 26.507-64.901 26.200Q-64.687 25.892-64.340 25.721Q-63.993 25.550-63.614 25.550Q-63.385 25.550-63.156 25.603Q-62.927 25.656-62.728 25.764Q-62.530 25.871-62.376 26.035L-62.082 25.595Q-62.059 25.550-62.018 25.550L-61.970 25.550Q-61.939 25.550-61.917 25.576Q-61.895 25.601-61.895 25.629L-61.895 27.204Q-61.895 27.225-61.918 27.252Q-61.942 27.280-61.970 27.280L-62.082 27.280Q-62.144 27.280-62.158 27.204Q-62.199 26.791-62.380 26.471Q-62.561 26.152-62.872 25.977Q-63.183 25.803-63.614 25.803Q-63.863 25.803-64.103 25.914Q-64.342 26.025-64.492 26.223Q-64.643 26.422-64.643 26.685Q-64.643 26.897-64.535 27.078Q-64.427 27.259-64.251 27.379Q-64.075 27.498-63.867 27.539L-62.838 27.768Q-62.520 27.840-62.253 28.045Q-61.987 28.250-61.835 28.544Q-61.683 28.838-61.683 29.170Q-61.683 29.563-61.888 29.898Q-62.093 30.233-62.438 30.422Q-62.783 30.612-63.180 30.612Q-63.600 30.612-63.979 30.499Q-64.359 30.387-64.629 30.137L-64.923 30.571Q-64.950 30.612-64.988 30.612L-65.036 30.612Q-65.063 30.612-65.089 30.587Q-65.114 30.561-65.114 30.534M-60.391 29.925Q-60.271 30.082-60.080 30.181Q-59.888 30.281-59.673 30.320Q-59.457 30.359-59.235 30.359Q-58.938 30.359-58.743 30.204Q-58.548 30.048-58.458 29.794Q-58.367 29.539-58.367 29.255Q-58.367 28.961-58.459 28.710Q-58.552 28.459-58.750 28.303Q-58.948 28.148-59.242 28.148L-59.758 28.148Q-59.786 28.148-59.811 28.122Q-59.837 28.097-59.837 28.073L-59.837 28.001Q-59.837 27.970-59.811 27.948Q-59.786 27.926-59.758 27.926L-59.317 27.895Q-58.955 27.895-58.735 27.538Q-58.514 27.180-58.514 26.791Q-58.514 26.463-58.709 26.259Q-58.904 26.056-59.235 26.056Q-59.522 26.056-59.775 26.140Q-60.028 26.223-60.192 26.411Q-60.045 26.411-59.945 26.526Q-59.844 26.640-59.844 26.791Q-59.844 26.941-59.950 27.051Q-60.056 27.160-60.213 27.160Q-60.374 27.160-60.483 27.051Q-60.592 26.941-60.592 26.791Q-60.592 26.466-60.384 26.247Q-60.175 26.029-59.859 25.926Q-59.543 25.824-59.235 25.824Q-58.917 25.824-58.589 25.928Q-58.261 26.032-58.034 26.254Q-57.807 26.476-57.807 26.791Q-57.807 27.225-58.094 27.550Q-58.381 27.874-58.815 28.021Q-58.504 28.086-58.224 28.252Q-57.943 28.418-57.766 28.676Q-57.588 28.934-57.588 29.255Q-57.588 29.665-57.832 29.975Q-58.077 30.284-58.458 30.448Q-58.839 30.612-59.235 30.612Q-59.604 30.612-59.962 30.499Q-60.319 30.387-60.563 30.137Q-60.808 29.888-60.808 29.518Q-60.808 29.347-60.691 29.235Q-60.575 29.122-60.404 29.122Q-60.295 29.122-60.204 29.173Q-60.114 29.224-60.059 29.317Q-60.004 29.409-60.004 29.518Q-60.004 29.686-60.117 29.805Q-60.230 29.925-60.391 29.925\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M207.543-46.35h22.762v-17.072h-22.762Z\"\u002F>\u003Cg transform=\"translate(279.465 -83.103)\">\u003Cpath d=\"M-60.121 29.665L-64.954 29.665Q-65.022 29.655-65.068 29.609Q-65.114 29.563-65.114 29.491Q-65.114 29.426-65.068 29.380Q-65.022 29.334-64.954 29.324L-60.121 29.324Q-60.052 29.334-60.006 29.380Q-59.960 29.426-59.960 29.491Q-59.960 29.563-60.006 29.609Q-60.052 29.655-60.121 29.665M-60.121 28.127L-64.954 28.127Q-65.022 28.117-65.068 28.071Q-65.114 28.025-65.114 27.953Q-65.114 27.809-64.954 27.785L-60.121 27.785Q-59.960 27.809-59.960 27.953Q-59.960 28.025-60.006 28.071Q-60.052 28.117-60.121 28.127M-57.475 30.612Q-58.111 30.612-58.475 30.267Q-58.839 29.922-58.974 29.397Q-59.109 28.872-59.109 28.247Q-59.109 27.222-58.753 26.523Q-58.398 25.824-57.475 25.824Q-56.549 25.824-56.197 26.523Q-55.845 27.222-55.845 28.247Q-55.845 28.872-55.980 29.397Q-56.115 29.922-56.477 30.267Q-56.839 30.612-57.475 30.612M-57.475 30.387Q-57.038 30.387-56.824 30.012Q-56.610 29.638-56.561 29.171Q-56.511 28.705-56.511 28.127Q-56.511 27.574-56.561 27.146Q-56.610 26.719-56.822 26.384Q-57.034 26.049-57.475 26.049Q-57.817 26.049-58.020 26.256Q-58.224 26.463-58.311 26.775Q-58.398 27.088-58.420 27.404Q-58.442 27.721-58.442 28.127Q-58.442 28.544-58.420 28.886Q-58.398 29.228-58.309 29.576Q-58.220 29.925-58.015 30.156Q-57.810 30.387-57.475 30.387\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M207.543-15.052h22.762v-17.072h-22.762Z\"\u002F>\u003Cg transform=\"translate(279.465 -51.804)\">\u003Cpath d=\"M-60.121 29.665L-64.954 29.665Q-65.022 29.655-65.068 29.609Q-65.114 29.563-65.114 29.491Q-65.114 29.426-65.068 29.380Q-65.022 29.334-64.954 29.324L-60.121 29.324Q-60.052 29.334-60.006 29.380Q-59.960 29.426-59.960 29.491Q-59.960 29.563-60.006 29.609Q-60.052 29.655-60.121 29.665M-60.121 28.127L-64.954 28.127Q-65.022 28.117-65.068 28.071Q-65.114 28.025-65.114 27.953Q-65.114 27.809-64.954 27.785L-60.121 27.785Q-59.960 27.809-59.960 27.953Q-59.960 28.025-60.006 28.071Q-60.052 28.117-60.121 28.127M-57.147 29.324L-59.191 29.324L-59.191 29.043L-56.860 25.871Q-56.826 25.824-56.761 25.824L-56.624 25.824Q-56.580 25.824-56.552 25.851Q-56.525 25.878-56.525 25.923L-56.525 29.043L-55.763 29.043L-55.763 29.324L-56.525 29.324L-56.525 29.983Q-56.525 30.192-55.770 30.192L-55.770 30.472L-57.902 30.472L-57.902 30.192Q-57.147 30.192-57.147 29.983L-57.147 29.324M-57.099 26.599L-58.890 29.043L-57.099 29.043\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M207.543 21.936h22.762V4.864h-22.762Z\"\u002F>\u003Cg transform=\"translate(279.465 -14.816)\">\u003Cpath d=\"M-60.121 29.665L-64.954 29.665Q-65.022 29.655-65.068 29.609Q-65.114 29.563-65.114 29.491Q-65.114 29.426-65.068 29.380Q-65.022 29.334-64.954 29.324L-60.121 29.324Q-60.052 29.334-60.006 29.380Q-59.960 29.426-59.960 29.491Q-59.960 29.563-60.006 29.609Q-60.052 29.655-60.121 29.665M-60.121 28.127L-64.954 28.127Q-65.022 28.117-65.068 28.071Q-65.114 28.025-65.114 27.953Q-65.114 27.809-64.954 27.785L-60.121 27.785Q-59.960 27.809-59.960 27.953Q-59.960 28.025-60.006 28.071Q-60.052 28.117-60.121 28.127M-56.139 30.472L-59.023 30.472L-59.023 30.270Q-59.023 30.240-58.996 30.212L-57.749 28.995Q-57.677 28.920-57.634 28.878Q-57.591 28.835-57.513 28.756Q-57.099 28.343-56.868 27.985Q-56.638 27.628-56.638 27.204Q-56.638 26.972-56.716 26.769Q-56.795 26.565-56.937 26.415Q-57.079 26.264-57.273 26.184Q-57.468 26.104-57.701 26.104Q-58.012 26.104-58.270 26.263Q-58.528 26.422-58.658 26.699L-58.637 26.699Q-58.470 26.699-58.362 26.810Q-58.254 26.921-58.254 27.085Q-58.254 27.242-58.364 27.355Q-58.473 27.468-58.637 27.468Q-58.798 27.468-58.911 27.355Q-59.023 27.242-59.023 27.085Q-59.023 26.709-58.815 26.422Q-58.606 26.135-58.271 25.979Q-57.936 25.824-57.581 25.824Q-57.157 25.824-56.778 25.982Q-56.398 26.141-56.164 26.458Q-55.930 26.774-55.930 27.204Q-55.930 27.515-56.070 27.784Q-56.210 28.052-56.415 28.257Q-56.621 28.462-56.983 28.744Q-57.345 29.026-57.455 29.122L-58.309 29.850L-57.666 29.850Q-57.403 29.850-57.114 29.848Q-56.826 29.847-56.607 29.838Q-56.388 29.829-56.371 29.812Q-56.310 29.747-56.272 29.580Q-56.234 29.412-56.197 29.170L-55.930 29.170\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M207.543 53.234h22.762V36.162h-22.762Z\"\u002F>\u003Cg transform=\"translate(279.465 16.482)\">\u003Cpath d=\"M-60.121 29.665L-64.954 29.665Q-65.022 29.655-65.068 29.609Q-65.114 29.563-65.114 29.491Q-65.114 29.426-65.068 29.380Q-65.022 29.334-64.954 29.324L-60.121 29.324Q-60.052 29.334-60.006 29.380Q-59.960 29.426-59.960 29.491Q-59.960 29.563-60.006 29.609Q-60.052 29.655-60.121 29.665M-60.121 28.127L-64.954 28.127Q-65.022 28.117-65.068 28.071Q-65.114 28.025-65.114 27.953Q-65.114 27.809-64.954 27.785L-60.121 27.785Q-59.960 27.809-59.960 27.953Q-59.960 28.025-60.006 28.071Q-60.052 28.117-60.121 28.127M-58.668 29.925Q-58.548 30.082-58.357 30.181Q-58.165 30.281-57.950 30.320Q-57.735 30.359-57.513 30.359Q-57.215 30.359-57.020 30.204Q-56.826 30.048-56.735 29.794Q-56.645 29.539-56.645 29.255Q-56.645 28.961-56.737 28.710Q-56.829 28.459-57.027 28.303Q-57.226 28.148-57.520 28.148L-58.036 28.148Q-58.063 28.148-58.089 28.122Q-58.114 28.097-58.114 28.073L-58.114 28.001Q-58.114 27.970-58.089 27.948Q-58.063 27.926-58.036 27.926L-57.595 27.895Q-57.232 27.895-57.012 27.538Q-56.791 27.180-56.791 26.791Q-56.791 26.463-56.986 26.259Q-57.181 26.056-57.513 26.056Q-57.800 26.056-58.053 26.140Q-58.306 26.223-58.470 26.411Q-58.323 26.411-58.222 26.526Q-58.121 26.640-58.121 26.791Q-58.121 26.941-58.227 27.051Q-58.333 27.160-58.490 27.160Q-58.651 27.160-58.760 27.051Q-58.870 26.941-58.870 26.791Q-58.870 26.466-58.661 26.247Q-58.453 26.029-58.136 25.926Q-57.820 25.824-57.513 25.824Q-57.195 25.824-56.867 25.928Q-56.539 26.032-56.311 26.254Q-56.084 26.476-56.084 26.791Q-56.084 27.225-56.371 27.550Q-56.658 27.874-57.092 28.021Q-56.781 28.086-56.501 28.252Q-56.221 28.418-56.043 28.676Q-55.865 28.934-55.865 29.255Q-55.865 29.665-56.110 29.975Q-56.354 30.284-56.735 30.448Q-57.116 30.612-57.513 30.612Q-57.882 30.612-58.239 30.499Q-58.596 30.387-58.841 30.137Q-59.085 29.888-59.085 29.518Q-59.085 29.347-58.969 29.235Q-58.853 29.122-58.682 29.122Q-58.572 29.122-58.482 29.173Q-58.391 29.224-58.336 29.317Q-58.282 29.409-58.282 29.518Q-58.282 29.686-58.395 29.805Q-58.507 29.925-58.668 29.925\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"m96.959-8.535 43.005-23.458\"\u002F>\u003Cpath stroke=\"none\" d=\"m141.72-32.95-3.575.127 1.82.83-.288 1.98\"\u002F>\u003Cg transform=\"translate(182.47 -54.748)\">\u003Cpath d=\"M-61.255 30.472L-65.224 30.472L-65.224 30.192Q-64.502 30.192-64.502 29.983L-64.502 26.182Q-64.502 25.971-65.224 25.971L-65.224 25.690L-62.910 25.690L-62.910 25.971Q-63.228 25.971-63.520 26.006Q-63.812 26.042-63.812 26.182L-63.812 29.983Q-63.812 30.123-63.723 30.158Q-63.634 30.192-63.446 30.192L-62.824 30.192Q-62.411 30.192-62.132 30.089Q-61.853 29.987-61.688 29.785Q-61.522 29.583-61.440 29.296Q-61.358 29.009-61.313 28.602L-61.047 28.602\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"m96.959 1.193 43.005 23.457\"\u002F>\u003Cpath stroke=\"none\" d=\"m141.72 25.608-2.043-2.937.287 1.98-1.82.83\"\u002F>\u003Cg transform=\"translate(182.04 -8.755)\">\u003Cpath d=\"M-63.104 30.472L-65.210 30.472L-65.210 30.192Q-64.489 30.192-64.489 29.983L-64.489 26.182Q-64.489 25.971-65.210 25.971L-65.210 25.690L-62.872 25.690Q-62.561 25.690-62.218 25.764Q-61.874 25.837-61.553 25.993Q-61.231 26.148-61.030 26.394Q-60.828 26.640-60.828 26.965Q-60.828 27.252-61.016 27.483Q-61.204 27.714-61.481 27.862Q-61.758 28.011-62.048 28.093Q-61.713 28.196-61.479 28.428Q-61.245 28.660-61.201 28.989L-61.115 29.597Q-61.088 29.809-61.042 29.978Q-60.996 30.147-60.891 30.267Q-60.787 30.387-60.599 30.387Q-60.384 30.387-60.268 30.202Q-60.151 30.017-60.151 29.785Q-60.134 29.720-60.059 29.703L-59.967 29.703Q-59.885 29.723-59.885 29.805Q-59.885 30.011-59.975 30.197Q-60.066 30.383-60.232 30.498Q-60.397 30.612-60.599 30.612Q-60.937 30.612-61.231 30.511Q-61.525 30.410-61.710 30.188Q-61.895 29.966-61.895 29.618L-61.895 29.009Q-61.895 28.760-62.038 28.573Q-62.182 28.387-62.405 28.288Q-62.629 28.189-62.879 28.189L-63.826 28.189L-63.826 29.983Q-63.826 30.192-63.104 30.192L-63.104 30.472M-63.826 26.182L-63.826 27.967L-62.971 27.967Q-62.684 27.967-62.445 27.924Q-62.206 27.881-62.018 27.772Q-61.830 27.662-61.718 27.464Q-61.607 27.266-61.607 26.965Q-61.607 26.388-61.975 26.179Q-62.342 25.971-62.971 25.971L-63.460 25.971Q-63.648 25.971-63.737 26.005Q-63.826 26.039-63.826 26.182\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"m160.492-40.278 44.91-11.23\"\u002F>\u003Cpath stroke=\"none\" d=\"m207.343-51.993-3.492-.776 1.552 1.26-.776 1.844\"\u002F>\u003Cg transform=\"translate(258.761 -83.07)\">\u003Cpath d=\"M-61.255 30.472L-65.224 30.472L-65.224 30.192Q-64.502 30.192-64.502 29.983L-64.502 26.182Q-64.502 25.971-65.224 25.971L-65.224 25.690L-62.910 25.690L-62.910 25.971Q-63.228 25.971-63.520 26.006Q-63.812 26.042-63.812 26.182L-63.812 29.983Q-63.812 30.123-63.723 30.158Q-63.634 30.192-63.446 30.192L-62.824 30.192Q-62.411 30.192-62.132 30.089Q-61.853 29.987-61.688 29.785Q-61.522 29.583-61.440 29.296Q-61.358 29.009-61.313 28.602L-61.047 28.602\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"m160.582-35.743 44.803 9.336\"\u002F>\u003Cpath stroke=\"none\" d=\"m207.343-25.999-2.806-2.219.848 1.811-1.5 1.322\"\u002F>\u003Cg transform=\"translate(258.353 -50.59)\">\u003Cpath d=\"M-63.104 30.472L-65.210 30.472L-65.210 30.192Q-64.489 30.192-64.489 29.983L-64.489 26.182Q-64.489 25.971-65.210 25.971L-65.210 25.690L-62.872 25.690Q-62.561 25.690-62.218 25.764Q-61.874 25.837-61.553 25.993Q-61.231 26.148-61.030 26.394Q-60.828 26.640-60.828 26.965Q-60.828 27.252-61.016 27.483Q-61.204 27.714-61.481 27.862Q-61.758 28.011-62.048 28.093Q-61.713 28.196-61.479 28.428Q-61.245 28.660-61.201 28.989L-61.115 29.597Q-61.088 29.809-61.042 29.978Q-60.996 30.147-60.891 30.267Q-60.787 30.387-60.599 30.387Q-60.384 30.387-60.268 30.202Q-60.151 30.017-60.151 29.785Q-60.134 29.720-60.059 29.703L-59.967 29.703Q-59.885 29.723-59.885 29.805Q-59.885 30.011-59.975 30.197Q-60.066 30.383-60.232 30.498Q-60.397 30.612-60.599 30.612Q-60.937 30.612-61.231 30.511Q-61.525 30.410-61.710 30.188Q-61.895 29.966-61.895 29.618L-61.895 29.009Q-61.895 28.760-62.038 28.573Q-62.182 28.387-62.405 28.288Q-62.629 28.189-62.879 28.189L-63.826 28.189L-63.826 29.983Q-63.826 30.192-63.104 30.192L-63.104 30.472M-63.826 26.182L-63.826 27.967L-62.971 27.967Q-62.684 27.967-62.445 27.924Q-62.206 27.881-62.018 27.772Q-61.830 27.662-61.718 27.464Q-61.607 27.266-61.607 26.965Q-61.607 26.388-61.975 26.179Q-62.342 25.971-62.971 25.971L-63.460 25.971Q-63.648 25.971-63.737 26.005Q-63.826 26.039-63.826 26.182\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"m160.492 28.008 44.91-11.23\"\u002F>\u003Cpath stroke=\"none\" d=\"m207.343 16.293-3.492-.776 1.552 1.261-.776 1.843\"\u002F>\u003Cg transform=\"translate(258.761 -14.783)\">\u003Cpath d=\"M-61.255 30.472L-65.224 30.472L-65.224 30.192Q-64.502 30.192-64.502 29.983L-64.502 26.182Q-64.502 25.971-65.224 25.971L-65.224 25.690L-62.910 25.690L-62.910 25.971Q-63.228 25.971-63.520 26.006Q-63.812 26.042-63.812 26.182L-63.812 29.983Q-63.812 30.123-63.723 30.158Q-63.634 30.192-63.446 30.192L-62.824 30.192Q-62.411 30.192-62.132 30.089Q-61.853 29.987-61.688 29.785Q-61.522 29.583-61.440 29.296Q-61.358 29.009-61.313 28.602L-61.047 28.602\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"m160.582 32.544 44.803 9.335\"\u002F>\u003Cpath stroke=\"none\" d=\"m207.343 42.287-2.806-2.219.848 1.811-1.5 1.322\"\u002F>\u003Cg transform=\"translate(258.353 17.696)\">\u003Cpath d=\"M-63.104 30.472L-65.210 30.472L-65.210 30.192Q-64.489 30.192-64.489 29.983L-64.489 26.182Q-64.489 25.971-65.210 25.971L-65.210 25.690L-62.872 25.690Q-62.561 25.690-62.218 25.764Q-61.874 25.837-61.553 25.993Q-61.231 26.148-61.030 26.394Q-60.828 26.640-60.828 26.965Q-60.828 27.252-61.016 27.483Q-61.204 27.714-61.481 27.862Q-61.758 28.011-62.048 28.093Q-61.713 28.196-61.479 28.428Q-61.245 28.660-61.201 28.989L-61.115 29.597Q-61.088 29.809-61.042 29.978Q-60.996 30.147-60.891 30.267Q-60.787 30.387-60.599 30.387Q-60.384 30.387-60.268 30.202Q-60.151 30.017-60.151 29.785Q-60.134 29.720-60.059 29.703L-59.967 29.703Q-59.885 29.723-59.885 29.805Q-59.885 30.011-59.975 30.197Q-60.066 30.383-60.232 30.498Q-60.397 30.612-60.599 30.612Q-60.937 30.612-61.231 30.511Q-61.525 30.410-61.710 30.188Q-61.895 29.966-61.895 29.618L-61.895 29.009Q-61.895 28.760-62.038 28.573Q-62.182 28.387-62.405 28.288Q-62.629 28.189-62.879 28.189L-63.826 28.189L-63.826 29.983Q-63.826 30.192-63.104 30.192L-63.104 30.472M-63.826 26.182L-63.826 27.967L-62.971 27.967Q-62.684 27.967-62.445 27.924Q-62.206 27.881-62.018 27.772Q-61.830 27.662-61.718 27.464Q-61.607 27.266-61.607 26.965Q-61.607 26.388-61.975 26.179Q-62.342 25.971-62.971 25.971L-63.460 25.971Q-63.648 25.971-63.737 26.005Q-63.826 26.039-63.826 26.182\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-64.762 21.631L-64.762 19.734L-65.401 19.734L-65.401 19.512Q-65.083 19.512-64.866 19.302Q-64.649 19.092-64.549 18.782Q-64.448 18.473-64.448 18.165L-64.181 18.165L-64.181 19.454L-63.104 19.454L-63.104 19.734L-64.181 19.734L-64.181 21.618Q-64.181 21.894-64.077 22.093Q-63.973 22.291-63.713 22.291Q-63.556 22.291-63.450 22.187Q-63.344 22.082-63.294 21.929Q-63.245 21.775-63.245 21.618L-63.245 21.204L-62.978 21.204L-62.978 21.631Q-62.978 21.857-63.077 22.067Q-63.176 22.277-63.361 22.409Q-63.545 22.540-63.774 22.540Q-64.212 22.540-64.487 22.303Q-64.762 22.065-64.762 21.631M-60.418 22.472L-62.154 22.472L-62.154 22.192Q-61.925 22.192-61.777 22.158Q-61.628 22.123-61.628 21.983L-61.628 20.134Q-61.628 19.864-61.736 19.803Q-61.843 19.741-62.154 19.741L-62.154 19.461L-61.125 19.386L-61.125 20.093Q-60.996 19.785-60.753 19.586Q-60.510 19.386-60.192 19.386Q-59.974 19.386-59.803 19.510Q-59.632 19.635-59.632 19.847Q-59.632 19.984-59.731 20.083Q-59.830 20.182-59.963 20.182Q-60.100 20.182-60.199 20.083Q-60.298 19.984-60.298 19.847Q-60.298 19.707-60.199 19.608Q-60.490 19.608-60.690 19.804Q-60.890 20.001-60.982 20.295Q-61.074 20.589-61.074 20.869L-61.074 21.983Q-61.074 22.192-60.418 22.192L-60.418 22.472M-58.989 21.744Q-58.989 21.412-58.765 21.185Q-58.541 20.958-58.198 20.830Q-57.854 20.701-57.482 20.649Q-57.109 20.596-56.805 20.596L-56.805 20.343Q-56.805 20.138-56.913 19.958Q-57.020 19.779-57.202 19.676Q-57.383 19.574-57.591 19.574Q-57.998 19.574-58.234 19.666Q-58.145 19.703-58.099 19.787Q-58.053 19.871-58.053 19.973Q-58.053 20.069-58.099 20.148Q-58.145 20.226-58.225 20.271Q-58.306 20.315-58.395 20.315Q-58.545 20.315-58.646 20.218Q-58.747 20.120-58.747 19.973Q-58.747 19.351-57.591 19.351Q-57.379 19.351-57.130 19.415Q-56.880 19.478-56.679 19.597Q-56.477 19.717-56.351 19.902Q-56.224 20.086-56.224 20.329L-56.224 21.905Q-56.224 22.021-56.163 22.117Q-56.101 22.212-55.988 22.212Q-55.879 22.212-55.814 22.118Q-55.749 22.024-55.749 21.905L-55.749 21.457L-55.482 21.457L-55.482 21.905Q-55.482 22.175-55.710 22.340Q-55.937 22.506-56.217 22.506Q-56.426 22.506-56.562 22.352Q-56.699 22.199-56.723 21.983Q-56.870 22.250-57.152 22.395Q-57.434 22.540-57.759 22.540Q-58.036 22.540-58.319 22.465Q-58.603 22.390-58.796 22.211Q-58.989 22.031-58.989 21.744M-58.374 21.744Q-58.374 21.918-58.273 22.048Q-58.172 22.178-58.017 22.248Q-57.861 22.318-57.697 22.318Q-57.478 22.318-57.270 22.221Q-57.061 22.123-56.933 21.942Q-56.805 21.761-56.805 21.535L-56.805 20.807Q-57.130 20.807-57.496 20.898Q-57.861 20.989-58.118 21.201Q-58.374 21.412-58.374 21.744M-53.384 22.472L-55.018 22.472L-55.018 22.192Q-54.789 22.192-54.640 22.158Q-54.491 22.123-54.491 21.983L-54.491 20.134Q-54.491 19.864-54.599 19.803Q-54.707 19.741-55.018 19.741L-55.018 19.461L-53.958 19.386L-53.958 20.035Q-53.787 19.727-53.483 19.556Q-53.179 19.386-52.833 19.386Q-52.328 19.386-52.044 19.609Q-51.760 19.833-51.760 20.329L-51.760 21.983Q-51.760 22.120-51.612 22.156Q-51.463 22.192-51.237 22.192L-51.237 22.472L-52.868 22.472L-52.868 22.192Q-52.639 22.192-52.490 22.158Q-52.341 22.123-52.341 21.983L-52.341 20.343Q-52.341 20.008-52.461 19.808Q-52.581 19.608-52.895 19.608Q-53.165 19.608-53.399 19.744Q-53.633 19.881-53.772 20.115Q-53.910 20.349-53.910 20.623L-53.910 21.983Q-53.910 22.120-53.760 22.156Q-53.609 22.192-53.384 22.192L-53.384 22.472M-50.649 22.465L-50.649 21.402Q-50.649 21.378-50.622 21.351Q-50.595 21.324-50.571 21.324L-50.461 21.324Q-50.396 21.324-50.383 21.382Q-50.287 21.816-50.041 22.067Q-49.795 22.318-49.381 22.318Q-49.040 22.318-48.787 22.185Q-48.534 22.052-48.534 21.744Q-48.534 21.587-48.628 21.472Q-48.722 21.358-48.860 21.289Q-48.999 21.221-49.166 21.183L-49.747 21.084Q-50.103 21.016-50.376 20.795Q-50.649 20.575-50.649 20.233Q-50.649 19.984-50.538 19.809Q-50.427 19.635-50.241 19.536Q-50.055 19.437-49.839 19.394Q-49.624 19.351-49.381 19.351Q-48.968 19.351-48.687 19.533L-48.472 19.358Q-48.462 19.355-48.455 19.353Q-48.448 19.351-48.438 19.351L-48.387 19.351Q-48.359 19.351-48.335 19.375Q-48.311 19.399-48.311 19.427L-48.311 20.274Q-48.311 20.295-48.335 20.322Q-48.359 20.349-48.387 20.349L-48.499 20.349Q-48.527 20.349-48.552 20.324Q-48.578 20.298-48.578 20.274Q-48.578 20.038-48.684 19.874Q-48.790 19.710-48.973 19.628Q-49.156 19.546-49.388 19.546Q-49.716 19.546-49.973 19.649Q-50.229 19.751-50.229 20.028Q-50.229 20.223-50.046 20.332Q-49.863 20.442-49.634 20.483L-49.060 20.589Q-48.814 20.637-48.600 20.765Q-48.387 20.893-48.250 21.096Q-48.113 21.300-48.113 21.549Q-48.113 22.062-48.479 22.301Q-48.845 22.540-49.381 22.540Q-49.877 22.540-50.208 22.246L-50.475 22.520Q-50.496 22.540-50.523 22.540L-50.571 22.540Q-50.595 22.540-50.622 22.513Q-50.649 22.486-50.649 22.465M-45.868 22.472L-47.419 22.472L-47.419 22.192Q-47.194 22.192-47.045 22.158Q-46.896 22.123-46.896 21.983L-46.896 20.134Q-46.896 19.946-46.944 19.862Q-46.992 19.779-47.090 19.760Q-47.187 19.741-47.399 19.741L-47.399 19.461L-46.343 19.386L-46.343 21.983Q-46.343 22.123-46.211 22.158Q-46.080 22.192-45.868 22.192L-45.868 22.472M-47.139 18.165Q-47.139 17.994-47.016 17.875Q-46.893 17.755-46.722 17.755Q-46.555 17.755-46.432 17.875Q-46.309 17.994-46.309 18.165Q-46.309 18.340-46.432 18.463Q-46.555 18.586-46.722 18.586Q-46.893 18.586-47.016 18.463Q-47.139 18.340-47.139 18.165M-44.695 21.631L-44.695 19.734L-45.334 19.734L-45.334 19.512Q-45.017 19.512-44.800 19.302Q-44.582 19.092-44.482 18.782Q-44.381 18.473-44.381 18.165L-44.114 18.165L-44.114 19.454L-43.038 19.454L-43.038 19.734L-44.114 19.734L-44.114 21.618Q-44.114 21.894-44.010 22.093Q-43.906 22.291-43.646 22.291Q-43.489 22.291-43.383 22.187Q-43.277 22.082-43.227 21.929Q-43.178 21.775-43.178 21.618L-43.178 21.204L-42.911 21.204L-42.911 21.631Q-42.911 21.857-43.010 22.067Q-43.109 22.277-43.294 22.409Q-43.478 22.540-43.707 22.540Q-44.145 22.540-44.420 22.303Q-44.695 22.065-44.695 21.631M-40.484 22.472L-42.036 22.472L-42.036 22.192Q-41.811 22.192-41.662 22.158Q-41.513 22.123-41.513 21.983L-41.513 20.134Q-41.513 19.946-41.561 19.862Q-41.609 19.779-41.706 19.760Q-41.804 19.741-42.016 19.741L-42.016 19.461L-40.959 19.386L-40.959 21.983Q-40.959 22.123-40.828 22.158Q-40.696 22.192-40.484 22.192L-40.484 22.472M-41.756 18.165Q-41.756 17.994-41.633 17.875Q-41.510 17.755-41.339 17.755Q-41.171 17.755-41.048 17.875Q-40.925 17.994-40.925 18.165Q-40.925 18.340-41.048 18.463Q-41.171 18.586-41.339 18.586Q-41.510 18.586-41.633 18.463Q-41.756 18.340-41.756 18.165M-39.879 20.989Q-39.879 20.647-39.744 20.348Q-39.609 20.049-39.370 19.825Q-39.131 19.601-38.813 19.476Q-38.495 19.351-38.164 19.351Q-37.719 19.351-37.319 19.567Q-36.919 19.782-36.685 20.160Q-36.451 20.537-36.451 20.989Q-36.451 21.330-36.593 21.614Q-36.735 21.898-36.979 22.105Q-37.224 22.311-37.533 22.426Q-37.842 22.540-38.164 22.540Q-38.594 22.540-38.996 22.339Q-39.397 22.137-39.638 21.785Q-39.879 21.433-39.879 20.989M-38.164 22.291Q-37.562 22.291-37.338 21.913Q-37.114 21.535-37.114 20.903Q-37.114 20.291-37.348 19.932Q-37.582 19.574-38.164 19.574Q-39.216 19.574-39.216 20.903Q-39.216 21.535-38.991 21.913Q-38.765 22.291-38.164 22.291M-34.175 22.472L-35.809 22.472L-35.809 22.192Q-35.580 22.192-35.431 22.158Q-35.282 22.123-35.282 21.983L-35.282 20.134Q-35.282 19.864-35.390 19.803Q-35.498 19.741-35.809 19.741L-35.809 19.461L-34.749 19.386L-34.749 20.035Q-34.578 19.727-34.274 19.556Q-33.970 19.386-33.624 19.386Q-33.119 19.386-32.835 19.609Q-32.551 19.833-32.551 20.329L-32.551 21.983Q-32.551 22.120-32.403 22.156Q-32.254 22.192-32.028 22.192L-32.028 22.472L-33.659 22.472L-33.659 22.192Q-33.430 22.192-33.281 22.158Q-33.132 22.123-33.132 21.983L-33.132 20.343Q-33.132 20.008-33.252 19.808Q-33.372 19.608-33.686 19.608Q-33.956 19.608-34.190 19.744Q-34.424 19.881-34.563 20.115Q-34.701 20.349-34.701 20.623L-34.701 21.983Q-34.701 22.120-34.551 22.156Q-34.400 22.192-34.175 22.192L-34.175 22.472M-31.440 22.465L-31.440 21.402Q-31.440 21.378-31.413 21.351Q-31.386 21.324-31.362 21.324L-31.252 21.324Q-31.187 21.324-31.174 21.382Q-31.078 21.816-30.832 22.067Q-30.586 22.318-30.172 22.318Q-29.831 22.318-29.578 22.185Q-29.325 22.052-29.325 21.744Q-29.325 21.587-29.419 21.472Q-29.513 21.358-29.651 21.289Q-29.790 21.221-29.957 21.183L-30.538 21.084Q-30.894 21.016-31.167 20.795Q-31.440 20.575-31.440 20.233Q-31.440 19.984-31.329 19.809Q-31.218 19.635-31.032 19.536Q-30.846 19.437-30.630 19.394Q-30.415 19.351-30.172 19.351Q-29.759 19.351-29.478 19.533L-29.263 19.358Q-29.253 19.355-29.246 19.353Q-29.239 19.351-29.229 19.351L-29.178 19.351Q-29.150 19.351-29.126 19.375Q-29.103 19.399-29.103 19.427L-29.103 20.274Q-29.103 20.295-29.126 20.322Q-29.150 20.349-29.178 20.349L-29.290 20.349Q-29.318 20.349-29.343 20.324Q-29.369 20.298-29.369 20.274Q-29.369 20.038-29.475 19.874Q-29.581 19.710-29.764 19.628Q-29.947 19.546-30.179 19.546Q-30.507 19.546-30.764 19.649Q-31.020 19.751-31.020 20.028Q-31.020 20.223-30.837 20.332Q-30.654 20.442-30.425 20.483L-29.851 20.589Q-29.605 20.637-29.391 20.765Q-29.178 20.893-29.041 21.096Q-28.904 21.300-28.904 21.549Q-28.904 22.062-29.270 22.301Q-29.636 22.540-30.172 22.540Q-30.668 22.540-30.999 22.246L-31.266 22.520Q-31.287 22.540-31.314 22.540L-31.362 22.540Q-31.386 22.540-31.413 22.513Q-31.440 22.486-31.440 22.465\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-22.966 23.152L-22.966 20.896L-25.215 20.896Q-25.283 20.886-25.329 20.840Q-25.375 20.794-25.375 20.722Q-25.375 20.578-25.215 20.555L-22.966 20.555L-22.966 18.299Q-22.955 18.230-22.909 18.184Q-22.863 18.138-22.791 18.138Q-22.648 18.138-22.624 18.299L-22.624 20.555L-20.382 20.555Q-20.221 20.578-20.221 20.722Q-20.221 20.794-20.267 20.840Q-20.313 20.886-20.382 20.896L-22.624 20.896L-22.624 23.152Q-22.648 23.313-22.791 23.313Q-22.863 23.313-22.909 23.267Q-22.955 23.221-22.966 23.152\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-14.967 22.472L-16.703 22.472L-16.703 22.192Q-16.474 22.192-16.325 22.158Q-16.177 22.123-16.177 21.983L-16.177 20.134Q-16.177 19.864-16.284 19.803Q-16.392 19.741-16.703 19.741L-16.703 19.461L-15.674 19.386L-15.674 20.093Q-15.544 19.785-15.302 19.586Q-15.059 19.386-14.741 19.386Q-14.522 19.386-14.351 19.510Q-14.180 19.635-14.180 19.847Q-14.180 19.984-14.280 20.083Q-14.379 20.182-14.512 20.182Q-14.649 20.182-14.748 20.083Q-14.847 19.984-14.847 19.847Q-14.847 19.707-14.748 19.608Q-15.038 19.608-15.238 19.804Q-15.438 20.001-15.531 20.295Q-15.623 20.589-15.623 20.869L-15.623 21.983Q-15.623 22.192-14.967 22.192L-14.967 22.472M-13.637 20.937Q-13.637 20.616-13.512 20.327Q-13.387 20.038-13.162 19.815Q-12.936 19.591-12.641 19.471Q-12.345 19.351-12.027 19.351Q-11.699 19.351-11.437 19.451Q-11.176 19.550-11 19.732Q-10.824 19.915-10.730 20.173Q-10.636 20.431-10.636 20.763Q-10.636 20.855-10.718 20.876L-12.974 20.876L-12.974 20.937Q-12.974 21.525-12.690 21.908Q-12.406 22.291-11.839 22.291Q-11.518 22.291-11.250 22.098Q-10.981 21.905-10.892 21.590Q-10.885 21.549-10.810 21.535L-10.718 21.535Q-10.636 21.559-10.636 21.631Q-10.636 21.638-10.643 21.665Q-10.756 22.062-11.126 22.301Q-11.497 22.540-11.921 22.540Q-12.359 22.540-12.759 22.332Q-13.158 22.123-13.398 21.756Q-13.637 21.389-13.637 20.937M-12.967 20.667L-11.152 20.667Q-11.152 20.390-11.250 20.138Q-11.347 19.885-11.545 19.729Q-11.743 19.574-12.027 19.574Q-12.304 19.574-12.518 19.732Q-12.731 19.891-12.849 20.146Q-12.967 20.401-12.967 20.667M-8.660 22.445L-9.641 19.946Q-9.703 19.803-9.821 19.768Q-9.939 19.734-10.154 19.734L-10.154 19.454L-8.674 19.454L-8.674 19.734Q-9.053 19.734-9.053 19.895Q-9.053 19.905-9.040 19.946L-8.325 21.778L-7.652 20.073Q-7.683 20.001-7.683 19.973Q-7.683 19.946-7.710 19.946Q-7.772 19.799-7.890 19.767Q-8.008 19.734-8.219 19.734L-8.219 19.454L-6.822 19.454L-6.822 19.734Q-7.198 19.734-7.198 19.895Q-7.198 19.926-7.191 19.946L-6.435 21.884L-5.748 20.134Q-5.728 20.083-5.728 20.028Q-5.728 19.888-5.841 19.811Q-5.953 19.734-6.093 19.734L-6.093 19.454L-4.873 19.454L-4.873 19.734Q-5.078 19.734-5.234 19.840Q-5.389 19.946-5.461 20.134L-6.367 22.445Q-6.401 22.540-6.514 22.540L-6.582 22.540Q-6.692 22.540-6.729 22.445L-7.512 20.442L-8.298 22.445Q-8.332 22.540-8.445 22.540L-8.513 22.540Q-8.623 22.540-8.660 22.445\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-4.484 21.744Q-4.484 21.412-4.261 21.185Q-4.037 20.958-3.693 20.830Q-3.350 20.701-2.977 20.649Q-2.605 20.596-2.300 20.596L-2.300 20.343Q-2.300 20.138-2.408 19.958Q-2.516 19.779-2.697 19.676Q-2.878 19.574-3.086 19.574Q-3.493 19.574-3.729 19.666Q-3.640 19.703-3.594 19.787Q-3.548 19.871-3.548 19.973Q-3.548 20.069-3.594 20.148Q-3.640 20.226-3.721 20.271Q-3.801 20.315-3.890 20.315Q-4.040 20.315-4.141 20.218Q-4.242 20.120-4.242 19.973Q-4.242 19.351-3.086 19.351Q-2.875 19.351-2.625 19.415Q-2.376 19.478-2.174 19.597Q-1.972 19.717-1.846 19.902Q-1.719 20.086-1.719 20.329L-1.719 21.905Q-1.719 22.021-1.658 22.117Q-1.596 22.212-1.483 22.212Q-1.374 22.212-1.309 22.118Q-1.244 22.024-1.244 21.905L-1.244 21.457L-0.978 21.457L-0.978 21.905Q-0.978 22.175-1.205 22.340Q-1.432 22.506-1.712 22.506Q-1.921 22.506-2.058 22.352Q-2.194 22.199-2.218 21.983Q-2.365 22.250-2.647 22.395Q-2.929 22.540-3.254 22.540Q-3.531 22.540-3.815 22.465Q-4.098 22.390-4.291 22.211Q-4.484 22.031-4.484 21.744M-3.869 21.744Q-3.869 21.918-3.768 22.048Q-3.668 22.178-3.512 22.248Q-3.357 22.318-3.192 22.318Q-2.974 22.318-2.765 22.221Q-2.557 22.123-2.429 21.942Q-2.300 21.761-2.300 21.535L-2.300 20.807Q-2.625 20.807-2.991 20.898Q-3.357 20.989-3.613 21.201Q-3.869 21.412-3.869 21.744M1.189 22.472L-0.547 22.472L-0.547 22.192Q-0.318 22.192-0.169 22.158Q-0.021 22.123-0.021 21.983L-0.021 20.134Q-0.021 19.864-0.128 19.803Q-0.236 19.741-0.547 19.741L-0.547 19.461L0.482 19.386L0.482 20.093Q0.612 19.785 0.854 19.586Q1.097 19.386 1.415 19.386Q1.634 19.386 1.805 19.510Q1.976 19.635 1.976 19.847Q1.976 19.984 1.876 20.083Q1.777 20.182 1.644 20.182Q1.507 20.182 1.408 20.083Q1.309 19.984 1.309 19.847Q1.309 19.707 1.408 19.608Q1.118 19.608 0.918 19.804Q0.718 20.001 0.625 20.295Q0.533 20.589 0.533 20.869L0.533 21.983Q0.533 22.192 1.189 22.192L1.189 22.472M2.560 20.961Q2.560 20.623 2.700 20.332Q2.840 20.042 3.085 19.828Q3.329 19.615 3.633 19.500Q3.937 19.386 4.262 19.386Q4.532 19.386 4.795 19.485Q5.059 19.584 5.250 19.762L5.250 18.364Q5.250 18.094 5.142 18.032Q5.035 17.971 4.724 17.971L4.724 17.690L5.800 17.615L5.800 21.799Q5.800 21.987 5.855 22.070Q5.910 22.154 6.010 22.173Q6.111 22.192 6.327 22.192L6.327 22.472L5.219 22.540L5.219 22.123Q4.802 22.540 4.177 22.540Q3.746 22.540 3.373 22.328Q3.001 22.117 2.780 21.756Q2.560 21.395 2.560 20.961M4.235 22.318Q4.443 22.318 4.630 22.246Q4.816 22.175 4.970 22.038Q5.123 21.901 5.219 21.723L5.219 20.114Q5.134 19.967 4.988 19.847Q4.843 19.727 4.674 19.668Q4.505 19.608 4.324 19.608Q3.763 19.608 3.495 19.997Q3.226 20.387 3.226 20.968Q3.226 21.539 3.461 21.929Q3.695 22.318 4.235 22.318\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M11.361 22.472L9.727 22.472L9.727 