**5** Mathematics and natural sciences

**5(091)** History of science

*References: 5*

**5-051** Practitioners of science. Scientists

*References: 5*

**502/504** Environmental science. Conservation of natural resources. Threats to the environment and protection against them

**502** The environment and its protection

**502.1** The environment and society. Conservation and protection in general

**502.11** Interaction, interdependence of environment and society. Mutual benefits, etc.

**502.12** Environmental awareness. 'Green' outlook. Green issues. 'Greenness'

**502.13** Conservation measures and management

**502.131** Development

**502.131.1** Sustainable development

*References: 332.146.2*

**502.132** Improvement

**502.14** Social, administrative, legislative measures on environmental conservation

**502.14:061** Organizations to promote conservation

*References: 502.14*

**502.14:329.63** 'Green' parties

*References: 502.14*

**502.15** Environment in relation to planning and development

*References: 71*

**502.17** Protection of the environment in general

*References: 620.9*

**502.171** Protection, rational use and renewal of natural resources

**502.171:531.62** Energy conservation

*References: 502.171*

**502.171:546.212** Water conservation

*References: 502.171*

**502.171:662.997** Conservation of geothermal energy

*References: 502.171*

**502.172** Protection of endangered living elements

**502.172:502.211** Protection of endangered species. 'Red data'

*References: 502.172*

**502.173** Control of usage, avoidance of waste, rationing of resources

**502.174** Restoration, salvage, reclamation, rescue measures. Wasteless and low-waste technology

*References: 620.97*

**502.174:627.533** Land reclamation

*References: 502.174*

**502.174.1** Recycling

*References: 628.39*

**502.174.2** Use of biodegradable materials

**502.174.3** Use of renewable energy resources

**502.174.3:627.8.09** Use of hydroelectric power

*References: 502.174.3*

**502.175** Control of environmental quality. Control of pollution

*References: 504.5 , 628.19 , 628.39 , 628.5*

**502.2** The environment as a whole

**502.21** Natural resources and energy

**502.21:523.9** Solar energy

*References: 502.21*

**502.21:621.039** Nuclear energy

*References: 502.21*

**502.211** The living world. The biosphere

**502.211:061** Organizations. Wildlife protection associations (e.g. World Wildlife Fund, WWF)

*References: 502.211*

**502.211:579** Protection of microbial kingdom

*References: 502.211*

**502.211:582** Protection of vegetation, plant kingdom

*References: 502.211*

**502.211:592/599(204** Protection of aquatic wildlife

*References: 502.211*

**502.22** The man-made environment

**502.3** Atmospheric environment

**502.3:504.5:621.43.064** Atmospheric pollution by exhaust fumes from traffic

*References: 502.3*

**502.3:504.5:662.6/.7** Atmospheric pollution by fossil fuels

*References: 502.3*

**502.3:504.7** Global warming of atmosphere

*References: 502.3*

