% % 91SUB.TEX Renamed msc.new EDITED FOR SCREEN USE 11-28-90 % 11-29,12-3 % This file contains the entire 1991 MR Subject Classification, % as revised. % %NOTE: The cs \MajorSub appears at the beginning of each major subdivision % I have not changed any line starting with \MajorSub 1991 MATHEMATICS SUBJECT CLASSIFICATION \MajorSub 00-XX General 00-01 Instructional exposition (textbooks, tutorial papers, etc.) 00-02 Research exposition (monographs, survey articles) 00Axx General and miscellaneous specific topics 00A05 General mathematics 00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.) 00A07 Problem books 00A08 Recreational mathematics 00A15 Bibliographies 00A20 Dictionaries and other general reference works 00A22 Formularies 00A30 Philosophy of mathematics, See also {03A05} 00A35 Methodology of mathematics, didactics 00A69 General applied mathematics, {For physics, See 00A79 and Sections 70 through 86} 00A71 Theory of mathematical modeling 00A72 General methods of simulation 00A73 Dimensional analysis 00A79 Physics (use more specific entries from Sections 70 through 86 when possible) 00A99 Miscellaneous topics 00Bxx Conference proceedings and collections of papers 00B05 Collections of abstracts of lectures 00B10 Collections of articles of general interest 00B15 Collections of articles of miscellaneous specific content 00B20 Proceedings of conferences of general interest 00B25 Proceedings of conferences of miscellaneous specific interest 00B30 Festschriften 00B50 Volumes of selected translations 00B55 Miscellaneous volumes of translations 00B60 Collections of reprinted articles, See also {01A75} \MajorSub 01-XX History and biography {See also the classification number --03 in the other sections} 01-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 01-01 Instructional exposition (textbooks, tutorial papers, etc.) 01-02 Research exposition (monographs, survey articles) 01-06 Proceedings, conferences, collections, etc. 01-08 Computational methods 01Axx History of mathematics and mathematicians 01A05 General histories, source books 01A07 Ethnomathematics, general 01A10 Paleolithic, Neolithic 01A12 Indigenous cultures of the Americas 01A13 Other indigenous cultures (non-European) 01A15 Indigenous European cultures (pre-Greek, etc.) 01A16 Egyptian 01A17 Babylonian 01A20 Greek, Roman 01A25 China 01A27 Japan 01A29 Southeast Asia 01A30 Islam (Medieval) 01A32 India 01A35 Medieval 01A40 15th and 16th centuries, Renaissance 01A45 17th century 01A50 18th century 01A55 19th century 01A60 20th century 01A65 Contemporary 01A67 Future prospectives 01A70 Biographies, obituaries, personalia, bibliographies 01A72 Schools of mathematics 01A73 Universities 01A74 Other institutions and academies 01A75 Collected or selected works; reprintings or translations of classics, See also {00B60} 01A80 Sociology (and profession) of mathematics 01A99 Miscellaneous topics \MajorSub 03-XX Mathematical logic and foundations 03-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 03-01 Instructional exposition (textbooks, tutorial papers, etc.) 03-02 Research exposition (monographs, survey articles) 03-03 Historical (must be assigned at least one Dclassification number from Section 01) 03-04 Explicit machine computation and programs (not the theory of computation or programming) 03-06 Proceedings, conferences, collections, etc. 03A05 Philosophical and critical, {For philosophy of mathematics, See 00A30} 03Bxx General logic 03B05 Classical propositional logic 03B10 Classical first-order logic 03B15 Higher-order logic and type-theory 03B20 Subsystems of classical logic (including intuitionistic logic) 03B22 Abstract deductive systems 03B25 Decidability of theories and sets of sentences, See also {11U05, 12L05, 20F10} 03B30 Foundations and axiomatics of classical theories 03B35 Mechanization of proofs and logical operations, See also {68T15} 03B40 Combinatory logic and $lambda$-calculus 03B45 Modal and tense logic, {For provability logics, See also 03F40} 03B46 Relevance and entailment 03B48 Probability and inductive logic, See also {60A05} 03B50 Many-valued logic 03B52 Fuzzy logic, See also {94D05} 03B53 Paraconsistent logic 03B55 Intermediate logics 03B60 Other nonclassical logic 03B65 Logic of natural languages, See also {68S05, 92K20} 03B70 Logic of programming, See also {68Q55, 68Q60} 03B80 Other applications of logic 03B99 None of the above but in this section 03Cxx Model theory 03C05 Equational classes, universal algebra, See also {08Axx} 03C07 Basic properties of first-order languages and structures 03C10 Quantifier elimination and related topics 03C13 Finite structures 03C15 Denumerable structures 03C20 Ultraproducts and related constructions 03C25 Model-theoretic forcing 03C30 Other model constructions 03C35 Categoricity and completeness of theories 03C40 Interpolation, preservation, definability 03C45 Stability and related concepts 03C50 Models with special properties (saturated, rigid, etc.) 03C52 Properties of classes of models 03C55 Set-theoretic model theory 03C57 Recursion-theoretic model theory, See also {03D45} 03C60 Model-theoretic algebra, See also {08C10, 12Lxx, 13L05} 03C62 Models of arithmetic and set theory, See also {03Hxx} 03C65 Models of other mathematical theories 03C68 Other classical first-order model theory 03C70 Logic on admissible sets 03C75 Other infinitary logic 03C80 Logic with extra quantifiers and operators, See also {03B45} 03C85 Second- and higher-order model theory 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) 03C95 Abstract model theory 03C99 None of the above but in this section 03Dxx Recursion theory 03D03 Thue and Post systems, etc. 03D05 Automata and formal grammars in connection with logical questions, See also {68Qxx} 03D10 Turing machines and related notions, See also {68Q05} 03D15 Complexity of computation, See also {68Q15} 03D20 Recursive functions and relations, subrecursive hierarchies 03D25 Recursively enumerable sets and degrees 03D30 Other degrees; reducibilities 03D35 Undecidability and degrees of sets of sentences 03D40 Word problems, etc., See also {06B25, 08A50, 20F10} 03D45 Theory of numerations, effectively presented structures, See also {03C57} 03D50 Recursive equivalence types of sets and structures, isols 03D55 Hierarchies 03D60 Recursion theory on ordinals, admissible sets, etc. 03D65 Higher-type and set recursion theory 03D70 Inductive definability 03D75 Abstract and axiomatic recursion theory 03D80 Applications of recursion theory 03D99 None of the above but in this section 03Exx Set theory, See also {04-XX} 03E05 Combinatorial set theory, See also {04A20} 03E10 Ordinal and cardinal numbers, See also {04A10} 03E15 Descriptive set theory, See also {04A15, 28A05, 54H05} 03E20 Other classical set theory 03E25 Axiom of choice and related propositions, See also {04A25} 03E30 Axiomatics of classical set theory and its fragments 03E35 Consistency and independence results 03E40 Other aspects of forcing and Boolean-valued models 03E45 Constructibility, ordinal definability, and related notions 03E47 Other notions of set-theoretic definability 03E50 Continuum hypothesis and Martin's axiom, See also { 04A30, 54A25} 03E55 Large cardinals 03E60 Determinacy and related principles which contradict the axiom of choice 03E65 Other hypotheses and axioms 03E70 Nonclassical and second-order set theories 03E72 Fuzzy sets See mainly{ 04A72} 03E75 Applications 03E99 None of the above but in this section 03Fxx Proof theory and constructive mathematics 03F03 Proof theory, general 03F05 Cut-elimination and normal-form theorems 03F07 Structure of proofs 03F10 Functionals in proof theory 03F15 Recursive ordinals and ordinal notations 03F20 Complexity of proofs 03F25 Relative consistency and interpretations 03F30 First-order arithmetic and fragments 03F35 Second- and higher-order arithmetic and fragments, See also {03E30, 03E70} 03F40 Godel numberings in proof theory 03F50 Metamathematics of constructive systems 03F55 Intuitionistic mathematics 03F60 Constructive and recursive analysis, See also {26E40, 46S30, 47S30} 03F65 Other constructive mathematics, See also {26E40, 46S30, 47S30} 03F99 None of the above but in this section 03Gxx Algebraic logic 03G05 Boolean algebras, See also {06Exx} 03G10 Lattices and related structures, See also {06Bxx} 03G12 Quantum logic, See also {81P10} 03G15 Cylindric and polyadic algebras; relation algebras 03G20 Lukasiewicz and Post algebras, See also {06D25, 06D30} 03G25 Other algebras related to logic, See also {06F35} 03G30 Categorical logic, topoi, See also {18B25} 03G99 None of the above but in this section 03Hxx Nonstandard models, See also {03C62} 03H05 Nonstandard models in mathematics, See also {26E35, 28E05, 30G06, 46S20, 47S20, 54J05} 03H10 Other applications of nonstandard models (economics, physics, etc.) 03H15 Nonstandard models of arithmetic, See also {11U10, 12L15, 13L05} 03H99 None of the above but in this section \MajorSub 04-XX Set theory, See also {03Exx} 04-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 04-01 Instructional exposition (textbooks, tutorial papers, etc.) 04-02 Research exposition (monographs, survey articles) 04-03 Historical (must be assigned at least one classification number from Section 01) 04-04 Explicit machine computation and programs (not the theory of computation or programming) 04-06 Proceedings, conferences, collections, etc. 04A03 Set algebra 04A05 Relations, functions, See also {08A02} 04A10 Ordinal and cardinal numbers; generalizations, See Also { 03E10} 04A15 Descriptive set theory; Borel classifications, Suslin schemes, etc., See also {03E15, 26A21, 28A05, 54H05} 04A20 Combinatorial set theory, See also {03E05, 05A05}; filters 04A25 Axiom of choice and equivalent propositions (Zorn's lemma, etc.), See also {03E25} 04A30 Continuum hypothesis, generalized continuum hypothesis, See also {03E50, 54A25} 04A72 Fuzzy sets, fuzzy relations, See also {03E72, 94D05}, {For fuzzy versions, See specific sections} 04A99 Miscellaneous topics \MajorSub 05-XX Combinatorics, {For finite fields, See 11Txx} 05-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 05-01 Instructional exposition (textbooks, tutorial papers, etc.) 05-02 Research exposition (monographs, survey articles) 05-03 Historical (must be assigned at least one classification number from Section 01) 05-04 Explicit machine computation and programs (not the theory of computation or programming) 05-06 Proceedings, conferences, collections, etc. 05Axx Classical combinatorial problems 05A05 Combinatorial choice problems (subsets, representatives, permutations) 05A10 Factorials, binomial coefficients, combinatorial functions, See also {11B65, 33Cxx} 05A15 Exact enumeration problems, generating functions, See Also {33Cxx, 33Dxx} 05A16 Asymptotic enumeration 05A17 Partitions of integers, See also {11P81, 11P82, 11P83} 05A18 Partitions of sets 05A19 Combinatorial identities 05A20 Combinatorial inequalities 05A30 $q$-calculus and related topics, See also {03Dxx} 05A40 Umbral calculus 05A99 None of the above but in this section 05Bxx Designs and configurations, {For applications of design theory, See 94C30} 05B05 Block designs, See also {51E05, 62K10} 05B07 Triple systems 05B10 Difference sets (number-theoretic, group-theoretic, etc.), See also {11B13} 05B15 Orthogonal arrays, Latin squares, Room squares 05B20 Matrices (incidence, Hadamard, etc.) 05B25 Finite geometries, See also {51D20, 51Exx} 05B30 Other designs, configurations, See also {51E30} 05B35 Matroids, geometric lattices, See also {52B40, 90C27} 05B40 Packing and covering, See also {11H31, 52C15, 52C17} 05B45 Tessellation and tiling problems, See also {52C20, 52C22} 05B50 Polyominoes 05B99 None of the above but in this section 05Cxx Graph theory, {For applications of graphs, See 68Q90, 68R10, 90C35, 94C15} 05C05 Trees 05C10 Topological graph theory, imbedding, See also {57M15, 57M25} 05C12 Distance in graphs 05C15 Chromatic theory of graphs and maps 05C20 Directed graphs (digraphs), tournaments 05C25 Graphs and groups, See also {20F32} 05C30 Enumeration of graphs and maps 05C35 Extremal problems, See also {90C35} 05C38 Paths and cycles, See also {90B10} 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs 05C50 Graphs and matrices 05C55 Generalized Ramsey theory 05C60 Isomorphism problems (reconstruction conjecture, perfect graphs, etc.) 05C65 Hypergraphs 05C70 Factorization, matching, covering and packing 05C75 Structural characterization of types of graphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C80 Random graphs 05C85 Graph algorithms, See also {68Q20, 68R10} 05C90 Applications 05C99 None of the above but in this section 05Dxx Extremal combinatorics 05D05 Extremal set theory 05D10 Ramsey theory 05D15 Transversal (matching) theory 05D99 None of the above but in this section 05Exx Algebraic combinatorics 05E05 Symmetric functions 05E10 Tableaux, representations of the symmetric group, See also {20C30} 05E15 Combinatorial problems concerning the classical groups, See also {22E45, 33C80} 05E20 Group actions on designs, geometries and codes 05E25 Group actions on posets and homology groups of posets, See also {06A09} 05E30 Association schemes, strongly regular graphs 05E35 Orthogonal polynomials, See also {33C45, 33C50, 33D45} 05E99 None of the above but in this section \MajorSub 06-XX Order, lattices, ordered algebraic structures, See also {18B35} 06-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 06-01 Instructional exposition (textbooks, tutorial papers, etc.) 06-02 Research exposition (monographs, survey articles) 06-03 Historical (must be assigned at least one classification number from Section 01) 06-04 Explicit machine computation and programs (not the theory of computation or programming) 06-06 Proceedings, conferences, collections, etc. 06Axx Ordered sets 06A05 Total order 06A06 Partial order, general 06A07 Combinatorics of partially ordered sets 06A08 Shellable posets, Cohen-Macaulay posets, See also {52B20} 06A09 Cohomology of posets, See also {05E25} 06A12 Semilattices, See also {20M10} 06A15 Galois correspondences, closure operators 06A23 Complete lattices, completions 06A99 None of the above but in this section 06Bxx Lattices, See also {03G10} 06B05 Structure theory 06B10 Ideals, congruence relations 06B15 Representation theory 06B20 Varieties of lattices 06B25 Free lattices, projective lattices, word problems, See also {03D40, 08A50, 20F10} 06B30 Topological lattices, order topologies, See also {06F30, 22A26, 54F05, 54H12} 06B35 Continuous lattices, generalizations, applications, See also {06B30, 06D10, 06F30, 18B35, 22A26, 68Q55} 06B99 None of the above but in this section 06Cxx Modular lattices, complemented lattices 06C05 Modular lattices, Desarguesian lattices 06C10 Semimodular lattices, geometric lattices 06C15 Complemented lattices, orthocomplemented lattices 06C20 Complemented modular lattices, continuous geometries 06C99 None of the above but in this section 06Dxx Distributive lattices 06D05 Structure and representation theory 06D10 Complete distributivity 06D15 Pseudocomplemented lattices 06D20 Heyting algebras, See also {03Gxx} 06D25 Post algebras, See also {03G20} 06D30 De Morgan algebras, Lukasiewicz algebras, See also {03G20} 06D99 None of the above but in this section 06Exx Boolean algebras (Boolean rings), See also {03G05} 06E05 Structure theory 06E10 Chain conditions, complete algebras 06E15 Stone space and related constructions 06E20 Ring-theoretic properties, See also {16E50, 16G30} 06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.) 06E30 Boolean functions, See also {94C10} 06E99 None of the above but in this section 06Fxx Ordered structures 06F05 Ordered semigroups, See also {20Mxx} 06F10 Noether lattices 06F15 Ordered groups, See also {20F60} 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces, See also {46A40} 06F25 Ordered rings, algebras, modules, {For ordered fields, See 12J15; See also 13J25, 16W80} 06F30 Topological lattices, order topologies, See also {06B30, 22A26, 54F05, 54H12} 06F35 BCK-algebras, BCI-algebras, See also {03G25} 06F99 None of the above but in this section \MajorSub 08-XX General algebraic systems 08-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 08-01 Instructional exposition (textbooks, tutorial papers, etc.) 08-02 Research exposition (monographs, survey articles) 08-03 Historical (must be assigned at least one classification number from Section 01) 08-04 Explicit machine computation and programs (not the theory of computation or programming) 08-06 Proceedings, conferences, collections, etc. 08Axx Algebraic structures, See also {03C05} 08A02 Relational systems, laws of composition 08A05 Structure theory 08A30 Subalgebras, congruence relations 08A35 Automorphisms, endomorphisms 08A40 Operations, polynomials, primal algebras 08A45 Equational compactness 08A50 Word problems, See also {03D40, 06B25, 20F10, 68R15} 08A55 Partial algebras 08A60 Unary algebras 08A62 Finitary algebras 08A65 Infinitary algebras 08A70 Applications of universal algebra in computer science 08A99 None of the above but in this section 08Bxx Varieties 08B05 Equational logic, Malcev (Maltsev) conditions 08B10 Congruence modularity, congruence distributivity 08B15 Lattices of varieties 08B20 Free algebras 08B25 Products, amalgamated products, and other kinds of limits and colimits, See also {18A30} 08B26 Subdirect products and subdirect irreducibility 08B30 Injectives, projectives 08B99 None of the above but in this section 08Cxx Other classes of algebras 08C05 Categories of algebras, See also {18C05} 08C10 Axiomatic model classes, See also {03Cxx, in particular 03C60} 08C15 Quasivarieties 08C99 None of the above but in this section \MajorSub 11-XX Number theory 11-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 11-01 Instructional exposition (textbooks, tutorial papers, etc.) 11-02 Research exposition (monographs, survey articles) 11-03 Historical (must be assigned at least one classification number from Section 01) 11-04 Explicit machine computation and programs (not the theory of computation or programming) 11-06 Proceedings, conferences, collections, etc. 11Axx Elementary number theory, {For analogues in number fields, See 11R04} 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11A07 Congruences; primitive roots; residue systems 11A15 Power residues, reciprocity 11A25 Arithmetic functions; related numbers; inversion formulas 11A41 Primes 11A51 Factorization; primality 11A55 Continued fractions, {For approximation results, See 11J70; See also 11K50, 30B70, 40A15} 11A63 Radix representation; digital problems, {For metric results, See 11K16} 11A67 Other representations 11A99 None of the above but in this section 11Bxx Sequences and sets 11B05 Density, gaps, topology 11B13 Additive bases, See also {05B10} 11B25 Arithmetic progressions, See also {11N13} 11B34 Representation functions 11B37 Recurrences, {For applications to special functions, See 33-XX} 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B50 Sequences (mod m) 11B57 Farey sequences; the sequences ${1^k, 2^k, ... }$ 11B65 Binomial coefficients; factorials; $q$-identities, See also {05A10, 05A30} 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 11B75 Other combinatorial number theory 11B83 Special sequences and polynomials 11B85 Automata sequences 11B99 None of the above but in this section 11Cxx Polynomials and matrices 11C08 Polynomials, See also {13F20} 11C20 Matrices, determinants, See also {15A36} 11C99 None of the above but in this section 11Dxx Diophantine equations, See also {11Gxx, 14Gxx} 11D04 Linear equations 11D09 Quadratic and bilinear equations 11D25 Cubic and quartic equations 11D41 Higher degree equations; Fermat's equation 11D57 Multiplicative and norm form equations 11D61 Exponential equations 11D68 Rational numbers as sums of fractions 11D72 Equations in many variables, See also {11P55} 11D75 Diophantine inequalities, See also {11J25} 11D79 Congruences in many variables 11D85 Representation problems, See also {11P55} 11D88 $p$-adic and power series fields 11D99 None of the above but in this section 11Exx Forms and linear algebraic groups, See also {19Gxx}, {For quadratic forms in linear algebra, See 15A63} 11E04 Quadratic forms over general fields 11E08 Quadratic forms over local rings and fields 11E10 Forms over real fields 11E12 Quadratic forms over global rings and fields 11E16 General binary quadratic forms 11E20 General ternary and quaternary quadratic forms; forms of more than two variables 11E25 Sums of squares and representations by other particular quadratic forms 11E39 Bilinear and Hermitian forms 11E41 Class numbers of quadratic and Hermitian forms 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11E57 Classical groups, See also {14Lxx, 20Gxx} 11E70 $K$-theory of quadratic and Hermitian forms 11E72 Galois cohomology of linear algebraic groups, See also {20G10} 11E76 Forms of degree higher than two 11E81 Algebraic theory of quadratic forms; Witt groups and rings, See also {19G12, 19G24} 11E88 Quadratic spaces; Clifford algebras, See also {15A63, 15A66} 11E95 $p$-adic theory 11E99 None of the above but in this section 11Fxx Discontinuous groups and automorphic forms, See also {11R39, 11S37, { 14-XX, 22Exx} { 14Gxx, 14Kxx, 22E50, 22E55}, 30F35, 32Nxx; for relations with quadratic forms, See 11E45} 11F03 Modular and automorphic functions 11F06 Structure of modular groups and generalizations; arithmetic groups, See also {20H05, 20H10, 22E40} 11F11 Modular forms, one variable 11F12 Automorphic forms, one variable 11F20 Dedekind eta function, Dedekind sums 11F22 Relationship to Lie algebras and finite simple groups 11F25 Hecke-Petersson operators, differential operators (one variable) 11F27 Theta series; Weil representation 11F30 Fourier coefficients of automorphic forms 11F32 Modular correspondences, etc. 11F33 Congruences for modular and $p$-adic modular forms, See also {14G20, 22E50} 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F41 Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, See also {14J20} 11F46 Siegel modular groups and their modular and automorphic forms 11F55 Other groups and their modular and automorphic forms (several variables) 11F60 Hecke-Petersson operators, differential operators (several variables) 11F66 Dirichlet series and functional equations in connection with modular forms 11F67 Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; Selberg trace formula 11F75 Cohomology of arithmetic groups 11F80 Galois properties 11F85 $p$-adic theory, local fields, See also {14G20, 22E50} 11F99 None of the above but in this section 11Gxx Arithmetic algebraic geometry (Diophantine geometry), See also {11Dxx, { 14-XX} { 14Gxx, 14Kxx}} 11G05 Elliptic curves over global fields, See also {14H52} 11G07 Elliptic curves over local fields, See also {14G20, 14H52} 11G09 Drinfeld modules; higher-dimensional motives, etc., See also {14L05} 11G10 Abelian varieties of dimension $\gtr 1$, See also {14Kxx} 11G15 Complex multiplication and moduli of abelian varieties, See also {14K22} 11G16 Elliptic and modular units, See also {11R27} 11G18 Arithmetic aspects of modular and Shimura varieties, See also {14G35} 11G20 Curves over finite and local fields, See also {14H25} 11G25 Varieties over finite and local fields, See also {14G15, 14G20} 11G30 Curves of arbitrary genus or genus $\ne 1$ over global fields, See also {14H25} 11G35 Varieties over global fields, See also { 14G25} 11G40 $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, See also {14G10} 11G45 Geometric class field theory, See also {11R37, 14C35, 19F05} 11G99 None of the above but in this section 11Hxx Geometry of numbers, {For applications in coding theory, See 94B75} 11H06 Lattices and convex bodies, See also {11P21, 52C05, 52C07} 11H16 Nonconvex bodies 11H31 Lattice packing and covering, See also {05B40, 52C15, 52C17} 11H46 Products of linear forms 11H50 Minima of forms 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11H56 Automorphism groups of lattices 11H60 Mean value and transfer theorems 11H99 None of the above but in this section 11Jxx Diophantine approximation, transcendental number theory, See also {11K60} 11J04 Homogeneous approximation to one number 11J06 Markov and Lagrange spectra and generalizations 11J13 Simultaneous homogeneous approximation, linear forms 11J17 Approximation by numbers from a fixed field 11J20 Inhomogeneous linear forms 11J25 Diophantine inequalities, See also {11D75} 11J54 Small fractional parts of polynomials and generalizations 11J61 Approximation in non-Archimedean valuations 11J68 Approximation to algebraic numbers 11J70 Continued fractions and generalizations, See also {11A55, 11K50} 11J71 Distribution modulo one, See also {11K06} 11J72 Irrationality; linear independence over a field 11J81 Transcendence (general theory) 11J82 Measures of irrationality and of transcendence 11J83 Metric theory 11J85 Algebraic independence; Gelfond's method 11J86 Linear forms in logarithms; Baker's method 11J89 Transcendence theory of elliptic and abelian functions 11J91 Transcendence theory of other special functions 11J99 None of the above but in this section 11Kxx Probabilistic theory: distribution modulo $1$; metric theory of algorithms 11K06 General theory of distribution modulo $1$, See also {11J71} 11K16 Normal numbers, radix expansions, etc., See also {11A63} 11K31 Special sequences 11K36 Well-distributed sequences and other variations 11K38 Irregularities of distribution, discrepancy, See also {11Nxx} 11K41 Continuous, $p$-adic and abstract analogues 11K45 Pseudo-random numbers; Monte Carlo methods 11K50 Metric theory of continued fractions, See also {11A55, 11J70} 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension, See also {11N99, 28Dxx} 11K60 Diophantine approximation, See also {11Jxx} 11K65 Arithmetic functions, See also {11Nxx} 11K70 Harmonic analysis and almost periodicity 11K99 None of the above but in this section 11Lxx Exponential sums and character sums, {For finite fields, See 11Txx} 11L03 Trigonometric and exponential sums, general 11L05 Gauss and Kloosterman sums; generalizations 11L07 Estimates on exponential sums 11L10 Jacobsthal and Brewer sums; other complete character sums 11L15 Weyl sums 11L20 Sums over primes 11L26 Sums over arbitrary intervals 11L40 Estimates on character sums 11L99 None of the above but in this section 11Mxx Zeta and $L$-functions: analytic theory 11M06 $zeta (s)$ and $L(s, chi)$ 11M20 Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11M26 Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M35 Hurwitz and Lerch zeta functions 11M41 Other Dirichlet series and zeta functions, {For local and global ground fields, See 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, See 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M45 Tauberian theorems, See also {40E05} 11M99 None of the above but in this section 11Nxx Multiplicative number theory 11N05 Distribution of primes 11N13 Primes in progressions, See also {11B25} 11N25 Distribution of integers with specified multiplicative constraints 11N30 Turan theory, See also {30Bxx} 11N32 Primes represented by polynomials; other multiplicative structure of polynomial values 11N35 Sieves 11N36 Applications of sieve methods 11N37 Asymptotic results on arithmetic functions 11N45 Asymptotic results on counting functions for algebraic and topological structures 11N56 Rate of growth of arithmetic functions 11N60 Distribution functions associated with additive and positive multiplicative functions 11N64 Other results on the distribution of values or the characterization of arithmetic functions 11N69 Distribution of integers in special residue classes 11N75 Applications of automorphic functions and forms to multiplicative problems, See also {11Fxx} 11N80 Generalized primes and integers 11N99 None of the above but in this section 11Pxx Additive number theory; partitions 11P05 Waring's problem and variants 11P21 Lattice points in specified regions 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method, See also {11D85} 11P81 Elementary theory of partitions, See also {05A17} 11P82 Analytic theory of partitions 11P83 Partitions; congruences and congruential restrictions 11P99 None of the above but in this section 11Rxx Algebraic number theory: global fields, {For complex multiplication, See 11G15} 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers 11R09 Polynomials (irreducibility, etc.) 