Free Encyclopedia of Mathematics 0.0.1by the PlanetMath authors Aatu, ack, akrowne, alek thiery, alinabi, almann,alozano, antizeus, antonio, aparna, ariels, armbrusterb, AxelBoldt, basseykay,bbukh, benjaminfjones, bhaire, brianbirgen, bs, bshanks, bwebste, cryo, danielm,Daume, debosberg, deiudi, digitalis, djao, Dr Absentius, draisma, drini, drum-mond, dublisk, Evandar, bonaci, ynnheiss, gabor sz, GaloisRadical, gantsich,gaurminirick, gholmes74, giri, greg, grouprly, gumau, Gunnar, Henry, iddo, igor,imran, jamika chris, jarino, jay, jgade, jihemme, Johan, karteef, karthik, kemy-ers3, Kevin OBryant, kidburla2003, KimJ, Koro, lha, lieven, livetoad, liyang, Lo-gan, Luci, m759, mathcam, mathwizard, matte, mclase, mhale, mike, mikestao-gan, mps, msihl, muqabala, n3o, nerdy2, nobody, npolys, Oblomov, ottocolori,paolini, patrickwonders, pbruin, petervr, PhysBrain, quadrate, quincynoodles,ratboy, RevBobo, Riemann, rmilson, ruiyang, Sabean, saforres, saki, say 10,scanez, scineram, seonyoung, slash, sleske, slider142, sprocketboy, sucrose, super-higgs, tensorking, thedagit, Thomas Heye, thouis, Timmy, tobix, tromp, tz26, un-lord, uriw, urz, vampyr, vernondalhart, vitriol, vladm, volator, vypertd, wberry,Wkbj79, wombat, x bas, xiaoyanggu, XJamRastare, xriso, yark et al.edited by Joe Corneli & Aaron KrowneCopyright c _ 2004 PlanetMath.org authors. Permission is granted to copy, dis-tribute and/or modify this document under the terms of the GNU Free Documen-tation License, Version 1.2 or any later version published by the Free SoftwareFoundation; with no Invariant Sections, with no Front-Cover Texts, and with noBack-Cover Texts. A copy of the license is included in the section entitled GNUFree Documentation License.IntroductionWelcome to the PlanetMath One Big Book compilation, the Free Encyclopedia of Math-ematics. This book gathers in a single document the best of the hundreds of authors andthousands of other contributors from the PlanetMath.org web site, as of January 4, 2004.The purpose of this compilation is to help the eorts of these people reach a wider audienceand allow the benets of their work to be accessed in a greater breadth of situations.We want to emphasize is that the Free Encyclopedia of Mathematics will always be a workin progress. Producing a book-format encycopedia from the amorphous web of interlinkedand multidimensionally-organized entries on PlanetMath is not easy. The print mediumdemands a linear presentation, and to boil the web site down into this format is a dicult,and in some ways lossy, transformation. A major part of our editorial eorts are going intomaking this transformation. We hope the organization weve chosen for now is useful toreaders, and in future editions you can expect continuing improvements.The linearization of PlanetMath.org is not the only editorial task we must perform.Throughout the millenia, readers have come to expect a strict standard of consistency andcorrectness from print books, and we must strive to meet this standard in the PlanetMathBook as closely as possible. This means applying more editorial control to the book formof PlanetMath than is applied to the web site. We hope you will agree that there is signi-cant value to be gained from unifying style, correcting errors, and ltering out not-yet-readycontent, so we will continue to do these things.For more details on planned improvements to this book, see the TODO le that came withthis archive. Remember that you can help us to improve this work by joining PlanetMath.organd ling corrections, adding entries, or just participating in the community. We are alsolooking for volunteers to help edit this book, or help with programming related to its pro-duction, or to help work on Noosphere, the PlanetMath software. To send us commentsabout the book, use the e-mail address pmbook@planetmath.org. For general commentsand queries, use feedback@planetmath.org.Happy mathing,Joe CorneliAaron KrowneTuesday, January 27, 2004iTop-level Math SubjectClassiciations00 General01 History and biography03 Mathematical logic and foundations05 Combinatorics06 Order, lattices, ordered algebraic structures08 General algebraic systems11 Number theory12 Field theory and polynomials13 Commutative rings and algebras14 Algebraic geometry15 Linear and multilinear algebra; matrix theory16 Associative rings and algebras17 Nonassociative rings and algebras18 Category theory; homological algebra19 $K$-theory20 Group theory and generalizations22 Topological groups, Lie groups26 Real functions28 Measure and integration30 Functions of a complex variable31 Potential theory32 Several complex variables and analytic spaces33 Special functions34 Ordinary differential equations35 Partial differential equations37 Dynamical systems and ergodic theory39 Difference and functional equations40 Sequences, series, summability41 Approximations and expansions42 Fourier analysis43 Abstract harmonic analysis44 Integral transforms, operational calculusii45 Integral equations46 Functional analysis47 Operator theory49 Calculus of variations and optimal control; optimization51 Geometry52 Convex and discrete geometry53 Differential geometry54 General topology55 Algebraic topology57 Manifolds and cell complexes58 Global analysis, analysis on manifolds60 Probability theory and stochastic processes62 Statistics65 Numerical analysis68 Computer science70 Mechanics of particles and systems74 Mechanics of deformable solids76 Fluid mechanics78 Optics, electromagnetic theory80 Classical thermodynamics, heat transfer81 Quantum theory82 Statistical mechanics, structure of matter83 Relativity and gravitational theory85 Astronomy and astrophysics86 Geophysics90 Operations research, mathematical programming91 Game theory, economics, social and behavioral sciences92 Biology and other natural sciences93 Systems theory; control94 Information and communication, circuits97 Mathematics educationiiiTable of ContentsIntroduction iTop-level Math Subject Classiciations iiTable of Contents ivGNU Free Documentation License liiUNCLA Unclassied 1Golomb ruler 1Hesse conguration 1Jordans Inequality 2Lagranges theorem 2Laurent series 3Lebesgue measure 3Leray spectral sequence 4Mobius transformation 4Mordell-Weil theorem 4Plateaus Problem 5Poisson random variable 5Shannons theorem 6Shapiro inequality 9Sylow p-subgroups 9Tchirnhaus transformations 9Wallis formulae 10ascending chain condition 10bounded 10bounded operator 11complex projective line 12converges uniformly 12descending chain condition 13diamond theorem 13equivalently oriented bases 13nitely generated R-module 14fraction 14group of covering transformations 15idempotent 15isolated 17isolated singularity 17isomorphic groups 17joint continuous density function 18joint cumulative distribution function 18joint discrete density function 19left function notation 20lift of a submanifold 20limit of a real function exits at a point 20lipschitz function 21lognormal random variable 21lowest upper bound 22marginal distribution 22measurable space 23measure zero 23minimum spanning tree 23minimum weighted path length 24mod 2 intersection number 25moment generating function 27monoid 27monotonic operator 27multidimensional Gaussian integral 28multiindex 29near operators 30negative binomial random variable 36normal random variable 37normalizer of a subset of a group 38nth root 38null tree 40open ball 40opposite ring 40orbit-stabilizer theorem 41orthogonal 41permutation group on a set 41prime element 42product measure 43projective line 43projective plane 43proof of calculus theorem used in the Lagrangemethod 44proof of orbit-stabilizer theorem 45proof of power rule 45proof of primitive element theorem 47proof of product rule 47proof of sum rule 48proof that countable unions are countable 48quadrature 48quotient module 49regular expression 49regular language 50right function notation 51ring homomorphism 51scalar 51schrodinger operator 51ivselection sort 52semiring 53simple function 54simple path 54solutions of an equation 54spanning tree 54square root 55stable sorting algorithm 56standard deviation 56stochastic independence 56substring 57successor 57sum rule 58superset 58symmetric polynomial 59the argument principle 59torsion-free module 59total order 60tree traversals 60trie 63unit vector 64unstable xed point 65weak* convergence in normed linear space 65well-ordering principle for natural numbers 6500-01 Instructional exposition (textbooks,tutorial papers, etc.) 66dimension 66toy theorem 6700-XX General 68method of exhaustion 6800A05 General mathematics 69Conways chained arrow notation 69Knuths up arrow notation 70arithmetic progression 70arity 71introducing 0th power 71lemma 71property 72saddle point approximation 72singleton 73subsequence 73surreal number 7300A07 Problem books 76Nesbitts inequality 76proof of Nesbitts inequality 7600A20 Dictionaries and other generalreference works 78completing the square 7800A99 Miscellaneous topics 80QED 80TFAE 80WLOG 81order of operations 8101A20 Greek, Roman 84Roman numerals 8401A55 19th century 85Poincar, Jules Henri 8501A60 20th century 90Bourbaki, Nicolas 90Erds Number 9703-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 98Burali-Forti paradox 98Cantors paradox 98Russells paradox 99biconditional 99bijection 100cartesian product 100chain 100characteristic function 101concentric circles 101conjunction 102disjoint 102empty set 102even number 103xed point 103innite 103injective function 104integer 104inverse function 105linearly ordered 106operator 106ordered pair 106ordering relation 106partition 107pullback 107set closed under an operation 108signature of a permutation 109subset 109surjective 110vtransposition 110truth table 11103-XX Mathematical logic and founda-tions 112standard enumeration 11203B05 Classical propositional logic 113CNF 113Proof that contrapositive statement is true usinglogical equivalence 113contrapositive 114disjunction 114equivalent 114implication 115propositional logic 115theory 