22.192Q9.956 22.192 10.105 22.158Q10.254 22.123 10.254 21.983L10.254 20.134Q10.254 19.864 10.146 19.803Q10.038 19.741 9.727 19.741L9.727 19.461L10.787 19.386L10.787 20.035Q10.958 19.727 11.262 19.556Q11.566 19.386 11.911 19.386Q12.311 19.386 12.588 19.526Q12.865 19.666 12.950 20.014Q13.118 19.721 13.417 19.553Q13.716 19.386 14.061 19.386Q14.567 19.386 14.851 19.609Q15.135 19.833 15.135 20.329L15.135 21.983Q15.135 22.120 15.283 22.156Q15.432 22.192 15.657 22.192L15.657 22.472L14.027 22.472L14.027 22.192Q14.253 22.192 14.403 22.156Q14.553 22.120 14.553 21.983L14.553 20.343Q14.553 20.008 14.434 19.808Q14.314 19.608 14 19.608Q13.730 19.608 13.496 19.744Q13.261 19.881 13.123 20.115Q12.985 20.349 12.985 20.623L12.985 21.983Q12.985 22.120 13.133 22.156Q13.282 22.192 13.508 22.192L13.508 22.472L11.877 22.472L11.877 22.192Q12.106 22.192 12.255 22.158Q12.404 22.123 12.404 21.983L12.404 20.343Q12.404 20.008 12.284 19.808Q12.164 19.608 11.850 19.608Q11.580 19.608 11.346 19.744Q11.112 19.881 10.973 20.115Q10.835 20.349 10.835 20.623L10.835 21.983Q10.835 22.120 10.985 22.156Q11.136 22.192 11.361 22.192L11.361 22.472M16.204 20.989Q16.204 20.647 16.339 20.348Q16.474 20.049 16.714 19.825Q16.953 19.601 17.271 19.476Q17.589 19.351 17.920 19.351Q18.365 19.351 18.764 19.567Q19.164 19.782 19.398 20.160Q19.633 20.537 19.633 20.989Q19.633 21.330 19.491 21.614Q19.349 21.898 19.105 22.105Q18.860 22.311 18.551 22.426Q18.241 22.540 17.920 22.540Q17.490 22.540 17.088 22.339Q16.686 22.137 16.445 21.785Q16.204 21.433 16.204 20.989M17.920 22.291Q18.522 22.291 18.746 21.913Q18.969 21.535 18.969 20.903Q18.969 20.291 18.735 19.932Q18.501 19.574 17.920 19.574Q16.867 19.574 16.867 20.903Q16.867 21.535 17.093 21.913Q17.319 22.291 17.920 22.291\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M20.450 20.961Q20.450 20.623 20.591 20.332Q20.731 20.042 20.975 19.828Q21.219 19.615 21.524 19.500Q21.828 19.386 22.153 19.386Q22.423 19.386 22.686 19.485Q22.949 19.584 23.140 19.762L23.140 18.364Q23.140 18.094 23.033 18.032Q22.925 17.971 22.614 17.971L22.614 17.690L23.691 17.615L23.691 21.799Q23.691 21.987 23.745 22.070Q23.800 22.154 23.901 22.173Q24.002 22.192 24.217 22.192L24.217 22.472L23.110 22.540L23.110 22.123Q22.693 22.540 22.067 22.540Q21.636 22.540 21.264 22.328Q20.891 22.117 20.671 21.756Q20.450 21.395 20.450 20.961M22.125 22.318Q22.334 22.318 22.520 22.246Q22.706 22.175 22.860 22.038Q23.014 21.901 23.110 21.723L23.110 20.114Q23.024 19.967 22.879 19.847Q22.734 19.727 22.564 19.668Q22.395 19.608 22.214 19.608Q21.654 19.608 21.385 19.997Q21.117 20.387 21.117 20.968Q21.117 21.539 21.351 21.929Q21.585 22.318 22.125 22.318M24.825 20.937Q24.825 20.616 24.950 20.327Q25.075 20.038 25.301 19.815Q25.526 19.591 25.822 19.471Q26.117 19.351 26.435 19.351Q26.763 19.351 27.025 19.451Q27.286 19.550 27.462 19.732Q27.638 19.915 27.732 20.173Q27.826 20.431 27.826 20.763Q27.826 20.855 27.744 20.876L25.489 20.876L25.489 20.937Q25.489 21.525 25.772 21.908Q26.056 22.291 26.623 22.291Q26.945 22.291 27.213 22.098Q27.481 21.905 27.570 21.590Q27.577 21.549 27.652 21.535L27.744 21.535Q27.826 21.559 27.826 21.631Q27.826 21.638 27.820 21.665Q27.707 22.062 27.336 22.301Q26.965 22.540 26.541 22.540Q26.104 22.540 25.704 22.332Q25.304 22.123 25.065 21.756Q24.825 21.389 24.825 20.937M25.495 20.667L27.310 20.667Q27.310 20.390 27.213 20.138Q27.115 19.885 26.917 19.729Q26.719 19.574 26.435 19.574Q26.158 19.574 25.945 19.732Q25.731 19.891 25.613 20.146Q25.495 20.401 25.495 20.667M30.082 22.472L28.479 22.472L28.479 22.192Q28.705 22.192 28.854 22.158Q29.002 22.123 29.002 21.983L29.002 18.364Q29.002 18.094 28.895 18.032Q28.787 17.971 28.479 17.971L28.479 17.690L29.556 17.615L29.556 21.983Q29.556 22.120 29.706 22.156Q29.857 22.192 30.082 22.192L30.082 22.472M31.176 23.702Q31.176 23.668 31.203 23.641Q31.473 23.412 31.622 23.089Q31.771 22.766 31.771 22.410L31.771 22.373Q31.661 22.472 31.497 22.472Q31.316 22.472 31.197 22.352Q31.077 22.233 31.077 22.052Q31.077 21.877 31.197 21.758Q31.316 21.638 31.497 21.638Q31.754 21.638 31.873 21.877Q31.993 22.117 31.993 22.410Q31.993 22.810 31.824 23.181Q31.655 23.552 31.357 23.808Q31.326 23.829 31.299 23.829Q31.258 23.829 31.217 23.788Q31.176 23.747 31.176 23.702\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-52.685 30.465L-52.685 29.402Q-52.685 29.378-52.657 29.351Q-52.630 29.324-52.606 29.324L-52.497 29.324Q-52.432 29.324-52.418 29.382Q-52.322 29.816-52.076 30.067Q-51.830 30.318-51.416 30.318Q-51.075 30.318-50.822 30.185Q-50.569 30.052-50.569 29.744Q-50.569 29.587-50.663 29.472Q-50.757 29.358-50.895 29.289Q-51.034 29.221-51.201 29.183L-51.782 29.084Q-52.138 29.016-52.411 28.795Q-52.685 28.575-52.685 28.233Q-52.685 27.984-52.573 27.809Q-52.462 27.635-52.276 27.536Q-52.090 27.437-51.874 27.394Q-51.659 27.351-51.416 27.351Q-51.003 27.351-50.723 27.533L-50.507 27.358Q-50.497 27.355-50.490 27.353Q-50.483 27.351-50.473 27.351L-50.422 27.351Q-50.395 27.351-50.371 27.375Q-50.347 27.399-50.347 27.427L-50.347 28.274Q-50.347 28.295-50.371 28.322Q-50.395 28.349-50.422 28.349L-50.535 28.349Q-50.562 28.349-50.588 28.324Q-50.613 28.298-50.613 28.274Q-50.613 28.038-50.719 27.874Q-50.825 27.710-51.008 27.628Q-51.191 27.546-51.423 27.546Q-51.751 27.546-52.008 27.649Q-52.264 27.751-52.264 28.028Q-52.264 28.223-52.081 28.332Q-51.898 28.442-51.669 28.483L-51.095 28.589Q-50.849 28.637-50.635 28.765Q-50.422 28.893-50.285 29.096Q-50.148 29.300-50.148 29.549Q-50.148 30.062-50.514 30.301Q-50.880 30.540-51.416 30.540Q-51.912 30.540-52.244 30.246L-52.510 30.520Q-52.531 30.540-52.558 30.540L-52.606 30.540Q-52.630 30.540-52.657 30.513Q-52.685 30.486-52.685 30.465M-47.903 30.472L-49.455 30.472L-49.455 30.192Q-49.229 30.192-49.080 30.158Q-48.932 30.123-48.932 29.983L-48.932 28.134Q-48.932 27.946-48.979 27.862Q-49.027 27.779-49.125 27.760Q-49.222 27.741-49.434 27.741L-49.434 27.461L-48.378 27.386L-48.378 29.983Q-48.378 30.123-48.246 30.158Q-48.115 30.192-47.903 30.192L-47.903 30.472M-49.174 26.165Q-49.174 25.994-49.051 25.875Q-48.928 25.755-48.757 25.755Q-48.590 25.755-48.467 25.875Q-48.344 25.994-48.344 26.165Q-48.344 26.340-48.467 26.463Q-48.590 26.586-48.757 26.586Q-48.928 26.586-49.051 26.463Q-49.174 26.340-49.174 26.165M-45.575 30.472L-47.209 30.472L-47.209 30.192Q-46.980 30.192-46.831 30.158Q-46.683 30.123-46.683 29.983L-46.683 28.134Q-46.683 27.864-46.790 27.803Q-46.898 27.741-47.209 27.741L-47.209 27.461L-46.149 27.386L-46.149 28.035Q-45.978 27.727-45.674 27.556Q-45.370 27.386-45.025 27.386Q-44.625 27.386-44.348 27.526Q-44.071 27.666-43.986 28.014Q-43.818 27.721-43.519 27.553Q-43.220 27.386-42.875 27.386Q-42.369 27.386-42.085 27.609Q-41.802 27.833-41.802 28.329L-41.802 29.983Q-41.802 30.120-41.653 30.156Q-41.504 30.192-41.279 30.192L-41.279 30.472L-42.909 30.472L-42.909 30.192Q-42.684 30.192-42.533 30.156Q-42.383 30.120-42.383 29.983L-42.383 28.343Q-42.383 28.008-42.502 27.808Q-42.622 27.608-42.937 27.608Q-43.207 27.608-43.441 27.744Q-43.675 27.881-43.813 28.115Q-43.952 28.349-43.952 28.623L-43.952 29.983Q-43.952 30.120-43.803 30.156Q-43.654 30.192-43.429 30.192L-43.429 30.472L-45.059 30.472L-45.059 30.192Q-44.830 30.192-44.681 30.158Q-44.533 30.123-44.533 29.983L-44.533 28.343Q-44.533 28.008-44.652 27.808Q-44.772 27.608-45.086 27.608Q-45.356 27.608-45.591 27.744Q-45.825 27.881-45.963 28.115Q-46.102 28.349-46.102 28.623L-46.102 29.983Q-46.102 30.120-45.951 30.156Q-45.801 30.192-45.575 30.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-40.324 29.638L-40.324 28.134Q-40.324 27.864-40.432 27.803Q-40.540 27.741-40.851 27.741L-40.851 27.461L-39.743 27.386L-39.743 29.618L-39.743 29.638Q-39.743 29.918-39.692 30.062Q-39.641 30.205-39.499 30.262Q-39.357 30.318-39.070 30.318Q-38.817 30.318-38.612 30.178Q-38.407 30.038-38.291 29.812Q-38.174 29.587-38.174 29.337L-38.174 28.134Q-38.174 27.864-38.282 27.803Q-38.390 27.741-38.701 27.741L-38.701 27.461L-37.593 27.386L-37.593 29.799Q-37.593 29.990-37.540 30.072Q-37.487 30.154-37.387 30.173Q-37.286 30.192-37.070 30.192L-37.070 30.472L-38.147 30.540L-38.147 29.976Q-38.256 30.158-38.402 30.281Q-38.547 30.404-38.733 30.472Q-38.920 30.540-39.121 30.540Q-40.324 30.540-40.324 29.638M-34.815 30.472L-36.418 30.472L-36.418 30.192Q-36.192 30.192-36.043 30.158Q-35.895 30.123-35.895 29.983L-35.895 26.364Q-35.895 26.094-36.002 26.032Q-36.110 25.971-36.418 25.971L-36.418 25.690L-35.341 25.615L-35.341 29.983Q-35.341 30.120-35.191 30.156Q-35.040 30.192-34.815 30.192L-34.815 30.472M-34.162 29.744Q-34.162 29.412-33.938 29.185Q-33.714 28.958-33.370 28.830Q-33.027 28.701-32.654 28.649Q-32.282 28.596-31.978 28.596L-31.978 28.343Q-31.978 28.138-32.085 27.958Q-32.193 27.779-32.374 27.676Q-32.555 27.574-32.764 27.574Q-33.171 27.574-33.406 27.666Q-33.317 27.703-33.271 27.787Q-33.225 27.871-33.225 27.973Q-33.225 28.069-33.271 28.148Q-33.317 28.226-33.398 28.271Q-33.478 28.315-33.567 28.315Q-33.717 28.315-33.818 28.218Q-33.919 28.120-33.919 27.973Q-33.919 27.351-32.764 27.351Q-32.552 27.351-32.302 27.415Q-32.053 27.478-31.851 27.597Q-31.650 27.717-31.523 27.902Q-31.397 28.086-31.397 28.329L-31.397 29.905Q-31.397 30.021-31.335 30.117Q-31.274 30.212-31.161 30.212Q-31.051 30.212-30.986 30.118Q-30.921 30.024-30.921 29.905L-30.921 29.457L-30.655 29.457L-30.655 29.905Q-30.655 30.175-30.882 30.340Q-31.109 30.506-31.390 30.506Q-31.598 30.506-31.735 30.352Q-31.872 30.199-31.896 29.983Q-32.043 30.250-32.325 30.395Q-32.607 30.540-32.931 30.540Q-33.208 30.540-33.492 30.465Q-33.776 30.390-33.969 30.211Q-34.162 30.031-34.162 29.744M-33.546 29.744Q-33.546 29.918-33.446 30.048Q-33.345 30.178-33.189 30.248Q-33.034 30.318-32.870 30.318Q-32.651 30.318-32.442 30.221Q-32.234 30.123-32.106 29.942Q-31.978 29.761-31.978 29.535L-31.978 28.807Q-32.302 28.807-32.668 28.898Q-33.034 28.989-33.290 29.201Q-33.546 29.412-33.546 29.744M-29.712 29.631L-29.712 27.734L-30.351 27.734L-30.351 27.512Q-30.033 27.512-29.816 27.302Q-29.599 27.092-29.498 26.782Q-29.397 26.473-29.397 26.165L-29.130 26.165L-29.130 27.454L-28.054 27.454L-28.054 27.734L-29.130 27.734L-29.130 29.618Q-29.130 29.894-29.026 30.093Q-28.922 30.291-28.662 30.291Q-28.505 30.291-28.399 30.187Q-28.293 30.082-28.244 29.929Q-28.194 29.775-28.194 29.618L-28.194 29.204L-27.927 29.204L-27.927 29.631Q-27.927 29.857-28.026 30.067Q-28.126 30.277-28.310 30.409Q-28.495 30.540-28.724 30.540Q-29.161 30.540-29.436 30.303Q-29.712 30.065-29.712 29.631M-27.158 28.937Q-27.158 28.616-27.034 28.327Q-26.909 28.038-26.683 27.815Q-26.458 27.591-26.162 27.471Q-25.866 27.351-25.548 27.351Q-25.220 27.351-24.959 27.451Q-24.697 27.550-24.521 27.732Q-24.345 27.915-24.251 28.173Q-24.157 28.431-24.157 28.763Q-24.157 28.855-24.239 28.876L-26.495 28.876L-26.495 28.937Q-26.495 29.525-26.212 29.908Q-25.928 30.291-25.360 30.291Q-25.039 30.291-24.771 30.098Q-24.503 29.905-24.414 29.590Q-24.407 29.549-24.332 29.535L-24.239 29.535Q-24.157 29.559-24.157 29.631Q-24.157 29.638-24.164 29.665Q-24.277 30.062-24.648 30.301Q-25.019 30.540-25.442 30.540Q-25.880 30.540-26.280 30.332Q-26.680 30.123-26.919 29.756Q-27.158 29.389-27.158 28.937M-26.488 28.667L-24.673 28.667Q-24.673 28.390-24.771 28.138Q-24.868 27.885-25.067 27.729Q-25.265 27.574-25.548 27.574Q-25.825 27.574-26.039 27.732Q-26.253 27.891-26.370 28.146Q-26.488 28.401-26.488 28.667\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-20.330 29.631L-20.330 27.734L-20.969 27.734L-20.969 27.512Q-20.651 27.512-20.434 27.302Q-20.217 27.092-20.117 26.782Q-20.016 26.473-20.016 26.165L-19.749 26.165L-19.749 27.454L-18.672 27.454L-18.672 27.734L-19.749 27.734L-19.749 29.618Q-19.749 29.894-19.645 30.093Q-19.541 30.291-19.281 30.291Q-19.124 30.291-19.018 30.187Q-18.912 30.082-18.862 29.929Q-18.813 29.775-18.813 29.618L-18.813 29.204L-18.546 29.204L-18.546 29.631Q-18.546 29.857-18.645 30.067Q-18.744 30.277-18.929 30.409Q-19.113 30.540-19.342 30.540Q-19.780 30.540-20.055 30.303Q-20.330 30.065-20.330 29.631M-17.777 28.989Q-17.777 28.647-17.642 28.348Q-17.507 28.049-17.268 27.825Q-17.028 27.601-16.711 27.476Q-16.393 27.351-16.061 27.351Q-15.617 27.351-15.217 27.567Q-14.817 27.782-14.583 28.160Q-14.349 28.537-14.349 28.989Q-14.349 29.330-14.491 29.614Q-14.632 29.898-14.877 30.105Q-15.121 30.311-15.431 30.426Q-15.740 30.540-16.061 30.540Q-16.492 30.540-16.893 30.339Q-17.295 30.137-17.536 29.785Q-17.777 29.433-17.777 28.989M-16.061 30.291Q-15.460 30.291-15.236 29.913Q-15.012 29.535-15.012 28.903Q-15.012 28.291-15.246 27.932Q-15.480 27.574-16.061 27.574Q-17.114 27.574-17.114 28.903Q-17.114 29.535-16.888 29.913Q-16.663 30.291-16.061 30.291\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-10.245 30.472L-10.512 30.472L-10.512 26.364Q-10.512 26.094-10.619 26.032Q-10.727 25.971-11.038 25.971L-11.038 25.690L-9.958 25.615L-9.958 27.785Q-9.749 27.594-9.464 27.490Q-9.178 27.386-8.881 27.386Q-8.563 27.386-8.266 27.507Q-7.969 27.628-7.746 27.844Q-7.524 28.059-7.398 28.344Q-7.271 28.630-7.271 28.961Q-7.271 29.406-7.511 29.770Q-7.750 30.134-8.143 30.337Q-8.536 30.540-8.980 30.540Q-9.175 30.540-9.365 30.484Q-9.554 30.428-9.715 30.323Q-9.876 30.219-10.016 30.058L-10.245 30.472M-9.930 28.127L-9.930 29.744Q-9.794 30.004-9.553 30.161Q-9.312 30.318-9.035 30.318Q-8.741 30.318-8.529 30.211Q-8.317 30.103-8.184 29.911Q-8.051 29.720-7.992 29.481Q-7.934 29.242-7.934 28.961Q-7.934 28.602-8.028 28.298Q-8.122 27.994-8.350 27.801Q-8.577 27.608-8.943 27.608Q-9.243 27.608-9.510 27.744Q-9.777 27.881-9.930 28.127\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M-6.461 28.937Q-6.461 28.616-6.336 28.327Q-6.211 28.038-5.985 27.815Q-5.760 27.591-5.464 27.471Q-5.169 27.351-4.851 27.351Q-4.523 27.351-4.261 27.451Q-4 27.550-3.824 27.732Q-3.648 27.915-3.554 28.173Q-3.460 28.431-3.460 28.763Q-3.460 28.855-3.542 28.876L-5.797 28.876L-5.797 28.937Q-5.797 29.525-5.514 29.908Q-5.230 30.291-4.663 30.291Q-4.341 30.291-4.073 30.098Q-3.805 29.905-3.716 29.590Q-3.709 29.549-3.634 29.535L-3.542 29.535Q-3.460 29.559-3.460 29.631Q-3.460 29.638-3.466 29.665Q-3.579 30.062-3.950 30.301Q-4.321 30.540-4.745 30.540Q-5.182 30.540-5.582 30.332Q-5.982 30.123-6.221 29.756Q-6.461 29.389-6.461 28.937M-5.791 28.667L-3.976 28.667Q-3.976 28.390-4.073 28.138Q-4.171 27.885-4.369 27.729Q-4.567 27.574-4.851 27.574Q-5.128 27.574-5.341 27.732Q-5.555 27.891-5.673 28.146Q-5.791 28.401-5.791 28.667M-2.872 30.465L-2.872 29.402Q-2.872 29.378-2.844 29.351Q-2.817 29.324-2.793 29.324L-2.684 29.324Q-2.619 29.324-2.605 29.382Q-2.509 29.816-2.263 30.067Q-2.017 30.318-1.604 30.318Q-1.262 30.318-1.009 30.185Q-0.756 30.052-0.756 29.744Q-0.756 29.587-0.850 29.472Q-0.944 29.358-1.082 29.289Q-1.221 29.221-1.388 29.183L-1.969 29.084Q-2.325 29.016-2.598 28.795Q-2.872 28.575-2.872 28.233Q-2.872 27.984-2.761 27.809Q-2.650 27.635-2.463 27.536Q-2.277 27.437-2.062 27.394Q-1.846 27.351-1.604 27.351Q-1.190 27.351-0.910 27.533L-0.694 27.358Q-0.684 27.355-0.677 27.353Q-0.671 27.351-0.660 27.351L-0.609 27.351Q-0.582 27.351-0.558 27.375Q-0.534 27.399-0.534 27.427L-0.534 28.274Q-0.534 28.295-0.558 28.322Q-0.582 28.349-0.609 28.349L-0.722 28.349Q-0.749 28.349-0.775 28.324Q-0.800 28.298-0.800 28.274Q-0.800 28.038-0.906 27.874Q-1.012 27.710-1.195 27.628Q-1.378 27.546-1.610 27.546Q-1.939 27.546-2.195 27.649Q-2.451 27.751-2.451 28.028Q-2.451 28.223-2.268 28.332Q-2.086 28.442-1.857 28.483L-1.282 28.589Q-1.036 28.637-0.823 28.765Q-0.609 28.893-0.472 29.096Q-0.336 29.300-0.336 29.549Q-0.336 30.062-0.701 30.301Q-1.067 30.540-1.604 30.540Q-2.099 30.540-2.431 30.246L-2.697 30.520Q-2.718 30.540-2.745 30.540L-2.793 30.540Q-2.817 30.540-2.844 30.513Q-2.872 30.486-2.872 30.465M0.820 29.631L0.820 27.734L0.181 27.734L0.181 27.512Q0.498 27.512 0.715 27.302Q0.933 27.092 1.033 26.782Q1.134 26.473 1.134 26.165L1.401 26.165L1.401 27.454L2.477 27.454L2.477 27.734L1.401 27.734L1.401 29.618Q1.401 29.894 1.505 30.093Q1.609 30.291 1.869 30.291Q2.026 30.291 2.132 30.187Q2.238 30.082 2.288 29.929Q2.337 29.775 2.337 29.618L2.337 29.204L2.604 29.204L2.604 29.631Q2.604 29.857 2.505 30.067Q2.406 30.277 2.221 30.409Q2.037 30.540 1.808 30.540Q1.370 30.540 1.095 30.303Q0.820 30.065 0.820 29.631\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.966 49.115)\">\u003Cpath d=\"M6.083 31.005Q6.083 30.759 6.280 30.575Q6.477 30.390 6.733 30.311Q6.596 30.199 6.524 30.038Q6.453 29.877 6.453 29.696Q6.453 29.375 6.664 29.129Q6.330 28.831 6.330 28.421Q6.330 27.960 6.719 27.673Q7.109 27.386 7.587 27.386Q8.059 27.386 8.394 27.632Q8.568 27.478 8.779 27.396Q8.989 27.314 9.218 27.314Q9.382 27.314 9.503 27.421Q9.624 27.529 9.624 27.693Q9.624 27.789 9.553 27.861Q9.481 27.932 9.389 27.932Q9.289 27.932 9.219 27.859Q9.149 27.785 9.149 27.686Q9.149 27.632 9.163 27.601L9.170 27.587Q9.177 27.567 9.185 27.556Q9.194 27.546 9.197 27.539Q8.842 27.539 8.555 27.762Q8.842 28.055 8.842 28.421Q8.842 28.736 8.657 28.968Q8.473 29.201 8.184 29.329Q7.895 29.457 7.587 29.457Q7.386 29.457 7.194 29.407Q7.003 29.358 6.825 29.248Q6.733 29.375 6.733 29.518Q6.733 29.700 6.861 29.835Q6.989 29.970 7.174 29.970L7.806 29.970Q8.254 29.970 8.623 30.041Q8.992 30.113 9.252 30.342Q9.512 30.571 9.512 31.005Q9.512 31.326 9.216 31.528Q8.920 31.730 8.517 31.819Q8.114 31.908 7.799 31.908Q7.481 31.908 7.078 31.819Q6.675 31.730 6.379 31.528Q6.083 31.326 6.083 31.005M6.538 31.005Q6.538 31.234 6.757 31.383Q6.976 31.532 7.268 31.600Q7.560 31.668 7.799 31.668Q7.963 31.668 8.172 31.632Q8.380 31.597 8.587 31.516Q8.794 31.436 8.925 31.308Q9.057 31.180 9.057 31.005Q9.057 30.653 8.676 30.559Q8.295 30.465 7.792 30.465L7.174 30.465Q6.935 30.465 6.736 30.616Q6.538 30.766 6.538 31.005M7.587 29.218Q8.254 29.218 8.254 28.421Q8.254 27.621 7.587 27.621Q6.917 27.621 6.917 28.421Q6.917 29.218 7.587 29.218M10.065 28.989Q10.065 28.647 10.200 28.348Q10.335 28.049 10.575 27.825Q10.814 27.601 11.132 27.476Q11.450 27.351 11.781 27.351Q12.226 27.351 12.625 27.567Q13.025 27.782 13.259 28.160Q13.494 28.537 13.494 28.989Q13.494 29.330 13.352 29.614Q13.210 29.898 12.966 30.105Q12.721 30.311 12.412 30.426Q12.102 30.540 11.781 30.540Q11.351 30.540 10.949 30.339Q10.547 30.137 10.306 29.785Q10.065 29.433 10.065 28.989M11.781 30.291Q12.383 30.291 12.607 29.913Q12.831 29.535 12.831 28.903Q12.831 28.291 12.596 27.932Q12.362 27.574 11.781 27.574Q10.728 27.574 10.728 28.903Q10.728 29.535 10.954 29.913Q11.180 30.291 11.781 30.291M14.146 29.744Q14.146 29.412 14.370 29.185Q14.594 28.958 14.938 28.830Q15.281 28.701 15.654 28.649Q16.026 28.596 16.331 28.596L16.331 28.343Q16.331 28.138 16.223 27.958Q16.115 27.779 15.934 27.676Q15.753 27.574 15.544 27.574Q15.138 27.574 14.902 27.666Q14.991 27.703 15.037 27.787Q15.083 27.871 15.083 27.973Q15.083 28.069 15.037 28.148Q14.991 28.226 14.910 28.271Q14.830 28.315 14.741 28.315Q14.591 28.315 14.490 28.218Q14.389 28.120 14.389 27.973Q14.389 27.351 15.544 27.351Q15.756 27.351 16.006 27.415Q16.255 27.478 16.457 27.597Q16.659 27.717 16.785 27.902Q16.912 28.086 16.912 28.329L16.912 29.905Q16.912 30.021 16.973 30.117Q17.035 30.212 17.147 30.212Q17.257 30.212 17.322 30.118Q17.387 30.024 17.387 29.905L17.387 29.457L17.653 29.457L17.653 29.905Q17.653 30.175 17.426 30.340Q17.199 30.506 16.918 30.506Q16.710 30.506 16.573 30.352Q16.436 30.199 16.413 29.983Q16.266 30.250 15.984 30.395Q15.702 30.540 15.377 30.540Q15.100 30.540 14.816 30.465Q14.533 30.390 14.340 30.211Q14.146 30.031 14.146 29.744M14.762 29.744Q14.762 29.918 14.862 30.048Q14.963 30.178 15.119 30.248Q15.274 30.318 15.438 30.318Q15.657 30.318 15.866 30.221Q16.074 30.123 16.202 29.942Q16.331 29.761 16.331 29.535L16.331 28.807Q16.006 28.807 15.640 28.898Q15.274 28.989 15.018 29.201Q14.762 29.412 14.762 29.744M19.738 30.472L18.135 30.472L18.135 30.192Q18.361 30.192 18.509 30.158Q18.658 30.123 18.658 29.983L18.658 26.364Q18.658 26.094 18.550 26.032Q18.443 25.971 18.135 25.971L18.135 25.690L19.212 25.615L19.212 29.983Q19.212 30.120 19.362 30.156Q19.513 30.192 19.738 30.192\" 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\">A model-free rat caches an action value per state-action pair (left); a model-based rat holds a transition tree plus a reward model and simulates paths to the best goal box (right). With rewards 0, 4, 2, 3, both pick L then R from S1 for a return of 4.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:303.784px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 227.838 159.146\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M-23.090 35.580Q-23.090 35.040-22.657 34.706Q-22.224 34.372-21.618 34.233Q-21.011 34.095-20.480 34.095L-20.480 33.761Q-20.480 33.502-20.598 33.258Q-20.717 33.014-20.926 32.867Q-21.134 32.719-21.407 32.719Q-21.969 32.719-22.281 32.917Q-22.132 32.944-22.040 33.062Q-21.947 33.181-21.947 33.339Q-21.947 33.515-22.073 33.645Q-22.198 33.774-22.378 33.774Q-22.567 33.774-22.694 33.647Q-22.822 33.519-22.822 33.339Q-22.822 32.869-22.382 32.662Q-21.943 32.456-21.407 32.456Q-21.117 32.456-20.829 32.544Q-20.541 32.632-20.308 32.792Q-20.075 32.952-19.926 33.192Q-19.776 33.431-19.776 33.726L-19.776 35.761Q-19.776 35.914-19.702 36.053Q-19.627 36.191-19.482 36.191Q-19.328 36.191-19.256 36.055Q-19.183 35.919-19.183 35.761L-19.183 35.185L-18.897 35.185L-18.897 35.761Q-18.897 36.094-19.146 36.319Q-19.394 36.543-19.724 36.543Q-19.983 36.543-20.165 36.349Q-20.348 36.156-20.392 35.879Q-20.559 36.200-20.893 36.396Q-21.227 36.591-21.596 36.591Q-22.149 36.591-22.620 36.343Q-23.090 36.094-23.090 35.580M-22.334 35.580Q-22.334 35.892-22.092 36.110Q-21.851 36.327-21.534 36.327Q-21.099 36.327-20.789 36.018Q-20.480 35.708-20.480 35.277L-20.480 34.350Q-20.897 34.350-21.323 34.477Q-21.750 34.605-22.042 34.882Q-22.334 35.158-22.334 35.580M-16.300 36.490L-18.533 36.490L-18.533 36.174Q-18.225 36.174-18.034 36.121Q-17.843 36.068-17.843 35.879L-17.843 32.926L-18.533 32.926L-18.533 32.610L-17.843 32.610L-17.843 31.542Q-17.843 31.159-17.630 30.836Q-17.417 30.513-17.063 30.329Q-16.709 30.144-16.331 30.144Q-16.010 30.144-15.760 30.322Q-15.509 30.500-15.509 30.812Q-15.509 30.988-15.628 31.107Q-15.747 31.225-15.922 31.225Q-16.103 31.225-16.226 31.107Q-16.349 30.988-16.349 30.812Q-16.349 30.571-16.133 30.443Q-16.239 30.408-16.375 30.408Q-16.643 30.408-16.825 30.590Q-17.008 30.773-17.100 31.043Q-17.192 31.313-17.192 31.577L-17.192 32.610L-16.142 32.610L-16.142 32.926L-17.166 32.926L-17.166 35.879Q-17.166 36.068-16.909 36.121Q-16.652 36.174-16.300 36.174L-16.300 36.490M-15.079 35.418L-15.079 32.926L-15.843 32.926L-15.843 32.667Q-15.439 32.667-15.173 32.401Q-14.907 32.135-14.786 31.735Q-14.666 31.335-14.666 30.953L-14.376 30.953L-14.376 32.610L-13.088 32.610L-13.088 32.926L-14.376 32.926L-14.376 35.383Q-14.376 35.752-14.250 36.026Q-14.125 36.301-13.800 36.301Q-13.501 36.301-13.363 36.007Q-13.224 35.712-13.224 35.383L-13.224 34.860L-12.938 34.860L-12.938 35.418Q-12.938 35.695-13.048 35.967Q-13.158 36.240-13.371 36.415Q-13.584 36.591-13.866 36.591Q-14.226 36.591-14.499 36.453Q-14.771 36.314-14.925 36.051Q-15.079 35.787-15.079 35.418M-10.113 36.591Q-10.671 36.591-11.143 36.308Q-11.616 36.024-11.890 35.547Q-12.165 35.071-12.165 34.517Q-12.165 34.121-12.022 33.746Q-11.879 33.370-11.622 33.082Q-11.365 32.794-11.007 32.625Q-10.649 32.456-10.245 32.456Q-9.700 32.456-9.328 32.693Q-8.957 32.930-8.770 33.348Q-8.584 33.765-8.584 34.302Q-8.584 34.354-8.608 34.392Q-8.632 34.429-8.680 34.429L-11.352 34.429L-11.352 34.508Q-11.352 35.255-11.040 35.778Q-10.728 36.301-10.029 36.301Q-9.625 36.301-9.304 36.044Q-8.983 35.787-8.860 35.383Q-8.843 35.303-8.759 35.303L-8.680 35.303Q-8.641 35.303-8.612 35.334Q-8.584 35.365-8.584 35.409L-8.584 35.444Q-8.689 35.787-8.911 36.046Q-9.133 36.305-9.447 36.448Q-9.761 36.591-10.113 36.591M-11.343 34.178L-9.230 34.178Q-9.230 33.910-9.282 33.664Q-9.335 33.418-9.456 33.196Q-9.577 32.974-9.774 32.847Q-9.972 32.719-10.245 32.719Q-10.587 32.719-10.840 32.944Q-11.093 33.168-11.218 33.506Q-11.343 33.844-11.343 34.178M-5.828 36.490L-8.061 36.490L-8.061 36.174Q-7.749 36.174-7.557 36.121Q-7.366 36.068-7.366 35.879L-7.366 33.431Q-7.366 33.190-7.437 33.082Q-7.507 32.974-7.641 32.950Q-7.775 32.926-8.061 32.926L-8.061 32.610L-6.747 32.513L-6.747 33.374Q-6.584 32.983-6.316 32.748Q-6.048 32.513-5.657 32.513Q-5.384 32.513-5.169 32.676Q-4.954 32.838-4.954 33.097Q-4.954 33.273-5.072 33.392Q-5.191 33.511-5.367 33.511Q-5.547 33.511-5.666 33.392Q-5.784 33.273-5.784 33.097Q-5.784 32.882-5.630 32.772L-5.648 32.772Q-6.026 32.772-6.259 33.034Q-6.492 33.295-6.591 33.682Q-6.689 34.069-6.689 34.429L-6.689 35.879Q-6.689 36.068-6.432 36.121Q-6.175 36.174-5.828 36.174\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M0.655 36.591Q0.110 36.591-0.334 36.308Q-0.778 36.024-1.032 35.552Q-1.287 35.079-1.287 34.548Q-1.287 33.994-1.011 33.528Q-0.734 33.062-0.261 32.788Q0.211 32.513 0.756 32.513Q1.090 32.513 1.393 32.645Q1.697 32.777 1.916 33.014L1.916 31.164Q1.916 30.922 1.846 30.814Q1.776 30.707 1.642 30.683Q1.508 30.658 1.222 30.658L1.222 30.342L2.589 30.245L2.589 35.673Q2.589 35.910 2.659 36.018Q2.729 36.125 2.865 36.149Q3.002 36.174 3.283 36.174L3.283 36.490L1.890 36.591L1.890 36.042Q1.639 36.305 1.321 36.448Q1.002 36.591 0.655 36.591M0.717 36.327Q1.086 36.327 1.395 36.116Q1.705 35.906 1.890 35.563L1.890 33.431Q1.718 33.128 1.437 32.950Q1.156 32.772 0.818 32.772Q0.128 32.772-0.176 33.293Q-0.479 33.814-0.479 34.556Q-0.479 35.277-0.209 35.802Q0.062 36.327 0.717 36.327M5.854 36.591Q5.296 36.591 4.823 36.308Q4.351 36.024 4.076 35.547Q3.801 35.071 3.801 34.517Q3.801 34.121 3.944 33.746Q4.087 33.370 4.344 33.082Q4.601 32.794 4.959 32.625Q5.318 32.456 5.722 32.456Q6.267 32.456 6.638 32.693Q7.009 32.930 7.196 33.348Q7.383 33.765 7.383 34.302Q7.383 34.354 7.359 34.392Q7.335 34.429 7.286 34.429L4.614 34.429L4.614 34.508Q4.614 35.255 4.926 35.778Q5.238 36.301 5.937 36.301Q6.342 36.301 6.662 36.044Q6.983 35.787 7.106 35.383Q7.124 35.303 7.207 35.303L7.286 35.303Q7.326 35.303 7.354 35.334Q7.383 35.365 7.383 35.409L7.383 35.444Q7.278 35.787 7.056 36.046Q6.834 36.305 6.520 36.448Q6.205 36.591 5.854 36.591M4.623 34.178L6.737 34.178Q6.737 33.910 6.684 33.664Q6.632 33.418 6.511 33.196Q6.390 32.974 6.192 32.847Q5.994 32.719 5.722 32.719Q5.379 32.719 5.126 32.944Q4.874 33.168 4.749 33.506Q4.623 33.844 4.623 34.178M9.888 36.472L8.556 33.225Q8.468 33.027 8.304 32.977Q8.139 32.926 7.836 32.926L7.836 32.610L9.760 32.610L9.760 32.926Q9.268 32.926 9.268 33.141Q9.268 33.163 9.286 33.225L10.301 35.699L11.211 33.475Q11.246 33.392 11.246 33.295Q11.246 33.124 11.123 33.025Q11 32.926 10.833 32.926L10.833 32.610L12.344 32.610L12.344 32.926Q12.059 32.926 11.846 33.069Q11.633 33.212 11.527 33.475L10.292 36.472Q10.248 36.591 10.121 36.591L10.059 36.591Q9.932 36.591 9.888 36.472\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M12.396 35.580Q12.396 35.040 12.829 34.706Q13.262 34.372 13.868 34.233Q14.475 34.095 15.006 34.095L15.006 33.761Q15.006 33.502 14.888 33.258Q14.769 33.014 14.560 32.867Q14.352 32.719 14.079 32.719Q13.517 32.719 13.205 32.917Q13.354 32.944 13.446 33.062Q13.539 33.181 13.539 33.339Q13.539 33.515 13.413 33.645Q13.288 33.774 13.108 33.774Q12.919 33.774 12.792 33.647Q12.664 33.519 12.664 33.339Q12.664 32.869 13.104 32.662Q13.543 32.456 14.079 32.456Q14.369 32.456 14.657 32.544Q14.945 32.632 15.178 32.792Q15.411 32.952 15.560 33.192Q15.710 33.431 15.710 33.726L15.710 35.761Q15.710 35.914 15.784 36.053Q15.859 36.191 16.004 36.191Q16.158 36.191 16.230 36.055Q16.303 35.919 16.303 35.761L16.303 35.185L16.589 35.185L16.589 35.761Q16.589 36.094 16.340 36.319Q16.092 36.543 15.762 36.543Q15.503 36.543 15.321 36.349Q15.138 36.156 15.094 35.879Q14.927 36.200 14.593 36.396Q14.259 36.591 13.890 36.591Q13.337 36.591 12.866 36.343Q12.396 36.094 12.396 35.580M13.152 35.580Q13.152 35.892 13.394 36.110Q13.635 36.327 13.952 36.327Q14.387 36.327 14.697 36.018Q15.006 35.708 15.006 35.277L15.006 34.350Q14.589 34.350 14.163 34.477Q13.736 34.605 13.444 34.882Q13.152 35.158 13.152 35.580M19.014 36.490L16.953 36.490L16.953 36.174Q17.261 36.174 17.452 36.121Q17.643 36.068 17.643 35.879L17.643 31.164Q17.643 30.922 17.573 30.814Q17.503 30.707 17.369 30.683Q17.235 30.658 16.953 30.658L16.953 30.342L18.320 30.245L18.320 35.879Q18.320 36.068 18.513 36.121Q18.707 36.174 19.014 36.174L19.014 36.490M20.201 35.418L20.201 33.431Q20.201 33.190 20.131 33.082Q20.060 32.974 19.926 32.950Q19.792 32.926 19.511 32.926L19.511 32.610L20.904 32.513L20.904 35.383Q20.904 35.761 20.950 35.947Q20.996 36.134 21.168 36.231Q21.339 36.327 21.695 36.327Q22.029 36.327 22.268 36.136Q22.508 35.945 22.633 35.642Q22.758 35.339 22.758 35.022L22.758 33.431Q22.758 33.190 22.688 33.082Q22.618 32.974 22.484 32.950Q22.350 32.926 22.064 32.926L22.064 32.610L23.462 32.513L23.462 35.673Q23.462 35.910 23.532 36.018Q23.602 36.125 23.736 36.149Q23.870 36.174 24.152 36.174L24.152 36.490L22.785 36.591L22.785 35.870Q22.618 36.196 22.312 36.393Q22.007 36.591 21.651 36.591Q20.992 36.591 20.596 36.321Q20.201 36.051 20.201 35.418M24.718 35.580Q24.718 35.040 25.151 34.706Q25.584 34.372 26.191 34.233Q26.797 34.095 27.329 34.095L27.329 33.761Q27.329 33.502 27.210 33.258Q27.091 33.014 26.883 32.867Q26.674 32.719 26.402 32.719Q25.839 32.719 25.527 32.917Q25.676 32.944 25.769 33.062Q25.861 33.181 25.861 33.339Q25.861 33.515 25.736 33.645Q25.610 33.774 25.430 33.774Q25.241 33.774 25.114 33.647Q24.986 33.519 24.986 33.339Q24.986 32.869 25.426 32.662Q25.865 32.456 26.402 32.456Q26.692 32.456 26.979 32.544Q27.267 32.632 27.500 32.792Q27.733 32.952 27.882 33.192Q28.032 33.431 28.032 33.726L28.032 35.761Q28.032 35.914 28.107 36.053Q28.181 36.191 28.326 36.191Q28.480 36.191 28.553 36.055Q28.625 35.919 28.625 35.761L28.625 35.185L28.911 35.185L28.911 35.761Q28.911 36.094 28.662 36.319Q28.414 36.543 28.085 36.543Q27.825 36.543 27.643 36.349Q27.461 36.156 27.417 35.879Q27.250 36.200 26.916 36.396Q26.582 36.591 26.213 36.591Q25.659 36.591 25.189 36.343Q24.718 36.094 24.718 35.580M25.474 35.580Q25.474 35.892 25.716 36.110Q25.958 36.327 26.274 36.327Q26.709 36.327 27.019 36.018Q27.329 35.708 27.329 35.277L27.329 34.350Q26.911 34.350 26.485 34.477Q26.059 34.605 25.767 34.882Q25.474 35.158 25.474 35.580M29.904 35.418L29.904 32.926L29.139 32.926L29.139 32.667Q29.544 32.667 29.809 32.401Q30.075 32.135 30.196 31.735Q30.317 31.335 30.317 30.953L30.607 30.953L30.607 32.610L31.895 32.610L31.895 32.926L30.607 32.926L30.607 35.383Q30.607 35.752 30.732 36.026Q30.858 36.301 31.183 36.301Q31.482 36.301 31.620 36.007Q31.758 35.712 31.758 35.383L31.758 34.860L32.044 34.860L32.044 35.418Q32.044 35.695 31.934 35.967Q31.824 