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11R18 Cyclotomic extensions 11R20 Other abelian and metabelian extensions 11R21 Other number fields 11R23 Iwasawa theory 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11R32 Galois theory 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers, See Also {20C10} 11R34 Galois cohomology, See also {12Gxx, 16H05, 19A31} 11R37 Class field theory 11R39 Langlands-Weil conjectures, nonabelian class field theory, See also {11Fxx, 22E55} 11R42 Zeta functions and $L$-functions of number fields, See also {11M41, 19F27} 11R44 Distribution of prime ideals, See also {11N05} 11R45 Density theorems 11R47 Other analytic theory, See also {11Nxx} 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11R54 Other algebras and orders, and their zeta and $L$-functions, See also {11S45, 16H05, 16Kxx} 11R56 Adele rings and groups 11R58 Arithmetic theory of algebraic function fields, See also {14-XX} 11R65 Class groups and Picard groups of orders 11R70 $K$-theory of global fields, See also {19Fxx} 11R80 Totally real and totally positive fields, See also {12J15} 11R99 None of the above but in this section 11Sxx Algebraic number theory: local and $p$-adic fields 11S05 Polynomials 11S15 Ramification and extension theory 11S20 Galois theory 11S23 Integral representations 11S25 Galois cohomology, See also {12Gxx, 16H05} 11S31 Class field theory; $p$-adic formal groups, See also {14L05} 11S37 Langlands-Weil conjectures, nonabelian class field theory, See also {11Fxx, 22E50} 11S40 Zeta functions and $L$-functions, See also {11M41, 19F27} 11S45 Algebras and orders, and their zeta functions, See Also {11R52, 11R54, 16H05, 16Kxx} 11S70 $K$-theory of local fields, See also {19Fxx} 11S80 Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 11S85 Other nonanalytic theory 11S99 None of the above but in this section 11Txx Finite fields and commutative rings (number-theoretic aspects) 11T06 Polynomials 11T22 Cyclotomy 11T23 Exponential sums 11T24 Other character sums and Gauss sums 11T30 Structure theory 11T55 Arithmetic theory of polynomial rings over finite fields 11T71 Algebraic coding theory; cryptography 11T99 None of the above but in this section 11Uxx Connections with logic 11U05 Decidability, See also {03B25} 11U07 Ultraproducts, See also {03C20} 11U09 Model theory, See also {03Cxx} 11U10 Nonstandard arithmetic, See also {03H15} 11U99 None of the above but in this section 11Yxx Computational number theory, See also {11-04} 11Y05 Factorization 11Y11 Primality 11Y16 Algorithms; complexity, See also {68Q25} 11Y35 Analytic computations 11Y40 Algebraic number theory computations 11Y50 Computer solution of Diophantine equations 11Y55 Calculation of integer sequences 11Y60 Evaluation of constants 11Y65 Continued fraction calculations 11Y70 Values of arithmetic functions; tables 11Y99 None of the above but in this section 11Z50 Miscellaneous applications of number theory \MajorSub 12-XX Field theory and polynomials 12-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 12-01 Instructional exposition (textbooks, tutorial papers, etc.) 12-02 Research exposition (monographs, survey articles) 12-03 Historical (must be assigned at least one classification number from Section 01) 12-04 Explicit machine computation and programs (not the theory of computation or programming) 12-06 Proceedings, conferences, collections, etc. 12Dxx Real and complex fields 12D05 Polynomials: factorization 12D10 Polynomials: location of zeros (algebraic theorems), {For the analytic theory, See 26C10, 30C15} 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), See also {11Exx} 12D99 None of the above but in this section 12Exx General field theory 12E05 Polynomials (irreducibility, etc.) 12E10 Special polynomials 12E12 Equations 12E15 Skew fields, division rings, See also {11R52, 11R54, 11S45, 16Kxx} 12E20 Finite fields (field-theoretic aspects) 12E25 Hilbertian fields; Hilbert's irreducibility theorem 12E99 None of the above but in this section 12Fxx Field extensions 12F05 Algebraic extensions 12F10 Separable extensions, Galois theory 12F12 Inverse Galois theory 12F15 Inseparable extensions 12F20 Transcendental extensions 12F99 None of the above but in this section 12Gxx Homological methods (field theory) 12G05 Galois cohomology, See also {13A20, 16H05} 12G10 Cohomological dimension 12G99 None of the above but in this section 12Hxx Differential and difference algebra 12H05 Differential algebra, See also {13Nxx} 12H10 Difference algebra, See also {39Axx} 12H20 Abstract differential equations, See also {34Gxx} 12H25 $p$-adic differential equations, See also {11S80, 14G20, 34Gxx} 12H99 None of the above but in this section 12Jxx Topological fields 12J05 Normed fields 12J10 Valued fields 12J12 Formally $p$-adic fields 12J15 Ordered fields 12J17 Topological semifields 12J20 General valuation theory 12J25 Non-Archimedean valued fields, See also {30G06, 32P05, 46S10, 47S10} 12J27 Krasner-Tate algebras See mainly{ 32P05; See also 46S10, 47S10} 12J99 None of the above but in this section 12Kxx Generalizations of fields 12K05 Near-fields, See also {16Y30} 12K10 Semifields, See also {16Y60} 12K99 None of the above but in this section 12Lxx Connections with logic 12L05 Decidability, See also {03B25} 12L10 Ultraproducts, See also {03C20} 12L12 Model theory, See also {03C60} 12L15 Nonstandard arithmetic, See also {03H15} 12L99 None of the above but in this section 12Y05 Computational aspects of field theory and polynomials \MajorSub 13-XX Commutative rings and algebras 13-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 13-01 Instructional exposition (textbooks, tutorial papers, etc.) 13-02 Research exposition (monographs, survey articles) 13-03 Historical (must be assigned at least one classification number from Section 01) 13-04 Explicit machine computation and programs (not the theory of computation or programming) 13-06 Proceedings, conferences, collections, etc. 13Axx General commutative ring theory 13A02 Graded rings, See also {16W50} 13A05 Divisibility 13A10 Radical theory 13A15 Ideals; multiplicative ideal theory 13A18 Valuations and their generalizations 13A20 Brauer groups, See also {12Gxx, 16H05} 13A30 Associated graded rings of ideals (Rees ring, form ring) and related topics 13A35 Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$, See also {13Mxx} 13A50 Invariant theory, See also {14D25} 13A99 None of the above but in this section 13Bxx Ring extensions and related topics 13B02 Extension theory 13B05 Galois theory (commutative rings) 13B10 Morphisms and derivations 13B15 Ramification theory 13B21 Integral dependence 13B22 Integral closure; integrally closed rings, related rings (Japanese, etc.) 13B24 Going up; going down; going between 13B25 Polynomials over commutative rings 13B30 Quotients and localization 13B35 Completion, See also {13J10} 13B40 Etale extensions and Henselization; Artin approximation, See also {13J15, 14B12} 13B99 None of the above but in this section 13Cxx Theory of modules and ideals 13C05 Structure, classification theorems 13C10 Projective and free modules and ideals, See also {18G05, 19A13} 13C11 Injective and flat modules and ideals, See also {18G05} 13C12 Torsion modules and ideals 13C13 Other special types 13C14 Cohen-Macaulay modules, See also {13H10} 13C15 Dimension theory, depth, related rings (catenary, etc.) 13C20 Class groups 13C40 Linkage, complete intersections and determinantal ideals, See also {14M12} 13C99 None of the above but in this section 13Dxx (Co)homological methods 13D02 Syzygies 13D03 (Co)homology of commutative rings and algebras 13D05 (Co)homological dimension, See also {18G20} 13D10 Deformations and infinitesimal methods, See also {14B12, 14D15, 16S80, 32Gxx} 13D15 Grothendieck groups, $K$-theory, See also {14C35, 18F30, 19-XX} 13D25 Complexes 13D30 Torsion theory, See also {13C12, 18E40} 13D40 Hilbert-Samuel functions and Poincare series 13D45 Local cohomology, See also {14B15} 13D99 None of the above but in this section 13Exx Chain conditions, finiteness conditions 13E05 Noetherian rings and modules 13E10 Artinian rings and modules, finite-dimensional algebras 13E15 Rings and modules of finite generation or presentation 13E99 None of the above but in this section 13Fxx Arithmetic rings and other special rings, See also {12-XX} 13F05 Dedekind and Prufer rings and their generalizations 13F07 Euclidean rings and generalizations 13F10 Principal ideal rings 13F15 Factorial rings, unique factorization domains, See Also { 14M05} 13F20 Polynomial rings and ideals, See also {11C08} 13F25 Formal power series rings, See also {13J05} 13F30 Valuation rings 13F40 Excellent rings 13F45 Seminormal rings 13F50 Rings with straightening laws, Hodge algebras 13F99 None of the above but in this section 13G05 Integral domains 13Hxx Local rings and semilocal rings 13H05 Regular local rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), See also {14M05} 13H15 Multiplicity theory and related topies 13H99 None of the above but in this section 13Jxx Topological rings and modules, See also {16W60, 16W80} 13J05 Power series rings, See also {13F25} 13J07 Analytical algebras and rings, See also {32B05} 13J10 Complete rings, completion, See also {13B35} 13J15 Henselian rings, See also {13B40} 13J20 Global topological rings 13J25 Ordered rings, See also {06F25} 13J99 None of the above but in this section 13K05 Witt vectors and related rings 13L05 Applications of logic to commutative algebra, See Also { 03Cxx, 03Hxx} 13Mxx Finite commutative rings, {For number-theoretic aspects, See 11Txx} 13M05 Structure 13M10 Polynomials (commutative rings) 13M99 None of the above but in this section 13Nxx Differential algebra, See also {12H05, 14F10} 13N05 Modules of differentials, See also {16S32} 13N10 Rings of differential operators, See also {16S32, 32C38} 13N99 None of the above but in this section 13Pxx Computational aspects of commutative algebra, See Also { 68Q40} 13P05 Polynomials, factorization, See also {12Y05} 13P10 Polynomial ideals, Grobner bases 13P99 None of the above but in this section \MajorSub 14-XX Algebraic geometry 14-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 14-01 Instructional exposition (textbooks, tutorial papers, etc.) 14-02 Research exposition (monographs, survey articles) 14-03 Historical (must be assigned at least one classification number from Section 01) 14-04 Explicit machine computation and programs (not the theory of computation or programming) 14-06 Proceedings, conferences, collections, etc. 14Axx Foundations 14A05 Relevant commutative algebra, See also {13-XX} 14A10 Varieties 14A15 Schemes 14A20 Generalizations (algebraic spaces, motifs) 14A22 Noncommutative algebraic geometry; algebraic supervarieties, See also {14M30, 32C11, 58A50} 14A25 Elementary questions 14A99 None of the above but in this section 14Bxx Local theory, See also {32Sxx} 14B05 Singularities, See also {14E15, 14H20, 32Sxx, 58C27} 14B07 Deformations of singularities, See also {14D15, 32Sxx} 14B10 Infinitesimal methods, See also {13D10} 14B12 Local deformation theory, Artin approximation, etc., See also {13B40, 13D10} 14B15 Local cohomology, See also {13D45, 32C36} 14B20 Formal neighborhoods 14B99 None of the above but in this section 14Cxx Cycles and subschemes 14C05 Parametrization (Chow and Hilbert schemes) 14C10 Equivalence relations 14C15 Rational equivalence rings 14C17 Intersection theory 14C20 Divisors, linear systems, invertible sheaves 14C21 Pencils, nets, webs, See also {53A60} 14C22 Picard groups 14C25 Algebraic cycles 14C30 Transcendental methods, Hodge theory, See also {14D07, 32G20, 32J25, 32S35}, Hodge conjecture 14C34 Torelli problem, See also {32G20} 14C35 Applications of methods of algebraic $K$-theory, See Also { 14F05, 19Exx} 14C40 Riemann-Roch theorems, See also {19E20, 19L10} 14C99 None of the above but in this section 14Dxx Families, fibrations 14D05 Structure of families (Picard-Lefschetz, Picard-Fuchs theory, etc.) 14D07 Variation of Hodge structures 14D10 Arithmetic ground fields (finite, local, global) 14D15 Formal methods; deformations, See also {13D10, 14B07, 16S80 32Gxx} 14D20 Algebraic moduli problems, moduli of vector bundles, {For analytic moduli problems, See 32G13} 14D22 Fine and coarse moduli spaces 14D25 Geometric invariants, See also {14L30} 14D99 None of the above but in this section 14Exx Mappings and correspondences 14E05 Rational maps, birational correspondences 14E07 Birational automorphisms, Cremona group and generalizations, See also {32G20} 14E09 Automorphisms, See also {14J50, 14L27} 14E10 General correspondences 14E15 Global theory of singularities, resolution, See Also { 14B05, 32S20, 32S45} 14E20 Coverings, fundamental group (mappings) 14E22 Ramification problems, See also {11S15} 14E25 Imbeddings 14E30 Minimal models 14E35 Results in dimension $\leq 3$ 14E40 Local structure of maps: etale, flat, etc., See Also { 13-XX, 14F20} 14E99 None of the above but in this section 14Fxx (Co)homology theory, See also {13Dxx} 14F05 Vector bundles, sheaves, related construction, See Also { 18F20, 32Lxx, 46M20} 14F10 Differentials and other special sheaves, See also {32C38} 14F17 Vanishing theorems, See also {32L20} 14F20 Etale and other Grothendieck topologies and cohomologies 14F25 Classical real and complex cohomology 14F30 $p$-adic cohomology, crystalline cohomology 14F32 Intersection (co)homology, See also {32S60} 14F35 Homotopy theory; fundamental groups, See also {14E20, 14H30} 14F40 de Rham cohomology, See also {14C30, 32C35, 32L10} 14F45 Topological properties 14F99 None of the above but in this section 14Gxx Arithmetic problems. Diophantine geometry, See also {11Dxx, 11Gxx} 14G05 Rationality questions, rational points 14G10 Zeta-functions and related questions, See also {11G40} (Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields 14G20 $p$-adic ground fields 14G25 Global ground fields 14G27 Nonalgebraically closed ground fields 14G35 Modular and Shimura varieties, See also {11F41, 11F46, 11G18} 14G40 Arithmetic varieties and schemes; Arakelov theory 14G99 None of the above but in this section 14Hxx Curves 14H05 Algebraic functions; function fields, See also {11R58} 14H10 Families, moduli (algebraic) 14H15 Families, moduli (analytic), See also {30F10, 32Gxx} 14H20 Singularities, local rings, See also {13Hxx} 14H25 Arithmetic ground fields, See also {11Dxx, 11G05, 14Gxx} 14H30 Coverings, fundamental group, See also {14E20, 14F35} 14H35 Correspondences, See also {14Exx} 14H40 Jacobians, See also {32G20} 14H42 Theta functions; Schottky problem, See also {14K25, 32G20} 14H45 Special curves and curves of low genus 14H50 Space curves 14H52 Elliptic curves, See also {11G05, 11G07, 14Kxx} 14H55 Riemann surfaces; Weierstrass points; gap sequences, See also {30Fxx} 14H60 Vector bundles on curves, See also {14F05} 14H99 None of the above but in this section 14Jxx Surfaces and higher-dimensional varieties, {For analytic theory, See 32Jxx} 14J05 Picard group, See also {14C22, 19A49, 32L05} 14J10 Families, moduli, classification: algebraic theory 14J15 Moduli, classification: analytic theory, See also {32G13, 32J15} 14J17 Singularities of surfaces 14J20 Arithmetic ground fields, See also {11Dxx, 11G25, 11G35, 14Gxx} 14J25 Special surfaces, {For Hilbert modular surfaces, See 14G35} 14J26 Rational and ruled surfaces 14J27 Elliptic surfaces 14J28 $K3$ surfaces and Enriques surfaces 14J29 Surfaces of general type 14J30 Special $3$-folds, See also {14E05} 14J35 Special $4$-folds, See also {14E05} 14J40 Special $n$-folds 14J45 Fano varieties 14J50 Automorphisms of surfaces and higher-dimensional varieties, See also {14E09} 14J60 Vector bundles on surfaces and higher-dimensional varieties, See also {14F05, 32Lxx} 14J70 Hypersurfaces 14J99 None of the above but in this section 14Kxx Abelian varieties and schemes 14K02 Isogeny 14K05 Algebraic theory 14K10 Algebraic moduli, classification 14K15 Arithmetic ground fields, See also {11Dxx, 11Fxx, 11Gxx, 14Gxx} 14K20 Analytic theory; abelian integrals and differentials 14K22 Complex multiplication, See also {11G15} 14K25 Theta functions 14K30 Picard schemes, higher Jacobians, See also {14H40, 32G20} 14K99 None of the above but in this section 14Lxx Group schemes, {For linear algebraic groups, See 20Gxx. For Lie algebras, See 17B45} 14L05 Formal groups, $p$-divisible groups, See also {55N22} 14L10 Group varieties 14L15 Group schemes 14L17 Affine algebraic groups, hyperalgebra constructions, See also {17B45, 18D35} 14L27 Automorphism groups, See also {14E09} 14L30 Group actions on varieties or schemes (quotients), See also {14D25} 14L35 Classical groups (geometric aspects), See also {20Gxx, 51N30} 14L40 Other algebraic groups (geometric aspects) 14L99 None of the above but in this section 14Mxx Special varieties 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), See also {13C14, 13F45, 13H10} 14M06 Linkage, See also {13C40} 14M07 Low codimension problems, See also {14Cxx} 14M10 Complete intersections, See also {13C40} 14M12 Determinantal varieties, See also {13C40} 14M15 Grassmannians, Schubert varieties, flag manifolds, See also {32M10, 51M35} 14M17 Homogeneous spaces and generalizations, See also {32M10, 53C30, 57T15} 14M20 Rational varieties 14M25 Toric varieties, Newton polyhedra 14M30 Supervarieties, See also {14A22, 32C11, 58A50} 14M99 None of the above but in this section 14Nxx Classical methods and problems, See also {51-XX} 14N05 Projective techniques, See also {51N35} 14N10 Enumerative problems (combinatorial problems) 14N99 None of the above but in this section 14Pxx Real algebraic and real analytic geometry 14P05 Real algebraic sets, See also {12Dxx} 14P10 Semialgebraic sets and related spaces 14P15 Real analytic and semianalytic sets, See also {32B20, 32C05} 14P20 Nash functions and manifolds, See also {32C07, 58A07} 14P25 Topology of real algebraic varieties 14P99 None of the above but in this section 14Qxx Computational aspects in algebraic geometry, See also {12-04, 68Q40} 14Q05 Curves 14Q10 Surfaces, hypersurfaces 14Q15 Higher-dimensional varieties 14Q20 Effectivity 14Q99 None of the above but in this section \MajorSub 15-XX Linear and multilinear algebra; matrix theory {(finite and infinite)} 15-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 15-01 Instructional exposition (textbooks, tutorial papers, etc.) 15-02 Research exposition (monographs, survey articles) 15-03 Historical (must be assigned at least one classification number from Section 01) 15-04 Explicit machine computation and programs (not the theory of computation or programming) 15-06 Proceedings, conferences, collections, etc. 15A03 Vector spaces, linear dependence, rank 15A04 Linear transformations, semilinear transformations 15A06 Linear equations 15A09 Matrix inversion, generalized inverses 15A12 Conditioning of matrices, See also {65F35} 15A15 Determinants, permanents, other special matrix functions, See also {19B10, 19B14} 15A18 Eigenvalues, singular values, and eigenvectors 15A21 Canonical forms, reductions, classification 15A22 Matrix pencils, See also {47A56} 15A23 Factorization of matrices 15A24 Matrix equations and identities 15A27 Commutativity 15A30 Algebraic systems of matrices, See also {16S50, 20Gxx, 20Hxx} 15A33 Matrices over special rings (quaternions, finite fields, etc.) 15A36 Matrices of integers, See also {11C20} 15A39 Linear inequalities 15A42 Inequalities involving eigenvalues and eigenvectors 15A45 Miscellaneous inequalities involving matrices 15A48 Positive matrices and their generalizations; cones of matrices 15A51 Stochastic matrices 15A52 Random matrices 15A54 Matrices over function rings in one or more variables 15A57 Other types of matrices (Hermitian, skew-Hermitian, etc.) 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory, See also {65F35, 65J05} 15A63 Quadratic and bilinear forms, inner products See mainly{ 11Exx} 15A66 Clifford algebras, spinors 15A69 Multilinear algebra, tensor products 15A72 Vector and tensor algebra, theory of invariants, See Also { 13A50, 14D25} 15A75 Exterior algebra, Grassmann algebras 15A78 Other algebras built from modules 15A90 Applications of matrix theory to physics 15A99 Miscellaneous topics \MajorSub 16-XX Associative rings and algebras, {For the commutative case, See 13-XX} 16-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 16-01 Instructional exposition (textbooks, tutorial papers, etc.) 16-02 Research exposition (monographs, survey articles) 16-03 Historical (must be assigned at least one classification number from Section 01) 16-04 Explicit machine computation and programs (not the theory of computation or programming) 16-06 Proceedings, conferences, collections, etc. 16Bxx General and miscellaneous 16B50 Category-theoretic methods and results (except as in 16D90, 16E10), See also {18-XX} 16B70 Applications of logic, See also {03Cxx} 16B99 None of the above but in this section 16Dxx Modules, bimodules and ideals 16D10 General module theory 16D15 1-sided ideals 16D20 Bimodules 16D25 2-sided ideals 16D30 Maximal and prime 2-sided ideals, See also {16N60, 16D60}, simple rings (except as in 16Kxx) 16D40 Free, projective, and flat modules and ideals, See Also { 18G05, 19A13} 16D50 Injective modules, self-injective rings, See also {16L60, 18G05} 16D60 Simple and semisimple modules, primitive rings and ideals 16D70 Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation 16D80 Other classes of modules and ideals, See also {16G60} 16D90 Module categories, See also {16Exx, 16Gxx, 16S90}; module theory in a category-theoretic context; Morita equivalence and duality 16D99 None of the above but in this section 16Exx Homological methods and results, See also {18Gxx} 16E10 Homological dimension 16E20 Grothendieck groups, $K$-theory, etc., See also {18F30, 19-XX} 16E30 Homological functors on modules 16E40 Hochschild and other homology and cohomology theories for rings 16E50 von Neumann regular rings and generalizations 16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. 16E70 Other rings of low global or flat dimension 16E99 None of the above but in this section 16Gxx Representation theory of rings and algebras 16G10 Representations of Artinian rings 16G20 Representations of quivers and partially ordered sets 16G30 Representations of orders, lattices, algebras over commutative rings, See also {16H05} 16G50 Cohen-Macaulay modules 16G60 Representation type (finite, tame, wild, etc.) 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16G99 None of the above but in this section 16H05 Orders and arithmetic, separable algebras, Azumaya algebras, See also {11R52, 11R54, 11S45, 13A20} 16Kxx Division rings and semisimple Artin rings, See also {12E15, 15A30} 16K20 Finite-dimensional, {For Brauer group theory, See 12Gxx, 13A20; for crossed products, See 16S35} 16K40 Infinite-dimensional and general 16K99 None of the above but in this section 16Lxx Local rings and generalizations 16L30 Noncommutative local and semilocal rings, perfect rings 16L60 Quasi-Frobenius rings, See also {16D50} 16L99 None of the above but in this section 16Nxx Radicals and radical properties of rings 16N20 Jacobson radical, quasimultiplication 16N40 Nil and nilpotent radicals, sets, ideals, rings 16N60 Prime and semiprime rings, See also {16D30, 16D60, 16U10} 16N80 General radicals and rings, {For radicals in module categories, See 16S90} 16N99 None of the above but in this section 16Pxx Chain conditions, growth conditions, and other forms of finiteness 16P10 Finite rings and finite-dimensional algebras, {For semisimple, See 16K20; for commutative, See 11Txx, 13Mxx} 16P20 Artinian rings and modules 16P40 Noetherian rings and modules 16P50 Localization and Noetherian rings, See also {16U20} 16P60 Chain conditions on annihilators and summands: Goldie type conditions, See also {16U20}, Krull dimension 16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 16P90 Growth rate, Gelfand-Kirillov dimension 16P99 None of the above but in this section 16Rxx Rings with polynomial identity 16R10 $T$-ideals, identities, varieties of rings and algebras 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16R30 Trace rings and invariant theory 16R40 Identities other than those of matrices over commutative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution) 16R99 None of the above but in this section 16Sxx Rings and algebras arising under various constructions 16S10 Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 16S20 Centralizing and normalizing extensions 16S30 Universal enveloping algebras of Lie algebras See mainly{ 17B35} 16S32 Rings of differential operators, See also {13N10, 32C38} 16S34 Group rings, See also {20C05, 20C07}, Laurent polynomial rings 16S35 Twisted and skew group rings, crossed products 16S36 Ordinary and skew polynomial rings and semigroup rings, See also {20M25} 16S40 Smash products of general Hopf actions, See also {16W30} 16S50 Endomorphism rings: matrix rings, See also {15-XX} 16S60 Rings of functions, subdirect products, sheaves of rings 16S70 Extensions of rings by ideals 16S80 Deformations of rings, See also {14D15} 16S90 Maximal ring of quotients, torsion theories, radicals on module categories, See also {13D30, 18E40}, {For radicals of rings, See { 16Nxx}} 16S99 None of the above but in this section 16Uxx Conditions on elements (including elements of matrix rings, etc.) 16U10 Integral domains 16U20 Ore rings, multiplicative sets, Ore localization 16U30 Divisibility, noncommutative UFDs 16U50 Algebraicity and local finiteness, See also {16N40} 16U60 Units, groups of units, general linear groups 16U70 Center, normalizer (invariant elements) 16U80 Generalizations of commutativity 16U99 None of the above but in this section 16Wxx Rings and algebras with additional structure 16W10 Rings with involution: Lie, Jordan and other nonassociative structures, See also {17B60, 17C50, 46Kxx} 16W20 Automorphisms and endomorphisms, actions of groups and semigroups and their fixed rings 16W25 Derivations, actions of Lie algebras 16W30 Coalgebras, bialgebras, Hopf algebras, See also {57T05, 16S30-- 16S40}; rings, modules, etc. on which these act 16W50 Graded rings and modules 16W55 ``Super'' (or ``skew'') structure, See also {17A70, 17B70, 17C70}, {For exterior algebras, See { 15A75}; for Clifford algebras, See 11E88, 15A66} 16W60 Filtrations and valuations, generalizations of Euclidean algorithm, completions, formal power series and related constructions, See also {13Jxx} 16W80 Topological and ordered rings and modules, See also {13Jxx} 16W99 None of the above but in this section 16Yxx Generalizations, {For nonassociative rings, See 17-xx} 16Y30 Near-rings, See also {12K05} 16Y60 Semirings, See also {12K10} 16Y99 None of the above but in this section \MajorSub 17-XX Nonassociative rings and algebras 17-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 17-01 Instructional exposition (textbooks, tutorial papers, etc.) 17-02 Research exposition (monographs, survey articles) 17-03 Historical (must be assigned at least one classification number from Section 01) 17-04 Explicit machine computation and programs (not the theory of computation or programming) 17-06 Proceedings, conferences, collections, etc. 17-08 Computational methods 17Axx General nonassociative rings 17A01 General theory 17A05 Power-associative 17A10 Commutative power-associative 17A15 Noncommutative Jordan algebras 17A20 Flexible algebras 17A25 Nodal algebras 17A30 Algebras satisfying other identities 17A35 Division algebras 17A36 Automorphisms, derivatives, other operators 17A40 Ternary compositions 17A42 Other $n$-ary compositions 17A45 Quadratic algebras (but not quadratic Jordan algebras) 17A50 Free algebras 17A60 Structure theory 17A65 Radical theory 17A70 Superalgebras 17A75 Composition algebras 17A80 Valued algebras 17A99 None of the above but in this section 17Bxx Lie algebras, {For Lie groups, See 22Exx} 17B01 Identities, free Lie algebras 17B05 Structure theory 17B10 Representations, algebraic theory (weights) 17B15 Representations, analytic theory 17B20 Simple, semisimple, reductive algebras (roots) 17B25 Exceptional algebras 17B30 Solvable, nilpotent algebras 17B35 Universal enveloping algebras, See also {16S30} 17B37 Quantum groups and related deformations, See also {16W30, 81R50, 82B23} 17B40 Automorphisms, derivations, other operators 17B45 Lie algebras of linear algebraic groups, See also {14Lxx and 20Gxx} 17B50 Modular Lie algebras 17B55 Homological methods in Lie algebras 17B56 Cohomology of Lie algebras 17B60 Lie rings associated with other structures (associative, Jordan, etc.), See also {15A30, 16W10, 17C40, 17C50} 17B65 Infinite-dimensional Lie algebras, See also {22E65} 17B66 Lie algebras of vector fields and related algebras 17B67 Kac-Moody algebras (structure and representation theory) 17B68 Virasoro and related algebras 17B70 Graded Lie algebras 17B81 Applications to physics 17B99 None of the above but in this section 17Cxx Jordan algebras (algebras, triples and pairs) 17C05 Identities and free Jordan structures 17C10 Structure theory 17C17 Radicals 17C20 Simple, semisimple algebras 17C27 Idempotents, Peirce decompositions 17C30 Associated groups, automorphisms 17C36 Associated manifolds 17C37 Associated geometries 17C40 Exceptional Jordan structures 17C50 Jordan structures associated with other structures, See also {16W10} 17C55 Finite-dimensional structures 17C60 Division algebras 17C65 Jordan structures on Banach spaces and algebras, See Also { 46H70, 46L70} 17C70 Super structures 17C90 Applications to physics 17C99 None of the above but in this section 17Dxx Other nonassociative rings and algebras 17D05 Alternative rings 17D10 Malcev (Maltsev) rings and algebras 17D15 Right alternative rings 17D20 $(gamma, delta)$-rings, including $(1,-1)$-rings 17D25 Lie-admissible algebras 17D92 Genetic algebras 17D99 None of the above but in this section \MajorSub 18-XX Category theory, homological algebra 18-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 18-01 Instructional exposition (textbooks, tutorial papers, etc.) 18-02 Research exposition (monographs, survey articles) 18-03 Historical (must be assigned at least one classification number from Section 01) 18-04 Explicit machine computation and programs (not the theory of computation or programming) 18-06 Proceedings, conferences, collections, etc. 18Axx General theory of categories and functors 18A05 Definitions, generalizations 18A10 Graphs, diagram schemes, precategories, neocategories, See also {20Lxx} 18A15 Foundations, relations to logic and deductive systems, See also {03-XX} 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms, factorization (bicategories) 18A22 Special properties of functors (faithful, full, etc.) 18A23 Natural morphisms, dinatural morphisms 18A25 Functor categories, comma categories 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18A32 Factorization of morphisms (via images, coimages, dominions, codominions), substructures, quotient structures, congruences, amalgams 18A35 Categories admitting limits (complete categories), functors commuting with limits, continuous functors, completions 18A40 Adjoint functors (representable functors, universal constructions, reflective subcategories, reflections, etc.), constructions of adjoints (Kan extensions, etc.) 18A99 None of the above but in this section 18Bxx Special categories 18B05 Category of sets, characterizations, See also {03-XX} 18B10 Category of relations, additive relations 18B15 Embedding theorems, universal categories, See also {18E20} 18B20 Categories of machines, automata, operative