**502.3:551.510.534** Ozone layer

*References: 502.3*

**502.3:551.510.534:504.12** Ozone layer depletion

*References: 502.3*

**502.3:551.515** Atmospheric layers

*References: 502.3*

**502.3:613.15** Air quality. Including: Clean air

*References: 502.3*

**502.3/.7** Parts of the environment

**502.5** Earth's surface. Landscape. Scenery

**502.51** Hydrospheric environment

**502.51:504.5** Hydrospheric pollution. Water pollution

*References: 502.51*

**502.51(26)** Ocean and sea environment. Including: Sea floor. Sea water

*References: 502.51*

**502.51(26):504.5:665.6** Oil spillage at sea

*References: 502.51*

**502.51(28)** Freshwater environment. Inland surface

*References: 502.51*

**502.51(282)** Fluvial environment. Including: Rivers. Streams

*References: 502.51*

**502.51(282.02)** Groundwater environment. Including: Springs. Aquifers

*References: 502.51*

**502.51(282.05)** Fluvio-marine environment. Including: River mouths. Estuaries

*References: 502.51*

**502.51(285)** Lakes. Reservoirs. Ponds

*References: 502.51*

**502.51(285.3)** Swamps. Marshes

*References: 502.51*

**502.52** Lithospheric environment

**502.52:551.21:502.13** Conservation of volcanic features

*References: 502.52*

**502.52:551.312.2:502.13** Conservation of marshes, moors

*References: 502.52*

**502.52:551.435.11:502.13** Conservation of gorges, canyons

*References: 502.52*

**502.52(210):502.13** Conservation of land formations. Coasts, etc.

*References: 502.52*

**502.52(23):502.13** Conservation of mountains, hills, slopes

*References: 502.52*

**502.521** Soil

*References: 631.4*

**502.6** Glacial environment

*References: 551.32*

**502.7** Earth's interior

**502.7:662.765.4** Conservation of oil deposits

*References: 502.7*

**504** Threats to the environment

**504.1** Direct damage. Depredation. Threat of depredation

**504.11** Mass depredation, despoliation. Destruction

**504.12** Depletion or exhaustion by over-exploitation or misuse

**504.12:502.7** Depletion of deposits

*References: 504.12*

**504.12:502.7:662.765.4** Depletion of oil deposits

*References: 504.12*

**504.12:502.7:662.767** Depletion of natural gas reserves

*References: 504.12*

**504.12:639.2.058** Overfishing

*References: 504.12*

**504.12:662.66** Depletion of coal deposits

*References: 504.12*

**504.121** Erosion

**504.121:551.311.21** Erosion by flowing water

*References: 504.121*

**504.122** Deforestation

**504.123** Desertification

**504.2** Accidental damage by a natural or unknown agent

**504.2:614.84** Damage by fire (especially open-air fire)

*References: 504.2*

**504.3** Damage by a known natural or biological agent

**504.3:632.5** Plant depredation. Including: Weeds. Epiphytes

*References: 504.3*

**504.3:632.6/.7** Animal depredation. Including: Vermin attack. Stampede

*References: 504.3*

**504.35** Genetic contamination

*References: 575.222.7*

**504.35:599.742.73** Genetic contamination of Felidae spp., e.g. of wild cat Felis silvestris by hybridization with feral domestic cat Felis catus