116transitive 116truth function 11703B10 Classical rst-order logic 1181 bootstrapping 118Boolean 119Godel numbering 120Godels incompleteness theorems 120Lindenbaum algebra 127Lindstroms theorem 128Pressburger arithmetic 129R-minimal element 129Skolemization 129arithmetical hierarchy 129arithmetical hierarchy is a proper hierarchy 130atomic formula 131creating an innite model 131criterion for consistency of sets of formulas 132deductions are 1 132example of Godel numbering 134example of well-founded induction 135rst order language 136rst order logic 137rst order theories 138free and bound variables 138generalized quantier 139logic 140proof of compactness theorem for rst order logic141proof of principle of transnite induction 141proof of the well-founded induction principle 141quantier 141quantier free 144subformula 144syntactic compactness theorem for rst order logic144transnite induction 144universal relation 145universal relations exist for each level of the arith-metical hierarchy 145well-founded induction 146well-founded induction on formulas 14703B15 Higher-order logic and type the-ory 143Hartigs quantier 143Russells theory of types 143analytic hierarchy 145game-theoretical quantier 146logical language 147second order logic 14803B40 Combinatory logic and lambda-calculus 150Church integer 150combinatory logic 150lambda calculus 15103B48 Probability and inductive logic154conditional probability 15403B99 Miscellaneous 155Beth property 155Hofstadters MIU system 155IF-logic 157Tarskis result on the undenability of Truth 160axiom 161compactness 164consistent 164interpolation property 164sentence 16503Bxx General logic 166Banach-Tarski paradox 16603C05 Equational classes, universal al-gebra 168congruence 168every congruence is the kernel of a homomor-phism 168homomorphic image of a -structure is a -structurevi169kernel 169kernel of a homomorphism is a congruence 169quotient structure 17003C07 Basic properties of rst-order lan-guages and structures 171Models constructed from constants 171Stone space 172alphabet 173axiomatizable theory 174denable 174denable type 175downward Lowenheim-Skolem theorem 176example of denable type 176example of strongly minimal 177rst isomorphism theorem 177language 178length of a string 179proof of homomorphic image of a -structure isa -structure 179satisfaction relation 180signature 181strongly minimal 181structure preserving mappings 181structures 182substructure 183type 183upward Lowenheim-Skolem theorem 18303C15 Denumerable structures 185random graph (innite) 18503C35 Categoricity and completeness oftheories 187-categorical 187Vaughts test 187proof of Vaughts test 18703C50 Models with special properties(saturated, rigid, etc.) 189example of universal structure 189homogeneous 191universal structure 19103C52 Properties of classes of models192amalgamation property 19203C64 Model theory of ordered struc-tures; o-minimality 193innitesimal 193o-minimality 194real closed elds 19403C68 Other classical rst-order modeltheory 196imaginaries 19603C90 Nonclassical models (Boolean-valued,sheaf, etc.) 198Boolean valued model 19803C99 Miscellaneous 199axiom of foundation 199elementarily equivalent 199elementary embedding 200model 200proof equivalence of formulation of foundation20103D10 Turing machines and related no-tions 203Turing machine 20303D20 Recursive functions and relations,subrecursive hierarchies 206primitive recursive 20603D25 Recursively (computably) enu-merable sets and degrees 207recursively enumerable 20703D75 Abstract and axiomatic computabil-ity and recursion theory 208Ackermann function 208halting problem 20903E04 Ordered sets and their conali-ties; pcf theory 211another denition of conality 211conality 211maximal element 212partitions less than conality 213well ordered set 213pigeonhole principle 213proof of pigeonhole principle 213tree (set theoretic) 214-complete 215Cantors diagonal argument 215Fodors lemma 216Schroeder-Bernstein theorem 216Veblen function 216additively indecomposable, 217viicardinal number 217cardinal successor 217cardinality 218cardinality of a countable union 218cardinality of the rationals 219classes of ordinals and enumerating functions 219club 219club lter 220countable 220countably innite 221nite 221xed points of normal functions 221height of an algebraic number 221if A is innite and B is a nite subset of A, thenA B is innite 222limit cardinal 222natural number 223ordinal arithmetic 224ordinal number 225power set 225proof of Fodors lemma 225proof of Schroeder-Bernstein theorem 225proof of xed points of normal functions 226proof of the existence of transcendental numbers226proof of theorems in aditively indecomposable227proof that the rationals are countable 228stationary set 228successor cardinal 229uncountable 229von Neumann integer 229von Neumann ordinal 230weakly compact cardinal 231weakly compact cardinals and the tree property231Cantors theorem 232proof of Cantors theorem 232additive 232antisymmetric 233constant function 233direct image 234domain 234dynkin system 234equivalence class 235bre 235ltration 236nite character 236x (transformation actions) 236function 237functional 237generalized cartesian product 238graph 238identity map 238inclusion mapping 239inductive set 239invariant 240inverse function theorem 240inverse image 241mapping 242mapping of period n is a bijection 242partial function 242partial mapping 243period of mapping 243pi-system 244proof of inverse function theorem 244proper subset 246range 246reexive 246relation 246restriction of a mapping 247set dierence 247symmetric 247symmetric dierence 248the inverse image commutes with set operations248transformation 249transitive 250transitive 250transitive closure 250Hausdors maximum principle 250Kuratowskis lemma 251Tukeys lemma 251Zermelos postulate 251Zermelos well-ordering theorem 251Zorns lemma 252axiom of choice 252equivalence of Hausdors maximum principle,Zorns lemma and the well-ordering theorem 252equivalence of Zorns lemma and the axiom ofviiichoice 253maximality principle 254principle of nite induction 254principle of nite induction proven from well-ordering principle 255proof of Tukeys lemma 255proof of Zermelos well-ordering theorem 255axiom of extensionality 256axiom of innity 256axiom of pairing 257axiom of power set 258axiom of union 258axiom schema of separation 259de Morgans laws 260de Morgans laws for sets (proof) 261set theory 261union 264universe 264von Neumann-Bernays-Gdel set theory 265FS iterated forcing preserves chain condition 267chain condition 268composition of forcing notions 268composition preserves chain condition 268equivalence of forcing notions 269forcing relation 270forcings are equivalent if one is dense in the other270iterated forcing 272iterated forcing and composition 273name 273partial order with chain condition does not col-lapse cardinals 274proof of partial order with chain condition doesnot collapse cardinals 274proof that forcing notions are equivalent to theircomposition 275complete partial orders do not add small subsets280proof of complete partial orders do not add smallsubsets 280Q is equivalent to and continuum hypothesis281Levy collapse 281proof of Q is equivalent to and continuum hy-pothesis 282Martins axiom 283Martins axiom and the continuum hypothesis283Martins axiom is consistent 284a shorter proof: Martins axiom and the contin-uum hypothesis 287continuum hypothesis 288forcing 288generalized continuum hypothesis 289inaccessible cardinals 290Q 290 290Dedekind innite 291Zermelo-Fraenkel axioms 291class 291complement 293delta system 293delta system lemma 293diagonal intersection 293intersection 294multiset 294proof of delta system lemma 294rational number 295saturated (set) 295separation and doubletons axiom 295set 29603Exx Set theory 299intersection 29903F03 Proof theory, general 300NJp 300NKp 300natural deduction 301sequent 301sound,, complete 30203F07 Structure of proofs 303induction 30303F30 First-order arithmetic and frag-ments 307Elementary Functional Arithmetic 307PA 308Peano arithmetic 30803F35 Second- and higher-order arith-metic and fragments 310ACA0 310ixRCA0 310Z2 310comprehension axiom 311induction axiom 31103G05 Boolean algebras 313Boolean algebra 313M. H. Stones representation theorem 31303G10 Lattices and related structures314Boolean lattice 314complete lattice 314lattice 31503G99 Miscellaneous 316Chu space 316Chu transform 316biextensional collapse 317example of Chu space 317property of a Chu space 31805-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 319example of pigeonhole principle 319multi-index derivative of a power 319multi-index notation 32005A10 Factorials, binomial coecients,combinatorial functions 322Catalan numbers 322Levi-Civita permutation symbol 323Pascals rule (bit string proof) 325Pascals rule proof 326Pascals triangle 326Upper and lower bounds to binomial coecient328binomial coecient 328double factorial 329factorial 329falling factorial 330inductive proof of binomial theorem 331multinomial theorem 332multinomial theorem (proof) 333proof of upper and lower bounds to binomial co-ecient 33405A15 Exact enumeration problems, gen-erating functions 336Stirling numbers of the rst kind 336Stirling numbers of the second kind 33805A19 Combinatorial identities 342Pascals rule 34205A99 Miscellaneous 343principle of inclusion-exclusion 343principle of inclusion-exclusion proof 34405B15 Orthogonal arrays, Latin squares,Room squares 346example of Latin squares 346graeco-latin squares 346latin square 347magic square 34705B35 Matroids, geometric lattices 348matroid 348polymatroid 35305C05 Trees 354AVL tree 354Aronszajn tree 354Suslin tree 354antichain 355balanced tree 355binary tree 355branch 356child node (of a tree) 356complete binary tree 357digital search tree 357digital tree 358example of Aronszajn tree 358example of tree (set theoretic) 359extended binary tree 359external path length 360internal node (of a tree) 360leaf node (of a tree) 361parent node (in a tree) 361proof that has the tree