categories, See also {03D05, 68Qxx} 18B25 Topoi, See also {03G30} 18B30 Categories of topological spaces and continuous mappings, See also {54-XX} 18B35 Preorders, orders and lattices (viewed as categories), See also {06-XX} 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories), See also {20Axx, 20Lxx, 20Mxx} 18B99 None of the above but in this section 18Cxx Categories and algebraic theories 18C05 Equational categories, See also {03C05, 08C05} 18C10 Theories (e.g. algebraic theories), structure, and semantics, See also {03G30} 18C15 Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples, See also {18Gxx} 18C20 Algebras and Kleisli categories associated with monads 18C99 None of the above but in this section 18Dxx Categories with structure 18D05 Double categories, $2$-categories, bicategories, hypercategories 18D10 Monoidal categories (= multiplicative categories), See also {19D23} 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 18D20 Enriched categories (over closed or monoidal categories) 18D25 Strong functors, strong adjunctions 18D30 Fibered categories 18D35 Structured objects in a category (group objects, etc.) 18D99 None of the above but in this section 18Exx Abelian categories 18E05 Preadditive, additive categories 18E10 Exact categories, abelian categories 18E15 Grothendieck categories 18E20 Embedding theorems, See also {18B15} 18E25 Derived functors and satellites 18E30 Derived categories, triangulated categories 18E35 Localization of categories 18E40 Torsion theories, radicals, See also {13D30, 16S90} 18E99 None of the above but in this section 18Fxx Categories and geometry 18F05 Local categories and functors 18F10 Grothendieck topologies, See also {14F20} 18F15 Abstract manifolds and fiber bundles, See also {55Rxx, 57Pxx} 18F20 Presheaves and sheaves, See also {14F05, 32C35, 32L10, 54B40, 55N30} 18F25 Algebraic $K$-theory and $L$-theory, See also {11Exx, 11R70, 11S70, 12-XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67} 18F30 Grothendieck groups, See also {13D15, 16E20, 19Axx} 18F99 None of the above but in this section 18Gxx Homological algebra, See also {13Dxx, 16Exx, 55Uxx} 18G05 Projectives and injectives, See also {13C10, 13C11, 16D40, 16D50} 18G10 Resolutions; derived functors, See also {18E25} 18G15 Ext and Tor, generalizations, Kunneth formula, See Also { 55U25} 18G20 Homological dimension, See also {13D05, 16E10} 18G25 Relative homological algebra, projective classes 18G30 Simplicial sets, simplicial objects (in a category), See also {55U10} 18G35 Chain complexes, See also {18E30, 55U15} 18G40 Spectral sequences, hypercohomology, See also {55Txx} 18G50 Nonabelian homological algebra 18G55 Nonabelian homotopical algebra 18G60 Other (co)homology theories (cyclic, dihedral, etc.), See also {19D55, 46L80, 58B30, 58G12} 18G99 None of the above but in this section \MajorSub 19-XX $K$-theory, See also {16E20, 18F25} 19-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 19-01 Instructional exposition (textbooks, tutorial papers, etc.) 19-02 Research exposition (monographs, survey articles) 19-03 Historical (must be assigned at least one classification number from Section 01) 19-04 Explicit machine computation and programs (not the theory of computation or programming) 19-06 Proceedings, conferences, collections, etc. 19Axx Grothendieck groups and $K_0$, See also {13D15, 18F30} 19A13 Stability for projective modules, See also {13C10} 19A15 Efficient generation 19A22 Frobenius induction, Burnside and representation rings 19A31 $K_0$ of group rings and orders 19A49 $K_0$ of other rings 19A99 None of the above but in this section 19Bxx Whitehead groups and $K_1$ 19B10 Stable range conditions 19B14 Stability for linear groups 19B28 $K_1$ of group rings and orders, See also {57Q10} 19B37 Congruence subgroup problems, See also {20H05} 19B99 None of the above but in this section 19Cxx Steinberg groups and $K_2$ 19C09 Central extensions and Schur multipliers 19C20 Symbols, presentations and stability of $K_2$ 19C30 $K_2$ and the Brauer group 19C40 Excision for $K_2$ 19C99 None of the above but in this section 19Dxx Higher algebraic $K$-theory 19D06 $Q$- and plus-constructions 19D10 Algebraic $K$-theory of spaces 19D23 Symmetric monoidal categories, See also {18D10} 19D25 Karoubi-Villamayor-Gersten $K$-theory 19D35 Negative $K$-theory, NK and Nil 19D45 Higher symbols, Milnor $K$-theory 19D50 Computations of higher $K$-theory of rings, See Also { 13D15, 16E20} 19D55 $K$-theory and homology; cyclic homology and cohomology, See also {18G60} 19D99 None of the above but in this section 19Exx $K$-theory in geometry 19E08 $K$-theory of schemes, See also {14C35} 19E15 Algebraic cycles, See also {14C25, 14C35} 19E20 Relations with cohomology theories, See also {14Fxx} 19E99 None of the above but in this section 19Fxx $K$-theory in number theory, See also {11R70, 11S70} 19F05 Generalized class field theory, See also {11G45} 19F15 Symbols and arithmetic, See also {11R37} 19F27 Etale cohomology, higher regulators, zeta and $L$-functions, See also {11G40, 11R42, 11S40, 14F20, 14G10} 19F99 None of the above but in this section 19Gxx $K$-theory of forms, See also {11Exx} 19G05 Stability for quadratic modules 19G12 Witt groups of rings, See also {11E81} 19G24 $L$-theory of group rings, See also {11E81} 19G38 Hermitian $K$-theory, relations with $K$-theory of rings 19G99 None of the above but in this section 19Jxx Obstructions from topology 19J05 Finiteness and other obstructions in $K_0$ 19J10 Whitehead (and related) torsion 19J25 Surgery obstructions, See also {57R67} 19J35 Obstructions to group actions 19J99 None of the above but in this section 19Kxx $K$-theory and operator algebras See mainly{ 46L80, and also 46M20} 19K14 $K_0$ as an ordered group, traces 19K33 EXT and $K$-homology, See also {55N22} 19K35 Kasparov theory ($KK$-theory), See also {58G12} 19K56 Index theory, See also {58G12} 19K99 None of the above but in this section 19Lxx Topological $K$-theory, See also {55N15, 55R50, 55S25} 19L10 Riemann-Roch theorems, Chern characters 19L20 $J$-homomorphism, Adams operations, See also {55Q50} 19L41 Connective $K$-theory, cobordism, See also {55N22} 19L47 Equivariant $K$-theory, See also {55N91, 55P91, 55Q91, 55R91, 55S91} 19L64 Computations, geometric applications 19L99 None of the above but in this section 19M05 Miscellaneous applications of $K$-theory \MajorSub 20-XX Group theory and generalizations 20-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 20-01 Instructional exposition (textbooks, tutorial papers, etc.) 20-02 Research exposition (monographs, survey articles) 20-03 Historical (must be assigned at least one classification number from Section 01) 20-04 Explicit machine computation and programs (not the theory of computation or programming) 20-06 Proceedings, conferences, collections, etc. 20Axx Foundations 20A05 Axiomatics and elementary properties 20A10 Metamathematical considerations, {For word problems, See 20F10} 20A15 Applications of logic to group theory 20A99 None of the above but in this section 20Bxx Permutation groups 20B05 General theory for finite groups 20B07 General theory for infinite groups 20B10 Characterization theorems 20B15 Primitive groups 20B20 Multiply transitive finite groups 20B22 Multiply transitive infinite groups 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures, See also {05Bxx, 12F10, 20G40, 20H30, 51-XX} 20B27 Infinite automorphism groups, See also {12F10} 20B30 Symmetric groups 20B35 Subgroups of symmetric groups 20B40 Computational methods 20B99 None of the above but in this section 20Cxx Representation theory of groups, See also {19A22 (for representation rings and Burnside rings)} 20C05 Group rings of finite groups and their modules, See Also { 16S34} 20C07 Group rings of infinite groups and their modules, See Also { 16S34} 20C10 Integral representations of finite groups 20C11 $p$-adic representations of finite groups 20C12 Integral representations of infinite groups 20C15 Ordinary representations and characters 20C20 Modular representations and characters 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups 20C32 Representations of infinite symmetric groups 20C33 Representations of finite groups of Lie type 20C34 Representations of sporadic groups 20C35 Applications of group representations to physics 20C40 Computational methods 20C99 None of the above but in this section 20Dxx Abstract finite groups 20D05 Classification of simple and nonsolvable groups 20D06 Simple groups: alternating groups and groups of Lie type, See also {20Gxx, 22Exx} 20D08 Simple groups: sporadic groups 20D10 Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks, See also {20F17} 20D15 Nilpotent groups, $p$-groups 20D20 Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20D25 Special subgroups (Frattini, Fitting, etc.) 20D30 Series and lattices of subgroups 20D35 Subnormal subgroups 20D40 Products of subgroups 20D45 Automorphisms 20D50 Covering of subgroups 20D60 Arithmetic and combinatorial problems 20D99 None of the above but in this section 20Exx Structure and classification of infinite or finite groups 20E05 Free nonabelian groups 20E06 Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 Subgroup theorems 20E08 Groups acting on trees 20E10 Quasivarieties and varieties of groups 20E15 Chains and lattices of subgroups, subnormal subgroups, See also {20F22} 20E18 Limits, profinite groups 20E22 Extensions, wreath products, and other compositions, See also {20J05} 20E25 Local properties 20E26 Residual properties and generalizations 20E28 Maximal subgroups 20E32 Simple groups, See also {20D05} 20E34 General structure theorems 20E36 General theorems concerning automorphisms of groups 20E42 Groups with a $BN$-pair; buildings, See also {51E24} 20E99 None of the above but in this section 20Fxx Special aspects of infinite or finite groups 20F05 Generators, relations, and presentations 20F06 Cancellation theory; application of van Kampen diagrams, See also {57M05} 20F10 Word problems, other decision problems, connections with logic and automata, See also {03B25, 03D05, 03D40, 06B25, 08A50, 68Qxx} 20F12 Commutator calculus 20F14 Derived series, central series, and generalizations 20F16 Solvable groups, supersolvable groups 20F17 Formations of groups, Fitting classes, See also {20D10} 20F18 Nilpotent groups 20F19 Generalizations of solvable and nilpotent groups 20F22 Other classes of groups defined by subgroup chains 20F24 FC-groups and their generalizations 20F28 Automorphism groups of groups, See also {20E36} 20F29 Representations of groups as automorphism groups of algebraic systems 20F32 Geometric group theory, See also {05C25, 20Exx, 20Gxx} 20F34 Fundamental groups and their automorphisms, See Also { 57M05, 57Sxx, 22E40} 20F36 Braid groups; Artin groups 20F38 Other groups related to topology or analysis 20F40 Associated Lie structures 20F45 Engel conditions 20F50 Periodic groups; locally finite groups 20F55 Coxeter groups, See also {22E40} 20F60 Ordered groups See mainly{ 06F15} 20F99 None of the above but in this section 20Gxx Linear algebraic groups (classical groups), {For arithmetic theory, See 11E57, 11H06; for geometric theory, See 14Lxx, 22Exx; for other methods in representation theory, See { 15AQ30, 22Exx} { 15A30, 22E45, 22E46, 22E47, 22E50, 22E55}} 20G05 Representation theory 20G10 Cohomology theory 20G15 Linear algebraic groups over arbitrary fields 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 20G25 Linear algebraic groups over local fields and their integers 20G30 Linear algebraic groups over global fields and their integers 20G35 Linear algebraic groups over adeles and other rings and schemes 20G40 Linear algebraic groups over finite fields 20G45 Applications to physics; explicit representations 20G99 None of the above but in this section 20Hxx Other groups of matrices, See also {15A30} 20H05 Unimodular groups, congruence subgroups, See also {11F06, 19B37, 22E40, 51F20} 20H10 Fuchsian groups and their generalizations, See also {11F06, 22E40, 30F35, 32Nxx} 20H15 Other geometric groups, including crystallographic groups, See also {51-XX, especially 51F15, and 82D25} 20H20 Other matrix groups over fields 20H25 Other matrix groups over rings 20H30 Other matrix groups over finite fields 20H99 None of the above but in this section 20Jxx Connections with homological algebra and category theory 20J05 Homological methods in group theory 20J06 Cohomology of finite groups 20J10 Groups arising as cohomology groups 20J15 Category of groups 20J99 None of the above but in this section 20Kxx Abelian groups 20K01 Finite abelian groups 20K05 Finitely generated groups 20K10 Torsion groups, primary groups and generalized primary groups 20K12 Ulm sequences 20K15 Torsion free groups, finite rank 20K20 Torsion free groups, infinite rank 20K21 Mixed groups 20K25 Direct sums, direct products, etc. 20K26 Indecomposable groups 20K27 Subgroups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. 20K35 Extensions 20K40 Homological and categorical methods 20K45 Topological methods, See also {22A05, 22B05} 20K99 None of the above but in this section 20Lxx Groupoids (i.e. small categories in which all morphisms are isomorphisms), {For sets with a single binary operation, See 20N02; for topological groupoids, See 22A22, 58H05} 20L05 General theory 20L10 Connections with group theory 20L13 Mappings of groupoids 20L15 Connections with topology 20L17 Connections with category theory 20L99 None of the above but in this section 20Mxx Semigroups 20M05 Free semigroups, generators and relations, word problems 20M07 Varieties of semigroups 20M10 General structure theory 20M11 Radical theory 20M12 Ideal theory 20M14 Commutative semigroups 20M15 Mappings of semigroups 20M17 Regular semigroups 20M18 Inverse semigroups 20M19 Orthodox semigroups 20M20 Semigroups of transformations, etc., See also {47D03, 47H20, 54H15} 20M25 Semigroup rings, multiplicative semigroups of rings, See also {16S36, 16Y60} 20M30 Representation of semigroups 20M35 Semigroups in automata theory, linguistics, etc., See Also { 03D05, 68Qxx, 68S05} 20M50 Connections of semigroups with homological algebra and category theory 20M99 None of the above but in this section 20Nxx Other generalizations of groups 20N02 Sets with a single binary operation (groupoids) 20N05 Loops, quasigroups, See also {05Bxx} 20N07 Mappings of loops 20N10 Ternary systems (heaps, semiheaps, heapoids, etc.) 20N15 $n$-ary systems 20N20 Hypergroups 20N25 Fuzzy groups, See also {04A72} 20N99 None of the above but in this section 20P05 Probabilistic methods in group theory \MajorSub 22-XX Topological groups, Lie groups, {For transformation groups, See 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, See 43-XX} 22-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 22-01 Instructional exposition (textbooks, tutorial papers, etc.) 22-02 Research exposition (monographs, survey articles) 22-03 Historical (must be assigned at least one classification number from Section 01) 22-04 Explicit machine computation and programs (not the theory of computation or programming) 22-06 Proceedings, conferences, collections, etc. 22Axx Topological and differentiable algebraic systems, {For topological rings and fields, See 12Jxx, 13Jxx, 16W80; for dual spaces of operator algebras and topological groups, See 47D35} 22A05 Structure of general topological groups 22A10 Analysis on general topological groups 22A15 Structure of topological semigroups 22A20 Analysis on topological semigroups 22A22 Topological groupoids (including differentiable and Lie groupoids) 22A25 Representations of general topological groups and semigroups 22A26 Topological semilattices, lattices and applications, See also {06B30, 06B35, 06F30} 22A30 Other topological algebraic systems and their representations 22A99 None of the above but in this section 22Bxx Locally compact abelian groups (LCA groups) 22B05 General properties and structure of LCA groups 22B10 Structure of group algebras of LCA groups 22B99 None of the above but in this section 22C05 Compact groups 22Dxx Locally compact groups and their algebras 22D05 General properties and structure of locally compact groups 22D10 Unitary representations of locally compact groups 22D12 Other representations of locally compact groups 22D15 Group algebras of locally compact groups 22D20 Representations of group algebras 22D25 $C$*-algebras and $W$*-algebras arising from group representations, See also {46Lxx} 22D30 Induced representations 22D35 Duality theorems 22D40 Ergodic theory on groups, See also {28Dxx, 43A60} 22D45 Automorphism groups of locally compact groups 22D99 None of the above but in this section 22Exx Lie groups, {For the topology of Lie groups and homogeneous spaces, See {57-XX, 57Sxx, 57Txx}; for analysis thereon, See {43-XX, 43A80, 43A85, 43A90} 22E05 Local Lie groups, See also {34-XX, 35-XX, 58H05} 22E10 General properties and structure of complex Lie groups, See also {32M05} 22E15 General properties and structure of real Lie groups 22E20 General properties and structure of other Lie groups 22E25 Nilpotent and solvable Lie groups 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 22E30 Analysis on real and complex Lie groups, See also {33C80, 43-XX} 22E35 Analysis on $p$-adic Lie groups, See also {11R56} 22E40 Discrete subgroups of Lie groups, See also {20Hxx, 32Nxx} 22E41 Continuous cohomology, See also {57R32, 57Txx, 58H10} 22E43 Structure and representation of the Lorentz group 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods, {For the purely algebraic theory, See 20G05} 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), See also {17B35} 22E50 Representations of Lie and linear algebraic groups over local fields 22E55 Representations of Lie and linear algebraic groups over global fields and adele rings, See also {20G05} 22E60 Lie algebras of Lie groups, {For the algebraic theory of Lie algebras, See 17Bxx} 22E65 Infinite-dimensional Lie groups and their Lie algebras, See also {17B65, 58B25, 58H05} 22E67 Loop groups and related constructions, group-theoretic treatment, See also {58D05} 22E70 Applications of Lie groups to physics; explicit representations, See also {81R05, 81R10} 22E99 None of the above but in this section \MajorSub 26-XX Real functions, See also {54C30} 26-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 26-01 Instructional exposition (textbooks, tutorial papers, etc.) 26-02 Research exposition (monographs, survey articles) 26-03 Historical (must be assigned at least one classification number from Section 01) 26-04 Explicit machine computation and programs (not the theory of computation or programming) 26-06 Proceedings, conferences, collections, etc. 26Axx Functions of one variable 26A03 Foundations: limits and generalizations, elementary topology of the line 26A06 One-variable calculus 26A09 Elementary functions 26A12 Rate of growth of functions, orders of infinity, slowly varying functions, See also {26A48} 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.), {For properties determined by Fourier coefficients, See 42A16; for those determined by approximation properties, See 41A25, 41A27} 26A16 Lipschitz (Holder) classes 26A18 Iteration, See also {39B12, 47H10, 54H25, 58F08, 58F13} 26A21 Classification of real functions; Baire classification of sets and functions, See also {04A15, 28A05, 54C50} 26A24 Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems, See also {28A15} 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A30 Singular functions, Cantor functions, functions with other special properties 26A33 Fractional derivatives and integrals 26A36 Antidifferentiation 26A39 Denjoy and Perron integrals, other special integrals 26A42 Integrals of Riemann, Stieltjes and Lebesgue type, See also {28-XX} 26A45 Functions of bounded variation, generalizations 26A46 Absolutely continuous functions 26A48 Monotonic functions, generalizations 26A51 Convexity, generalizations 26A99 None of the above but in this section 26Bxx Functions of several variables 26B05 Continuity and differentiation questions 26B10 Implicit function theorems, Jacobians, transformations with several variables 26B12 Calculus of vector functions 26B15 Integration: length, area, volume, See also {28A75, 51M25} 26B20 Integral formulas (Stokes, Gauss, Green, etc.) 26B25 Convexity, generalizations 26B30 Absolutely continuous functions, functions of bounded variation 26B35 Special properties of functions of several variables, Holder conditions, etc. 26B40 Representation and superposition of functions 26B99 None of the above but in this section 26Cxx Polynomials, rational functions 26C05 Polynomials: analytic properties, etc., See also {12Dxx, 12Exx} 26C10 Polynomials: location of zeros, See also {12D10, 30C15, 65H05} 26C15 Rational functions, See also {14Pxx} 26C99 None of the above but in this section 26Dxx Inequalities, {For maximal function inequalities, See 42B25; for functional inequalities, See 39B72; for probabilistic inequalities, See 60E15} 26D05 Inequalities for trigonometric functions and polynomials 26D07 Inequalities involving other types of functions 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals 26D20 Other analytical inequalities 26D99 None of the above but in this section 26Exx Miscellaneous topics, See also {58Cxx} 26E05 Real-analytic functions, See also {32B05, 32C05} 26E10 $C^\infty$-functions, quasi-analytic functions, See also {58C25} 26E15 Calculus of functions on infinite-dimensional spaces, See also {46G05, 58Cxx} 26E20 Calculus of functions taking values in infinite-dimensional spaces, See also {46E40, 46G10, 58Cxx} 26E25 Set-valued functions, See also {28B20, 54C60}, {For nonsmooth analysis, See 49J52, 58Cxx, 90Cxx} 26E30 Non-Archimedean analysis, See also {12J25} 26E35 Nonstandard analysis, See also {03H05, 28E05, 54J05} 26E40 Constructive real analysis, See also {03F60, 03F65} 26E50 Fuzzy real analysis, See also {04A72, 28E10} 26E99 None of the above but in this section \MajorSub 28-XX Measure and integration, {For analysis on manifolds, See 58-XX} 28-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 28-01 Instructional exposition (textbooks, tutorial papers, etc.) 28-02 Research exposition (monographs, survey articles) 28-03 Historical (must be assigned at least one classification number from Section 01) 28-04 Explicit machine computation and programs (not the theory of computation or programming) 28-06 Proceedings, conferences, collections, etc. 28Axx Classical measure theory 28A05 Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets, See Also { 03E15, 04A15, 26A21, 54H05} 28A10 Real- or complex-valued set functions 28A12 Contents, measures, outer measures, capacities 28A15 Abstract differentiation theory, differentiation of set functions, See also {26A24} 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A25 Integration with respect to measures and other set functions 28A33 Spaces of measures, convergence of measures, See Also { 46E27, 60Bxx} 28A35 Measures and integrals in product spaces 28A50 Integration and disintegration of measures 28A51 Lifting theory, See also {46G15} 28A60 Measures on Boolean rings, measure algebras, See Also { 54H10} 28A75 Length, area, volume, other geometric measure theory, See also {26B15, 49Q15} 28A78 Hausdorff measures 28A80 Fractals, See also {58Fxx} 28A99 None of the above but in this section 28Bxx Set functions, measures and integrals with values in abstract spaces 28B05 Vector-valued set functions, measures and integrals, See also {46G10} 28B10 Group- or semigroup-valued set functions, measures and integrals 28B15 Set functions, measures and integrals with values in ordered spaces 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections, See Also { 26E25, 54C60, 54C65, 90A14} 28B99 None of the above but in this section 28Cxx Set functions and measures on spaces with additional structure, See also {46G12, 58C35, 58D20} 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 28C10 Set functions and measures on topological groups, Haar measures, invariant measures, See also {22Axx, 43A05} 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.), See also {46G12, 58C35, 58D20, 60B11} 28C99 None of the above but in this section 28Dxx Measure-theoretic ergodic theory, See also {11K50, 11K55, 22D40, 47A35, 54H20, 58Fxx, 60Fxx, 60G10} 28D05 Measure-preserving transformations 28D10 One-parameter continuous families of measure-preserving transformations 28D15 General groups of measure-preserving transformations 28D20 Entropy and other invariants 28D99 None of the above but in this section 28Exx Miscellaneous topics in measure theory 28E05 Nonstandard measure theory, See also {03H05, 26E35} 28E10 Fuzzy measure theory, See also {04A72, 26E50, 94D05} 28E15 Other connections with logic and set theory 28E99 None of the above but in this section \MajorSub 30-XX Functions of a complex variable, {For analysis on manifolds, See 58-XX} 30-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 30-01 Instructional exposition (textbooks, tutorial papers, etc.) 30-02 Research exposition (monographs, survey articles) 30-03 Historical (must be assigned at least one classification number from Section 01) 30-04 Explicit machine computation and programs (not the theory of computation or programming) 30-06 Proceedings, conferences, collections, etc. 30Axx General properties 30A05 Monogenic properties of complex functions (including polygenic and areolar monogenic functions) 30A10 Inequalities in the complex domain 30A99 None of the above but in this section 30Bxx Series expansions 30B10 Power series (including lacunary series) 30B20 Random power series 30B30 Boundary behavior of power series, over-convergence 30B40 Analytic continuation 30B50 Dirichlet series and other series expansions, exponential series, See also {11M41, 42-XX} 30B60 Completeness problems, closure of a system of functions 30B70 Continued fractions, See also {11A55, 40A15} 30B99 None of the above but in this section 30Cxx Geometric function theory 30C10 Polynomials 30C15 Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), {For algebraic theory, See 12D10; for real methods, See 26C10} 30C20 Conformal mappings of special domains 30C25 Covering theorems in conformal mapping theory 30C30 Numerical methods in conformal mapping theory, See Also { 65E05} 30C35 General theory of conformal mappings 30C40 Kernel functions and applications 30C45 Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions 30C55 General theory of univalent and multivalent functions 30C62 Quasiconformal mappings in the plane 30C65 Quasiconformal mappings in $R^n$, other generalizations 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30C80 Maximum principle; Schwarz's lemma, Lindelof principle, analogues and generalizations; subordination 30C85 Capacity and harmonic measure in the complex plane, See also {31A15} 30C99 None of the above but in this section 30Dxx Entire and meromorphic functions, and related topics 30D05 Functional equations in the complex domain, iteration and composition of analytic functions, See also {34A20, 39-XX, 58F05, 58F23} 30D10 Representations of entire functions by series and integrals 30D15 Special classes of entire functions and growth estimates 30D20 Entire functions, general theory 30D30 Meromorphic functions, general theory 30D35 Distribution of values, Nevanlinna theory 30D40 Cluster sets, prime ends, boundary behavior 30D45 Bloch functions, normal functions, normal families 30D50 Blashke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part 30D55 ${H}^p$-classes 30D60 Quasi-analytic and other classes of functions 30D99 None of the above but in this section 30Exx Miscellaneous topics of analysis in the complex domain 30E05 Moment problems, interpolation problems 30E10 Approximation in the complex domain 30E15 Asymptotic representations in the complex domain 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions, See also {45Exx} 30E25 Boundary value problems, See also {45Exx} 30E99 None of the above but in this section 30Fxx Riemann surfaces 30F10 Compact Riemann surfaces and uniformization, See Also {14H15, 32G15} 30F15 Harmonic functions on Riemann surfaces 30F20 Classification theory of Riemann surfaces 30F25 Ideal boundary theory 30F30 Differentials on Riemann surfaces 30F35 Fuchsian groups and automorphic functions, See also {11Fxx, 20H10, 22E40, 32Gxx, 32Nxx} 30F40 Kleinian groups, See also {20H10} 30F45 Conformal metrics (hyperbolic, Poincare, distance functions) 30F50 Klein surfaces 30F60 Teichmuller theory, See also {32G15} 30F99 None of the above but in this section 30Gxx Generalized function theory 30G06 Non-Archimedean function theory, See also {12J25}; nonstandard function theory, See also {03H05} 30G12 Finely holomorphic functions and topological function theory 30G15 Topological function theory 30G20 Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.) 