数学主题分类 - 英文（下）
* 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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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}
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 01-XX)
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 01-XX)
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 01-XX)
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, 65Y10}
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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 01-XX)
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}