property 362root (of a tree) 362tree 363weight-balanced binary trees are ultrametric 364weighted path length 36605C10 Topological graph theory, imbed-ding 367Heawood number 367Kuratowskis theorem 368Szemeredi-Trotter theorem 368crossing lemma 369crossing number 369xgraph topology 369planar graph 370proof of crossing lemma 37005C12 Distance in graphs 372Hamming distance 37205C15 Coloring of graphs and hyper-graphs 373bipartite graph 373chromatic number 374chromatic number and girth 375chromatic polynomial 375colouring problem 376complete bipartite graph 377complete k-partite graph 378four-color conjecture 378k-partite graph 379property B 38005C20 Directed graphs (digraphs), tour-naments 381cut 381de Bruijn digraph 381directed graph 382ow 383maximum ow/minimum cut theorem 384tournament 38505C25 Graphs and groups 387Cayley graph 38705C38 Paths and cycles 388Euler path 388Veblens theorem 388acyclic graph 389bridges of Knigsberg 389cycle 390girth 391path 391proof of Veblens theorem 39205C40 Connectivity 393k-connected graph 393Thomassens theorem on 3-connected graphs 393Tuttes wheel theorem 394connected graph 394cutvertex 39505C45 Eulerian and Hamiltonian graphs396Bondy and Chvtal theorem 396Dirac theorem 396Euler circuit 397Fleurys algorithm 397Hamiltonian cycle 398Hamiltonian graph 398Hamiltonian path 398Ores theorem 398Petersen graph 399hypohamiltonian 399traceable 39905C60 Isomorphism problems (reconstruc-tion conjecture, etc.) 400graph isomorphism 40005C65 Hypergraphs 402Steiner system 402nite plane 402hypergraph 403linear space 40405C69 Dominating sets, independent sets,cliques 405Mantels theorem 405clique 405proof of Mantels theorem 40505C70 Factorization, matching, coveringand packing 407Petersen theorem 407Tutte theorem 407bipartite matching 407edge covering 409matching 409maximal bipartite matching algorithm 410maximal matching/minimal edge covering theo-rem 41105C75 Structural characterization of typesof graphs 413multigraph 413pseudograph 41305C80 Random graphs 414examples of probabilistic proofs 414probabilistic method 41505C90 Applications 417Hasse diagram 41705C99 Miscellaneous 419Eulers polyhedron theorem 419Poincare formula 419xiTurans theorem 419Wagners theorem 420block 420bridge 420complete graph 420degree (of a vertex) 421distance (in a graph) 421edge-contraction 421graph 422graph minor theorem 422graph theory 423homeomorphism 424loop 424minor (of a graph) 424neighborhood (of a vertex) 425null graph 425order (of a graph) 425proof of Eulers polyhedron theorem 426proof of Turans theorem 427realization 427size (of a graph) 428subdivision 428subgraph 429wheel graph 42905D05 Extremal set theory 431LYM inequality 431Sperners theorem 43205D10 Ramsey theory 433Erdos-Rado theorem 433Ramseys theorem 433Ramseys theorem 434arrows 435coloring 436proof of Ramseys theorem 43705D15 Transversal (matching) theory 438Halls marriage theorem 438proof of Halls marriage theorem 438saturate 440system of distinct representatives 44005E05 Symmetric functions 441elementary symmetric polynomial 441reduction algorithm for symmetric polynomials44106-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 443equivalence relation 44306-XX Order, lattices, ordered algebraicstructures 445join 445meet 44506A06 Partial order, general 446directed set 446inmum 446sets that do not have an inmum 447supremum 447upper bound 44806A99 Miscellaneous 449dense (in a poset) 449partial order 449poset 450quasi-order 450well quasi ordering 45006B10 Ideals, congruence relations 452order in an algebra 45206C05 Modular lattices, Desarguesianlattices 453modular lattice 45306D99 Miscellaneous 454distributive 454distributive lattice 45406E99 Miscellaneous 455Boolean ring 45508A40 Operations, polynomials, primalalgebras 456coecients of a polynomial 45608A99 Miscellaneous 457binary operation 457ltered algebra 45711-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 459Euler phi-function 459Euler-Fermat theorem 460Fermats little theorem 460Fermats theorem proof 460Goldbachs conjecture 460Jordans totient function 461Legendre symbol 461Pythagorean triplet 462Wilsons theorem 462arithmetic mean 462xiiceiling 463computation of powers using Fermats little the-orem 463congruences 464coprime 464cube root 464oor 465geometric mean 465googol 466googolplex 467greatest common divisor 467group theoretic proof of Wilsons theorem 467harmonic mean 467mean 468number eld 468pi 468proof of Wilsons theorem 470proof of fundamental theorem of arithmetic 471root of unity 47111-01 Instructional exposition (textbooks,tutorial papers, etc.) 472base 47211-XX Number theory 474Lehmers Conjecture 474Sierpinski conjecture 474prime triples conjecture 47511A05 Multiplicative structure; Euclideanalgorithm; greatest common divisors 476Bezouts lemma (number theory) 476Euclids algorithm 476Euclids lemma 478Euclids lemma proof 478fundamental theorem of arithmetic 479perfect number 479smooth number 48011A07 Congruences; primitive roots; residuesystems 481Antons congruence 481Fermats Little Theorem proof (Inductive) 482Jacobi symbol 483Shanks-Tonelli algorithm 483Wieferich prime 483Wilsons theorem for prime powers 484factorial module prime powers 485proof of Euler-Fermat theorem 485proof of Lucass theorem 48611A15 Power residues, reciprocity 487Eulers criterion 487Gauss lemma 487Zolotarevs lemma 489cubic reciprocity law 491proof of Eulers criterion 493proof of quadratic reciprocity rule 494quadratic character of 2 495quadratic reciprocity for polynomials 496quadratic reciprocity rule 497quadratic residue 49711A25 Arithmetic functions; related num-bers; inversion formulas 498Dirichlet character 498Liouville function 498Mangoldt function 499Mertens rst theorem 499Moebius function 499Moebius in version 500arithmetic function 502multiplicative function 503non-multiplicative function 505totient 507unit 50711A41 Primes 508Chebyshev functions 508Euclids proof of the innitude of primes 509Mangoldt summatory function 509Mersenne numbers 510Thues lemma 510composite number 511prime 511prime counting function 511prime dierence function 512prime number theorem 512prime number theorem result 513proof of Thues Lemma 514semiprime 515sieve of Eratosthenes 516test for primality of Mersenne numbers 51611A51 Factorization; primality 517Fermat Numbers 517Fermat compositeness test 517Zsigmondys theorem 518xiiidivisibility 518division algorithm for integers 519proof of division algorithm for integers 519square-free number 520squarefull number 520the prime power dividing a factorial 52111A55 Continued fractions 523Stern-Brocot tree 523continued fraction 52411A63 Radix representation; digital prob-lems 527Kummers theorem 527corollary of Kummers theorem 52811A67 Other representations 529Sierpinski Erdos egyptian fraction conjecture 529adjacent fraction 529any rational number is a sum of unit fractions530conjecture on fractions with odd denominators532unit fraction 53211A99 Miscellaneous 533ABC conjecture 533Suranyi theorem 533irrational to an irrational power can be rational534triangular numbers 53411B05 Density, gaps, topology 536Cauchy-Davenport theorem 536Manns theorem 536Schnirelmann density 537Sidon set 537asymptotic density 538discrete space 538essential component 539normal order 53911B13 Additive bases 541Erdos-Turan conjecture 541additive basis 542asymptotic basis 542base con version 542sumset 54611B25 Arithmetic progressions 547Behrends construction 547Freimans theorem 548Szemeredis theorem 548multidimensional arithmetic progression 54911B34 Representation functions 550Erdos-Fuchs theorem 55011B37 Recurrences 551Collatz problem 551recurrence relation 55111B39 Fibonacci and Lucas numbers andpolynomials and generalizations 553Fibonacci sequence 553Hogatts theorem 554Lucas numbers 554golden ratio 55411B50 Sequences (mod m) 556Erdos-Ginzburg-Ziv theorem 55611B57 Farey sequences; the sequences ?557Farey sequence 55711B65 Binomial coecients; factorials;q-identities 559Lucass Theorem 559binomial theorem 55911B68 Bernoulli and Euler numbers andpolynomials 561Bernoulli number 561Bernoulli periodic function 561Bernoulli polynomial 562generalized Bernoulli number 56211B75 Other combinatorial number the-ory 563Erdos-Heilbronn conjecture 563Freiman isomorphism 563sum-free 56411B83 Special sequences and polynomi-als 565Beatty sequence 565Beattys theorem 566Fraenkels partition theorem 566Sierpinski numbers 567palindrome 567proof of Beattys theorem 568square-free sequence 569superincreasing sequence 56911B99 Miscellaneous 570Lychrel number 570xivclosed form 57111C08 Polynomials 573content of a polynomial 573cyclotomic polynomial 573height of a polynomial 574length of a polynomial 574proof of Eisenstein criterion 574proof that the cyclotomic polynomial is irreducible57511D09 Quadratic and bilinear equations577Pells equation and simple continued fractions57711D41 Higher degree equations; Fermatsequation 578Beal conjecture 578Euler quartic conjecture 579Fermats last theorem 58011D79 Congruences in many variables582Chinese remainder theorem 582Chinese remainder theorem proof 58311D85 Representation problems 586polygonal number 58611D99 Miscellaneous 588Diophantine equation 58811E39 Bilinear and Hermitian forms 590Hermitian form 590non-degenerate bilinear form 590positive denite form 591symmetric bilinear form 591Cliord algebra 59111Exx Forms and linear algebraic groups593quadratic function associated with a linear func-tional 59311F06 Structure of modular groups andgeneralizations; arithmetic groups 594Taniyama-Shimura theorem 59411F30 Fourier coecients of automor-phic forms 597Fourier coecients 59711F67 Special values of automorphic L-series, periods of modular forms, cohomol-ogy, modular symbols 598Schanuels conjecutre 598period 59811G05 Elliptic curves over global elds600complex multiplication 60011H06 Lattices and convex bodies 602Minkowskis theorem 602lattice in Rn60211H46 Products of linear forms 604triple scalar product 60411J04 Homogeneous approximation toone number 605Dirichlets approximation theorem 