30G25 Discrete analytic functions 30G30 Other generalizations of analytic functions (including abstract-valued functions) 30G35 Functions of hypercomplex variables and generalized variables 30G99 None of the above but in this section 30H05 Spaces and algebras of analytic functions, See also {32E25, 46Exx, 46J15} \MajorSub 31-XX Potential theory, {For probabilistic potential theory, See 60J45} 31-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 31-01 Instructional exposition (textbooks, tutorial papers, etc.) 31-02 Research exposition (monographs, survey articles) 31-03 Historical (must be assigned at least one classification number from Section 01) 31-04 Explicit machine computation and programs (not the theory of computation or programming) 31-06 Proceedings, conferences, collections, etc. 31Axx Two-dimensional theory 31A05 Harmonic, subharmonic, superharmonic functions 31A10 Integral representations, integral operators, integral equations methods 31A15 Potentials and capacity, harmonic measure, extremal length, See also {30C85} 31A20 Boundary behavior (theorems of Fatou type, etc.) 31A25 Boundary value and inverse problems 31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation 31A35 Connections with differential equations 31A99 None of the above but in this section 31Bxx Higher-dimensional theory 31B05 Harmonic, subharmonic, superharmonic functions 31B10 Integral representations, integral operators, integral equations methods 31B15 Potentials and capacities, extremal length 31B20 Boundary value and inverse problems 31B25 Boundary behavior 31B30 Biharmonic and polyharmonic equations and functions 31B35 Connections with differential equations 31B99 None of the above but in this section 31Cxx Other generalizations 31C05 Harmonic, subharmonic, superharmonic functions 31C10 Pluriharmonic and plurisubharmonic functions, See Also { 32F05} 31C12 Potential theory on Riemannian manifolds, See also {53C20; for Hodge theory, See 58A14} 31C15 Potentials and capacities 31C20 Discrete potential theory and numerical methods 31C25 Dirichlet spaces 31C35 Martin boundary theory, See also {60J50} 31C40 Fine potential theory 31C45 Other generalizations (nonlinear potential theory, etc.) 31C99 None of the above but in this section 31D05 Axiomatic potential theory \MajorSub 32-XX Several complex variables and analytic spaces, {For infinite-dimensional holomorphy, See also 46G20, 58B12} 32-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 32-01 Instructional exposition (textbooks, tutorial papers, etc.) 32-02 Research exposition (monographs, survey articles) 32-03 Historical (must be assigned at least one classification number from Section 01) 32-04 Explicit machine computation and programs (not the theory of computation or programming) 32-06 Proceedings, conferences, collections, etc. 32Axx Holomorphic functions of several complex variables 32A05 Power series, series of functions 32A07 Special domains (Reinhardt, Hartogs, tube domains, etc.) 32A10 Holomorphic functions 32A15 Entire functions 32A17 Special families of functions (e.g. normal families) 32A20 Meromorphic functions 32A22 Nevanlinna theory (local); growth estimates; other inequalities, {For geometric theory, See 32H25, 32H30} 32A25 Integral representation 32A27 Local theory of residues, See also {32C30} 32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30), {For functions of several hypercomplex variables, See 30G35} 32A35 ${H}^p$-spaces, See also {32M15, 42B30, 43A85, 46J15} 32A37 Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA) in $n$ dimensions), See also {46Exx} 32A40 Boundary behavior 32A45 Hyperfunctions, See also {46F15} 32A99 None of the above but in this section 32Bxx Local analytic geometry, See also {13-XX and 14-XX} 32B05 Analytic algebras and generalizations, preparation theorems 32B10 Germs of analytic sets 32B15 Analytic subsets of affine space 32B20 Semi-analytic sets and subanalytic sets, See also {14P15} 32B25 Triangulation and related questions 32B99 None of the above but in this section 32Cxx General theory of analytic spaces 32C05 Real-analytic manifolds, real-analytic spaces, See Also {14Pxx, 58A07} 32C07 Real-analytic sets, complex Nash functions, See Also {14P15, 14P20} 32C10 Complex manifolds, {For almost complex manifolds, See 53C15} 32C11 Complex supergeometry, See also {14A22, 14M30, 58A50} 32C15 Complex spaces 32C16 CR-manifolds 32C17 Kahler geometry, {For differential-geometric methods, See 53C55} 32C18 Topology of analytic spaces 32C20 Normal analytic spaces 32C25 Analytic subsets and submanifolds 32C30 Integration on analytic sets and spaces, currents, {For local theory, See 32A25 or 32A27} 32C35 Analytic sheaves and cohomology groups, See also {14Fxx, 18F20, 55N30} 32C36 Local cohomology of analytic spaces 32C37 Duality theorems 32C38 Sheaves of differential operators and their modules, See also {14F10, 16S32, 35A27, 58G07} 32C81 Applications to physics 32C99 None of the above but in this section 32Dxx Analytic continuation 32D05 Domains of holomorphy 32D10 Envelopes of holomorphy 32D15 Continuation of analytic objects 32D20 Removable singularities 32D99 None of the above but in this section 32Exx Holomorphic convexity 32E05 Holomorphically convex complex spaces, reduction theory 32E10 Stein spaces, Stein manifolds 32E20 Polynomial convexity 32E25 Algebras of holomorphic functions, See also {30H05, 46J10, 46J15} 32E30 Holomorphic and polynomial approximation, Runge pairs, interpolation 32E35 Global boundary behavior of holomorphic functions 32E99 None of the above but in this section 32Fxx Geometric convexity, partial differential operators 32F05 Plurisubharmonic functions and generalizations, See Also { 31C10} 32F07 Complex Monge-Ampere operator 32F10 $q$-convexity, $q$-concavity 32F15 Pseudoconvex domains 32F20 $\overline\partial$- and $\overline\partial_b$-Neumann problems, See also {35N15} 32F25 Real submanifolds in complex manifolds 32F30 Pseudoconvex manifolds 32F40 CR structures, (tangential) CR operators and generalizations 32F99 None of the above but in this section 32Gxx Deformations of analytic structures 32G05 Deformations of complex structures, See also {13D10, 16S80, 58H10, 58H15} 32G07 Deformations of special (e.g. CR) structures 32G08 Deformations of fiber bundles 32G10 Deformations of submanifolds and subspaces 32G13 Analytic moduli problems, {For algebraic moduli problems, See 14D20, 14D22, 14H10, 14J10}, See also {14H15, 14J15} 32G15 Moduli of Riemann surfaces, Teichmuller theory, See Also { 14H15, 30Fxx} 32G20 Period matrices, variation of Hodge structure; degenerations, See also {14D05, 14D07, 14K30} 32G34 Moduli and deformations for ordinary differential equations, See also {34A20} 32G81 Applications to physics 32G99 None of the above but in this section 32Hxx Holomorphic mappings and correspondences 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions 32H04 Meromorphic mappings 32H10 Bergman kernel function, representative domains 32H15 Invariant metrics and pseudodistances 32H20 Hyperbolic complex manifolds 32H25 Picard-type theorems and generalizations, {For function-theoretic properties, See 32A22} 32H30 Value distribution theory in higher dimensions, {For function-theoretic properties, See 32A22} 32H35 Proper mappings, finiteness theorems 32H40 Boundary regularity of holomorphic maps 32H50 Iteration problems 32H99 None of the above but in this section 32Jxx Compact analytic spaces, {For Riemann surfaces, See 14Hxx, 30Fxx; for algebraic theory, See 14Jxx} 32J05 Compactification of analytic spaces 32J10 Algebraic dependence theorems 32J15 Compact surfaces 32J17 Compact $3$-folds 32J18 Compact $n$-folds $(n \ge 4)$ 32J20 Algebraicity criteria 32J25 Transcendental methods of algebraic geometry, See Also { 14C30} 32J27 Compact Kahler manifolds: generalizations, classification 32J81 Applications to physics 32J99 None of the above but in this section 32Kxx Generalizations of analytic spaces {(should also be assigned at least one other classification number in this section)} 32K05 Banach analytic spaces, See also {58Bxx} 32K07 Formal and graded complex spaces, See also {58C50} 32K15 Differentiable functions on analytic spaces, differentiable spaces, See also {58C25} 32K99 None of the above but in this section 32Lxx Holomorphic fiber spaces, See also {55Rxx} 32L05 Holomorphic fiber bundles and generalizations 32L07 Hermite-Einstein bundles; Kahler-Einstein bundles, See also {53C07} 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results, See also {14F05, 18F20, 55N30} 32L15 Bundle convexity, See also {32F10} 32L20 Vanishing theorems 32L25 Twistor theory, double fibrations 32L30 Holomorphic foliations, See also {58F18} 32L81 Applications to physics 32L99 None of the above but in this section 32Mxx Complex spaces with a group of automorphisms 32M05 Complex Lie groups, automorphism groups of complex spaces, See also {22E10} 32M10 Homogeneous complex manifolds, See also {14M17, 57T15} 32M12 Almost homogeneous manifolds and spaces, See also {14M17} 32M15 Hermitian symmetric spaces, bounded symmetric domains, See also {22E10, 22E40, 53C35, 57T15} 32M99 None of the above but in this section 32Nxx Automorphic functions, See also {11Fxx, 20H10, 22E40, 30F35} 32N05 General theory of automorphic functions of several complex variables 32N10 Automorphic forms 32N15 Automorphic functions in symmetric domains 32N99 None of the above but in this section 32P05 Non-Archimedean complex analysis (should also be assigned at least one other classification number from Section $32$ describing the type of problem) 32Sxx Singularities 32S05 Local singularities, See also {14B05} 32S10 Invariants of analytic local rings 32S15 Equisingularity (topological and analytic), See Also { 14E15} 32S20 Global theory of singularities; cohomological properties, See also {14E15} 32S25 (Hyper-) surface singularities, See also {14J17} 32S30 Deformations of singularities; vanishing cycles, See Also { 14B07} 32S35 Mixed Hodge theory of singular varieties, See also {14C30, 14D07} 32S40 Monodromy; relations with differential equations and $D$-modules 32S45 Modifications; resolution of singularities, See Also { 14E15} 32S50 Topological aspects: Lefschetz theorems, topological classification, invariants 32S55 Milnor fibration; relations with knot theory, See Also { 57M25, 57Q45} 32S60 Stratifications; constructible sheaves; intersection cohomology, See also {58C27} 32S65 Singularities of holomorphic vector fields 32S70 Other operations on singularities \MajorSub 33-XX Special functions, {33-XX deals with the properties of functions as functions. For orthogonal functions, See also 42Cxx; for aspects of combinatorics, See 05Axx; for number-theoretic aspects, See 11-XX; for representation theory, See 22Exx} 33-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 33-01 Instructional exposition (textbooks, tutorial papers, etc.) 33-02 Research exposition (monographs, survey articles) 33-03 Historical (must be assigned at least one classification number from Section 01) 33-04 Explicit machine computation and programs (not the theory of computation or programming) 33-06 Proceedings, conferences, collections, etc. 33Bxx Elementary classical functions 33B10 Exponential and trigonometric functions 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 33B99 None of the above but in this section 33Cxx Hypergeometric functions 33C05 Classical hypergeometric functions, $_2F_1$ 33C10 Bessel and Airy functions, cylinder functions, $_0F_1$ 33C15 Confluent hypergeometric functions, Whittaker functions, $_1F_1$ 33C20 Generalized hypergeometric series, $_pF_q$ 33C45 Orthogonal polynomials and functions (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.) 33C50 Orthogonal polynomials and functions in several variables 33C55 Spherical functions, spherical harmonics, ultraspherical polynomials 33C60 Hypergeometric integrals and functions defined by them ($E$, $G$ and ${H}$ functions) 33C65 Appell, Horn and Lauricella functions 33C70 Other hypergeometric functions and integrals in several variables 33C75 Elliptic integrals as hypergeometric functions 33C80 Connections with groups, algebras, root systems and related topics 33C90 Applications 33C99 None of the above but in this section 33Dxx Basic hypergeometric functions 33D05 $q$-gamma functions, $q$-beta functions and integrals 33D10 Basic theta functions 33D15 Basic hypergeometric functions in one variable 33D20 Generalized basic hypergeometric series 33D45 Basic orthogonal polynomials and functions in one and several variables 33D55 Basic spherical functions, spherical harmonics (continuous and discrete) 33D60 Basic hypergeometric integrals and functions defined by them 33D65 Bibasic functions and multiple bases 33D70 Other basic hypergeometric functions and integrals in several variables 33D80 Connections with groups, algebras, and related topics 33D90 Applications 33D99 None of the above but in this section 33Exx Other special functions 33E05 Elliptic functions and integrals 33E10 Lame, Mathieu, and spheroidal wave functions 33E15 Other wave functions 33E20 Other functions defined by series and integrals 33E30 Other functions coming from differential, difference and integral equations 33E99 None of the above but in this section \MajorSub 34-XX Ordinary differential equations 34-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 34-01 Instructional exposition (textbooks, tutorial papers, etc.) 34-02 Research exposition (monographs, survey articles) 34-03 Historical (must be assigned at least one classification number from Section 01) 34-04 Explicit machine computation and programs (not the theory of computation or programming) 34-06 Proceedings, conferences, collections, etc. 34Axx General theory 34A05 Explicit solutions and reductions 34A09 Implicit equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34A20 Differential equations in the complex domain, See Also {30D05, 32G34} 34A25 Analytical theory: series, transformations, transforms, operational calculus, etc., See also {44-XX, 47E05} 34A26 Geometric methods in differential equations 34A30 Linear equations and systems 34A34 Nonlinear equations and systems, general 34A35 Differential equations of infinite order 34A37 Differential equations with impulses 34A40 Differential inequalities 34A45 Theoretical approximation of solutions 34A46 Theoretical solution methods other than approximations 34A47 Bifurcation 34A50 Numerical approximation of solutions, {For numerical analysis, See 65Lxx} 34A55 Inverse problems 34A60 Equations with multivalued right-hand sides, See Also {49J24, 49K24} 34A65 Stiff equations 34A99 None of the above but in this section 34Bxx Boundary value problems, {For ordinary differential operators, See 34Lxx} 34B05 Linear equations 34B10 Multipoint boundary value problems 34B15 Nonlinear boundary value problems 34B20 Weyl theory and its generalizations 34B24 Sturm-Liouville theory, See also {34Lxx} 34B27 Green functions 34B30 Special equations (Mathieu, Hill, Bessel, etc.) 34B99 None of the above but in this section 34Cxx Qualitative theory, See also {58Fxx} 34C05 Location of integral curves, singular points, limit cycles 34C10 Oscillation theory, zeros, disconjugacy and comparison theory 34C11 Growth, boundedness, comparison of solutions 34C15 Nonlinear oscillations 34C20 Transformation and reduction of equations and systems, normal forms 34C23 Bifurcation See mainly{58F14} 34C25 Periodic solutions 34C27 Almost-periodic solutions 34C28 Other types of ``recurrent'' solutions 34C29 Averaging method 34C30 Manifolds of solutions 34C35 Dynamical systems, See also {54H20, 58Fxx, 70-XX} 34C37 Homoclinic and heteroclinic solutions, See also {58F15} 34C40 Equations and systems on manifolds See mainly{58Fxx, 58Gxx} 34C45 Method of integral manifolds 34C50 Method of accelerated convergence 34C99 None of the above but in this section 34Dxx Stability theory, See also {58F10, 93Dxx} 34D05 Asymptotic properties 34D08 Lyapunov exponents 34D10 Perturbations 34D15 Singular perturbations 34D20 Lyapunov stability 34D25 Popov-type stability 34D30 Structural stability and analogous concepts , See Also {58F10, 58F12} 34D35 Stability of manifolds of solutions 34D40 Ultimate boundedness 34D45 Attractors 34D99 None of the above but in this section 34Exx Asymptotic theory 34E05 Asymptotic expansions 34E10 Perturbations, asymptotics 34E15 Singular perturbations, general theory 34E20 Singular perturbations, turning point theory, WKB methods 34E99 None of the above but in this section 34F05 Equations and systems with randomness, See also {34K50, 60H10, 93E03} 34Gxx Differential equations in abstract spaces, See also {58D25} 34G10 Linear equations, See also {47Axx, 47Bxx, 47D06, 47D09} 34G20 Nonlinear equations, See also {47Hxx} 34G99 None of the above but in this section 34H05 Control problems, See also {49J25, 49K25, 93C15} 34Kxx Functional-differential and differential-difference equations with or without deviating arguments 34K05 General theory 34K10 Boundary value problems 34K15 Qualitative theory 34K20 Stability theory 34K25 Asymptotic theory 34K30 Equations in abstract spaces, See also {34Gxx} 34K35 Control problems, See also {49J25, 49K25, 93C15} 34K40 Neutral equations 34K50 Stochastic delay equations, See also {34F05, 60Hxx} 34K99 None of the above but in this section 34Lxx Ordinary differential operators, See also {47E05} 34L05 General spectral theory 34L10 Eigenfunction expansions, completeness of eigenfunctions 34L15 Estimation of eigenvalues, upper and lower bounds 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 34L25 Scattering theory 34L30 Nonlinear ordinary differential operators 34L40 Particular operators (Dirac, one-dimensional Schrodinger, etc.) 34L99 None of the above but in this section \MajorSub 35-XX Partial differential equations 35-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 35-01 Instructional exposition (textbooks, tutorial papers, etc.) 35-02 Research exposition (monographs, survey articles) 35-03 Historical (must be assigned at least one classification number from Section 01) 35-04 Explicit machine computation and programs (not the theory of computation or programming) 35-06 Proceedings, conferences, collections, etc. 35Axx General theory 35A05 General existence and uniqueness theorems 35A07 Local existence and uniqueness theorems, See also {35H05, 35Sxx} 35A08 Fundamental solutions 35A10 Cauchy-Kovalevskaya theorems 35A15 Variational methods 35A20 Analytic methods, singularities 35A22 Transform methods (e.g. integral transforms) 35A25 Other special methods 35A27 Microlocal methods; methods of sheaf theory and homological algebra in PDE, See also {32C38, 58G07} 35A30 Geometric theory, characteristics, transformations, See also {58G35, 58G37} 35A35 Theoretical approximation to solutions 35A40 Numerical approximation to solutions, {For numerical analysis, See 65Mxx, 65Nxx, 65P05} 35A99 None of the above but in this section 35Bxx Qualitative properties of solutions 35B05 General behavior of solutions of PDE (comparison theorems; oscillation, zeros and growth of solutions; mean value theorems) 35B10 Periodic solutions 35B15 Almost periodic solutions 35B20 Perturbations 35B25 Singular perturbations 35B27 Homogenization; partial differential equations in media with periodic structure, See also {73B27, 76D30} 35B30 Dependence of solutions of PDE on initial and boundary data, parameters, See also {58F14} 35B32 Bifurcation, See also {58F14} 35B35 Stability, boundedness 35B37 PDE in connection with control problems, See also {49J20, 49K20, 93C20} 35B40 Asymptotic behavior of solutions 35B45 A priori estimates 35B50 Maximum principles 35B60 Continuation and prolongation of solutions of PDE, See also {58A15, 58A17, 58Hxx} 35B65 Smoothness/regularity of solutions of PDE 35B99 None of the above but in this section 35Cxx Representations of solutions 35C05 Solutions in closed form 35C10 Series solutions, expansion theorems 35C15 Integral representations of solutions of PDE 35C20 Asymptotic expansions 35C99 None of the above but in this section 35Dxx Generalized solutions of partial differential equations 35D05 Existence of generalized solutions 35D10 Regularity of generalized solutions 35D99 None of the above but in this section 35Exx Equations and systems with constant coefficients, See Also { 35N05} 35E05 Fundamental solutions 35E10 Convexity properties 35E15 Initial value problems 35E20 General theory 35E99 None of the above but in this section 35Fxx General first-order equations and systems 35F05 General theory of linear first-order PDE 35F10 Initial value problems for linear first-order PDE, linear evolution equations 35F15 Boundary value problems for linear first-order PDE 35F20 General theory of nonlinear first-order PDE 35F25 Initial value problems for nonlinear first-order PDE, nonlinear evolution equations 35F30 Boundary value problems for nonlinear first-order PDE 35F99 None of the above but in this section 35Gxx General higher-order equations and systems 35G05 General theory of linear higher-order PDE 35G10 Initial value problems for linear higher-order PDE, linear evolution equations 35G15 Boundary value problems for linear higher-order PDE 35G20 General theory of nonlinear higher-order PDE 35G25 Initial value problems for nonlinear higher-order PDE, nonlinear evolution equations 35G30 Boundary value problems for nonlinear higher-order PDE 35G99 None of the above but in this section 35H05 Hypoelliptic equations and systems, See also {58Gxx} 35Jxx Partial differential equations of elliptic type, See also {{ 58Gxx} {58G05, 58G10}} 35J05 Laplace equation, reduced wave equation (Helmholtz), Poisson equation, See also {31Axx, 31Bxx} 35J10 Schrodinger operator, See also {35Pxx} 35J15 General theory of second-order, elliptic equations 35J20 Variational methods for second-order, elliptic equations 35J25 Boundary value problems for second-order, elliptic equations 35J30 General theory of higher-order, elliptic equations , See also {31A30, 31B30} 35J35 Variational methods for higher-order, elliptic equations 35J40 Boundary value problems for higher-order, elliptic equations 35J45 General theory of elliptic systems of PDE 35J50 Variational methods for elliptic systems 35J55 Boundary value problems for elliptic systems 35J60 Nonlinear PDE of elliptic type 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE 35J67 Boundary values of solutions to elliptic PDE 35J70 Elliptic partial differential equations of degenerate type 35J85 Unilateral problems and variational inequalities for elliptic PDE, See also {35R35, 49J40} 35J99 None of the above but in this section 35Kxx Parabolic equations and systems, See also {35Bxx, 35Dxx, 35R30, 35R35, 58G11} 35K05 Heat equation 35K10 General theory of second-order, parabolic equations 35K15 Initial value problems for second-order, parabolic equations 35K20 Boundary value problems for second-order, parabolic equations 35K22 Evolution equations (any order in the spatial derivatives), See also {58D25} 35K25 General theory of higher-order, parabolic equations 35K30 Initial value problems for higher-order, parabolic equations 35K35 Boundary value problems for higher-order, parabolic equations 35K40 General theory of parabolic systems of PDE 35K45 Initial value problems for pararabolic systems 35K50 Boundary value problems for parabolic systems 35K55 Nonlinear PDE of parabolic type 35K57 Reaction-diffusion equations 35K60 Nonlinear boundary value problems for linear parabolic PDE; boundary value problems for nonlinear parabolic PDE 35K65 Parabolic partial differential equations of degenerate type 35K70 Ultraparabolic, pseudoparabolic PDE, etc. 35K85 Unilateral problems and variational inequalities for parabolic PDE, See also {35R35, 49J40} 35K99 None of the above but in this section 35Lxx Partial differential equations of hyperbolic type, See Also { 58G16} 35L05 Wave equation 35L10 General theory of second-order, hyperbolic equations 35L15 Initial value problems for second-order, hyperbolic equations 35L20 Boundary value problems for second-order, hyperbolic equations 35L25 General theory of higher-order, hyperbolic equations 35L30 Initial value problems for higher-order, hyperbolic equations 35L35 Boundary value problems for higher-order, hyperbolic equations 35L40 General theory of hyperbolic systems of first-order PDE 35L45 Initial value problems for hyperbolic systems of first-order PDE 35L50 Boundary value problems for hyperbolic systems of first-order PDE 35L55 Hyperbolic systems of higher-order PDE 35L60 Nonlinear first-order PDE of hyperbolic type 35L65 Conservation laws 35L67 Shocks and singularities, See also {58C27, 76L05} 35L70 Nonlinear second-order PDE of hyperbolic type 35L75 Nonlinear hyperabolic PDE of higher ($\gtr 2$) order 35L80 Hyperbolic PDE of degenerate type 35L85 Unilateral problems; variational inequalities for hyperbolic PDE, See also {35R35, 49J40} 35L99 None of the above but in this section 35Mxx Partial differential equations of special type (mixed, composite, etc.), {For degenerate types, See 35J70, 35K65, 35L80} 35M10 PDE of mixed type 35M20 PDE of composite type 35M99 None of the above but in this section 35Nxx Overdetermined systems, See also {{58Gxx} {58G05, 58G07, 58Hxx}} 35N05 Overdetermined systems with constant coefficients 35N10 Overdetermined systems with variable coefficients (general) 35N15 $\overline\partial$-Neumann problem and generalizations; formal complexes, See also {32F20 and 58G05} 35N99 None of the above but in this section 35Pxx Spectral theory and eigenvalue problems for partial differential operators, See also {47Axx, 47Bxx, 47F05} 35P05 General spectral theory of PDE 35P10 Completeness of eigenfunctions, eigenfunction expansions for PDO 35P15 Estimation of eigenvalues, upper and lower bounds 35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PDO 35P25 Scattering theory for PDE, See also {47A40} 35P30 Nonlinear eigenvalue problems, nonlinear spectral theory for PDO 35P99 None of the above but in this section 35Qxx Equations of mathematical physics and other areas of application, See also {35J05, 35J10, 35K05, 35L05} 35Q05 Euler-Poisson-Darboux equation and generalizations 35Q15 Riemann-Hilbert problems, See also {30E25, 31A25, 31B20} 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10} 35Q35 Other equations arising in fluid mechanics 35Q40 Equations from quantum mechanics 35Q51 Solitons, See also {58F07} 35Q53 KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.), See also {58F07} 35Q55 NLS-like (nonlinear Schrodinger) equations, See Also { 58F07} 35Q58 Other completely integrable equations, See also {58F07} 35Q60 Equations of electromagnetic theory and optics 35Q72 Other equations from mechanics 35Q75 PDE in relativity 35Q80 Applications of PDE in areas other than physics 35Q99 None of the above but in this section 35Rxx Miscellaneous topics involving partial differential equations, {For equations on manifolds, See 58Gxx; for manifolds of solutions, See 58Bxx; for stochastic PDEs, See also 60H15} 35R05 PDE with discontinuous coefficients or data 35R10 Partial functional-differential or differential-difference equations, with or without deviating arguments 35R15 Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables), See also {46Gxx, 58D25} 35R20 Partial operator-differential equations (i.e. PDE on finite-dimensional spaces for abstract space valued functions), See also {34Gxx, 47A50, 47D03, 47D06, 47D09, 47H15, 47H20} 35R25 Improperly posed problems for PDE 35R30 Inverse problems (undetermined coefficients, etc.) for PDE 35R35 Free boundary problems for PDE 35R45 Partial differential inequalities 35R50 Partial differential equations of infinite order 35R60 Partial differential equations with randomness, See Also { 60H15} 35R70 PDE with multivalued right-hand sides 35R99 None of the above but in this section 35Sxx Pseudodifferential operators and other generalizations of partial differential operators, See also {47G30, 58G15} 35S05 General theory of $\Psi$DO 35S10 Initial value problems for $\Psi$DO 35S15 Boundary value problems for $\Psi$DO 35S30 Fourier integral operators 35S35 Topological aspects: intersection cohomology, stratified sets, etc., See also {32C38, 32S40, 32S60, 58G07} 35S50 Paradifferential operators 35S99 None of the above but in this section \MajorSub 39-XX Finite differences and functional equations 39-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 39-01 Instructional exposition (textbooks, tutorial papers, etc.) 39-02 Research exposition (monographs, survey articles) 39-03 Historical (must be assigned at least one classification number from Section 01) 39-04 Explicit machine computation and programs (not the theory of computation or programming) 39-06 Proceedings, conferences, collections, etc. 39Axx Difference equations, {For dynamical systems, See 58Fxx} 39A05 General 39A10 Difference equations, See also {33Dxx} 39A11 Stability of difference equations 39A12 Discrete version of topics in analysis 39A70 Difference operators, See also {47B39} 39A99 None of the above but in this section 39Bxx Functional equations, See also {30D05} 39B05 General 39B12 Iteration theory, iterative and composite equations, See also {26A18, 30D05, 58F08} 39B22 Equations for real functions 39B32 Equations for complex functions, See also {30D05} 39B42 Matrix and operator equations 39B52 Equations for functions with more general domains and/or ranges 39B62 Systems of functional equations 39B72 Inequalities involving unknown functions, See also {26A51, 26Dxx} 39B99 None of the above but in this section \MajorSub 40-XX Sequences, series, summability 40-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 40-01 Instructional exposition (textbooks, tutorial papers, etc.) 40-02 Research exposition (monographs, survey articles) 40-03 Historical (must be assigned at least one classification number from Section 01) 40-04 Explicit machine computation and programs (not the theory of computation or programming) 40-06 Proceedings, conferences, collections, etc. 40Axx Convergence and divergence of infinite limiting processes 40A05 Convergence and divergence of series and sequences 40A10 Convergence and divergence of integrals 40A15 Convergence and divergence of continued fractions, See also {30B70} 40A20 Convergence and divergence of infinite products 40A25 Approximation to limiting values (summation of series, etc.), {For the Euler-Maclaurin summation formula, See 65B15} 40A30 Convergence and divergence of series and sequences of functions 40A99 None of the above but in this section 40B05 Multiple sequences and series {(should also be assigned at least one other classification number in this section)} 40Cxx General summability methods 40C05 Matrix methods 40C10 Integral methods 40C15 Function-theoretic methods (including power series methods and semicontinuous methods) 40C99 None of the above but in this section 40Dxx Direct theorems on summability 40D05 General theorems 40D09 Structure of summability fields 40D10 Tauberian constants and oscillation limits 40D15 Convergence factors and summability