36.240 31.611 36.415Q31.398 36.591 31.117 36.591Q30.756 36.591 30.484 36.453Q30.212 36.314 30.058 36.051Q29.904 35.787 29.904 35.418M34.852 36.490L32.866 36.490L32.866 36.174Q33.173 36.174 33.365 36.121Q33.556 36.068 33.556 35.879L33.556 33.431Q33.556 33.185 33.490 33.080Q33.424 32.974 33.299 32.950Q33.173 32.926 32.901 32.926L32.901 32.610L34.233 32.513L34.233 35.879Q34.233 36.073 34.397 36.123Q34.562 36.174 34.852 36.174L34.852 36.490M33.253 30.966Q33.253 30.760 33.402 30.610Q33.551 30.461 33.754 30.461Q33.885 30.461 34.002 30.531Q34.118 30.601 34.189 30.718Q34.259 30.834 34.259 30.966Q34.259 31.168 34.110 31.318Q33.960 31.467 33.754 31.467Q33.551 31.467 33.402 31.318Q33.253 31.168 33.253 30.966M35.384 34.583Q35.384 34.016 35.656 33.528Q35.929 33.040 36.399 32.748Q36.869 32.456 37.436 32.456Q37.858 32.456 38.234 32.625Q38.610 32.794 38.886 33.086Q39.163 33.379 39.321 33.774Q39.480 34.170 39.480 34.583Q39.480 35.132 39.201 35.594Q38.922 36.055 38.454 36.323Q37.985 36.591 37.436 36.591Q36.882 36.591 36.412 36.323Q35.942 36.055 35.663 35.594Q35.384 35.132 35.384 34.583M37.436 36.301Q37.933 36.301 38.210 36.040Q38.486 35.778 38.579 35.374Q38.671 34.969 38.671 34.473Q38.671 33.998 38.572 33.609Q38.473 33.220 38.201 32.970Q37.928 32.719 37.436 32.719Q36.724 32.719 36.461 33.214Q36.197 33.708 36.197 34.473Q36.197 35.273 36.452 35.787Q36.707 36.301 37.436 36.301M42.130 36.490L40.042 36.490L40.042 36.174Q40.350 36.174 40.541 36.121Q40.732 36.068 40.732 35.879L40.732 33.431Q40.732 33.190 40.662 33.082Q40.591 32.974 40.457 32.950Q40.323 32.926 40.042 32.926L40.042 32.610L41.382 32.513L41.382 33.348Q41.580 32.966 41.934 32.739Q42.288 32.513 42.714 32.513Q43.993 32.513 43.993 33.726L43.993 35.879Q43.993 36.068 44.184 36.121Q44.375 36.174 44.683 36.174L44.683 36.490L42.595 36.490L42.595 36.174Q42.907 36.174 43.099 36.121Q43.290 36.068 43.290 35.879L43.290 33.761Q43.290 33.502 43.246 33.280Q43.202 33.058 43.057 32.915Q42.912 32.772 42.652 32.772Q42.310 32.772 42.028 32.961Q41.747 33.150 41.591 33.462Q41.435 33.774 41.435 34.121L41.435 35.879Q41.435 36.068 41.629 36.121Q41.822 36.174 42.130 36.174L42.130 36.490M45.654 35.985Q45.654 35.857 45.722 35.741Q45.790 35.624 45.907 35.554Q46.023 35.484 46.159 35.484Q46.361 35.484 46.513 35.631Q46.665 35.778 46.665 35.985Q46.665 36.191 46.515 36.341Q46.366 36.490 46.159 36.490Q45.948 36.490 45.801 36.338Q45.654 36.187 45.654 35.985M45.654 33.115Q45.654 32.917 45.799 32.763Q45.944 32.610 46.159 32.610Q46.296 32.610 46.412 32.678Q46.528 32.746 46.597 32.862Q46.665 32.979 46.665 33.115Q46.665 33.317 46.513 33.469Q46.361 33.620 46.159 33.620Q45.953 33.620 45.803 33.467Q45.654 33.313 45.654 33.115\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M54.079 36.490L51.846 36.490L51.846 36.174Q52.158 36.174 52.350 36.121Q52.541 36.068 52.541 35.879L52.541 33.431Q52.541 33.190 52.471 33.082Q52.400 32.974 52.266 32.950Q52.132 32.926 51.846 32.926L51.846 32.610L53.160 32.513L53.160 33.374Q53.323 32.983 53.591 32.748Q53.859 32.513 54.250 32.513Q54.523 32.513 54.738 32.676Q54.953 32.838 54.953 33.097Q54.953 33.273 54.835 33.392Q54.716 33.511 54.540 33.511Q54.360 33.511 54.242 33.392Q54.123 33.273 54.123 33.097Q54.123 32.882 54.277 32.772L54.259 32.772Q53.881 32.772 53.648 33.034Q53.415 33.295 53.316 33.682Q53.218 34.069 53.218 34.429L53.218 35.879Q53.218 36.068 53.475 36.121Q53.732 36.174 54.079 36.174L54.079 36.490M57.524 36.591Q56.966 36.591 56.494 36.308Q56.021 36.024 55.747 35.547Q55.472 35.071 55.472 34.517Q55.472 34.121 55.615 33.746Q55.758 33.370 56.015 33.082Q56.272 32.794 56.630 32.625Q56.988 32.456 57.392 32.456Q57.937 32.456 58.309 32.693Q58.680 32.930 58.867 33.348Q59.054 33.765 59.054 34.302Q59.054 34.354 59.029 34.392Q59.005 34.429 58.957 34.429L56.285 34.429L56.285 34.508Q56.285 35.255 56.597 35.778Q56.909 36.301 57.608 36.301Q58.012 36.301 58.333 36.044Q58.654 35.787 58.777 35.383Q58.794 35.303 58.878 35.303L58.957 35.303Q58.996 35.303 59.025 35.334Q59.054 35.365 59.054 35.409L59.054 35.444Q58.948 35.787 58.726 36.046Q58.504 36.305 58.190 36.448Q57.876 36.591 57.524 36.591M56.294 34.178L58.408 34.178Q58.408 33.910 58.355 33.664Q58.302 33.418 58.181 33.196Q58.060 32.974 57.863 32.847Q57.665 32.719 57.392 32.719Q57.050 32.719 56.797 32.944Q56.544 33.168 56.419 33.506Q56.294 33.844 56.294 34.178M61.892 34.842L59.427 34.842L59.427 34.249L61.892 34.249L61.892 34.842M64.740 38.235L62.653 38.235L62.653 37.923Q62.965 37.923 63.156 37.872Q63.347 37.822 63.347 37.624L63.347 33.286Q63.347 33.045 63.173 32.985Q63 32.926 62.653 32.926L62.653 32.610L64.024 32.513L64.024 33.053Q64.283 32.785 64.617 32.649Q64.951 32.513 65.320 32.513Q65.720 32.513 66.076 32.680Q66.432 32.847 66.687 33.128Q66.942 33.409 67.085 33.774Q67.227 34.139 67.227 34.548Q67.227 35.110 66.950 35.576Q66.674 36.042 66.199 36.316Q65.724 36.591 65.175 36.591Q64.863 36.591 64.573 36.457Q64.283 36.323 64.050 36.077L64.050 37.624Q64.050 37.822 64.241 37.872Q64.432 37.923 64.740 37.923L64.740 38.235M64.050 33.467L64.050 35.598Q64.208 35.923 64.492 36.125Q64.775 36.327 65.118 36.327Q65.531 36.327 65.825 36.051Q66.120 35.774 66.267 35.356Q66.414 34.939 66.414 34.548Q66.414 34.170 66.280 33.761Q66.146 33.352 65.872 33.075Q65.597 32.799 65.219 32.799Q64.977 32.799 64.762 32.875Q64.547 32.952 64.356 33.113Q64.164 33.273 64.050 33.467M69.908 36.490L67.847 36.490L67.847 36.174Q68.155 36.174 68.346 36.121Q68.537 36.068 68.537 35.879L68.537 31.164Q68.537 30.922 68.467 30.814Q68.396 30.707 68.262 30.683Q68.128 30.658 67.847 30.658L67.847 30.342L69.214 30.245L69.214 35.879Q69.214 36.068 69.407 36.121Q69.600 36.174 69.908 36.174L69.908 36.490M70.475 35.580Q70.475 35.040 70.908 34.706Q71.341 34.372 71.947 34.233Q72.554 34.095 73.085 34.095L73.085 33.761Q73.085 33.502 72.967 33.258Q72.848 33.014 72.639 32.867Q72.430 32.719 72.158 32.719Q71.596 32.719 71.284 32.917Q71.433 32.944 71.525 33.062Q71.617 33.181 71.617 33.339Q71.617 33.515 71.492 33.645Q71.367 33.774 71.187 33.774Q70.998 33.774 70.870 33.647Q70.743 33.519 70.743 33.339Q70.743 32.869 71.182 32.662Q71.622 32.456 72.158 32.456Q72.448 32.456 72.736 32.544Q73.024 32.632 73.257 32.792Q73.490 32.952 73.639 33.192Q73.788 33.431 73.788 33.726L73.788 35.761Q73.788 35.914 73.863 36.053Q73.938 36.191 74.083 36.191Q74.237 36.191 74.309 36.055Q74.382 35.919 74.382 35.761L74.382 35.185L74.667 35.185L74.667 35.761Q74.667 36.094 74.419 36.319Q74.171 36.543 73.841 36.543Q73.582 36.543 73.399 36.349Q73.217 36.156 73.173 35.879Q73.006 36.200 72.672 36.396Q72.338 36.591 71.969 36.591Q71.415 36.591 70.945 36.343Q70.475 36.094 70.475 35.580M71.231 35.580Q71.231 35.892 71.472 36.110Q71.714 36.327 72.031 36.327Q72.466 36.327 72.775 36.018Q73.085 35.708 73.085 35.277L73.085 34.350Q72.668 34.350 72.242 34.477Q71.815 34.605 71.523 34.882Q71.231 35.158 71.231 35.580M77.111 36.490L75.023 36.490L75.023 36.174Q75.331 36.174 75.522 36.121Q75.713 36.068 75.713 35.879L75.713 33.431Q75.713 33.190 75.643 33.082Q75.573 32.974 75.439 32.950Q75.304 32.926 75.023 32.926L75.023 32.610L76.364 32.513L76.364 33.348Q76.561 32.966 76.915 32.739Q77.269 32.513 77.695 32.513Q78.974 32.513 78.974 33.726L78.974 35.879Q78.974 36.068 79.165 36.121Q79.356 36.174 79.664 36.174L79.664 36.490L77.576 36.490L77.576 36.174Q77.888 36.174 78.080 36.121Q78.271 36.068 78.271 35.879L78.271 33.761Q78.271 33.502 78.227 33.280Q78.183 33.058 78.038 32.915Q77.893 32.772 77.634 32.772Q77.291 32.772 77.010 32.961Q76.728 33.150 76.572 33.462Q76.416 33.774 76.416 34.121L76.416 35.879Q76.416 36.068 76.610 36.121Q76.803 36.174 77.111 36.174\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M85.507 36.490L83.275 36.490L83.275 36.174Q83.582 36.174 83.773 36.121Q83.965 36.068 83.965 35.879L83.965 32.926L83.275 32.926L83.275 32.610L83.965 32.610L83.965 31.542Q83.965 31.159 84.178 30.836Q84.391 30.513 84.745 30.329Q85.098 30.144 85.476 30.144Q85.797 30.144 86.048 30.322Q86.298 30.500 86.298 30.812Q86.298 30.988 86.179 31.107Q86.061 31.225 85.885 31.225Q85.705 31.225 85.582 31.107Q85.459 30.988 85.459 30.812Q85.459 30.571 85.674 30.443Q85.569 30.408 85.432 30.408Q85.164 30.408 84.982 30.590Q84.800 30.773 84.707 31.043Q84.615 31.313 84.615 31.577L84.615 32.610L85.665 32.610L85.665 32.926L84.641 32.926L84.641 35.879Q84.641 36.068 84.898 36.121Q85.155 36.174 85.507 36.174L85.507 36.490M88.276 36.490L86.043 36.490L86.043 36.174Q86.355 36.174 86.546 36.121Q86.738 36.068 86.738 35.879L86.738 33.431Q86.738 33.190 86.667 33.082Q86.597 32.974 86.463 32.950Q86.329 32.926 86.043 32.926L86.043 32.610L87.357 32.513L87.357 33.374Q87.520 32.983 87.788 32.748Q88.056 32.513 88.447 32.513Q88.719 32.513 88.935 32.676Q89.150 32.838 89.150 33.097Q89.150 33.273 89.031 33.392Q88.913 33.511 88.737 33.511Q88.557 33.511 88.438 33.392Q88.320 33.273 88.320 33.097Q88.320 32.882 88.473 32.772L88.456 32.772Q88.078 32.772 87.845 33.034Q87.612 33.295 87.513 33.682Q87.414 34.069 87.414 34.429L87.414 35.879Q87.414 36.068 87.671 36.121Q87.928 36.174 88.276 36.174L88.276 36.490M89.669 34.583Q89.669 34.016 89.941 33.528Q90.214 33.040 90.684 32.748Q91.154 32.456 91.721 32.456Q92.143 32.456 92.519 32.625Q92.894 32.794 93.171 33.086Q93.448 33.379 93.606 33.774Q93.764 34.170 93.764 34.583Q93.764 35.132 93.485 35.594Q93.206 36.055 92.738 36.323Q92.270 36.591 91.721 36.591Q91.167 36.591 90.697 36.323Q90.227 36.055 89.948 35.594Q89.669 35.132 89.669 34.583M91.721 36.301Q92.217 36.301 92.494 36.040Q92.771 35.778 92.863 35.374Q92.956 34.969 92.956 34.473Q92.956 33.998 92.857 33.609Q92.758 33.220 92.486 32.970Q92.213 32.719 91.721 32.719Q91.009 32.719 90.745 33.214Q90.482 33.708 90.482 34.473Q90.482 35.273 90.737 35.787Q90.991 36.301 91.721 36.301M96.414 36.490L94.327 36.490L94.327 36.174Q94.634 36.174 94.826 36.121Q95.017 36.068 95.017 35.879L95.017 33.431Q95.017 33.190 94.946 33.082Q94.876 32.974 94.742 32.950Q94.608 32.926 94.327 32.926L94.327 32.610L95.667 32.513L95.667 33.348Q95.865 32.970 96.225 32.741Q96.586 32.513 97.008 32.513Q98.053 32.513 98.238 33.322Q98.440 32.952 98.798 32.733Q99.156 32.513 99.574 32.513Q100.198 32.513 100.523 32.807Q100.848 33.102 100.848 33.726L100.848 35.879Q100.848 36.068 101.042 36.121Q101.235 36.174 101.543 36.174L101.543 36.490L99.455 36.490L99.455 36.174Q99.763 36.174 99.956 36.121Q100.150 36.068 100.150 35.879L100.150 33.761Q100.150 33.330 100.022 33.051Q99.895 32.772 99.508 32.772Q99.165 32.772 98.882 32.961Q98.598 33.150 98.442 33.462Q98.286 33.774 98.286 34.121L98.286 35.879Q98.286 36.068 98.477 36.121Q98.669 36.174 98.976 36.174L98.976 36.490L96.889 36.490L96.889 36.174Q97.201 36.174 97.392 36.121Q97.583 36.068 97.583 35.879L97.583 33.761Q97.583 33.502 97.539 33.280Q97.495 33.058 97.350 32.915Q97.205 32.772 96.946 32.772Q96.410 32.772 96.065 33.179Q95.720 33.585 95.720 34.121L95.720 35.879Q95.720 36.068 95.913 36.121Q96.107 36.174 96.414 36.174\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M105.087 34.583Q105.087 34.016 105.360 33.528Q105.632 33.040 106.102 32.748Q106.573 32.456 107.140 32.456Q107.561 32.456 107.937 32.625Q108.313 32.794 108.590 33.086Q108.867 33.379 109.025 33.774Q109.183 34.170 109.183 34.583Q109.183 35.132 108.904 35.594Q108.625 36.055 108.157 36.323Q107.689 36.591 107.140 36.591Q106.586 36.591 106.116 36.323Q105.645 36.055 105.366 35.594Q105.087 35.132 105.087 34.583M107.140 36.301Q107.636 36.301 107.913 36.040Q108.190 35.778 108.282 35.374Q108.374 34.969 108.374 34.473Q108.374 33.998 108.276 33.609Q108.177 33.220 107.904 32.970Q107.632 32.719 107.140 32.719Q106.428 32.719 106.164 33.214Q105.900 33.708 105.900 34.473Q105.900 35.273 106.155 35.787Q106.410 36.301 107.140 36.301M111.833 36.490L109.745 36.490L109.745 36.174Q110.053 36.174 110.244 36.121Q110.435 36.068 110.435 35.879L110.435 33.431Q110.435 33.190 110.365 33.082Q110.295 32.974 110.161 32.950Q110.027 32.926 109.745 32.926L109.745 32.610L111.086 32.513L111.086 33.348Q111.284 32.966 111.637 32.739Q111.991 32.513 112.417 32.513Q113.696 32.513 113.696 33.726L113.696 35.879Q113.696 36.068 113.887 36.121Q114.078 36.174 114.386 36.174L114.386 36.490L112.299 36.490L112.299 36.174Q112.611 36.174 112.802 36.121Q112.993 36.068 112.993 35.879L112.993 33.761Q112.993 33.502 112.949 33.280Q112.905 33.058 112.760 32.915Q112.615 32.772 112.356 32.772Q112.013 32.772 111.732 32.961Q111.451 33.150 111.295 33.462Q111.139 33.774 111.139 34.121L111.139 35.879Q111.139 36.068 111.332 36.121Q111.525 36.174 111.833 36.174L111.833 36.490M116.895 36.591Q116.337 36.591 115.865 36.308Q115.392 36.024 115.118 35.547Q114.843 35.071 114.843 34.517Q114.843 34.121 114.986 33.746Q115.129 33.370 115.386 33.082Q115.643 32.794 116.001 32.625Q116.359 32.456 116.764 32.456Q117.308 32.456 117.680 32.693Q118.051 32.930 118.238 33.348Q118.425 33.765 118.425 34.302Q118.425 34.354 118.401 34.392Q118.376 34.429 118.328 34.429L115.656 34.429L115.656 34.508Q115.656 35.255 115.968 35.778Q116.280 36.301 116.979 36.301Q117.383 36.301 117.704 36.044Q118.025 35.787 118.148 35.383Q118.165 35.303 118.249 35.303L118.328 35.303Q118.368 35.303 118.396 35.334Q118.425 35.365 118.425 35.409L118.425 35.444Q118.319 35.787 118.097 36.046Q117.875 36.305 117.561 36.448Q117.247 36.591 116.895 36.591M115.665 34.178L117.779 34.178Q117.779 33.910 117.726 33.664Q117.673 33.418 117.552 33.196Q117.432 32.974 117.234 32.847Q117.036 32.719 116.764 32.719Q116.421 32.719 116.168 32.944Q115.915 33.168 115.790 33.506Q115.665 33.844 115.665 34.178\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M124.102 36.591Q123.552 36.591 123.093 36.312Q122.634 36.033 122.366 35.563Q122.098 35.093 122.098 34.548Q122.098 34.135 122.247 33.757Q122.396 33.379 122.671 33.084Q122.946 32.790 123.315 32.623Q123.684 32.456 124.102 32.456Q124.436 32.456 124.756 32.522Q125.077 32.588 125.306 32.774Q125.534 32.961 125.534 33.286Q125.534 33.462 125.407 33.590Q125.279 33.717 125.103 33.717Q124.919 33.717 124.789 33.592Q124.660 33.467 124.660 33.286Q124.660 33.150 124.734 33.038Q124.809 32.926 124.941 32.873Q124.633 32.746 124.102 32.746Q123.666 32.746 123.398 33.023Q123.130 33.300 123.018 33.715Q122.906 34.130 122.906 34.548Q122.906 34.978 123.045 35.380Q123.183 35.782 123.478 36.042Q123.772 36.301 124.211 36.301Q124.633 36.301 124.936 36.051Q125.240 35.800 125.345 35.391Q125.354 35.361 125.378 35.336Q125.402 35.312 125.433 35.312L125.543 35.312Q125.631 35.312 125.631 35.427Q125.486 35.963 125.073 36.277Q124.660 36.591 124.102 36.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M128.030 36.490L125.943 36.490L125.943 36.174Q126.250 36.174 126.442 36.121Q126.633 36.068 126.633 35.879L126.633 31.164Q126.633 30.922 126.562 30.814Q126.492 30.707 126.358 30.683Q126.224 30.658 125.943 30.658L125.943 30.342L127.310 30.245L127.310 33.295Q127.507 32.939 127.861 32.726Q128.215 32.513 128.615 32.513Q129.894 32.513 129.894 33.726L129.894 35.879Q129.894 36.068 130.085 36.121Q130.276 36.174 130.583 36.174L130.583 36.490L128.496 36.490L128.496 36.174Q128.808 36.174 128.999 36.121Q129.190 36.068 129.190 35.879L129.190 33.761Q129.190 33.502 129.146 33.280Q129.102 33.058 128.957 32.915Q128.812 32.772 128.553 32.772Q128.210 32.772 127.929 32.961Q127.648 33.150 127.492 33.462Q127.336 33.774 127.336 34.121L127.336 35.879Q127.336 36.068 127.529 36.121Q127.723 36.174 128.030 36.174L128.030 36.490M131.150 35.580Q131.150 35.040 131.583 34.706Q132.016 34.372 132.623 34.233Q133.229 34.095 133.761 34.095L133.761 33.761Q133.761 33.502 133.642 33.258Q133.523 33.014 133.315 32.867Q133.106 32.719 132.833 32.719Q132.271 32.719 131.959 32.917Q132.108 32.944 132.201 33.062Q132.293 33.181 132.293 33.339Q132.293 33.515 132.168 33.645Q132.042 33.774 131.862 33.774Q131.673 33.774 131.546 33.647Q131.418 33.519 131.418 33.339Q131.418 32.869 131.858 32.662Q132.297 32.456 132.833 32.456Q133.123 32.456 133.411 32.544Q133.699 32.632 133.932 32.792Q134.165 32.952 134.314 33.192Q134.464 33.431 134.464 33.726L134.464 35.761Q134.464 35.914 134.539 36.053Q134.613 36.191 134.758 36.191Q134.912 36.191 134.985 36.055Q135.057 35.919 135.057 35.761L135.057 35.185L135.343 35.185L135.343 35.761Q135.343 36.094 135.094 36.319Q134.846 36.543 134.517 36.543Q134.257 36.543 134.075 36.349Q133.893 36.156 133.849 35.879Q133.682 36.200 133.348 36.396Q133.014 36.591 132.644 36.591Q132.091 36.591 131.621 36.343Q131.150 36.094 131.150 35.580M131.906 35.580Q131.906 35.892 132.148 36.110Q132.390 36.327 132.706 36.327Q133.141 36.327 133.451 36.018Q133.761 35.708 133.761 35.277L133.761 34.350Q133.343 34.350 132.917 34.477Q132.491 34.605 132.198 34.882Q131.906 35.158 131.906 35.580M137.786 36.490L135.699 36.490L135.699 36.174Q136.006 36.174 136.197 36.121Q136.389 36.068 136.389 35.879L136.389 33.431Q136.389 33.190 136.318 33.082Q136.248 32.974 136.114 32.950Q135.980 32.926 135.699 32.926L135.699 32.610L137.039 32.513L137.039 33.348Q137.237 32.966 137.591 32.739Q137.944 32.513 138.371 32.513Q139.649 32.513 139.649 33.726L139.649 35.879Q139.649 36.068 139.841 36.121Q140.032 36.174 140.339 36.174L140.339 36.490L138.252 36.490L138.252 36.174Q138.564 36.174 138.755 36.121Q138.946 36.068 138.946 35.879L138.946 33.761Q138.946 33.502 138.902 33.280Q138.858 33.058 138.713 32.915Q138.568 32.772 138.309 32.772Q137.966 32.772 137.685 32.961Q137.404 33.150 137.248 33.462Q137.092 33.774 137.092 34.121L137.092 35.879Q137.092 36.068 137.285 36.121Q137.478 36.174 137.786 36.174L137.786 36.490M140.796 37.184Q140.796 36.872 141.025 36.633Q141.253 36.393 141.574 36.292Q141.394 36.152 141.300 35.943Q141.205 35.734 141.205 35.501Q141.205 35.088 141.482 34.754Q141.069 34.350 141.069 33.836Q141.069 33.445 141.289 33.146Q141.508 32.847 141.860 32.680Q142.211 32.513 142.589 32.513Q143.143 32.513 143.561 32.825Q143.749 32.632 144.011 32.522Q144.272 32.412 144.558 32.412Q144.760 32.412 144.894 32.555Q145.028 32.698 145.028 32.891Q145.028 33.018 144.945 33.097Q144.861 33.177 144.738 33.177Q144.620 33.177 144.536 33.097Q144.453 33.018 144.453 32.891Q144.453 32.816 144.461 32.790Q144.479 32.755 144.501 32.722Q144.523 32.689 144.532 32.676Q144.088 32.676 143.741 32.979Q144.101 33.361 144.101 33.836Q144.101 34.130 143.974 34.376Q143.846 34.622 143.633 34.798Q143.420 34.974 143.141 35.071Q142.862 35.167 142.589 35.167Q142.080 35.167 141.671 34.899Q141.543 35.071 141.543 35.277Q141.543 35.519 141.702 35.690Q141.860 35.862 142.093 35.862L142.857 35.862Q143.402 35.862 143.848 35.958Q144.294 36.055 144.593 36.345Q144.892 36.635 144.892 37.184Q144.892 37.764 144.207 38.054Q143.521 38.344 142.849 38.344Q142.176 38.344 141.486 38.054Q140.796 37.764 140.796 37.184M141.337 37.184Q141.337 37.479 141.590 37.677Q141.842 37.874 142.203 37.969Q142.563 38.063 142.849 38.063Q143.350 38.063 143.851 37.837Q144.352 37.611 144.352 37.184Q144.352 36.727 143.910 36.595Q143.468 36.464 142.857 36.464L142.093 36.464Q141.899 36.464 141.721 36.558Q141.543 36.653 141.440 36.817Q141.337 36.982 141.337 37.184M142.581 34.886L142.589 34.886Q143.372 34.886 143.372 33.836Q143.372 32.790 142.589 32.790Q141.798 32.790 141.798 33.836Q141.798 34.886 142.581 34.886M147.467 36.591Q146.909 36.591 146.437 36.308Q145.964 36.024 145.690 35.547Q145.415 35.071 145.415 34.517Q145.415 34.121 145.558 33.746Q145.701 33.370 145.958 33.082Q146.215 32.794 146.573 32.625Q146.931 32.456 147.335 32.456Q147.880 32.456 148.252 32.693Q148.623 32.930 148.810 33.348Q148.997 33.765 148.997 34.302Q148.997 34.354 148.972 34.392Q148.948 34.429 148.900 34.429L146.228 34.429L146.228 34.508Q146.228 35.255 146.540 35.778Q146.852 36.301 147.551 36.301Q147.955 36.301 148.276 36.044Q148.597 35.787 148.720 35.383Q148.737 35.303 148.821 35.303L148.900 35.303Q148.939 35.303 148.968 35.334Q148.997 35.365 148.997 35.409L148.997 35.444Q148.891 35.787 148.669 36.046Q148.447 36.305 148.133 36.448Q147.819 36.591 147.467 36.591M146.237 34.178L148.351 34.178Q148.351 33.910 148.298 33.664Q148.245 33.418 148.124 33.196Q148.003 32.974 147.806 32.847Q147.608 32.719 147.335 32.719Q146.993 32.719 146.740 32.944Q146.487 33.168 146.362 33.506Q146.237 33.844 146.237 34.178M151.519 36.591Q150.974 36.591 150.530 36.308Q150.086 36.024 149.832 35.552Q149.577 35.079 149.577 34.548Q149.577 33.994 149.853 33.528Q150.130 33.062 150.603 32.788Q151.075 32.513 151.620 32.513Q151.954 32.513 152.257 32.645Q152.561 32.777 152.780 33.014L152.780 31.164Q152.780 30.922 152.710 30.814Q152.640 30.707 152.506 30.683Q152.372 30.658 152.086 30.658L152.086 30.342L153.453 30.245L153.453 35.673Q153.453 35.910 153.523 36.018Q153.593 36.125 153.729 36.149Q153.866 36.174 154.147 36.174L154.147 36.490L152.754 36.591L152.754 36.042Q152.503 36.305 152.185 36.448Q151.866 36.591 151.519 36.591M151.581 36.327Q151.950 36.327 152.259 36.116Q152.569 35.906 152.754 35.563L152.754 33.431Q152.582 33.128 152.301 32.950Q152.020 32.772 151.682 32.772Q150.992 32.772 150.688 33.293Q150.385 33.814 150.385 34.556Q150.385 35.277 150.655 35.802Q150.926 36.327 151.581 36.327\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M159.992 36.490L157.759 36.490L157.759 36.174Q158.071 36.174 158.263 36.121Q158.454 36.068 158.454 35.879L158.454 33.431Q158.454 33.190 158.384 33.082Q158.313 32.974 158.179 32.950Q158.045 32.926 157.759 32.926L157.759 32.610L159.073 32.513L159.073 33.374Q159.236 32.983 159.504 32.748Q159.772 32.513 160.163 32.513Q160.436 32.513 160.651 32.676Q160.866 32.838 160.866 33.097Q160.866 33.273 160.748 33.392Q160.629 33.511 160.453 33.511Q160.273 33.511 160.155 33.392Q160.036 33.273 160.036 33.097Q160.036 32.882 160.190 32.772L160.172 32.772Q159.794 32.772 159.561 33.034Q159.328 33.295 159.229 33.682Q159.131 34.069 159.131 34.429L159.131 35.879Q159.131 36.068 159.388 36.121Q159.645 36.174 159.992 36.174L159.992 36.490M163.437 36.591Q162.879 36.591 162.407 36.308Q161.934 36.024 161.660 35.547Q161.385 35.071 161.385 34.517Q161.385 34.121 161.528 33.746Q161.671 33.370 161.928 33.082Q162.185 32.794 162.543 32.625Q162.901 32.456 163.305 32.456Q163.850 32.456 164.222 32.693Q164.593 32.930 164.780 33.348Q164.967 33.765 164.967 34.302Q164.967 34.354 164.942 34.392Q164.918 34.429 164.870 34.429L162.198 34.429L162.198 34.508Q162.198 35.255 162.510 35.778Q162.822 36.301 163.521 36.301Q163.925 36.301 164.246 36.044Q164.567 35.787 164.690 35.383Q164.707 35.303 164.791 35.303L164.870 35.303Q164.909 35.303 164.938 35.334Q164.967 35.365 164.967 35.409L164.967 35.444Q164.861 35.787 164.639 36.046Q164.417 36.305 164.103 36.448Q163.789 36.591 163.437 36.591M162.207 34.178L164.321 34.178Q164.321 33.910 164.268 33.664Q164.215 33.418 164.094 33.196Q163.973 32.974 163.776 32.847Q163.578 32.719 163.305 32.719Q162.963 32.719 162.710 32.944Q162.457 33.168 162.332 33.506Q162.207 33.844 162.207 34.178M167.265 36.472L166.074 33.225Q165.999 33.027 165.854 32.977Q165.709 32.926 165.419 32.926L165.419 32.610L167.300 32.610L167.300 32.926Q166.777 32.926 166.777 33.150Q166.777 33.181 166.795 33.225L167.660 35.607L168.416 33.519L168.306 33.225Q168.232 33.027 168.084 32.977Q167.937 32.926 167.652 32.926L167.652 32.610L169.440 32.610L169.440 32.926Q168.930 32.926 168.930 33.150Q168.930 33.203 168.939 33.225L169.858 35.725L170.675 33.475Q170.693 33.431 170.693 33.330Q170.693 33.137 170.541 33.032Q170.389 32.926 170.187 32.926L170.187 32.610L171.730 32.610L171.730 32.926Q171.462 32.926 171.266 33.075Q171.071 33.225 170.983 33.475L169.884 36.472Q169.853 36.591 169.721 36.591L169.660 36.591Q169.541 36.591 169.497 36.472L168.579 33.941L167.652 36.472Q167.608 36.591 167.489 36.591L167.427 36.591Q167.296 36.591 167.265 36.472\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.811 -99.177)\">\u003Cpath d=\"M172.034 35.580Q172.034 35.040 172.467 34.706Q172.900 34.372 173.506 34.233Q174.113 34.095 174.644 34.095L174.644 33.761Q174.644 33.502 174.526 33.258Q174.407 33.014 174.198 32.867Q173.990 32.719 173.717 32.719Q173.155 32.719 172.843 32.917Q172.992 32.944 173.084 33.062Q173.177 33.181 173.177 33.339Q173.177 33.515 173.051 33.645Q172.926 33.774 172.746 33.774Q172.557 33.774 172.430 33.647Q172.302 33.519 172.302 33.339Q172.302 32.869 172.742 32.662Q173.181 32.456 173.717 32.456Q174.007 32.456 174.295 32.544Q174.583 32.632 174.816 32.792Q175.049 32.952 175.198 33.192Q175.348 33.431 175.348 33.726L175.348 35.761Q175.348 35.914 175.422 36.053Q175.497 36.191 175.642 36.191Q175.796 36.191 175.868 36.055Q175.941 35.919 175.941 35.761L175.941 35.185L176.227 35.185L176.227 35.761Q176.227 36.094 175.978 36.319Q175.730 36.543 175.400 36.543Q175.141 36.543 174.959 36.349Q174.776 36.156 174.732 35.879Q174.565 36.200 174.231 36.396Q173.897 36.591 173.528 36.591Q172.975 36.591 172.504 36.343Q172.034 36.094 172.034 35.580M172.790 35.580Q172.790 35.892 173.032 36.110Q173.273 36.327 173.590 36.327Q174.025 36.327 174.335 36.018Q174.644 35.708 174.644 35.277L174.644 34.350Q174.227 34.350 173.801 34.477Q173.374 34.605 173.082 34.882Q172.790 35.158 172.790 35.580M178.767 36.490L176.534 36.490L176.534 36.174Q176.846 36.174 177.037 36.121Q177.228 36.068 177.228 35.879L177.228 33.431Q177.228 33.190 177.158 33.082Q177.088 32.974 176.954 32.950Q176.820 32.926 176.534 32.926L176.534 32.610L177.848 32.513L177.848 33.374Q178.011 32.983 178.279 32.748Q178.547 32.513 178.938 32.513Q179.210 32.513 179.426 32.676Q179.641 32.838 179.641 33.097Q179.641 33.273 179.522 33.392Q179.404 33.511 179.228 33.511Q179.048 33.511 178.929 33.392Q178.811 33.273 178.811 33.097Q178.811 32.882 178.964 32.772L178.947 32.772Q178.569 32.772 178.336 33.034Q178.103 33.295 178.004 33.682Q177.905 34.069 177.905 34.429L177.905 35.879Q177.905 36.068 178.162 36.121Q178.419 36.174 178.767 36.174L178.767 36.490M182.150 36.591Q181.605 36.591 181.162 36.308Q180.718 36.024 180.463 35.552Q180.208 35.079 180.208 34.548Q180.208 33.994 180.485 33.528Q180.762 33.062 181.234 32.788Q181.707 32.513 182.251 32.513Q182.585 32.513 182.889 32.645Q183.192 32.777 183.412 33.014L183.412 31.164Q183.412 30.922 183.341 30.814Q183.271 30.707 183.137 30.683Q183.003 30.658 182.717 30.658L182.717 30.342L184.084 30.245L184.084 35.673Q184.084 35.910 184.154 36.018Q184.225 36.125 184.361 36.149Q184.497 36.174 184.778 36.174L184.778 36.490L183.385 36.591L183.385 36.042Q183.135 36.305 182.816 36.448Q182.498 36.591 182.150 36.591M182.212 36.327Q182.581 36.327 182.891 36.116Q183.201 35.906 183.385 35.563L183.385 33.431Q183.214 33.128 182.933 32.950Q182.651 32.772 182.313 32.772Q181.623 32.772 181.320 33.293Q181.017 33.814 181.017 34.556Q181.017 35.277 181.287 35.802Q181.557 36.327 182.212 36.327\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-13.5 5.192c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.958-4.458 9.958-9.958Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(-4.201 -28.907)\">\u003Cpath d=\"M-22.970 36.552L-22.970 34.979Q-22.970 34.952-22.945 34.926Q-22.919 34.901-22.892 34.901L-22.779 34.901Q-22.751 34.901-22.728 34.928Q-22.704 34.955-22.704 34.979Q-22.704 35.324-22.572 35.588Q-22.440 35.851-22.211 36.020Q-21.982 36.189-21.680 36.270Q-21.377 36.350-21.036 36.350Q-20.769 36.350-20.533 36.222Q-20.297 36.094-20.152 35.871Q-20.007 35.649-20.007 35.383Q-20.007 35.160-20.113 34.964Q-20.219 34.767-20.400 34.632Q-20.581 34.497-20.807 34.446L-21.835 34.214Q-22.146 34.142-22.406 33.956Q-22.666 33.769-22.818 33.498Q-22.970 33.226-22.970 32.911Q-22.970 32.525-22.757 32.218Q-22.543 31.910-22.196 31.739Q-21.849 31.568-21.470 31.568Q-21.241 31.568-21.012 31.621Q-20.783 31.674-20.584 31.782Q-20.386 31.889-20.232 32.053L-19.938 31.613Q-19.915 31.568-19.874 31.568L-19.826 31.568Q-19.795 31.568-19.773 31.594Q-19.751 31.619-19.751 31.647L-19.751 33.222Q-19.751 33.243-19.774 33.270Q-19.798 33.298-19.826 33.298L-19.938 33.298Q-20 33.298-20.014 33.222Q-20.055 32.809-20.236 32.489Q-20.417 32.170-20.728 31.995Q-21.039 31.821-21.470 31.821Q-21.719 31.821-21.959 31.932Q-22.198 32.043-22.348 32.241Q-22.499 32.440-22.499 32.703Q-22.499 32.915-22.391 33.096Q-22.283 33.277-22.107 33.397Q-21.931 33.516-21.723 33.557L-20.694 33.786Q-20.376 33.858-20.109 34.063Q-19.843 34.268-19.691 34.562Q-19.539 34.856-19.539 35.188Q-19.539 35.581-19.744 35.916Q-19.949 36.251-20.294 36.440Q-20.639 36.630-21.036 36.630Q-21.456 36.630-21.835 36.517Q-22.215 36.405-22.485 36.155L-22.779 36.589Q-22.806 36.630-22.844 36.630L-22.892 36.630Q-22.919 36.630-22.945 36.605Q-22.970 36.579-22.970 36.552M-15.717 36.490L-18.247 36.490L-18.247 36.210Q-17.279 36.210-17.279 36.001L-17.279 32.382Q-17.672 32.570-18.294 32.570L-18.294 32.289Q-17.877 32.289-17.513 32.188Q-17.149 32.088-16.893 31.842L-16.767 31.842Q-16.702 31.859-16.685 31.927L-16.685 36.001Q-16.685 36.210-15.717 36.210\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M54.786-26.106c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(64.085 -60.204)\">\u003Cpath d=\"M-22.970 36.552L-22.970 34.979Q-22.970 34.952-22.945 34.926Q-22.919 34.901-22.892 34.901L-22.779 34.901Q-22.751 34.901-22.728 34.928Q-22.704 34.955-22.704 34.979Q-22.704 35.324-22.572 35.588Q-22.440 35.851-22.211 36.020Q-21.982 36.189-21.680 36.270Q-21.377 36.350-21.036 36.350Q-20.769 36.350-20.533 36.222Q-20.297 36.094-20.152 35.871Q-20.007 35.649-20.007 35.383Q-20.007 35.160-20.113 34.964Q-20.219 34.767-20.400 34.632Q-20.581 34.497-20.807 34.446L-21.835 34.214Q-22.146 34.142-22.406 33.956Q-22.666 33.769-22.818 33.498Q-22.970 33.226-22.970 32.911Q-22.970 32.525-22.757 32.218Q-22.543 31.910-22.196 31.739Q-21.849 31.568-21.470 31.568Q-21.241 31.568-21.012 31.621Q-20.783 31.674-20.584 31.782Q-20.386 31.889-20.232 32.053L-19.938 31.613Q-19.915 31.568-19.874 31.568L-19.826 31.568Q-19.795 31.568-19.773 31.594Q-19.751 31.619-19.751 31.647L-19.751 33.222Q-19.751 33.243-19.774 33.270Q-19.798 33.298-19.826 33.298L-19.938 33.298Q-20 33.298-20.014 33.222Q-20.055 32.809-20.236 32.489Q-20.417 32.170-20.728 31.995Q-21.039 31.821-21.470 31.821Q-21.719 31.821-21.959 31.932Q-22.198 32.043-22.348 32.241Q-22.499 32.440-22.499 32.703Q-22.499 32.915-22.391 33.096Q-22.283 33.277-22.107 33.397Q-21.931 33.516-21.723 33.557L-20.694 33.786Q-20.376 33.858-20.109 34.063Q-19.843 34.268-19.691 34.562Q-19.539 34.856-19.539 35.188Q-19.539 35.581-19.744 35.916Q-19.949 36.251-20.294 36.440Q-20.639 36.630-21.036 36.630Q-21.456 36.630-21.835 36.517Q-22.215 36.405-22.485 36.155L-22.779 36.589Q-22.806 36.630-22.844 36.630L-22.892 36.630Q-22.919 36.630-22.945 36.605Q-22.970 36.579-22.970 36.552M-15.717 36.490L-18.602 36.490L-18.602 36.288Q-18.602 36.258-18.575 36.230L-17.327 35.013Q-17.255 