60511J68 Approximation to algebraic num-bers 606Davenport-Schmidt theorem 606Liouville approximation theorem 606proof of Liouville approximation theorem 60711J72 Irrationality; linear independenceover a eld 609nth root of 2 is irrational for n 3 (proof usingFermats last theorem) 609e is irrational (proof) 610irrational 610square root of 2 is irrational 61111J81 Transcendence (general theory)612Fundamental Theorem of Transcendence 612Gelfonds theorem 612four exponentials conjecture 612six exponentials theorem 613transcendental number 61411K16 Normal numbers, radix expan-sions, etc. 615absolutely normal 61511K45 Pseudo-random numbers; MonteCarlo methods 617pseudorandom numbers 617quasirandom numbers 618random numbers 619truly random numbers 61911L03 Trigonometric and exponential sums,general 620Ramanujan sum 62011L05 Gauss and Kloosterman sums; gen-xveralizations 622Gauss sum 622Kloosterman sum 623Landsberg-Schaar relation 623derivation of Gauss sum up to a sign 62411L40 Estimates on character sums 625Plya-Vinogradov inequality 62511M06 (s) and L(s, ) 627Aperys constant 627Dedekind zeta function 627Dirichlet L-series 628Riemann -function 629Riemann Xi function 630Riemann omega function 630functional equation for the Riemann Xi function630functional equation for the Riemann theta func-tion 631generalized Riemann hypothesis 631proof of functional equation for the Riemann thetafunction 63111M99 Miscellaneous 633Riemann zeta function 633formulae for zeta in the critical strip 636functional equation of the Riemann zeta function638value of the Riemann zeta function at s = 2 63811N05 Distribution of primes 640Bertrands conjecture 640Bruns constant 640proof of Bertrands conjecture 640twin prime conjecture 64211N13 Primes in progressions 643primes in progressions 64811N32 Primes represented by polynomi-als; other multiplicative structure of poly-nomial values 644Euler four-square identity 64411N56 Rate of growth of arithmetic func-tions 645highly composite number 64511N99 Miscellaneous 646Chinese remainder theorem 646proof of chinese remainder theorem 64611P05 Warings problem and variants648Lagranges four-square theorem 648Warings problem 648proof of Lagranges four-square theorem 64911P81 Elementary theory of partitions651pentagonal number theorem 65111R04 Algebraic numbers; rings of alge-braic integers 653Dedekind domain 653Dirichlets unit theorem 653Eisenstein integers 654Galois representation 654Gaussian integer 658algebraic conjugates 659algebraic integer 659algebraic number 659algebraic number eld 659calculating the splitting of primes 660characterization in terms of prime ideals 661ideal classes form an abelian group 661integral basis 661integrally closed 662transcendental root theorem 66211R06 PV-numbers and generalizations;other special algebraic numbers 663Salem number 66311R11 Quadratic extensions 664prime ideal decomposition in quadratic exten-sions of 66411R18 Cyclotomic extensions 666Kronecker-Weber theorem 666examples of regular primes 667prime ideal decomposition in cyclotomic exten-sions of 668regular prime 66911R27 Units and factorization 670regulator 67011R29 Class numbers, class groups, dis-criminants 672Existence of Hilbert Class Field 672class number formula 673discriminant 673ideal class 674xviray class group 67511R32 Galois theory 676Galois criterion for solvability of a polynomial byradicals 67611R34 Galois cohomology 677Hilbert Theorem 90 67711R37 Class eld theory 678Artin map 678Tchebotarev density theorem 679modulus 679multiplicative congruence 680ray class eld 68011R56 Ad`ele rings and groups 682adle 682idle 682restricted direct product 68311R99 Miscellaneous 684Henselian eld 684valuation 68511S15 Ramication and extension the-ory 686decomposition group 686examples of prime ideal decomposition in num-ber elds 688inertial degree 691ramication index 692unramied action 69711S31 Class eld theory; p-adic formalgroups 699Hilbert symbol 69911S99 Miscellaneous 700p-adic integers 700local eld 70111Y05 Factorization 703Pollards rho method 703quadratic sieve 70611Y55 Calculation of integer sequences709Kolakoski sequence 70911Z05 Miscellaneous applications of num-ber theory 711 function 711arithmetic derivative 711example of arithmetic derivative 712proof that (n) is the number of positive divisorsof n 71212-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 714monomial 714order and degree of polynomial 71512-XX Field theory and polynomials 716homogeneous polynomial 716subeld 71612D05 Polynomials: factorization 717factor theorem 717proof of factor theorem 717proof of rational root theorem 718rational root theorem 719sextic equation 71912D10 Polynomials: location of zeros(algebraic theorems) 720Cardanos derivation of the cubic formula 720Ferrari-Cardano derivation of the quartic formula721Galois-theoretic derivation of the cubic formula722Galois-theoretic derivation of the quartic formula724cubic formula 728derivation of quadratic formula 728quadratic formula 729quartic formula 730reciprocal polynomial 730root 731variant of Cardanos derivation 73212D99 Miscellaneous 733Archimedean property 733complex 734complex conjugate 735complex number 737examples of totally real elds 738fundamental theorem of algebra 739imaginary 739imaginary unit 739indeterminate form 739inequalities for real numbers 740interval 742modulus of complex number 743proof of fundamental theorem of algebra 744proof of the fundamental theorem of algebra 744xviireal and complex embeddings 744real number 746totally real and imaginary elds 74712E05 Polynomials (irreducibility, etc.)748Gausss Lemma I 748Gausss Lemma II 749discriminant 749polynomial ring 751resolvent 751de Moivre identity 754monic 754Wedderburns Theorem 754proof of Wedderburns theorem 755second proof of Wedderburns theorem 756nite eld 757Frobenius automorphism 760characteristic 761characterization of eld 761example of an innite eld of nite characteristic762examples of elds 762eld 764eld homomorphism 764prime subeld 76512F05 Algebraic extensions 766a nite extension of elds is an algebraic exten-sion 766algebraic closure 767algebraic extension 767algebraically closed 767algebraically dependent 768existence of the minimal polynomial 768nite extension 769minimal polynomial 769norm 770primitive element theorem 770splitting eld 770the eld extension R/ is not nite 771trace 77112F10 Separable extensions, Galois the-ory 772Abelian extension 772Fundamental Theorem of Galois Theory 772Galois closure 773Galois conjugate 773Galois extension 773Galois group 773absolute Galois group 774cyclic extension 774example of nonperfect eld 774xed eld 774innite Galois theory 774normal closure 776normal extension 776perfect eld 777radical extension 777separable 777separable closure 77812F20 Transcendental extensions 779transcendence degree 77912F99 Miscellaneous 780composite eld 780extension eld 78012J15 Ordered elds 782ordered eld 78213-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 783absolute value 783associates 784cancellation ring 784comaximal 784every prime ideal is radical 784module 785radical of an ideal 786ring 786subring 787tensor product 78713-XX Commutative rings and algebras789commutative ring 78913A02 Graded rings 790graded ring 79013A05 Divisibility 791Eisenstein criterion 79113A10 Radical theory 792Hilberts Nullstellensatz 792nilradical 792radical of an integer 79313A15 Ideals; multiplicative ideal theoryxviii794contracted ideal 794existence of maximal ideals 794extended ideal 795fractional ideal 796homogeneous ideal 797ideal 797maximal ideal 797principal ideal 798the set of prime ideals of a commutative ringwith identity 79813A50 Actions of groups on commuta-tive rings; invariant theory 799Schwarz (1975) theorem 799invariant polynomial 80013A99 Miscellaneous 801Lagranges identity 801characteristic 802cyclic ring 802proof of Euler four-square identity 803proof that every subring of a cyclic ring is a cyclicring 804proof that every subring of a cyclic ring is anideal 804zero ring 80513B02 Extension theory 806algebraic 806module-nite 80613B05 Galois theory 807algebraic 80713B21 Integral dependence 808integral 80813B22 Integral closure of rings and ide-als ; integrally closed rings, related rings(Japanese, etc.) 809integral closure 80913B30 Quotients and localization 810fraction eld 810localization 810multiplicative set 81113C10 Projective and free modules andideals 812example of free module 81213C12 Torsion modules and ideals 813torsion element 81313C15 Dimension theory, depth, relatedrings (catenary, etc.) 814Krulls principal ideal theorem 81413C99 Miscellaneous 815Artin-Rees theorem 815Nakayamas lemma 815prime ideal 815proof of Nakayamas lemma 816proof of Nakayamas lemma 817support 81713E05 Noetherian rings and modules 818Hilbert basis theorem 818Noetherian module 818proof of Hilbert basis theorem 819nitely generated modules over a principal idealdomain 81913F07 Euclidean rings and generaliza-tions 821Euclidean domain 821Euclidean valuation 821proof of Bezouts Theorem 822proof that an Euclidean domain is a PID 82213F10 Principal ideal rings 823Smith normalform 82313F25 Formal power series rings 825formal power series 82513F30 Valuation rings 831discrete valuation 831discrete valuation ring 83113G05 Integral domains 833Dedekind-Hasse valuation 833PID 834UFD 834a nite integral domain is a eld 835an artinian integral domain is a eld 835example of PID 835eld of quotients 836integral domain 836irreducible 837motivation for Euclidean domains 837zero divisor 83813H05 Regular local rings 839regular local ring 83913H99 Miscellaneous 840local ring 840xixsemi-local ring 84113J10 Complete rings, completion 842completion 84213J25 Ordered rings 844ordered ring 84413J99 Miscellaneous 845topological ring 84513N15 Derivations 846derivation 84613P10 Polynomial ideals, Grobner bases847Grobner basis 84714-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 849Picard group 849ane space 849ane variety 849dual isogeny 850nite morphism 850isogeny 851line bundle 851nonsingular