[{"data":1,"prerenderedAt":1227},["ShallowReactive",2],{"subject:relativity":3,"course-wordcounts":75,"nav:relativity":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F03.relativity\u002Findex.md","Relativity","Special and general relativity from the two postulates to curved spacetime —\nthe Lorentz group, relativistic dynamics, the Schwarzschild solution, black\nholes, and gravitational waves.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Relativity is what you get by taking one experimental fact — that the speed of\nlight is the same for every observer — completely seriously. The sequence starts\nwith the two postulates of special relativity and the Lorentz transformation they\nforce, then builds the geometry of Minkowski spacetime: the invariant interval,\nthe light cone, and the paradoxes of time dilation, length contraction, and\nsimultaneity that are really just features of that geometry. From there it\nrebuilds mechanics into relativistic dynamics, unifies mass with energy, and\nrecasts electromagnetism in manifestly covariant form. The second half turns to\ngravity: the equivalence principle, curved spacetime and the Einstein field\nequation, the Schwarzschild solution and its orbits, the classical tests, black\nholes and their horizons, gravitational waves, and finally the bridge to\ncosmology. It follows Tipler & Llewellyn for the foundations and Hartle, Carroll,\nand Schutz for the general theory, and each topic rests on the geometry laid down\nbefore it.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,42,46,48,52,54],{"p":20},"An observer's clock and ruler are not absolute. Special relativity starts\nfrom two postulates — the laws of physics look the same in every inertial\nframe, and light travels at the same speed \u003Cem>c\u003C\u002Fem> for everyone — and\nfollows them without flinching to conclusions that overturn Newton's\nseparate space and time.\n",{"fig":22,"n":23,"caption":24,"large":25},"rel-lightcone","001","The light cone through an event splits spacetime into causal future, past,\nand an unreachable elsewhere; a worldline threads the interior.\n",true,{"fig":27,"n":28,"caption":29},"rel-lightclock","002","A moving light clock runs slow: the photon's diagonal path is longer, so\neach tick takes longer.\n",{"p":31},"Space and time merge into a single four-dimensional\n\u003Cstrong>spacetime\u003C\u002Fstrong>. The one quantity every observer agrees on is\nthe interval between events, and the light cone it defines fixes what can\ncause what — sorting every event into an absolute future, past, or the\nspacelike elsewhere no signal can reach.\n",{"p":33},"The bridge between frames is the \u003Cstrong>Lorentz transformation\u003C\u002Fstrong>.\nIt mixes space and time while leaving the speed of light invariant, and its\nconsequences — time dilation, length contraction, the relativity of\nsimultaneity — are geometry, not illusion.\n",{"fig":35,"n":36,"caption":37},"rel-boost","003","A Lorentz boost tilts the moving frame's space and time axes toward the\nlight line — which is why simultaneity is relative.\n",{"p":39},"Dynamics is rebuilt to match. Momentum and energy join into a single\nfour-vector, mass reveals itself as a form of energy through\n\u003Cem>E = mc²\u003C\u002Fem>, and Maxwell's electromagnetism turns out to have been\nrelativistic all along.\n",{"p":41},"Then gravity. The \u003Cstrong>equivalence principle\u003C\u002Fstrong> — that free fall\nis indistinguishable from floating in empty space — says gravity is not a\nforce but the curvature of spacetime, and matter is what does the curving.\n",{"fig":43,"n":44,"caption":45},"rel-curvature","004","Mass curves the spacetime grid, and a free body orbits by following the\nstraightest available path through it.\n",{"p":47},"The \u003Cstrong>Einstein field equation\u003C\u002Fstrong> ties that curvature to energy\nand momentum. Its first exact solution, the Schwarzschild metric, describes\nspacetime around a star and predicts the bending of light, the precession\nof Mercury, and clocks that tick faster with altitude.\n",{"fig":49,"n":50,"caption":51},"rel-gravwave","005","A gravitational wave stretches and squeezes a ring of free test masses as\nit passes.\n",{"p":53},"Push a mass small enough and an \u003Cstrong>event horizon\u003C\u002Fstrong> forms: a\nblack hole, a region from which not even light escapes. Ripples in the\ncurvature — \u003Cstrong>gravitational waves\u003C\u002Fstrong> — travel outward at\n\u003Cem>c\u003C\u002Fem> and were first caught stretching a detector in 2015.\n",{"p":55},"Turn the same equations on the universe as a whole and they give an\nexpanding cosmos with a finite past. Relativity is the geometry underneath\nall of it.\n","physics","Relativity rebuilt space and time around a single invariant speed. This course\nruns from the postulates of special relativity, the Lorentz transformation, and\nspacetime geometry through relativistic dynamics and covariant electromagnetism,\nthen into general relativity: the equivalence principle, curved spacetime and the\nEinstein equation, the Schwarzschild solution and its orbits, the classical tests,\nblack holes, gravitational waves, and the bridge to cosmology. It follows Tipler &\nLlewellyn for the foundations and Hartle, Carroll, and Schutz for the general theory.\n",false,"md",{},"\u002Frelativity",[],"---\ntitle: Relativity\nstatus: available\ncategory: physics\nblurb: |\n  Special and general relativity from the two postulates to curved spacetime —\n  the Lorentz group, relativistic dynamics, the Schwarzschild solution, black\n  holes, and gravitational waves.\ndescription: |\n  Relativity rebuilt space and time around a single invariant speed. This course\n  runs from the postulates of special relativity, the Lorentz transformation, and\n  spacetime geometry through relativistic dynamics and covariant electromagnetism,\n  then into general relativity: the equivalence principle, curved spacetime and the\n  Einstein equation, the Schwarzschild solution and its orbits, the classical tests,\n  black holes, gravitational waves, and the bridge to cosmology. It follows Tipler &\n  Llewellyn for the foundations and Hartle, Carroll, and Schutz for the general theory.\nbrief:\n  - p: |\n      An observer's clock and ruler are not absolute. Special relativity starts\n      from two postulates — the laws of physics look the same in every inertial\n      frame, and light travels at the same speed \u003Cem>c\u003C\u002Fem> for everyone — and\n      follows them without flinching to conclusions that overturn Newton's\n      separate space and time.\n  - fig: rel-lightcone\n    n: \"001\"\n    caption: |\n      The light cone through an event splits spacetime into causal future, past,\n      and an unreachable elsewhere; a worldline threads the interior.\n    large: true\n  - fig: rel-lightclock\n    n: \"002\"\n    caption: |\n      A moving light clock runs slow: the photon's diagonal path is longer, so\n      each tick takes longer.\n  - p: |\n      Space and time merge into a single four-dimensional\n      \u003Cstrong>spacetime\u003C\u002Fstrong>. The one quantity every observer agrees on is\n      the interval between events, and the light cone it defines fixes what can\n      cause what — sorting every event into an absolute future, past, or the\n      spacelike elsewhere no signal can reach.\n  - p: |\n      The bridge between frames is the \u003Cstrong>Lorentz transformation\u003C\u002Fstrong>.\n      It mixes space and time while leaving the speed of light invariant, and its\n      consequences — time dilation, length contraction, the relativity of\n      simultaneity — are geometry, not illusion.\n  - fig: rel-boost\n    n: \"003\"\n    caption: |\n      A Lorentz boost tilts the moving frame's space and time axes toward the\n      light line — which is why simultaneity is relative.\n  - p: |\n      Dynamics is rebuilt to match. Momentum and energy join into a single\n      four-vector, mass reveals itself as a form of energy through\n      \u003Cem>E = mc²\u003C\u002Fem>, and Maxwell's electromagnetism turns out to have been\n      relativistic all along.\n  - p: |\n      Then gravity. The \u003Cstrong>equivalence principle\u003C\u002Fstrong> — that free fall\n      is indistinguishable from floating in empty space — says gravity is not a\n      force but the curvature of spacetime, and matter is what does the curving.\n  - fig: rel-curvature\n    n: \"004\"\n    caption: |\n      Mass curves the spacetime grid, and a free body orbits by following the\n      straightest available path through it.\n  - p: |\n      The \u003Cstrong>Einstein field equation\u003C\u002Fstrong> ties that curvature to energy\n      and momentum. Its first exact solution, the Schwarzschild metric, describes\n      spacetime around a star and predicts the bending of light, the precession\n      of Mercury, and clocks that tick faster with altitude.\n  - fig: rel-gravwave\n    n: \"005\"\n    caption: |\n      A gravitational wave stretches and squeezes a ring of free test masses as\n      it passes.\n  - p: |\n      Push a mass small enough and an \u003Cstrong>event horizon\u003C\u002Fstrong> forms: a\n      black hole, a region from which not even light escapes. Ripples in the\n      curvature — \u003Cstrong>gravitational waves\u003C\u002Fstrong> — travel outward at\n      \u003Cem>c\u003C\u002Fem> and were first caught stretching a detector in 2015.\n  - p: |\n      Turn the same equations on the universe as a whole and they give an\n      expanding cosmos with a finite past. Relativity is the geometry underneath\n      all of it.\n---\n\nRelativity is what you get by taking one experimental fact — that the speed of\nlight is the same for every observer — completely seriously. The sequence starts\nwith the two postulates of special relativity and the Lorentz transformation they\nforce, then builds the geometry of Minkowski spacetime: the invariant interval,\nthe light cone, and the paradoxes of time dilation, length contraction, and\nsimultaneity that are really just features of that geometry. From there it\nrebuilds mechanics into relativistic dynamics, unifies mass with energy, and\nrecasts electromagnetism in manifestly covariant form. The second half turns to\ngravity: the equivalence principle, curved spacetime and the Einstein field\nequation, the Schwarzschild solution and its orbits, the classical tests, black\nholes and their horizons, gravitational waves, and finally the bridge to\ncosmology. It follows Tipler & Llewellyn for the foundations and Hartle, Carroll,\nand Schutz for the general theory, and each topic rests on the geometry laid down\nbefore it.