This Mathematics Subject Classification was compiled by the Editorial Offices of both Mathematical Reviews and Zentralblatt für Mathematik / Mathematics Abstracts in 1991.

See also the table of main areas.

**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}# 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# 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# 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# 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# 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# 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# 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## 11Z05 Miscellaneous applications of number theory

# 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

# 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# 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# 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# 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# 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# 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# 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

# 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

# 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# 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# 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# 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}

# 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

# 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# 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# 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# 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# 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# 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}

# 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# 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# 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# 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# 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}

# 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# 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# 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

# 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)

# 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# 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# 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}

# 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# 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# 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

# 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# 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

# 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# 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# 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}

# 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# 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# 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# 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# 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# 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# 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

# 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# 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# 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# 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# 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# 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}