34.938-17.213 34.896Q-17.170 34.853-17.091 34.774Q-16.678 34.361-16.447 34.003Q-16.216 33.646-16.216 33.222Q-16.216 32.990-16.295 32.787Q-16.374 32.583-16.515 32.433Q-16.657 32.282-16.852 32.202Q-17.047 32.122-17.279 32.122Q-17.590 32.122-17.848 32.281Q-18.106 32.440-18.236 32.717L-18.216 32.717Q-18.048 32.717-17.941 32.828Q-17.833 32.939-17.833 33.103Q-17.833 33.260-17.942 33.373Q-18.052 33.486-18.216 33.486Q-18.376 33.486-18.489 33.373Q-18.602 33.260-18.602 33.103Q-18.602 32.727-18.394 32.440Q-18.185 32.153-17.850 31.997Q-17.515 31.842-17.160 31.842Q-16.736 31.842-16.356 32Q-15.977 32.159-15.743 32.476Q-15.509 32.792-15.509 33.222Q-15.509 33.533-15.649 33.802Q-15.789 34.070-15.994 34.275Q-16.199 34.480-16.562 34.762Q-16.924 35.044-17.033 35.140L-17.888 35.868L-17.245 35.868Q-16.982 35.868-16.693 35.866Q-16.404 35.865-16.186 35.856Q-15.967 35.847-15.950 35.830Q-15.888 35.765-15.851 35.598Q-15.813 35.430-15.775 35.188L-15.509 35.188\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M54.786 36.49c0-5.5-4.459-9.958-9.959-9.958S34.87 30.99 34.87 36.49s4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(64.085 2.392)\">\u003Cpath d=\"M-22.970 36.552L-22.970 34.979Q-22.970 34.952-22.945 34.926Q-22.919 34.901-22.892 34.901L-22.779 34.901Q-22.751 34.901-22.728 34.928Q-22.704 34.955-22.704 34.979Q-22.704 35.324-22.572 35.588Q-22.440 35.851-22.211 36.020Q-21.982 36.189-21.680 36.270Q-21.377 36.350-21.036 36.350Q-20.769 36.350-20.533 36.222Q-20.297 36.094-20.152 35.871Q-20.007 35.649-20.007 35.383Q-20.007 35.160-20.113 34.964Q-20.219 34.767-20.400 34.632Q-20.581 34.497-20.807 34.446L-21.835 34.214Q-22.146 34.142-22.406 33.956Q-22.666 33.769-22.818 33.498Q-22.970 33.226-22.970 32.911Q-22.970 32.525-22.757 32.218Q-22.543 31.910-22.196 31.739Q-21.849 31.568-21.470 31.568Q-21.241 31.568-21.012 31.621Q-20.783 31.674-20.584 31.782Q-20.386 31.889-20.232 32.053L-19.938 31.613Q-19.915 31.568-19.874 31.568L-19.826 31.568Q-19.795 31.568-19.773 31.594Q-19.751 31.619-19.751 31.647L-19.751 33.222Q-19.751 33.243-19.774 33.270Q-19.798 33.298-19.826 33.298L-19.938 33.298Q-20 33.298-20.014 33.222Q-20.055 32.809-20.236 32.489Q-20.417 32.170-20.728 31.995Q-21.039 31.821-21.470 31.821Q-21.719 31.821-21.959 31.932Q-22.198 32.043-22.348 32.241Q-22.499 32.440-22.499 32.703Q-22.499 32.915-22.391 33.096Q-22.283 33.277-22.107 33.397Q-21.931 33.516-21.723 33.557L-20.694 33.786Q-20.376 33.858-20.109 34.063Q-19.843 34.268-19.691 34.562Q-19.539 34.856-19.539 35.188Q-19.539 35.581-19.744 35.916Q-19.949 36.251-20.294 36.440Q-20.639 36.630-21.036 36.630Q-21.456 36.630-21.835 36.517Q-22.215 36.405-22.485 36.155L-22.779 36.589Q-22.806 36.630-22.844 36.630L-22.892 36.630Q-22.919 36.630-22.945 36.605Q-22.970 36.579-22.970 36.552M-18.247 35.943Q-18.127 36.100-17.936 36.199Q-17.744 36.299-17.529 36.338Q-17.313 36.377-17.091 36.377Q-16.794 36.377-16.599 36.222Q-16.404 36.066-16.314 35.812Q-16.223 35.557-16.223 35.273Q-16.223 34.979-16.315 34.728Q-16.408 34.477-16.606 34.321Q-16.804 34.166-17.098 34.166L-17.614 34.166Q-17.642 34.166-17.667 34.140Q-17.693 34.115-17.693 34.091L-17.693 34.019Q-17.693 33.988-17.667 33.966Q-17.642 33.944-17.614 33.944L-17.173 33.913Q-16.811 33.913-16.591 33.556Q-16.370 33.198-16.370 32.809Q-16.370 32.481-16.565 32.277Q-16.760 32.074-17.091 32.074Q-17.378 32.074-17.631 32.158Q-17.884 32.241-18.048 32.429Q-17.901 32.429-17.801 32.544Q-17.700 32.658-17.700 32.809Q-17.700 32.959-17.806 33.069Q-17.912 33.178-18.069 33.178Q-18.230 33.178-18.339 33.069Q-18.448 32.959-18.448 32.809Q-18.448 32.484-18.240 32.265Q-18.031 32.047-17.715 31.944Q-17.399 31.842-17.091 31.842Q-16.773 31.842-16.445 31.946Q-16.117 32.050-15.890 32.272Q-15.663 32.494-15.663 32.809Q-15.663 33.243-15.950 33.568Q-16.237 33.892-16.671 34.039Q-16.360 34.104-16.080 34.270Q-15.799 34.436-15.622 34.694Q-15.444 34.952-15.444 35.273Q-15.444 35.683-15.688 35.993Q-15.933 36.302-16.314 36.466Q-16.695 36.630-17.091 36.630Q-17.460 36.630-17.818 36.517Q-18.175 36.405-18.419 36.155Q-18.664 35.906-18.664 35.536Q-18.664 35.365-18.547 35.253Q-18.431 35.140-18.260 35.140Q-18.151 35.140-18.060 35.191Q-17.970 35.242-17.915 35.335Q-17.860 35.427-17.860 35.536Q-17.860 35.704-17.973 35.823Q-18.086 35.943-18.247 35.943\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M104.578-40.332h22.762v-17.072h-22.762Z\"\u002F>\u003Cg transform=\"translate(134.356 -83.103)\">\u003Cpath d=\"M-17.977 35.683L-22.810 35.683Q-22.878 35.673-22.924 35.627Q-22.970 35.581-22.970 35.509Q-22.970 35.444-22.924 35.398Q-22.878 35.352-22.810 35.342L-17.977 35.342Q-17.908 35.352-17.862 35.398Q-17.816 35.444-17.816 35.509Q-17.816 35.581-17.862 35.627Q-17.908 35.673-17.977 35.683M-17.977 34.145L-22.810 34.145Q-22.878 34.135-22.924 34.089Q-22.970 34.043-22.970 33.971Q-22.970 33.827-22.810 33.803L-17.977 33.803Q-17.816 33.827-17.816 33.971Q-17.816 34.043-17.862 34.089Q-17.908 34.135-17.977 34.145M-15.331 36.630Q-15.967 36.630-16.331 36.285Q-16.695 35.940-16.830 35.415Q-16.965 34.890-16.965 34.265Q-16.965 33.240-16.609 32.541Q-16.254 31.842-15.331 31.842Q-14.405 31.842-14.053 32.541Q-13.701 33.240-13.701 34.265Q-13.701 34.890-13.836 35.415Q-13.971 35.940-14.333 36.285Q-14.695 36.630-15.331 36.630M-15.331 36.405Q-14.894 36.405-14.680 36.030Q-14.466 35.656-14.417 35.189Q-14.367 34.723-14.367 34.145Q-14.367 33.592-14.417 33.164Q-14.466 32.737-14.678 32.402Q-14.890 32.067-15.331 32.067Q-15.673 32.067-15.876 32.274Q-16.080 32.481-16.167 32.793Q-16.254 33.106-16.276 33.422Q-16.298 33.739-16.298 34.145Q-16.298 34.562-16.276 34.904Q-16.254 35.246-16.165 35.594Q-16.076 35.943-15.871 36.174Q-15.666 36.405-15.331 36.405\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M104.578-3.344h22.762v-17.071h-22.762Z\"\u002F>\u003Cg transform=\"translate(134.356 -46.114)\">\u003Cpath d=\"M-17.977 35.683L-22.810 35.683Q-22.878 35.673-22.924 35.627Q-22.970 35.581-22.970 35.509Q-22.970 35.444-22.924 35.398Q-22.878 35.352-22.810 35.342L-17.977 35.342Q-17.908 35.352-17.862 35.398Q-17.816 35.444-17.816 35.509Q-17.816 35.581-17.862 35.627Q-17.908 35.673-17.977 35.683M-17.977 34.145L-22.810 34.145Q-22.878 34.135-22.924 34.089Q-22.970 34.043-22.970 33.971Q-22.970 33.827-22.810 33.803L-17.977 33.803Q-17.816 33.827-17.816 33.971Q-17.816 34.043-17.862 34.089Q-17.908 34.135-17.977 34.145M-13.995 36.490L-16.524 36.490L-16.524 36.210Q-15.557 36.210-15.557 36.001L-15.557 32.382Q-15.950 32.570-16.572 32.570L-16.572 32.289Q-16.155 32.289-15.791 32.188Q-15.427 32.088-15.170 31.842L-15.044 31.842Q-14.979 31.859-14.962 31.927L-14.962 36.001Q-14.962 36.210-13.995 36.210\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M104.578 27.954h22.762V10.882h-22.762Z\"\u002F>\u003Cg transform=\"translate(134.356 -14.816)\">\u003Cpath d=\"M-17.977 35.683L-22.810 35.683Q-22.878 35.673-22.924 35.627Q-22.970 35.581-22.970 35.509Q-22.970 35.444-22.924 35.398Q-22.878 35.352-22.810 35.342L-17.977 35.342Q-17.908 35.352-17.862 35.398Q-17.816 35.444-17.816 35.509Q-17.816 35.581-17.862 35.627Q-17.908 35.673-17.977 35.683M-17.977 34.145L-22.810 34.145Q-22.878 34.135-22.924 34.089Q-22.970 34.043-22.970 33.971Q-22.970 33.827-22.810 33.803L-17.977 33.803Q-17.816 33.827-17.816 33.971Q-17.816 34.043-17.862 34.089Q-17.908 34.135-17.977 34.145M-13.995 36.490L-16.879 36.490L-16.879 36.288Q-16.879 36.258-16.852 36.230L-15.605 35.013Q-15.533 34.938-15.490 34.896Q-15.447 34.853-15.369 34.774Q-14.955 34.361-14.724 34.003Q-14.494 33.646-14.494 33.222Q-14.494 32.990-14.572 32.787Q-14.651 32.583-14.793 32.433Q-14.935 32.282-15.129 32.202Q-15.324 32.122-15.557 32.122Q-15.868 32.122-16.126 32.281Q-16.384 32.440-16.514 32.717L-16.493 32.717Q-16.326 32.717-16.218 32.828Q-16.110 32.939-16.110 33.103Q-16.110 33.260-16.220 33.373Q-16.329 33.486-16.493 33.486Q-16.654 33.486-16.767 33.373Q-16.879 33.260-16.879 33.103Q-16.879 32.727-16.671 32.440Q-16.462 32.153-16.127 31.997Q-15.792 31.842-15.437 31.842Q-15.013 31.842-14.634 32Q-14.254 32.159-14.020 32.476Q-13.786 32.792-13.786 33.222Q-13.786 33.533-13.926 33.802Q-14.066 34.070-14.271 34.275Q-14.477 34.480-14.839 34.762Q-15.201 35.044-15.311 35.140L-16.165 35.868L-15.522 35.868Q-15.259 35.868-14.970 35.866Q-14.682 35.865-14.463 35.856Q-14.244 35.847-14.227 35.830Q-14.166 35.765-14.128 35.598Q-14.090 35.430-14.053 35.188L-13.786 35.188\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M104.578 62.098h22.762V45.026h-22.762Z\"\u002F>\u003Cg transform=\"translate(134.356 19.327)\">\u003Cpath d=\"M-17.977 35.683L-22.810 35.683Q-22.878 35.673-22.924 35.627Q-22.970 35.581-22.970 35.509Q-22.970 35.444-22.924 35.398Q-22.878 35.352-22.810 35.342L-17.977 35.342Q-17.908 35.352-17.862 35.398Q-17.816 35.444-17.816 35.509Q-17.816 35.581-17.862 35.627Q-17.908 35.673-17.977 35.683M-17.977 34.145L-22.810 34.145Q-22.878 34.135-22.924 34.089Q-22.970 34.043-22.970 33.971Q-22.970 33.827-22.810 33.803L-17.977 33.803Q-17.816 33.827-17.816 33.971Q-17.816 34.043-17.862 34.089Q-17.908 34.135-17.977 34.145M-16.524 35.943Q-16.404 36.100-16.213 36.199Q-16.021 36.299-15.806 36.338Q-15.591 36.377-15.369 36.377Q-15.071 36.377-14.876 36.222Q-14.682 36.066-14.591 35.812Q-14.501 35.557-14.501 35.273Q-14.501 34.979-14.593 34.728Q-14.685 34.477-14.883 34.321Q-15.082 34.166-15.376 34.166L-15.892 34.166Q-15.919 34.166-15.945 34.140Q-15.970 34.115-15.970 34.091L-15.970 34.019Q-15.970 33.988-15.945 33.966Q-15.919 33.944-15.892 33.944L-15.451 33.913Q-15.088 33.913-14.868 33.556Q-14.647 33.198-14.647 32.809Q-14.647 32.481-14.842 32.277Q-15.037 32.074-15.369 32.074Q-15.656 32.074-15.909 32.158Q-16.162 32.241-16.326 32.429Q-16.179 32.429-16.078 32.544Q-15.977 32.658-15.977 32.809Q-15.977 32.959-16.083 33.069Q-16.189 33.178-16.346 33.178Q-16.507 33.178-16.616 33.069Q-16.726 32.959-16.726 32.809Q-16.726 32.484-16.517 32.265Q-16.309 32.047-15.992 31.944Q-15.676 31.842-15.369 31.842Q-15.051 31.842-14.723 31.946Q-14.395 32.050-14.167 32.272Q-13.940 32.494-13.940 32.809Q-13.940 33.243-14.227 33.568Q-14.514 33.892-14.948 34.039Q-14.637 34.104-14.357 34.270Q-14.077 34.436-13.899 34.694Q-13.721 34.952-13.721 35.273Q-13.721 35.683-13.966 35.993Q-14.210 36.302-14.591 36.466Q-14.972 36.630-15.369 36.630Q-15.738 36.630-16.095 36.517Q-16.452 36.405-16.697 36.155Q-16.941 35.906-16.941 35.536Q-16.941 35.365-16.825 35.253Q-16.709 35.140-16.538 35.140Q-16.428 35.140-16.338 35.191Q-16.247 35.242-16.192 35.335Q-16.138 35.427-16.138 35.536Q-16.138 35.704-16.251 35.823Q-16.363 35.943-16.524 35.943\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m-14.225 9.424 46.91 21.5\"\u002F>\u003Cpath stroke=\"none\" d=\"m35.593 32.258-3.587-4.46.678 3.126-2.811 1.528\"\u002F>\u003Cg transform=\"translate(31.24 -6.933)\">\u003Cpath d=\"M-20.960 36.490L-23.066 36.490L-23.066 36.210Q-22.345 36.210-22.345 36.001L-22.345 32.200Q-22.345 31.989-23.066 31.989L-23.066 31.708L-20.728 31.708Q-20.417 31.708-20.074 31.782Q-19.730 31.855-19.409 32.011Q-19.087 32.166-18.886 32.412Q-18.684 32.658-18.684 32.983Q-18.684 33.270-18.872 33.501Q-19.060 33.732-19.337 33.880Q-19.614 34.029-19.904 34.111Q-19.569 34.214-19.335 34.446Q-19.101 34.678-19.057 35.007L-18.971 35.615Q-18.944 35.827-18.898 35.996Q-18.852 36.165-18.747 36.285Q-18.643 36.405-18.455 36.405Q-18.240 36.405-18.124 36.220Q-18.007 36.035-18.007 35.803Q-17.990 35.738-17.915 35.721L-17.823 35.721Q-17.741 35.741-17.741 35.823Q-17.741 36.029-17.831 36.215Q-17.922 36.401-18.088 36.516Q-18.253 36.630-18.455 36.630Q-18.793 36.630-19.087 36.529Q-19.381 36.428-19.566 36.206Q-19.751 35.984-19.751 35.636L-19.751 35.027Q-19.751 34.778-19.894 34.591Q-20.038 34.405-20.261 34.306Q-20.485 34.207-20.735 34.207L-21.682 34.207L-21.682 36.001Q-21.682 36.210-20.960 36.210L-20.960 36.490M-21.682 32.200L-21.682 33.985L-20.827 33.985Q-20.540 33.985-20.301 33.942Q-20.062 33.899-19.874 33.790Q-19.686 33.680-19.574 33.482Q-19.463 33.284-19.463 32.983Q-19.463 32.406-19.831 32.197Q-20.198 31.989-20.827 31.989L-21.316 31.989Q-21.504 31.989-21.593 32.023Q-21.682 32.057-21.682 32.200\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m-14.225.96 48-22\"\u002F>\u003Cpath stroke=\"none\" d=\"m35.593-21.874-3.575-.12 1.757.954-.424 1.954\"\u002F>\u003Cg transform=\"translate(31.671 -50.48)\">\u003Cpath d=\"M-19.111 36.490L-23.080 36.490L-23.080 36.210Q-22.358 36.210-22.358 36.001L-22.358 32.200Q-22.358 31.989-23.080 31.989L-23.080 31.708L-20.766 31.708L-20.766 31.989Q-21.084 31.989-21.376 32.024Q-21.668 32.060-21.668 32.200L-21.668 36.001Q-21.668 36.141-21.579 36.176Q-21.490 36.210-21.302 36.210L-20.680 36.210Q-20.267 36.210-19.988 36.107Q-19.709 36.005-19.544 35.803Q-19.378 35.601-19.296 35.314Q-19.214 35.027-19.169 34.620L-18.903 34.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m54.502-29.202 47.971-15.352\"\u002F>\u003Cpath stroke=\"none\" d=\"m104.378-45.164-3.535-.548 1.63 1.158-.655 1.89\"\u002F>\u003Cg transform=\"translate(112.896 -81.196)\">\u003Cpath d=\"M-19.111 36.490L-23.080 36.490L-23.080 36.210Q-22.358 36.210-22.358 36.001L-22.358 32.200Q-22.358 31.989-23.080 31.989L-23.080 31.708L-20.766 31.708L-20.766 31.989Q-21.084 31.989-21.376 32.024Q-21.668 32.060-21.668 32.200L-21.668 36.001Q-21.668 36.141-21.579 36.176Q-21.490 36.210-21.302 36.210L-20.680 36.210Q-20.267 36.210-19.988 36.107Q-19.709 36.005-19.544 35.803Q-19.378 35.601-19.296 35.314Q-19.214 35.027-19.169 34.620L-18.903 34.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"m54.788-24.114 47.04 9.41\"\u002F>\u003Cpath stroke=\"none\" d=\"m104.378-14.195-3.67-2.855 1.12 2.345-1.937 1.734\"\u002F>\u003Cg transform=\"translate(112.537 -44.648)\">\u003Cpath d=\"M-20.960 36.490L-23.066 36.490L-23.066 36.210Q-22.345 36.210-22.345 36.001L-22.345 32.200Q-22.345 31.989-23.066 31.989L-23.066 31.708L-20.728 31.708Q-20.417 31.708-20.074 31.782Q-19.730 31.855-19.409 32.011Q-19.087 32.166-18.886 32.412Q-18.684 32.658-18.684 32.983Q-18.684 33.270-18.872 33.501Q-19.060 33.732-19.337 33.880Q-19.614 34.029-19.904 34.111Q-19.569 34.214-19.335 34.446Q-19.101 34.678-19.057 35.007L-18.971 35.615Q-18.944 35.827-18.898 35.996Q-18.852 36.165-18.747 36.285Q-18.643 36.405-18.455 36.405Q-18.240 36.405-18.124 36.220Q-18.007 36.035-18.007 35.803Q-17.990 35.738-17.915 35.721L-17.823 35.721Q-17.741 35.741-17.741 35.823Q-17.741 36.029-17.831 36.215Q-17.922 36.401-18.088 36.516Q-18.253 36.630-18.455 36.630Q-18.793 36.630-19.087 36.529Q-19.381 36.428-19.566 36.206Q-19.751 35.984-19.751 35.636L-19.751 35.027Q-19.751 34.778-19.894 34.591Q-20.038 34.405-20.261 34.306Q-20.485 34.207-20.735 34.207L-21.682 34.207L-21.682 36.001Q-21.682 36.210-20.960 36.210L-20.960 36.490M-21.682 32.200L-21.682 33.985L-20.827 33.985Q-20.540 33.985-20.301 33.942Q-20.062 33.899-19.874 33.790Q-19.686 33.680-19.574 33.482Q-19.463 33.284-19.463 32.983Q-19.463 32.406-19.831 32.197Q-20.198 31.989-20.827 31.989L-21.316 31.989Q-21.504 31.989-21.593 32.023Q-21.682 32.057-21.682 32.200\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m54.705 34.12 47.729-11.457\"\u002F>\u003Cpath stroke=\"none\" d=\"m104.378 22.197-3.485-.81 1.54 1.276-.793 1.836\"\u002F>\u003Cg transform=\"translate(112.947 -14.846)\">\u003Cpath d=\"M-19.111 36.490L-23.080 36.490L-23.080 36.210Q-22.358 36.210-22.358 36.001L-22.358 32.200Q-22.358 31.989-23.080 31.989L-23.080 31.708L-20.766 31.708L-20.766 31.989Q-21.084 31.989-21.376 32.024Q-21.668 32.060-21.668 32.200L-21.668 36.001Q-21.668 36.141-21.579 36.176Q-21.490 36.210-21.302 36.210L-20.680 36.210Q-20.267 36.210-19.988 36.107Q-19.709 36.005-19.544 35.803Q-19.378 35.601-19.296 35.314Q-19.214 35.027-19.169 34.620L-18.903 34.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m54.705 38.86 46.562 11.177\"\u002F>\u003Cpath stroke=\"none\" d=\"m104.378 50.783-4.38-3.684 1.269 2.938-2.465 2.04\"\u002F>\u003Cg transform=\"translate(112.516 20.029)\">\u003Cpath d=\"M-20.960 36.490L-23.066 36.490L-23.066 36.210Q-22.345 36.210-22.345 36.001L-22.345 32.200Q-22.345 31.989-23.066 31.989L-23.066 31.708L-20.728 31.708Q-20.417 31.708-20.074 31.782Q-19.730 31.855-19.409 32.011Q-19.087 32.166-18.886 32.412Q-18.684 32.658-18.684 32.983Q-18.684 33.270-18.872 33.501Q-19.060 33.732-19.337 33.880Q-19.614 34.029-19.904 34.111Q-19.569 34.214-19.335 34.446Q-19.101 34.678-19.057 35.007L-18.971 35.615Q-18.944 35.827-18.898 35.996Q-18.852 36.165-18.747 36.285Q-18.643 36.405-18.455 36.405Q-18.240 36.405-18.124 36.220Q-18.007 36.035-18.007 35.803Q-17.990 35.738-17.915 35.721L-17.823 35.721Q-17.741 35.741-17.741 35.823Q-17.741 36.029-17.831 36.215Q-17.922 36.401-18.088 36.516Q-18.253 36.630-18.455 36.630Q-18.793 36.630-19.087 36.529Q-19.381 36.428-19.566 36.206Q-19.751 35.984-19.751 35.636L-19.751 35.027Q-19.751 34.778-19.894 34.591Q-20.038 34.405-20.261 34.306Q-20.485 34.207-20.735 34.207L-21.682 34.207L-21.682 36.001Q-21.682 36.210-20.960 36.210L-20.960 36.490M-21.682 32.200L-21.682 33.985L-20.827 33.985Q-20.540 33.985-20.301 33.942Q-20.062 33.899-19.874 33.790Q-19.686 33.680-19.574 33.482Q-19.463 33.284-19.463 32.983Q-19.463 32.406-19.831 32.197Q-20.198 31.989-20.827 31.989L-21.316 31.989Q-21.504 31.989-21.593 32.023Q-21.682 32.057-21.682 32.200\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(43.314 -70.299)\">\u003Cpath d=\"M-21.555 36.463L-22.683 33.964Q-22.755 33.817-22.885 33.785Q-23.015 33.752-23.244 33.752L-23.244 33.472L-21.730 33.472L-21.730 33.752Q-22.082 33.752-22.082 33.899Q-22.082 33.944-22.071 33.964L-21.207 35.882L-20.427 34.152Q-20.393 34.084-20.393 34.005Q-20.393 33.892-20.477 33.822Q-20.561 33.752-20.680 33.752L-20.680 33.472L-19.484 33.472L-19.484 33.752Q-19.703 33.752-19.874 33.855Q-20.044 33.957-20.133 34.152L-21.169 36.463Q-21.217 36.558-21.323 36.558L-21.401 36.558Q-21.507 36.558-21.555 36.463M-13.776 35.683L-18.609 35.683Q-18.677 35.673-18.723 35.627Q-18.770 35.581-18.770 35.509Q-18.770 35.444-18.723 35.398Q-18.677 35.352-18.609 35.342L-13.776 35.342Q-13.708 35.352-13.661 35.398Q-13.615 35.444-13.615 35.509Q-13.615 35.581-13.661 35.627Q-13.708 35.673-13.776 35.683M-13.776 34.145L-18.609 34.145Q-18.677 34.135-18.723 34.089Q-18.770 34.043-18.770 33.971Q-18.770 33.827-18.609 33.803L-13.776 33.803Q-13.615 33.827-13.615 33.971Q-13.615 34.043-13.661 34.089Q-13.708 34.135-13.776 34.145M-9.794 36.490L-12.323 36.490L-12.323 36.210Q-11.356 36.210-11.356 36.001L-11.356 32.382Q-11.749 32.570-12.371 32.570L-12.371 32.289Q-11.954 32.289-11.590 32.188Q-11.226 32.088-10.970 31.842L-10.843 31.842Q-10.778 31.859-10.761 31.927L-10.761 36.001Q-10.761 36.210-9.794 36.210\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(43.314 12.214)\">\u003Cpath d=\"M-21.555 36.463L-22.683 33.964Q-22.755 33.817-22.885 33.785Q-23.015 33.752-23.244 33.752L-23.244 33.472L-21.730 33.472L-21.730 33.752Q-22.082 33.752-22.082 33.899Q-22.082 33.944-22.071 33.964L-21.207 35.882L-20.427 34.152Q-20.393 34.084-20.393 34.005Q-20.393 33.892-20.477 33.822Q-20.561 33.752-20.680 33.752L-20.680 33.472L-19.484 33.472L-19.484 33.752Q-19.703 33.752-19.874 33.855Q-20.044 33.957-20.133 34.152L-21.169 36.463Q-21.217 36.558-21.323 36.558L-21.401 36.558Q-21.507 36.558-21.555 36.463M-13.776 35.683L-18.609 35.683Q-18.677 35.673-18.723 35.627Q-18.770 35.581-18.770 35.509Q-18.770 35.444-18.723 35.398Q-18.677 35.352-18.609 35.342L-13.776 35.342Q-13.708 35.352-13.661 35.398Q-13.615 35.444-13.615 35.509Q-13.615 35.581-13.661 35.627Q-13.708 35.673-13.776 35.683M-13.776 34.145L-18.609 34.145Q-18.677 34.135-18.723 34.089Q-18.770 34.043-18.770 33.971Q-18.770 33.827-18.609 33.803L-13.776 33.803Q-13.615 33.827-13.615 33.971Q-13.615 34.043-13.661 34.089Q-13.708 34.135-13.776 34.145M-12.323 35.943Q-12.204 36.100-12.012 36.199Q-11.821 36.299-11.605 36.338Q-11.390 36.377-11.168 36.377Q-10.871 36.377-10.676 36.222Q-10.481 36.066-10.390 35.812Q-10.300 35.557-10.300 35.273Q-10.300 34.979-10.392 34.728Q-10.484 34.477-10.683 34.321Q-10.881 34.166-11.175 34.166L-11.691 34.166Q-11.718 34.166-11.744 34.140Q-11.770 34.115-11.770 34.091L-11.770 34.019Q-11.770 33.988-11.744 33.966Q-11.718 33.944-11.691 33.944L-11.250 33.913Q-10.888 33.913-10.667 33.556Q-10.447 33.198-10.447 32.809Q-10.447 32.481-10.642 32.277Q-10.836 32.074-11.168 32.074Q-11.455 32.074-11.708 32.158Q-11.961 32.241-12.125 32.429Q-11.978 32.429-11.877 32.544Q-11.776 32.658-11.776 32.809Q-11.776 32.959-11.882 33.069Q-11.988 33.178-12.146 33.178Q-12.306 33.178-12.416 33.069Q-12.525 32.959-12.525 32.809Q-12.525 32.484-12.316 32.265Q-12.108 32.047-11.792 31.944Q-11.476 31.842-11.168 31.842Q-10.850 31.842-10.522 31.946Q-10.194 32.050-9.967 32.272Q-9.739 32.494-9.739 32.809Q-9.739 33.243-10.026 33.568Q-10.313 33.892-10.748 34.039Q-10.437 34.104-10.156 34.270Q-9.876 34.436-9.698 34.694Q-9.521 34.952-9.521 35.273Q-9.521 35.683-9.765 35.993Q-10.009 36.302-10.390 36.466Q-10.771 36.630-11.168 36.630Q-11.537 36.630-11.894 36.517Q-12.251 36.405-12.496 36.155Q-12.740 35.906-12.740 35.536Q-12.740 35.365-12.624 35.253Q-12.508 35.140-12.337 35.140Q-12.228 35.140-12.137 35.191Q-12.046 35.242-11.992 35.335Q-11.937 35.427-11.937 35.536Q-11.937 35.704-12.050 35.823Q-12.163 35.943-12.323 35.943\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M-21.463 36.490L-23.097 36.490L-23.097 36.210Q-22.868 36.210-22.719 36.176Q-22.570 36.141-22.570 36.001L-22.570 34.152Q-22.570 33.882-22.678 33.821Q-22.786 33.759-23.097 33.759L-23.097 33.479L-22.037 33.404L-22.037 34.053Q-21.866 33.745-21.562 33.574Q-21.258 33.404-20.913 33.404Q-20.407 33.404-20.123 33.627Q-19.839 33.851-19.839 34.347L-19.839 36.001Q-19.839 36.138-19.691 36.174Q-19.542 36.210-19.316 36.210L-19.316 36.490L-20.947 36.490L-20.947 36.210Q-20.718 36.210-20.569 36.176Q-20.420 36.141-20.420 36.001L-20.420 34.361Q-20.420 34.026-20.540 33.826Q-20.660 33.626-20.974 33.626Q-21.244 33.626-21.478 33.762Q-21.712 33.899-21.851 34.133Q-21.989 34.367-21.989 34.641L-21.989 36.001Q-21.989 36.138-21.839 36.174Q-21.688 36.210-21.463 36.210L-21.463 36.490M-18.770 34.955Q-18.770 34.634-18.645 34.345Q-18.520 34.056-18.294 33.833Q-18.069 33.609-17.773 33.489Q-17.478 33.369-17.160 33.369Q-16.832 33.369-16.570 33.469Q-16.309 33.568-16.133 33.750Q-15.957 33.933-15.863 34.191Q-15.769 34.449-15.769 34.781Q-15.769 34.873-15.851 34.894L-18.106 34.894L-18.106 34.955Q-18.106 35.543-17.823 35.926Q-17.539 36.309-16.972 36.309Q-16.650 36.309-16.382 36.116Q-16.114 35.923-16.025 35.608Q-16.018 35.567-15.943 35.553L-15.851 35.553Q-15.769 35.577-15.769 35.649Q-15.769 35.656-15.775 35.683Q-15.888 36.080-16.259 36.319Q-16.630 36.558-17.054 36.558Q-17.491 36.558-17.891 36.350Q-18.291 36.141-18.530 35.774Q-18.770 35.407-18.770 34.955M-18.100 34.685L-16.285 34.685Q-16.285 34.408-16.382 34.156Q-16.480 33.903-16.678 33.747Q-16.876 33.592-17.160 33.592Q-17.437 33.592-17.650 33.750Q-17.864 33.909-17.982 34.164Q-18.100 34.419-18.100 34.685M-13.793 36.463L-14.774 33.964Q-14.835 33.821-14.953 33.786Q-15.071 33.752-15.287 33.752L-15.287 33.472L-13.807 33.472L-13.807 33.752Q-14.186 33.752-14.186 33.913Q-14.186 33.923-14.172 33.964L-13.458 35.796L-12.785 34.091Q-12.815 34.019-12.815 33.991Q-12.815 33.964-12.843 33.964Q-12.904 33.817-13.022 33.785Q-13.140 33.752-13.352 33.752L-13.352 33.472L-11.954 33.472L-11.954 33.752Q-12.330 33.752-12.330 33.913Q-12.330 33.944-12.323 33.964L-11.568 35.902L-10.881 34.152Q-10.860 34.101-10.860 34.046Q-10.860 33.906-10.973 33.829Q-11.086 33.752-11.226 33.752L-11.226 33.472L-10.006 33.472L-10.006 33.752Q-10.211 33.752-10.366 33.858Q-10.522 33.964-10.594 34.152L-11.500 36.463Q-11.534 36.558-11.646 36.558L-11.715 36.558Q-11.824 36.558-11.862 36.463L-12.645 34.460L-13.431 36.463Q-13.465 36.558-13.578 36.558L-13.646 36.558Q-13.755 36.558-13.793 36.463\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M-5.126 37.847L-6.756 37.847L-6.756 37.567Q-6.527 37.567-6.378 37.532Q-6.230 37.498-6.230 37.358L-6.230 34.012Q-6.230 33.841-6.366 33.800Q-6.503 33.759-6.756 33.759L-6.756 33.479L-5.676 33.404L-5.676 33.810Q-5.454 33.609-5.167 33.506Q-4.879 33.404-4.572 33.404Q-4.145 33.404-3.781 33.617Q-3.417 33.831-3.203 34.195Q-2.989 34.559-2.989 34.979Q-2.989 35.424-3.229 35.788Q-3.468 36.152-3.861 36.355Q-4.254 36.558-4.698 36.558Q-4.965 36.558-5.213 36.458Q-5.460 36.357-5.648 36.176L-5.648 37.358Q-5.648 37.495-5.500 37.531Q-5.351 37.567-5.126 37.567L-5.126 37.847M-5.648 34.159L-5.648 35.769Q-5.515 36.022-5.272 36.179Q-5.030 36.336-4.753 36.336Q-4.425 36.336-4.172 36.135Q-3.919 35.933-3.786 35.615Q-3.652 35.297-3.652 34.979Q-3.652 34.750-3.717 34.521Q-3.782 34.292-3.910 34.094Q-4.039 33.896-4.233 33.776Q-4.428 33.657-4.661 33.657Q-4.955 33.657-5.223 33.786Q-5.491 33.916-5.648 34.159M-0.686 36.490L-2.289 36.490L-2.289 36.210Q-2.063 36.210-1.914 36.176Q-1.766 36.141-1.766 36.001L-1.766 32.382Q-1.766 32.112-1.873 32.050Q-1.981 31.989-2.289 31.989L-2.289 31.708L-1.212 31.633L-1.212 36.001Q-1.212 36.138-1.062 36.174Q-0.911 36.210-0.686 36.210L-0.686 36.490M-0.033 35.762Q-0.033 35.430 0.191 35.203Q0.415 34.976 0.759 34.848Q1.102 34.719 1.475 34.667Q1.847 34.614 2.151 34.614L2.151 34.361Q2.151 34.156 2.044 33.976Q1.936 33.797 1.755 33.694Q1.574 33.592 1.365 33.592Q0.958 33.592 0.723 33.684Q0.812 33.721 0.858 33.805Q0.904 33.889 0.904 33.991Q0.904 34.087 0.858 34.166Q0.812 34.244 0.731 34.289Q0.651 34.333 0.562 34.333Q0.412 34.333 0.311 34.236Q0.210 34.138 0.210 33.991Q0.210 33.369 1.365 33.369Q1.577 33.369 1.827 33.433Q2.076 33.496 2.278 33.615Q2.479 33.735 2.606 33.920Q2.732 34.104 2.732 34.347L2.732 35.923Q2.732 36.039 2.794 36.135Q2.855 36.230 2.968 36.230Q3.078 36.230 3.143 36.136Q3.208 36.042 3.208 35.923L3.208 35.475L3.474 35.475L3.474 35.923Q3.474 36.193 3.247 36.358Q3.020 36.524 2.739 36.524Q2.531 36.524 2.394 36.370Q2.257 36.217 2.233 36.001Q2.086 36.268 1.804 36.413Q1.522 36.558 1.198 36.558Q0.921 36.558 0.637 36.483Q0.354 36.408 0.160 36.229Q-0.033 36.049-0.033 35.762M0.583 35.762Q0.583 35.936 0.683 36.066Q0.784 36.196 0.940 36.266Q1.095 36.336 1.259 36.336Q1.478 36.336 1.687 36.239Q1.895 36.141 2.023 35.960Q2.151 35.779 2.151 35.553L2.151 34.825Q1.827 34.825 1.461 34.916Q1.095 35.007 0.839 35.219Q0.583 35.430 0.583 35.762M5.573 36.490L3.939 36.490L3.939 36.210Q4.168 36.210 4.317 36.176Q4.465 36.141 4.465 36.001L4.465 34.152Q4.465 33.882 4.358 33.821Q4.250 33.759 3.939 33.759L3.939 33.479L4.999 33.404L4.999 34.053Q5.169 33.745 5.474 33.574Q5.778 33.404 6.123 33.404Q6.629 33.404 6.913 33.627Q7.196 33.851 7.196 34.347L7.196 36.001Q7.196 36.138 7.345 36.174Q7.494 36.210 7.719 36.210L7.719 36.490L6.089 36.490L6.089 36.210Q6.318 36.210 6.467 36.176Q6.615 36.141 6.615 36.001L6.615 34.361Q6.615 34.026 6.496 33.826Q6.376 33.626 6.062 33.626Q5.791 33.626 5.557 33.762Q5.323 33.899 5.185 34.133Q5.046 34.367 5.046 34.641L5.046 36.001Q5.046 36.138 5.197 36.174Q5.347 36.210 5.573 36.210L5.573 36.490M8.707 36.070Q8.707 35.902 8.830 35.779Q8.953 35.656 9.127 35.656Q9.295 35.656 9.418 35.779Q9.541 35.902 9.541 36.070Q9.541 36.244 9.418 36.367Q9.295 36.490 9.127 36.490Q8.953 36.490 8.830 36.367Q8.707 36.244 8.707 36.070M8.707 33.886Q8.707 33.718 8.830 33.595Q8.953 33.472 9.127 33.472Q9.295 33.472 9.418 33.595Q9.541 33.718 9.541 33.886Q9.541 34.060 9.418 34.183Q9.295 34.306 9.127 34.306Q8.953 34.306 8.830 34.183Q8.707 34.060 8.707 33.886\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M16.318 36.490L14.212 36.490L14.212 36.210Q14.933 36.210 14.933 36.001L14.933 32.200Q14.933 31.989 14.212 31.989L14.212 31.708L16.550 31.708Q16.861 31.708 17.204 31.782Q17.548 31.855 17.869 32.011Q18.191 32.166 18.392 32.412Q18.594 32.658 18.594 32.983Q18.594 33.270 18.406 33.501Q18.218 33.732 17.941 33.880Q17.664 34.029 17.374 34.111Q17.709 34.214 17.943 34.446Q18.177 34.678 18.221 35.007L18.307 35.615Q18.334 35.827 18.380 35.996Q18.426 36.165 18.531 36.285Q18.635 36.405 18.823 36.405Q19.038 36.405 19.154 36.220Q19.271 36.035 19.271 35.803Q19.288 35.738 19.363 35.721L19.455 35.721Q19.537 35.741 19.537 35.823Q19.537 36.029 19.447 36.215Q19.356 36.401 19.190 36.516Q19.025 36.630 18.823 36.630Q18.485 36.630 18.191 36.529Q17.897 36.428 17.712 36.206Q17.527 35.984 17.527 35.636L17.527 35.027Q17.527 34.778 17.384 34.591Q17.240 34.405 17.017 34.306Q16.793 34.207 16.543 34.207L15.596 34.207L15.596 36.001Q15.596 36.210 16.318 36.210L16.318 36.490M15.596 32.200L15.596 33.985L16.451 33.985Q16.738 33.985 16.977 33.942Q17.216 33.899 17.404 33.790Q17.592 33.680 17.704 33.482Q17.815 33.284 17.815 32.983Q17.815 32.406 17.447 32.197Q17.080 31.989 16.451 31.989L15.962 31.989Q15.774 31.989 15.685 32.023Q15.596 32.057 15.596 32.200\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M23.160 35.649L23.160 33.752L22.521 33.752L22.521 33.530Q22.839 33.530 23.056 33.320Q23.273 33.110 23.373 32.800Q23.474 32.491 23.474 32.183L23.741 32.183L23.741 33.472L24.818 33.472L24.818 33.752L23.741 33.752L23.741 35.636Q23.741 35.912 23.845 36.111Q23.949 36.309 24.209 36.309Q24.366 36.309 24.472 36.205Q24.578 36.100 24.628 35.947Q24.677 35.793 24.677 35.636L24.677 35.222L24.944 35.222L24.944 35.649Q24.944 35.875 24.845 36.085Q24.746 36.295 24.561 36.427Q24.377 36.558 24.148 36.558Q23.710 36.558 23.435 36.321Q23.160 36.083 23.160 35.649M27.436 36.490L25.802 36.490L25.802 36.210Q26.031 36.210 26.180 36.176Q26.328 36.141 26.328 36.001L26.328 32.382Q26.328 32.112 26.221 32.050Q26.113 31.989 25.802 31.989L25.802 31.708L26.882 31.633L26.882 34.019Q26.988 33.834 27.166 33.692Q27.343 33.551 27.552 33.477Q27.760 33.404 27.986 33.404Q28.492 33.404 28.776 33.627Q29.059 33.851 29.059 34.347L29.059 36.001Q29.059 36.138 29.208 36.174Q29.357 36.210 29.582 36.210L29.582 36.490L27.952 36.490L27.952 36.210Q28.181 36.210 28.329 36.176Q28.478 36.141 28.478 36.001L28.478 34.361Q28.478 34.026 28.359 33.826Q28.239 33.626 27.924 33.626Q27.654 33.626 27.420 33.762Q27.186 33.899 27.048 34.133Q26.909 34.367 26.909 34.641L26.909 36.001Q26.909 36.138 27.060 36.174Q27.210 36.210 27.436 36.210L27.436 36.490M30.129 34.955Q30.129 34.634 30.254 34.345Q30.379 34.056 30.604 33.833Q30.830 33.609 31.125 33.489Q31.421 33.369 31.739 33.369Q32.067 33.369 32.329 33.469Q32.590 33.568 32.766 33.750Q32.942 33.933 33.036 34.191Q33.130 34.449 33.130 34.781Q33.130 34.873 33.048 34.894L30.792 34.894L30.792 34.955Q30.792 35.543 31.076 35.926Q31.360 36.309 31.927 36.309Q32.248 36.309 32.517 36.116Q32.785 35.923 32.874 35.608Q32.881 35.567 32.956 35.553L33.048 35.553Q33.130 35.577 33.130 35.649Q33.130 35.656 33.123 35.683Q33.010 36.080 32.640 36.319Q32.269 36.558 31.845 36.558Q31.407 36.558 31.007 36.350Q30.608 36.141 