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min read",0.785,47100,157,{"title":5,"description":57},[],"available","03.relativity\u002Findex",[],"4A1Epz_2pD1YkUfvy0bKo3Dn9akPQHPsN5NqXXMNwg8",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":76,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":77,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":78,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":79,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":80,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":81,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":82,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":83,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":84,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":85,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":86,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":87,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":88,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":89,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":90,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":91,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":92,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":93,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":94,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":95,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":96,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":97,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":98,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":99,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":100,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":101,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":102,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":103,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":104,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":105,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":106,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":107,"\u002Falgorithms\u002Fsequences\u002Ftries":108,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":109,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":110,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":111,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":112,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":113,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":114,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":115,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":116,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":117,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":118,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":119,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":120,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":121,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":122,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":123,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":124,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":125,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":126,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":127,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":128,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":129,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":130,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":131,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":132,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":133,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":134,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":135,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":136,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":137,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":138,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":139,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":140,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":141,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":142,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":143,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":144,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":145,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":146,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":147,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":148,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":149,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":150,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":151,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":152,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":153,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":154,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":155,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":156,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":157,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":158,"\u002Falgorithms":159,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":160,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":161,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":162,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":163,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":164,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":165,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":166,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":167,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":168,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":169,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":170,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":171,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":172,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":173,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":174,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":175,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":176,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":177,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":178,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":179,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":180,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":181,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":182,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":183,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":184,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":185,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":186,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":187,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":188,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":189,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":190,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":191,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":192,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":193,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":174,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":194,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":195,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":196,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":164,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":197,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":198,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":199,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":200,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":201,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":202,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":203,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":204,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":205,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":206,"\u002Fcalculus":207,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":208,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":209,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":210,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":211,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":212,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":213,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":214,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":215,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":216,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":217,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":218,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":219,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":220,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":221,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":222,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":223,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":224,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":225,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":226,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":227,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":228,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":229,"\u002Fmechanics\u002Frotation\u002Frolling-motion":230,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":231,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":232,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":233,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":234,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":235,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":236,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":237,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":238,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":239,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":240,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":241,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":242,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":243,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":244,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":245,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":246,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":247,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":248,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":249,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":250,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":251,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":252,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":253,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":254,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":255,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":256,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":257,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":258,"\u002Fmechanics":259,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":260,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":261,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":262,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":263,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":264,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":265,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":266,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":267,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":268,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":269,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":270,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":271,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":248,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":272,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":273,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":274,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":244,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":109,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":275,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":235,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":276,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":277,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":278,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":279,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":280,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":281,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":282,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":283,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":284,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":209,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":285,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":286,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":287,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":288,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":289,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":290,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":291,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":292,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":227,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":226,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":293,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":294,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":295,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":296,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":297,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":298,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":299,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":300,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":301,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":253,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":251,"\u002Felectricity-and-magnetism":302,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":303,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