MSC2010
MSC2010
This document is a printed form of MSC2010, an MSC revision produced
jointly by the editorial staffs of Mathematical Reviews (MR) and Zentralblatt fu¨r
Mathematik (Zbl) in consultation with the mathematical community. The goals of
this revision of the Mathematics Subject Classification (MSC) were set out in the
announcement of it and call for comments by the Executive Editor of MR and the
Chief Editor of Zbl in August 2006. This document results from the MSC revision
process that has been going on since then. MSC2010 will be fully deployed from
July 2010.
The editors of MR and Zbl deploying this revision therefore ask for feedback on
remaining errors to help in this work, which should be given, preferably, on the Web
site at http://msc2010.org or, if the internet is not available, through e-mail to
feedback@msc2010.org. They are grateful for the many suggestions that were
received previously which have much influenced what we have.
How to use the
Mathematics Subject Classification [MSC]
The main purpose of the classification of items in the mathematical literature
using the Mathematics Subject Classification scheme is to help users find the
items of present or potential interest to them as readily as possible—in products
derived from the Mathematical Reviews Database (MRDB) such as MathSciNet, in
Zentralblatt MATH (ZMATH), or anywhere else where this classification scheme is
used. An item in the mathematical literature should be classified so as to attract the
attention of all those possibly interested in it. The item may be something which
falls squarely within one clear area of the MSC, or it may involve several areas.
Ideally, the MSC codes attached to an item should represent the subjects to which
the item contains a contribution. The classification should serve both those closely
concerned with specific subject areas, and those familiar enough with subjects to
apply their results and methods elsewhere, inside or outside of mathematics. It will
be extremely useful
for both users and classifiers to familiarize themselves with the entire classification
system and thus to become aware of all the classifications of possible interest to
them.
Every item in the MRDB or ZMATH receives precisely one primary
classification, which is simply the MSC code that describes its principal
contribution. When an item contains several principal contributions to different
areas, the primary classification should cover the most important among them. A
paper or book may be assigned one or several secondary classification numbers to
cover any remaining principal contributions, ancillary results, motivation or origin of
the matters discussed, intended or potential field of application, or other significant
aspects worthy of notice.
The principal contribution is meant to be the one including the most important
part of the work actually done in the item. For example, a paper whose main overall
content is the solution of a problem in graph theory, which arose in computer
science and whose solution is (perhaps) at present only of interest to computer
scientists, would have a primary classification in 05C (Graph Theory) with one
or more secondary classifications in 68 (Computer Science); conversely, a paper
whose overall content lies mainly in computer science should receive a primary
classification in 68, even if it makes heavy use of graph theory and proves several
new graph-theoretic results along the way.
There are two types of cross-references given at the end of many of the entries
in the MSC. The first type is in braces: “{For A, see X}”; if this appears in section
Y, it means that contributions described by A should usually be assigned the
classification code X, not Y. The other type of cross-reference merely points out
related classifications; it is in brackets: “[See also . . . ]”, “[See mainly . . . ]”, etc.,
and the classification codes listed in the brackets may, but need not, be included in
the classification codes of a paper, or they may be used in place of the classification
where the cross-reference is given. The classifier must judge which classification is
the most appropriate for the paper at hand.
00-XX GENERAL
00-01 Instructional exposition (textbooks, tutorial papers, etc.)
00-02 Research exposition (monographs, survey articles)
00Axx General and miscellaneous specific topics
00A05 General mathematics
00A06 Mathematics for nonmathematicians (engineering, social sciences,
etc.)
00A07 Problem books
00A08 Recreational mathematics [See also 97A20]
00A09 Popularization of mathematics
00A15 Bibliographies
00A17 External book reviews
00A20 Dictionaries and other general reference works
00A22 Formularies
00A30 Philosophy of mathematics [See also 03A05]
00A35 Methodology of mathematics, didactics [See also 97Cxx, 97Dxx]
00A65 Mathematics and music
00A66 Mathematics and visual arts, visualization
00A67 Mathematics and architecture
00A69 General applied mathematics {For physics, see 00A79 and Sections
70 through 86}
00A71 Theory of mathematical modeling
00A72 General methods of simulation
00A73 Dimensional analysis
00A79 Physics (use more specific entries from Sections 70 through 86 when
possible)
00A99 Miscellaneous topics
00Bxx Conference proceedings and collections of papers
00B05 Collections of abstracts of lectures
00B10 Collections of articles of general interest
00B15 Collections of articles of miscellaneous specific content
00B20 Proceedings of conferences of general interest
00B25 Proceedings of conferences of miscellaneous specific interest
00B30 Festschriften
00B50 Volumes of selected translations
00B55 Miscellaneous volumes of translations
00B60 Collections of reprinted articles [See also 01A75]
00B99 None of the above, but in this section
01-XX HISTORY AND BIOGRAPHY [See also the classification
number–03 in the other sections]
01-00 General reference works (handbooks, dictionaries, bibliographies,
etc.)