factors 40D20 Summability and bounded fields of methods 40D25 Inclusion and equivalence theorems 40D99 None of the above but in this section 40Exx Inversion theorems 40E05 Tauberian theorems, general 40E10 Growth estimates 40E15 Lacunary inversion theorems 40E20 Tauberian constants 40E99 None of the above but in this section 40F05 Absolute and strong summability 40Gxx Special methods of summability 40G05 Cesaro, Euler, Norlund and Hausdorff methods 40G10 Abel, Borel and power series methods 40G99 None of the above but in this section 40H05 Functional analytic methods in summability 40J05 Summability in abstract structures, See also {43A55, 46A35, 46B15} \MajorSub 41-XX Approximations and expansions, {For all approximation theory in the complex domain, See {30Exx} {30E05 and 30E10}; for all trigonometric approximation and interpolation, See {42Axx} {42A10 and 42A15}; for numerical approximation, See 65Dxx} 41-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 41-01 Instructional exposition (textbooks, tutorial papers, etc.) 41-02 Research exposition (monographs, survey articles) 41-03 Historical (must be assigned at least one classification number from Section 01) 41-04 Explicit machine computation and programs (not the theory of computation or programming) 41-06 Proceedings, conferences, collections, etc. 41A05 Interpolation, See also {42A15 and 65D05} 41A10 Approximation by polynomials, {For approximation by trigonometric polynomials, See 42A10} 41A15 Spline approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities) 41A20 Approximation by rational functions 41A21 Pade approximation 41A25 Rate of convergence, degree of approximation 41A27 Inverse theorems 41A28 Simultaneous approximation 41A29 Approximation with constraints 41A30 Approximation by other special function classes 41A35 Approximation by operators (in particular, by integral operators) 41A36 Approximation by positive operators 41A40 Saturation 41A44 Best constants 41A45 Approximation by arbitrary linear expressions 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A50 Best approximation, Chebyshev systems 41A52 Uniqueness of best approximation 41A55 Approximate quadratures 41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also {30E15} 41A63 Multidimensional problems (should also be assigned at least one other classification number in this section) 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A80 Remainders in approximation formulas 41A99 Miscellaneous topics \MajorSub 42-XX Fourier analysis 42-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 42-01 Instructional exposition (textbooks, tutorial papers, etc.) 42-02 Research exposition (monographs, survey articles) 42-03 Historical (must be assigned at least one classification number from Section 01) 42-04 Explicit machine computation and programs (not the theory of computation or programming) 42-06 Proceedings, conferences, collections, etc. 42Axx Fourier analysis in one variable 42A05 Trigonometric polynomials, inequalities, extremal problems 42A10 Trigonometric approximation 42A15 Trigonometric interpolation 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series, {For automorphic theory, See mainly 11F30} 42A20 Convergence of Fourier and trigonometric series 42A24 Summability of Fourier and trigonometric series 42A28 Absolute convergence, absolute summability 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A45 Multipliers 42A50 Conjugate functions, conjugate series, singular integrals 42A55 Lacunary series of trigonometric and other functions; Riesz products 42A61 Probabilistic methods 42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization 42A65 Completeness of sets of functions 42A70 Trigonometric moment problems 42A75 Classical almost periodic functions, mean periodic functions, See also {43A60} 42A82 Positive definite functions 42A85 Convolution, factorization 42A99 None of the above but in this section 42Bxx Fourier analysis in several variables, {For automorphic theory, See mainly 11F30} 42B05 Fourier series and coefficients 42B08 Summability 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B15 Multipliers 42B20 Singular integrals (Calderon-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B30 $H^p$-spaces 42B99 None of the above but in this section 42Cxx Nontrigonometric Fourier analysis 42C05 Orthogonal functions and polynomials, general theory, See also {33A65, 33C45, 33C50, 33D45} 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions 42C20 Rearrangements and other transformations of Fourier and other orthogonal series 42C25 Uniqueness and localization for orthogonal series 42C30 Completeness of sets of functions 42C99 None of the above but in this section \MajorSub 43-XX Abstract harmonic analysis, {For other analysis on topological and Lie groups, See 22Exx} 43-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 43-01 Instructional exposition (textbooks, tutorial papers, etc.) 43-02 Research exposition (monographs, survey articles) 43-03 Historical (must be assigned at least one classification number from Section 01) 43-04 Explicit machine computation and programs (not the theory of computation or programming) 43-06 Proceedings, conferences, collections, etc. 43A05 Measures on groups and semigroups, etc. 43A07 Means on groups, semigroups, etc.; amenable groups 43A10 Measure algebras on groups, semigroups, etc. 43A15 $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A17 Analysis on ordered groups, ${H}^p$-theory 43A20 $L^1$-algebras on groups, semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A25 Fourier and Fourier-Stieltjes transforms on locally compact abelian groups 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A32 Other transforms and operators of Fourier type 43A35 Positive definite functions on groups, semigroups, etc. 43A40 Character groups and dual objects 43A45 Spectral synthesis on groups, semigroups, etc. 43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 43A50 Convergence of Fourier series and of inverse transforms 43A55 Summability methods on groups, semigroups, etc., See Also { 40J05} 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 43A62 Hypergroups 43A65 Representations of groups, semigroups, etc., See Also { 22A10, 22A20, 22Dxx, 22E45} 43A70 Analysis on specific locally compact abelian groups, See also {11R56, 22B05} 43A75 Analysis on specific compact groups 43A77 Analysis on general compact groups 43A80 Analysis on other specific Lie groups, See also {22Exx} 43A85 Analysis on homogeneous spaces 43A90 Spherical functions, See also {22E45, 22E46, 33C65, 33D55} 43A95 Categorical methods, See also {46Mxx} 43A99 Miscellaneous topics \MajorSub 44-XX Integral transforms, operational calculus, {For fractional derivatives and integrals, See 26A33. For Fourier transforms, See 42A38, 42B10. For integral transforms in distribution spaces, See 46F12. For numerical methods, See 65R10} 44-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 44-01 Instructional exposition (textbooks, tutorial papers, etc.) 44-02 Research exposition (monographs, survey articles) 44-03 Historical (must be assigned at least one classification number from Section 01) 44-04 Explicit machine computation and programs (not the theory of computation or programming) 44-06 Proceedings, conferences, collections, etc. 44A05 General transforms, See also {42A38} 44A10 Laplace transform 44A12 Radon transform, See also {92C55} 44A15 Special transforms (Legendre, Hilbert, etc.) 44A20 Transforms of special functions 44A30 Multiple transforms 44A35 Convolution 44A40 Calculus of Mikusinski and other operational calculi 44A45 Classical operational calculus 44A55 Discrete operational calculus 44A60 Moment problems 44A99 Miscellaneous topics \MajorSub 45-XX Integral equations 45-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 45-01 Instructional exposition (textbooks, tutorial papers, etc.) 45-02 Research exposition (monographs, survey articles) 45-03 Historical (must be assigned at least one classification number from Section 01) 45-04 Explicit machine computation and programs (not the theory of computation or programming) 45-06 Proceedings, conferences, collections, etc. 45A05 Linear integral equations 45B05 Fredholm integral equations 45C05 Eigenvalue problems, See also {34Lxx, 35Pxx, 45P05, 47A75} 45D05 Volterra integral equations, See also {34A12} 45Exx Singular integral equations, See also {{30Exx, 44-XX} { 30E20, 30E25, 44A15, 44A35}} 45E05 Integral equations with kernels of Cauchy type, See Also { 35J15} 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type), See also {47B35} 45E99 None of the above but in this section 45Fxx Systems of linear integral equations 45F05 Systems of nonsingular linear integral equations 45F10 Dual, triple, etc., integral and series equations 45F15 Systems of singular linear integral equations 45F99 None of the above but in this section 45Gxx Nonlinear integral equations, See also {47H15} 45G05 Singular nonlinear integral equations 45G10 Other nonlinear integral equations 45G15 Systems of nonlinear integral equations 45H05 Miscellaneous special kernels, See also {44A15} 45J05 Integro-ordinary differential equations 45K05 Integro-partial differential equations 45Lxx Approximation of solutions 45L05 Theoretical approximation of solutions 45L10 Numerical approximation of solutions, {For numerical analysis, See 65R20} 45L99 None of the above but in this section 45Mxx Qualitative behavior 45M05 Asymptotics 45M10 Stability theory 45M15 Periodic solutions 45M20 Positive solutions 45M99 None of the above but in this section 45N05 Abstract integral equations, integral equations in abstract spaces 45P05 Integral operators, See also {47B38, 47G10} \MajorSub 46-XX Functional analysis, {For manifolds modeled on topological linear spaces, See 57N20, 58Bxx} 46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 46-01 Instructional exposition (textbooks, tutorial papers, etc.) 46-02 Research exposition (monographs, survey articles) 46-03 Historical (must be assigned at least one classification number from Section 01) 46-04 Explicit machine computation and programs (not the theory of computation or programming) 46-06 Proceedings, conferences, collections, etc. 46Axx Topological linear spaces and related structures, {For function spaces, See 46Exx} 46A03 General theory of locally convex spaces 46A04 Locally convex Frechet spaces and (DF)-spaces 46A08 Barrelled spaces, bornological spaces 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.), See also {46M40} 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46A17 Bornologies and related structures; Mackey convergence, etc. 46A19 Other ``topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${R}$, etc.) 46A20 Duality theory 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators, See also {46M10} 46A25 Reflexivity and semi-reflexivity, See also {46B10} 46A30 Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness) 46A32 Spaces of linear operators; topological tensor products; approximation properties, See also {46B28, 46M05, 47D15, 47D50} 46A35 Summability and bases, See also {46B15} 46A40 Ordered topological linear spaces, vector lattices, See also {06F20, 46B40, 46B42} 46A45 Sequence spaces (including Kothe sequence spaces), See also {46B45} 46A50 Compactness in topological linear spaces; angelic spaces, etc. 46A55 Convex sets in topological linear spaces; Choquet theory, See also {52A07} 46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.) 46A80 Modular spaces 46A99 None of the above but in this section 46Bxx Normed linear spaces and Banach spaces; Banach lattices, {For function spaces, See 46Exx} 46B03 Isomorphic theory (including renorming) of Banach spaces 46B04 Isometric theory of Banach spaces 46B07 Local theory of Banach spaces 46B08 Ultraproduct techniques in Banach space theory, See Also { 46M07} 46B09 Probabilistic methods in Banach space theory, See Also { 60Bxx} 46B10 Duality and reflexivity, See also {46A25} 46B15 Summability and bases, See also {46A35} 46B20 Geometry and structure of normed linear spaces 46B22 Radon-Nikodym, Kreuin-Milman and related properties, See also {46G10} 46B25 Classical Banach spaces in the general theory 46B26 Nonseparable Banach spaces 46B28 Spaces of operators; tensor products; approximation properties, See also {46A32, 46M05, 47D15, 47D50} 46B40 Ordered normed spaces, See also {46A40, 46B42} 46B42 Banach lattices, See also {46A40, 46B40} 46B45 Banach sequence spaces, See also {46A45} 46B70 Interpolation between normed linear spaces, See Also { 46M35} 46B99 None of the above but in this section 46Cxx Inner product spaces and their generalizations, Hilbert spaces , {For function spaces, See 46Exx} 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 46C15 Characterizations of Hilbert spaces 46C20 Spaces with indefinite inner product 46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.) 46C99 None of the above but in this section 46Exx Linear function spaces and their duals, See also {30H05, 32E25, 46F05; for function algebras, See 46J10} 46E05 Lattices of continuous, differentiable or analytic functions 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 46E20 Hilbert spaces of continuous, differentiable or analytic functions 46E22 Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces) 46E25 Rings and algebras of continuous, differentiable or analytic functions, {For Banach function algebras, See 46J10, 46J15} 46E27 Spaces of measures, See also {28A33, 46Gxx} 46E30 Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems 46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables 46E40 Spaces of vector- and operator-valued functions 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces, See also {46G20} 46E99 None of the above but in this section 46Fxx Distributions, generalized functions, distribution spaces, {For distribution theory on nonlinear spaces, See 58Cxx} 46F05 Topological linear spaces of test functions, distributions and ultradistributions, See also {46E10, 46E35} 46F10 Operations with distributions 46F12 Integral transforms in distribution spaces, See Also { 42-XX, 44-XX} 46F15 Hyperfunctions, analytic functionals, See also {32A25, 32A45, 32C35, 58G07} 46F20 Distributions and ultradistributions as boundary values of analytic functions, See also {30D40, 30E25, 32A40} 46F25 Distributions on infinite-dimensional spaces, See Also { 58C35} 46F99 None of the above but in this section 46Gxx Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces), See Also {28-XX} {For nonlinear functional analysis, See 47Hxx, 58-XX, especially 58Cxx} 46G05 Derivatives, See also {58C20, 58C25} 46G10 Vector-valued measures and integration, See also {28Bxx, 46B22} 46G12 Measures and integration on abstract linear spaces, See also {28C20} 46G15 Functional analytic lifting theory, See also {28A51} 46G20 Infinite-dimensional holomorphy, See also {32-XX, 46E50, 58B12, 58C10} 46G99 None of the above but in this section 46Hxx Topological algebras, normed rings and algebras, Banach algebras, {For group algebras, convolution algebras and measure algebras, See 43A10, 43A20} 46H05 General theory of topological algebras 46H10 Ideals and subalgebras 46H15 Representations of topological algebras 46H20 Structure, classification of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46H30 Functional calculus in topological algebras, See Also { 47A60} 46H35 Topological algebras of operators [See mainly 47D30, and also 47D25, 47D50] 46H40 Automatic continuity 46H70 Nonassociative topological algebras, See also {46K70, 46L70} 46H99 None of the above but in this section 46Jxx Commutative Banach algebras and commutative topological algebras, See also {46E25} 46J05 General theory of commutative topological algebras 46J10 Banach algebras of continuous functions, function algebras, See also {46E25} 46J15 Banach algebras of differentiable or analytic functions, ${H}^p$-spaces, See also {30D55, 30H05, 32A35, 32A37, 32E25, 42B30} 46J20 Ideals, maximal ideals, boundaries 46J25 Representations of commutative topological algebras 46J30 Subalgebras 46J40 Structure, classification of commutative topological algebras 46J45 Radical Banach algebras 46J99 None of the above but in this section 46Kxx Topological (rings and) algebras with an involution, See also {16W10} 46K05 General theory of topological algebras with involution 46K10 Representations of topological algebras with involution 46K15 Hilbert algebras 46K50 Nonselfadjoint (sub)algebras in algebras with involution 46K70 Nonassociative topological algebras with an involution, See also {46H70, 46L70} 46K99 None of the above but in this section 46Lxx Selfadjoint operator algebras ($C$*-algebras, von Neumann ($W$*-) algebras, etc.), See also {22D25} 46L05 General theory of $C$*-algebras 46L10 General theory of von Neumann algebras 46L30 States 46L35 Classifications of $C^*$-algebras, factors 46L37 Subfactors and their classification 46L40 Automorphisms 46L45 Decomposition theory for $C^*$-algebras 46L50 Noncommutative measure, integration and probability 46L55 Noncommutative dynamical systems, See also {28Dxx, 54H20, 58Fxx} 46L57 Derivations, dissipations and positive semigroups in $C^*$-algebras 46L60 Applications of selfadjoint operator algebras to physics, See also {46N50, 46N55, 47D45, 81T05, 82B10, 82C10} 46L70 Nonassociative selfadjoint operator algebras, See Also { 46H70, 46K70} 46L80 $K$-theory and operator algebras (including cyclic theory), See also {18F25, 19Kxx, 46M20, 55Rxx, 58G12} 46L85 Noncommutative topology, See also {58B30, 58G12} 46L87 Noncommutative differential geometry, See also {58B30, 58G12} 46L89 Other ``noncommutative'' mathematics based on $C^*$-algebra theory, See also {58B30, 58G12} 46L99 None of the above but in this section 46Mxx Methods of category theory in functional analysis, See Also { 18-XX} 46M05 Tensor products, See also {46A32, 46B28, 47A80} 46M07 Ultraproducts, See also {46B08, 46S20} 46M10 Projective and injective objects, See also {46A22} 46M15 Functors, {For $K$-theory and EXT, See 19K33, 46L80, 46M20} 46M20 Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.), See also {14F05, 18Fxx, 19Kxx, 32Cxx, 32Lxx, 46L80, 55Rxx} 46M35 Abstract interpolation of topological vector spaces, See also {46B70} 46M40 Inductive and projective limits, See also {46A13} 46M99 None of the above but in this section 46Nxx Miscellaneous applications of functional analysis, See Also { 47Nxx} 46N10 Applications in optimization, convex analysis, mathematical programming, economics 46N20 Applications to differential and integral equations 46N30 Applications in probability theory and statistics 46N40 Applications in numerical analysis, See also {65Jxx} 46N50 Applications in quantum physics 46N55 Applications in statistical physics 46N60 Applications in biology and other sciences 46N99 None of the above but in this section 46Sxx Other (nonclassical) types of functional analysis, See Also { 47Sxx} 46S10 Functional analysis over fields other than ${R}$ or ${C}$ or the quaternions; non-Archimedean functional analysis, See also {12J25, 32P05} 46S20 Nonstandard functional analysis, See also {03H05} 46S30 Constructive functional analysis, See also {03F65} 46S40 Fuzzy functional analysis, See also {04A72} 46S50 Functional analysis in probabilistic metric linear spaces 46S99 None of the above but in this section \MajorSub 47-XX Operator theory 47-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 47-01 Instructional exposition (textbooks, tutorial papers, etc.) 47-02 Research exposition (monographs, survey articles) 47-03 Historical (must be assigned at least one classification number from Section 01) 47-04 Explicit machine computation and programs (not the theory of computation or programming) 47-06 Proceedings, conferences, collections, etc. 47Axx General theory of linear operators 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A10 Spectrum, resolvent 47A11 Local spectral properties 47A12 Numerical range 47A13 Several-variable spectral theory 47A15 Invariant subspaces 47A20 Dilations, extensions, compressions 47A25 Spectral sets 47A30 Norms (inequalities, more than one norm, etc.) 47A35 Ergodic theory, See also {28Dxx} 47A40 Scattering theory, See also {34L25, 35P25, 81Uxx} 47A45 Canonical models for contractions and nonselfadjoint operators 47A48 Operator colligations (= nodes) 47A50 Equations and inequalities involving linear operators, with vector unknowns 47A53 (Semi-) Fredholm operators; index theories, See Also { 58B10, 58G12} 47A55 Perturbation theory 47A56 Functions whose values are linear operators (operator and matrix pencils, etc.) 47A57 Operator methods in interpolation, moment and extension problems, See also {30E05, 42A70, 42A82} 47A58 Operator approximation theory 47A60 Functional calculus 47A62 Equations involving linear operators, with operator unknowns 47A63 Operator inequalities, operator means, shorted operators, etc. 47A65 Structure theory 47A66 (Non)quasitriangular, (non)quasidiagonal operators 47A67 Representation theory 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) 47A70 (Generalized) eigenfunction expansions; rigged Hilbert spaces 47A75 Eigenvalue problems, See also {49Rxx} 47A80 Tensor products of operators, See also {46M05} 47A99 None of the above but in this section 47Bxx Special classes of linear operators 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 47B07 Operators defined by compactness properties 47B10 Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.), See also {47D50} 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B20 Subnormal operators, hyponormal operators, etc. 47B25 Symmetric and selfadjoint operators (unbounded) 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators, See also {45P05, 47G10 for other integral operators; See also 32H10, 32M15} 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B38 Operators on function spaces (including composition operators, kernel operators) 47B39 Difference operators, See also {39A70} 47B40 Spectral operators, decomposable operators, well-bounded operators, etc. 47B44 Accretive operators, dissipative operators, etc. 47B47 Commutators, derivations, elementary operators, etc. 47B48 Operators on Banach algebras 47B49 Transformers (= operators on spaces of operators) 47B50 Operators on spaces with an indefinite metric 47B60 Operators on ordered spaces 47B65 Positive operators and order-bounded operators 47B80 Random operators, See also {60H25} 47B99 None of the above but in this section 47Cxx Individual linear operators as elements of algebraic systems 47C05 Operators in algebras 47C10 Operators in *-algebras 47C15 Operators in $C$*- or von Neumann algebras 47C99 None of the above but in this section 47Dxx Algebraic systems of linear operators, See also {46Lxx} 47D03 (Semi)groups of linear operators, {For nonlinear operators, See 47H20; See also 20M20} 47D06 One-parameter semigroups and linear evolution equations 47D07 Markov semigroups, {For Markov processes, See 60Jxx} 47D09 Operator cosine functions and higher-order Cauchy problems 47D15 Linear spaces of operators, See also {46A32, 46B28} 47D20 Convex sets and cones of operators, See also {90C25, 90C31} 47D25 Operator algebras on Hilbert space 47D27 Dual operator algebras 47D30 Operator algebras on Banach spaces and other linear topological spaces 47D35 Dual spaces of operator algebras and topological groups 47D40 Algebras of unbounded operators 47D45 Applications of operator algebras to physics, See Also { 46L60, 46N50, 46N55, 47N50, 47N55} 47D50 Operator ideals 47D99 None of the above but in this section 47E05 Ordinary differential operators, See also {34Bxx, 34Lxx, 58F19} 47F05 Partial differential operators, See also {35Pxx, 58G05} 47Gxx Integral, integro-differential, and pseudodifferential operators 47G10 Integral operators, See also {45P05} 47G20 Integro-differential operators, See also {45J05, 45K05} 47G30 Pseudodifferential operators, See also {35Sxx, 58G15} 47G99 None of the above but in this section 47Hxx Nonlinear operators, {For global and geometric aspects, See 58-XX, especially 58Cxx} 47H04 Set-valued operators, See also {28B20, 54C60, 58C06} 47H05 Monotone operators (with respect to duality) 47H06 Accretive operators, dissipative operators, etc. 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H09 Nonexpansive mappings, and their generalizations (ultimately compact mappings, measures of noncompactness, $A$-proper mappings, $K$-set contractions, etc.) 47H10 Fixed-point theorems, See also {54H25, 55M20, 58C30} 47H11 Degree theory, See also {55M25, 58C30} 47H12 Spectral theory of nonlinear operators, See also {58C40} 47H15 Equations involving nonlinear operators, See also {58E07 for abstract bifurcation theory} 47H17 Methods for solving equations involving nonlinear operators, See also {58C15}, {For numerical analysis, See 65J15} 47H19 Inequalities involving nonlinear operators, See Also { 49J27, 49J40, 49K27} 47H20 Semigroups of nonlinear operators and nonlinear evolution equations, See also {58D07} 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiui, Uryson, hysteresis operators, etc.), See also {45P05} 47H40 Random operators, See also {60H25} 47H99 None of the above but in this section 47Nxx Miscellaneous applications of operator theory, See Also { 46Nxx} 47N10 Applications in optimization, convex analysis, mathematical programming, economics 47N20 Applications to differential and integral equations 47N30 Applications in probability theory and statistics 47N40 Applications in numerical analysis, See also {65Jxx} 47N50 Applications in quantum physics 47N55 Applications in statistical physics 47N60 Applications in biology and other sciences 47N70 Applications in systems theory, circuits, etc. 47N99 None of the above but in this section 47Sxx Other (nonclassical) types of operator theory, See Also { 46Sxx} 47S10 Operator theory over fields other than ${R}$, ${C}$ or the quaternions; non-Archimedean operator theory 47S20 Nonstandard operator theory, See also {03H05} 47S30 Constructive operator theory, See also {03F65} 47S40 Fuzzy operator theory, See also {04A72} 47S50 Operator theory in probabilistic metric linear spaces 47S99 None of the above but in this section \MajorSub 49-XX Calculus of variations and optimal control; optimization, See also {34H05, 65Kxx, 90Cxx, 93-XX} 49-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 49-01 Instructional exposition (textbooks, tutorial papers, etc.) 49-02 Research exposition (monographs, survey articles) 49-03 Historical (must be assigned at least one classification number from Section 01) 49-04 Explicit machine computation and programs (not the theory of computation or programming) 49-06 Proceedings, conferences, collections, etc. 49Jxx Existence theory for optimal solutions 49J05 Free problems in one independent variable 49J10 Free problems in two or more independent variables 49J15 Optimal control problems involving ordinary differential equations 49J20 Optimal control problems involving partial differential equations 49J22 Optimal control problems involving integral equations 49J24 Optimal control problems involving differential inclusions 49J25 Optimal control problems involving equations with retarded arguments, See also {34K35} 49J27 Problems in abstract spaces [see 90C48] 49J30 Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49J35 Minimax problems 49J40 Variational methods including variational inequalities 49J45 Problems involving semicontinuity and convergence 49J50 Frechet and Gateaux differentiability, See also {46G05, 58C20} 49J52 Nonsmooth analysis (other weak concepts of optimality), See also {58C20, 90C48} 49J55 Problems involving randomness, See also {93E20} 49J99 None of the above but in this section 49Kxx Necessary conditions and sufficient conditions for optimality 49K05 Free problems in one independent variable 49K10 Free problems in two or more independent variables 49K15 Problems involving ordinary differential equations 49K20 Problems involving partial differential equations 49K22 Problems involving integral equations 49K24 Problems involving differential inclusions 49K25 Problems involving equations with retarded arguments, See also {34K35} 49K27 Problems in abstract spaces [See 90C48, 93C25] 49K30 Optimal solutions belonging to restricted classes 49K35 Minimax problems 49K40 Sensitivity of optimal solutions in the presence of perturbations 49K45 Problems involving randomness, See also {93E20} 49K99 None of the above but in this section 49Lxx Caratheodory, Hamilton-Jacobi theories, including dynamic programming 49L05 Free problems and problems involving ordinary differential equations 49L10 Free problems and problems involving partial differential equations 49L15 Problems in abstract spaces or problems involving functional relations other than differential equations 49L20 Dynamic programming method 49L25 Viscosity solutions 49L99 None of the above but in this section 49Mxx Methods of successive approximations, {For discrete problems, See 90Cxx; See also 65Kxx} 49M05 Methods based on necessary conditions 49M07 Gradient methods 49M10 Methods of steepest descent type 49M15 Methods of Newton-Raphson, Galerkin and Ritz types 49M20 Methods of relaxation type 49M25 Finite difference methods 49M27 Decomposition methods 49M29 Multiplier methods 49M30 Other methods, not based on necessary conditions (penalty function, etc.) 49M35 Methods of linear programming type, See also {90C05} 49M37 Nonlinear programming, See also {90C30} 49M39 Semi-infinite programming, See also {90C34} 49M40 Methods of quadratic programming type, See also {90C20} 49M45 Methods of convex programming type, See also {90C25} 49M49 Geometric programming, See also {90C28} 49M99 None of the above but in this section 49Nxx Miscellaneous topics 49N05 Linear optimal control problems, See also {93C05} 49N10 Linear-quadratic problems 49N15 Duality theory 49N20 Periodic optimization 49N25 Impulsive optimal control problems, See also {93C57} 