":304,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":305,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":306,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":307,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":308,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":309,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":310,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":311,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":161,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":312,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":313,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":165,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":314,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":315,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":316,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":317,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":318,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":319,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":320,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":321,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":322,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":323,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":324,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":325,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":326,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":327,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":328,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":329,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":330,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":331,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":332,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":333,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":200,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":334,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":335,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":336,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":337,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":338,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":339,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":340,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":341,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":342,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":343,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":344,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":345,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":346,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":347,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":348,"\u002Flinear-algebra":349,"\u002Ftheory-of-computation":350,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":351,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":352,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":353,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":354,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":355,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":356,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":357,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":358,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":359,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":360,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":361,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":362,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":363,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":364,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":365,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":366,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":367,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":368,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":369,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":370,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":371,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":372,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":373,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":374,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":375,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":376,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":377,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":378,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":379,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":380,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":381,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":382,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":383,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":384,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":385,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":386,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":387,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":388,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":389,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":390,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":391,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":392,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":393,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":394,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":395,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":396,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":397,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":398,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":399,"\u002Fcomputer-architecture":350,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":400,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":401,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":402,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":165,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":403,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":164,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":171,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":404,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":405,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":205,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":406,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":407,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":408,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":409,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":410,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":411,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":412,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":413,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":414,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":415,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":416,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":417,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":412,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":418,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":419,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":420,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":421,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":422,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":423,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":424,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":425,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":426,"\u002Fdifferential-equations":427,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":428,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":429,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":430,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":431,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":310,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":432,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":433,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":434,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":435,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":436,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":437,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":180,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":438,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":439,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":440,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":441,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":442,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":443,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":444,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":445,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":446,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":447,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":402,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":448,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":449,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":450,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":451,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":452,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":331,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":453,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":454,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":341,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":455,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":456,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":457,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":458,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":459,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":460,"\u002Frelativity":68,"\u002Fphysical-computing":350,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":461,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":441,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":462,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":463,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":464,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":465,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":466,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":467,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":468,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":417,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":341,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":469,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":470,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":471,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":472,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":468,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":473,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":449,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":202,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":474,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":475,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":182,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":476,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":477,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":478,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":479,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":480,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":481,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":482,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":483,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":484,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":441,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":485,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":486,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":487,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":474,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":163,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":488,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":489,"\u002Fquantum-mechanics":490,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":425,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