*References: 504.35*

**504.4** Damage from natural causes. Natural disasters. Natural hazards

**504.4:551.21** Volcanic damage

*References: 504.4*

**504.4:551.515.3** Hurricane damage

*References: 504.4*

**504.4:556.166** Flood damage

*References: 504.4*

**504.4:556.167** Drought damage

*References: 504.4*

**504.5** Damage from harmful materials. Pollution

**504.5:551.577.13** Pollution by acid rain

*References: 504.5*

**504.5:628.3** Pollution by sewage

*References: 504.5*

**504.5:628.4.047** Pollution by radioactive waste

*References: 504.5*

**504.5:661.16** Pollution by pesticides

*References: 504.5*

**504.6** Damage from physical factors

**504.6:534.322.3** Damage caused by noise

*References: 504.6*

**504.61** Damage by man to the environment

**504.61:355.01** Damage caused by war

*References: 504.61*

**504.61:62** Damage caused by engineering works

*References: 504.61*

**504.61:69** Damage caused by building works

*References: 504.61*

**504.7** Global warming. 'Greenhouse effect'

**504.7:582.26/.27** Algal blooming

*References: 504.7*

**504.8** Nuclear winter

**504.9** Wilful, wanton human damage to the environment. Vandalism

**51** Mathematics

**51-3** Computation techniques

**51-32** Calculation without aids

**51-33** Use of tables

*References: 519.66*

**51-35** Use of mechanical, graphical and other devices

*References: 519.67*

**51-37** Use of calculators, computers in mathematics

*References: 004*

**51-7** Mathematical studies and methods in other fields of knowledge

*References: 519.862*

**51-8** Mathematical games and recreations

**510** Fundamental and general considerations of mathematics

**510.2** General problems of mathematical logic. Foundations of mathematics

**510.2:1** Philosophical problems of mathematical logic

*References: 510.2*

**510.21** Foundations of mathematics

**510.215** Category-theoretic foundations of mathematics

**510.23** Formalism (mathematical aspects)

**510.24** Intuitionistic mathematics

**510.25** Constructive approach. Constructive mathematics

**510.252** Languages of constructive mathematics

**510.254** Constructive mathematical theories

**510.26** Logical positivism

*References: 165.731*

**510.27** Logical semantics; theory of truth

*References: 519.767*

**510.275** Logical and semantic antinomies

**510.28** Applications of mathematical logic

**510.28:510.28** Applications of mathematical logic in computer science

*References: 510.28*

**510.287** Logical analysis of natural languages

**510.29** Other problems of mathematical logic and foundations

**510.3** Set theory

**510.31** Naive set theory

**510.315** Set algebra; functions and relations

**510.32** Axiomatic set theory

**510.322** Basic axiomatic systems of set theory

**510.322.2** Zermelo-Fraenkel set theory and related systems

**510.322.4** Type theory and related systems

**510.322.5** Quine's set theory NF and related systems

**510.325** Set-theoretic axioms and principles

**510.325.2** Axiom of choice, axiom of determinacy, and related principles

**510.325.4** Continuum hypothesis, Martin's axiom, and related principles

**510.327** Consistency and independence in set theory

**510.329** Other axiomatic systems of set theory

**510.33** Set-theoretic constructions

**510.332** Forcing

**510.334** Boolean-valued models and related models

**510.336** Inner models of set theory

**510.34** Combinatorial set theory

**510.344** Partition calculus

**510.346** Filters, ultrafilters, pcf-theory

**510.35** Descriptive set theory

**510.36** Theory of order types, ordinal and cardinal numbers

**510.362** Theory of ordered sets

**510.363** Ordinal numbers

**510.365** Cardinal numbers

**510.367** Cardinal characteristics of the continuum

**510.368** Large cardinals

**510.37** Nonstandard set theory

**510.38** Nonclassical set theories

**510.383** Intuitionistic set theory

**510.384** Intuitionistic type theory

**510.39** Other problems of set theory

**510.393** Category-theoretic methods in set theory

**510.398** Applications of set theory

**510.4** Formalizations of traditional mathematical theories

*References: 511.526*

**510.42** Formal arithmetic

**510.422** Fragments and subsystems of arithmetic

**510.424** Second- and higher-order arithmetic

**510.426** Models of arithmetic

**510.427** Arithmetical definability

**510.429** Other problems in formalization of arithmetic

**510.43** Nonclassical systems of arithmetic

**510.435** Intuitionistic arithmetic

**510.437** Epistemic arithmetic

**510.46** Axiomatization of analysis

**510.47** Nonstandard and nonclassical analysis

**510.472** Nonstandard analysis

**510.475** Intuitionistic analysis

**510.477** Constructive and recursive analysis

**510.49** Other logic-mathematical theories

**510.495** Reverse mathematics

**510.5** Computability theory

*References: 519.712*

**510.51** Computability theory: general aspects

**510.52** Complexity theory

**510.522** Complexity of computation

**510.524** Algorithmic information theory. Kolmogorov complexity

**510.53** Algorithmic problems. Computable set theory

*References: 510.54 , 512.54.05*

**510.532** Hierarchies of algorithmic problems and classes of sets

**510.535** Algorithmic reducibilities and degrees

**510.54** Theory of numerations

*References: 510.53*