49N30 Problems with incomplete information, See also {93C41} 49N35 Closed-loop controls 49N40 Open-loop controls 49N45 Inverse problems in calculus of variations 49N50 Inverse problems in optimal control theory 49N55 Noneconomic applications of optimal control theory and differential games, See also {90D25} 49N60 Regularity of solutions in the calculus of variations 49N65 Applications of measurable selections to control theory, See also {28B20, 26E25} 49N99 None of the above but in this section 49Qxx Manifolds, See also {58Fxx} 49Q05 Minimal surfaces, See also {53A10, 58E12} 49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90} 49Q12 Sensitivity analysis 49Q15 Geometric measure and integration theory, integral and normal currents, See also {28A75, 32C30, 58A25, 58C35} 49Q20 Variational problems in a geometric measure-theoretic setting 49Q25 Surface area 49Q99 None of the above but in this section 49Rxx Variational methods, {For eigenvalues of operators, See 47A75} 49R05 Variational approach to eigenvalues 49R10 Rayleigh-Ritz methods 49R15 Weinstein and Aronszajn methods, intermediate problems 49R20 Linear operators in Hilbert spaces 49R99 None of the above but in this section 49S05 Variational principles of physics \MajorSub 51-XX Geometry, {For algebraic geometry, See 14-XX} 51-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 51-01 Instructional exposition (textbooks, tutorial papers, etc.) 51-02 Research exposition (monographs, survey articles) 51-03 Historical (must be assigned at least one classification number from Section 01) 51-04 Explicit machine computation and programs (not the theory of computation or programming) 51-06 Proceedings, conferences, collections, etc. 51Axx Linear incidence geometry 51A05 General theory and projective geometries 51A10 Homomorphism, automorphism and dualities 51A15 Structures with parallelism 51A20 Configuration theorems 51A25 Algebraization, See also {12Kxx, 20N05} 51A30 Desarguesian and Pappian geometries 51A35 Non-Desarguesian affine and projective planes 51A40 Translation planes and spreads 51A45 Incidence structures imbeddable into projective geometries 51A50 Polar geometry, symplectic spaces, orthogonal spaces 51A99 None of the above but in this section 51Bxx Nonlinear incidence geometry 51B05 General theory 51B10 Mobius geometries 51B15 Laguerre geometries 51B20 Minkowski geometries 51B25 Lie geometries 51B99 None of the above but in this section 51C05 Ring geometry (Hjelmslev, Barbilian, etc.) 51Dxx Geometric closure systems 51D05 Abstract (Maeda) geometries 51D10 Abstract geometries with exchange axiom 51D15 Abstract geometries with parallelism 51D20 Combinatorial geometries, See also {05B25, 05B35} 51D25 Lattices of subspaces, See also {05B35} 51D30 Continuous geometries and related topics, See also {06Cxx} 51D99 None of the above but in this section 51Exx Finite geometry and special incidence structures 51E05 General block designs, See also {05B05} 51E10 Steiner systems 51E12 Generalized quadrangles, generalized polygons 51E14 Finite partial geometries (general), nets, partial spreads 51E15 Affine and projective planes 51E20 Combinatorial structures in finite projective spaces, See also {05Bxx} 51E21 Blocking sets, ovals, $k$-arcs 51E22 Linear codes and caps in Galois spaces, See also {94B05} 51E23 Spreads and packing problems 51E24 Buildings and the geometry of diagrams 51E25 Other finite nonlinear geometries 51E26 Other finite linear geometries 51E30 Other finite incidence structures, See also {05B30} 51E99 None of the above but in this section 51Fxx Metric geometry 51F05 Absolute planes 51F10 Absolute spaces 51F15 Reflection groups, reflection geometries, See also {20H10, 20H15; for Coxeter groups, See 20F55} 51F20 Congruence and orthogonality, See also {20H05} 51F25 Orthogonal and unitary groups, See also {20H05} 51F99 None of the above but in this section 51G05 Ordered geometries (ordered incidence structures, etc.) 51Hxx Topological geometry 51H05 General theory 51H10 Topological linear incidence structures 51H15 Topological nonlinear incidence structures 51H20 Topological geometries on manifolds, See also {57-XX} 51H25 Geometries with differentiable structure, See also {53Cxx, 53C70} 51H30 Geometries with algebraic manifold structure, See Also { 14-XX} 51H99 None of the above but in this section 51Jxx Incidence groups 51J05 General theory 51J10 Projective incidence groups 51J15 Kinematic spaces 51J20 Representation by near-fields and near-algebras, See Also { 12K05, 16Y30} 51J99 None of the above but in this section 51Kxx Distance geometry 51K05 General theory 51K10 Synthetic differential geometry 51K99 None of the above but in this section 51Lxx Geometric order structures, See also {53C75} 51L05 Geometry of orders of nondifferentiable curves 51L10 Directly differentiable curves 51L15 $n$-vertex theorems via direct methods 51L20 Geometry of orders of surfaces 51L99 None of the above but in this section 51Mxx Real and complex geometry 51M04 Elementary problems in Euclidean geometries 51M05 Euclidean geometries (general) and generalizations 51M09 Elementary problems in hyperbolic and elliptic geometries 51M10 Hyperbolic and elliptic geometries (general) and generalizations 51M15 Geometric constructions 51M16 Inequalities and extremum problems, {For convex problems, See 52A40} 51M20 Polyhedra and polytopes; regular figures, division of spaces, See also {51F15} 51M25 Length, area and volume, See also {26B15} 51M30 Line geometries and their generalizations, See also {53A25} 51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), See also {14M15} 51M99 None of the above but in this section 51Nxx Analytic and descriptive geometry 51N05 Descriptive geometry, See also {65D17, 68U07} 51N10 Affine analytic geometry 51N15 Projective analytic geometry 51N20 Euclidean analytic geometry 51N25 Analytic geometry with other transformation groups 51N30 Geometry of classical groups, See also {20Gxx, 14L35} 51N35 Questions of classical algebraic geometry, See also {14Nxx} 51N99 None of the above but in this section 51P05 Geometry and physics (should also be assigned at least one other classification number from Sections 70--86) \MajorSub 52-XX Convex and discrete geometry 52-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 52-01 Instructional exposition (textbooks, tutorial papers, etc.) 52-02 Research exposition (monographs, survey articles) 52-03 Historical (must be assigned at least one classification number from Section 01) 52-04 Explicit machine computation and programs (not the theory of computation or programming) 52-06 Proceedings, conferences, collections, etc. 52Axx General convexity 52A01 Axiomatic and generalized convexity 52A05 Convex sets without dimension restrictions 52A07 Convex sets in topological vector spaces, See also {46A55} 52A10 Convex sets in $2$ dimensions (including convex curves), See also {53A04} 52A15 Convex sets in $3$ dimensions (including convex surfaces), See also {53A05, 53C45} 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces), See also {53A07, 53C45} 52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.), See also {46Bxx} 52A22 Random convex sets and integral geometry, See also {53C65, 60D05} 52A27 Approximation by convex sets 52A30 Variants of convex sets (star-shaped, ($m, n$)-convex, etc.) 52A35 Helly-type theorems 52A37 Other problems of combinatorial convexity 52A38 Length, area, volume, See also {26B15, 28A75, 49Q20} 52A39 Mixed volumes and related topics 52A40 Inequalities and extremum problems 52A41 Convex functions and convex programs, See also {26B25, 90C25} 52A55 Spherical and hyperbolic convexity 52A99 None of the above but in this section 52Bxx Polytopes and polyhedra 52B05 Combinatorial properties (number of faces, shortest paths, etc.), See also {05Cxx} 52B10 Three-dimensional polytopes 52B11 $n$-dimensional polytopes 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 52B15 Symmetry properties of polytopes 52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also {06A08, 13F20, 13Hxx} 52B30 Arrangements of hyperplanes 52B35 Gale and other diagrams 52B40 Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), See Also { 05B35} 52B45 Dissections and valuations (Hilbert's third problem, etc.) 52B55 Computational aspects related to convexity, {For computational geometry and algorithms, See 68Q20, 68Q25, 68U05; for numerical algorithms, See 65Yxx} 52B60 Isoperimetric problems for polytopes 52B70 Polyhedral manifolds 52B99 None of the above but in this section 52Cxx Discrete geometry 52C05 Lattices and convex bodies in $2$ dimensions, See Also { 11H06, 11H31, 11P21} 52C07 Lattices and convex bodies in $n$ dimensions, See Also { 11H06, 11H31, 11P21} 52C10 Erdos problems and related topics of discrete geometry, See also {11Hxx} 52C15 Packing and covering in $2$ dimensions, See also {05B40, 11H31} 52C17 Packing and covering in $n$ dimensions, See also {05B40, 11H31} 52C20 Tilings in $2$ dimensions, See also {05B45, 51M20} 52C22 Tilings in $n$ dimensions, See also {05B45, 51M20} 52C25 Rigidity and flexibility of structures, See also {70B15} 52C99 None of the above but in this section \MajorSub 53-XX Differential geometry, {For differential topology, See 57Rxx. For foundational questions of differentiable manifolds, See 58Axx} 53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 53-01 Instructional exposition (textbooks, tutorial papers, etc.) 53-02 Research exposition (monographs, survey articles) 53-03 Historical (must be assigned at least one classification number from Section 01) 53-04 Explicit machine computation and programs (not the theory of computation or programming) 53-06 Proceedings, conferences, collections, etc. 53Axx Classical differential geometry 53A04 Curves in Euclidean space 53A05 Surfaces in Euclidean space 53A07 Higher-dimension and -codimension surfaces in Euclidean $n$-space 53A10 Minimal surfaces, surfaces with prescribed mean curvature, See also {49Q05, 49Q10, 53C42} 53A15 Affine differential geometry 53A17 Kinematics 53A20 Projective differential geometry 53A25 Differential line geometry 53A30 Conformal differential geometry 53A35 Non-Euclidean differential geometry 53A40 Other special differential geometries 53A45 Vector and tensor analysis 53A50 Spinor analysis 53A55 Differential invariants (local theory), geometric objects 53A60 Geometry of webs, See also {14C21, 20N05} 53A99 None of the above but in this section 53Bxx Local differential geometry 53B05 Linear and affine connections 53B10 Projective connections 53B15 Other connections 53B20 Local Riemannian geometry 53B21 Methods of Riemannian geometry 53B25 Local submanifolds, See also {53C40} 53B30 Lorentz metrics, indefinite metrics 53B35 Hermitian and Kahlerian structures, See also {32Cxx} 53B40 Finsler spaces and generalizations (areal metrics) 53B50 Applications to physics 53B99 None of the above but in this section 53Cxx Global differential geometry, See also {51H25, 58-XX; for related bundle theory, See 55Rxx, 57Rxx} 53C05 Connections, general theory 53C07 Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills), See also {32L07} 53C10 $G$-structures 53C12 Foliations (differential geometric aspects), See Also { 57R30, 57R32} 53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.) 53C20 Global Riemannian geometry, including pinching, See Also { 31C12, 58B20} 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions, See also {58G30} 53C22 Geodesics, See also {58E10} 53C23 Global topological methods (a la Gromov) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Homogeneous manifolds, See also {14M15, 14M17, 32M10, 57T15} 53C35 Symmetric spaces, See also {32M15, 57T15} 53C40 Global submanifolds, See also {53B25} 53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42} 53C45 Global surface theory (convex surfaces a la A. D. Aleksandrov) 53C50 Lorentz manifolds, manifolds with indefinite metrics 53C55 Hermitian and Kahlerian manifolds, See also {32Cxx} 53C56 Other complex differential geometry, See also {32Cxx} 53C60 Finsler spaces and generalizations (areal metrics), See also {58B20} 53C65 Integral geometry, See also {52A22, 60D05}; differential forms, currents, etc. See mainly{58Axx} 53C70 Direct methods ($G$-spaces of Busemann, etc.) 53C75 Geometric orders, order geometry, See also {51Lxx} 53C80 Applications to physics 53C99 None of the above but in this section \MajorSub 54-XX General topology, {For the topology of manifolds of all dimensions, See 57Nxx} 54-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 54-01 Instructional exposition (textbooks, tutorial papers, etc.) 54-02 Research exposition (monographs, survey articles) 54-03 Historical (must be assigned at least one classification number from Section 01) 54-04 Explicit machine computation and programs (not the theory of computation or programming) 54-06 Proceedings, conferences, collections, etc. 54Axx Generalities 54A05 Topological spaces and generalizations (closure spaces, etc.) 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54A15 Syntopogeneous structures 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets), See also {03Exx, 04A30}, {For ultrafilters, See 54D80} 54A35 Consistency and independence results, See also {03E35} 54A40 Fuzzy topology, See also {04A72} 54A99 None of the above but in this section 54Bxx Basic constructions 54B05 Subspaces 54B10 Product spaces 54B15 Quotient spaces, decompositions 54B17 Adjunction spaces and similar constructions 54B20 Hyperspaces 54B30 Categorical methods, See also {18B30} 54B35 Spectra 54B40 Presheaves and sheaves, See also {18F20} 54B99 None of the above but in this section 54Cxx Maps and general types of spaces defined by maps 54C05 Continuous maps 54C08 Weak and generalized continuity 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C15 Retraction 54C20 Extension of maps 54C25 Embedding 54C30 Real-valued functions, See also {26-XX} 54C35 Function spaces, See also {46Exx, 58D15} 54C40 Algebraic properties of function spaces, See also {46J10} 54C45 $C$- and $C$*-embedding 54C50 Special sets defined by functions, See also {26A21} 54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties), See also {55M15} 54C56 Shape theory, See also {55P55, 57N25} 54C60 Set-valued maps, See also {26E25, 28B20, 47H04, 58C06} 54C65 Selections, See also {28B20} 54C70 Entropy 54C99 None of the above but in this section 54Dxx Fairly general properties 54D05 Connected and locally connected spaces (general aspects) 54D10 Lower separation axioms, ($T_0$--$T_3$, etc.) 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54D20 Noncompact covering properties (paracompact, Lindelof, etc.) 54D25 ``$P$-minimal'' and ``$P$-closed'' spaces 54D30 Compactness 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D40 Remainders 54D45 Local compactness, $sigma$-compactness 54D50 $k$-spaces 54D55 Sequential spaces 54D60 Realcompactness and realcompactification 54D65 Separability 54D70 Base properties 54D80 Special constructions of spaces (spaces of ultrafilters, etc.) 54D99 None of the above but in this section 54Exx Spaces with richer structures 54E05 Proximity structures and generalizations 54E15 Uniform structures and generalizations 54E17 Nearness spaces 54E18 $p$-spaces, $M$-spaces, $sigma$-spaces, etc. 54E20 Stratifiable spaces, cosmic spaces, etc. 54E25 Semimetric spaces 54E30 Moore spaces 54E35 Metric spaces, metrizability 54E40 Special maps on metric spaces 54E45 Compact (locally compact) metric spaces 54E50 Complete metric spaces 54E52 Baire category, Baire spaces 54E55 Bitopologies 54E70 Probabilistic metric spaces 54E99 None of the above but in this section 54Fxx Special properties 54F05 Ordered topological spaces, partially ordered spaces, See also {06B30, 06F30} 54F15 Continua and generalizations 54F35 Higher-dimensional local connectedness, See also {55Mxx, 55Nxx} 54F45 Dimension theory, See also {55M10} 54F50 Spaces of dimension $\leq 1$; curves, dendrites, See Also { 26A03} 54F55 Unicoherence, multicoherence 54F65 Topological characterizations of particular spaces 54F99 None of the above but in this section 54Gxx Peculiar spaces 54G05 Extremally disconnected spaces, $F$-spaces, etc. 54G10 $P$-spaces 54G12 Scattered spaces 54G15 Pathological spaces 54G20 Counterexamples 54G99 None of the above but in this section 54Hxx Connections with other structures, applications 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets), See also {03E15, 04A15, 26A21, 28A05} 54H10 Topological representations of algebraic systems, See Also { 22-XX} 54H11 Topological groups, See also {22A05} 54H12 Topological lattices, etc., See also {06B30, 06F30} 54H13 Topological fields, rings, etc., See also {12Jxx}, {For algebraic aspects, See 13Jxx, 16W80} 54H15 Transformation groups and semigroups, See also {20M20, 22-XX, 57Sxx} 54H20 Topological dynamics, See also {28Dxx, 34C35, 58Fxx} 54H25 Fixed-point and coincidence theorems, See also {47H10, 55M20} 54H99 None of the above but in this section 54J05 Nonstandard topology, See also {03H05} \MajorSub 55-XX Algebraic topology 55-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 55-01 Instructional exposition (textbooks, tutorial papers, etc.) 55-02 Research exposition (monographs, survey articles) 55-03 Historical (must be assigned at least one classification number from Section 01) 55-04 Explicit machine computation and programs (not the theory of computation or programming) 55-06 Proceedings, conferences, collections, etc. 55Mxx Classical topics, {For the topology of Euclidean spaces and manifolds, See {57Nxx} {57Nxx}} 55M05 Duality 55M10 Dimension theory, See also {54F45} 55M15 Absolute neighborhood retracts, See also {54C55} 55M20 Fixed points and coincidences, See also {54H25} 55M25 Degree, winding number 55M30 Ljusternik-Schnirelman (Lyusternik-Shnirelman) category of a space 55M35 Finite groups of transformations (including Smith theory), See also {57S17} 55M99 None of the above but in this section 55Nxx Homology and cohomology theories, See also {57Txx} 55N05 Cech types 55N07 Steenrod-Sitnikov homologies 55N10 Singular theory 55N15 $K$-theory, See also {19Lxx}, {For algebraic $K$-theory, See 18F25, 19-XX} 55N20 Generalized (extraordinary) homology and cohomology theories 55N22 Bordism and cobordism theories, formal group laws, See also {14L05, 19L41, 57R75, 57R77, 57R85, 57R90} 55N25 Homology with local coefficients, equivariant cohomology 55N30 Sheaf cohomology, See also {18F20, 32C35, 32L10} 55N33 Intersection homology and cohomology 55N35 Other homology theories 55N40 Axioms for homology theory and uniqueness theorems 55N45 Products and intersections 55N91 Equivariant homology and cohomology, See also {19L47} 55N99 None of the above but in this section 55Pxx Homotopy theory, {For simple homotopy type, See 57Q10} 55P05 Homotopy extension properties, cofibrations 55P10 Homotopy equivalences 55P15 Classification of homotopy type 55P20 Eilenberg-Mac Lane spaces 55P25 Spanier-Whitehead duality 55P30 Eckmann-Hilton duality 55P35 Loop spaces 55P40 Suspensions 55P42 Stable homotopy theory, spectra 55P45 ${H}$-spaces and duals 55P47 Infinite loop spaces 55P50 Category and cocategory, etc. 55P55 Shape theory, See also {54C56, 55Q07} 55P60 Localization and completion 55P62 Rational homotopy theory 55P65 Homotopy functors 55P91 Equivariant homotopy theory, See also {19L47} 55P99 None of the above but in this section 55Qxx Homotopy groups 55Q05 Homotopy groups, general; sets of homotopy classes 55Q07 Shape groups 55Q10 Stable homotopy groups 55Q15 Whitehead products and generalizations 55Q20 Homotopy groups of wedges, joins, and simple spaces 55Q25 Hopf invariants 55Q30 Homotopy groups of triads, $n$-ads 55Q35 Operations in homotopy groups 55Q40 Homotopy groups of spheres 55Q45 Stable homotopy of spheres 55Q50 $J$-morphism, See also {19L20} 55Q52 Homotopy groups of special spaces 55Q55 Cohomotopy groups 55Q70 Homotopy groups of special types, See also {55N05, 55N07} 55Q91 Equivariant homotopy groups, See also {19L47} 55Q99 None of the above but in this section 55Rxx Fiber spaces and bundles, See also {18F15, 32Lxx, 46M20, { 57Rxx} {57R20, 57R22, 57R25}} 55R05 Fiber spaces 55R10 Fiber bundles 55R12 Transfer 55R15 Classification 55R20 Spectral sequences and homology of fiber spaces, See Also { 55Txx} 55R25 Sphere bundles and vector space bundles 55R35 Classifying spaces of groups and ${H}$-spaces 55R40 Homology of classifying spaces, characteristic classes, See also {57Txx, 57R20} 55R45 Homology and homotopy of $BO$ and $BU$; Bott periodicity 55R50 Stable classes of vector space bundles, $K$-theory, See also {19Lxx}, {For algebraic $K$-theory, See 18F25, 19-XX} 55R55 Fiberings with singularities 55R60 Microbundles and block bundles, See also {57N55, 57Q50} 55R65 Generalizations of fiber spaces and bundles 55R91 Equivariant fiber spaces and bundles, See also {19L47} 55R99 None of the above but in this section 55Sxx Operations and obstructions 55S05 Primary cohomology operations 55S10 Steenrod algebra 55S12 Dyer-Lashof operations 55S15 Symmetric products, cyclic products 55S20 Secondary and higher cohomology operations 55S25 $K$-theory operations and generalized cohomology operations, See also {19D55, 19Lxx} 55S30 Massey products 55S35 Obstruction theory 55S36 Extension and compression of mappings 55S37 Classification of mappings 55S40 Sectioning fiber spaces and bundles 55S45 Postnikov systems, $k$-invariants 55S91 Equivariant operations and obstructions, See also {19L47} 55S99 None of the above but in this section 55Txx Spectral sequences, See also {18G40, 55R20} 55T05 General 55T10 Serre spectral sequences 55T15 Adams spectral sequences 55T20 Eilenberg-Moore spectral sequences, See also {57T35} 55T25 Generalized cohomology 55T99 None of the above but in this section 55Uxx Applied homological algebra and category theory, See Also { 18Gxx} 55U05 Abstract complexes 55U10 Semisimplicial complexes 55U15 Chain complexes 55U20 Universal coefficient theorems, Bockstein operator 55U25 Homology of a product, Kunneth formula 55U30 Duality 55U35 Abstract homotopy theory 55U40 Topological categories 55U99 None of the above but in this section \MajorSub 57-XX Manifolds and cell complexes, {For complex manifolds, See 32C10} 57-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 57-01 Instructional exposition (textbooks, tutorial papers, etc.) 57-02 Research exposition (monographs, survey articles) 57-03 Historical (must be assigned at least one classification number from Section 01) 57-04 Explicit machine computation and programs (not the theory of computation or programming) 57-06 Proceedings, conferences, collections, etc. 57Mxx Low-dimensional topology 57M05 Fundamental group, presentations, free differential calculus 57M07 Topological methods in group theory 57M10 Covering spaces 57M12 Special coverings, e.g. branched 57M15 Relations with graph theory, See also {05Cxx} 57M20 Two-dimensional complexes 57M25 Knots and links in $S^3$, {For higher dimensions, See 57Q45} 57M30 Wild knots and surfaces, etc., wild embeddings 57M35 Dehn's lemma, sphere theorem, loop theorem, asphericity 57M40 Characterizations of $E^3$ and $S^3$ (Poincare conjecture), See also {57N12} 57M50 Geometric structures on low-dimensional manifolds 57M60 Group actions in low dimensions 57M99 None of the above but in this section 57Nxx Topological manifolds 57N05 Topology of $E^2$, $2$-manifolds 57N10 Topology of general $3$-manifolds, See also {57Mxx} 57N12 Topology of $E^3$ and $S^3$, See also {57M40} 57N13 Topology of $E^4$, $4$-manifolds, See also {14Jxx, 32Jxx} 57N15 Topology of $E^n$, $n$-manifolds ($4 \less n \less \infty$) 57N17 Topology of topological vector spaces 57N20 Topology of infinite-dimensional manifolds, See Also { 58Bxx} 57N25 Shapes, See also {54C56, 55P55, 55Q07} 57N30 Engulfing 57N35 Embeddings and immersions 57N37 Isotopy and pseudo-isotopy 57N40 Neighborhoods of submanifolds 57N45 Flatness and tameness 57N50 $S^{n-1}\subset E^n$, Schoenflies problem 57N55 Microbundles and block bundles, See also {55R60, 57Q50} 57N60 Cellularity 57N65 Algebraic topology of manifolds 57N70 Cobordism and concordance 57N75 General position and transversality 57N80 Stratifications 57N99 None of the above but in this section 57Pxx Generalized manifolds, See also {18F15} 57P05 Local properties of generalized manifolds 57P10 Poincare duality spaces 57P99 None of the above but in this section 57Qxx PL-topology 57Q05 General topology of complexes 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., See also {19B28} 57Q12 Wall finiteness obstruction for CW-complexes 57Q15 Triangulating manifolds 57Q20 Cobordism 57Q25 Comparison of PL-structures: classification, Hauptvermutung 57Q30 Engulfing 57Q35 Embeddings and immersions 57Q37 Isotopy 57Q40 Regular neighborhoods 57Q45 Knots and links (in high dimensions), {For the low-dimensional case, See 57M25} 57Q50 Microbundles and block bundles, See also {55R60, 57N55} 57Q55 Approximations 57Q60 Cobordism and concordance 57Q65 General position and transversality 57Q91 Equivariant PL-topology 57Q99 None of the above but in this section 57Rxx Differential topology, {For foundational questions of differentiable manifolds, See 58Axx; for infinite-dimensional manifolds, See 58Bxx} 57R05 Triangulating 57R10 Smoothing 57R12 Smooth approximations 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57R19 Algebraic topology on manifolds 57R20 Characteristic classes and numbers 57R22 Topology of vector bundles and fiber bundles, See Also { 55Rxx} 57R25 Vector fields, frame fields 57R27 Controllability of vector fields on $C^\infty$ and real-analytic manifolds, See also {49Qxx, 58F40, 93B05} 57R30 Foliations; geometric theory 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology, See also {58H10} 57R35 Differentiable mappings 57R40 Embeddings 57R42 Immersions 57R45 Singularities of differentiable mappings 57R50 Diffeomorphisms 57R52 Isotopy 57R55 Differentiable structures 57R57 Applications of global analysis to structures on manifolds, See also {58-XX} 57R60 Homotopy spheres, Poincare conjecture 57R65 Surgery and handlebodies 57R67 Surgery obstructions, Wall groups, See also {19J25} 57R70 Critical points and critical submanifolds 57R75 O- and SO-cobordism 57R77 Complex cobordism (U- and SU-cobordism), See also {55N22} 57R80 $h$- and $s$-cobordism 57R85 Equivariant cobordism 57R90 Other types of cobordism See{55N22} 57R91 Equivariant algebraic topology of manifolds 57R95 Realizing cycles by submanifolds 57R99 None of the above but in this section 57Sxx Topological transformation groups, See also {20F34, 22-XX, 54H15, 58D05} 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 57S10 Compact groups of homeomorphisms 57S15 Compact Lie groups of differentiable transformations 57S17 Finite transformation groups 57S20 Noncompact Lie groups of transformations 57S25 Groups acting on specific manifolds 57S30 Discontinuous groups of transformations 57S99 None of the above but in this section 57Txx Homology and homotopy of topological groups and related structures 57T05 Hopf algebras, See also {16W30} 57T10 Homology and cohomology of Lie groups 57T15 Homology and cohomology of homogeneous spaces of Lie groups 57T20 Homotopy groups of topological groups and homogeneous spaces 57T25 Homology and cohomology of ${H}$-spaces 57T30 Bar and cobar constructions, See also {18G55, 55Uxx} 57T35 Applications of Eilenberg-Moore spectral sequences, See also {55R20, 55T20} 57T99 None of the above but in this section \MajorSub 58-XX Global analysis, analysis on manifolds, See also {{ 32-XX} {32Cxx, 32Fxx}, 46-XX, 47Hxx, 53Cxx; for geometric integration theory, See {49Fxx} {49Q15}} 58-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 58-01 Instructional exposition (textbooks, tutorial papers, etc.) 58-02 Research exposition (monographs, survey articles) 58-03 Historical (must be assigned at least one classification number from Section 01) 58-04 Explicit machine computation and programs (not the theory of computation or programming) 58-06 Proceedings, conferences, collections, etc. 58Axx General theory of differentiable manifolds 58A03 Topos-theoretic approach to differentiable manifolds 58A05 Differentiable manifolds, foundations 58A07 Real-analytic and Nash manifolds, See also {14P20, 32C07} 58A10 Differential forms 58A12 de Rham theory, See also {14Fxx} 58A14 Hodge theory, See also {14C30, 14Fxx, 32J25, 32S35} 58A15 Exterior differential systems (Cartan theory) 58A17 Pfaffian systems 58A20 Jets 58A25 Currents, See also {32C30, 53C65} 58A30 Vector distributions (subbundles of the tangent bundles) 58A35 Stratified sets, See also {32S60, 58C27} 58A40 Differential spaces 58A50 Supermanifolds and graded manifolds, See also {14A22, 32C11} 58A99 None of the above but in this section 58Bxx Infinite-dimensional manifolds 58B05 Homotopy and topological questions 58B10 Differentiability questions 58B12 Questions of holomorphy, See also {32-XX, 46G20} 58B15 Fredholm structures, See also {47A53} 58B20 Riemannian, Finsler and other geometric structures, See also {53C20, 53C60} 58B25 Group structures and generalizations on infinite-dimensional manifolds, See also {22E65, 58D05} 58B30 Noncommutative differential geometry and topology, See also {46L30, 46L87, 46L89} 58B99 None of the above but in this section 58Cxx Calculus on manifolds; nonlinear operators, See also {47Hxx} 58C05 Real-valued functions 58C06 Set valued and function-space valued mappings, See Also { 47H04, 54C60} 58C07 Continuity properties of mappings 58C10 Holomorphic maps, See also {32-XX} 58C15 Implicit function theorems; global Newton methods 58C20 Differentiation theory (Gateaux, Frechet, etc.), See also {26Exx, 46G05} 58C25 Differentiable maps 58C27 Singularities of differentiable maps, See also {14B05, 14E15, 32Sxx} 58C28 Catastrophes, See also {57R70, 58Exx} 58C30 Fixed point theorems on manifoldsSee also {47H10} 58C35 Integration on manifolds; measures on manifolds, See Also { 28Cxx} 58C40 Spectral theory; eigenvalue problems, See also {47H12, 58E07} 58C50 Analysis on supermanifolds or graded manifolds 58C99 None of the above but in this section 58Dxx Spaces and manifolds of mappings {(including nonlinear versions of 46Exx)} 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds, See also {22E65, 57S05} 58D07 Groups and semigroups of nonlinear operators, See Also { 17B65, 47D03, 47D06, 47H20} 58D10 Spaces of imbeddings and immersions 58D15 Manifolds of mappings, See also {54C35} 58D17 Manifolds of metrics (esp. Riemannian) 58D19 Group actions and symmetry properties 58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps See{28Cxx} 58D25 Equations in function spaces; evolution equations, See also {34Gxx, 35K22, 35R15, 47H15} 58D27 Moduli problems for differential geometric structures 58D29 Moduli problems for topological structures 58D30 Applications (in quantum mechanics (Feynman path integrals), relativity, fluid dynamics, etc.) 58D99 None of the above but in this section 58Exx Variational problems in infinite-dimensional spaces 58E05 Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirelman) theory, etc.) 58E07 Abstract bifurcation theory 58E09 Group-invariant bifurcation theory 58E10 Applications to the theory of geodesics (problems in one independent variable) 58E11 Critical metrics 58E12 Applications to minimal surfaces (problems in two independent variables) 58E15 Application to extremal problems in several variables; Yang-Mills functionals, See also {81T13}, etc. 58E17 Pareto optimality, etc., applications to economics 58E20 Harmonic maps 58E25 Applications to control theory (optimal and nonoptimal) 58E30 Variational principles 58E35 Variational inequalities (global problems) 58E40 Group actions 58E50 Applications 58E99 None of the above but in this section 58Fxx Ordinary differential equations on manifolds; dynamical systems, See also {28D10, 34Cxx, {34Cxx} {54H20}} 58F03 One-dimensional dynamics, general symbolic dynamics, See also {26A18} 58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10} 58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10} 58F07 Completely integrable systems (including systems with an infinite number of degrees of freedom) 58F08 Point-mapping properties, iterations, completeness; dynamics of cellular automata, See also {26A18, 30D05} 58F09 Morse-Smale systems 58F10 Stability theory 58F11 Ergodic theory; invariant measures, See also {28Dxx} 58F12 Structure of attractors (and repellors) 58F13 Strange attractors; chaos and other pathologies, See Also { 70K50} 58F14 Bifurcation theory and singularities 58F15 Hyperbolic structures (expanding maps, Anosov systems, etc.) 58F17 Geodesic and horocycle flows 58F18 Relations with foliations 58F19 Eigenvalue and spectral problems 58F20 Periodic points and zeta functions 58F21 Limit cycles, singular points, etc. 58F22 Periodic solutions 58F23 Holomorphic dynamics, See also {30D05} 58F25 Flows 58F27 Quasiperiodic flows 58F30 Perturbations 58F32 Functional-differential equations on manifolds 58F35 Invariance properties 58F36 Normal forms 58F37 Correspondences and other transformation methods (e.g. Lie-Backlund) 58F39 Dynamical systems treatment of PDE (should be assigned another number from 58F), See also {35B32, 35K57} 58F40 Applications 58F99 None of the above but in this section 58Gxx Partial differential equations on manifolds; differential operators, See also {35-XX} 58G03 Elliptic equations on manifolds, general theory 58G05 Differential complexes, See also {35Nxx}; elliptic complexes 58G07 Relations with hyperfunctions 58G10 Index theory and related fixed point theorems, See Also { 19K56, 46L80} 58G11 Heat and other parabolic equation methods 58G12 Exotic index theories, See also {19K56, 46L05, 46L10, 46L80, 46M20} 58G15 Pseudodifferential and Fourier integral operators on manifolds, See also {35Sxx} 58G16 Hyperbolic equations 58G17 Propagation of singularities; initial value problems 58G18 Perturbations; asymptotics 58G20 Boundary value problems on manifolds 58G25 Spectral problems; spectral geometry; scattering theory, See also {35Pxx} 58G26 Determinants and determinant bundles 58G28 Bifurcations, See also {35B32} 58G30 Relations with special manifold structures (Riemannian, Finsler, etc.) 58G32 Diffusion processes and stochastic analysis on manifolds 58G35 Invariance and symmetry properties, See also {35A30} 58G37 Correspondences and other transformation methods (e.g. Lie-Backlund), See also {35A22} 58G40 Applications 58G99 None of the above but in this section 58Hxx Pseudogroups, differentiable groupoids and general structures on manifolds 58H05 Pseudogroups and differentiable groupoids, See also {22A22, 22E65} 58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.), See also {57R32} 58H15 Deformations of structures, See also {32Gxx, 58G05} 58H99 None of the above but in this section 58Z05 Applications to physics \MajorSub 60-XX Probability theory and stochastic processes, {For additional applications, See 11Kxx, 62-XX, 90-XX, 92-XX, 93-XX, 94-XX. For numerical results, See 65U05} 60-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 60-01 Instructional exposition (textbooks, tutorial papers, etc.) 60-02 Research exposition (monographs, survey articles) 60-03 Historical (must be assigned at least one classification number from Section 01) 60-04 Explicit machine computation and programs (not the theory of computation or programming) 60-06 Proceedings, conferences, collections, etc. 60Axx Foundations of probability theory 60A05 Axioms; other general questions 60A10 Probabilistic measure theory, {For ergodic theory, See 28Dxx and 60Fxx} 60A99 None of the above but in this section 60Bxx Probability theory on algebraic and topological structures 60B05 Probability measures on topological spaces 60B10 Convergence of probability measures 60B11 Probability theory on linear topological spaces, See Also { 28C20} 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60B15 Probability measures on groups, Fourier transforms, factorization 60B99 None of the above but in this section 60C05 Combinatorial probability 60D05 Geometric probability, stochastic geometry, random sets, See also {52A22, 53C65} 60Exx Distribution theory, See also {62Exx, 62Hxx} 60E05 Distributions: general theory 60E07 Infinitely divisible distributions; stable distributions 60E10 Characteristic functions; other transforms 60E15 Inequalities (Chebyshev, Kolmogorov, etc.) 60E99 None of the above but in this section 60Fxx Limit theorems, See also {28Dxx, 60B12} 60F05 Central limit and other weak theorems 60F10 Large deviations 60F15 Strong theorems 60F17 Functional limit theorems; invariance principles 60F20 Zero-one laws 60F25 $L^p$-limit theorems 60F99 None of the above but in this section 60Gxx Stochastic processes 60G05 Foundations of stochastic processes 60G07 General theory of processes 60G09 Exchangeability 60G10 Stationary processes 60G12 General second-order processes 60G15 Gaussian processes 60G17 Sample path properties 60G18 Self-similar processes 60G20 Generalized stochastic processes 60G25 Prediction theory, See also {62M20} 60G30 Continuity and singularity of induced measures 60G35 Applications (signal detection, filtering, etc.), See Also { 62M20, 93E10, 93E11, 94Axx} 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60} 60G42 Martingales with discrete parameter 60G44 Martingales with continuous parameter 60G46 Martingales and classical analysis 60G48 Generalizations of martingales 60G50 Sums of independent random variables 60G55 Point processes 60G57 Random measures 60G60 Random fields 60G70 Extreme value theory; extremal processes 60G99 None of the above but in this section 60Hxx Stochastic analysis, See also {58G32} 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations, See Also { 34F05} 60H15 Stochastic partial differential equations, See also {35R60} 60H20 Stochastic integral equations 60H25 Random operators and equations, See also {47B80} 60H30 Applications of stochastic analysis (to PDE, etc.) 60H99 None of the above but in this section 60Jxx Markov processes 60J05 Markov processes with discrete parameter 60J10 Markov chains with discrete parameter 60J15 Random walks 60J20 Applications of discrete Markov processes (social mobility, learning theory, industrial processes, etc.), See Also { 90B30, 92H10, 92H35, 92J40} 60J25 Markov processes with continuous parameter 60J27 Markov chains with continuous parameter 60J30 Processes with independent increments 60J35 Transition functions, generators and resolvents, See Also { 47D03, 47D07} 60J40 Right processes 60J45 Probabilistic potential theory, See also {31Cxx, 31D05} 60J50 Boundary theory 60J55 Local time and additive functionals 60J57 Multiplicative functionals 60J60 Diffusion processes, See also {58G32} 60J65 Brownian motion, See also {58G32} 60J70 Applications of diffusion theory (population genetics, absorption problems, etc.), See also {92Dxx} 60J75 Jump processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J85 Applications of branching processes, See also {92Dxx} 60J99 None of the above but in this section 60Kxx Special processes 60K05 Renewal theory 60K10 Applications (reliability, demand theory, etc.) 60K15 Markov renewal processes, semi-Markov processes 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.), See also {90Bxx} 60K25 Queueing theory, See also {68M20, 90B22} 60K30 Applications (congestion, allocation, storage, traffic, etc.), See also {90Bxx} 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43} 60K40 Other physical applications of random processes 60K99 None of the above but in this section \MajorSub 62-XX Statistics, {For numerical methods, See 65U05} 62-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 62-01 Instructional exposition (textbooks, tutorial papers, etc.) 62-02 Research exposition (monographs, survey articles) 62-03 Historical (must be assigned at least one classification number from Section 01) 62-04 Explicit machine computation and programs (not the theory of computation or programming) 62-06 Proceedings, conferences, collections, etc. 62-07 Data analysis 62-09 Graphical methods 62Axx Foundations 62A05 Invariance and group considerations 62A10 The likelihood approach 62A15 The Bayesian approach 62A20 The classical approach 62A25 The structural approach 62A30 The fiducial approach 62A99 None of the above but in this section 62Bxx Sufficiency and information 62B05 Sufficient statistics and fields 62B10 Statistical information theory, See also {94A17} 62B15 Theory of statistical experiments 62B20 Measure-theoretic results, etc. 62B99 None of the above but in this section 62Cxx Decision theory, See also {90A05, 90B50; for game theory, See 90D35} 62C05 General considerations 62C07 Complete class results 62C10 Bayesian problems; characterization of Bayes procedures 62C12 Empirical decision procedures; empirical Bayes procedures 62C15 Admissibility 62C20 Minimax procedures 62C25 Compound decision problems 62C99 None of the above but in this section 62D05 Sampling theory, sample surveys 62Exx Distribution theory, See also {60Exx} 62E10 Characterization and structure theory 62E15 Exact distribution theory 62E17 Approximations to distributions (nonasymptotic) 62E20 Asymptotic distribution theory 62E25 Monte Carlo studies 62E30 Formal computational methods (polykays, etc.) 62E99 None of the above but in this section 62Fxx Parametric inference 62F03 Hypothesis testing 62F04 Small sample properties of tests 62F05 Asymptotic properties of tests 62F07 Ranking and selection 62F10 Point estimation 62F11 Small sample properties of estimators 62F12 Asymptotic properties of estimators 62F15 Bayesian inference 62F25 Tolerance and confidence regions 62F30 Inference under constraints 62F35 Robustness and adaptive procedures 62F99 None of the above but in this section 62Gxx Nonparametric inference 62G05 Estimation 62G07 Curve estimation (nonparametric regression, density estimation, etc.) 62G09 Resampling methods 62G10 Hypothesis testing 62G15 Tolerance and confidence regions 62G20 Asymptotic properties 62G30 Order statistics; empirical distribution functions 62G35 Robustness 62G99 None of the above but in this section 62Hxx Multivariate analysis, See also {60Exx} 62H05 Characterization and structure theory 62H10 Distribution of statistics 62H11 Directional data; spatial statistics 62H12 Estimation 62H15 Hypothesis testing 62H17 Contingency 62H20 Measures of association (correlation, canonical correlation, etc.) 62H25 Factor analysis and principal components; correspondence analysis 62H30 Classification and discrimination; cluster analysis, See also {68T10} 62H40 Projection pursuit 62H99 None of the above but in this section 62Jxx Linear inference, regression 62J02 General nonlinear regression 62J05 Linear regression 62J07 Ridge regression; James-Stein estimators 62J10 Analysis of variance and covariance 62J12 Generalized linear models 62J15 Paired and multiple comparisons 62J20 Diagnostics 62J99 None of the above but in this section 62Kxx Experimental design, See also {05Bxx} 62K05 Optimal designs 62K10 Block designs 62K15 Factorial designs 62K99 None of the above but in this section 62Lxx Sequential methods 62L05 Sequential design 62L10 Sequential analysis 62L12 Sequential estimation 62L15 Optimal stopping, See also {60G40, 90D60} 62L20 Stochastic approximation 62L99 None of the above but in this section 62Mxx Inference from stochastic processes 62M02 Markov processes: hypothesis testing 62M05 Markov processes: estimation 62M07 Non-Markovian processes: hypothesis testing 62M09 Non-Markovian processes: estimation 62M10 Time series, auto-correlation, regression, etc., See Also { 90A20} 62M15 Spectral analysis 62M20 Prediction, See also {60G25}; filtering, See also {60G35, 93E10, 93E11} 62M30 Spatial processes 62M40 Random fields 62M99 None of the above but in this section 62Nxx Engineering statistics 62N05 Reliability and life testing, See also {90B25} 62N10 Quality control 62N99 None of the above but in this section 62Pxx Applications, See also {90-XX, 92-XX} 62P05 Applications to actuarial sciences and financial mathematics 62P10 Applications to biology 62P15 Applications to psychology 62P20 Applications to economics, See also {90Axx} 62P25 Applications to social sciences 62P99 None of the above but in this section 62Q05 Statistical tables \MajorSub 65-XX Numerical analysis 65-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 65-01 Instructional exposition (textbooks, tutorial papers, etc.) 65-02 Research exposition (monographs, survey articles) 65-03 Historical (must be assigned at least one classification number from Section 01) 65-04 Explicit machine computation and programs (not the theory of computation or programming) 65-05 Experimental papers 65-06 Proceedings, conferences, collections, etc. 65A05 Tables 65Bxx Acceleration of convergence 65B05 Extrapolation to the limit, deferred corrections 65B10 Summation of series 65B15 Euler-Maclaurin formula 65B99 None of the above but in this section 65Cxx Numerical simulation, {For theoretical aspects, See 68U20} 65C05 Monte Carlo methods 65C10 Random number generation 65C20 Models, numerical methods 65C99 None of the above but in this section 65Dxx Numerical approximation, {Primarily algorithms; for theory, See 41-XX} 65D05 Interpolation 65D07 Splines 65D10 Smoothing, curve fitting 65D15 Algorithms for functional approximation 65D17 Computer aided design (modeling of curves and surfaces), See also {68U05} 65D20 Computation of special functions, construction of tables 65D25 Numerical differentiation 65D30 Numerical integration 65D32 Quadrature and cubature formulas 65D99 None of the above but in this section 65E05 Numerical methods in complex analysis (potential theory, etc.), {For numerical methods in conformal mapping, See 30C30} 65Fxx Numerical linear algebra 65F05 Direct methods for linear systems and matrix inversion 65F10 Iterative methods for linear systems, See also {65N22} 65F15 Eigenvalues, eigenvectors 65F20 Overdetermined systems, pseudoinverses 65F25 Orthogonalization 65F30 Other matrix algorithms 65F35 Matrix norms, conditioning, scaling, See also {15A12, 15A60} 65F40 Determinants 65F50 Sparse matrices 65F99 None of the above but in this section 65Gxx Error analysis 65G05 Roundoff error 65G10 Interval and finite arithmetic 65G99 None of the above but in this section 65Hxx Nonlinear algebraic or transcendental equations 65H05 Single equations 65H10 Systems of equations 65H17 Eigenvalues, eigenvectors and bifurcation problems, See also {58C40, 58E07, 58F14, 90C30} 65H20 Global methods, including homotopy approaches, See Also { 58C30, 90C30} 65H99 None of the above but in this section 65Jxx Numerical analysis in abstract spaces 65J05 General theory 65J10 Equations with linear operators (do not use 65Fxx) 65J15 Equations with nonlinear operators (do not use 65Hxx) 65J20 Improperly posed problems 65J99 None of the above but in this section 65Kxx Mathematical programming, optimization and variational techniques 65K05 Mathematical programming, See also {90Cxx} 65K10 Optimization and variational techniques, See also {49Mxx, 93B40} 65K99 None of the above but in this section 65Lxx Ordinary differential equations 65L05 Initial value problems 65L06 Multistep, Runge-Kutta and extrapolation methods 65L07 Numerical investigation of stability of solutions 65L08 Improperly posed problems 65L10 Boundary value problems 65L12 Finite difference methods 65L15 Eigenvalue problems 65L20 Stability of numerical methods 65L50 Mesh generation and refinement 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 65L70 Error bounds 65L99 None of the above but in this section 65Mxx Partial differential equations, initial value and time-dependent initial-boundary value problems 65M06 Finite difference methods 65M12 Stability and convergence of numerical methods 65M15 Error bounds 65M20 Method of lines 65M25 Method of characteristics 65M30 Improperly posed problems 65M50 Mesh generation and refinement 65M55 Multigrid methods; domain decomposition 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M70 Spectral, collocation and related methods 65M99 None of the above but in this section 65Nxx Partial differential equations, boundary value problems 65N06 Finite difference methods 65N12 Stability and convergence of numerical methods 65N15 Error bounds 65N22 Solution of discretized equations, See also {65Fxx, 65Hxx} 65N25 Eigenvalue problems 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N35 Spectral, collocation and related methods 65N38 Boundary element methods 65N40 Method of lines 65N45 Method of contraction of the boundary 65N50 Mesh generation and refinement 65N55 Multigrid methods; domain decomposition 65N99 None of the above but in this section 65P05 Partial differential equations, miscellaneous problems 65Q05 Difference and functional equations, recurrence relations 65Rxx Integral equations, integral transforms, See also {45Lxx} 65R10 Integral transforms 65R20 Integral equations 65R30 Improperly posed problems 65R99 None of the above but in this section 65S05 Graphical methods 65Txx Numerical methods in Fourier analysis 65T10 Trigonometric approximation and interpolation 65T20 Discrete and fast Fourier transforms 65T99 None of the above but in this section 65U05 Numerical methods in probability and statistics 65Yxx Computer aspects of numerical algorithms 65Y05 Parallel computation 65Y10 Algorithms for specific classes of architectures 65Y15 Packaged methods 65Y20 Complexity and performance of numerical algorithms, See also {68Q25} 65Y25 Computer graphics and computational geometry, See Also { 51N05, 68U05} 65Y99 None of the above but in this section \MajorSub 68-XX Computer science, {For papers involving machine computations and programs in a specific mathematical area, See {Section --04 in} that area} 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 68-01 Instructional exposition (textbooks, tutorial papers, etc.) 68-02 Research exposition (monographs, survey articles) 68-03 Historical (must be assigned at least one classification number from Section 01) 68-04 Explicit machine computation and programs (not the theory of computation or programming) 68-06 Proceedings, conferences, collections, etc. 68Mxx Computer system organization 68M05 General 68M07 Mathematical problems of computer architecture 68M10 Computer networks, See also {90B12} 68M15 Reliability and testing 68M20 Performance evaluation; queueing; scheduling, See Also { 60K25, 90Bxx} 68M99 None of the above but in this section 68Nxx Software 68N05 General theory of programming 68N15 Programming languages 68N17 Logic programming 68N20 Compilers and generators, See also {68Q52} 68N25 Monitors and operating systems 68N99 None of the above but in this section 68Pxx Theory of data 68P05 Data structures 68P10 Searching and sorting 68P15 Database theory 68P20 Information storage and retrieval 68P25 Data encryption, See also {94A60} 68P99 None of the above but in this section 68Qxx Theory of computing 68Q05 Models of computation (abstract processors, Turing machines, etc.), See also {03D10} 68Q10 Modes of computation (concurrent, parallel, nondeterministic, etc.), See also {68Q90} 68Q15 Complexity classes, See also {03D15} 68Q20 Nonnumerical algorithms, {For numerical algorithms, See 65-XX; for combinatorics and graph theory, See 68Rxx} 68Q22 Parallel and distributed algorithms, {For numerical algorithms, See 65Y05, 68Y10} 68Q25 Analysis of algorithms and problem complexity 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.) 68Q35 VLSI algorithms 68Q40 Symbolic computation, algebraic computation, See Also { 11Yxx, 12Y05, 13Pxx, 14Qxx, 16-08, 17-08} 68Q42 Rewriting systems 68Q45 Formal languages, See also {03D05, 20M35, 94A45} 68Q50 Grammars, See also {03D05} 68Q52 Parsing, See also {68N20} 68Q55 Semantics, See also {03B70, 06B35} 68Q60 Specification and verification of programs, See Also { 03B70} 68Q65 Abstract data types; algebraic specification 68Q68 Automata theory, general, See also {03D05} 68Q70 Algebraic theory of automata, See also {18B20, 20M35} 68Q75 Stochastic and nondeterministic automata 68Q80 Tessellation automata, iterative arrays, cellular structures 68Q90 Transition nets 68Q99 None of the above but in this section 68Rxx Discrete mathematics in relation to computer science 68R05 Combinatorics 68R10 Graph theory, See also {05Cxx, 90B10, 90B35, 90C35} 68R15 Combinatorics on words 68R99 None of the above but in this section 68S05 Mathematical linguistics, See also {03B65, 92K20} 68Txx Artificial intelligence, See also {92J40} 68T01 General 68T05 Learning and adaptive systems 68T10 Pattern recognition, speech recognition, {For cluster analysis, See 62H30} 68T15 Theorem proving, See also {03B35} 68T20 Problem solving 68T25 AI languages 68T27 AI logics 68T30 Knowledge representation 68T35 AI software systems (knowledge-based systems, expert systems, etc.) 68T99 None of the above but in this section 68Uxx Computing methodologies 68U05 Computer graphics; computational geometry, See also {65Y25} 68U07 Computer-aided design 68U10 Image processing 68U15 Text processing; mathematical typography 68U20 Simulation, See also {65Cxx} 68U30 Other applications 68U99 None of the above but in this section \MajorSub 70-XX Mechanics of particles and systems, {For relativistic mechanics, See {83-XX} {83A05 and 83C10}; for statistical mechanics, See 82-XX} 70-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 70-01 Instructional exposition (textbooks, tutorial papers, etc.) 70-02 Research exposition (monographs, survey articles) 70-03 Historical (must be assigned at least one classification number from Section 01) 70-04 Explicit machine computation and programs (not the theory of computation or programming) 70-05 Experimental papers 70-06 Proceedings, conferences, collections, etc. 70-08 Computational methods 70A05 Axiomatics, foundations 70Bxx Kinematics, See also {53A17} 70B05 Kinematics of a particle 70B10 Kinematics of a rigid body 70B15 Mechanisms, robots, See also {73K05} 70B99 None of the above but in this section 70C20 Statics 70Dxx Dynamics of a particle, See also {70Hxx} 70D05 Newtonian dynamics 70D10 Lagrangian dynamics 70D99 None of the above but in this section 70Exx Dynamics of a rigid body 70E05 Motion of the gyroscope 70E10 Motion of projectiles and rockets 70E15 Motion of rigid bodies 70E20 Perturbation methods for Euler's equations 70E99 None of the above but in this section 70Fxx Dynamics of a system of particles, including celestial mechanics 70F05 Two-body problem 70F07 Three-body problem 70F10 $n$-body problem 70F15 Celestial mechanics 70F20 Holonomic systems 70F25 Nonholonomic systems 70F30 Impulsive motion 70F35 Collisions 70F99 None of the above but in this section 70Gxx General representations of dynamical systems, See Also { 58Fxx} 70G05 Riemannian geometry, tensorial methods, See also {53A45, 53A50, 53B20} 70G10 Generalized coordinates 70G15 Space of events 70G20 Impulse-energy space 70G25 Configuration space 70G30 State space 70G35 Phase space 70G50 Classical field theories (general) 70G99 None of the above but in this section 70Hxx Hamiltonian and Lagrangian mechanics, See also {58F05} 70H05 Hamilton's equations 70H10 Liouville's theorem 70H15 Canonical transformations 70H20 Hamilton-Jacobi equations 70H25 Hamilton's principle 70H30 Other variational principles 70H33 Symmetries 70H35 Lagrange's equation of motion 70H40 Relativistic dynamics 70H99 None of the above but in this section 70Jxx Linear vibration theory, See also {73D30} 70J05 Finite degree of freedom systems 70J10 Modal analysis 70J25 Stability 70J30 Free motions 70J35 Forced motions 70J40 Parametric resonances 70J99 None of the above but in this section 70Kxx Nonlinear motions, See also {34Cxx, 58Fxx, 73D35, 73K12} 70K05 Phase plane analysis 70K10 Limit cycles 70K15 Lyapunov theorems 70K20 Stability 70K25 Free motions 70K28 Parametric resonances 70K30 Nonlinear resonances 70K40 Forced motions 70K50 Transition to stochasticity (chaotic behavior), See Also { 58F13} 70K99 None of the above but in this section 70L05 Random vibrations, See also {73K35, 93Exx} 70M20 Orbital mechanics 70P05 Variable mass, rockets 70Q05 Control of mechanical systems, See also {49-XX, 73K50, 93Cxx, 93Dxx} \MajorSub 73-XX Mechanics of solids 73-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 73-01 Instructional exposition (textbooks, tutorial papers, etc.) 73-02 Research exposition (monographs, survey articles) 73-03 Historical (must be assigned at least one classification number from Section 01) 73-04 Explicit machine computation and programs (not the theory of computation or programming) 73-05 Experimental papers 73-06 Proceedings, conferences, collections, etc. 73A05 Axiomatics, foundations of solid mechanics 73Bxx Continuum mechanics of solids (constitutive description and properties) 73B05 Constitutive equations 73B10 Symmetry groups 73B18 Nonlocal theories 73B25 Polar theories 73B27 Nonhomogeneous materials; homogenization 73B30 Thermodynamics of solids, {For gases and fluids, See 80-XX} 73B35 Random materials 73B40 Anisotropic materials 73B50 Stress concentrations 73B99 None of the above but in this section 73Cxx Elasticity, {For the biharmonic equation, See 31A30, 31B30} 73C02 Classical linear elasticity 73C05 Stress functions 73C10 Saint-Venant's principle 73C15 Uniqueness theorems 73C35 Mixed boundary value problems, See also {45F05} 73C50 Nonlinear elasticity 73C99 None of the above but in this section 73Dxx Wave propagation in and vibrations of solids 73D05 Impact and explosion problems, See also {76L05} 73D10 Integral transforms 73D15 Body waves 73D20 Surface waves 73D25 Wave diffraction and dispersion 73D30 Linear vibrations, See also {70Jxx} 73D35 Nonlinear vibrations, See also {70Kxx} 73D40 Singular surfaces 73D50 Inverse problems, See also {35Lxx, 35R30} 73D70 Random waves 73D99 None of the above but in this section 73Exx Plasticity 73E05 Constitutive specifications (yield criteria, flow rules, hardening, softening) 73E10 Method of successive approximations 73E20 Limit analysis 73E50 Time-dependent problems 73E60 Viscoplasticity 73E70 Plastic waves 73E99 None of the above but in this section 73Fxx Viscoelasticity 73F05 Creep and relaxation functions 73F10 Correspondence principle 73F15 Time-dependent problems 73F20 Aging of materials 73F25 Environmental-dependent materials 73F99 None of the above but in this section 73Gxx Finite deformations 73G05 Finite elasticity 73G20 Finite plasticity 73G25 Finite viscoelasticity 73G99 None of the above but in this section 73Hxx Stability (linear and nonlinear) 73H05 Buckling 73H10 Dynamic stability 73H99 None of the above but in this section 73Kxx Mechanics of structures 73K03 Strings 73K05 Beams, columns, rods 73K10 Plates, discs, membranes 73K12 Dynamics of structures 73K15 Shells 73K20 Composite structures and materials 73K35 Random vibrations 73K40 Optimization, See also {90C90} 73K50 Control of structures 73K70 Aero- or hydromechanic structure interactions 73K99 None of the above but in this section 73M25 Fracture mechanics 73N20 Geophysical solid mechanics, See also {86A15, 86A17} 73P05 Biomechanics of solids, See also {92C10} 73Q05 Soil and rock mechanics, See also {86A60} 73R05 Electromagnetic elasticity 73S10 Micromechanics of solids 73T05 Contact and surface mechanics 73Vxx Basic methods in solid mechanics, See also {65-xx} 73V05 Finite element methods 73V10 Boundary element methods 73V15 Finite difference methods 73V20 Other numerical methods 73V25 Variational methods 73V30 Stochastic analysis 73V35 Complex variable techniques 73V99 None of the above but in this section \MajorSub 76-XX Fluid mechanics, {For general continuum mechanics, See 73Bxx, or other parts of 73-XX} 76-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 76-01 Instructional exposition (textbooks, tutorial papers, etc.) 76-02 Research exposition (monographs, survey articles) 76-03 Historical (must be assigned at least one classification number from Section 01) 76-04 