":491,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":492,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":314,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":493,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":200,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":494,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":495,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":337,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":447,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":496,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":497,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":498,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":499,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":500,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":473,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":501,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":307,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":502,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":503,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":161,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":504,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":505,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":462,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":190,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":341,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":506,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":507,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":326,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":451,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":508,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":509,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":510,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":346,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":511,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":512,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":513,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":513,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":514,"\u002Freal-analysis":515,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":516,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":517,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":518,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":519,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":520,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":521,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":522,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":523,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":524,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":525,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":489,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":526,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":518,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":436,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":527,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":528,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":529,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":530,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":531,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":532,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":533,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":534,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":528,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":535,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":504,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":536,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":537,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":538,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":539,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":540,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":541,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":542,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":543,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":544,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":545,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":179,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":546,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":547,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":421,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":548,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":549,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":477,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":549,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":550,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":551,"\u002Fabstract-algebra":552,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":553,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":554,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":555,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":556,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":557,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":469,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":558,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":559,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":560,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":561,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":562,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":563,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":564,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":304,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":433,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":179,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":565,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":163,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":566,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":567,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":201,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":494,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":568,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":569,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":570,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":571,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":572,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":573,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":574,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":575,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":576,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":577,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":578,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":579,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":580,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":581,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":525,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":582,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":583,"\u002Fatomic-physics":584,"\u002Fdatabases":350,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":585,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":586,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":587,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":516,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":588,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":589,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":590,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":591,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":592,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":593,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":594,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":595,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":596,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":597,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":598,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":599,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":600,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":601,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":602,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":603,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":604,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":605,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":606,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":607,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":598,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":608,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":609,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":610,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":611,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":612,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":557,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":613,"\u002Fcategory-theory":614,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":615,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":616,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":617,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":618,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":619,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":620,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":579,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":621,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":622,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":623,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":624,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":625,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":626,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":627,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":628,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":629,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":630,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":631,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":632,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":633,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":634,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":635,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":636,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":637,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":638,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":639,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":640,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":641,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":642,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":643,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":644,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":645,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":646,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":647,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":591,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":648,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":649,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":650,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":651,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":652,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":653,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":654,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":144,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":655,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":656,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":380,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":657,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":658,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":659,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":660,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":661,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":662,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":663,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":664,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":665,