01-01 Instructional exposition (textbooks, tutorial papers, etc.)
01-02 Research exposition (monographs, survey articles)
01-06 Proceedings, conferences, collections, etc.
01-08 Computational methods
01Axx History of mathematics and mathematicians
01A05 General histories, source books
01A07 Ethnomathematics, general
01A10 Paleolithic, Neolithic
01A12 Indigenous cultures of the Americas
01A13 Other indigenous cultures (non-European)
01A15 Indigenous European cultures (pre-Greek, etc.)
01A16 Egyptian
01A17 Babylonian
01A20 Greek, Roman
01A25 China
01A27 Japan
01A29 Southeast Asia
01A30 Islam (Medieval)
01A32 India
01A35 Medieval
01A40 15th and 16th centuries, Renaissance
01A45 17th century
01A50 18th century
01A55 19th century
01A60 20th century
01A61 Twenty-first century
01A65 Contemporary
01A67 Future prospectives
01A70 Biographies, obituaries, personalia, bibliographies
01A72 Schools of mathematics
01A73 Universities
01A74 Other institutions and academies
01A75 Collected or selected works; reprintings or translations of classics
[See also 00B60]
01A80 Sociology (and profession) of mathematics
01A85 Historiography
01A90 Bibliographic studies
01A99 Miscellaneous topics
03-XX MATHEMATICAL LOGIC AND FOUNDATIONS
03-00 General reference works (handbooks, dictionaries, bibliographies,
etc.)
03-01 Instructional exposition (textbooks, tutorial papers, etc.)
03-02 Research exposition (monographs, survey articles)
03-03 Historical (must also be assigned at least one classification number
from Section 01)
[MSC Source Date: Monday 21 December 2009 09:49]
[Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.]
MSC201003-XX S2
03-04 Explicit machine computation and programs (not the theory of
computation or programming)
03-06 Proceedings, conferences, collections, etc.
03Axx Philosophical aspects of logic and foundations
03A05 Philosophical and critical {For philosophy of mathematics, see also
00A30}
03A10 Logic in the philosophy of science
03A99 None of the above, but in this section
03Bxx General logic
03B05 Classical propositional logic
03B10 Classical first-order logic
03B15 Higher-order logic and type theory
03B20 Subsystems of classical logic (including intuitionistic logic)
03B22 Abstract deductive systems
03B25 Decidability of theories and sets of sentences [See also 11U05, 12L05,
20F10]
03B30 Foundations of classical theories (including reverse mathematics)
[See also 03F35]
03B35 Mechanization of proofs and logical operations [See also 68T15]
03B40 Combinatory logic and lambda-calculus [See also 68N18]
03B42 Logics of knowledge and belief (including belief change)
03B44 Temporal logic
03B45 Modal logic (including the logic of norms) {For knowledge and belief,
see 03B42; for temporal logic, see 03B44; for provability logic, see
also 03F45}
03B47 Substructural logics (including relevance, entailment, linear logic,
Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects
see 03F52}
03B48 Probability and inductive logic [See also 60A05]
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05]
03B53 Paraconsistent logics
03B55 Intermediate logics
03B60 Other nonclassical logic
03B62 Combined logics
03B65 Logic of natural languages [See also 68T50, 91F20]
03B70 Logic in computer science [See also 68–XX]
03B80 Other applications of logic
03B99 None of the above, but in this section
03Cxx Model theory
03C05 Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05]
03C07 Basic properties of first-order languages and structures
03C10 Quantifier elimination, model completeness and related topics
03C13 Finite structures [See also 68Q15, 68Q19]
03C15 Denumerable structures
03C20 Ultraproducts and related constructions
03C25 Model-theoretic forcing
03C30 Other model constructions
03C35 Categoricity and completeness of theories
03C40 Interpolation, preservation, definability
03C45 Classification theory, stability and related concepts [See also 03C48]
03C48 Abstract elementary classes and related topics [See also 03C45]
03C50 Models with special properties (saturated, rigid, etc.)