**510.55** Generalized recursion theory

**510.552** Continuous and recursive functionals

**510.554** Inductive definability

**510.556** Computability on abstract structures: ordinals, admissible sets, etc.

**510.557** Ordinal notations; recursive ordinals

**510.56** General theory of calculi

**510.565** Term rewriting systems

**510.567** Formal grammars

**510.57** Computable, recursive functions

**510.575** Subrecursive hierarchies: classes of recursive functions

**510.58** Various models of computation

**510.582** Turing machines and related notions

**510.584** Lambda-calculus and combinatory logic

**510.586** Automata and their logical applications

*References: 519.713*

**510.589** Mathematical models of computational processes

**510.59** Other problems in the theory of computation

**510.598** Applications of theory of computation

**510.6** Mathematical logic

*References: 16*

**510.62** Languages of mathematical logic. Formal languages

*References: 519.76*

**510.623** Theory of logical consequence

**510.625** Logical calculi

**510.627** Formalizations of traditional logics

**510.627.3** Syllogistics

**510.628** Applications of logical calculi

**510.63** Classical logic

**510.633** Classical propositional logic

*References: 512.563*

**510.633.3** Boolean functions

*References: 512.563*

**510.634** Second-order propositional logic

**510.635** Classical first-order predicate logic

**510.635.5** Generalized quantifiers

**510.635.8** Algebraic Logic

**510.635.82** Abstract algebraic logic

**510.635.83** Algebras of relations (cylindric, polyadic, relation set algebras)

**510.635.84** Cylindric-like algebras

**510.635.85** Model theoretic properties of algebras related to logic

**510.636** Classical logical theories

**510.637** Higher-order predicate logics

**510.638** Infinitary languages and logics

**510.64** Nonclassical logics

**510.641** Implicative calculi

**510.642** Intuitionistic logic and superintuitionistic logics

**510.642.5** Subsystems of intuitionistic logic

**510.643** Modal logics

**510.643.3** Tense logics

**510.643.5** Dynamic logics

**510.643.7** Probability logics

**510.644** Many-valued logics

**510.644.4** Fuzzy logics and fuzzy sets

**510.644.7** Paraconsistent logics

**510.646** Inductive logic

*References: 162.3*

**510.647** Probabilistic logic

*References: 519.21*

**510.648** Substructural logics

**510.648.3** Relevant logics

**510.648.5** Linear logic

**510.648.6** Lambek calculus

**510.648.7** Categorical logic

**510.68** Applied logical systems

**510.683** Quantum logic

**510.685** Nonmonotonic logics

**510.687** Logics of knowledge

**510.69** Other logical systems

**510.7** Proof theory

**510.71** General problems of proof theory

**510.72** Various proof systems

**510.722** Hilbert type calculi

**510.724** Sequential calculi; cut-elimination

**510.725** Natural deduction systems; normalization

**510.727** Tableau method

**510.74** Complexity of proofs

**510.75** Proof search in logical calculi

**510.752** Resolution method

**510.755** Logic programming

**510.757** Equational reasoning

**510.76** Interpretations and relative consistency

**510.77** Applications of ordinals in proof theory. Ordinal analysis

**510.79** Other problems of proof theory

**510.8** Mathematical structures. Model theory

**510.81** General problems of model theory

**510.82** Axiomatic theories

**510.822** Completeness and categoricity of theories; model completeness

**510.825** Decidable and undecidable theories

**510.825.5** Quantifier elimination

**510.83** Models of mathematical theories

**510.84** Definability and related concepts, geometric stability theory

**510.85** Classification theory stability

**510.86** Model-theoretic constructions

**510.862** Realizing and omitting types

**510.863** Two-cardinal theorems

**510.864** Ultraproducts and related constructions

**510.865** Model-theoretic forcing

**510.87** Classes of models

**510.872** Models with special properties (saturated, etc.)

**510.874** Ordered models, o-minimality

**510.876** Denumerable models

**510.877** Finite model theory

**510.878** Recursion-theoretic model theory

**510.88** Nonclassical models

**510.89** Other problems of model theory

**510.893** Category-theoretic methods in model theory

**510.898** Applications of model theory

**511** Number theory

**511.1** Arithmetic. Elementary number theory

**511.11** Construction of number systems (natural numbers, integers, real numbers, complex numbers). Number notations

**511.12** Integers and their manipulation

**511.13** Factors. Fractions. Ratios. Proportions. Percentages

*References: 511.172*

**511.14** Roots and their extraction. Irrationals and other incommensurables (elementary treatment). Surds. Rationalization. Imaginaries. Complex numbers

*References: 511.2*

**511.17** Elementary number theory

**511.172** Multiplicative structure of the ring Z of integers (including divisibility, least common multiple, most common divider, congruences, power residue, quadratic residue)

*References: 511.13*

**511.174** Arithmetic functions (elementary theory)

**511.176** Number sequences (including Faray sequences). Special numbers and polynomials (including Bernoulli numbers and polynomials)