Explicit machine computation and programs (not the theory of computation or programming) 76-05 Experimental papers 76-06 Proceedings, conferences, collections, etc. 76Axx Foundations, constitutive equations, rheology 76A02 Foundations of fluid mechanics 76A05 Non-Newtonian fluids 76A10 Viscoelastic fluids 76A15 Liquid crystals, See also {82D30} 76A99 None of the above but in this section 76Bxx Incompressible inviscid fluids, potential theory 76B05 Airfoil theory 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, hydrofoil theory, sloshing 76B15 Water waves, gravity waves; dispersion and diffraction, nonlinear interaction, See also {35Q53} 76B20 Ship waves 76B25 Solitary and cnoidal waves, See also {35Q51} 76B35 Random waves, inviscid fluids 76B40 Added mass computations 76B45 Capillarity 76B99 None of the above but in this section 76Cxx Incompressible inviscid fluids, vorticity flows 76C05 Vorticity flows 76C10 Internal waves 76C15 Atmospheric waves 76C20 Rossby waves 76C99 None of the above but in this section 76Dxx Incompressible viscous fluids 76D05 Navier-Stokes equations, See also {35Q30} 76D07 Stokes flows 76D08 Lubrication theory 76D10 Boundary-layer theory 76D15 Boundary-layer separation and reattachment 76D20 Higher-order effects in boundary layers 76D25 Wakes and jets 76D30 Singular perturbation problems 76D33 Waves 76D35 Random waves, viscous fluids 76D45 Capillarity, See also {76B45} 76D99 None of the above but in this section 76Exx Hydrodynamic stability 76E05 Stability of parallel flows 76E10 Inertial instability 76E15 Convective instability 76E20 Instability of geophysical and astrophysical flows 76E25 Magnetohydrodynamic and electrohydrodynamic instabilities 76E30 Nonlinear effects 76E99 None of the above but in this section 76Fxx Turbulence, See also {58F13, 58F27, 60Gxx, 60Jxx} 76F05 Homogeneous isotropic turbulence 76F10 Shear flows 76F20 Turbulence via chaos techniques 76F99 None of the above but in this section 76G25 General aerodynamics and subsonic flows 76H05 Transonic flows 76J20 Supersonic flows 76K05 Hypersonic flows 76L05 Shock waves and blast waves, See also {35L67, 73D05} 76Mxx Basic methods in fluid mechanics, See also {65-XX} 76M10 Finite element methods 76M15 Boundary element methods 76M20 Finite difference methods 76M25 Other numerical methods 76M30 Variational methods 76M35 Stochastic analysis 76M99 None of the above but in this section 76Nxx Compressible fluids and gas dynamics, general 76N10 Compressible fluids, general, See also {35Q30} 76N15 Gas dynamics, general 76N20 Boundary-layer theory 76N99 None of the above but in this section 76P05 Rarefied gas flows, Boltzmann equation, See also {82B40, 82C40, 82D05} 76Q05 Hydrodynamic sound, acoustics 76Rxx Diffusion and convection 76R05 Forced convection 76R10 Free convection 76R50 Diffusion, See also {60J60} 76R99 None of the above but in this section 76S05 Flows in porous media; filtration; Seepage 76T05 Two-phase and multiphase flows 76U05 Rotating fluids 76V05 Stratified and reacting fluids 76W05 Magnetohydrodynamics and electrohydrodynamics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76Y05 Quantum hydrodynamics and relativistic hydrodynamics, See also {83C55, 85A30} 76Zxx Biological fluid mechanics, See also {92Cxx} 76Z05 Physiological flows, See also {92C35} 76Z10 Biopropulsion in water and in air 76Z99 None of the above but in this section \MajorSub 78-XX Optics, electromagnetic theory, {For quantum optics, See 81V80} 78-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 78-01 Instructional exposition (textbooks, tutorial papers, etc.) 78-02 Research exposition (monographs, survey articles) 78-03 Historical (must be assigned at least one classification number from Section 01) 78-04 Explicit machine computation and programs (not the theory of computation or programming) 78-05 Experimental papers 78-06 Proceedings, conferences, collections, etc. 78-08 Computational methods 78A02 Foundations 78A05 Geometric optics 78A10 Physical optics 78A15 Electron optics 78A20 Space charge waves 78A25 Electromagnetic theory, general 78A30 Electro- and magnetostatics 78A35 Motion of charged particles 78A40 Waves and radiation 78A45 Diffraction, scattering, See also {34E20 for WKB methods} 78A50 Antennas, wave-guides 78A55 Technical applications 78A60 Lasers, masers, optical bistability, nonlinear optics, See also {81V80} 78A70 Biological applications, See also {92C30, 92J30} 78A97 Mathematically heuristic optics and electromagnetic theory (must also be assigned at least one other classification number in this section) 78A99 Miscellaneous topics \MajorSub 80-XX Classical thermodynamics, heat transfer, {For thermodynamics of solids, See 73B30} 80-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 80-01 Instructional exposition (textbooks, tutorial papers, etc.) 80-02 Research exposition (monographs, survey articles) 80-03 Historical (must be assigned at least one classification number from Section 01) 80-04 Explicit machine computation and programs (not the theory of computation or programming) 80-05 Experimental papers 80-06 Proceedings, conferences, collections, etc. 80-08 Computational methods 80A05 Foundations 80A10 Classical thermodynamics 80A15 Thermodynamics of mixtures 80A20 Heat and mass transfer, heat flow 80A22 Stefan problems, phase changes, etc. 80A23 Inverse problems 80A25 Combustion, interior ballistics 80A30 Chemical kinetics, See also {92C45, 92E20} 80A32 Chemically reacting flows, See also {92C45, 92E20} 80A50 Chemistry (general) See mainly{92Exx} 80A97 Mathematically heuristic classical thermodynamics (must also be assigned at least one other classification number in this section) 80A99 Miscellaneous topics \MajorSub 81-XX Quantum Theory 81-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 81-01 Instructional exposition (textbooks, tutorial papers, etc.) 81-02 Research exposition (monographs, survey articles) 81-03 Historical (must be assigned at least one classification number from Section 01) 81-04 Explicit machine computation and programs (not the theory of computation or programming) 81-05 Experimental papers 81-06 Proceedings, conferences, collections, etc. 81-08 Computational methods 81Pxx Axiomatics, foundations, philosophy 81P05 General and philosophical 81P10 Logical foundations of quantum mechanics 81P15 Quantum measurement theory 81P20 Stochastic mechanics (including stochastic electrodynamics) 81P99 None of the above but in this section 81Qxx General mathematical topics and methods in quantum theory 81Q05 Closed and approximate solutions to the Schrodinger, Dirac, Klein-Gordon and other quantum-mechanical equations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q15 Perturbation theories for operators and differential equations 81Q20 Semiclassical techniques including WKB and Maslov methods 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, See also {14D05, 32S40} 81Q40 Bethe-Salpeter and other integral equations 81Q50 Quantum chaos, See also {58F13} 81Q60 Supersymmetric quantum mechanics 81Q99 None of the above but in this section 81Rxx Groups and algebras in quantum theory 81R05 Representations of finite-dimensional groups and algebras motivated by physics, See also {20C35, 22E70} 81R10 Representations of infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody and other current algebras, See also {17B65, 17B67, 22E65, 22E67, 22E70} 81R20 Covariant wave equations 81R25 Spinor and twistor methods, See also {32L25} 81R30 Coherent states, See also {22E45} 81R40 Symmetry breaking 81R50 Quantum groups and related algebraic methods, See Also { 16W30, 17B37} 81R99 None of the above but in this section 81Sxx General quantum mechanics and problems of quantization 81S05 Commutation relations 81S10 Geometric quantization, symplectic methods, See Also { 58F06} 81S20 Stochastic quantization 81S25 Quantum stochastic calculus 81S30 Phase space methods including Wigner distributions, etc. 81S40 Path integrals, See also {58D30} 81S99 None of the above but in this section 81Txx Quantum field theory; related classical field theories 81T05 Axiomatic quantum field theory; operator algebras 81T08 Constructive quantum field theory 81T10 Model quantum field theories 81T13 Yang-Mills and other gauge theories, See also {53C07, 58E15} 81T15 Perturbative methods of renormalization 81T16 Nonperturbative methods of renormalization 81T17 Renormalization group methods 81T18 Feynman diagrams 81T20 Quantum field theory on curved space backgrounds 81T25 Quantum field theory on lattices 81T27 Continuum limits 81T30 String and superstring theories; other extended objects , See also {83E30} 81T40 Two-dimensional field theories, conformal field theories, etc. 81T50 Anomalies 81T60 Supersymmetric field theories 81T70 Quantization in field theory; cohomological methods, See also {58F06} 81T80 Simulation and numerical modeling 81T99 None of the above but in this section 81Uxx Scattering theory, See also {47A40} 81U05 $2$-body potential scattering theory, See also {34E20 for WKB methods} 81U10 $n$-body potential scattering theory 81U20 $S$-matrix theory, etc. 81U30 Dispersion theory, dispersion relations 81U40 Inverse scattering problems, See also {58F07, 58F19} 81U99 None of the above but in this section 81Vxx Applications to specific physical systems 81V05 Strong interaction, including quantum chromodynamics 81V10 Electromagnetic interaction; quantum electrodynamics 81V15 Weak interaction 81V17 Gravitational interaction 81V19 Other fundamental interactions 81V22 Unified theories 81V25 Other elementary particle theory 81V35 Nuclear physics 81V45 Atomic physics 81V55 Molecular physics, See also {92E10} 81V70 Many-body theory 81V80 Quantum optics 81V99 None of the above but in this section \MajorSub 82-XX Statistical mechanics, structure of matter 82-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 82-01 Instructional exposition (textbooks, tutorial papers, etc.) 82-02 Research exposition (monographs, survey articles) 82-03 Historical (must be assigned at least one classification number from Section 01) 82-04 Explicit machine computation and programs (not the theory of computation or programming) 82-05 Experimental papers 82-06 Proceedings, conferences, collections, etc. 82-08 Computational methods 82Bxx Equilibrium statistical mechanics 82B03 Foundations 82B05 Classical equilibrium statistical mechanics (general) 82B10 Quantum equilibrium statistical mechanics (general) 82B20 Lattice systems (Ising, dimer, Potts, etc.) 82B21 Continuum models (systems of particles, etc.) 82B23 Exactly solvable models 82B24 Interface problems 82B26 Phase transitions (general) 82B27 Critical phenomena 82B28 Renormalization group methods, See also {81T17} 82B30 Statistical thermodynamics, See also {80-XX} 82B31 Stochastic methods 82B35 Irreversible thermodynamics, including Onsager-Machlup theory, See also {92E20} 82B40 Kinetic theory of gases 82B41 Random walks, random surfaces, lattice animals, etc., See also {60J15, 82C41} 82B43 Percolation, See also {60K35} 82B44 Disordered systems (random Ising models, random Schrodinger operators, etc.) 82B80 Numerical methods (Monte Carlo, series resummation, etc.), See also {65-XX, 81T80} 82B99 Miscellaneous topics 82Cxx Time-dependent statistical mechanics (dynamic and nonequilibrium) 82C03 Foundations 82C05 Classical dynamic and nonequilibrium statistical mechanics (general) 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 82C20 Dynamic lattice systems (kinetic Ising, etc.) 82C21 Dynamic continuum models (systems of particles, etc.) 82C22 Interacting particle systems, See also {60K35} 82C23 Exactly solvable dynamic models, See also {58F07} 82C24 Interface problems 82C26 Dynamic and nonequilibrium phase transitions (general) 82C27 Dynamic critical phenomena 82C28 Dynamic renormalization group methods, See also {81T17} 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.), See also {60H10} 82C32 Neural nets, See also {68T05, 92B20, 92J40} 82C35 Irreversible thermodynamics, including Onsager-Machlup theory 82C40 Kinetic theory of gases 82C41 Dynamics of random walks, random surfaces, lattice animals, etc., See also {60J15} 82C43 Time-dependent percolation, See also {60K35} 82C44 Dynamics of disordered systems (random Ising systems, etc.) 82C70 Transport processes 82C80 Numerical methods (Monte Carlo, series resummation, etc.) 82C99 None of the above but in this section 82Dxx Applications to specific types of physical systems 82D05 Gases 82D10 Plasmas 82D15 Liquids 82D20 Solids 82D25 Crystals, {For crystallographic group theory, See 20H15} 82D30 Random media, disordered materials (including liquid crystals and spin glasses) 82D35 Metals 82D40 Magnetic materials 82D45 Ferroelectrics 82D50 Superfluids 82D55 Superconductors 82D60 Polymers 82D75 Nuclear reactor theory 82D99 None of the above but in this section \MajorSub 83-XX Relativity and gravitational theory 83-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 83-01 Instructional exposition (textbooks, tutorial papers, etc.) 83-02 Research exposition (monographs, survey articles) 83-03 Historical (must be assigned at least one classification number from Section 01) 83-04 Explicit machine computation and programs (not the theory of computation or programming) 83-05 Experimental papers 83-06 Proceedings, conferences, collections, etc. 83-08 Computational methods 83A05 Special relativity 83B05 Observational and experimental questions 83Cxx General relativity 83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems) 83C10 Equations of motion 83C15 Exact solutions 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries 83C22 Electromagnetic fields, Einstein-Maxwell equations 83C25 Approximation procedures, weak fields 83C27 Lattice gravity, Regge calculus and other discrete methods 83C30 Asymptotic procedures (radiation, news functions, {H}-spaces, etc.) 83C35 Gravitational waves 83C40 Gravitational energy and conservation laws; groups of motions 83C45 Quantization of the gravitational field 83C47 Quantum field theory aspects, See also {81T20} 83C50 Electromagnetic fields 83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.) 83C57 Black holes 83C60 Spinor and twistor methods 83C75 Space-time singularities, cosmic censorship, etc. 83C99 None of the above but in this section 83D05 Relativistic gravitational theories other than Einstein's, including asymmetric field theories 83Exx Unified, higher-dimensional and super field theories 83E05 Geometrodynamics 83E15 Kaluza-Klein and other higher-dimensional theories 83E30 String and superstring theories, See also {81T30} 83E50 Supergravity 83E99 None of the above but in this section 83F05 Cosmology \MajorSub 85-XX Astronomy and astrophysics, {For celestial mechanics, See 70F15} 85-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 85-01 Instructional exposition (textbooks, tutorial papers, etc.) 85-02 Research exposition (monographs, survey articles) 85-03 Historical (must be assigned at least one classification number from Section 01) 85-04 Explicit machine computation and programs (not the theory of computation or programming) 85-05 Experimental papers 85-06 Proceedings, conferences, collections, etc. 85-08 Computational methods 85A04 General 85A05 Galactic and stellar dynamics 85A15 Galactic and stellar structure 85A20 Stellar atmospheres 85A25 Radiative transfer 85A30 Hydrodynamic and hydromagnetic problems, See also {76Y05} 85A35 Statistical astronomy 85A40 Cosmology, {For relativistic cosmology, See 83F05} 85A45 Radio astronomy 85A99 Miscellaneous topics \MajorSub 86-XX Geophysics, See also {73N05, 76U05, 76V05} 86-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 86-01 Instructional exposition (textbooks, tutorial papers, etc.) 86-02 Research exposition (monographs, survey articles) 86-03 Historical (must be assigned at least one classification number from Section 01) 86-04 Explicit machine computation and programs (not the theory of computation or programming) 86-05 Experimental papers 86-06 Proceedings, conferences, collections, etc. 86-08 Computational methods 86A04 General 86A05 Hydrology, hydrography, oceanography, See also {76B15, 76B20, 76B25, 76C15, 76E20, 76Q05, 76Rxx, 76U05} 86A10 Meteorology and atmospheric physics, See also {76Bxx, 76C15, 76C20, 76E20, 76N15, 76Q05, 76Rxx, 76U05, 76V05} 86A15 Seismology, See also {73Dxx, 73Fxx, 73M25, 73N20, 73Q05} 86A17 Global dynamics, earthquake problems 86A20 Potentials, prospecting, See also {76S05, 76W05} 86A22 Inverse problems, See also {35R30} 86A25 Geo-electricity and geomagnetism, See also {78A25} 86A30 Geodesy, mapping problems 86A32 Geostatistics 86A40 Glaciology 86A60 Geological problems 86A99 Miscellaneous topics \MajorSub 90-XX Economics, operations research, programming, games 90-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 90-01 Instructional exposition (textbooks, tutorial papers, etc.) 90-02 Research exposition (monographs, survey articles) 90-03 Historical (must be assigned at least one classification number from Section 01) 90-04 Explicit machine computation and programs (not the theory of computation or programming) 90-06 Proceedings, conferences, collections, etc. 90-08 Computational methods 90Axx Mathematical economics, {For econometrics, See 62P20} 90A05 Decision theory, See also {62Cxx, 90B50, 90D35} 90A06 Individual preferences 90A07 Group preferences 90A08 Social choice 90A09 Finance, portfolios, investment 90A10 Utility theory 90A11 Production theory, theory of the firm 90A12 Price theory and market structure 90A14 Equilibrium: general theory 90A15 General economic models, trade models 90A16 Dynamic economic models, growth models 90A17 Multisectoral models 90A19 Statistical models; economic indexes and measures 90A20 Economic time series analysis, See also {62M10} 90A25 Spatial models 90A27 Public goods 90A28 Voting theory 90A30 Environmental economics (natural resource models, harvesting, pollution, etc.) 90A35 Informational economics 90A36 Incentives theory 90A40 Consumer behavior, demand theory 90A43 Expected utility; risk-averse utility 90A46 Risk theory 90A50 Labor market 90A53 Special types of economies 90A56 Special types of equilibria 90A58 Models of real-world systems; general macro-economic models, etc. 90A60 Market models (auctions, bargaining, bidding, selling, etc.) 90A70 Macro-economic policy-making, taxation 90A80 Resource allocation 90A99 None of the above but in this section 90Bxx Operations research and management science 90B05 Inventory, storage, reservoirs 90B06 Transportation, logistics 90B10 Flows in networks, deterministic 90B12 Communication networks, See also {68M10, 94A05} 90B15 Flows in networks, probabilistic 90B20 Highway traffic 90B22 Queues and service, See also {60K25, 68M20} 90B25 Reliability, availability, maintenance, inspection, See also {60K10, 62N05} 90B30 Production models 90B35 Scheduling theory, See also {68M20} 90B40 Search theory 90B50 Management decision-making, including multiple objectives, See also {90A05, 90C31, 90D35} 90B60 Marketing, advertising, See also {90A60} 90B70 Theory of organizations, industrial and manpower planning, See also {92H35} 90B80 Discrete location and assignment, See also {90C10} 90B85 Continuous location 90B90 Case-oriented studies 90B99 None of the above but in this section 90Cxx Mathematical programming, See also {49Mxx, 65Kxx} 90C05 Linear programming 90C06 Large-scale problems 90C08 Special problems of linear programming (transportation, multi-index, etc.) 90C09 Boolean programming 90C10 Integer programming 90C11 Mixed integer programming 90C15 Stochastic programming 90C20 Quadratic programming 90C25 Convex programming 90C26 Nonconvex programming, quasiconvex programming 90C27 Combinatorial optimization 90C28 Geometric programming 90C29 Multi-objective and goal programming; vector optimization 90C30 Nonlinear programming 90C31 Sensitivity, stability, parametric optimization 90C32 Fractional programming 90C33 Complementarity problems 90C34 Semi-infinite programming 90C35 Programming involving graphs or networks 90C39 Dynamic programming, See also {49L20} 90C40 Markov and semi-Markov decision processes 90C42 Markov programming and Markov renewal programming 90C45 Continuous programming 90C48 Programming in abstract spaces 90C60 Abstract computational complexity for mathematical programming problems, See also {68Q25} 90C70 Fuzzy programming 90C90 Applications of mathematical programming 90C99 None of the above but in this section 90Dxx Game theory 90D05 $2$-person games 90D06 $n$-person games, $n>2$ 90D10 Noncooperative games 90D12 Cooperative games 90D13 Games with infinitely many players 90D15 Stochastic games, See also {93E05} 90D20 Multistage and repeated games 90D25 Differential games, See also {49N55} 90D26 Pursuit and evasion games 90D35 Decision theory for games, See also {62Cxx, 90A05, 90B50} 90D40 Game-theoretic models, See also {65Cxx} 90D42 Positional games 90D43 Games involving graphs 90D44 Games involving topology or set theory 90D46 Combinatorial games 90D50 Discrete-time games 90D55 Games of timing 90D60 Probabilistic games; gambling 90D65 Hierarchical games 90D70 Spaces of games 90D80 Applications of game theory 90D99 None of the above but in this section \MajorSub 92-XX Biology and other natural sciences, behavioral sciences 92-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 92-01 Instructional exposition (textbooks, tutorial papers, etc.) 92-02 Research exposition (monographs, survey articles) 92-03 Historical (must be assigned at least one classification number from Section 01) 92-04 Explicit machine computation and programs (not the theory of computation or programming) 92-06 Proceedings, conferences, collections, etc. 92-08 Computational methods 92Bxx Mathematical biology in general 92B05 General biology and biomathematics 92B10 Taxonomy, statistics 92B15 General biostatistics, See also {62P10} 92B20 Neural networks, See also {68T05, 82C32, 94Cxx} 92B99 None of the above but in this section 92Cxx Physiological, cellular and medical topics 92C05 Biophysics 92C10 Biomechanics, See also {73P05} 92C15 Developmental biology, pattern formation 92C20 Neural biology 92C30 Physiology (general) 92C35 Physiological flow, See also {76Z05} 92C40 Biochemistry, molecular biology 92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.), See also {80A30} 92C50 Medical applications (general) 92C55 Tomography, See also {44A12, 65R10} 92C60 Medical epidemiology 92C99 None of the above but in this section 92Dxx Genetics and population dynamics 92D10 Genetics, {For genetic algebras, See 17D92} 92D15 Problems related to evolution 92D20 Protein sequences, DNA sequences 92D25 Population dynamics (general) 92D30 Epidemiology 92D40 Ecology 92D50 Animal behavior 92D99 None of the above but in this section 92Exx Chemistry, {For biochemistry, See 92C40} 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.) 92E20 Classical flows, reactions, etc., See also {80A30, 80A32} 92E99 None of the above but in this section 92F05 Other natural sciences 92Gxx Social and behavioral sciences: methodology, {For statistics, See 62-XX} 92G05 Measurement theory 92G15 One- and multidimensional scaling 92G20 Test theory 92G25 Questionnaire analysis 92G30 Clustering, See also {62H30} 92G40 Q-analysis 92G99 None of the above but in this section 92Hxx Mathematical sociology (including anthropology) 92H10 Models of societies, social and urban evolution 92H20 Mathematical geography and demography 92H25 Spatial models, See also {90A25} 92H30 Social networks 92H35 Manpower systems, See also {90A50, 90B70} 92H99 None of the above but in this section 92Jxx Mathematical psychology 92J10 Cognitive psychology 92J30 Psychophysics and psychophysiology; perception 92J40 Memory and learning, See also {68T05} 92J45 Measurement and performance 92J99 None of the above but in this section 92Kxx Other social and behavioral sciences (mathematical treatment) 92K10 History, political science 92K20 Linguistics, See also {03B65, 68S05} 92K99 None of the above but in this section \MajorSub 93-XX Systems theory; control, {For optimal control, See 49-XX} 93-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 93-01 Instructional exposition (textbooks, tutorial papers, etc.) 93-02 Research exposition (monographs, survey articles) 93-03 Historical (must be assigned at least one classification number from Section 01) 93-04 Explicit machine computation and programs (not the theory of computation or programming) 93-06 Proceedings, conferences, collections, etc. 93Axx General 93A05 Axiomatic system theory 93A10 General systems 93A13 Hierarchical systems 93A14 Decentralized systems 93A15 Large scale systems 93A20 Cascaded systems 93A25 Input-output systems 93A30 Mathematical modeling (models of systems, model-matching, etc.) 93A99 None of the above but in this section 93Bxx Controllability, observability, and system structure 93B03 Attainable sets 93B05 Controllability 93B06 Relations between controllability and optimal solutions 93B07 Observability 93B10 Canonical structure 93B11 System structure simplification 93B12 Variable structure systems 93B15 Realizability of systems from input-output data 93B17 Transformation 93B18 Linearizability 93B20 Minimal systems representations 93B25 Algebraic methods 93B27 Geometric methods (including algebro-geometric) 93B28 Operator-theoretic methods, See also {47A57, 47N70} 93B29 Differential-geometric methods 93B30 System identification 93B35 Sensitivity (robustness) 93B36 ${H}^ \infty$-control 93B40 Computational methods 93B50 Synthesis problems 93B51 Design techniques (robust design, computer-aided design, etc.) 93B52 Feedback control 93B55 Pole and zero placement problems 93B60 Eigenvalue problems 93B99 None of the above but in this section 93Cxx Control systems, guided systems 93C05 Linear 93C10 Nonlinear 93C15 Systems governed by ordinary differential equations 93C20 Systems governed by partial differential equations 93C22 Systems governed by integral equations 93C25 Systems in abstract spaces 93C30 Systems governed by functional relations other than differential or integral equations 93C35 Multivariable, multidimensional systems 93C40 Adaptive control systems 93C41 Problems with incomplete information 93C42 Fuzzy control 93C45 Time-invariant 93C50 Time-dependent 93C55 Discrete-time 93C57 Sampled-data 93C60 Continuous-time 93C62 Digital systems 93C70 Time-scale analysis and related topics 93C73 Perturbations in control systems 93C80 Frequency-response methods 93C83 Control problems involving computers (process control, etc.) 93C85 Automated systems (robots, etc.) 93C90 Random disturbances in control systems 93C95 Applications of control theory 93C99 None of the above but in this section 93Dxx Stability 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.) 93D09 Robust stability 93D10 Popov-type stability of feedback systems 93D15 Stabilization of systems by feedback 93D20 Asymptotic stability 93D21 Adaptive and robust stabilization 93D22 Interrelations between stability problems and optimization problems 93D25 Input-output approaches 93D30 Scalar and vector Lyapunov functions 93D99 None of the above but in this section 93Exx Stochastic systems and control 93E03 Stochastic systems, general 93E05 Stochastic games, stochastic differential games, See Also { 90D15} 93E10 Estimation and detection, See also {60G35} 93E11 Filtering, See also {60G35} 93E12 System identification 93E14 Data smoothing 93E15 Stochastic stability 93E20 Optimal stochastic control 93E23 Stochastic gradient methods 93E24 Least squares and related methods 93E25 Other computational methods 93E30 Computer simulations of stochastic systems 93E35 Stochastic learning and adaptive control 93E99 None of the above but in this section \MajorSub 94-XX Information and communication, circuits 94-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 94-01 Instructional exposition (textbooks, tutorial papers, etc.) 94-02 Research exposition (monographs, survey articles) 94-03 Historical (must be assigned at least one classification number from Section 01) 94-04 Explicit machine computation and programs (not the theory of computation or programming) 94-06 Proceedings, conferences, collections, etc. 94Axx Communication, information 94A05 Communication theory, See also {60G35, 90B12} 94A11 Application of orthogonal functions in communication 94A12 Signal theory (characterization, reconstruction, etc.) 94A13 Detection theory 94A14 Modulation and demodulation 94A15 Information theory, general, See also {62B10} 94A17 Measures of information, entropy 94A24 Coding theorems (Shannon theory) 94A29 Source coding 94A34 Rate-distortion theory 94A40 Channel models 94A45 Prefix, length-variable, comma-free codes, See also {20M35, 68Q45} 94A50 Theory of questionnaires 94A55 Shift register sequences 94A60 Cryptography, See also {11T71, 68P25} 94A99 None of the above but in this section 94Bxx Theory of error-correcting codes 94B05 Linear codes, general 94B10 Convolutional codes 94B12 Combined modulation schemes (including trellis codes) 94B15 Cyclic codes 94B20 Burst-correcting codes 94B25 Combinatorial codes 94B27 Geometric methods (including applications of algebraic geometry), See also {11T71} 94B30 Majority codes 94B35 Decoding 94B40 Arithmetic codes, See also {11T71} 94B50 Synchronization error-correcting codes 94B60 Other types of codes 94B65 Bounds on codes 94B70 Error probability 94B75 Applications of the theory of convex sets and geometry of numbers (covering radius, etc.), See also {11H31} 94B99 None of the above but in this section 94Cxx Circuits, networks 94C05 Analytic circuit theory 94C10 Switching theory, application of Boolean algebra; Boolean functions, See also {06E30} 94C12 Fault detection; testing 94C15 Applications of graph theory, See also {05Cxx, 68R10} 94C30 Applications of design theory, See also {05Bxx} 94C99 None of the above but in this section 94D05 Fuzzy sets and logic {(in connection with questions of Section 94)}, See also {03B52, 03E72, 04A72, 28E10}