03C52 Properties of classes of models
03C55 Set-theoretic model theory
03C57 Effective and recursion-theoretic model theory [See also 03D45]
03C60 Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]
03C62 Models of arithmetic and set theory [See also 03Hxx]
03C64 Model theory of ordered structures; o-minimality
03C65 Models of other mathematical theories
03C68 Other classical first-order model theory
03C70 Logic on admissible sets
03C75 Other infinitary logic
03C80 Logic with extra quantifiers and operators [See also 03B42, 03B44,
03B45, 03B48]
03C85 Second- and higher-order model theory
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03C95 Abstract model theory
03C98 Applications of model theory [See also 03C60]
03C99 None of the above, but in this section
03Dxx Computability and recursion theory
03D03 Thue and Post systems, etc.
03D05 Automata and formal grammars in connection with logical questions
[See also 68Q45, 68Q70, 68R15]
03D10 Turing machines and related notions [See also 68Q05]
03D15 Complexity of computation (including implicit computational
complexity) [See also 68Q15, 68Q17]
03D20 Recursive functions and relations, subrecursive hierarchies
03D25 Recursively (computably) enumerable sets and degrees
03D28 Other Turing degree structures
03D30 Other degrees and reducibilities
03D32 Algorithmic randomness and dimension [See also 68Q30]
03D35 Undecidability and degrees of sets of sentences
03D40 Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15]
03D45 Theory of numerations, effectively presented structures
[See also 03C57; for intuitionistic and similar approaches see 03F55]
03D50 Recursive equivalence types of sets and structures, isols
03D55 Hierarchies
03D60 Computability and recursion theory on ordinals, admissible sets, etc.
03D65 Higher-type and set recursion theory
03D70 Inductive definability
03D75 Abstract and axiomatic computability and recursion theory
03D78 Computation over the reals {For constructive aspects, see 03F60}
03D80 Applications of computability and recursion theory
03D99 None of the above, but in this section
03Exx Set theory
03E02 Partition relations
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
03E15 Descriptive set theory [See also 28A05, 54H05]
03E17 Cardinal characteristics of the continuum
03E20 Other classical set theory (including functions, relations, and set
algebra)
03E25 Axiom of choice and related propositions
03E30 Axiomatics of classical set theory and its fragments
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
03E45 Inner models, including constructibility, ordinal definability, and core
models
03E47 Other notions of set-theoretic definability
03E50 Continuum hypothesis and Martin’s axiom [See also 03E57]
03E55 Large cardinals
03E57 Generic absoluteness and forcing axioms [See also 03E50]
03E60 Determinacy principles
03E65 Other hypotheses and axioms
03E70 Nonclassical and second-order set theories
03E72 Fuzzy set theory
03E75 Applications of set theory
03E99 None of the above, but in this section
03Fxx Proof theory and constructive mathematics
03F03 Proof theory, general
03F05 Cut-elimination and normal-form theorems
03F07 Structure of proofs
03F10 Functionals in proof theory
03F15 Recursive ordinals and ordinal notations
03F20 Complexity of proofs
03F25 Relative consistency and interpretations
03F30 First-order arithmetic and fragments
03F35 Second- and higher-order arithmetic and fragments [See also 03B30]
03F40 Go¨del numberings and issues of incompleteness
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
[See also 03B45, 03G25, 06E25]
03F50 Metamathematics of constructive systems
03F52 Linear logic and other substructural logics [See also 03B47]
03F55 Intuitionistic mathematics
03F60 Constructive and recursive analysis [See also 03B30, 03D45, 03D78,
26E40, 46S30, 47S30]
03F65 Other constructive mathematics [See also 03D45]
03F99 None of the above, but in this section
03Gxx Algebraic logic
03G05 Boolean algebras [See also 06Exx]
03G10 Lattices and related structures [See also 06Bxx]
03G12 Quantum logic [See also 06C15, 81P10]
03G15 Cylindric and polyadic algebras; relation algebras
03G20 Lukasiewicz and Post algebras [See also 06D25, 06D30]
03G25 Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35]
03G27 Abstract algebraic logic
03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10]
03G99 None of the above, but in this section
03Hxx Nonstandard models [See also 03C62]
03H05 Nonstandard models in mathematics [See also 26E35, 28E05, 30G06,
46S20, 47S20, 54J05]
03H10 Other applications of nonstandard models (economics, physics, etc.)
03H15 Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
03H99 None of the above, but in this section
[MSC Source Date: Monday 21 December 2009 09:49]
[Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.]
MSC2010S3 06Exx
05-XX COMBINATORICS {For finite fields, see 11Txx}
05-00 General reference works (handbooks, dictionaries, bibliographies,
etc.)
05-01 Instructional exposition (textbooks, tutorial papers, etc.)