variety 852projective space 852projective variety 854quasi-nite morphism 85414A10 Varieties and morphisms 855Zariski topology 855algebraic map 856algebraic sets and polynomial ideals 856noetherian topological space 857regular map 857structure sheaf 85814A15 Schemes and morphisms 859closed immersion 859coherent sheaf 859bre product 860prime spectrum 860scheme 863separated scheme 864singular set 86414A99 Miscellaneous 865Cartier divisor 865General position 865Serres twisting theorem 866ample 866height of a prime ideal 866invertible sheaf 866locally free 867normal irreducible varieties are nonsingular incodimension 1 867sheaf of meromorphic functions 867very ample 86714C20 Divisors, linear systems, invert-ible sheaves 869divisor 869Rational and birational maps 870general type 87014F05 Vector bundles, sheaves, relatedconstructions 871direct image (functor) 87114F20 Etale and other Grothendieck topolo-gies and cohomologies 872site 87214F25 Classical real and complex coho-mology 873Serre duality 873sheaf cohomology 87414G05 Rational points 875Hasse principle 87514H37 Automorphisms 876Frobenius morphism 87614H45 Special curves and curves of lowgenus 878Fermats spiral 878archimedean spiral 878folium of Descartes 879spiral 87914H50 Plane and space curves 880torsion (space curve) 88014H52 Elliptic curves 881Birch and Swinnerton-Dyer conjecture 881Hasses bound for elliptic curves over nite elds882L-series of an elliptic curve 882Mazurs theorem on torsion of elliptic curves 884Mordell curve 884Nagell-Lutz theorem 885Selmer group 886bad reduction 887conductor of an elliptic curve 890xxelliptic curve 890height function 894j-invariant 895rank of an elliptic curve 896supersingular 897the torsion subgroup of an elliptic curve injectsin the reduction of the curve 89714H99 Miscellaneous 900Riemann-Roch theorem 900genus 900projective curve 901proof of Riemann-Roch theorem 90114L17 Ane algebraic groups, hyperal-gebra constructions 902ane algebraic group 902algebraic torus 90214M05 Varieties dened by ring con-ditions (factorial, Cohen-Macaulay, semi-normal) 903normal 90314M15 Grassmannians, Schubert vari-eties, ag manifolds 904Borel-Bott-Weil theorem 904ag variety 90514R15 Jacobian problem 906Jacobian conjecture 90615-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 907Cholesky decomposition 907Hadamard matrix 908Hessenberg matrix 909If A Mn(k) and A is supertriangular thenAn= 0 910Jacobi determinant 910Jacobis Theorem 912Kronecker product 912LU decomposition 913Peetres inequality 914Schur decomposition 915antipodal 916conjugate transpose 916corollary of Schur decomposition 917covector 918diagonal matrix 918diagonalization 920diagonally dominant matrix 920eigenvalue (of a matrix) 921eigenvalue problem 922eigenvalues of orthogonal matrices 924eigenvector 925exactly determined 926free vector space over a set 926in a vector space, v = 0 if and only if = 0 orv is the zero vector 928invariant subspace 929least squares 929linear algebra 930linear least squares 932linear manifold 934matrix exponential 934matrix operations 935nilpotent matrix 938nilpotent transformation 938non-zero vector 939o-diagonal entry 940orthogonal matrices 940orthogonal vectors 941overdetermined 941partitioned matrix 941pentadiagonal matrix 942proof of Cayley-Hamilton theorem 942proof of Schur decomposition 943singular value decomposition 944skew-symmetric matrix 945square matrix 946strictly upper triangular matrix 946symmetric matrix 947theorem for normal triangular matrices 947triangular matrix 948tridiagonal matrix 949under determined 950unit triangular matrix 950unitary 951vector space 952vector subspace 953zero map 954zero vector in a vector space is unique 955zero vector space 95515-01 Instructional exposition (textbooks,tutorial papers, etc.) 956xxicirculant matrix 956matrix 95715-XX Linear and multilinear algebra;matrix theory 960linearly dependent functions 96015A03 Vector spaces, linear dependence,rank 961Sylvesters law 961basis 961complementary subspace 962dimension 963every vector space has a basis 964ag 964frame 965linear combination 968linear independence 968list vector 968nullity 969orthonormal basis 970physical vector 970proof of rank-nullity theorem 972rank 973rank-nullity theorem 973similar matrix 974span 975theorem for the direct sum of nite dimensionalvector spaces 976vector 97615A04 Linear transformations, semilin-ear transformations 980admissibility 980conductor of a vector 980cyclic decomposition theorem 981cyclic subspace 981dimension theorem for symplectic complement(proof) 981dual homomorphism 982dual homomorphism of the derivative 983image of a linear transformation 984invertible linear transformation 984kernel of a linear transformation 985linear transformation 985minimal polynomial (endomorphism) 986symplectic complement 987trace 98815A06 Linear equations 989Gaussian elimination 989nite-dimensional linear problem 991homogeneous linear problem 992linear problem 993reduced row echelon form 993row echelon form 994under-determined polynomial interpolation 99415A09 Matrix inversion, generalized in-verses 996matrix adjoint 996matrix inverse 99715A12 Conditioning of matrices 1000singular 100015A15 Determinants, permanents, otherspecial matrix functions 1001Cayley-Hamilton theorem 1001Cramers rule 1001cofactor expansion 1002determinant 1003determinant as a multilinear mapping 1005determinants of some matrices of special form1006example of Cramers rule 1006proof of Cramers rule 1008proof of cofactor expansion 1008resolvent matrix 100915A18 Eigenvalues, singular values, andeigenvectors 1010Jordan canonical form theorem 1010Lagrange multiplier method 1011Perron-Frobenius theorem 1011characteristic equation 1012eigenvalue 1012eigenvalue 101315A21 Canonical forms, reductions, clas-sication 1015companion matrix 1015eigenvalues of an involution 1015linear involution 1016normal matrix 1017projection 1018quadratic form 101915A23 Factorization of matrices 1021QR decomposition 1021xxii15A30 Algebraic systems of matrices 1023ideals in matrix algebras 102315A36 Matrices of integers 1025permutation matrix 102515A39 Linear inequalities 1026Farkas lemma 102615A42 Inequalities involving eigenvaluesand eigenvectors 1027Gershgorins circle theorem 1027Gershgorins circle theorem result 1027Shurs inequality 102815A48 Positive matrices and their gen-eralizations; cones of matrices 1029negative denite 1029negative semidenite 1029positive denite 1030positive semidenite 1030primitive matrix 1031reducible matrix 103115A51 Stochastic matrices 1032Birko-von Neumann theorem 1032proof of Birko-von Neumann theorem 103215A57 Other types of matrices (Hermi-tian, skew-Hermitian, etc.) 1035Hermitian matrix 1035direct sum of Hermitian and skew-Hermitian ma-trices 1036identity matrix 1037skew-Hermitian matrix 1037transpose 103815A60 Norms of matrices, numerical range,applications of functional analysis to ma-trix theory 1041Frobenius matrix norm 1041matrix p-norm 1042self consistent matrix norm 104315A63 Quadratic and bilinear forms, in-ner products 1044Cauchy-Schwarz inequality 1044adjoint endomorphism 1045anti-symmetric 1046bilinear map 1046dot product 1049every orthonormal set is linearly independent 1050inner product 1051inner product space 1051proof of Cauchy-Schwarz inequality 1052self-dual 1052skew-symmetric bilinear form 1053spectral theorem 105315A66 Cliord algebras, spinors 1056geometric algebra 105615A69 Multilinear algebra, tensor prod-ucts 1058Einstein summation convention 1058basic tensor 1059multi-linear 1061outer multiplication 1061tensor 1062tensor algebra 1065tensor array 1065tensor product (vector spaces) 1067tensor transformations 106915A72 Vector and tensor algebra, theoryof invariants 1072bac-cab rule 1072cross product 1072euclidean vector 1073rotational invariance of cross product 107415A75 Exterior algebra, Grassmann al-gebras 1076contraction 1076exterior algebra 107715A99 Miscellaneous topics 1081Kronecker delta 1081dual space 1081example of trace of a matrix 1083generalized Kronecker delta symbol 1083linear functional 1084modules are a generalization of vector spaces 1084proof of properties of trace of a matrix 1085quasipositive matrix 1086trace of a matrix 1086Volume 216-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1088direct product of modules 1088direct sum 1089xxiiiexact sequence 1089quotient ring 109016D10 General module theory 1091annihilator 1091annihilator is an ideal 1091artinian 1092composition series 1092conjugate module 1093modular law 1093module 1093proof of modular law 1094zero module 109416D20 Bimodules 1095bimodule 109516D25 Ideals 1096associated prime 1096nilpotent ideal 1096primitive ideal 1096product of ideals 1097proper ideal 1097semiprime ideal 1097zero ideal 109816D40 Free, projective, and at modulesand ideals 1099nitely generated projective module 1099at module 1099free module 1100free module 1100projective cover 1100projective module 110116D50 Injective modules, self-injectiverings 1102injective hull 1102injective module 110216D60 Simple and semisimple modules,primitive rings and ideals 1104central simple algebra 1104completely reducible 1104simple ring 110516D80 Other classes of modules and ide-als 1106essential submodule 1106faithful module 1106minimal prime ideal 1107module of nite rank 1107simple module 1107superuous submodule 1107uniform module 110816E05 Syzygies, resolutions, complexes1109n-chain 1109chain complex 1109at resolution 1110free resolution 1110injective resolution 1110projective resolution 1110short exact sequence 1111split short exact sequence 1111von Neumann regular 111116K20 Finite-dimensional 1112quaternion algebra 111216K50 Brauer groups 1113Brauer group 111316K99 Miscellaneous 1114division ring 111416N20 Jacobson radical, quasimultipli-cation 1115Jacobson radical 1115a ring modulo its Jacobson radical is semiprimi-tive 1116examples of semiprimitive rings 1116proof of Characterizations of the Jacobson radi-cal 1117properties of the Jacobson radical 1118quasi-regularity 