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":666,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":667,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":668,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":353,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":669,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":670,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":671,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":672,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":390,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":673,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":674,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":675,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":676,"\u002Fdeep-learning":350,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":677,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":474,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":678,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":679,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":680,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":681,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":682,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":683,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":684,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":685,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":521,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":686,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":687,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":688,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":409,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":202,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":689,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":690,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":691,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":336,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":692,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":693,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":414,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":341,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":694,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":310,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":180,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":196,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":695,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":696,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":697,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":328,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":698,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":190,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":699,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":700,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":701,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":702,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":523,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":703,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":704,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":705,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":706,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":469,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":707,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":448,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":708,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":309,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":709,"\u002Fstatistical-mechanics":710,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":711,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":175,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":435,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":712,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":713,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":714,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":715,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":716,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":717,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":308,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":718,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":719,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":720,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":721,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":451,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":722,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":723,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":724,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":725,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":726,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":505,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":450,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":727,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":728,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":729,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":730,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":441,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":731,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":732,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":677,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":577,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":305,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":733,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":171,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":734,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":735,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":316,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":736,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":570,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":737,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":163,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":738,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":174,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":739,"\u002Fcondensed-matter":490,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":740,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":741,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":742,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":743,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":188,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":744,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":745,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":188,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":746,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":600,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":747,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":748,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":749,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":747,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":750,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":751,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":752,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":753,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":754,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":755,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":756,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":757,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":678,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":758,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":751,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":759,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":760,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":394,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":761,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":578,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":762,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":763,"\u002Flogic":764,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":765,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":766,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":380,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":767,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":768,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":769,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":770,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":760,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":771,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":772,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":773,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":671,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":774,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":775,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":776,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":777,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":778,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":397,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":779,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":780,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":781,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":782,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":587,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":783,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":759,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":784,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":785,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":786,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":408,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":787,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":537,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":788,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":789,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":370,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":790,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":791,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":792,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":793,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":794,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":647,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":795,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":796,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":797,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":682,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":798,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":799,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":800,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":397,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":463,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":801,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":802,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":803,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":804,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":805,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":806,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":807,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":808,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":809,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":810,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":811,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":812,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":813,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":814,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":815,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":816,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":817,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":818,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":819,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":820,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":821,"\u002Freinforcement-learning":350,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