30.368 35.774Q30.129 35.407 30.129 34.955M30.799 34.685L32.614 34.685Q32.614 34.408 32.517 34.156Q32.419 33.903 32.221 33.747Q32.023 33.592 31.739 33.592Q31.462 33.592 31.248 33.750Q31.035 33.909 30.917 34.164Q30.799 34.419 30.799 34.685M35.400 36.490L33.766 36.490L33.766 36.210Q33.995 36.210 34.143 36.176Q34.292 36.141 34.292 36.001L34.292 34.152Q34.292 33.882 34.184 33.821Q34.077 33.759 33.766 33.759L33.766 33.479L34.825 33.404L34.825 34.053Q34.996 33.745 35.300 33.574Q35.605 33.404 35.950 33.404Q36.456 33.404 36.739 33.627Q37.023 33.851 37.023 34.347L37.023 36.001Q37.023 36.138 37.172 36.174Q37.320 36.210 37.546 36.210L37.546 36.490L35.916 36.490L35.916 36.210Q36.145 36.210 36.293 36.176Q36.442 36.141 36.442 36.001L36.442 34.361Q36.442 34.026 36.322 33.826Q36.203 33.626 35.888 33.626Q35.618 33.626 35.384 33.762Q35.150 33.899 35.012 34.133Q34.873 34.367 34.873 34.641L34.873 36.001Q34.873 36.138 35.024 36.174Q35.174 36.210 35.400 36.210\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M43.026 36.490L40.920 36.490L40.920 36.210Q41.641 36.210 41.641 36.001L41.641 32.200Q41.641 31.989 40.920 31.989L40.920 31.708L43.258 31.708Q43.569 31.708 43.912 31.782Q44.256 31.855 44.577 32.011Q44.899 32.166 45.100 32.412Q45.302 32.658 45.302 32.983Q45.302 33.270 45.114 33.501Q44.926 33.732 44.649 33.880Q44.372 34.029 44.082 34.111Q44.417 34.214 44.651 34.446Q44.885 34.678 44.929 35.007L45.015 35.615Q45.042 35.827 45.088 35.996Q45.134 36.165 45.239 36.285Q45.343 36.405 45.531 36.405Q45.746 36.405 45.862 36.220Q45.979 36.035 45.979 35.803Q45.996 35.738 46.071 35.721L46.163 35.721Q46.245 35.741 46.245 35.823Q46.245 36.029 46.155 36.215Q46.064 36.401 45.898 36.516Q45.733 36.630 45.531 36.630Q45.193 36.630 44.899 36.529Q44.605 36.428 44.420 36.206Q44.235 35.984 44.235 35.636L44.235 35.027Q44.235 34.778 44.092 34.591Q43.948 34.405 43.725 34.306Q43.501 34.207 43.251 34.207L42.304 34.207L42.304 36.001Q42.304 36.210 43.026 36.210L43.026 36.490M42.304 32.200L42.304 33.985L43.159 33.985Q43.446 33.985 43.685 33.942Q43.924 33.899 44.112 33.790Q44.300 33.680 44.412 33.482Q44.523 33.284 44.523 32.983Q44.523 32.406 44.155 32.197Q43.788 31.989 43.159 31.989L42.670 31.989Q42.482 31.989 42.393 32.023Q42.304 32.057 42.304 32.200M47.144 37.720Q47.144 37.686 47.172 37.659Q47.442 37.430 47.590 37.107Q47.739 36.784 47.739 36.428L47.739 36.391Q47.630 36.490 47.465 36.490Q47.284 36.490 47.165 36.370Q47.045 36.251 47.045 36.070Q47.045 35.895 47.165 35.776Q47.284 35.656 47.465 35.656Q47.722 35.656 47.841 35.895Q47.961 36.135 47.961 36.428Q47.961 36.828 47.792 37.199Q47.623 37.570 47.325 37.826Q47.295 37.847 47.267 37.847Q47.226 37.847 47.185 37.806Q47.144 37.765 47.144 37.720\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M53.355 36.490L51.619 36.490L51.619 36.210Q51.848 36.210 51.997 36.176Q52.145 36.141 52.145 36.001L52.145 34.152Q52.145 33.882 52.038 33.821Q51.930 33.759 51.619 33.759L51.619 33.479L52.648 33.404L52.648 34.111Q52.778 33.803 53.020 33.604Q53.263 33.404 53.581 33.404Q53.800 33.404 53.971 33.528Q54.142 33.653 54.142 33.865Q54.142 34.002 54.042 34.101Q53.943 34.200 53.810 34.200Q53.673 34.200 53.574 34.101Q53.475 34.002 53.475 33.865Q53.475 33.725 53.574 33.626Q53.284 33.626 53.084 33.822Q52.884 34.019 52.791 34.313Q52.699 34.607 52.699 34.887L52.699 36.001Q52.699 36.210 53.355 36.210L53.355 36.490M54.685 34.955Q54.685 34.634 54.810 34.345Q54.935 34.056 55.160 33.833Q55.386 33.609 55.681 33.489Q55.977 33.369 56.295 33.369Q56.623 33.369 56.885 33.469Q57.146 33.568 57.322 33.750Q57.498 33.933 57.592 34.191Q57.686 34.449 57.686 34.781Q57.686 34.873 57.604 34.894L55.348 34.894L55.348 34.955Q55.348 35.543 55.632 35.926Q55.916 36.309 56.483 36.309Q56.804 36.309 57.072 36.116Q57.341 35.923 57.430 35.608Q57.437 35.567 57.512 35.553L57.604 35.553Q57.686 35.577 57.686 35.649Q57.686 35.656 57.679 35.683Q57.566 36.080 57.196 36.319Q56.825 36.558 56.401 36.558Q55.963 36.558 55.563 36.350Q55.164 36.141 54.924 35.774Q54.685 35.407 54.685 34.955M55.355 34.685L57.170 34.685Q57.170 34.408 57.072 34.156Q56.975 33.903 56.777 33.747Q56.579 33.592 56.295 33.592Q56.018 33.592 55.804 33.750Q55.591 33.909 55.473 34.164Q55.355 34.419 55.355 34.685M58.800 35.649L58.800 33.752L58.161 33.752L58.161 33.530Q58.479 33.530 58.696 33.320Q58.913 33.110 59.014 32.800Q59.115 32.491 59.115 32.183L59.381 32.183L59.381 33.472L60.458 33.472L60.458 33.752L59.381 33.752L59.381 35.636Q59.381 35.912 59.486 36.111Q59.590 36.309 59.850 36.309Q60.007 36.309 60.113 36.205Q60.219 36.100 60.268 35.947Q60.318 35.793 60.318 35.636L60.318 35.222L60.584 35.222L60.584 35.649Q60.584 35.875 60.485 36.085Q60.386 36.295 60.202 36.427Q60.017 36.558 59.788 36.558Q59.351 36.558 59.075 36.321Q58.800 36.083 58.800 35.649M61.969 35.656L61.969 34.152Q61.969 33.882 61.861 33.821Q61.753 33.759 61.442 33.759L61.442 33.479L62.550 33.404L62.550 35.636L62.550 35.656Q62.550 35.936 62.601 36.080Q62.652 36.223 62.794 36.280Q62.936 36.336 63.223 36.336Q63.476 36.336 63.681 36.196Q63.886 36.056 64.002 35.830Q64.119 35.605 64.119 35.355L64.119 34.152Q64.119 33.882 64.011 33.821Q63.903 33.759 63.592 33.759L63.592 33.479L64.700 33.404L64.700 35.817Q64.700 36.008 64.753 36.090Q64.806 36.172 64.906 36.191Q65.007 36.210 65.223 36.210L65.223 36.490L64.146 36.558L64.146 35.994Q64.037 36.176 63.891 36.299Q63.746 36.422 63.560 36.490Q63.374 36.558 63.172 36.558Q61.969 36.558 61.969 35.656M67.561 36.490L65.824 36.490L65.824 36.210Q66.053 36.210 66.202 36.176Q66.351 36.141 66.351 36.001L66.351 34.152Q66.351 33.882 66.243 33.821Q66.135 33.759 65.824 33.759L65.824 33.479L66.853 33.404L66.853 34.111Q66.983 33.803 67.226 33.604Q67.468 33.404 67.786 33.404Q68.005 33.404 68.176 33.528Q68.347 33.653 68.347 33.865Q68.347 34.002 68.248 34.101Q68.148 34.200 68.015 34.200Q67.878 34.200 67.779 34.101Q67.680 34.002 67.680 33.865Q67.680 33.725 67.779 33.626Q67.489 33.626 67.289 33.822Q67.089 34.019 66.997 34.313Q66.904 34.607 66.904 34.887L66.904 36.001Q66.904 36.210 67.561 36.210L67.561 36.490M70.613 36.490L68.979 36.490L68.979 36.210Q69.208 36.210 69.357 36.176Q69.505 36.141 69.505 36.001L69.505 34.152Q69.505 33.882 69.398 33.821Q69.290 33.759 68.979 33.759L68.979 33.479L70.039 33.404L70.039 34.053Q70.209 33.745 70.514 33.574Q70.818 33.404 71.163 33.404Q71.669 33.404 71.953 33.627Q72.236 33.851 72.236 34.347L72.236 36.001Q72.236 36.138 72.385 36.174Q72.534 36.210 72.759 36.210L72.759 36.490L71.129 36.490L71.129 36.210Q71.358 36.210 71.507 36.176Q71.655 36.141 71.655 36.001L71.655 34.361Q71.655 34.026 71.536 33.826Q71.416 33.626 71.102 33.626Q70.832 33.626 70.597 33.762Q70.363 33.899 70.225 34.133Q70.086 34.367 70.086 34.641L70.086 36.001Q70.086 36.138 70.237 36.174Q70.387 36.210 70.613 36.210\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.844 39.692)\">\u003Cpath d=\"M76.546 35.943Q76.666 36.100 76.857 36.199Q77.049 36.299 77.264 36.338Q77.479 36.377 77.702 36.377Q77.999 36.377 78.194 36.222Q78.389 36.066 78.479 35.812Q78.570 35.557 78.570 35.273Q78.570 34.979 78.478 34.728Q78.385 34.477 78.187 34.321Q77.989 34.166 77.695 34.166L77.179 34.166Q77.151 34.166 77.126 34.140Q77.100 34.115 77.100 34.091L77.100 34.019Q77.100 33.988 77.126 33.966Q77.151 33.944 77.179 33.944L77.620 33.913Q77.982 33.913 78.202 33.556Q78.423 33.198 78.423 32.809Q78.423 32.481 78.228 32.277Q78.033 32.074 77.702 32.074Q77.415 32.074 77.162 32.158Q76.909 32.241 76.745 32.429Q76.892 32.429 76.992 32.544Q77.093 32.658 77.093 32.809Q77.093 32.959 76.987 33.069Q76.881 33.178 76.724 33.178Q76.563 33.178 76.454 33.069Q76.345 32.959 76.345 32.809Q76.345 32.484 76.553 32.265Q76.762 32.047 77.078 31.944Q77.394 31.842 77.702 31.842Q78.020 31.842 78.348 31.946Q78.676 32.050 78.903 32.272Q79.130 32.494 79.130 32.809Q79.130 33.243 78.843 33.568Q78.556 33.892 78.122 34.039Q78.433 34.104 78.713 34.270Q78.994 34.436 79.171 34.694Q79.349 34.952 79.349 35.273Q79.349 35.683 79.105 35.993Q78.860 36.302 78.479 36.466Q78.098 36.630 77.702 36.630Q77.333 36.630 76.975 36.517Q76.618 36.405 76.374 36.155Q76.129 35.906 76.129 35.536Q76.129 35.365 76.246 35.253Q76.362 35.140 76.533 35.140Q76.642 35.140 76.733 35.191Q76.823 35.242 76.878 35.335Q76.933 35.427 76.933 35.536Q76.933 35.704 76.820 35.823Q76.707 35.943 76.546 35.943\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The devaluation replan. Left: before, both agents pick L then R for return 4. Right: the model-based rat learns the S2-right box now pays 1 (not 4) from one direct visit, re-backs the tree - S2 becomes max(0,1)=1, S3 stays 3 - and flips to R then R for return 3. The changed numbers and the new best path are highlighted in red.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:449.904px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 337.428 65.012\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-65.403-34.881H8.574V-71.87h-73.977Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-18.800-61.376L-20.434-61.376L-20.434-61.656Q-20.205-61.656-20.056-61.690Q-19.907-61.725-19.907-61.865L-19.907-63.714Q-19.907-63.984-20.015-64.045Q-20.123-64.107-20.434-64.107L-20.434-64.387L-19.374-64.462L-19.374-63.813Q-19.203-64.121-18.899-64.292Q-18.595-64.462-18.250-64.462Q-17.850-64.462-17.573-64.322Q-17.296-64.182-17.211-63.834Q-17.043-64.127-16.744-64.295Q-16.445-64.462-16.100-64.462Q-15.594-64.462-15.310-64.239Q-15.026-64.015-15.026-63.519L-15.026-61.865Q-15.026-61.728-14.878-61.692Q-14.729-61.656-14.504-61.656L-14.504-61.376L-16.134-61.376L-16.134-61.656Q-15.908-61.656-15.758-61.692Q-15.608-61.728-15.608-61.865L-15.608-63.505Q-15.608-63.840-15.727-64.040Q-15.847-64.240-16.161-64.240Q-16.431-64.240-16.665-64.104Q-16.900-63.967-17.038-63.733Q-17.176-63.499-17.176-63.225L-17.176-61.865Q-17.176-61.728-17.028-61.692Q-16.879-61.656-16.653-61.656L-16.653-61.376L-18.284-61.376L-18.284-61.656Q-18.055-61.656-17.906-61.690Q-17.757-61.725-17.757-61.865L-17.757-63.505Q-17.757-63.840-17.877-64.040Q-17.997-64.240-18.311-64.240Q-18.581-64.240-18.815-64.104Q-19.049-63.967-19.188-63.733Q-19.326-63.499-19.326-63.225L-19.326-61.865Q-19.326-61.728-19.176-61.692Q-19.025-61.656-18.800-61.656L-18.800-61.376M-13.957-62.859Q-13.957-63.201-13.822-63.500Q-13.687-63.799-13.447-64.023Q-13.208-64.247-12.890-64.372Q-12.572-64.497-12.241-64.497Q-11.796-64.497-11.397-64.281Q-10.997-64.066-10.763-63.688Q-10.528-63.311-10.528-62.859Q-10.528-62.518-10.670-62.234Q-10.812-61.950-11.056-61.743Q-11.301-61.537-11.610-61.422Q-11.920-61.308-12.241-61.308Q-12.671-61.308-13.073-61.509Q-13.475-61.711-13.716-62.063Q-13.957-62.415-13.957-62.859M-12.241-61.557Q-11.639-61.557-11.415-61.935Q-11.192-62.313-11.192-62.945Q-11.192-63.557-11.426-63.916Q-11.660-64.274-12.241-64.274Q-13.294-64.274-13.294-62.945Q-13.294-62.313-13.068-61.935Q-12.842-61.557-12.241-61.557\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-9.711-62.887Q-9.711-63.225-9.570-63.516Q-9.430-63.806-9.186-64.020Q-8.942-64.233-8.637-64.348Q-8.333-64.462-8.008-64.462Q-7.738-64.462-7.475-64.363Q-7.212-64.264-7.021-64.086L-7.021-65.484Q-7.021-65.754-7.128-65.816Q-7.236-65.877-7.547-65.877L-7.547-66.158L-6.470-66.233L-6.470-62.049Q-6.470-61.861-6.416-61.778Q-6.361-61.694-6.260-61.675Q-6.159-61.656-5.944-61.656L-5.944-61.376L-7.051-61.308L-7.051-61.725Q-7.468-61.308-8.094-61.308Q-8.525-61.308-8.897-61.520Q-9.270-61.731-9.490-62.092Q-9.711-62.453-9.711-62.887M-8.036-61.530Q-7.827-61.530-7.641-61.602Q-7.455-61.673-7.301-61.810Q-7.147-61.947-7.051-62.125L-7.051-63.734Q-7.137-63.881-7.282-64.001Q-7.427-64.121-7.597-64.180Q-7.766-64.240-7.947-64.240Q-8.507-64.240-8.776-63.851Q-9.044-63.461-9.044-62.880Q-9.044-62.309-8.810-61.919Q-8.576-61.530-8.036-61.530M-5.336-62.911Q-5.336-63.232-5.211-63.521Q-5.086-63.810-4.860-64.033Q-4.635-64.257-4.339-64.377Q-4.044-64.497-3.726-64.497Q-3.398-64.497-3.136-64.397Q-2.875-64.298-2.699-64.116Q-2.523-63.933-2.429-63.675Q-2.335-63.417-2.335-63.085Q-2.335-62.993-2.417-62.972L-4.672-62.972L-4.672-62.911Q-4.672-62.323-4.389-61.940Q-4.105-61.557-3.538-61.557Q-3.216-61.557-2.948-61.750Q-2.680-61.943-2.591-62.258Q-2.584-62.299-2.509-62.313L-2.417-62.313Q-2.335-62.289-2.335-62.217Q-2.335-62.210-2.341-62.183Q-2.454-61.786-2.825-61.547Q-3.196-61.308-3.620-61.308Q-4.057-61.308-4.457-61.516Q-4.857-61.725-5.096-62.092Q-5.336-62.459-5.336-62.911M-4.666-63.181L-2.851-63.181Q-2.851-63.458-2.948-63.710Q-3.046-63.963-3.244-64.119Q-3.442-64.274-3.726-64.274Q-4.003-64.274-4.216-64.116Q-4.430-63.957-4.548-63.702Q-4.666-63.447-4.666-63.181M-0.079-61.376L-1.682-61.376L-1.682-61.656Q-1.456-61.656-1.307-61.690Q-1.159-61.725-1.159-61.865L-1.159-65.484Q-1.159-65.754-1.266-65.816Q-1.374-65.877-1.682-65.877L-1.682-66.158L-0.605-66.233L-0.605-61.865Q-0.605-61.728-0.455-61.692Q-0.304-61.656-0.079-61.656L-0.079-61.376M2.399-62.630L0.342-62.630L0.342-63.133L2.399-63.133L2.399-62.630M5-61.376L3.267-61.376L3.267-61.656Q3.493-61.656 3.642-61.690Q3.790-61.725 3.790-61.865L3.790-64.114L3.203-64.114L3.203-64.394L3.790-64.394L3.790-65.211Q3.790-65.529 3.968-65.777Q4.146-66.024 4.436-66.165Q4.727-66.305 5.038-66.305Q5.294-66.305 5.498-66.163Q5.701-66.021 5.701-65.778Q5.701-65.642 5.602-65.543Q5.503-65.443 5.366-65.443Q5.229-65.443 5.130-65.543Q5.031-65.642 5.031-65.778Q5.031-65.959 5.171-66.052Q5.093-66.079 4.994-66.079Q4.785-66.079 4.631-65.946Q4.477-65.813 4.397-65.609Q4.317-65.406 4.317-65.197L4.317-64.394L5.205-64.394L5.205-64.114L4.344-64.114L4.344-61.865Q4.344-61.656 5-61.656L5-61.376M7.431-61.376L5.694-61.376L5.694-61.656Q5.923-61.656 6.072-61.690Q6.221-61.725 6.221-61.865L6.221-63.714Q6.221-63.984 6.113-64.045Q6.005-64.107 5.694-64.107L5.694-64.387L6.723-64.462L6.723-63.755Q6.853-64.063 7.096-64.262Q7.338-64.462 7.656-64.462Q7.875-64.462 8.046-64.338Q8.217-64.213 8.217-64.001Q8.217-63.864 8.118-63.765Q8.018-63.666 7.885-63.666Q7.748-63.666 7.649-63.765Q7.550-63.864 7.550-64.001Q7.550-64.141 7.649-64.240Q7.359-64.240 7.159-64.044Q6.959-63.847 6.867-63.553Q6.774-63.259 6.774-62.979L6.774-61.865Q6.774-61.656 7.431-61.656L7.431-61.376M8.760-62.911Q8.760-63.232 8.885-63.521Q9.010-63.810 9.235-64.033Q9.461-64.257 9.756-64.377Q10.052-64.497 10.370-64.497Q10.698-64.497 10.960-64.397Q11.221-64.298 11.397-64.116Q11.573-63.933 11.667-63.675Q11.761-63.417 11.761-63.085Q11.761-62.993 11.679-62.972L9.423-62.972L9.423-62.911Q9.423-62.323 9.707-61.940Q9.991-61.557 10.558-61.557Q10.879-61.557 11.148-61.750Q11.416-61.943 11.505-62.258Q11.512-62.299 11.587-62.313L11.679-62.313Q11.761-62.289 11.761-62.217Q11.761-62.210 11.754-62.183Q11.642-61.786 11.271-61.547Q10.900-61.308 10.476-61.308Q10.038-61.308 9.639-61.516Q9.239-61.725 8.999-62.092Q8.760-62.459 8.760-62.911M9.430-63.181L11.245-63.181Q11.245-63.458 11.148-63.710Q11.050-63.963 10.852-64.119Q10.654-64.274 10.370-64.274Q10.093-64.274 9.880-64.116Q9.666-63.957 9.548-63.702Q9.430-63.447 9.430-63.181M12.308-62.911Q12.308-63.232 12.433-63.521Q12.558-63.810 12.783-64.033Q13.009-64.257 13.304-64.377Q13.600-64.497 13.918-64.497Q14.246-64.497 14.507-64.397Q14.769-64.298 14.945-64.116Q15.121-63.933 15.215-63.675Q15.309-63.417 15.309-63.085Q15.309-62.993 15.227-62.972L12.971-62.972L12.971-62.911Q12.971-62.323 13.255-61.940Q13.538-61.557 14.106-61.557Q14.427-61.557 14.695-61.750Q14.964-61.943 15.053-62.258Q15.059-62.299 15.135-62.313L15.227-62.313Q15.309-62.289 15.309-62.217Q15.309-62.210 15.302-62.183Q15.189-61.786 14.819-61.547Q14.448-61.308 14.024-61.308Q13.586-61.308 13.186-61.516Q12.787-61.725 12.547-62.092Q12.308-62.459 12.308-62.911M12.978-63.181L14.793-63.181Q14.793-63.458 14.695-63.710Q14.598-63.963 14.400-64.119Q14.202-64.274 13.918-64.274Q13.641-64.274 13.427-64.116Q13.214-63.957 13.096-63.702Q12.978-63.447 12.978-63.181\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-26.511-53.403L-27.639-55.902Q-27.711-56.049-27.841-56.081Q-27.971-56.114-28.200-56.114L-28.200-56.394L-26.686-56.394L-26.686-56.114Q-27.038-56.114-27.038-55.967Q-27.038-55.922-27.027-55.902L-26.163-53.984L-25.383-55.714Q-25.349-55.782-25.349-55.861Q-25.349-55.974-25.433-56.044Q-25.517-56.114-25.636-56.114L-25.636-56.394L-24.440-56.394L-24.440-56.114Q-24.659-56.114-24.830-56.011Q-25-55.909-25.089-55.714L-26.125-53.403Q-26.173-53.308-26.279-53.308L-26.357-53.308Q-26.463-53.308-26.511-53.403\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-24.271-54.104Q-24.271-54.436-24.048-54.663Q-23.824-54.890-23.480-55.018Q-23.137-55.147-22.764-55.199Q-22.392-55.252-22.087-55.252L-22.087-55.505Q-22.087-55.710-22.195-55.890Q-22.303-56.069-22.484-56.172Q-22.665-56.274-22.873-56.274Q-23.280-56.274-23.516-56.182Q-23.427-56.145-23.381-56.061Q-23.335-55.977-23.335-55.875Q-23.335-55.779-23.381-55.700Q-23.427-55.622-23.508-55.577Q-23.588-55.533-23.677-55.533Q-23.827-55.533-23.928-55.630Q-24.029-55.728-24.029-55.875Q-24.029-56.497-22.873-56.497Q-22.662-56.497-22.412-56.433Q-22.163-56.370-21.961-56.251Q-21.759-56.131-21.633-55.946Q-21.506-55.762-21.506-55.519L-21.506-53.943Q-21.506-53.827-21.445-53.731Q-21.383-53.636-21.270-53.636Q-21.161-53.636-21.096-53.730Q-21.031-53.824-21.031-53.943L-21.031-54.391L-20.765-54.391L-20.765-53.943Q-20.765-53.673-20.992-53.508Q-21.219-53.342-21.499-53.342Q-21.708-53.342-21.845-53.496Q-21.981-53.649-22.005-53.865Q-22.152-53.598-22.434-53.453Q-22.716-53.308-23.041-53.308Q-23.318-53.308-23.602-53.383Q-23.885-53.458-24.078-53.637Q-24.271-53.817-24.271-54.104M-23.656-54.104Q-23.656-53.930-23.555-53.800Q-23.455-53.670-23.299-53.600Q-23.144-53.530-22.979-53.530Q-22.761-53.530-22.552-53.627Q-22.344-53.725-22.216-53.906Q-22.087-54.087-22.087-54.313L-22.087-55.041Q-22.412-55.041-22.778-54.950Q-23.144-54.859-23.400-54.647Q-23.656-54.436-23.656-54.104M-18.680-53.376L-20.283-53.376L-20.283-53.656Q-20.057-53.656-19.908-53.690Q-19.760-53.725-19.760-53.865L-19.760-57.484Q-19.760-57.754-19.867-57.816Q-19.975-57.877-20.283-57.877L-20.283-58.158L-19.206-58.233L-19.206-53.865Q-19.206-53.728-19.056-53.692Q-18.905-53.656-18.680-53.656L-18.680-53.376M-17.511-54.210L-17.511-55.714Q-17.511-55.984-17.618-56.045Q-17.726-56.107-18.037-56.107L-18.037-56.387L-16.930-56.462L-16.930-54.230L-16.930-54.210Q-16.930-53.930-16.878-53.786Q-16.827-53.643-16.685-53.586Q-16.543-53.530-16.256-53.530Q-16.003-53.530-15.798-53.670Q-15.593-53.810-15.477-54.036Q-15.361-54.261-15.361-54.511L-15.361-55.714Q-15.361-55.984-15.468-56.045Q-15.576-56.107-15.887-56.107L-15.887-56.387L-14.780-56.462L-14.780-54.049Q-14.780-53.858-14.727-53.776Q-14.674-53.694-14.573-53.675Q-14.472-53.656-14.257-53.656L-14.257-53.376L-15.333-53.308L-15.333-53.872Q-15.443-53.690-15.588-53.567Q-15.733-53.444-15.920-53.376Q-16.106-53.308-16.308-53.308Q-17.511-53.308-17.511-54.210M-13.710-54.911Q-13.710-55.232-13.585-55.521Q-13.460-55.810-13.235-56.033Q-13.009-56.257-12.714-56.377Q-12.418-56.497-12.100-56.497Q-11.772-56.497-11.510-56.397Q-11.249-56.298-11.073-56.116Q-10.897-55.933-10.803-55.675Q-10.709-55.417-10.709-55.085Q-10.709-54.993-10.791-54.972L-13.047-54.972L-13.047-54.911Q-13.047-54.323-12.763-53.940Q-12.479-53.557-11.912-53.557Q-11.591-53.557-11.322-53.750Q-11.054-53.943-10.965-54.258Q-10.958-54.299-10.883-54.313L-10.791-54.313Q-10.709-54.289-10.709-54.217Q-10.709-54.210-10.716-54.183Q-10.829-53.786-11.199-53.547Q-11.570-53.308-11.994-53.308Q-12.432-53.308-12.831-53.516Q-13.231-53.725-13.471-54.092Q-13.710-54.459-13.710-54.911M-13.040-55.181L-11.225-55.181Q-11.225-55.458-11.322-55.710Q-11.420-55.963-11.618-56.119Q-11.816-56.274-12.100-56.274Q-12.377-56.274-12.591-56.116Q-12.804-55.957-12.922-55.702Q-13.040-55.447-13.040-55.181\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-5.769-52.019L-7.399-52.019L-7.399-52.299Q-7.170-52.299-7.021-52.334Q-6.873-52.368-6.873-52.508L-6.873-55.854Q-6.873-56.025-7.009-56.066Q-7.146-56.107-7.399-56.107L-7.399-56.387L-6.319-56.462L-6.319-56.056Q-6.097-56.257-5.810-56.360Q-5.522-56.462-5.215-56.462Q-4.788-56.462-4.424-56.249Q-4.060-56.035-3.846-55.671Q-3.632-55.307-3.632-54.887Q-3.632-54.442-3.872-54.078Q-4.111-53.714-4.504-53.511Q-4.897-53.308-5.341-53.308Q-5.608-53.308-5.856-53.408Q-6.103-53.509-6.291-53.690L-6.291-52.508Q-6.291-52.371-6.143-52.335Q-5.994-52.299-5.769-52.299L-5.769-52.019M-6.291-55.707L-6.291-54.097Q-6.158-53.844-5.915-53.687Q-5.673-53.530-5.396-53.530Q-5.068-53.530-4.815-53.731Q-4.562-53.933-4.429-54.251Q-4.295-54.569-4.295-54.887Q-4.295-55.116-4.360-55.345Q-4.425-55.574-4.553-55.772Q-4.682-55.970-4.876-56.090Q-5.071-56.209-5.304-56.209Q-5.598-56.209-5.866-56.080Q-6.134-55.950-6.291-55.707\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M-2.822-54.911Q-2.822-55.232-2.697-55.521Q-2.572-55.810-2.346-56.033Q-2.121-56.257-1.825-56.377Q-1.530-56.497-1.212-56.497Q-0.884-56.497-0.622-56.397Q-0.361-56.298-0.185-56.116Q-0.009-55.933 0.085-55.675Q0.179-55.417 0.179-55.085Q0.179-54.993 0.097-54.972L-2.158-54.972L-2.158-54.911Q-2.158-54.323-1.875-53.940Q-1.591-53.557-1.024-53.557Q-0.702-53.557-0.434-53.750Q-0.166-53.943-0.077-54.258Q-0.070-54.299 0.005-54.313L0.097-54.313Q0.179-54.289 0.179-54.217Q0.179-54.210 0.173-54.183Q0.060-53.786-0.311-53.547Q-0.682-53.308-1.106-53.308Q-1.543-53.308-1.943-53.516Q-2.343-53.725-2.582-54.092Q-2.822-54.459-2.822-54.911M-2.152-55.181L-0.337-55.181Q-0.337-55.458-0.434-55.710Q-0.532-55.963-0.730-56.119Q-0.928-56.274-1.212-56.274Q-1.489-56.274-1.702-56.116Q-1.916-55.957-2.034-55.702Q-2.152-55.447-2.152-55.181M2.517-53.376L0.781-53.376L0.781-53.656Q1.010-53.656 1.159-53.690Q1.307-53.725 1.307-53.865L1.307-55.714Q1.307-55.984 1.200-56.045Q1.092-56.107 0.781-56.107L0.781-56.387L1.810-56.462L1.810-55.755Q1.940-56.063 2.182-56.262Q2.425-56.462 2.743-56.462Q2.962-56.462 3.133-56.338Q3.303-56.213 3.303-56.001Q3.303-55.864 3.204-55.765Q3.105-55.666 2.972-55.666Q2.835-55.666 2.736-55.765Q2.637-55.864 2.637-56.001Q2.637-56.141 2.736-56.240Q2.446-56.240 2.246-56.044Q2.046-55.847 1.953-55.553Q1.861-55.259 1.861-54.979L1.861-53.865Q1.861-53.656 2.517-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.827 5.75)\">\u003Cpath d=\"M6.594-53.383L6.594-54.446Q6.594-54.470 6.622-54.497Q6.649-54.524 6.673-54.524L6.782-54.524Q6.847-54.524 6.861-54.466Q6.957-54.032 7.203-53.781Q7.449-53.530 7.863-53.530Q8.204-53.530 8.457-53.663Q8.710-53.796 8.710-54.104Q8.710-54.261 8.616-54.376Q8.522-54.490 8.384-54.559Q8.245-54.627 8.078-54.665L7.497-54.764Q7.141-54.832 6.868-55.053Q6.594-55.273 6.594-55.615Q6.594-55.864 6.706-56.039Q6.817-56.213 7.003-56.312Q7.189-56.411 7.405-56.454Q7.620-56.497 7.863-56.497Q8.276-56.497 8.556-56.315L8.772-56.490Q8.782-56.493 8.789-56.495Q8.796-56.497 8.806-56.497L8.857-56.497Q8.884-56.497 8.908-56.473Q8.932-56.449 8.932-56.421L8.932-55.574Q8.932-55.553 8.908-55.526Q8.884-55.499 8.857-55.499L8.744-55.499Q8.717-55.499 8.691-55.524Q8.666-55.550 8.666-55.574Q8.666-55.810 8.560-55.974Q8.454-56.138 8.271-56.220Q8.088-56.302 7.856-56.302Q7.528-56.302 7.271-56.199Q7.015-56.097 7.015-55.820Q7.015-55.625 7.198-55.516Q7.381-55.406 7.610-55.365L8.184-55.259Q8.430-55.211 8.644-55.083Q8.857-54.955 8.994-54.752Q9.131-54.548 9.131-54.299Q9.131-53.786 8.765-53.547Q8.399-53.308 7.863-53.308Q7.367-53.308 7.035-53.602L6.769-53.328Q6.748-53.308 6.721-53.308L6.673-53.308Q6.649-53.308 6.622-53.335Q6.594-53.362 6.594-53.383M10.286-54.217L10.286-56.114L9.647-56.114L9.647-56.336Q9.965-56.336 10.182-56.546Q10.399-56.756 10.499-57.066Q10.600-57.375 10.600-57.683L10.867-57.683L10.867-56.394L11.944-56.394L11.944-56.114L10.867-56.114L10.867-54.230Q10.867-53.954 10.971-53.755Q11.075-53.557 11.335-53.557Q11.492-53.557 11.598-53.661Q11.704-53.766 11.754-53.919Q11.803-54.073 11.803-54.230L11.803-54.644L12.070-54.644L12.070-54.217Q12.070-53.991 11.971-53.781Q11.872-53.571 11.687-53.439Q11.503-53.308 11.274-53.308Q10.836-53.308 10.561-53.545Q10.286-53.783 10.286-54.217M12.938-54.104Q12.938-54.436 13.162-54.663Q13.386-54.890 13.729-55.018Q14.073-55.147 14.446-55.199Q14.818-55.252 15.122-55.252L15.122-55.505Q15.122-55.710 15.015-55.890Q14.907-56.069 14.726-56.172Q14.545-56.274 14.336-56.274Q13.929-56.274 13.694-56.182Q13.782-56.145 13.829-56.061Q13.875-55.977 13.875-55.875Q13.875-55.779 13.829-55.700Q13.782-55.622 13.702-55.577Q13.622-55.533 13.533-55.533Q13.383-55.533 13.282-55.630Q13.181-55.728 13.181-55.875Q13.181-56.497 14.336-56.497Q14.548-56.497 14.798-56.433Q15.047-56.370 15.249-56.251Q15.450-56.131 15.577-55.946Q15.703-55.762 15.703-55.519L15.703-53.943Q15.703-53.827 15.765-53.731Q15.826-53.636 15.939-53.636Q16.049-53.636 16.113-53.730Q16.178-53.824 16.178-53.943L16.178-54.391L16.445-54.391L16.445-53.943Q16.445-53.673 16.218-53.508Q15.990-53.342 15.710-53.342Q15.502-53.342 15.365-53.496Q15.228-53.649 15.204-53.865Q15.057-53.598 14.775-53.453Q14.493-53.308 14.169-53.308Q13.892-53.308 13.608-53.383Q13.324-53.458 13.131-53.637Q12.938-53.817 12.938-54.104M13.553-54.104Q13.553-53.930 13.654-53.800Q13.755-53.670 13.911-53.600Q14.066-53.530 14.230-53.530Q14.449-53.530 14.657-53.627Q14.866-53.725 14.994-53.906Q15.122-54.087 15.122-54.313L15.122-55.041Q14.798-55.041 14.432-54.950Q14.066-54.859 13.810-54.647Q13.553-54.436 13.553-54.104M17.388-54.217L17.388-56.114L16.749-56.114L16.749-56.336Q17.067-56.336 17.284-56.546Q17.501-56.756 17.602-57.066Q17.703-57.375 17.703-57.683L17.969-57.683L17.969-56.394L19.046-56.394L19.046-56.114L17.969-56.114L17.969-54.230Q17.969-53.954 18.074-53.755Q18.178-53.557 18.438-53.557Q18.595-53.557 18.701-53.661Q18.807-53.766 18.856-53.919Q18.906-54.073 18.906-54.230L18.906-54.644L19.173-54.644L19.173-54.217Q19.173-53.991 19.073-53.781Q18.974-53.571 18.790-53.439Q18.605-53.308 18.376-53.308Q17.939-53.308 17.664-53.545Q17.388-53.783 17.388-54.217M19.942-54.911Q19.942-55.232 20.066-55.521Q20.191-55.810 20.417-56.033Q20.642-56.257 20.938-56.377Q21.234-56.497 21.551-56.497Q21.880-56.497 22.141-56.397Q22.403-56.298 22.579-56.116Q22.755-55.933 22.849-55.675Q22.943-55.417 22.943-55.085Q22.943-54.993 22.861-54.972L20.605-54.972L20.605-54.911Q20.605-54.323 20.888-53.940Q21.172-53.557 21.739-53.557Q22.061-53.557 22.329-53.750Q22.597-53.943 22.686-54.258Q22.693-54.299 22.768-54.313L22.861-54.313Q22.943-54.289 22.943-54.217Q22.943-54.210 22.936-54.183Q22.823-53.786 22.452-53.547Q22.081-53.308 21.657-53.308Q21.220-53.308 20.820-53.516Q20.420-53.725 20.181-54.092Q19.942-54.459 19.942-54.911M20.612-55.181L22.426-55.181Q22.426-55.458 22.329-55.710Q22.232-55.963 22.033-56.119Q21.835-56.274 21.551-56.274Q21.275-56.274 21.061-56.116Q20.847-55.957 20.729-55.702Q20.612-55.447 20.612-55.181\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M59.789-34.881h73.977V-71.87H59.789Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(95.133 4.826)\">\u003Cpath d=\"M-21.919-61.383L-21.919-62.446Q-21.919-62.470-21.891-62.497Q-21.864-62.524-21.840-62.524L-21.731-62.524Q-21.666-62.524-21.652-62.466Q-21.556-62.032-21.310-61.781Q-21.064-61.530-20.650-61.530Q-20.309-61.530-20.056-61.663Q-19.803-61.796-19.803-62.104Q-19.803-62.261-19.897-62.376Q-19.991-62.490-20.129-62.559Q-20.268-62.627-20.435-62.665L-21.016-62.764Q-21.372-62.832-21.645-63.053Q-21.919-63.273-21.919-63.615Q-21.919-63.864-21.807-64.039Q-21.696-64.213-21.510-64.312Q-21.324-64.411-21.108-64.454Q-20.893-64.497-20.650-64.497Q-20.237-64.497-19.957-64.315L-19.741-64.490Q-19.731-64.493-19.724-64.495Q-19.717-64.497-19.707-64.497L-19.656-64.497Q-19.629-64.497-19.605-64.473Q-19.581-64.449-19.581-64.421L-19.581-63.574Q-19.581-63.553-19.605-63.526Q-19.629-63.499-19.656-63.499L-19.769-63.499Q-19.796-63.499-19.822-63.524Q-19.847-63.550-19.847-63.574Q-19.847-63.810-19.953-63.974Q-20.059-64.138-20.242-64.220Q-20.425-64.302-20.657-64.302Q-20.985-64.302-21.242-64.199Q-21.498-64.097-21.498-63.820Q-21.498-63.625-21.315-63.516Q-21.132-63.406-20.903-63.365L-20.329-63.259Q-20.083-63.211-19.869-63.083Q-19.656-62.955-19.519-62.752Q-19.382-62.548-19.382-62.299Q-19.382-61.786-19.748-61.547Q-20.114-61.308-20.650-61.308Q-21.146-61.308-21.478-61.602L-21.744-61.328Q-21.765-61.308-21.792-61.308L-21.840-61.308Q-21.864-61.308-21.891-61.335Q-21.919-61.362-21.919-61.383M-18.179-62.210L-18.179-63.714Q-18.179-63.984-18.287-64.045Q-18.395-64.107-18.706-64.107L-18.706-64.387L-17.598-64.462L-17.598-62.230L-17.598-62.210Q-17.598-61.930-17.547-61.786Q-17.496-61.643-17.354-61.586Q-17.212-61.530-16.925-61.530Q-16.672-61.530-16.467-61.670Q-16.262-61.810-16.146-62.036Q-16.029-62.261-16.029-62.511L-16.029-63.714Q-16.029-63.984-16.137-64.045Q-16.245-64.107-16.556-64.107L-16.556-64.387L-15.448-64.462L-15.448-62.049Q-15.448-61.858-15.395-61.776Q-15.342-61.694-15.242-61.675Q-15.141-61.656-14.925-61.656L-14.925-61.376L-16.002-61.308L-16.002-61.872Q-16.111-61.690-16.257-61.567Q-16.402-61.444-16.588-61.376Q-16.775-61.308-16.976-61.308Q-18.179-61.308-18.179-62.210M-14.337-62.887Q-14.337-63.215-14.202-63.516Q-14.067-63.816-13.832-64.037Q-13.596-64.257-13.292-64.377Q-12.987-64.497-12.663-64.497Q-12.157-64.497-11.808-64.394Q-11.460-64.292-11.460-63.916Q-11.460-63.769-11.557-63.668Q-11.654-63.567-11.801-63.567Q-11.955-63.567-12.054-63.666Q-12.153-63.765-12.153-63.916Q-12.153-64.104-12.013-64.196Q-12.215-64.247-12.656-64.247Q-13.011-64.247-13.240-64.051Q-13.469-63.854-13.570-63.545Q-13.671-63.235-13.671-62.887Q-13.671-62.538-13.545-62.232Q-13.418-61.926-13.163-61.742Q-12.909-61.557-12.553-61.557Q-12.331-61.557-12.147-61.641Q-11.962-61.725-11.827-61.880Q-11.692-62.036-11.634-62.244Q-11.620-62.299-11.566-62.299L-11.453-62.299Q-11.422-62.299-11.400-62.275Q-11.378-62.251-11.378-62.217L-11.378-62.196Q-11.463-61.909-11.651-61.711Q-11.839-61.513-12.104-61.410Q-12.369-61.308-12.663-61.308Q-13.093-61.308-13.481-61.514Q-13.869-61.721-14.103-62.084Q-14.337-62.446-14.337-62.887M-10.790-62.887Q-10.790-63.215-10.655-63.516Q-10.520-63.816-10.284-64.037Q-10.048-64.257-9.744-64.377Q-9.440-64.497-9.115-64.497Q-8.609-64.497-8.260-64.394Q-7.912-64.292-7.912-63.916Q-7.912-63.769-8.009-63.668Q-8.107-63.567-8.254-63.567Q-8.407-63.567-8.506-63.666Q-8.606-63.765-8.606-63.916Q-8.606-64.104-8.465-64.196Q-8.667-64.247-9.108-64.247Q-9.463-64.247-9.692-64.051Q-9.921-63.854-10.022-63.545Q-10.123-63.235-10.123-62.887Q-10.123-62.538-9.997-62.232Q-9.870-61.926-9.616-61.742Q-9.361-61.557-9.005-61.557Q-8.783-61.557-8.599-61.641Q-8.414-61.725-8.279-61.880Q-8.144-62.036-8.086-62.244Q-8.072-62.299-8.018-62.299L-7.905-62.299Q-7.874-62.299-7.852-62.275Q-7.830-62.251-7.830-62.217L-7.830-62.196Q-7.915-61.909-8.103-61.711Q-8.291-61.513-8.556-61.410Q-8.821-61.308-9.115-61.308Q-9.546-61.308-9.933-61.514Q-10.321-61.721-10.556-62.084Q-10.790-62.446-10.790-62.887M-7.283-62.911Q-7.283-63.232-7.158-63.521Q-7.033-63.810-6.808-64.033Q-6.582-64.257-6.286-64.377Q-5.991-64.497-5.673-64.497Q-5.345-64.497-5.083-64.397Q-4.822-64.298-4.646-64.116Q-4.470-63.933-4.376-63.675Q-4.282-63.417-4.282-63.085Q-4.282-62.993-4.364-62.972L-6.620-62.972L-6.620-62.911Q-6.620-62.323-6.336-61.940Q-6.052-61.557-5.485-61.557Q-5.164-61.557-4.895-61.750Q-4.627-61.943-4.538-62.258Q-4.531-62.299-4.456-62.313L-4.364-62.313Q-4.282-62.289-4.282-62.217Q-4.282-62.210-4.289-62.183Q-4.401-61.786-4.772-61.547Q-5.143-61.308-5.567-61.308Q-6.004-61.308-6.404-61.516Q-6.804-61.725-7.044-62.092Q-7.283-62.459-7.283-62.911M-6.613-63.181L-4.798-63.181Q-4.798-63.458-4.895-63.710Q-4.993-63.963-5.191-64.119Q-5.389-64.274-5.673-64.274Q-5.950-64.274-6.163-64.116Q-6.377-63.957-6.495-63.702Q-6.613-63.447-6.613-63.181M-3.694-61.383L-3.694-62.446Q-3.694-62.470-3.667-62.497Q-3.639-62.524-3.615-62.524L-3.506-62.524Q-3.441-62.524-3.427-62.466Q-3.332-62.032-3.086-61.781Q-2.839-61.530-2.426-61.530Q-2.084-61.530-1.831-61.663Q-1.578-61.796-1.578-62.104Q-1.578-62.261-1.672-62.376Q-1.766-62.490-1.905-62.559Q-2.043-62.627-2.211-62.665L-2.792-62.764Q-3.147-62.832-3.421-63.053Q-3.694-63.273-3.694-63.615Q-3.694-63.864-3.583-64.039Q-3.472-64.213-3.285-64.312Q-3.099-64.411-2.884-64.454Q-2.669-64.497-2.426-64.497Q-2.012-64.497-1.732-64.315L-1.517-64.490Q-1.506-64.493-1.500-64.495Q-1.493-64.497-1.483-64.497L-1.431-64.497Q-1.404-64.497-1.380-64.473Q-1.356-64.449-1.356-64.421L-1.356-63.574Q-1.356-63.553-1.380-63.526Q-1.404-63.499-1.431-63.499L-1.544-63.499Q-1.571-63.499-1.597-63.524Q-1.623-63.550-1.623-63.574Q-1.623-63.810-1.729-63.974Q-1.835-64.138-2.017-64.220Q-2.200-64.302-2.433-64.302Q-2.761-64.302-3.017-64.199Q-3.274-64.097-3.274-63.820Q-3.274-63.625-3.091-63.516Q-2.908-63.406-2.679-63.365L-2.105-63.259Q-1.858-63.211-1.645-63.083Q-1.431-62.955-1.295-62.752Q-1.158-62.548-1.158-62.299Q-1.158-61.786-1.524-61.547Q-1.889-61.308-2.426-61.308Q-2.921-61.308-3.253-61.602L-3.520-61.328Q-3.540-61.308-3.567-61.308L-3.615-61.308Q-3.639-61.308-3.667-61.335Q-3.694-61.362-3.694-61.383M-0.529-61.383L-0.529-62.446Q-0.529-62.470-0.502-62.497Q-0.474-62.524-0.450-62.524L-0.341-62.524Q-0.276-62.524-0.262-62.466Q-0.167-62.032 