1119semiprimitive ring 112016N40 Nil and nilpotent radicals, sets,ideals, rings 1121Koethe conjecture 1121nil and nilpotent ideals 112116N60 Prime and semiprime rings 1123prime ring 112316N80 General radicals and rings 1124prime radical 1124radical theory 112416P40 Noetherian rings and modules 1126Noetherian ring 1126noetherian 112616P60 Chain conditions on annihilatorsand summands: Goldie-type conditions ,xxivKrull dimension 1128Goldie ring 1128uniform dimension 112816S10 Rings determined by universal prop-erties (free algebras, coproducts, adjunc-tion of inverses, etc.) 1130Ore domain 113016S34 Group rings , Laurent polynomialrings 1131support 113116S36 Ordinary and skew polynomial ringsand semigroup rings 1132Gaussian polynomials 1132q skew derivation 1133q skew polynomial ring 1133sigma derivation 1133sigma, delta constant 1133skew derivation 1133skew polynomial ring 113416S99 Miscellaneous 1135algebra 1135algebra (module) 113516U10 Integral domains 1137Pr ufer domain 1137valuation domain 113716U20 Ore rings, multiplicative sets, Orelocalization 1139Goldies Theorem 1139Ore condition 1139Ores theorem 1140classical ring of quotients 1140saturated 114116U70 Center, normalizer (invariant el-ements) 1142center (rings) 114216U99 Miscellaneous 1143anti-idempotent 114316W20 Automorphisms and endomor-phisms 1144ring of endomorphisms 114416W30 Coalgebras, bialgebras, Hopf al-gebras ; rings, modules, etc. on whichthese act 1146Hopf algebra 1146almost cocommutative bialgebra 1147bialgebra 1148coalgebra 1148coinvariant 1149comodule 1149comodule algebra 1149comodule coalgebra 1150module algebra 1150module coalgebra 115016W50 Graded rings and modules 1151graded algebra 1151graded module 1151supercommutative 115116W55 Super (or skew) structure1153super tensor product 1153superalgebra 1153supernumber 115416W99 Miscellaneous 1155Hamiltonian quaternions 115516Y30 Near-rings 1158near-ring 115817A01 General theory 1159commutator bracket 115917B05 Structure theory 1161Killing form 1161Levis theorem 1161nilradical 1161radical 116217B10 Representations, algebraic theory(weights) 1163Ados theorem 1163Lie algebra representation 1163adjoint representation 1164examples of non-matrix Lie groups 1165isotropy representation 116517B15 Representations, analytic theory1166invariant form (Lie algebras) 116617B20 Simple, semisimple, reductive (su-per)algebras (roots) 1167Borel subalgebra 1167Borel subgroup 1167Cartan matrix 1168Cartan subalgebra 1168Cartans criterion 1168xxvCasimir operator 1168Dynkin diagram 1169Verma module 1169Weyl chamber 1170Weyl group 1170Weyls theorem 1170classication of nite-dimensional representationsof semi-simple Lie algebras 1171cohomology of semi-simple Lie algebras 1171nilpotent cone 1171parabolic subgroup 1172pictures of Dynkin diagrams 1172positive root 1175rank 1175root lattice 1175root system 1176simple and semi-simple Lie algebras 1177simple root 1178weight (Lie algebras) 1178weight lattice 117817B30 Solvable, nilpotent (super)algebras1179Engels theorem 1179Lies theorem 1182solvable Lie algebra 118317B35 Universal enveloping (super)algebras1184Poincare-Birkho-Witt theorem 1184universal enveloping algebra 118517B56 Cohomology of Lie (super)algebras1187Lie algebra cohomology 118717B67 Kac-Moody (super)algebras (struc-ture and representation theory) 1188Kac-Moody algebra 1188generalized Cartan matrix 118817B99 Miscellaneous 1190Jacobi identity interpretations 1190Lie algebra 1190real form 119218-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1193Grothendieck spectral sequence 1193category of sets 1194functor 1194monic 1194natural equivalence 1195representable functor 1195supplemental axioms for an Abelian category 119518A05 Denitions, generalizations 1197autofunctor 1197automorphism 1197category 1198category example (arrow category) 1199commutative diagram 1199double dual embedding 1200dual category 1201duality principle 1201endofunctor 1202examples of initial objects, terminal objects andzero objects 1202forgetful functor 1204isomorphism 1205natural transformation 1205types of homomorphisms 1205zero object 120618A22 Special properties of functors (faith-ful, full, etc.) 1208exact functor 120818A25 Functor categories, comma cate-gories 1210Yoneda embedding 121018A30 Limits and colimits (products, sums,directed limits, pushouts, ber products,equalizers, kernels, ends and coends, etc.)1211categorical direct product 1211categorical direct sum 1211kernel 121218A40 Adjoint functors (universal con-structions, reective subcategories, Kan ex-tensions, etc.) 1213adjoint functor 1213equivalence of categories 121418B40 Groupoids, semigroupoids, semi-groups, groups (viewed as categories) 1215groupoid (category theoretic) 121518E10 Exact categories, abelian cate-gories 1216abelian category 1216xxviexact sequence 1217derived category 1218enough injectives 121818F20 Presheaves and sheaves 1219locally ringed space 1219presheaf 1220sheaf 1220sheacation 1225stalk 122618F30 Grothendieck groups 1228Grothendieck group 122818G10 Resolutions; derived functors 1229derived functor 122918G15 Ext and Tor, generalizations, K unnethformula 1231Ext 123118G30 Simplicial sets, simplicial objects(in a category) 1232nerve 1232simplicial category 1232simplicial object 123318G35 Chain complexes 12355-lemma 12359-lemma 1236Snake lemma 1236chain homotopy 1237chain map 1237homology (chain complex) 123718G40 Spectral sequences, hypercoho-mology 1238spectral sequence 123819-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1239Algebraic K-theory 1239K-theory 1240examples of algebraic K-theory groups 124119K33 EXT and K-homology 1242Fredholm module 1242K-homology 124319K99 Miscellaneous 1244examples of K-theory groups 124420-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1245alternating group is a normal subgroup of thesymmetric group 1245associative 1245canonical projection 1246centralizer 1246commutative 1247examples of groups 1247group 1250quotient group 125020-02 Research exposition (monographs,survey articles) 1252length function 125220-XX Group theory and generalizations1253free product with amalgamated subgroup 1253nonabelian group 125420A05 Axiomatics and elementary prop-erties 1255Feit-Thompson theorem 1255Proof: The orbit of any element of a group is asubgroup 1255center 1256characteristic subgroup 1256class function 1257conjugacy class 1258conjugacy class formula 1258conjugate stabilizer subgroups 1258coset 1259cyclic group 1259derived subgroup 1260equivariant 1260examples of nite simple groups 1261nitely generated group 1262rst isomorphism theorem 1262fourth isomorphism theorem 1262generator 1263group actions and homomorphisms 1263group homomorphism 1265homogeneous space 1265identity element 1268inner automorphism 1268kernel 1269maximal 1269normal subgroup 1269normality of subgroups is not transitive 1269normalizer 1270order (of a group) 1271xxviipresentation of a group 1271proof of rst isomorphism theorem 1272proof of second isomorphism theorem 1273proof that all cyclic groups are abelian 1274proof that all cyclic groups of the same order areisomorphic to each other 1274proof that all subgroups of a cyclic group arecyclic 1274regular group action 1275second isomorphism theorem 1275simple group 1276solvable group 1276subgroup 1276third isomorphism theorem 127720A99 Miscellaneous 1279Cayley table 1279proper subgroup 1280quaternion group 128020B05 General theory for nite groups1282cycle notation 1282permutation group 128320B15 Primitive groups 1284primitive transitive permutation group 128420B20 Multiply transitive nite groups1286Jordans theorem (multiply transitive groups) 1286multiply transitive 1286sharply multiply transitive 128720B25 Finite automorphism groups of al-gebraic, geometric, or combinatorial struc-tures 1288diamond theory 128820B30 Symmetric groups 1289symmetric group 1289symmetric group 128920B35 Subgroups of symmetric groups1290Cayleys theorem 129020B99 Miscellaneous 1291(p, q) shue 1291Frobenius group 1291permutation 1292proof of Cayleys theorem 129220C05 Group rings of nite groups andtheir modules 1294group ring 129420C15 Ordinary representations and char-acters 1295Maschkes theorem 1295a representation which is not completely reducible1295orthogonality relations 129620C30 Representations of nite symmet-ric groups 1299example of immanent 1299immanent 1299permanent 129920C99 Miscellaneous 1301Frobenius reciprocity 1301Schurs lemma 1301character 1302group representation 1303induced representation 1303regular representation 1304restriction representation 130420D05 Classication of simple and non-solvable groups 1305Burnside p q theorem 1305classication of semisimple groups 1305semisimple group 130520D08 Simple groups: sporadic groups1307Janko groups 130720D10 Solvable groups, theory of for-mations, Schunck classes, Fitting classes,-length, ranks 1308Cuhinins Theorem 1308separable 1308supersolvable group 130920D15 Nilpotent groups, p-groups 1310Burnside basis theorem 131020D20 Sylow subgroups, Sylow proper-ties, -groups, -structure 1311-groups and t-groups 1311p-subgroup 1311Burnside normal complement theorem 1312Frattini argument 1312Sylow p-subgroup 1312Sylow theorems 1312xxviiiSylows rst theorem 1313Sylows third theorem 1313application of Sylows theorems to groups of or-der pq 1313p-primary component 1314proof of Frattini argument 1314proof of Sylow theorems 1314subgroups containing the normalizers of Sylowsubgroups normalize themselves 131620D25 Special subgroups (Frattini, Fit-ting, etc.) 1317Fittings theorem 1317characteristically simple group 1317the Frattini subgroup is nilpotent 131720D30 Series and lattices of subgroups1319maximal condition 1319minimal condition 1319subnormal series 132020D35 Subnormal subgroups 1321subnormal subgroup 132120D99 Miscellaneous 1322Cauchys theorem 1322Lagranges theorem 1322exponent 1322fully invariant subgroup 1323proof of Cauchys theorem 1323proof of Lagranges theorem 1324proof of the converse of Lagranges theorem fornite cyclic groups 1324proof that