*References: 512.622*

**511.178** Combinatorial number theory (including elementary methods of partitions)

**511.2** Algebraic number theory

*References: 511.14 , 511.331 , 511.38 , 511.522 , 511.525 , 511.8 , 512.62*

**511.22** Local fields

*References: 512.625*

**511.221** Multiplicative group and ring of integers of local field as Galois modules

**511.222** Filtrations and ramification

*References: 512.625.23*

**511.223** Local class field theory. Reciprocity laws

*References: 512.625.3*

**511.224** Galois representations for local fields

*References: 512.547*

**511.225** Non-Archimedean analysis

**511.225.1** p-adic analytic functions

*References: 517.53*

**511.225.2** p-adic Banach algebras

*References: 517.986.2*

**511.225.3** Special p-adic functions

*References: 517.58*

**511.23** Algebraic number fields

**511.231** Isolated questions of algebraic number fields theory

**511.232** Asymptotic distribution of fields

**511.233** Discriminants of algebraic number fields

**511.234** Divisor class group. Divisor class number

**511.235** Special fields

**511.236** Class field theory

*References: 512.626.35 , 512.627 , 512.627.1*

**511.237** Non-Abelian generalizations of class fields theory

**511.238** Galois theory of algebraic number fields

*References: 512.627.4 , 512.627.46*

**511.24** Algebraic function fields of one variable over finite field

*References: 512.772*

**511.26** Special algebraic numbers (including PV-numbers)

**511.3** Analytic number theory

*References: 511.524*

**511.31** Elementary methods in analytic number theory

*References: 511.337*

**511.313** Distribution of primes (elementary methods)

**511.315** Sequences. Densities of sequences. Shnirelman method

**511.317** Estimates on arithmetic functions. Extremal values of arithmetic functions

**511.32** Estimates on trigonometric sums and character sums

**511.33** Multiplicative number theory

*References: 511.38*

**511.331** Arithmetic Dirichlet series

*References: 511.2 , 512.75 , 517.537.6*

**511.333** Distribution of primes

**511.333.2** Distribution of primes in commutative semigroups

**511.335** Dirichlet series and asymptotics of number functions

**511.337** Sieve methods. Large sieve method

**511.338** Lattice points in regions

**511.34** Additive number theory. Analytic methods

*References: 511.524*

**511.344** Partitions

**511.346** Waring problem. Additive equations and systems

**511.348** Goldbach problem. Additive equations and systems on primes

**511.35** Asymptotics and estimates of number functions

**511.352** Multiplicative and additive functions

**511.38** Automorphic forms

*References: 511.2 , 511.33 , 512.7*

**511.381** Automorphic forms of one variable. Modular forms of one variable

**511.382** Hilbert modular forms

**511.383** Siegel modular forms

**511.384** General approach to automorphic forms

*References: 512.817 , 515.178 , 517.986*

**511.386** p-adic automorphic forms

**511.4** Diophantine approximations and inequalities. Theory of transcendental numbers. Geometry of numbers

**511.41** Continued fractions. Generalizations of continued fraction algorithms

*References: 511.72*

**511.42** Diophantine approximations: general theory

*References: 511.46*

**511.43** Analytic theory of Diophantine inequalities (circular method). Fractional parts. Uniform distributions. Applications to numerical analysis

**511.44** Approximation by rational and algebraic numbers to algebraic numbers (including Thue inequality and its generalizations)