05-02 Research exposition (monographs, survey articles)
05-03 Historical (must also be assigned at least one classification number
from Section 01)
05-04 Explicit machine computation and programs (not the theory of
computation or programming)
05-06 Proceedings, conferences, collections, etc.
05Axx Enumerative combinatorics {For enumeration in graph theory, see
05C30}
05A05 Permutations, words, matrices
05A10 Factorials, binomial coefficients, combinatorial functions
[See also 11B65, 33Cxx]
05A15 Exact enumeration problems, generating functions [See also 33Cxx,
33Dxx]
05A16 Asymptotic enumeration
05A17 Partitions of integers [See also 11P81, 11P82, 11P83]
05A18 Partitions of sets
05A19 Combinatorial identities, bijective combinatorics
05A20 Combinatorial inequalities
05A30 q-calculus and related topics [See also 33Dxx]
05A40 Umbral calculus
05A99 None of the above, but in this section
05Bxx Designs and configurations {For applications of design theory, see
94C30}
05B05 Block designs [See also 51E05, 62K10]
05B07 Triple systems
05B10 Difference sets (number-theoretic, group-theoretic, etc.)
[See also 11B13]
05B15 Orthogonal arrays, Latin squares, Room squares
05B20 Matrices (incidence, Hadamard, etc.)
05B25 Finite geometries [See also 51D20, 51Exx]
05B30 Other designs, configurations [See also 51E30]
05B35 Matroids, geometric lattices [See also 52B40, 90C27]
05B40 Packing and covering [See also 11H31, 52C15, 52C17]
05B45 Tessellation and tiling problems [See also 52C20, 52C22]
05B50 Polyominoes
05B99 None of the above, but in this section
05Cxx Graph theory {For applications of graphs, see 68R10, 81Q30, 81T15,
82B20, 82C20, 90C35, 92E10, 94C15}
05C05 Trees
05C07 Vertex degrees [See also 05E30]
05C10 Planar graphs; geometric and topological aspects of graph theory
[See also 57M15, 57M25]
05C12 Distance in graphs
05C15 Coloring of graphs and hypergraphs
05C17 Perfect graphs
05C20 Directed graphs (digraphs), tournaments
05C21 Flows in graphs
05C22 Signed and weighted graphs
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
[See also 20F65]
05C30 Enumeration in graph theory
05C31 Graph polynomials
05C35 Extremal problems [See also 90C35]
05C38 Paths and cycles [See also 90B10]
05C40 Connectivity
05C42 Density (toughness, etc.)
05C45 Eulerian and Hamiltonian graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C51 Graph designs and isomomorphic decomposition [See also 05B30]
05C55 Generalized Ramsey theory [See also 05D10]
05C57 Games on graphs [See also 91A43, 91A46]
05C60 Isomorphism problems (reconstruction conjecture, etc.) and
homomorphisms (subgraph embedding, etc.)
05C62 Graph representations (geometric and intersection representations,
etc.) For graph drawing, see also 68R10
05C63 Infinite graphs
05C65 Hypergraphs
05C69 Dominating sets, independent sets, cliques
05C70 Factorization, matching, partitioning, covering and packing
05C72 Fractional graph theory, fuzzy graph theory
05C75 Structural characterization of families of graphs
05C76 Graph operations (line graphs, products, etc.)
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C80 Random graphs [See also 60B20]
05C81 Random walks on graphs
05C82 Small world graphs, complex networks [See also 90Bxx, 91D30]
05C83 Graph minors
05C85 Graph algorithms [See also 68R10, 68W05]
05C90 Applications [See also 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35,
92E10, 94C15]
05C99 None of the above, but in this section
05Dxx Extremal combinatorics
05D05 Extremal set theory
05D10 Ramsey theory [See also 05C55]
05D15 Transversal (matching) theory
05D40 Probabilistic methods
05D99 None of the above, but in this section
05Exx Algebraic combinatorics
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory [See also 20C30]
05E15 Combinatorial aspects of groups and algebras [See also 14Nxx,
22E45, 33C80]
05E18 Group actions on combinatorial structures
05E30 Association schemes, strongly regular graphs
05E40 Combinatorial aspects of commutative algebra
05E45 Combinatorial aspects of simplicial complexes
05E99 None of the above, but in this section
06-XX ORDER, LATTICES, ORDERED ALGEBRAIC STRUCTURES
[See also 18B35]
06-00 General reference works (handbooks, dictionaries, bibliographies,
etc.)
06-01 Instructional exposition (textbooks, tutorial papers, etc.)
06-02 Research exposition (monographs, survey articles)
06-03 Historical (must also be assig