expG divides [G[ 1324proof that [g[ divides expG 1325proof that every group of prime order is cyclic132520E05 Free nonabelian groups 1326Nielsen-Schreier theorem 1326Scheier index formula 1326free group 1326proof of Nielsen-Schreier theorem and Schreierindex formula 1327Jordan-Holder decomposition 1328pronite group 1328extension 1329holomorph 1329proof of the Jordan Holder decomposition theo-rem 1329semidirect product of groups 1330wreath product 1333Jordan-Hlder decomposition theorem 1334simplicity of the alternating groups 1334abelian groups of order 120 1337fundamental theorem of nitely generated abeliangroups 1337conjugacy class 1338Frattini subgroup 1338non-generator 133820Exx Structure and classication of in-nite or nite groups 1339faithful group action 133920F18 Nilpotent groups 1340classication of nite nilpotent groups 1340nilpotent group 134020F22 Other classes of groups dened bysubgroup chains 1342inverse limit 134220F28 Automorphism groups of groups1344outer automorphism group 134420F36 Braid groups; Artin groups 1345braid group 134520F55 Reection and Coxeter groups 1347cycle 1347dihedral group 134820F65 Geometric group theory 1349groups that act freely on trees are free 134920F99 Miscellaneous 1350perfect group 135020G15 Linear algebraic groups over ar-bitrary elds 1351Nagaos theorem 1351computation of the order of GL(n, Fq) 1351general linear group 1352order of the general linear group over a nite eld1352special linear group 135220G20 Linear algebraic groups over thereals, the complexes, the quaternions 1353orthogonal group 135320G25 Linear algebraic groups over localelds and their integers 1354xxixIharas theorem 135420G40 Linear algebraic groups over -nite elds 1355SL2(F3) 135520J06 Cohomology of groups 1356group cohomology 1356stronger Hilbert theorem 90 135720J15 Category of groups 1359variety of groups 135920K01 Finite abelian groups 1360Schinzels theorem 136020K10 Torsion groups, primary groupsand generalized primary groups 1361torsion 136120K25 Direct sums, direct products, etc.1362direct product of groups 136220K99 Miscellaneous 1363Klein 4-group 1363divisible group 1364example of divisible group 1364locally cyclic group 136420Kxx Abelian groups 1366abelian group 136620M10 General structure theory 1367existence of maximal semilattice decomposition1367semilattice decomposition of a semigroup 1368simple semigroup 136820M12 Ideal theory 1370Rees factor 1370ideal 137020M14 Commutative semigroups 1372Archimedean semigroup 1372commutative semigroup 137220M20 Semigroups of transformations,etc. 1373semigroup of transformations 137320M30 Representation of semigroups; ac-tions of semigroups on sets 1375counting theorem 1375example of group action 1375group action 1376orbit 1377proof of counting theorem 1377stabilizer 137820M99 Miscellaneous 1379a semilattice is a commutative band 1379adjoining an identity to a semigroup 1379band 1380bicyclic semigroup 1380congruence 1381cyclic semigroup 1381idempotent 1382null semigroup 1383semigroup 1383semilattice 1383subsemigroup,, submonoid,, and subgroup 1384zero elements 138420N02 Sets with a single binary opera-tion (groupoids) 1386groupoid 1386idempotency 1386left identity and right identity 138720N05 Loops, quasigroups 1388Moufang loop 1388loop and quasigroup 138922-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1390xed-point subspace 139022-XX Topological groups, Lie groups1391Cantor space 139122A05 Structure of general topologicalgroups 1392topological group 139222C05 Compact groups 1393n-torus 1393reductive 139322D05 General properties and structureof locally compact groups 1394-simple 139422D15 Group algebras of locally com-pact groups 1395group C-algebra 139522E10 General properties and structureof complex Lie groups 1396existence and uniqueness of compact real form1396maximal torus 1397xxxLie group 1397complexication 1399Hilbert-Weyl theorem 1400the connection between Lie groups and Lie alge-bras 140126-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1402derivative notation 1402fundamental theorems of calculus 1403logarithm 1404proof of the rst fundamental theorem of calcu-lus 1405proof of the second fundamental theorem of cal-culus 1405root-mean-square 1406square 140626-XX Real functions 1408abelian function 1408full-width at half maximum 140826A03 Foundations: limits and general-izations, elementary topology of the line1410Cauchy sequence 1410Dedekind cuts 1410binomial proof of positive integer power rule 1413exponential 1414interleave sequence 1415limit inferior 1415limit superior 1416power rule 1417properties of the exponential 1417squeeze rule 141826A06 One-variable calculus 1420Darbouxs theorem (analysis) 1420Fermats Theorem (stationary points) 1420Heaviside step function 1421Leibniz rule 1421Rolles theorem 1422binomial formula 1422chain rule 1422complex Rolles theorem 1423complex mean-value theorem 1423denite integral 1424derivative of even/odd function (proof) 1425direct sum of even/odd functions (example) 1425even/odd function 1426example of chain rule 1427example of increasing/decreasing/monotone func-tion 1428extended mean-value theorem 1428increasing/decreasing/monotone function 1428intermediate value theorem 1429limit 1429mean value theorem 1430mean-value theorem 1430monotonicity criterion 1431nabla 1431one-sided limit 1432product rule 1432proof of Darbouxs theorem 1433proof of Fermats Theorem (stationary points)1434proof of Rolles theorem 1434proof of Taylors Theorem 1435proof of binomial formula 1436proof of chain rule 1436proof of extended mean-value theorem 1437proof of intermediate value theorem 1437proof of mean value theorem 1438proof of monotonicity criterion 1439proof of quotient rule 1439quotient rule 1440signum function 144026A09 Elementary functions 1443denitions in trigonometry 1443hyperbolic functions 144426A12 Rate of growth of functions, or-ders of innity, slowly varying functions1446Landau notation 144626A15 Continuity and related questions(modulus of continuity, semicontinuity, dis-continuities, etc.) 1448Dirichlets function 1448semi-continuous 1448semicontinuous 1449uniformly continuous 145026A16 Lipschitz (H older) classes 1451Lipschitz condition 1451Lipschitz condition and dierentiability 1452xxxiLipschitz condition and dierentiability result 145326A18 Iteration 1454iteration 1454periodic point 145426A24 Dierentiation (functions of onevariable): general theory, generalized deriva-tives, mean-value theorems 1455Leibniz notation 1455derivative 1456lHpitals rule 1460proof of De lHpitals rule 1461related rates 146226A27 Nondierentiability (nondieren-tiable functions, points of nondierentia-bility), discontinuous derivatives 1464Weierstrass function 146426A36 Antidierentiation 1465antiderivative 1465integration by parts 1465integrations by parts for the Lebesgue integral146626A42 Integrals of Riemann, Stieltjesand Lebesgue type 1468Riemann sum 1468Riemann-Stieltjes integral 1469continuous functions are Riemann integrable 1469generalized Riemann integral 1469proof of Continuous functions are Riemann inte-grable 147026A51 Convexity, generalizations 1471concave function 147126Axx Functions of one variable 1472function centroid 147226B05 Continuity and dierentiation ques-tions 1473C0 (U) is not empty 1473Rademachers Theorem 1474smooth functions with compact support 147526B10 Implicit function theorems, Jaco-bians, transformations with several vari-ables 1477Jacobian matrix 1477directional derivative 1477gradient 1478implicit dierentiation 1481implicit function theorem 1481proof of implicit function theorem 148226B12 Calculus of vector functions 1484Clairauts theorem 1484Fubinis Theorem 1484Generalised N-dimensional Riemann Sum 1485Generalized N-dimensional Riemann Integral 1485Helmholtz equation 1486Hessian matrix 1487Jordan Content of an N-cell 1487Laplace equation 1487chain rule (several variables) 1488divergence 1489extremum 1490irrotational eld 1490partial derivative 1491plateau 1492proof of Greens theorem 1492relations between Hessian matrix and local ex-trema 1493solenoidal eld 149426B15 Integration: length, area, volume1495arc length 149526B20 Integral formulas (Stokes, Gauss,Green, etc.) 1497Greens theorem 149726B25 Convexity, generalizations 1499convex function 1499extremal value of convex/concave functions 150026B30 Absolutely continuous functions,functions of bounded variation 1502absolutely continuous function 1502total variation 150326B99 Miscellaneous 1505derivation of zeroth weighted power mean 1505weighted power mean 150626C15 Rational functions 1507rational function 150726C99 Miscellaneous 1508Laguerre Polynomial 150826D05 Inequalities for trigonometric func-tions and polynomials 1509Weierstrass product inequality 1509proof of Jordans Inequality 1509xxxii26D10 Inequalities involving derivativesand dierential and integral operators 1511Gronwalls lemma 1511proof of Gronwalls lemma 151126D15 Inequalities for sums, series andintegrals 1513Carlemans inequality 1513Chebyshevs inequality 1513MacLaurins Inequality 1514Minkowski inequality 1514Muirheads theorem 1515Schurs inequality 1515Youngs inequality 1515arithmetic-geometric-harmonic means inequality1516general means inequality 1516power mean 1517proof of Chebyshevs inequality 1517proof of Minkowski inequality 1518proof of arithmetic-geometric-harmonic means in-equality 1519proof of general means inequality 1521proof of rearrangement inequality 1522rearrangement inequality 152326D99 Miscellaneous 1524Bernoullis inequality 1524proof of Bernoullis inequality 152426E35 Nonstandard analysis 1526hyperreal 1526e is not a quadratic irrational 1527zero of a function 152828-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1530extended real numbers 153028-XX Measure and integration 1532Riemann integral 1532martingale 153228A05 Classes of sets (Borel elds, -rings, etc.), measurable sets, Suslin sets,analytic sets 1534Borel -algebra 153428A10 Real- or complex-valued set func-tions 1535-nite 1535Argand diagram 1535Hahn-Kolmogorov theorem 