0.079-61.781Q0.326-61.530 0.739-61.530Q1.081-61.530 1.334-61.663Q1.587-61.796 1.587-62.104Q1.587-62.261 1.493-62.376Q1.399-62.490 1.260-62.559Q1.122-62.627 0.954-62.665L0.373-62.764Q0.018-62.832-0.255-63.053Q-0.529-63.273-0.529-63.615Q-0.529-63.864-0.418-64.039Q-0.307-64.213-0.120-64.312Q0.066-64.411 0.281-64.454Q0.496-64.497 0.739-64.497Q1.153-64.497 1.433-64.315L1.648-64.490Q1.659-64.493 1.665-64.495Q1.672-64.497 1.683-64.497L1.734-64.497Q1.761-64.497 1.785-64.473Q1.809-64.449 1.809-64.421L1.809-63.574Q1.809-63.553 1.785-63.526Q1.761-63.499 1.734-63.499L1.621-63.499Q1.594-63.499 1.568-63.524Q1.542-63.550 1.542-63.574Q1.542-63.810 1.436-63.974Q1.330-64.138 1.148-64.220Q0.965-64.302 0.732-64.302Q0.404-64.302 0.148-64.199Q-0.108-64.097-0.108-63.820Q-0.108-63.625 0.074-63.516Q0.257-63.406 0.486-63.365L1.060-63.259Q1.307-63.211 1.520-63.083Q1.734-62.955 1.871-62.752Q2.007-62.548 2.007-62.299Q2.007-61.786 1.642-61.547Q1.276-61.308 0.739-61.308Q0.244-61.308-0.088-61.602L-0.355-61.328Q-0.375-61.308-0.402-61.308L-0.450-61.308Q-0.474-61.308-0.502-61.335Q-0.529-61.362-0.529-61.383M2.595-62.859Q2.595-63.201 2.730-63.500Q2.865-63.799 3.104-64.023Q3.344-64.247 3.662-64.372Q3.979-64.497 4.311-64.497Q4.755-64.497 5.155-64.281Q5.555-64.066 5.789-63.688Q6.023-63.311 6.023-62.859Q6.023-62.518 5.882-62.234Q5.740-61.950 5.495-61.743Q5.251-61.537 4.942-61.422Q4.632-61.308 4.311-61.308Q3.880-61.308 3.479-61.509Q3.077-61.711 2.836-62.063Q2.595-62.415 2.595-62.859M4.311-61.557Q4.913-61.557 5.136-61.935Q5.360-62.313 5.360-62.945Q5.360-63.557 5.126-63.916Q4.892-64.274 4.311-64.274Q3.258-64.274 3.258-62.945Q3.258-62.313 3.484-61.935Q3.709-61.557 4.311-61.557M8.368-61.376L6.632-61.376L6.632-61.656Q6.861-61.656 7.009-61.690Q7.158-61.725 7.158-61.865L7.158-63.714Q7.158-63.984 7.050-64.045Q6.943-64.107 6.632-64.107L6.632-64.387L7.661-64.462L7.661-63.755Q7.790-64.063 8.033-64.262Q8.276-64.462 8.594-64.462Q8.812-64.462 8.983-64.338Q9.154-64.213 9.154-64.001Q9.154-63.864 9.055-63.765Q8.956-63.666 8.823-63.666Q8.686-63.666 8.587-63.765Q8.488-63.864 8.488-64.001Q8.488-64.141 8.587-64.240Q8.296-64.240 8.096-64.044Q7.896-63.847 7.804-63.553Q7.712-63.259 7.712-62.979L7.712-61.865Q7.712-61.656 8.368-61.656\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.133 4.826)\">\u003Cpath d=\"M14.224-61.376L12.488-61.376L12.488-61.656Q12.717-61.656 12.866-61.690Q13.014-61.725 13.014-61.865L13.014-63.714Q13.014-63.984 12.907-64.045Q12.799-64.107 12.488-64.107L12.488-64.387L13.517-64.462L13.517-63.755Q13.647-64.063 13.889-64.262Q14.132-64.462 14.450-64.462Q14.669-64.462 14.840-64.338Q15.011-64.213 15.011-64.001Q15.011-63.864 14.911-63.765Q14.812-63.666 14.679-63.666Q14.542-63.666 14.443-63.765Q14.344-63.864 14.344-64.001Q14.344-64.141 14.443-64.240Q14.153-64.240 13.953-64.044Q13.753-63.847 13.660-63.553Q13.568-63.259 13.568-62.979L13.568-61.865Q13.568-61.656 14.224-61.656L14.224-61.376M15.554-62.911Q15.554-63.232 15.679-63.521Q15.804-63.810 16.029-64.033Q16.255-64.257 16.550-64.377Q16.846-64.497 17.164-64.497Q17.492-64.497 17.754-64.397Q18.015-64.298 18.191-64.116Q18.367-63.933 18.461-63.675Q18.555-63.417 18.555-63.085Q18.555-62.993 18.473-62.972L16.217-62.972L16.217-62.911Q16.217-62.323 16.501-61.940Q16.785-61.557 17.352-61.557Q17.673-61.557 17.941-61.750Q18.210-61.943 18.299-62.258Q18.306-62.299 18.381-62.313L18.473-62.313Q18.555-62.289 18.555-62.217Q18.555-62.210 18.548-62.183Q18.435-61.786 18.065-61.547Q17.694-61.308 17.270-61.308Q16.832-61.308 16.432-61.516Q16.033-61.725 15.793-62.092Q15.554-62.459 15.554-62.911M16.224-63.181L18.039-63.181Q18.039-63.458 17.941-63.710Q17.844-63.963 17.646-64.119Q17.448-64.274 17.164-64.274Q16.887-64.274 16.673-64.116Q16.460-63.957 16.342-63.702Q16.224-63.447 16.224-63.181M20.787-60.019L19.157-60.019L19.157-60.299Q19.386-60.299 19.534-60.334Q19.683-60.368 19.683-60.508L19.683-63.854Q19.683-64.025 19.546-64.066Q19.410-64.107 19.157-64.107L19.157-64.387L20.237-64.462L20.237-64.056Q20.459-64.257 20.746-64.360Q21.033-64.462 21.341-64.462Q21.768-64.462 22.132-64.249Q22.496-64.035 22.710-63.671Q22.923-63.307 22.923-62.887Q22.923-62.442 22.684-62.078Q22.445-61.714 22.052-61.511Q21.659-61.308 21.214-61.308Q20.948-61.308 20.700-61.408Q20.452-61.509 20.264-61.690L20.264-60.508Q20.264-60.371 20.413-60.335Q20.561-60.299 20.787-60.299L20.787-60.019M20.264-63.707L20.264-62.097Q20.397-61.844 20.640-61.687Q20.883-61.530 21.160-61.530Q21.488-61.530 21.741-61.731Q21.993-61.933 22.127-62.251Q22.260-62.569 22.260-62.887Q22.260-63.116 22.195-63.345Q22.130-63.574 22.002-63.772Q21.874-63.970 21.679-64.090Q21.484-64.209 21.252-64.209Q20.958-64.209 20.690-64.080Q20.421-63.950 20.264-63.707M23.959-61.796Q23.959-61.964 24.082-62.087Q24.205-62.210 24.379-62.210Q24.547-62.210 24.670-62.087Q24.793-61.964 24.793-61.796Q24.793-61.622 24.670-61.499Q24.547-61.376 24.379-61.376Q24.205-61.376 24.082-61.499Q23.959-61.622 23.959-61.796\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.133 4.826)\">\u003Cpath d=\"M-26.303-53.376L-28.036-53.376L-28.036-53.656Q-27.810-53.656-27.661-53.690Q-27.513-53.725-27.513-53.865L-27.513-56.114L-28.101-56.114L-28.101-56.394L-27.513-56.394L-27.513-57.211Q-27.513-57.529-27.335-57.777Q-27.157-58.024-26.867-58.165Q-26.576-58.305-26.265-58.305Q-26.009-58.305-25.805-58.163Q-25.602-58.021-25.602-57.778Q-25.602-57.642-25.701-57.543Q-25.800-57.443-25.937-57.443Q-26.074-57.443-26.173-57.543Q-26.272-57.642-26.272-57.778Q-26.272-57.959-26.132-58.052Q-26.210-58.079-26.310-58.079Q-26.518-58.079-26.672-57.946Q-26.826-57.813-26.906-57.609Q-26.986-57.406-26.986-57.197L-26.986-56.394L-26.098-56.394L-26.098-56.114L-26.959-56.114L-26.959-53.865Q-26.959-53.656-26.303-53.656L-26.303-53.376M-25.048-54.210L-25.048-55.714Q-25.048-55.984-25.156-56.045Q-25.264-56.107-25.575-56.107L-25.575-56.387L-24.467-56.462L-24.467-54.230L-24.467-54.210Q-24.467-53.930-24.416-53.786Q-24.365-53.643-24.223-53.586Q-24.081-53.530-23.794-53.530Q-23.541-53.530-23.336-53.670Q-23.131-53.810-23.015-54.036Q-22.898-54.261-22.898-54.511L-22.898-55.714Q-22.898-55.984-23.006-56.045Q-23.114-56.107-23.425-56.107L-23.425-56.387L-22.317-56.462L-22.317-54.049Q-22.317-53.858-22.264-53.776Q-22.211-53.694-22.111-53.675Q-22.010-53.656-21.794-53.656L-21.794-53.376L-22.871-53.308L-22.871-53.872Q-22.980-53.690-23.126-53.567Q-23.271-53.444-23.457-53.376Q-23.644-53.308-23.845-53.308Q-25.048-53.308-25.048-54.210M-20.680-54.217L-20.680-56.114L-21.319-56.114L-21.319-56.336Q-21.001-56.336-20.784-56.546Q-20.567-56.756-20.467-57.066Q-20.366-57.375-20.366-57.683L-20.099-57.683L-20.099-56.394L-19.022-56.394L-19.022-56.114L-20.099-56.114L-20.099-54.230Q-20.099-53.954-19.995-53.755Q-19.891-53.557-19.631-53.557Q-19.474-53.557-19.368-53.661Q-19.262-53.766-19.212-53.919Q-19.163-54.073-19.163-54.230L-19.163-54.644L-18.896-54.644L-18.896-54.217Q-18.896-53.991-18.995-53.781Q-19.094-53.571-19.279-53.439Q-19.463-53.308-19.692-53.308Q-20.130-53.308-20.405-53.545Q-20.680-53.783-20.680-54.217M-17.512-54.210L-17.512-55.714Q-17.512-55.984-17.619-56.045Q-17.727-56.107-18.038-56.107L-18.038-56.387L-16.931-56.462L-16.931-54.230L-16.931-54.210Q-16.931-53.930-16.879-53.786Q-16.828-53.643-16.686-53.586Q-16.544-53.530-16.257-53.530Q-16.004-53.530-15.799-53.670Q-15.594-53.810-15.478-54.036Q-15.362-54.261-15.362-54.511L-15.362-55.714Q-15.362-55.984-15.469-56.045Q-15.577-56.107-15.888-56.107L-15.888-56.387L-14.781-56.462L-14.781-54.049Q-14.781-53.858-14.728-53.776Q-14.675-53.694-14.574-53.675Q-14.473-53.656-14.258-53.656L-14.258-53.376L-15.334-53.308L-15.334-53.872Q-15.444-53.690-15.589-53.567Q-15.734-53.444-15.921-53.376Q-16.107-53.308-16.309-53.308Q-17.512-53.308-17.512-54.210M-11.920-53.376L-13.656-53.376L-13.656-53.656Q-13.427-53.656-13.279-53.690Q-13.130-53.725-13.130-53.865L-13.130-55.714Q-13.130-55.984-13.238-56.045Q-13.345-56.107-13.656-56.107L-13.656-56.387L-12.627-56.462L-12.627-55.755Q-12.498-56.063-12.255-56.262Q-12.012-56.462-11.694-56.462Q-11.476-56.462-11.305-56.338Q-11.134-56.213-11.134-56.001Q-11.134-55.864-11.233-55.765Q-11.332-55.666-11.465-55.666Q-11.602-55.666-11.701-55.765Q-11.800-55.864-11.800-56.001Q-11.800-56.141-11.701-56.240Q-11.992-56.240-12.192-56.044Q-12.392-55.847-12.484-55.553Q-12.576-55.259-12.576-54.979L-12.576-53.865Q-12.576-53.656-11.920-53.656L-11.920-53.376M-10.590-54.911Q-10.590-55.232-10.466-55.521Q-10.341-55.810-10.115-56.033Q-9.890-56.257-9.594-56.377Q-9.298-56.497-8.980-56.497Q-8.652-56.497-8.391-56.397Q-8.129-56.298-7.953-56.116Q-7.777-55.933-7.683-55.675Q-7.589-55.417-7.589-55.085Q-7.589-54.993-7.671-54.972L-9.927-54.972L-9.927-54.911Q-9.927-54.323-9.644-53.940Q-9.360-53.557-8.792-53.557Q-8.471-53.557-8.203-53.750Q-7.935-53.943-7.846-54.258Q-7.839-54.299-7.764-54.313L-7.671-54.313Q-7.589-54.289-7.589-54.217Q-7.589-54.210-7.596-54.183Q-7.709-53.786-8.080-53.547Q-8.451-53.308-8.874-53.308Q-9.312-53.308-9.712-53.516Q-10.112-53.725-10.351-54.092Q-10.590-54.459-10.590-54.911M-9.920-55.181L-8.105-55.181Q-8.105-55.458-8.203-55.710Q-8.300-55.963-8.498-56.119Q-8.697-56.274-8.980-56.274Q-9.257-56.274-9.471-56.116Q-9.685-55.957-9.802-55.702Q-9.920-55.447-9.920-55.181\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.133 4.826)\">\u003Cpath d=\"M-4.329-54.859Q-4.329-55.201-4.194-55.500Q-4.059-55.799-3.819-56.023Q-3.580-56.247-3.262-56.372Q-2.944-56.497-2.613-56.497Q-2.168-56.497-1.769-56.281Q-1.369-56.066-1.134-55.688Q-0.900-55.311-0.900-54.859Q-0.900-54.518-1.042-54.234Q-1.184-53.950-1.428-53.743Q-1.673-53.537-1.982-53.422Q-2.291-53.308-2.613-53.308Q-3.043-53.308-3.445-53.509Q-3.847-53.711-4.088-54.063Q-4.329-54.415-4.329-54.859M-2.613-53.557Q-2.011-53.557-1.787-53.935Q-1.563-54.313-1.563-54.945Q-1.563-55.557-1.798-55.916Q-2.032-56.274-2.613-56.274Q-3.665-56.274-3.665-54.945Q-3.665-54.313-3.440-53.935Q-3.214-53.557-2.613-53.557\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.133 4.826)\">\u003Cpath d=\"M-0.087-54.887Q-0.087-55.215 0.048-55.516Q0.183-55.816 0.419-56.037Q0.655-56.257 0.959-56.377Q1.264-56.497 1.588-56.497Q2.094-56.497 2.443-56.394Q2.791-56.292 2.791-55.916Q2.791-55.769 2.694-55.668Q2.597-55.567 2.450-55.567Q2.296-55.567 2.197-55.666Q2.098-55.765 2.098-55.916Q2.098-56.104 2.238-56.196Q2.036-56.247 1.595-56.247Q1.240-56.247 1.011-56.051Q0.782-55.854 0.681-55.545Q0.580-55.235 0.580-54.887Q0.580-54.538 0.706-54.232Q0.833-53.926 1.088-53.742Q1.342-53.557 1.698-53.557Q1.920-53.557 2.104-53.641Q2.289-53.725 2.424-53.880Q2.559-54.036 2.617-54.244Q2.631-54.299 2.685-54.299L2.798-54.299Q2.829-54.299 2.851-54.275Q2.873-54.251 2.873-54.217L2.873-54.196Q2.788-53.909 2.600-53.711Q2.412-53.513 2.147-53.410Q1.882-53.308 1.588-53.308Q1.158-53.308 0.770-53.514Q0.382-53.721 0.148-54.084Q-0.087-54.446-0.087-54.887M3.461-54.887Q3.461-55.215 3.596-55.516Q3.731-55.816 3.967-56.037Q4.203-56.257 4.507-56.377Q4.811-56.497 5.136-56.497Q5.642-56.497 5.991-56.394Q6.339-56.292 6.339-55.916Q6.339-55.769 6.242-55.668Q6.144-55.567 5.997-55.567Q5.844-55.567 5.745-55.666Q5.645-55.765 5.645-55.916Q5.645-56.104 5.786-56.196Q5.584-56.247 5.143-56.247Q4.787-56.247 4.558-56.051Q4.329-55.854 4.229-55.545Q4.128-55.235 4.128-54.887Q4.128-54.538 4.254-54.232Q4.381-53.926 4.635-53.742Q4.890-53.557 5.245-53.557Q5.468-53.557 5.652-53.641Q5.837-53.725 5.972-53.880Q6.107-54.036 6.165-54.244Q6.179-54.299 6.233-54.299L6.346-54.299Q6.377-54.299 6.399-54.275Q6.421-54.251 6.421-54.217L6.421-54.196Q6.336-53.909 6.148-53.711Q5.960-53.513 5.695-53.410Q5.430-53.308 5.136-53.308Q4.705-53.308 4.318-53.514Q3.930-53.721 3.695-54.084Q3.461-54.446 3.461-54.887M7.583-54.210L7.583-55.714Q7.583-55.984 7.476-56.045Q7.368-56.107 7.057-56.107L7.057-56.387L8.164-56.462L8.164-54.230L8.164-54.210Q8.164-53.930 8.216-53.786Q8.267-53.643 8.409-53.586Q8.551-53.530 8.838-53.530Q9.091-53.530 9.296-53.670Q9.501-53.810 9.617-54.036Q9.733-54.261 9.733-54.511L9.733-55.714Q9.733-55.984 9.626-56.045Q9.518-56.107 9.207-56.107L9.207-56.387L10.314-56.462L10.314-54.049Q10.314-53.858 10.367-53.776Q10.420-53.694 10.521-53.675Q10.622-53.656 10.837-53.656L10.837-53.376L9.761-53.308L9.761-53.872Q9.651-53.690 9.506-53.567Q9.361-53.444 9.174-53.376Q8.988-53.308 8.787-53.308Q7.583-53.308 7.583-54.210M13.069-52.019L11.439-52.019L11.439-52.299Q11.668-52.299 11.817-52.334Q11.965-52.368 11.965-52.508L11.965-55.854Q11.965-56.025 11.828-56.066Q11.692-56.107 11.439-56.107L11.439-56.387L12.519-56.462L12.519-56.056Q12.741-56.257 13.028-56.360Q13.315-56.462 13.623-56.462Q14.050-56.462 14.414-56.249Q14.778-56.035 14.992-55.671Q15.205-55.307 15.205-54.887Q15.205-54.442 14.966-54.078Q14.727-53.714 14.334-53.511Q13.941-53.308 13.496-53.308Q13.230-53.308 12.982-53.408Q12.734-53.509 12.546-53.690L12.546-52.508Q12.546-52.371 12.695-52.335Q12.844-52.299 13.069-52.299L13.069-52.019M12.546-55.707L12.546-54.097Q12.680-53.844 12.922-53.687Q13.165-53.530 13.442-53.530Q13.770-53.530 14.023-53.731Q14.276-53.933 14.409-54.251Q14.542-54.569 14.542-54.887Q14.542-55.116 14.477-55.345Q14.412-55.574 14.284-55.772Q14.156-55.970 13.961-56.090Q13.766-56.209 13.534-56.209Q13.240-56.209 12.972-56.080Q12.703-55.950 12.546-55.707M15.899-54.104Q15.899-54.436 16.123-54.663Q16.347-54.890 16.691-55.018Q17.034-55.147 17.407-55.199Q17.779-55.252 18.083-55.252L18.083-55.505Q18.083-55.710 17.976-55.890Q17.868-56.069 17.687-56.172Q17.506-56.274 17.297-56.274Q16.891-56.274 16.655-56.182Q16.744-56.145 16.790-56.061Q16.836-55.977 16.836-55.875Q16.836-55.779 16.790-55.700Q16.744-55.622 16.663-55.577Q16.583-55.533 16.494-55.533Q16.344-55.533 16.243-55.630Q16.142-55.728 16.142-55.875Q16.142-56.497 17.297-56.497Q17.509-56.497 17.759-56.433Q18.008-56.370 18.210-56.251Q18.412-56.131 18.538-55.946Q18.664-55.762 18.664-55.519L18.664-53.943Q18.664-53.827 18.726-53.731Q18.787-53.636 18.900-53.636Q19.010-53.636 19.075-53.730Q19.140-53.824 19.140-53.943L19.140-54.391L19.406-54.391L19.406-53.943Q19.406-53.673 19.179-53.508Q18.952-53.342 18.671-53.342Q18.463-53.342 18.326-53.496Q18.189-53.649 18.165-53.865Q18.018-53.598 17.736-53.453Q17.454-53.308 17.130-53.308Q16.853-53.308 16.569-53.383Q16.286-53.458 16.092-53.637Q15.899-53.817 15.899-54.104M16.515-54.104Q16.515-53.930 16.615-53.800Q16.716-53.670 16.872-53.600Q17.027-53.530 17.191-53.530Q17.410-53.530 17.619-53.627Q17.827-53.725 17.955-53.906Q18.083-54.087 18.083-54.313L18.083-55.041Q17.759-55.041 17.393-54.950Q17.027-54.859 16.771-54.647Q16.515-54.436 16.515-54.104M21.505-53.376L19.871-53.376L19.871-53.656Q20.100-53.656 20.249-53.690Q20.397-53.725 20.397-53.865L20.397-55.714Q20.397-55.984 20.290-56.045Q20.182-56.107 19.871-56.107L19.871-56.387L20.931-56.462L20.931-55.813Q21.101-56.121 21.406-56.292Q21.710-56.462 22.055-56.462Q22.561-56.462 22.845-56.239Q23.128-56.015 23.128-55.519L23.128-53.865Q23.128-53.728 23.277-53.692Q23.426-53.656 23.651-53.656L23.651-53.376L22.021-53.376L22.021-53.656Q22.250-53.656 22.399-53.690Q22.547-53.725 22.547-53.865L22.547-55.505Q22.547-55.840 22.428-56.040Q22.308-56.240 21.994-56.240Q21.724-56.240 21.489-56.104Q21.255-55.967 21.117-55.733Q20.978-55.499 20.978-55.225L20.978-53.865Q20.978-53.728 21.129-53.692Q21.279-53.656 21.505-53.656L21.505-53.376M24.239-54.887Q24.239-55.215 24.374-55.516Q24.509-55.816 24.745-56.037Q24.981-56.257 25.285-56.377Q25.589-56.497 25.914-56.497Q26.420-56.497 26.768-56.394Q27.117-56.292 27.117-55.916Q27.117-55.769 27.020-55.668Q26.922-55.567 26.775-55.567Q26.621-55.567 26.522-55.666Q26.423-55.765 26.423-55.916Q26.423-56.104 26.563-56.196Q26.362-56.247 25.921-56.247Q25.565-56.247 25.336-56.051Q25.107-55.854 25.006-55.545Q24.906-55.235 24.906-54.887Q24.906-54.538 25.032-54.232Q25.159-53.926 25.413-53.742Q25.668-53.557 26.023-53.557Q26.245-53.557 26.430-53.641Q26.615-53.725 26.750-53.880Q26.885-54.036 26.943-54.244Q26.956-54.299 27.011-54.299L27.124-54.299Q27.155-54.299 27.177-54.275Q27.199-54.251 27.199-54.217L27.199-54.196Q27.114-53.909 26.926-53.711Q26.738-53.513 26.473-53.410Q26.208-53.308 25.914-53.308Q25.483-53.308 25.095-53.514Q24.707-53.721 24.473-54.084Q24.239-54.446 24.239-54.887M28.122-52.241Q28.252-52.173 28.389-52.173Q28.559-52.173 28.710-52.262Q28.860-52.351 28.971-52.496Q29.082-52.641 29.161-52.809L29.424-53.376L28.255-55.902Q28.180-56.049 28.050-56.081Q27.920-56.114 27.688-56.114L27.688-56.394L29.209-56.394L29.209-56.114Q28.860-56.114 28.860-55.967Q28.864-55.946 28.865-55.929Q28.867-55.912 28.867-55.902L29.725-54.043L30.497-55.714Q30.532-55.782 30.532-55.861Q30.532-55.974 30.448-56.044Q30.364-56.114 30.251-56.114L30.251-56.394L31.448-56.394L31.448-56.114Q31.229-56.114 31.056-56.010Q30.884-55.905 30.791-55.714L29.455-52.809Q29.284-52.439 29.014-52.193Q28.744-51.947 28.389-51.947Q28.119-51.947 27.900-52.113Q27.681-52.279 27.681-52.542Q27.681-52.679 27.773-52.768Q27.866-52.856 28.006-52.856Q28.142-52.856 28.231-52.768Q28.320-52.679 28.320-52.542Q28.320-52.439 28.267-52.361Q28.214-52.282 28.122-52.241\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M184.98-34.881h73.978V-71.87h-73.977Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(221.595 6.43)\">\u003Cpath d=\"M-19.252-61.376L-20.886-61.376L-20.886-61.656Q-20.657-61.656-20.508-61.690Q-20.359-61.725-20.359-61.865L-20.359-63.714Q-20.359-63.984-20.467-64.045Q-20.575-64.107-20.886-64.107L-20.886-64.387L-19.826-64.462L-19.826-63.813Q-19.655-64.121-19.351-64.292Q-19.047-64.462-18.702-64.462Q-18.302-64.462-18.025-64.322Q-17.748-64.182-17.663-63.834Q-17.495-64.127-17.196-64.295Q-16.897-64.462-16.552-64.462Q-16.046-64.462-15.762-64.239Q-15.478-64.015-15.478-63.519L-15.478-61.865Q-15.478-61.728-15.330-61.692Q-15.181-61.656-14.956-61.656L-14.956-61.376L-16.586-61.376L-16.586-61.656Q-16.360-61.656-16.210-61.692Q-16.060-61.728-16.060-61.865L-16.060-63.505Q-16.060-63.840-16.179-64.040Q-16.299-64.240-16.613-64.240Q-16.883-64.240-17.117-64.104Q-17.352-63.967-17.490-63.733Q-17.628-63.499-17.628-63.225L-17.628-61.865Q-17.628-61.728-17.480-61.692Q-17.331-61.656-17.105-61.656L-17.105-61.376L-18.736-61.376L-18.736-61.656Q-18.507-61.656-18.358-61.690Q-18.209-61.725-18.209-61.865L-18.209-63.505Q-18.209-63.840-18.329-64.040Q-18.449-64.240-18.763-64.240Q-19.033-64.240-19.267-64.104Q-19.501-63.967-19.640-63.733Q-19.778-63.499-19.778-63.225L-19.778-61.865Q-19.778-61.728-19.628-61.692Q-19.477-61.656-19.252-61.656L-19.252-61.376M-14.409-62.859Q-14.409-63.201-14.274-63.500Q-14.139-63.799-13.899-64.023Q-13.660-64.247-13.342-64.372Q-13.024-64.497-12.693-64.497Q-12.248-64.497-11.849-64.281Q-11.449-64.066-11.215-63.688Q-10.980-63.311-10.980-62.859Q-10.980-62.518-11.122-62.234Q-11.264-61.950-11.508-61.743Q-11.753-61.537-12.062-61.422Q-12.372-61.308-12.693-61.308Q-13.123-61.308-13.525-61.509Q-13.927-61.711-14.168-62.063Q-14.409-62.415-14.409-62.859M-12.693-61.557Q-12.091-61.557-11.867-61.935Q-11.644-62.313-11.644-62.945Q-11.644-63.557-11.878-63.916Q-12.112-64.274-12.693-64.274Q-13.746-64.274-13.746-62.945Q-13.746-62.313-13.520-61.935Q-13.294-61.557-12.693-61.557\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(221.595 6.43)\">\u003Cpath d=\"M-10.163-62.887Q-10.163-63.225-10.022-63.516Q-9.882-63.806-9.638-64.020Q-9.394-64.233-9.089-64.348Q-8.785-64.462-8.460-64.462Q-8.190-64.462-7.927-64.363Q-7.664-64.264-7.473-64.086L-7.473-65.484Q-7.473-65.754-7.580-65.816Q-7.688-65.877-7.999-65.877L-7.999-66.158L-6.922-66.233L-6.922-62.049Q-6.922-61.861-6.868-61.778Q-6.813-61.694-6.712-61.675Q-6.611-61.656-6.396-61.656L-6.396-61.376L-7.503-61.308L-7.503-61.725Q-7.920-61.308-8.546-61.308Q-8.977-61.308-9.349-61.520Q-9.722-61.731-9.942-62.092Q-10.163-62.453-10.163-62.887M-8.488-61.530Q-8.279-61.530-8.093-61.602Q-7.907-61.673-7.753-61.810Q-7.599-61.947-7.503-62.125L-7.503-63.734Q-7.589-63.881-7.734-64.001Q-7.879-64.121-8.049-64.180Q-8.218-64.240-8.399-64.240Q-8.959-64.240-9.228-63.851Q-9.496-63.461-9.496-62.880Q-9.496-62.309-9.262-61.919Q-9.028-61.530-8.488-61.530M-5.788-62.911Q-5.788-63.232-5.663-63.521Q-5.538-63.810-5.312-64.033Q-5.087-64.257-4.791-64.377Q-4.496-64.497-4.178-64.497Q-3.850-64.497-3.588-64.397Q-3.327-64.298-3.151-64.116Q-2.975-63.933-2.881-63.675Q-2.787-63.417-2.787-63.085Q-2.787-62.993-2.869-62.972L-5.124-62.972L-5.124-62.911Q-5.124-62.323-4.841-61.940Q-4.557-61.557-3.990-61.557Q-3.668-61.557-3.400-61.750Q-3.132-61.943-3.043-62.258Q-3.036-62.299-2.961-62.313L-2.869-62.313Q-2.787-62.289-2.787-62.217Q-2.787-62.210-2.793-62.183Q-2.906-61.786-3.277-61.547Q-3.648-61.308-4.072-61.308Q-4.509-61.308-4.909-61.516Q-5.309-61.725-5.548-62.092Q-5.788-62.459-5.788-62.911M-5.118-63.181L-3.303-63.181Q-3.303-63.458-3.400-63.710Q-3.498-63.963-3.696-64.119Q-3.894-64.274-4.178-64.274Q-4.455-64.274-4.668-64.116Q-4.882-63.957-5-63.702Q-5.118-63.447-5.118-63.181M-0.531-61.376L-2.134-61.376L-2.134-61.656Q-1.908-61.656-1.759-61.690Q-1.611-61.725-1.611-61.865L-1.611-65.484Q-1.611-65.754-1.718-65.816Q-1.826-65.877-2.134-65.877L-2.134-66.158L-1.057-66.233L-1.057-61.865Q-1.057-61.728-0.907-61.692Q-0.756-61.656-0.531-61.656L-0.531-61.376M1.947-62.630L-0.110-62.630L-0.110-63.133L1.947-63.133L1.947-62.630M3.557-61.376L3.291-61.376L3.291-65.484Q3.291-65.754 3.183-65.816Q3.075-65.877 2.764-65.877L2.764-66.158L3.844-66.233L3.844-64.063Q4.053-64.254 4.338-64.358Q4.624-64.462 4.921-64.462Q5.239-64.462 5.536-64.341Q5.834-64.220 6.056-64.004Q6.278-63.789 6.404-63.504Q6.531-63.218 6.531-62.887Q6.531-62.442 6.292-62.078Q6.052-61.714 5.659-61.511Q5.266-61.308 4.822-61.308Q4.627-61.308 4.437-61.364Q4.248-61.420 4.087-61.525Q3.926-61.629 3.786-61.790L3.557-61.376M3.872-63.721L3.872-62.104Q4.008-61.844 4.249-61.687Q4.490-61.530 4.767-61.530Q5.061-61.530 5.273-61.637Q5.485-61.745 5.618-61.937Q5.752-62.128 5.810-62.367Q5.868-62.606 5.868-62.887Q5.868-63.246 5.774-63.550Q5.680-63.854 5.452-64.047Q5.225-64.240 4.859-64.240Q4.559-64.240 4.292-64.104Q4.025-63.967 3.872-63.721M7.225-62.104Q7.225-62.436 7.449-62.663Q7.672-62.890 8.016-63.018Q8.359-63.147 8.732-63.199Q9.105-63.252 9.409-63.252L9.409-63.505Q9.409-63.710 9.301-63.890Q9.193-64.069 9.012-64.172Q8.831-64.274 8.623-64.274Q8.216-64.274 7.980-64.182Q8.069-64.145 8.115-64.061Q8.161-63.977 8.161-63.875Q8.161-63.779 8.115-63.700Q8.069-63.622 7.989-63.577Q7.908-63.533 7.819-63.533Q7.669-63.533 7.568-63.630Q7.467-63.728 7.467-63.875Q7.467-64.497 8.623-64.497Q8.835-64.497 9.084-64.433Q9.334-64.370 9.535-64.251Q9.737-64.131 9.863-63.946Q9.990-63.762 9.990-63.519L9.990-61.943Q9.990-61.827 10.051-61.731Q10.113-61.636 10.226-61.636Q10.335-61.636 10.400-61.730Q10.465-61.824 10.465-61.943L10.465-62.391L10.731-62.391L10.731-61.943Q10.731-61.673 10.504-61.508Q10.277-61.342 9.997-61.342Q9.788-61.342 9.651-61.496Q9.515-61.649 9.491-61.865Q9.344-61.598 9.062-61.453Q8.780-61.308 8.455-61.308Q8.178-61.308 7.895-61.383Q7.611-61.458 7.418-61.637Q7.225-61.817 7.225-62.104M7.840-62.104Q7.840-61.930 7.941-61.800Q8.042-61.670 8.197-61.600Q8.353-61.530 8.517-61.530Q8.735-61.530 8.944-61.627Q9.152-61.725 9.281-61.906Q9.409-62.087 9.409-62.313L9.409-63.041Q9.084-63.041 8.718-62.950Q8.353-62.859 8.096-62.647Q7.840-62.436 7.840-62.104M11.148-61.383L11.148-62.446Q11.148-62.470 11.176-62.497Q11.203-62.524 11.227-62.524L11.336-62.524Q11.401-62.524 11.415-62.466Q11.511-62.032 11.757-61.781Q12.003-61.530 12.417-61.530Q12.758-61.530 13.011-61.663Q13.264-61.796 13.264-62.104Q13.264-62.261 13.170-62.376Q13.076-62.490 12.938-62.559Q12.799-62.627 12.632-62.665L12.051-62.764Q11.695-62.832 11.422-63.053Q11.148-63.273 11.148-63.615Q11.148-63.864 11.260-64.039Q11.371-64.213 11.557-64.312Q11.743-64.411 11.959-64.454Q12.174-64.497 12.417-64.497Q12.830-64.497 13.110-64.315L13.326-64.490Q13.336-64.493 13.343-64.495Q13.350-64.497 13.360-64.497L13.411-64.497Q13.439-64.497 13.462-64.473Q13.486-64.449 13.486-64.421L13.486-63.574Q13.486-63.553 13.462-63.526Q13.439-63.499 13.411-63.499L13.298-63.499Q13.271-63.499 13.245-63.524Q13.220-63.550 13.220-63.574Q13.220-63.810 13.114-63.974Q13.008-64.138 12.825-64.220Q12.642-64.302 12.410-64.302Q12.082-64.302 11.825-64.199Q11.569-64.097 11.569-63.820Q11.569-63.625 11.752-63.516Q11.935-63.406 12.164-63.365L12.738-63.259Q12.984-63.211 13.198-63.083Q13.411-62.955 13.548-62.752Q13.685-62.548 13.685-62.299Q13.685-61.786 13.319-61.547Q12.953-61.308 12.417-61.308Q11.921-61.308 11.589-61.602L11.323-61.328Q11.302-61.308 11.275-61.308L11.227-61.308Q11.203-61.308 11.176-61.335Q11.148-61.362 11.148-61.383M14.273-62.911Q14.273-63.232 14.397-63.521Q14.522-63.810 14.748-64.033Q14.973-64.257 15.269-64.377Q15.565-64.497 15.882-64.497Q16.210-64.497 16.472-64.397Q16.733-64.298 16.909-64.116Q17.085-63.933 17.179-63.675Q17.273-63.417 17.273-63.085Q17.273-62.993 17.191-62.972L14.936-62.972L14.936-62.911Q14.936-62.323 15.219-61.940Q15.503-61.557 16.070-61.557Q16.392-61.557 16.660-61.750Q16.928-61.943 17.017-62.258Q17.024-62.299 17.099-62.313L17.191-62.313Q17.273-62.289 17.273-62.217Q17.273-62.210 17.267-62.183Q17.154-61.786 16.783-61.547Q16.412-61.308 15.988-61.308Q15.551-61.308 15.151-61.516Q14.751-61.725 14.512-62.092Q14.273-62.459 14.273-62.911M14.942-63.181L16.757-63.181Q16.757-63.458 16.660-63.710Q16.563-63.963 16.364-64.119Q16.166-64.274 15.882-64.274Q15.606-64.274 15.392-64.116Q15.178-63.957 15.060-63.702Q14.942-63.447 14.942-63.181M17.861-62.887Q17.861-63.225 18.002-63.516Q18.142-63.806 18.386-64.020Q18.630-64.233 18.935-64.348Q19.239-64.462 19.564-64.462Q19.834-64.462 20.097-64.363Q20.360-64.264 20.551-64.086L20.551-65.484Q20.551-65.754 20.444-65.816Q20.336-65.877 20.025-65.877L20.025-66.158L21.102-66.233L21.102-62.049Q21.102-61.861 21.156-61.778Q21.211-61.694 21.312-61.675Q21.413-61.656 21.628-61.656L21.628-61.376L20.521-61.308L20.521-61.725Q20.104-61.308 19.478-61.308Q19.047-61.308 18.675-61.520Q18.302-61.731 18.082-62.092Q17.861-62.453 17.861-62.887M19.536-61.530Q19.745-61.530 19.931-61.602Q20.117-61.673 20.271-61.810Q20.425-61.947 20.521-62.125L20.521-63.734Q20.435-63.881 20.290-64.001Q20.145-64.121 19.975-64.180Q19.806-64.240 19.625-64.240Q19.065-64.240 18.796-63.851Q18.528-63.461 18.528-62.880Q18.528-62.309 18.762-61.919Q18.996-61.530 19.536-61.530\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(221.595 6.43)\">\u003Cpath