":822,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":823,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":824,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":825,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":826,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":827,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":828,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":829,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":830,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":831,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":832,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":833,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":834,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":676,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":531,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":835,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":836,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":837,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":838,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":839,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":840,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":658,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":841,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":842,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":843,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":844,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":845,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":846,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":847,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":848,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":849,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":850,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":851,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":852,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":590,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":842,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":853,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":131,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":854,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":855,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":856,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":857,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":858,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":639,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":859,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":860,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":861,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":862,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":863,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":864,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":865,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":866,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":867,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":868,"\u002Fartificial-intelligence":350,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":606,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":869,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":167,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":165,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":700,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":870,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":193,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":502,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":871,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":872,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":873,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":562,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":874,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":875,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":876,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":761,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":877,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":878,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":177,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":406,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":879,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":506,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":880,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":881,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":402,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":489,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":882,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":883,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":884,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":172,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":546,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":409,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":885,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":457,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":520,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":886,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":887,"\u002Fnuclear-physics":888,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":889,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":890,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":527,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":891,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":892,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":893,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":376,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":894,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":895,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":672,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":896,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":841,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":897,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":898,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":899,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":900,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":901,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":902,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":903,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":354,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":904,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":905,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":847,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":906,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":907,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":908,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":909,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":910,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":911,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":912,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":913,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":771,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":914,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":915,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":916,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":917,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":918,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":919,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":920,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":921,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":922,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":923,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":924,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":925,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":625,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":792,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":382,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":926,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":927,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":928,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":929,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":639,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":930,"\u002Fnatural-language-processing":350,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":931,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":547,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":932,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":692,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":933,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":934,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":882,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":935,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":936,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":497,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":200,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":937,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":452,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":938,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":484,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":939,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":738,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":940,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":941,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":707,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":942,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":171,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":943,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":944,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":945,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":488,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":704,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":946,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":947,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":563,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":948,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":608,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":949,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":950,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":497,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":530,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":951,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":952,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":953,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":586,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":954,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":328,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":955,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":431,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":956,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":957,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":754,"\u002Fparticle-physics":958,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":548,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":702,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":959,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":487,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":960,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":547,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":460,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":961,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":962,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":963,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":964,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":965,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":753,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":684,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":966,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":967,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":968,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":969,