1536measure 1536outer measure 1536properties for measure 153828A12 Contents, measures, outer mea-sures, capacities 1540Hahn decomposition theorem 1540Jordan decomposition 1540Lebesgue decomposition theorem 1541Lebesgue outer measure 1541absolutely continuous 1542counting measure 1543measurable set 1543outer regular 1543signed measure 1543singular measure 154428A15 Abstract dierentiation theory,dierentiation of set functions 1545Hardy-Littlewood maximal theorem 1545Lebesgue dierentiation theorem 1545Radon-Nikodym theorem 1546integral depending on a parameter 154728A20 Measurable and nonmeasurablefunctions, sequences of measurable func-tions, modes of convergence 1549Egorovs theorem 1549Fatous lemma 1549Fatou-Lebesgue theorem 1550dominated convergence theorem 1550measurable function 1550monotone convergence theorem 1551proof of Egorovs theorem 1551proof of Fatous lemma 1552proof of Fatou-Lebesgue theorem 1552proof of dominated convergence theorem 1553proof of monotone convergence theorem 155328A25 Integration with respect to mea-sures and other set functions 1555L(X, d) 1555Hardy-Littlewood maximal operator 1555Lebesgue integral 155628A60 Measures on Boolean rings, mea-sure algebras 1558-algebra 1558-algebra 1558xxxiiialgebra 1559measurable set (for outer measure) 155928A75 Length, area, volume, other geo-metric measure theory 1561Lebesgue density theorem 156128A80 Fractals 1562Cantor set 1562Hausdor dimension 1565Koch curve 1566Sierpinski gasket 1567fractal 156728Axx Classical measure theory 1569Vitalis Theorem 1569proof of Vitalis Theorem 156928B15 Set functions, measures and inte-grals with values in ordered spaces 1571Lp-space 1571locally integrable function 157228C05 Integration theory via linear func-tionals (Radon measures, Daniell integrals,etc.), representing set functions and mea-sures 1573Haar integral 157328C10 Set functions and measures ontopological groups, Haar measures, invari-ant measures 1575Haar measure 157528C20 Set functions and measures andintegrals in innite-dimensional spaces (Wienermeasure, Gaussian measure, etc.) 1577essential supremum 157728D05 Measure-preserving transforma-tions 1578measure-preserving 157830-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1579domain 1579region 1579regular region 1580topology of the complex plane 158030-XX Functions of a complex variable1581z0 is a pole of f 158130A99 Miscellaneous 1582Riemann mapping theorem 1582Runges theorem 1582Weierstrass M-test 1583annulus 1583conformally equivalent 1583contour integral 1584orientation 1585proof of Weierstrass M-test 1585unit disk 1586upper half plane 1586winding number and fundamental group 158630B10 Power series (including lacunaryseries) 1587Euler relation 1587analytic 1588existence of power series 1588innitely-dierentiable function that is not ana-lytic 1590power series 1591proof of radius of convergence 1592radius of convergence 159330B50 Dirichlet series and other seriesexpansions, exponential series 1594Dirichlet series 159430C15 Zeros of polynomials, rational func-tions, and other analytic functions (e.g.zeros of functions with bounded Dirichletintegral) 1596Mason-Stothers theorem 1596zeroes of analytic functions are isolated 159630C20 Conformal mappings of specialdomains 1598automorphisms of unit disk 1598unit disk upper half plane conformal equivalencetheorem 159830C35 General theory of conformal map-pings 1599proof of conformal mapping theorem 159930C80 Maximum principle; Schwarzs lemma,Lindelof principle, analogues and general-izations; subordination 1601Schwarz lemma 1601maximum principle 1601proof of Schwarz lemma 160230D20 Entire functions, general theory1603xxxivLiouvilles theorem 1603Moreras theorem 1603entire 1604holomorphic 1604proof of Liouvilles theorem 160430D30 Meromorphic functions, generaltheory 1606Casorati-Weierstrass theorem 1606Mittag-Leers theorem 1606Riemanns removable singularity theorem 1607essential singularity 1607meromorphic 1607pole 1607proof of Casorati-Weierstrass theorem 1608proof of Riemanns removable singularity theo-rem 1608residue 1609simple pole 161030E20 Integration, integrals of Cauchytype, integral representations of analyticfunctions 1611Cauchy integral formula 1611Cauchy integral theorem 1612Cauchy residue theorem 1613Gauss mean value theorem 1614Mobius circle transformation theorem 1614Mobius transformation cross-ratio preservationtheorem 1614Rouchs theorem 1614absolute convergence implies convergence for aninnite product 1615absolute convergence of innite product 1615closed curve theorem 1615conformal Mobius circle map theorem 1615conformal mapping 1616conformal mapping theorem 1616convergence/divergence for an innite product1616example of conformal mapping 1616examples of innite products 1617link between innite products and sums 1617proof of Cauchy integral formula 1618proof of Cauchy residue theorem 1619proof of Gauss mean value theorem 1620proof of Goursats theorem 1620proof of Mobius circle transformation theorem1622proof of Simultaneous converging or diverging ofproduct and sum theorem 1623proof of absolute convergence implies convergencefor an innite product 1624proof of closed curve theorem 1624proof of conformal Mobius circle map theorem1624simultaneous converging or diverging of productand sum theorem 1625Cauchy-Riemann equations 1625Cauchy-Riemann equations (polar coordinates)1626proof of the Cauchy-Riemann equations 1626removable singularity 162730F40 Kleinian groups 1629Klein 4-group 162931A05 Harmonic, subharmonic, super-harmonic functions 1630a harmonic function on a graph which is boundedbelow and nonconstant 1630example of harmonic functions on graphs 1630examples of harmonic functions on Rn1631harmonic function 163231B05 Harmonic, subharmonic, super-harmonic functions 1633Laplacian 163332A05 Power series, series of functions1634exponential function 163432C15 Complex spaces 1637Riemann sphere 163732F99 Miscellaneous 1638star-shaped region 163832H02 Holomorphic mappings, (holomor-phic) embeddings and related questions 1639Blochs theorem 1639Hartogs theorem 163932H25 Picard-type theorems and gener-alizations 1640Picards theorem 1640little Picard theorem 164033-XX Special functions 1641beta function 1641xxxv33B10 Exponential and trigonometric func-tions 1642natural logarithm 164233B15 Gamma, beta and polygamma func-tions 1643Bohr-Mollerup theorem 1643gamma function 1643proof of Bohr-Mollerup theorem 164533B30 Higher logarithm functions 1647Lambert W function 164733B99 Miscellaneous 1648natural log base 164833D45 Basic orthogonal polynomials andfunctions (Askey-Wilson polynomials, etc.)1649orthogonal polynomials 164933E05 Elliptic functions and integrals1651Weierstrass sigma function 1651elliptic function 1652elliptic integrals and Jacobi elliptic functions 1652examples of elliptic functions 1654modular discriminant 165434-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1656Liapunov function 1656Lorenz equation 1657Wronskian determinant 1659dependence on initial conditions of solutions ofordinary dierential equations 1660dierential equation 1661existence and uniqueness of solution of ordinarydierential equations 1662maximal interval of existence of ordinary dier-ential equations 1663method of undetermined coecients 1663natural symmetry of the Lorenz equation 1664symmetry of a solution of an ordinary dieren-tial equation 1665symmetry of an ordinary dierential equation166534-01 Instructional exposition (textbooks,tutorial papers, etc.) 1667second order linear dierential equation with con-stant coecients 166734A05 Explicit solutions and reductions1669separation of variables 1669variation of parameters 167034A12 Initial value problems, existence,uniqueness, continuous dependence and con-tinuation of solutions 1672initial value problem 167234A30 Linear equations and systems, gen-eral 1674Chebyshev equation 167434A99 Miscellaneous 1676autonomous system 167634B24 Sturm-Liouville theory 1677eigenfunction 167734C05 Location of integral curves, sin-gular points, limit cycles 1678Hopf bifurcation theorem 1678Poincare-Bendixson theorem 1679omega limit set 167934C07 Theory of limit cycles of polyno-mial and analytic vector elds (existence,uniqueness, bounds, Hilberts 16th prob-lem and ramif 1680Hilberts 16th problem for quadratic vector elds168034C23 Bifurcation 1682equivariant branching lemma 168234C25 Periodic solutions 1683Bendixsons negative criterion 1683Dulacs criteria 1683proof of Bendixsons negative criterion 168434C99 Miscellaneous 1685Hartman-Grobman theorem 1685equilibrium point 1685stable manifold theorem 168634D20 Lyapunov stability 1687Lyapunov stable 1687neutrally stable xed point 1687stable xed point 168734L05 General spectral theory 1688Gelfand spectral radius theorem 168834L15 Estimation of eigenvalues, upperand lower bounds 1689Rayleigh quotient 1689xxxvi34L40 Particular operators (Dirac, one-dimensional Schrodinger, etc.) 1690Dirac delta function 1690construction of Dirac delta function 169135-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 1692dierential operator 169235J05 Laplace equation, reduced waveequation (Helmholtz), Poisson equation 1694Poissons equation 169435L05 Wave equation 1695wave equation 169535Q53 KdV-like equations (Korteweg-deVries, Burgers, sine-Gordon, sinh-Gordon,etc.) 1697Korteweg - de Vries equation 169735Q99 Miscellaneous 1698heat equation 169837-00 General reference works (hand-books, dictionaries, bibliographies, etc.) 169937A30 Ergodic theorems, spectral the-ory, Markov operators 1700ergodic 1700fundamental theorem of demography 1700proof of fundamental theorem of demography 170137B05 Transformations and group ac-tions with special properties (minimality,distality, proximality, etc.) 1703discontinuous action 170337B20 Notions of recurrence 1704nonwandering set 170437B99 Miscellaneous 1705-limit set 1705asymptotically stable 1706expansive 1706the only compact metric spaces that admit a pos-itively expansive homeomorphism are discrete spaces1707topological conjugation 1708topolo