d=\"M-27.574-54.217L-27.574-56.114L-28.213-56.114L-28.213-56.336Q-27.895-56.336-27.678-56.546Q-27.461-56.756-27.361-57.066Q-27.260-57.375-27.260-57.683L-26.993-57.683L-26.993-56.394L-25.916-56.394L-25.916-56.114L-26.993-56.114L-26.993-54.230Q-26.993-53.954-26.889-53.755Q-26.785-53.557-26.525-53.557Q-26.368-53.557-26.262-53.661Q-26.156-53.766-26.106-53.919Q-26.057-54.073-26.057-54.230L-26.057-54.644L-25.790-54.644L-25.790-54.217Q-25.790-53.991-25.889-53.781Q-25.988-53.571-26.173-53.439Q-26.357-53.308-26.586-53.308Q-27.024-53.308-27.299-53.545Q-27.574-53.783-27.574-54.217M-23.230-53.376L-24.966-53.376L-24.966-53.656Q-24.737-53.656-24.589-53.690Q-24.440-53.725-24.440-53.865L-24.440-55.714Q-24.440-55.984-24.548-56.045Q-24.655-56.107-24.966-56.107L-24.966-56.387L-23.937-56.462L-23.937-55.755Q-23.808-56.063-23.565-56.262Q-23.322-56.462-23.004-56.462Q-22.786-56.462-22.615-56.338Q-22.444-56.213-22.444-56.001Q-22.444-55.864-22.543-55.765Q-22.642-55.666-22.775-55.666Q-22.912-55.666-23.011-55.765Q-23.110-55.864-23.110-56.001Q-23.110-56.141-23.011-56.240Q-23.302-56.240-23.502-56.044Q-23.702-55.847-23.794-55.553Q-23.886-55.259-23.886-54.979L-23.886-53.865Q-23.886-53.656-23.230-53.656L-23.230-53.376M-21.801-54.104Q-21.801-54.436-21.577-54.663Q-21.353-54.890-21.010-55.018Q-20.666-55.147-20.294-55.199Q-19.921-55.252-19.617-55.252L-19.617-55.505Q-19.617-55.710-19.725-55.890Q-19.832-56.069-20.014-56.172Q-20.195-56.274-20.403-56.274Q-20.810-56.274-21.046-56.182Q-20.957-56.145-20.911-56.061Q-20.865-55.977-20.865-55.875Q-20.865-55.779-20.911-55.700Q-20.957-55.622-21.037-55.577Q-21.118-55.533-21.207-55.533Q-21.357-55.533-21.458-55.630Q-21.559-55.728-21.559-55.875Q-21.559-56.497-20.403-56.497Q-20.191-56.497-19.942-56.433Q-19.692-56.370-19.491-56.251Q-19.289-56.131-19.163-55.946Q-19.036-55.762-19.036-55.519L-19.036-53.943Q-19.036-53.827-18.975-53.731Q-18.913-53.636-18.800-53.636Q-18.691-53.636-18.626-53.730Q-18.561-53.824-18.561-53.943L-18.561-54.391L-18.294-54.391L-18.294-53.943Q-18.294-53.673-18.522-53.508Q-18.749-53.342-19.029-53.342Q-19.238-53.342-19.374-53.496Q-19.511-53.649-19.535-53.865Q-19.682-53.598-19.964-53.453Q-20.246-53.308-20.571-53.308Q-20.848-53.308-21.131-53.383Q-21.415-53.458-21.608-53.637Q-21.801-53.817-21.801-54.104M-21.186-54.104Q-21.186-53.930-21.085-53.800Q-20.984-53.670-20.829-53.600Q-20.673-53.530-20.509-53.530Q-20.290-53.530-20.082-53.627Q-19.873-53.725-19.745-53.906Q-19.617-54.087-19.617-54.313L-19.617-55.041Q-19.942-55.041-20.308-54.950Q-20.673-54.859-20.930-54.647Q-21.186-54.436-21.186-54.104M-16.196-53.376L-17.830-53.376L-17.830-53.656Q-17.601-53.656-17.452-53.690Q-17.303-53.725-17.303-53.865L-17.303-55.714Q-17.303-55.984-17.411-56.045Q-17.519-56.107-17.830-56.107L-17.830-56.387L-16.770-56.462L-16.770-55.813Q-16.599-56.121-16.295-56.292Q-15.991-56.462-15.645-56.462Q-15.140-56.462-14.856-56.239Q-14.572-56.015-14.572-55.519L-14.572-53.865Q-14.572-53.728-14.424-53.692Q-14.275-53.656-14.049-53.656L-14.049-53.376L-15.680-53.376L-15.680-53.656Q-15.451-53.656-15.302-53.690Q-15.153-53.725-15.153-53.865L-15.153-55.505Q-15.153-55.840-15.273-56.040Q-15.393-56.240-15.707-56.240Q-15.977-56.240-16.211-56.104Q-16.445-55.967-16.584-55.733Q-16.722-55.499-16.722-55.225L-16.722-53.865Q-16.722-53.728-16.572-53.692Q-16.421-53.656-16.196-53.656L-16.196-53.376M-13.461-53.383L-13.461-54.446Q-13.461-54.470-13.434-54.497Q-13.407-54.524-13.383-54.524L-13.273-54.524Q-13.208-54.524-13.195-54.466Q-13.099-54.032-12.853-53.781Q-12.607-53.530-12.193-53.530Q-11.852-53.530-11.599-53.663Q-11.346-53.796-11.346-54.104Q-11.346-54.261-11.440-54.376Q-11.534-54.490-11.672-54.559Q-11.811-54.627-11.978-54.665L-12.559-54.764Q-12.915-54.832-13.188-55.053Q-13.461-55.273-13.461-55.615Q-13.461-55.864-13.350-56.039Q-13.239-56.213-13.053-56.312Q-12.867-56.411-12.651-56.454Q-12.436-56.497-12.193-56.497Q-11.780-56.497-11.499-56.315L-11.284-56.490Q-11.274-56.493-11.267-56.495Q-11.260-56.497-11.250-56.497L-11.199-56.497Q-11.171-56.497-11.147-56.473Q-11.123-56.449-11.123-56.421L-11.123-55.574Q-11.123-55.553-11.147-55.526Q-11.171-55.499-11.199-55.499L-11.311-55.499Q-11.339-55.499-11.364-55.524Q-11.390-55.550-11.390-55.574Q-11.390-55.810-11.496-55.974Q-11.602-56.138-11.785-56.220Q-11.968-56.302-12.200-56.302Q-12.528-56.302-12.785-56.199Q-13.041-56.097-13.041-55.820Q-13.041-55.625-12.858-55.516Q-12.675-55.406-12.446-55.365L-11.872-55.259Q-11.626-55.211-11.412-55.083Q-11.199-54.955-11.062-54.752Q-10.925-54.548-10.925-54.299Q-10.925-53.786-11.291-53.547Q-11.657-53.308-12.193-53.308Q-12.689-53.308-13.020-53.602L-13.287-53.328Q-13.308-53.308-13.335-53.308L-13.383-53.308Q-13.407-53.308-13.434-53.335Q-13.461-53.362-13.461-53.383M-8.680-53.376L-10.231-53.376L-10.231-53.656Q-10.006-53.656-9.857-53.690Q-9.708-53.725-9.708-53.865L-9.708-55.714Q-9.708-55.902-9.756-55.986Q-9.804-56.069-9.902-56.088Q-9.999-56.107-10.211-56.107L-10.211-56.387L-9.155-56.462L-9.155-53.865Q-9.155-53.725-9.023-53.690Q-8.892-53.656-8.680-53.656L-8.680-53.376M-9.951-57.683Q-9.951-57.854-9.828-57.973Q-9.705-58.093-9.534-58.093Q-9.367-58.093-9.244-57.973Q-9.121-57.854-9.121-57.683Q-9.121-57.508-9.244-57.385Q-9.367-57.262-9.534-57.262Q-9.705-57.262-9.828-57.385Q-9.951-57.508-9.951-57.683M-7.507-54.217L-7.507-56.114L-8.146-56.114L-8.146-56.336Q-7.829-56.336-7.612-56.546Q-7.394-56.756-7.294-57.066Q-7.193-57.375-7.193-57.683L-6.926-57.683L-6.926-56.394L-5.850-56.394L-5.850-56.114L-6.926-56.114L-6.926-54.230Q-6.926-53.954-6.822-53.755Q-6.718-53.557-6.458-53.557Q-6.301-53.557-6.195-53.661Q-6.089-53.766-6.039-53.919Q-5.990-54.073-5.990-54.230L-5.990-54.644L-5.723-54.644L-5.723-54.217Q-5.723-53.991-5.822-53.781Q-5.921-53.571-6.106-53.439Q-6.290-53.308-6.519-53.308Q-6.957-53.308-7.232-53.545Q-7.507-53.783-7.507-54.217M-3.296-53.376L-4.848-53.376L-4.848-53.656Q-4.623-53.656-4.474-53.690Q-4.325-53.725-4.325-53.865L-4.325-55.714Q-4.325-55.902-4.373-55.986Q-4.421-56.069-4.518-56.088Q-4.616-56.107-4.828-56.107L-4.828-56.387L-3.771-56.462L-3.771-53.865Q-3.771-53.725-3.640-53.690Q-3.508-53.656-3.296-53.656L-3.296-53.376M-4.568-57.683Q-4.568-57.854-4.445-57.973Q-4.322-58.093-4.151-58.093Q-3.983-58.093-3.860-57.973Q-3.737-57.854-3.737-57.683Q-3.737-57.508-3.860-57.385Q-3.983-57.262-4.151-57.262Q-4.322-57.262-4.445-57.385Q-4.568-57.508-4.568-57.683M-2.691-54.859Q-2.691-55.201-2.556-55.500Q-2.421-55.799-2.182-56.023Q-1.943-56.247-1.625-56.372Q-1.307-56.497-0.976-56.497Q-0.531-56.497-0.131-56.281Q0.269-56.066 0.503-55.688Q0.737-55.311 0.737-54.859Q0.737-54.518 0.595-54.234Q0.453-53.950 0.209-53.743Q-0.036-53.537-0.345-53.422Q-0.654-53.308-0.976-53.308Q-1.406-53.308-1.808-53.509Q-2.209-53.711-2.450-54.063Q-2.691-54.415-2.691-54.859M-0.976-53.557Q-0.374-53.557-0.150-53.935Q0.074-54.313 0.074-54.945Q0.074-55.557-0.160-55.916Q-0.394-56.274-0.976-56.274Q-2.028-56.274-2.028-54.945Q-2.028-54.313-1.803-53.935Q-1.577-53.557-0.976-53.557M3.013-53.376L1.379-53.376L1.379-53.656Q1.608-53.656 1.757-53.690Q1.906-53.725 1.906-53.865L1.906-55.714Q1.906-55.984 1.798-56.045Q1.690-56.107 1.379-56.107L1.379-56.387L2.439-56.462L2.439-55.813Q2.610-56.121 2.914-56.292Q3.218-56.462 3.564-56.462Q4.069-56.462 4.353-56.239Q4.637-56.015 4.637-55.519L4.637-53.865Q4.637-53.728 4.785-53.692Q4.934-53.656 5.160-53.656L5.160-53.376L3.529-53.376L3.529-53.656Q3.758-53.656 3.907-53.690Q4.056-53.725 4.056-53.865L4.056-55.505Q4.056-55.840 3.936-56.040Q3.816-56.240 3.502-56.240Q3.232-56.240 2.998-56.104Q2.764-55.967 2.625-55.733Q2.487-55.499 2.487-55.225L2.487-53.865Q2.487-53.728 2.637-53.692Q2.788-53.656 3.013-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(221.595 6.43)\">\u003Cpath d=\"M10.152-53.376L8.518-53.376L8.518-53.656Q8.747-53.656 8.896-53.690Q9.045-53.725 9.045-53.865L9.045-55.714Q9.045-55.984 8.937-56.045Q8.829-56.107 8.518-56.107L8.518-56.387L9.578-56.462L9.578-55.813Q9.749-56.121 10.053-56.292Q10.357-56.462 10.702-56.462Q11.102-56.462 11.379-56.322Q11.656-56.182 11.741-55.834Q11.909-56.127 12.208-56.295Q12.507-56.462 12.852-56.462Q13.358-56.462 13.642-56.239Q13.926-56.015 13.926-55.519L13.926-53.865Q13.926-53.728 14.074-53.692Q14.223-53.656 14.448-53.656L14.448-53.376L12.818-53.376L12.818-53.656Q13.044-53.656 13.194-53.692Q13.344-53.728 13.344-53.865L13.344-55.505Q13.344-55.840 13.225-56.040Q13.105-56.240 12.791-56.240Q12.521-56.240 12.287-56.104Q12.052-55.967 11.914-55.733Q11.776-55.499 11.776-55.225L11.776-53.865Q11.776-53.728 11.924-53.692Q12.073-53.656 12.299-53.656L12.299-53.376L10.668-53.376L10.668-53.656Q10.897-53.656 11.046-53.690Q11.195-53.725 11.195-53.865L11.195-55.505Q11.195-55.840 11.075-56.040Q10.955-56.240 10.641-56.240Q10.371-56.240 10.137-56.104Q9.903-55.967 9.764-55.733Q9.626-55.499 9.626-55.225L9.626-53.865Q9.626-53.728 9.776-53.692Q9.927-53.656 10.152-53.656L10.152-53.376M14.995-54.859Q14.995-55.201 15.130-55.500Q15.265-55.799 15.505-56.023Q15.744-56.247 16.062-56.372Q16.380-56.497 16.711-56.497Q17.156-56.497 17.555-56.281Q17.955-56.066 18.189-55.688Q18.424-55.311 18.424-54.859Q18.424-54.518 18.282-54.234Q18.140-53.950 17.896-53.743Q17.651-53.537 17.342-53.422Q17.032-53.308 16.711-53.308Q16.281-53.308 15.879-53.509Q15.477-53.711 15.236-54.063Q14.995-54.415 14.995-54.859M16.711-53.557Q17.313-53.557 17.537-53.935Q17.760-54.313 17.760-54.945Q17.760-55.557 17.526-55.916Q17.292-56.274 16.711-56.274Q15.658-56.274 15.658-54.945Q15.658-54.313 15.884-53.935Q16.110-53.557 16.711-53.557\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(221.595 6.43)\">\u003Cpath d=\"M19.241-54.887Q19.241-55.225 19.382-55.516Q19.522-55.806 19.766-56.020Q20.010-56.233 20.315-56.348Q20.619-56.462 20.944-56.462Q21.214-56.462 21.477-56.363Q21.740-56.264 21.931-56.086L21.931-57.484Q21.931-57.754 21.824-57.816Q21.716-57.877 21.405-57.877L21.405-58.158L22.482-58.233L22.482-54.049Q22.482-53.861 22.536-53.778Q22.591-53.694 22.692-53.675Q22.793-53.656 23.008-53.656L23.008-53.376L21.901-53.308L21.901-53.725Q21.484-53.308 20.858-53.308Q20.427-53.308 20.055-53.520Q19.682-53.731 19.462-54.092Q19.241-54.453 19.241-54.887M20.916-53.530Q21.125-53.530 21.311-53.602Q21.497-53.673 21.651-53.810Q21.805-53.947 21.901-54.125L21.901-55.734Q21.815-55.881 21.670-56.001Q21.525-56.121 21.355-56.180Q21.186-56.240 21.005-56.240Q20.445-56.240 20.176-55.851Q19.908-55.461 19.908-54.880Q19.908-54.309 20.142-53.919Q20.376-53.530 20.916-53.530M23.616-54.911Q23.616-55.232 23.741-55.521Q23.866-55.810 24.092-56.033Q24.317-56.257 24.613-56.377Q24.908-56.497 25.226-56.497Q25.554-56.497 25.816-56.397Q26.077-56.298 26.253-56.116Q26.429-55.933 26.523-55.675Q26.617-55.417 26.617-55.085Q26.617-54.993 26.535-54.972L24.280-54.972L24.280-54.911Q24.280-54.323 24.563-53.940Q24.847-53.557 25.414-53.557Q25.736-53.557 26.004-53.750Q26.272-53.943 26.361-54.258Q26.368-54.299 26.443-54.313L26.535-54.313Q26.617-54.289 26.617-54.217Q26.617-54.210 26.611-54.183Q26.498-53.786 26.127-53.547Q25.756-53.308 25.332-53.308Q24.895-53.308 24.495-53.516Q24.095-53.725 23.856-54.092Q23.616-54.459 23.616-54.911M24.286-55.181L26.101-55.181Q26.101-55.458 26.004-55.710Q25.906-55.963 25.708-56.119Q25.510-56.274 25.226-56.274Q24.949-56.274 24.736-56.116Q24.522-55.957 24.404-55.702Q24.286-55.447 24.286-55.181M28.873-53.376L27.270-53.376L27.270-53.656Q27.496-53.656 27.645-53.690Q27.793-53.725 27.793-53.865L27.793-57.484Q27.793-57.754 27.686-57.816Q27.578-57.877 27.270-57.877L27.270-58.158L28.347-58.233L28.347-53.865Q28.347-53.728 28.497-53.692Q28.648-53.656 28.873-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M8.774-53.376h48.615\"\u002F>\u003Cpath stroke=\"none\" d=\"m59.389-53.376-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cpath fill=\"none\" d=\"M134.166-53.376h48.615\"\u002F>\u003Cpath stroke=\"none\" d=\"m184.78-53.376-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-24.978 35.424)\">\u003Cpath d=\"M-26.303-53.376L-28.036-53.376L-28.036-53.656Q-27.810-53.656-27.661-53.690Q-27.513-53.725-27.513-53.865L-27.513-56.114L-28.101-56.114L-28.101-56.394L-27.513-56.394L-27.513-57.211Q-27.513-57.529-27.335-57.777Q-27.157-58.024-26.867-58.165Q-26.576-58.305-26.265-58.305Q-26.009-58.305-25.805-58.163Q-25.602-58.021-25.602-57.778Q-25.602-57.642-25.701-57.543Q-25.800-57.443-25.937-57.443Q-26.074-57.443-26.173-57.543Q-26.272-57.642-26.272-57.778Q-26.272-57.959-26.132-58.052Q-26.210-58.079-26.310-58.079Q-26.518-58.079-26.672-57.946Q-26.826-57.813-26.906-57.609Q-26.986-57.406-26.986-57.197L-26.986-56.394L-26.098-56.394L-26.098-56.114L-26.959-56.114L-26.959-53.865Q-26.959-53.656-26.303-53.656L-26.303-53.376M-25.564-54.104Q-25.564-54.436-25.341-54.663Q-25.117-54.890-24.773-55.018Q-24.430-55.147-24.057-55.199Q-23.685-55.252-23.380-55.252L-23.380-55.505Q-23.380-55.710-23.488-55.890Q-23.596-56.069-23.777-56.172Q-23.958-56.274-24.166-56.274Q-24.573-56.274-24.809-56.182Q-24.720-56.145-24.674-56.061Q-24.628-55.977-24.628-55.875Q-24.628-55.779-24.674-55.700Q-24.720-55.622-24.800-55.577Q-24.881-55.533-24.970-55.533Q-25.120-55.533-25.221-55.630Q-25.322-55.728-25.322-55.875Q-25.322-56.497-24.166-56.497Q-23.955-56.497-23.705-56.433Q-23.456-56.370-23.254-56.251Q-23.052-56.131-22.926-55.946Q-22.799-55.762-22.799-55.519L-22.799-53.943Q-22.799-53.827-22.738-53.731Q-22.676-53.636-22.563-53.636Q-22.454-53.636-22.389-53.730Q-22.324-53.824-22.324-53.943L-22.324-54.391L-22.058-54.391L-22.058-53.943Q-22.058-53.673-22.285-53.508Q-22.512-53.342-22.792-53.342Q-23.001-53.342-23.138-53.496Q-23.274-53.649-23.298-53.865Q-23.445-53.598-23.727-53.453Q-24.009-53.308-24.334-53.308Q-24.611-53.308-24.894-53.383Q-25.178-53.458-25.371-53.637Q-25.564-53.817-25.564-54.104M-24.949-54.104Q-24.949-53.930-24.848-53.800Q-24.748-53.670-24.592-53.600Q-24.436-53.530-24.272-53.530Q-24.054-53.530-23.845-53.627Q-23.637-53.725-23.509-53.906Q-23.380-54.087-23.380-54.313L-23.380-55.041Q-23.705-55.041-24.071-54.950Q-24.436-54.859-24.693-54.647Q-24.949-54.436-24.949-54.104M-21.641-53.383L-21.641-54.446Q-21.641-54.470-21.613-54.497Q-21.586-54.524-21.562-54.524L-21.453-54.524Q-21.388-54.524-21.374-54.466Q-21.278-54.032-21.032-53.781Q-20.786-53.530-20.373-53.530Q-20.031-53.530-19.778-53.663Q-19.525-53.796-19.525-54.104Q-19.525-54.261-19.619-54.376Q-19.713-54.490-19.851-54.559Q-19.990-54.627-20.157-54.665L-20.738-54.764Q-21.094-54.832-21.367-55.053Q-21.641-55.273-21.641-55.615Q-21.641-55.864-21.530-56.039Q-21.418-56.213-21.232-56.312Q-21.046-56.411-20.831-56.454Q-20.615-56.497-20.373-56.497Q-19.959-56.497-19.679-56.315L-19.463-56.490Q-19.453-56.493-19.446-56.495Q-19.439-56.497-19.429-56.497L-19.378-56.497Q-19.351-56.497-19.327-56.473Q-19.303-56.449-19.303-56.421L-19.303-55.574Q-19.303-55.553-19.327-55.526Q-19.351-55.499-19.378-55.499L-19.491-55.499Q-19.518-55.499-19.544-55.524Q-19.569-55.550-19.569-55.574Q-19.569-55.810-19.675-55.974Q-19.781-56.138-19.964-56.220Q-20.147-56.302-20.379-56.302Q-20.707-56.302-20.964-56.199Q-21.220-56.097-21.220-55.820Q-21.220-55.625-21.037-55.516Q-20.854-55.406-20.625-55.365L-20.051-55.259Q-19.805-55.211-19.592-55.083Q-19.378-54.955-19.241-54.752Q-19.104-54.548-19.104-54.299Q-19.104-53.786-19.470-53.547Q-19.836-53.308-20.373-53.308Q-20.868-53.308-21.200-53.602L-21.466-53.328Q-21.487-53.308-21.514-53.308L-21.562-53.308Q-21.586-53.308-21.613-53.335Q-21.641-53.362-21.641-53.383M-17.949-54.217L-17.949-56.114L-18.588-56.114L-18.588-56.336Q-18.270-56.336-18.053-56.546Q-17.836-56.756-17.736-57.066Q-17.635-57.375-17.635-57.683L-17.368-57.683L-17.368-56.394L-16.291-56.394L-16.291-56.114L-17.368-56.114L-17.368-54.230Q-17.368-53.954-17.264-53.755Q-17.160-53.557-16.900-53.557Q-16.743-53.557-16.637-53.661Q-16.531-53.766-16.481-53.919Q-16.432-54.073-16.432-54.230L-16.432-54.644L-16.165-54.644L-16.165-54.217Q-16.165-53.991-16.264-53.781Q-16.363-53.571-16.548-53.439Q-16.732-53.308-16.961-53.308Q-17.399-53.308-17.674-53.545Q-17.949-53.783-17.949-54.217M-14.856-52.146Q-14.856-52.180-14.829-52.207Q-14.559-52.436-14.410-52.759Q-14.261-53.082-14.261-53.438L-14.261-53.475Q-14.371-53.376-14.535-53.376Q-14.716-53.376-14.835-53.496Q-14.955-53.615-14.955-53.796Q-14.955-53.971-14.835-54.090Q-14.716-54.210-14.535-54.210Q-14.278-54.210-14.159-53.971Q-14.039-53.731-14.039-53.438Q-14.039-53.038-14.208-52.667Q-14.377-52.296-14.675-52.040Q-14.706-52.019-14.733-52.019Q-14.774-52.019-14.815-52.060Q-14.856-52.101-14.856-52.146\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-24.978 35.424)\">\u003Cpath d=\"M-8.767-53.376L-10.319-53.376L-10.319-53.656Q-10.093-53.656-9.944-53.690Q-9.796-53.725-9.796-53.865L-9.796-55.714Q-9.796-55.902-9.844-55.986Q-9.891-56.069-9.989-56.088Q-10.086-56.107-10.298-56.107L-10.298-56.387L-9.242-56.462L-9.242-53.865Q-9.242-53.725-9.110-53.690Q-8.979-53.656-8.767-53.656L-8.767-53.376M-10.038-57.683Q-10.038-57.854-9.915-57.973Q-9.792-58.093-9.621-58.093Q-9.454-58.093-9.331-57.973Q-9.208-57.854-9.208-57.683Q-9.208-57.508-9.331-57.385Q-9.454-57.262-9.621-57.262Q-9.792-57.262-9.915-57.385Q-10.038-57.508-10.038-57.683M-6.439-53.376L-8.073-53.376L-8.073-53.656Q-7.844-53.656-7.695-53.690Q-7.547-53.725-7.547-53.865L-7.547-55.714Q-7.547-55.984-7.654-56.045Q-7.762-56.107-8.073-56.107L-8.073-56.387L-7.013-56.462L-7.013-55.813Q-6.843-56.121-6.538-56.292Q-6.234-56.462-5.889-56.462Q-5.383-56.462-5.099-56.239Q-4.816-56.015-4.816-55.519L-4.816-53.865Q-4.816-53.728-4.667-53.692Q-4.518-53.656-4.293-53.656L-4.293-53.376L-5.923-53.376L-5.923-53.656Q-5.694-53.656-5.545-53.690Q-5.397-53.725-5.397-53.865L-5.397-55.505Q-5.397-55.840-5.516-56.040Q-5.636-56.240-5.950-56.240Q-6.220-56.240-6.455-56.104Q-6.689-55.967-6.827-55.733Q-6.966-55.499-6.966-55.225L-6.966-53.865Q-6.966-53.728-6.815-53.692Q-6.665-53.656-6.439-53.656L-6.439-53.376M-1.907-53.376L-3.640-53.376L-3.640-53.656Q-3.414-53.656-3.266-53.690Q-3.117-53.725-3.117-53.865L-3.117-56.114L-3.705-56.114L-3.705-56.394L-3.117-56.394L-3.117-57.211Q-3.117-57.529-2.939-57.777Q-2.761-58.024-2.471-58.165Q-2.180-58.305-1.869-58.305Q-1.613-58.305-1.410-58.163Q-1.206-58.021-1.206-57.778Q-1.206-57.642-1.305-57.543Q-1.405-57.443-1.541-57.443Q-1.678-57.443-1.777-57.543Q-1.876-57.642-1.876-57.778Q-1.876-57.959-1.736-58.052Q-1.815-58.079-1.914-58.079Q-2.122-58.079-2.276-57.946Q-2.430-57.813-2.510-57.609Q-2.591-57.406-2.591-57.197L-2.591-56.394L-1.702-56.394L-1.702-56.114L-2.563-56.114L-2.563-53.865Q-2.563-53.656-1.907-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-24.978 35.424)\">\u003Cpath d=\"M1.003-53.376L-0.600-53.376L-0.600-53.656Q-0.374-53.656-0.225-53.690Q-0.077-53.725-0.077-53.865L-0.077-57.484Q-0.077-57.754-0.184-57.816Q-0.292-57.877-0.600-57.877L-0.600-58.158L0.477-58.233L0.477-53.865Q0.477-53.728 0.627-53.692Q0.778-53.656 1.003-53.656L1.003-53.376M1.557-54.911Q1.557-55.232 1.682-55.521Q1.807-55.810 2.032-56.033Q2.258-56.257 2.553-56.377Q2.849-56.497 3.167-56.497Q3.495-56.497 3.757-56.397Q4.018-56.298 4.194-56.116Q4.370-55.933 4.464-55.675Q4.558-55.417 4.558-55.085Q4.558-54.993 4.476-54.972L2.220-54.972L2.220-54.911Q2.220-54.323 2.504-53.940Q2.788-53.557 3.355-53.557Q3.676-53.557 3.945-53.750Q4.213-53.943 4.302-54.258Q4.309-54.299 4.384-54.313L4.476-54.313Q4.558-54.289 4.558-54.217Q4.558-54.210 4.551-54.183Q4.438-53.786 4.068-53.547Q3.697-53.308 3.273-53.308Q2.835-53.308 2.436-53.516Q2.036-53.725 1.796-54.092Q1.557-54.459 1.557-54.911M2.227-55.181L4.042-55.181Q4.042-55.458 3.945-55.710Q3.847-55.963 3.649-56.119Q3.451-56.274 3.167-56.274Q2.890-56.274 2.677-56.116Q2.463-55.957 2.345-55.702Q2.227-55.447 2.227-55.181M6.329-53.376L5.006-53.376L5.006-53.656Q5.566-53.656 5.946-54.056L6.660-54.853L5.748-55.902Q5.611-56.049 5.462-56.081Q5.313-56.114 5.047-56.114L5.047-56.394L6.547-56.394L6.547-56.114Q6.356-56.114 6.356-55.980Q6.356-55.950 6.387-55.902L6.981-55.218L7.422-55.714Q7.535-55.844 7.535-55.960Q7.535-56.022 7.498-56.068Q7.460-56.114 7.402-56.114L7.402-56.394L8.718-56.394L8.718-56.114Q8.157-56.114 7.778-55.714L7.156-55.013L8.150-53.865Q8.250-53.766 8.350-53.721Q8.451-53.677 8.562-53.667Q8.673-53.656 8.851-53.656L8.851-53.376L7.357-53.376L7.357-53.656Q7.422-53.656 7.482-53.690Q7.542-53.725 7.542-53.790Q7.542-53.837 7.511-53.865L6.834-54.651L6.301-54.056Q6.188-53.926 6.188-53.810Q6.188-53.745 6.229-53.701Q6.271-53.656 6.329-53.656L6.329-53.376M10.963-53.376L9.412-53.376L9.412-53.656Q9.637-53.656 9.786-53.690Q9.935-53.725 9.935-53.865L9.935-55.714Q9.935-55.902 9.887-55.986Q9.839-56.069 9.741-56.088Q9.644-56.107 9.432-56.107L9.432-56.387L10.488-56.462L10.488-53.865Q10.488-53.725 10.620-53.690Q10.751-53.656 10.963-53.656L10.963-53.376M9.692-57.683Q9.692-57.854 9.815-57.973Q9.938-58.093 10.109-58.093Q10.276-58.093 10.399-57.973Q10.522-57.854 10.522-57.683Q10.522-57.508 10.399-57.385Q10.276-57.262 10.109-57.262Q9.938-57.262 9.815-57.385Q9.692-57.508 9.692-57.683M12.416-53.376L12.149-53.376L12.149-57.484Q12.149-57.754 12.042-57.816Q11.934-57.877 11.623-57.877L11.623-58.158L12.703-58.233L12.703-56.063Q12.912-56.254 13.197-56.358Q13.482-56.462 13.780-56.462Q14.098-56.462 14.395-56.341Q14.692-56.220 14.915-56.004Q15.137-55.789 15.263-55.504Q15.390-55.218 15.390-54.887Q15.390-54.442 15.150-54.078Q14.911-53.714 14.518-53.511Q14.125-53.308 13.681-53.308Q13.486-53.308 13.296-53.364Q13.106-53.420 12.946-53.525Q12.785-53.629 12.645-53.790L12.416-53.376M12.730-55.721L12.730-54.104Q12.867-53.844 13.108-53.687Q13.349-53.530 13.626-53.530Q13.920-53.530 14.132-53.637Q14.344-53.745 14.477-53.937Q14.610-54.128 14.668-54.367Q14.727-54.606 14.727-54.887Q14.727-55.246 14.633-55.550Q14.539-55.854 14.311-56.047Q14.084-56.240 13.718-56.240Q13.417-56.240 13.151-56.104Q12.884-55.967 12.730-55.721M17.693-53.376L16.090-53.376L16.090-53.656Q16.316-53.656 16.465-53.690Q16.613-53.725 16.613-53.865L16.613-57.484Q16.613-57.754 16.506-57.816Q16.398-57.877 16.090-57.877L16.090-58.158L17.167-58.233L17.167-53.865Q17.167-53.728 17.317-53.692Q17.468-53.656 17.693-53.656L17.693-53.376M18.247-54.911Q18.247-55.232 18.372-55.521Q18.497-55.810 18.722-56.033Q18.948-56.257 19.243-56.377Q19.539-56.497 19.857-56.497Q20.185-56.497 20.447-56.397Q20.708-56.298 20.884-56.116Q21.060-55.933 21.154-55.675Q21.248-55.417 21.248-55.085Q21.248-54.993 21.166-54.972L18.910-54.972L18.910-54.911Q18.910-54.323 19.194-53.940Q19.478-53.557 20.045-53.557Q20.366-53.557 20.635-53.750Q20.903-53.943 20.992-54.258Q20.999-54.299 21.074-54.313L21.166-54.313Q21.248-54.289 21.248-54.217Q21.248-54.210 21.241-54.183Q21.128-53.786 20.758-53.547Q20.387-53.308 19.963-53.308Q19.525-53.308 19.125-53.516Q18.726-53.725 18.486-54.092Q18.247-54.459 18.247-54.911M18.917-55.181L20.732-55.181Q20.732-55.458 20.635-55.710Q20.537-55.963 20.339-56.119Q20.141-56.274 19.857-56.274Q19.580-56.274 19.366-56.116Q19.153-55.957 19.035-55.702Q18.917-55.447 18.917-55.181\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M-26.457-52.019L-28.087-52.019L-28.087-52.299Q-27.858-52.299-27.709-52.334Q-27.561-52.368-27.561-52.508L-27.561-55.854Q-27.561-56.025-27.697-56.066Q-27.834-56.107-28.087-56.107L-28.087-56.387L-27.007-56.462L-27.007-56.056Q-26.785-56.257-26.498-56.360Q-26.210-56.462-25.903-56.462Q-25.476-56.462-25.112-56.249Q-24.748-56.035-24.534-55.671Q-24.320-55.307-24.320-54.887Q-24.320-54.442-24.560-54.078Q-24.799-53.714-25.192-53.511Q-25.585-53.308-26.029-53.308Q-26.296-53.308-26.544-53.408Q-26.791-53.509-26.979-53.690L-26.979-52.508Q-26.979-52.371-26.831-52.335Q-26.682-52.299-26.457-52.299L-26.457-52.019M-26.979-55.707L-26.979-54.097Q-26.846-53.844-26.603-53.687Q-26.361-53.530-26.084-53.530Q-25.756-53.530-25.503-53.731Q-25.250-53.933-25.117-54.251Q-24.983-54.569-24.983-54.887Q-24.983-55.116-25.048-55.345Q-25.113-55.574-25.241-55.772Q-25.370-55.970-25.564-56.090Q-25.759-56.209-25.992-56.209Q-26.286-56.209-26.554-56.080Q-26.822-55.950-26.979-55.707M-21.935-53.376L-23.671-53.376L-23.671-53.656Q-23.442-53.656-23.293-53.690Q-23.144-53.725-23.144-53.865L-23.144-55.714Q-23.144-55.984-23.252-56.045Q-23.360-56.107-23.671-56.107L-23.671-56.387L-22.642-56.462L-22.642-55.755Q-22.512-56.063-22.269-56.262Q-22.027-56.462-21.709-56.462Q-21.490-56.462-21.319-56.338Q-21.148-56.213-21.148-56.001Q-21.148-55.864-21.248-55.765Q-21.347-55.666-21.480-55.666Q-21.617-55.666-21.716-55.765Q-21.815-55.864-21.815-56.001Q-21.815-56.141-21.716-56.240Q-22.006-56.240-22.206-56.044Q-22.406-55.847-22.498-55.553Q-22.591-55.259-22.591-54.979L-22.591-53.865Q-22.591-53.656-21.935-53.656L-21.935-53.376M-20.605-54.911Q-20.605-55.232-20.480-55.521Q-20.355-55.810-20.130-56.033Q-19.904-56.257-19.609-56.377Q-19.313-56.497-18.995-56.497Q-18.667-56.497-18.405-56.397Q-18.144-56.298-17.968-56.116Q-17.792-55.933-17.698-55.675Q-17.604-55.417-17.604-55.085Q-17.604-54.993-17.686-54.972L-19.942-54.972L-19.942-54.911Q-19.942-54.323-19.658-53.940Q-19.374-53.557-18.807-53.557Q-18.486-53.557-18.217-53.750Q-17.949-53.943-17.860-54.258Q-17.853-54.299-17.778-54.313L-17.686-54.313Q-17.604-54.289-17.604-54.217Q-17.604-54.210-17.611-54.183Q-17.724-53.786-18.094-53.547Q-18.465-53.308-18.889-53.308Q-19.327-53.308-19.727-53.516Q-20.126-53.725-20.366-54.092Q-20.605-54.459-20.605-54.911M-19.935-55.181L-18.120-55.181Q-18.120-55.458-18.217-55.710Q-18.315-55.963-18.513-56.119Q-18.711-56.274-18.995-56.274Q-19.272-56.274-19.486-56.116Q-19.699-55.957-19.817-55.702Q-19.935-55.447-19.935-55.181M-17.016-54.887Q-17.016-55.225-16.876-55.516Q-16.736-55.806-16.491-56.020Q-16.247-56.233-15.943-56.348Q-15.639-56.462-15.314-56.462Q-15.044-56.462-14.781-56.363Q-14.518-56.264-14.326-56.086L-14.326-57.484Q-14.326-57.754-14.434-57.816Q-14.541-57.877-14.852-57.877L-14.852-58.158L-13.776-58.233L-13.776-54.049Q-13.776-53.861-13.721-53.778Q-13.666-53.694-13.566-53.675Q-13.465-53.656-13.249-53.656L-13.249-53.376L-14.357-53.308L-14.357-53.725Q-14.774-53.308-15.399-53.308Q-15.830-53.308-16.203-53.520Q-16.575-53.731-16.796-54.092Q-17.016-54.453-17.016-54.887M-15.341-53.530Q-15.133-53.530-14.946-53.602Q-14.760-53.673-14.606-53.810Q-14.453-53.947-14.357-54.125L-14.357-55.734Q-14.442-55.881-14.588-56.001Q-14.733-56.121-14.902-56.180Q-15.071-56.240-15.252-56.240Q-15.813-56.240-16.081-55.851Q-16.350-55.461-16.350-54.880Q-16.350-54.309-16.115-53.919Q-15.881-53.530-15.341-53.530M-10.983-53.376L-12.535-53.376L-12.535-53.656Q-12.310-53.656-12.161-53.690Q-12.012-53.725-12.012-53.865L-12.012-55.714Q-12.012-55.902-12.060-55.986Q-12.108-56.069-12.205-56.088Q-12.303-56.107-12.515-56.107L-12.515-56.387L-11.458-56.462L-11.458-53.865Q-11.458-53.725-11.327-53.690Q-11.195-53.656-10.983-53.656L-10.983-53.376M-12.255-57.683Q-12.255-57.854-12.132-57.973Q-12.009-58.093-11.838-58.093Q-11.670-58.093-11.547-57.973Q-11.424-57.854-11.424-57.683Q-11.424-57.508-11.547-57.385Q-11.670-57.262-11.838-57.262Q-12.009-57.262-12.132-57.385Q-12.255-57.508-12.255-57.683M-10.337-54.887Q-10.337-55.215-10.202-55.516Q-10.067-55.816-9.832-56.037Q-9.596-56.257-9.291-56.377Q-8.987-56.497-8.663-56.497Q-8.157-56.497-7.808-56.394Q-7.459-56.292-7.459-55.916Q-7.459-55.769-7.557-55.668Q-7.654-55.567-7.801-55.567Q-7.955-55.567-8.054-55.666Q-8.153-55.765-8.153-55.916Q-8.153-56.104-8.013-56.196Q-8.215-56.247-8.656-56.247Q-9.011-56.247-9.240-56.051Q-9.469-55.854-9.570-55.545Q-9.671-55.235-9.671-54.887Q-9.671-54.538-9.544-54.232Q-9.418-53.926-9.163-53.742Q-8.909-53.557-8.553-53.557Q-8.331-53.557-8.146-53.641Q-7.962-53.725-7.827-53.880Q-7.692-54.036-7.634-54.244Q-7.620-54.299-7.565-54.299L-7.453-54.299Q-7.422-54.299-7.400-54.275Q-7.377-54.251-7.377-54.217L-7.377-54.196Q-7.463-53.909-7.651-53.711Q-7.839-53.513-8.104-53.410Q-8.369-53.308-8.663-53.308Q-9.093-53.308-9.481-53.514Q-9.869-53.721-10.103-54.084Q-10.337-54.446-10.337-54.887M-6.263-54.217L-6.263-56.114L-6.902-56.114L-6.902-56.336Q-6.584-56.336-6.367-56.546Q-6.150-56.756-6.050-57.066Q-5.949-57.375-5.949-57.683L-5.682-57.683L-5.682-56.394L-4.605-56.394L-4.605-56.114L-5.682-56.114L-5.682-54.230Q-5.682-53.954-5.578-53.755Q-5.474-53.557-5.214-53.557Q-5.057-53.557-4.951-53.661Q-4.845-53.766-4.795-53.919Q-4.746-54.073-4.746-54.230L-4.746-54.644L-4.479-54.644L-4.479-54.217Q-4.479-53.991-4.578-53.781Q-4.677-53.571-4.862-53.439Q-5.046-53.308-5.275-53.308Q-5.713-53.308-5.988-53.545Q-6.263-53.783-6.263-54.217M-2.052-53.376L-3.604-53.376L-3.604-53.656Q-3.378-53.656-3.230-53.690Q-3.081-53.725-3.081-53.865L-3.081-55.714Q-3.081-55.902-3.129-55.986Q-3.177-56.069-3.274-56.088Q-3.372-56.107-3.583-56.107L-3.583-56.387L-2.527-56.462L-2.527-53.865Q-2.527-53.725-2.396-53.690Q-2.264-53.656-2.052-53.656L-2.052-53.376M-3.324-57.683Q-3.324-57.854-3.201-57.973Q-3.078-58.093-2.907-58.093Q-2.739-58.093-2.616-57.973Q-2.493-57.854-2.493-57.683Q-2.493-57.508-2.616-57.385Q-2.739-57.262-2.907-57.262Q-3.078-57.262-3.201-57.385Q-3.324-57.508-3.324-57.683M0.183-53.403L-0.945-55.902Q-1.017-56.049-1.146-56.081Q-1.276-56.114-1.505-56.114L-1.505-56.394L0.009-56.394L0.009-56.114Q-0.343-56.114-0.343-55.967Q-0.343-55.922-0.333-55.902L0.532-53.984L1.311-55.714Q1.345-55.782 1.345-55.861Q1.345-55.974 1.262-56.044Q1.178-56.114 1.058-56.114L1.058-56.394L2.254-56.394L2.254-56.114Q2.036-56.114 1.865-56.011Q1.694-55.909 1.605-55.714L0.569-53.403Q0.522-53.308 0.416-53.308L0.337-53.308Q0.231-53.308 0.183-53.403\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M2.567-54.911Q2.567-55.232 2.692-55.521Q2.817-55.810 3.043-56.033Q3.268-56.257 3.564-56.377Q3.859-56.497 4.177-56.497Q4.505-56.497 4.767-56.397Q5.028-56.298 5.204-56.116Q5.380-55.933 5.474-55.675Q5.568-55.417 5.568-55.085Q5.568-54.993 5.486-54.972L3.231-54.972L3.231-54.911Q3.231-54.323 3.514-53.940Q3.798-53.557 4.365-53.557Q4.687-53.557 4.955-53.750Q5.223-53.943 5.312-54.258Q5.319-54.299 5.394-54.313L5.486-54.313Q5.568-54.289 5.568-54.217Q5.568-54.210 5.562-54.183Q5.449-53.786 5.078-53.547Q4.707-53.308 4.283-53.308Q3.846-53.308 3.446-53.516Q3.046-53.725 2.807-54.092Q2.567-54.459 2.567-54.911M3.237-55.181L5.052-55.181Q5.052-55.458 4.955-55.710Q4.857-55.963 4.659-56.119Q4.461-56.274 4.177-56.274Q3.900-56.274 3.687-56.116Q3.473-55.957 3.355-55.702Q3.237-55.447 3.237-55.181M6.655-52.146Q6.655-52.180 6.683-52.207Q6.953-52.436 7.101-52.759Q7.250-53.082 7.250-53.438L7.250-53.475Q7.141-53.376 6.977-53.376Q6.795-53.376 6.676-53.496Q6.556-53.615 6.556-53.796Q6.556-53.971 6.676-54.090Q6.795-54.210 6.977-54.210Q7.233-54.210 7.353-53.971Q7.472-53.731 7.472-53.438Q7.472-53.038 7.303-52.667Q7.134-52.296 6.836-52.040Q6.806-52.019 6.778-52.019Q6.737-52.019 6.696-52.060Q6.655-52.101 6.655-52.146\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M12.872-53.376L11.136-53.376L11.136-53.656Q11.365-53.656 11.514-53.690Q11.662-53.725 11.662-53.865L11.662-55.714Q11.662-55.984 11.555-56.045Q11.447-56.107 11.136-56.107L11.136-56.387L12.165-56.462L12.165-55.755Q12.295-56.063 12.537-56.262Q12.780-56.462 13.098-56.462Q13.317-56.462 13.488-56.338Q13.659-56.213 13.659-56.001Q13.659-55.864 13.559-55.765Q13.460-55.666 13.327-55.666Q13.190-55.666 13.091-55.765Q12.992-55.864 12.992-56.001Q12.992-56.141 13.091-56.240Q12.801-56.240 12.601-56.044Q12.401-55.847 12.308-55.553Q12.216-55.259 12.216-54.979L12.216-53.865Q12.216-53.656 12.872-53.656L12.872-53.376M14.202-54.911Q14.202-55.232 14.327-55.521Q14.452-55.810 14.677-56.033Q14.903-56.257 15.198-56.377Q15.494-56.497 15.812-56.497Q16.140-56.497 16.402-56.397Q16.663-56.298 16.839-56.116Q17.015-55.933 17.109-55.675Q17.203-55.417 17.203-55.085Q17.203-54.993 17.121-54.972L14.865-54.972L14.865-54.911Q14.865-54.323 15.149-53.940Q15.433-53.557 16-53.557Q16.321-53.557 16.589-53.750Q16.858-53.943 16.947-54.258Q16.954-54.299 17.029-54.313L17.121-54.313Q17.203-54.289 17.203-54.217Q17.203-54.210 17.196-54.183Q17.083-53.786 16.713-53.547Q16.342-53.308 15.918-53.308Q15.480-53.308 15.080-53.516Q14.681-53.725 14.441-54.092Q14.202-54.459 14.202-54.911M14.872-55.181L16.687-55.181Q16.687-55.458 16.589-55.710Q16.492-55.963 16.294-56.119Q16.096-56.274 15.812-56.274Q15.535-56.274 15.321-56.116Q15.108-55.957 14.990-55.702Q14.872-55.447 14.872-55.181M19.674-54.630L17.617-54.630L17.617-55.133L19.674-55.133L19.674-54.630M22.067-53.403L20.939-55.902Q20.867-56.049 20.737-56.081Q20.607-56.114 20.378-56.114L20.378-56.394L21.892-56.394L21.892-56.114Q21.540-56.114 21.540-55.967Q21.540-55.922 21.551-55.902L22.415-53.984L23.195-55.714Q23.229-55.782 23.229-55.861Q23.229-55.974 23.145-56.044Q23.061-56.114 22.942-56.114L22.942-56.394L24.138-56.394L24.138-56.114Q23.919-56.114 23.748-56.011Q23.578-55.909 23.489-55.714L22.453-53.403Q22.405-53.308 22.299-53.308L22.221-53.308Q22.115-53.308 22.067-53.403\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M24.327-54.104Q24.327-54.436 24.550-54.663Q24.774-54.890 25.118-55.018Q25.461-55.147 25.834-55.199Q26.206-55.252 26.511-55.252L26.511-55.505Q26.511-55.710 26.403-55.890Q26.295-56.069 