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":881,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":970,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":444,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":506,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":488,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":971,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":572,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":189,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":972,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":973,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":703,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":974,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":975,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":976,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":977,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":170,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":978,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":979,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":431,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":980,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":981,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":696,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":352,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":982,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":983,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":608,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":594,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":606,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":703,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":984,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":169,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":361,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":985,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":448,"\u002Fastrophysics-cosmology":552,"\u002Fcolophon":986,"\u002F":350},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,[988,1021,1045,1069,1093,1128,1147,1172,1192,1212],{"module":989,"moduleNumber":990,"slug":991,"lessons":992},"Foundations of Relativity",1,"foundations",[993,998,1003,1009,1015],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"The Postulates of Special Relativity","\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates",[989],"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",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"The Lorentz Transformation and Spacetime","\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime",[989],"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",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Time Dilation, Length Contraction, and Paradoxes","\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction",3,[989],"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",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Relativistic Momentum and Energy","\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy",4,[989],"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",{"title":1016,"path":1017,"lessonNumber":1018,"topics":1019,"summary":1020},"A Taste of General Relativity","\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity",5,[989],"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",{"module":1022,"moduleNumber":16,"slug":1023,"lessons":1024},"Spacetime and the Lorentz Group","spacetime-and-the-lorentz-group",[1025,1030,1035,1040],{"title":1026,"path":1027,"lessonNumber":990,"topics":1028,"summary":1029},"Minkowski Spacetime and the Interval","\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval",[1022],"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",{"title":1031,"path":1032,"lessonNumber":16,"topics":1033,"summary":1034},"Four-Vectors and Index Notation","\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation",[1022],"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",{"title":1036,"path":1037,"lessonNumber":1006,"topics":1038,"summary":1039},"The Lorentz Group and Rapidity","\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity",[1022],"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",{"title":1041,"path":1042,"lessonNumber":1012,"topics":1043,"summary":1044},"Doppler, Aberration, and Appearance","\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance",[1022],"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",{"module":1046,"moduleNumber":1006,"slug":1047,"lessons":1048},"Relativistic Dynamics","relativistic-dynamics",[1049,1054,1059,1064],{"title":1050,"path":1051,"lessonNumber":990,"topics":1052,"summary":1053},"Four-Momentum, Four-Force, and Accelerated Motion","\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion",[1046],"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",{"title":1055,"path":1056,"lessonNumber":16,"topics":1057,"summary":1058},"Particle Decays and Two-Body Kinematics","\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics",[1046],"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",{"title":1060,"path":1061,"lessonNumber":1006,"topics":1062,"summary":1063},"Relativistic Collisions and Threshold Energies","\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame",[1046],"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",{"title":1065,"path":1066,"lessonNumber":1012,"topics":1067,"summary":1068},"Mandelstam Variables and Lorentz Invariants","\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants",[1046],"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",{"module":1070,"moduleNumber":1012,"slug":1071,"lessons":1072},"Covariant Electromagnetism","covariant-electrodynamics",[1073,1078,1083,1088],{"title":1074,"path":1075,"lessonNumber":990,"topics":1076,"summary":1077},"The Four-Current and Four-Potential","\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential",[1070],"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",{"title":1079,"path":1080,"lessonNumber":16,"topics":1081,"summary":1082},"The Electromagnetic Field Tensor","\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor",[1070],"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",{"title":1084,"path":1085,"lessonNumber":1006,"topics":1086,"summary":1087},"How E and B Transform","\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields",[1070],"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",{"title":1089,"path":1090,"lessonNumber":1012,"topics":1091,"summary":1092},"Covariant Maxwell and the Stress–Energy Tensor","\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor",[1070],"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",{"module":1094,"moduleNumber":1018,"slug":1095,"lessons":1096},"Curved Spacetime","curved-spacetime",[1097,1102,1107,1112,1117,1122],{"title":1098,"path":1099,"lessonNumber":990,"topics":1100,"summary":1101},"The Equivalence Principle","\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized",[1094],"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",{"title":1103,"path":1104,"lessonNumber":16,"topics":1105,"summary":1106},"Manifolds, Vectors, and the Metric","\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric",[1094],"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",{"title":1108,"path":1109,"lessonNumber":1006,"topics":1110,"summary":1111},"Parallel Transport and the Covariant Derivative","\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols",[1094],"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",{"title":1113,"path":1114,"lessonNumber":1012,"topics":1115,"summary":1116},"Geodesics and the Newtonian Limit","\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation",[1094],"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",{"title":1118,"path":1119,"lessonNumber":1018,"topics":1120,"summary":1121},"Curvature and the Riemann Tensor","\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation",[1094],"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",{"title":1123,"path":1124,"lessonNumber":1125,"topics":1126,"summary":1127},"The Einstein Field Equations","\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations",6,[1094],"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",{"module":1129,"moduleNumber":1125,"slug":1130,"lessons":1131},"The Schwarzschild Solution","the-schwarzschild-solution",[1132,1137,1142],{"title":1133,"path":1134,"lessonNumber":990,"topics":1135,"summary":1136},"The Schwarzschild Metric","\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric",[1129],"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",{"title":1138,"path":1139,"lessonNumber":16,"topics":1140,"summary":1141},"Orbits in the Schwarzschild Geometry","\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild",[1129],"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",{"title":1143,"path":1144,"lessonNumber":1006,"topics":1145,"summary":1146},"Null Geodesics and the Photon Sphere","\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics",[1129],"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",{"module":1148,"moduleNumber":1149,"slug":1150,"lessons":1151},"Tests of General Relativity",7,"tests-of-general-relativity",[1152,1157,1162,1167],{"title":1153,"path":1154,"lessonNumber":990,"topics":1155,"summary":1156},"The Perihelion Precession of Mercury","\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury",[1148],"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",{"title":1158,"path":1159,"lessonNumber":16,"topics":1160,"summary":1161},"Light Deflection and Gravitational Lensing","\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing",[1148],"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",{"title":1163,"path":1164,"lessonNumber":1006,"topics":1165,"summary":1166},"Gravitational Redshift and the Shapiro Delay","\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay",[1148],"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",{"title":1168,"path":1169,"lessonNumber":1012,"topics":1170,"summary":1171},"Relativity and the Global Positioning System","\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps",[1148],"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",{"module":1173,"moduleNumber":1174,"slug":1175,"lessons":1176},"Black Holes",8,"black-holes",[1177,1182,1187],{"title":1178,"path":1179,"lessonNumber":990,"topics":1180,"summary":1181},"Horizons and Coordinate Singularities","\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities",[1173],"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",{"title":1183,"path":1184,"lessonNumber":16,"topics":1185,"summary":1186},"Rotating and Charged Black Holes","\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes",[1173],"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",{"title":1188,"path":1189,"lessonNumber":1006,"topics":1190,"summary":1191},"Black-Hole Thermodynamics","\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics",[1173],"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",{"module":1193,"moduleNumber":1194,"slug":1195,"lessons":1196},"Gravitational Waves",9,"gravitational-waves",[1197,1202,1207],{"title":1198,"path":1199,"lessonNumber":990,"topics":1200,"summary":1201},"Linearized Gravity and Wave Solutions","\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions",[1193],"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",{"title":1203,"path":1204,"lessonNumber":16,"topics":1205,"summary":1206},"The Quadrupole Formula","\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula",[1193],"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",{"title":1208,"path":1209,"lessonNumber":1006,"topics":1210,"summary":1211},"LIGO and the First Detections","\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events",[1193],"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",{"module":1213,"moduleNumber":1214,"slug":1215,"lessons":1216},"A Bridge to Cosmology",10,"cosmological-bridge",[1217,1222],{"title":1218,"path":1219,"lessonNumber":990,"topics":1220,"summary":1221},"The Cosmological Principle and the FLRW Metric","\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric",[1213],"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",{"title":1223,"path":1224,"lessonNumber":16,"topics":1225,"summary":1226},"The Friedmann Equations and Cosmic Dynamics","\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics",[1213],"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",1786059477561]