26.114-56.172Q25.933-56.274 25.725-56.274Q25.318-56.274 25.082-56.182Q25.171-56.145 25.217-56.061Q25.263-55.977 25.263-55.875Q25.263-55.779 25.217-55.700Q25.171-55.622 25.090-55.577Q25.010-55.533 24.921-55.533Q24.771-55.533 24.670-55.630Q24.569-55.728 24.569-55.875Q24.569-56.497 25.725-56.497Q25.936-56.497 26.186-56.433Q26.435-56.370 26.637-56.251Q26.839-56.131 26.965-55.946Q27.092-55.762 27.092-55.519L27.092-53.943Q27.092-53.827 27.153-53.731Q27.215-53.636 27.328-53.636Q27.437-53.636 27.502-53.730Q27.567-53.824 27.567-53.943L27.567-54.391L27.833-54.391L27.833-53.943Q27.833-53.673 27.606-53.508Q27.379-53.342 27.099-53.342Q26.890-53.342 26.753-53.496Q26.617-53.649 26.593-53.865Q26.446-53.598 26.164-53.453Q25.882-53.308 25.557-53.308Q25.280-53.308 24.996-53.383Q24.713-53.458 24.520-53.637Q24.327-53.817 24.327-54.104M24.942-54.104Q24.942-53.930 25.043-53.800Q25.143-53.670 25.299-53.600Q25.454-53.530 25.619-53.530Q25.837-53.530 26.046-53.627Q26.254-53.725 26.382-53.906Q26.511-54.087 26.511-54.313L26.511-55.041Q26.186-55.041 25.820-54.950Q25.454-54.859 25.198-54.647Q24.942-54.436 24.942-54.104M29.918-53.376L28.315-53.376L28.315-53.656Q28.541-53.656 28.690-53.690Q28.838-53.725 28.838-53.865L28.838-57.484Q28.838-57.754 28.731-57.816Q28.623-57.877 28.315-57.877L28.315-58.158L29.392-58.233L29.392-53.865Q29.392-53.728 29.542-53.692Q29.693-53.656 29.918-53.656L29.918-53.376M31.087-54.210L31.087-55.714Q31.087-55.984 30.980-56.045Q30.872-56.107 30.561-56.107L30.561-56.387L31.668-56.462L31.668-54.230L31.668-54.210Q31.668-53.930 31.720-53.786Q31.771-53.643 31.913-53.586Q32.055-53.530 32.342-53.530Q32.595-53.530 32.800-53.670Q33.005-53.810 33.121-54.036Q33.237-54.261 33.237-54.511L33.237-55.714Q33.237-55.984 33.130-56.045Q33.022-56.107 32.711-56.107L32.711-56.387L33.818-56.462L33.818-54.049Q33.818-53.858 33.871-53.776Q33.924-53.694 34.025-53.675Q34.126-53.656 34.341-53.656L34.341-53.376L33.265-53.308L33.265-53.872Q33.155-53.690 33.010-53.567Q32.865-53.444 32.678-53.376Q32.492-53.308 32.290-53.308Q31.087-53.308 31.087-54.210M34.888-54.911Q34.888-55.232 35.013-55.521Q35.138-55.810 35.363-56.033Q35.589-56.257 35.884-56.377Q36.180-56.497 36.498-56.497Q36.826-56.497 37.088-56.397Q37.349-56.298 37.525-56.116Q37.701-55.933 37.795-55.675Q37.889-55.417 37.889-55.085Q37.889-54.993 37.807-54.972L35.551-54.972L35.551-54.911Q35.551-54.323 35.835-53.940Q36.119-53.557 36.686-53.557Q37.007-53.557 37.276-53.750Q37.544-53.943 37.633-54.258Q37.640-54.299 37.715-54.313L37.807-54.313Q37.889-54.289 37.889-54.217Q37.889-54.210 37.882-54.183Q37.769-53.786 37.399-53.547Q37.028-53.308 36.604-53.308Q36.166-53.308 35.767-53.516Q35.367-53.725 35.127-54.092Q34.888-54.459 34.888-54.911M35.558-55.181L37.373-55.181Q37.373-55.458 37.276-55.710Q37.178-55.963 36.980-56.119Q36.782-56.274 36.498-56.274Q36.221-56.274 36.007-56.116Q35.794-55.957 35.676-55.702Q35.558-55.447 35.558-55.181M38.477-53.383L38.477-54.446Q38.477-54.470 38.504-54.497Q38.532-54.524 38.556-54.524L38.665-54.524Q38.730-54.524 38.744-54.466Q38.839-54.032 39.085-53.781Q39.331-53.530 39.745-53.530Q40.087-53.530 40.340-53.663Q40.593-53.796 40.593-54.104Q40.593-54.261 40.499-54.376Q40.405-54.490 40.266-54.559Q40.128-54.627 39.960-54.665L39.379-54.764Q39.024-54.832 38.750-55.053Q38.477-55.273 38.477-55.615Q38.477-55.864 38.588-56.039Q38.699-56.213 38.885-56.312Q39.072-56.411 39.287-56.454Q39.502-56.497 39.745-56.497Q40.159-56.497 40.439-56.315L40.654-56.490Q40.664-56.493 40.671-56.495Q40.678-56.497 40.688-56.497L40.740-56.497Q40.767-56.497 40.791-56.473Q40.815-56.449 40.815-56.421L40.815-55.574Q40.815-55.553 40.791-55.526Q40.767-55.499 40.740-55.499L40.627-55.499Q40.600-55.499 40.574-55.524Q40.548-55.550 40.548-55.574Q40.548-55.810 40.442-55.974Q40.336-56.138 40.153-56.220Q39.971-56.302 39.738-56.302Q39.410-56.302 39.154-56.199Q38.897-56.097 38.897-55.820Q38.897-55.625 39.080-55.516Q39.263-55.406 39.492-55.365L40.066-55.259Q40.312-55.211 40.526-55.083Q40.740-54.955 40.876-54.752Q41.013-54.548 41.013-54.299Q41.013-53.786 40.647-53.547Q40.282-53.308 39.745-53.308Q39.249-53.308 38.918-53.602L38.651-53.328Q38.631-53.308 38.603-53.308L38.556-53.308Q38.532-53.308 38.504-53.335Q38.477-53.362 38.477-53.383\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M44.311-54.859Q44.311-55.201 44.446-55.500Q44.581-55.799 44.821-56.023Q45.060-56.247 45.378-56.372Q45.696-56.497 46.027-56.497Q46.472-56.497 46.871-56.281Q47.271-56.066 47.506-55.688Q47.740-55.311 47.740-54.859Q47.740-54.518 47.598-54.234Q47.456-53.950 47.212-53.743Q46.967-53.537 46.658-53.422Q46.349-53.308 46.027-53.308Q45.597-53.308 45.195-53.509Q44.793-53.711 44.552-54.063Q44.311-54.415 44.311-54.859M46.027-53.557Q46.629-53.557 46.853-53.935Q47.077-54.313 47.077-54.945Q47.077-55.557 46.842-55.916Q46.608-56.274 46.027-56.274Q44.975-56.274 44.975-54.945Q44.975-54.313 45.200-53.935Q45.426-53.557 46.027-53.557M50.016-53.376L48.382-53.376L48.382-53.656Q48.611-53.656 48.760-53.690Q48.909-53.725 48.909-53.865L48.909-55.714Q48.909-55.984 48.801-56.045Q48.693-56.107 48.382-56.107L48.382-56.387L49.442-56.462L49.442-55.813Q49.613-56.121 49.917-56.292Q50.221-56.462 50.566-56.462Q51.072-56.462 51.356-56.239Q51.640-56.015 51.640-55.519L51.640-53.865Q51.640-53.728 51.788-53.692Q51.937-53.656 52.163-53.656L52.163-53.376L50.532-53.376L50.532-53.656Q50.761-53.656 50.910-53.690Q51.059-53.725 51.059-53.865L51.059-55.505Q51.059-55.840 50.939-56.040Q50.819-56.240 50.505-56.240Q50.235-56.240 50.001-56.104Q49.767-55.967 49.628-55.733Q49.490-55.499 49.490-55.225L49.490-53.865Q49.490-53.728 49.640-53.692Q49.790-53.656 50.016-53.656\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M57.200-53.376L55.464-53.376L55.464-53.656Q55.693-53.656 55.842-53.690Q55.990-53.725 55.990-53.865L55.990-55.714Q55.990-55.984 55.883-56.045Q55.775-56.107 55.464-56.107L55.464-56.387L56.493-56.462L56.493-55.755Q56.623-56.063 56.865-56.262Q57.108-56.462 57.426-56.462Q57.645-56.462 57.816-56.338Q57.987-56.213 57.987-56.001Q57.987-55.864 57.887-55.765Q57.788-55.666 57.655-55.666Q57.518-55.666 57.419-55.765Q57.320-55.864 57.320-56.001Q57.320-56.141 57.419-56.240Q57.129-56.240 56.929-56.044Q56.729-55.847 56.636-55.553Q56.544-55.259 56.544-54.979L56.544-53.865Q56.544-53.656 57.200-53.656L57.200-53.376M58.530-54.911Q58.530-55.232 58.655-55.521Q58.780-55.810 59.005-56.033Q59.231-56.257 59.526-56.377Q59.822-56.497 60.140-56.497Q60.468-56.497 60.730-56.397Q60.991-56.298 61.167-56.116Q61.343-55.933 61.437-55.675Q61.531-55.417 61.531-55.085Q61.531-54.993 61.449-54.972L59.193-54.972L59.193-54.911Q59.193-54.323 59.477-53.940Q59.761-53.557 60.328-53.557Q60.649-53.557 60.917-53.750Q61.186-53.943 61.275-54.258Q61.282-54.299 61.357-54.313L61.449-54.313Q61.531-54.289 61.531-54.217Q61.531-54.210 61.524-54.183Q61.411-53.786 61.041-53.547Q60.670-53.308 60.246-53.308Q59.808-53.308 59.408-53.516Q59.009-53.725 58.769-54.092Q58.530-54.459 58.530-54.911M59.200-55.181L61.015-55.181Q61.015-55.458 60.917-55.710Q60.820-55.963 60.622-56.119Q60.424-56.274 60.140-56.274Q59.863-56.274 59.649-56.116Q59.436-55.957 59.318-55.702Q59.200-55.447 59.200-55.181M63.507-53.403L62.526-55.902Q62.464-56.045 62.346-56.080Q62.228-56.114 62.013-56.114L62.013-56.394L63.493-56.394L63.493-56.114Q63.114-56.114 63.114-55.953Q63.114-55.943 63.127-55.902L63.842-54.070L64.515-55.775Q64.484-55.847 64.484-55.875Q64.484-55.902 64.457-55.902Q64.395-56.049 64.277-56.081Q64.159-56.114 63.948-56.114L63.948-56.394L65.345-56.394L65.345-56.114Q64.969-56.114 64.969-55.953Q64.969-55.922 64.976-55.902L65.732-53.964L66.419-55.714Q66.439-55.765 66.439-55.820Q66.439-55.960 66.326-56.037Q66.214-56.114 66.073-56.114L66.073-56.394L67.294-56.394L67.294-56.114Q67.089-56.114 66.933-56.008Q66.778-55.902 66.706-55.714L65.800-53.403Q65.766-53.308 65.653-53.308L65.585-53.308Q65.475-53.308 65.438-53.403L64.655-55.406L63.869-53.403Q63.835-53.308 63.722-53.308L63.654-53.308Q63.544-53.308 63.507-53.403\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M67.682-54.104Q67.682-54.436 67.905-54.663Q68.129-54.890 68.473-55.018Q68.816-55.147 69.189-55.199Q69.561-55.252 69.866-55.252L69.866-55.505Q69.866-55.710 69.758-55.890Q69.650-56.069 69.469-56.172Q69.288-56.274 69.080-56.274Q68.673-56.274 68.437-56.182Q68.526-56.145 68.572-56.061Q68.618-55.977 68.618-55.875Q68.618-55.779 68.572-55.700Q68.526-55.622 68.445-55.577Q68.365-55.533 68.276-55.533Q68.126-55.533 68.025-55.630Q67.924-55.728 67.924-55.875Q67.924-56.497 69.080-56.497Q69.291-56.497 69.541-56.433Q69.790-56.370 69.992-56.251Q70.194-56.131 70.320-55.946Q70.447-55.762 70.447-55.519L70.447-53.943Q70.447-53.827 70.508-53.731Q70.570-53.636 70.683-53.636Q70.792-53.636 70.857-53.730Q70.922-53.824 70.922-53.943L70.922-54.391L71.188-54.391L71.188-53.943Q71.188-53.673 70.961-53.508Q70.734-53.342 70.454-53.342Q70.245-53.342 70.108-53.496Q69.972-53.649 69.948-53.865Q69.801-53.598 69.519-53.453Q69.237-53.308 68.912-53.308Q68.635-53.308 68.351-53.383Q68.068-53.458 67.875-53.637Q67.682-53.817 67.682-54.104M68.297-54.104Q68.297-53.930 68.398-53.800Q68.498-53.670 68.654-53.600Q68.809-53.530 68.974-53.530Q69.192-53.530 69.401-53.627Q69.609-53.725 69.737-53.906Q69.866-54.087 69.866-54.313L69.866-55.041Q69.541-55.041 69.175-54.950Q68.809-54.859 68.553-54.647Q68.297-54.436 68.297-54.104M73.355-53.376L71.619-53.376L71.619-53.656Q71.848-53.656 71.997-53.690Q72.145-53.725 72.145-53.865L72.145-55.714Q72.145-55.984 72.038-56.045Q71.930-56.107 71.619-56.107L71.619-56.387L72.648-56.462L72.648-55.755Q72.778-56.063 73.020-56.262Q73.263-56.462 73.581-56.462Q73.800-56.462 73.971-56.338Q74.142-56.213 74.142-56.001Q74.142-55.864 74.042-55.765Q73.943-55.666 73.810-55.666Q73.673-55.666 73.574-55.765Q73.475-55.864 73.475-56.001Q73.475-56.141 73.574-56.240Q73.284-56.240 73.084-56.044Q72.884-55.847 72.791-55.553Q72.699-55.259 72.699-54.979L72.699-53.865Q72.699-53.656 73.355-53.656L73.355-53.376M74.726-54.887Q74.726-55.225 74.866-55.516Q75.006-55.806 75.251-56.020Q75.495-56.233 75.799-56.348Q76.103-56.462 76.428-56.462Q76.698-56.462 76.961-56.363Q77.225-56.264 77.416-56.086L77.416-57.484Q77.416-57.754 77.308-57.816Q77.201-57.877 76.890-57.877L76.890-58.158L77.966-58.233L77.966-54.049Q77.966-53.861 78.021-53.778Q78.076-53.694 78.176-53.675Q78.277-53.656 78.493-53.656L78.493-53.376L77.385-53.308L77.385-53.725Q76.968-53.308 76.343-53.308Q75.912-53.308 75.539-53.520Q75.167-53.731 74.946-54.092Q74.726-54.453 74.726-54.887M76.401-53.530Q76.609-53.530 76.796-53.602Q76.982-53.673 77.136-53.810Q77.289-53.947 77.385-54.125L77.385-55.734Q77.300-55.881 77.154-56.001Q77.009-56.121 76.840-56.180Q76.671-56.240 76.490-56.240Q75.929-56.240 75.661-55.851Q75.392-55.461 75.392-54.880Q75.392-54.309 75.627-53.919Q75.861-53.530 76.401-53.530\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M81.846-54.887Q81.846-55.215 81.981-55.516Q82.116-55.816 82.352-56.037Q82.588-56.257 82.892-56.377Q83.197-56.497 83.521-56.497Q84.027-56.497 84.376-56.394Q84.724-56.292 84.724-55.916Q84.724-55.769 84.627-55.668Q84.530-55.567 84.383-55.567Q84.229-55.567 84.130-55.666Q84.031-55.765 84.031-55.916Q84.031-56.104 84.171-56.196Q83.969-56.247 83.528-56.247Q83.173-56.247 82.944-56.051Q82.715-55.854 82.614-55.545Q82.513-55.235 82.513-54.887Q82.513-54.538 82.639-54.232Q82.766-53.926 83.021-53.742Q83.275-53.557 83.631-53.557Q83.853-53.557 84.037-53.641Q84.222-53.725 84.357-53.880Q84.492-54.036 84.550-54.244Q84.564-54.299 84.618-54.299L84.731-54.299Q84.762-54.299 84.784-54.275Q84.806-54.251 84.806-54.217L84.806-54.196Q84.721-53.909 84.533-53.711Q84.345-53.513 84.080-53.410Q83.815-53.308 83.521-53.308Q83.091-53.308 82.703-53.514Q82.315-53.721 82.081-54.084Q81.846-54.446 81.846-54.887\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.368 35.424)\">\u003Cpath d=\"M86.868-53.376L85.234-53.376L85.234-53.656Q85.463-53.656 85.612-53.690Q85.761-53.725 85.761-53.865L85.761-57.484Q85.761-57.754 85.653-57.816Q85.545-57.877 85.234-57.877L85.234-58.158L86.314-58.233L86.314-55.847Q86.420-56.032 86.598-56.174Q86.776-56.315 86.984-56.389Q87.193-56.462 87.418-56.462Q87.924-56.462 88.208-56.239Q88.492-56.015 88.492-55.519L88.492-53.865Q88.492-53.728 88.640-53.692Q88.789-53.656 89.015-53.656L89.015-53.376L87.384-53.376L87.384-53.656Q87.613-53.656 87.762-53.690Q87.911-53.725 87.911-53.865L87.911-55.505Q87.911-55.840 87.791-56.040Q87.671-56.240 87.357-56.240Q87.087-56.240 86.853-56.104Q86.619-55.967 86.480-55.733Q86.342-55.499 86.342-55.225L86.342-53.865Q86.342-53.728 86.492-53.692Q86.643-53.656 86.868-53.656L86.868-53.376M89.661-54.104Q89.661-54.436 89.884-54.663Q90.108-54.890 90.452-55.018Q90.795-55.147 91.168-55.199Q91.540-55.252 91.845-55.252L91.845-55.505Q91.845-55.710 91.737-55.890Q91.629-56.069 91.448-56.172Q91.267-56.274 91.059-56.274Q90.652-56.274 90.416-56.182Q90.505-56.145 90.551-56.061Q90.597-55.977 90.597-55.875Q90.597-55.779 90.551-55.700Q90.505-55.622 90.424-55.577Q90.344-55.533 90.255-55.533Q90.105-55.533 90.004-55.630Q89.903-55.728 89.903-55.875Q89.903-56.497 91.059-56.497Q91.270-56.497 91.520-56.433Q91.769-56.370 91.971-56.251Q92.173-56.131 92.299-55.946Q92.426-55.762 92.426-55.519L92.426-53.943Q92.426-53.827 92.487-53.731Q92.549-53.636 92.662-53.636Q92.771-53.636 92.836-53.730Q92.901-53.824 92.901-53.943L92.901-54.391L93.167-54.391L93.167-53.943Q93.167-53.673 92.940-53.508Q92.713-53.342 92.433-53.342Q92.224-53.342 92.087-53.496Q91.951-53.649 91.927-53.865Q91.780-53.598 91.498-53.453Q91.216-53.308 90.891-53.308Q90.614-53.308 90.330-53.383Q90.047-53.458 89.854-53.637Q89.661-53.817 89.661-54.104M90.276-54.104Q90.276-53.930 90.377-53.800Q90.477-53.670 90.633-53.600Q90.789-53.530 90.953-53.530Q91.171-53.530 91.380-53.627Q91.588-53.725 91.716-53.906Q91.845-54.087 91.845-54.313L91.845-55.041Q91.520-55.041 91.154-54.950Q90.789-54.859 90.532-54.647Q90.276-54.436 90.276-54.104M95.266-53.376L93.632-53.376L93.632-53.656Q93.861-53.656 94.010-53.690Q94.159-53.725 94.159-53.865L94.159-55.714Q94.159-55.984 94.051-56.045Q93.943-56.107 93.632-56.107L93.632-56.387L94.692-56.462L94.692-55.813Q94.863-56.121 95.167-56.292Q95.471-56.462 95.816-56.462Q96.322-56.462 96.606-56.239Q96.890-56.015 96.890-55.519L96.890-53.865Q96.890-53.728 97.038-53.692Q97.187-53.656 97.413-53.656L97.413-53.376L95.782-53.376L95.782-53.656Q96.011-53.656 96.160-53.690Q96.309-53.725 96.309-53.865L96.309-55.505Q96.309-55.840 96.189-56.040Q96.069-56.240 95.755-56.240Q95.485-56.240 95.251-56.104Q95.017-55.967 94.878-55.733Q94.740-55.499 94.740-55.225L94.740-53.865Q94.740-53.728 94.890-53.692Q95.040-53.656 95.266-53.656L95.266-53.376M97.959-52.843Q97.959-53.089 98.156-53.273Q98.352-53.458 98.609-53.537Q98.472-53.649 98.400-53.810Q98.329-53.971 98.329-54.152Q98.329-54.473 98.540-54.719Q98.205-55.017 98.205-55.427Q98.205-55.888 98.595-56.175Q98.985-56.462 99.463-56.462Q99.935-56.462 100.270-56.216Q100.444-56.370 100.654-56.452Q100.865-56.534 101.094-56.534Q101.258-56.534 101.379-56.427Q101.500-56.319 101.500-56.155Q101.500-56.059 101.429-55.987Q101.357-55.916 101.265-55.916Q101.165-55.916 101.095-55.989Q101.025-56.063 101.025-56.162Q101.025-56.216 101.039-56.247L101.046-56.261Q101.053-56.281 101.061-56.292Q101.070-56.302 101.073-56.309Q100.718-56.309 100.431-56.086Q100.718-55.793 100.718-55.427Q100.718-55.112 100.533-54.880Q100.349-54.647 100.060-54.519Q99.771-54.391 99.463-54.391Q99.262-54.391 99.070-54.441Q98.879-54.490 98.701-54.600Q98.609-54.473 98.609-54.330Q98.609-54.148 98.737-54.013Q98.865-53.878 99.050-53.878L99.682-53.878Q100.130-53.878 100.499-53.807Q100.868-53.735 101.128-53.506Q101.388-53.277 101.388-52.843Q101.388-52.522 101.092-52.320Q100.796-52.118 100.393-52.029Q99.990-51.940 99.675-51.940Q99.357-51.940 98.954-52.029Q98.551-52.118 98.255-52.320Q97.959-52.522 97.959-52.843M98.414-52.843Q98.414-52.614 98.633-52.465Q98.851-52.316 99.144-52.248Q99.436-52.180 99.675-52.180Q99.839-52.180 100.048-52.216Q100.256-52.251 100.463-52.332Q100.670-52.412 100.801-52.540Q100.933-52.668 100.933-52.843Q100.933-53.195 100.552-53.289Q100.171-53.383 99.668-53.383L99.050-53.383Q98.810-53.383 98.612-53.232Q98.414-53.082 98.414-52.843M99.463-54.630Q100.130-54.630 100.130-55.427Q100.130-56.227 99.463-56.227Q98.793-56.227 98.793-55.427Q98.793-54.630 99.463-54.630M101.941-54.911Q101.941-55.232 102.066-55.521Q102.191-55.810 102.416-56.033Q102.642-56.257 102.938-56.377Q103.233-56.497 103.551-56.497Q103.879-56.497 104.141-56.397Q104.402-56.298 104.578-56.116Q104.754-55.933 104.848-55.675Q104.942-55.417 104.942-55.085Q104.942-54.993 104.860-54.972L102.604-54.972L102.604-54.911Q102.604-54.323 102.888-53.940Q103.172-53.557 103.739-53.557Q104.060-53.557 104.329-53.750Q104.597-53.943 104.686-54.258Q104.693-54.299 104.768-54.313L104.860-54.313Q104.942-54.289 104.942-54.217Q104.942-54.210 104.935-54.183Q104.823-53.786 104.452-53.547Q104.081-53.308 103.657-53.308Q103.220-53.308 102.820-53.516Q102.420-53.725 102.181-54.092Q101.941-54.459 101.941-54.911M102.611-55.181L104.426-55.181Q104.426-55.458 104.329-55.710Q104.231-55.963 104.033-56.119Q103.835-56.274 103.551-56.274Q103.274-56.274 103.061-56.116Q102.847-55.957 102.729-55.702Q102.611-55.447 102.611-55.181\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(217.705 35.424)\">\u003Cpath d=\"M-26.303-53.376L-28.036-53.376L-28.036-53.656Q-27.810-53.656-27.661-53.690Q-27.513-53.725-27.513-53.865L-27.513-56.114L-28.101-56.114L-28.101-56.394L-27.513-56.394L-27.513-57.211Q-27.513-57.529-27.335-57.777Q-27.157-58.024-26.867-58.165Q-26.576-58.305-26.265-58.305Q-26.009-58.305-25.805-58.163Q-25.602-58.021-25.602-57.778Q-25.602-57.642-25.701-57.543Q-25.800-57.443-25.937-57.443Q-26.074-57.443-26.173-57.543Q-26.272-57.642-26.272-57.778Q-26.272-57.959-26.132-58.052Q-26.210-58.079-26.310-58.079Q-26.518-58.079-26.672-57.946Q-26.826-57.813-26.906-57.609Q-26.986-57.406-26.986-57.197L-26.986-56.394L-26.098-56.394L-26.098-56.114L-26.959-56.114L-26.959-53.865Q-26.959-53.656-26.303-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(217.705 35.424)\">\u003Cpath d=\"M-23.395-53.376L-24.998-53.376L-24.998-53.656Q-24.772-53.656-24.623-53.690Q-24.475-53.725-24.475-53.865L-24.475-57.484Q-24.475-57.754-24.582-57.816Q-24.690-57.877-24.998-57.877L-24.998-58.158L-23.921-58.233L-23.921-53.865Q-23.921-53.728-23.771-53.692Q-23.620-53.656-23.395-53.656L-23.395-53.376M-22.841-54.911Q-22.841-55.232-22.716-55.521Q-22.591-55.810-22.366-56.033Q-22.140-56.257-21.845-56.377Q-21.549-56.497-21.231-56.497Q-20.903-56.497-20.641-56.397Q-20.380-56.298-20.204-56.116Q-20.028-55.933-19.934-55.675Q-19.840-55.417-19.840-55.085Q-19.840-54.993-19.922-54.972L-22.178-54.972L-22.178-54.911Q-22.178-54.323-21.894-53.940Q-21.610-53.557-21.043-53.557Q-20.722-53.557-20.453-53.750Q-20.185-53.943-20.096-54.258Q-20.089-54.299-20.014-54.313L-19.922-54.313Q-19.840-54.289-19.840-54.217Q-19.840-54.210-19.847-54.183Q-19.960-53.786-20.330-53.547Q-20.701-53.308-21.125-53.308Q-21.563-53.308-21.962-53.516Q-22.362-53.725-22.602-54.092Q-22.841-54.459-22.841-54.911M-22.171-55.181L-20.356-55.181Q-20.356-55.458-20.453-55.710Q-20.551-55.963-20.749-56.119Q-20.947-56.274-21.231-56.274Q-21.508-56.274-21.721-56.116Q-21.935-55.957-22.053-55.702Q-22.171-55.447-22.171-55.181M-18.069-53.376L-19.392-53.376L-19.392-53.656Q-18.832-53.656-18.452-54.056L-17.738-54.853L-18.650-55.902Q-18.787-56.049-18.936-56.081Q-19.085-56.114-19.351-56.114L-19.351-56.394L-17.851-56.394L-17.851-56.114Q-18.042-56.114-18.042-55.980Q-18.042-55.950-18.011-55.902L-17.417-55.218L-16.976-55.714Q-16.863-55.844-16.863-55.960Q-16.863-56.022-16.900-56.068Q-16.938-56.114-16.996-56.114L-16.996-56.394L-15.680-56.394L-15.680-56.114Q-16.241-56.114-16.620-55.714L-17.242-55.013L-16.248-53.865Q-16.148-53.766-16.048-53.721Q-15.947-53.677-15.836-53.667Q-15.725-53.656-15.547-53.656L-15.547-53.376L-17.041-53.376L-17.041-53.656Q-16.976-53.656-16.916-53.690Q-16.856-53.725-16.856-53.790Q-16.856-53.837-16.887-53.865L-17.564-54.651L-18.097-54.056Q-18.210-53.926-18.210-53.810Q-18.210-53.745-18.169-53.701Q-18.127-53.656-18.069-53.656L-18.069-53.376M-13.435-53.376L-14.986-53.376L-14.986-53.656Q-14.761-53.656-14.612-53.690Q-14.463-53.725-14.463-53.865L-14.463-55.714Q-14.463-55.902-14.511-55.986Q-14.559-56.069-14.657-56.088Q-14.754-56.107-14.966-56.107L-14.966-56.387L-13.910-56.462L-13.910-53.865Q-13.910-53.725-13.778-53.690Q-13.647-53.656-13.435-53.656L-13.435-53.376M-14.706-57.683Q-14.706-57.854-14.583-57.973Q-14.460-58.093-14.289-58.093Q-14.122-58.093-13.999-57.973Q-13.876-57.854-13.876-57.683Q-13.876-57.508-13.999-57.385Q-14.122-57.262-14.289-57.262Q-14.460-57.262-14.583-57.385Q-14.706-57.508-14.706-57.683M-11.982-53.376L-12.249-53.376L-12.249-57.484Q-12.249-57.754-12.356-57.816Q-12.464-57.877-12.775-57.877L-12.775-58.158L-11.695-58.233L-11.695-56.063Q-11.486-56.254-11.201-56.358Q-10.916-56.462-10.618-56.462Q-10.300-56.462-10.003-56.341Q-9.706-56.220-9.483-56.004Q-9.261-55.789-9.135-55.504Q-9.008-55.218-9.008-54.887Q-9.008-54.442-9.248-54.078Q-9.487-53.714-9.880-53.511Q-10.273-53.308-10.717-53.308Q-10.912-53.308-11.102-53.364Q-11.292-53.420-11.452-53.525Q-11.613-53.629-11.753-53.790L-11.982-53.376M-11.668-55.721L-11.668-54.104Q-11.531-53.844-11.290-53.687Q-11.049-53.530-10.772-53.530Q-10.478-53.530-10.266-53.637Q-10.054-53.745-9.921-53.937Q-9.788-54.128-9.730-54.367Q-9.671-54.606-9.671-54.887Q-9.671-55.246-9.765-55.550Q-9.859-55.854-10.087-56.047Q-10.314-56.240-10.680-56.240Q-10.981-56.240-11.247-56.104Q-11.514-55.967-11.668-55.721M-6.705-53.376L-8.308-53.376L-8.308-53.656Q-8.082-53.656-7.933-53.690Q-7.785-53.725-7.785-53.865L-7.785-57.484Q-7.785-57.754-7.892-57.816Q-8-57.877-8.308-57.877L-8.308-58.158L-7.231-58.233L-7.231-53.865Q-7.231-53.728-7.081-53.692Q-6.930-53.656-6.705-53.656L-6.705-53.376M-6.151-54.911Q-6.151-55.232-6.026-55.521Q-5.901-55.810-5.676-56.033Q-5.450-56.257-5.155-56.377Q-4.859-56.497-4.541-56.497Q-4.213-56.497-3.951-56.397Q-3.690-56.298-3.514-56.116Q-3.338-55.933-3.244-55.675Q-3.150-55.417-3.150-55.085Q-3.150-54.993-3.232-54.972L-5.488-54.972L-5.488-54.911Q-5.488-54.323-5.204-53.940Q-4.920-53.557-4.353-53.557Q-4.032-53.557-3.763-53.750Q-3.495-53.943-3.406-54.258Q-3.399-54.299-3.324-54.313L-3.232-54.313Q-3.150-54.289-3.150-54.217Q-3.150-54.210-3.157-54.183Q-3.270-53.786-3.640-53.547Q-4.011-53.308-4.435-53.308Q-4.873-53.308-5.273-53.516Q-5.672-53.725-5.912-54.092Q-6.151-54.459-6.151-54.911M-5.481-55.181L-3.666-55.181Q-3.666-55.458-3.763-55.710Q-3.861-55.963-4.059-56.119Q-4.257-56.274-4.541-56.274Q-4.818-56.274-5.032-56.116Q-5.245-55.957-5.363-55.702Q-5.481-55.447-5.481-55.181M-2.063-52.146Q-2.063-52.180-2.036-52.207Q-1.766-52.436-1.617-52.759Q-1.468-53.082-1.468-53.438L-1.468-53.475Q-1.578-53.376-1.742-53.376Q-1.923-53.376-2.043-53.496Q-2.162-53.615-2.162-53.796Q-2.162-53.971-2.043-54.090Q-1.923-54.210-1.742-54.210Q-1.485-54.210-1.366-53.971Q-1.246-53.731-1.246-53.438Q-1.246-53.038-1.415-52.667Q-1.585-52.296-1.882-52.040Q-1.913-52.019-1.940-52.019Q-1.981-52.019-2.022-52.060Q-2.063-52.101-2.063-52.146\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(217.705 35.424)\">\u003Cpath d=\"M4.099-53.376L2.465-53.376L2.465-53.656Q2.694-53.656 2.843-53.690Q2.992-53.725 2.992-53.865L2.992-55.714Q2.992-55.984 2.884-56.045Q2.776-56.107 2.465-56.107L2.465-56.387L3.525-56.462L3.525-55.813Q3.696-56.121 4-56.292Q4.304-56.462 4.649-56.462Q5.049-56.462 5.326-56.322Q5.603-56.182 5.688-55.834Q5.856-56.127 6.155-56.295Q6.454-56.462 6.799-56.462Q7.305-56.462 7.589-56.239Q7.873-56.015 7.873-55.519L7.873-53.865Q7.873-53.728 8.021-53.692Q8.170-53.656 8.395-53.656L8.395-53.376L6.765-53.376L6.765-53.656Q6.991-53.656 7.141-53.692Q7.291-53.728 7.291-53.865L7.291-55.505Q7.291-55.840 7.172-56.040Q7.052-56.240 6.738-56.240Q6.468-56.240 6.234-56.104Q5.999-55.967 5.861-55.733Q5.723-55.499 5.723-55.225L5.723-53.865Q5.723-53.728 5.871-53.692Q6.020-53.656 6.246-53.656L6.246-53.376L4.615-53.376L4.615-53.656Q4.844-53.656 4.993-53.690Q5.142-53.725 5.142-53.865L5.142-55.505Q5.142-55.840 5.022-56.040Q4.902-56.240 4.588-56.240Q4.318-56.240 4.084-56.104Q3.850-55.967 3.711-55.733Q3.573-55.499 3.573-55.225L3.573-53.865Q3.573-53.728 3.723-53.692Q3.874-53.656 4.099-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(217.705 35.424)\">\u003Cpath d=\"M9.346-54.210L9.346-55.714Q9.346-55.984 9.238-56.045Q9.130-56.107 8.819-56.107L8.819-56.387L9.927-56.462L9.927-54.230L9.927-54.210Q9.927-53.930 9.978-53.786Q10.029-53.643 10.171-53.586Q10.313-53.530 10.600-53.530Q10.853-53.530 11.058-53.670Q11.263-53.810 11.379-54.036Q11.496-54.261 11.496-54.511L11.496-55.714Q11.496-55.984 11.388-56.045Q11.280-56.107 10.969-56.107L10.969-56.387L12.077-56.462L12.077-54.049Q12.077-53.858 12.130-53.776Q12.183-53.694 12.283-53.675Q12.384-53.656 12.600-53.656L12.600-53.376L11.523-53.308L11.523-53.872Q11.414-53.690 11.268-53.567Q11.123-53.444 10.937-53.376Q10.750-53.308 10.549-53.308Q9.346-53.308 9.346-54.210M13.187-53.383L13.187-54.446Q13.187-54.470 13.215-54.497Q13.242-54.524 13.266-54.524L13.375-54.524Q13.440-54.524 13.454-54.466Q13.550-54.032 13.796-53.781Q14.042-53.530 14.456-53.530Q14.797-53.530 15.050-53.663Q15.303-53.796 15.303-54.104Q15.303-54.261 15.209-54.376Q15.115-54.490 14.977-54.559Q14.838-54.627 14.671-54.665L14.090-54.764Q13.734-54.832 13.461-55.053Q13.187-55.273 13.187-55.615Q13.187-55.864 13.299-56.039Q13.410-56.213 13.596-56.312Q13.782-56.411 13.998-56.454Q14.213-56.497 14.456-56.497Q14.869-56.497 15.149-56.315L15.365-56.490Q15.375-56.493 15.382-56.495Q15.389-56.497 15.399-56.497L15.450-56.497Q15.478-56.497 15.501-56.473Q15.525-56.449 15.525-56.421L15.525-55.574Q15.525-55.553 15.501-55.526Q15.478-55.499 15.450-55.499L15.337-55.499Q15.310-55.499 15.284-55.524Q15.259-55.550 15.259-55.574Q15.259-55.810 15.153-55.974Q15.047-56.138 14.864-56.220Q14.681-56.302 14.449-56.302Q14.121-56.302 13.864-56.199Q13.608-56.097 13.608-55.820Q13.608-55.625 13.791-55.516Q13.974-55.406 14.203-55.365L14.777-55.259Q15.023-55.211 15.237-55.083Q15.450-54.955 15.587-54.752Q15.724-54.548 15.724-54.299Q15.724-53.786 15.358-53.547Q14.992-53.308 14.456-53.308Q13.960-53.308 13.628-53.602L13.362-53.328Q13.341-53.308 13.314-53.308L13.266-53.308Q13.242-53.308 13.215-53.335Q13.187-53.362 13.187-53.383M16.879-54.217L16.879-56.114L16.240-56.114L16.240-56.336Q16.558-56.336 16.775-56.546Q16.992-56.756 17.092-57.066Q17.193-57.375 17.193-57.683L17.460-57.683L17.460-56.394L18.537-56.394L18.537-56.114L17.460-56.114L17.460-54.230Q17.460-53.954 17.564-53.755Q17.668-53.557 17.928-53.557Q18.085-53.557 18.191-53.661Q18.297-53.766 18.347-53.919Q18.396-54.073 18.396-54.230L18.396-54.644L18.663-54.644L18.663-54.217Q18.663-53.991 18.564-53.781Q18.465-53.571 18.280-53.439Q18.096-53.308 17.867-53.308Q17.429-53.308 17.154-53.545Q16.879-53.783 16.879-54.217\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(217.705 35.424)\">\u003Cpath d=\"M23.819-52.019L22.189-52.019L22.189-52.299Q22.418-52.299 22.567-52.334Q22.715-52.368 22.715-52.508L22.715-55.854Q22.715-56.025 22.579-56.066Q22.442-56.107 22.189-56.107L22.189-56.387L23.269-56.462L23.269-56.056Q23.491-56.257 23.778-56.360Q24.066-56.462 24.373-56.462Q24.800-56.462 25.164-56.249Q25.528-56.035 25.742-55.671Q25.956-55.307 25.956-54.887Q25.956-54.442 25.716-54.078Q25.477-53.714 25.084-53.511Q24.691-53.308 24.247-53.308Q23.980-53.308 23.732-53.408Q23.485-53.509 23.297-53.690L23.297-52.508Q23.297-52.371 23.445-52.335Q23.594-52.299 23.819-52.299L23.819-52.019M23.297-55.707L23.297-54.097Q23.430-53.844 23.673-53.687Q23.915-53.530 24.192-53.530Q24.520-53.530 24.773-53.731Q25.026-53.933 25.159-54.251Q25.293-54.569 25.293-54.887Q25.293-55.116 25.228-55.345Q25.163-55.574 25.035-55.772Q24.906-55.970 24.712-56.090Q24.517-56.209 24.284-56.209Q23.990-56.209 23.722-56.080Q23.454-55.950 23.297-55.707M28.259-53.376L26.656-53.376L26.656-53.656Q26.882-53.656 27.031-53.690Q27.179-53.725 27.179-53.865L27.179-57.484Q27.179-57.754 27.072-57.816Q26.964-57.877 26.656-57.877L26.656-58.158L27.733-58.233L27.733-53.865Q27.733-53.728 27.883-53.692Q28.034-53.656 28.259-53.656L28.259-53.376M28.912-54.104Q28.912-54.436 29.136-54.663Q29.360-54.890 29.704-55.018Q30.047-55.147 30.420-55.199Q30.792-55.252 31.096-55.252L31.096-55.505Q31.096-55.710 30.989-55.890Q30.881-56.069 30.700-56.172Q30.519-56.274 30.310-56.274Q29.903-56.274 29.668-56.182Q29.757-56.145 29.803-56.061Q29.849-55.977 29.849-55.875Q29.849-55.779 29.803-55.700Q29.757-55.622 29.676-55.577Q29.596-55.533 29.507-55.533Q29.357-55.533 29.256-55.630Q29.155-55.728 29.155-55.875Q29.155-56.497 30.310-56.497Q30.522-56.497 30.772-56.433Q31.021-56.370 31.223-56.251Q31.424-56.131 31.551-55.946Q31.677-55.762 31.677-55.519L31.677-53.943Q31.677-53.827 31.739-53.731Q31.800-53.636 31.913-53.636Q32.023-53.636 32.088-53.730Q32.153-53.824 32.153-53.943L32.153-54.391L32.419-54.391L32.419-53.943Q32.419-53.673 32.192-53.508Q31.965-53.342 31.684-53.342Q31.476-53.342 31.339-53.496Q31.202-53.649 31.178-53.865Q31.031-53.598 30.749-53.453Q30.467-53.308 30.143-53.308Q29.866-53.308 29.582-53.383Q29.299-53.458 29.105-53.637Q28.912-53.817 28.912-54.104M29.528-54.104Q29.528-53.930 29.628-53.800Q29.729-53.670 29.885-53.600Q30.040-53.530 30.204-53.530Q30.423-53.530 30.632-53.627Q30.840-53.725 30.968-53.906Q31.096-54.087 31.096-54.313L31.096-55.041Q30.772-55.041 30.406-54.950Q30.040-54.859 29.784-54.647Q29.528-54.436 29.528-54.104M34.518-53.376L32.884-53.376L32.884-53.656Q33.113-53.656 33.262-53.690Q33.410-53.725 33.410-53.865L33.410-55.714Q33.410-55.984 33.303-56.045Q33.195-56.107 32.884-56.107L32.884-56.387L33.944-56.462L33.944-55.813Q34.114-56.121 34.419-56.292Q34.723-56.462 35.068-56.462Q35.574-56.462 35.858-56.239Q36.141-56.015 36.141-55.519L36.141-53.865Q36.141-53.728 36.290-53.692Q36.439-53.656 36.664-53.656L36.664-53.376L35.034-53.376L35.034-53.656Q35.263-53.656 35.412-53.690Q35.560-53.725 35.560-53.865L35.560-55.505Q35.560-55.840 35.441-56.040Q35.321-56.240 35.007-56.240Q34.736-56.240 34.502-56.104Q34.268-55.967 34.130-55.733Q33.991-55.499 33.991-55.225L33.991-53.865Q33.991-53.728 34.142-53.692Q34.292-53.656 34.518-53.656\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The successor representation as a middle ground. Model-free caches a scalar value per state (fast, but reward and dynamics are fused). The SR caches expected future occupancy (a predictive map, re-valued instantly when reward changes). Model-based holds the transition model and plans (flexible, but must search).\u003C\u002Ffigcaption>",1785117802285]