*References: 511.523*

**511.46** Theory of transcendental numbers

*References: 511.42*

**511.462** Irrationality and linear independence of numbers, estimates of their measures

**511.464** Transcendence and algebraic independence of numbers, estimates of their measures

**511.48** Geometry of numbers

**511.5** Diophantine equations. Theory of forms

*References: 512.647*

**511.52** Algebraic Diophantine equations

**511.522** Algebraic number theory methods in Diophantine equations. Norm equation

*References: 511.2*

**511.523** Diophantine approximation methods in Diophantine equations (including Thue method and its generalizations)

*References: 511.44*

**511.524** Analytic theory of Diophantine equations

*References: 511.3*

**511.525** Equations over local number fields and rings

**511.526** Mathematical logic methods in Diophantine equations

*References: 510.4*

**511.528** Special Diophantine equations

**511.53** Nonalgebraic Diophantine equations

**511.55** Theory of quadratic forms

*References: 512.646*

**511.57** Theory of higher-degree forms and systems of forms

*References: 512.647 , 512.75 , 512.772 , 512.774*

**511.7** Metric number theory. Probabilistic number theory

**511.72** Metric number theory

*References: 511.41*

**511.75** Probabilistic number theory

**511.8** Noncommutative arithmetic

*References: 511.2 , 512.55 , 512.743*

**511.82** Arithmetic of matrices

**511.84** Arithmetic of quaternions

**511.86** Arithmetic of central simple algebras

**511.88** Arithmetic of general algebras and rings

**512** Algebra

**512.1** Elementary algebra: operations, formulae

**512.12** Addition. Subtraction. Multiplication. Division. Progressions

**512.13** Identities and inequalities

**512.14** Binomial formula. Sum of powers of first n numbers etc. Powers of polynomials

**512.5** General algebra

**512.53** Semigroups

**512.531** Semigroups with finiteness conditions

**512.532** Generating sets, relations and identities of semigroups

**512.532.2** Free semigroups, defining relations

**512.532.3** Commutative semigroups

**512.532.5** Idempotent semigroups

**512.533** Equivalences and complexes in semigroups. Homomorphisms

**512.533.1** One-side regular equivalences

**512.533.2** Homomorphisms of semigroups

**512.533.4** Special elements and complexes in semigroups

**512.533.5** Ideals of semigroups

**512.533.7** Subsemigroups

**512.533.8** Lattices of subsemigroups, ideals and congruences of semigroups

**512.534** Semigroups of transformations

**512.534.1** Semigroups of multivalued transformations (binary relations)

**512.534.2** Semigroups of single-value transformations

**512.534.3** Representation of semigroups by transformations

**512.534.4** Restrictive semigroups of mappings

**512.534.5** Semigroups of transformations of sets equipped with structures

**512.534.7** Matrix semigroups. Linear semigroups

**512.535** Inverse semigroups (generalized groups)

**512.536** Semigroups with additional structures

**512.536.1** Semigroups with operators. Omega semigroups

**512.536.2** Connection with theory of rings. Multiplicative semigroups of rings

**512.536.4** Semigroups with involution

**512.536.5** Quasi-ordered and ordered semigroups

**512.536.7** Topological semigroups

**512.538** Various generalizations of associativity

**512.54** Groups. Group theory

**512.54.01** Axiomatizable classes of groups

**512.54.03** Elementary theory of various classes of groups

**512.54.05** Algorithmic problems in group theory. The word problem

**512.541** Abelian groups

**512.541.1** Purity and its generalizations

**512.541.2** High subgroups

**512.541.3** Direct and subdirect sums

**512.541.4** Extensions of Abelian groups

**512.541.5** Mappings of a group into itself and into other groups

**512.541.6** Primary Abelian groups

**512.541.7** Torsion-free Abelian groups

**512.541.8** Systems of generators. Factorization

**512.542** Finite groups

**512.542.1** Generators and defining relations

**512.542.2** Automorphisms of finite groups

**512.542.3** Finite p-groups

**512.542.4** Finite solvable groups

**512.542.5** Finite simple groups

**512.542.6** Arithmetic structure and normal structure of finite groups

**512.542.7** Permutation groups

*References: 519.115.1*

**512.543** Relations between elements of groups

**512.543.1** Systems of generators

**512.543.2** Varieties of groups

**512.543.5** Operations on groups

**512.543.7** Equations in groups. Embedding theorems in group theory

**512.544** Relations between subgroups. Generalized solvable groups and finiteness conditions

**512.544.2** Lattices of subgroups

**512.544.3** Normal series and systems

**512.544.4** Characteristic subgroups, automorphisms, endomorphisms

**512.544.5** Locally finite groups

**512.544.7** Approximations of groups

**512.545** Ordered groups

**512.546** Topological groups

**512.546.1** General theory of topological groups

**512.546.2** Abelian topological groups

**512.546.3** Locally compact groups

**512.546.4** Representations of topological groups

**512.546.6** Relations between subgroups in topological groups. Finiteness and related conditions

**512.546.8** Generalizations of topological groups

**512.547** Linear representations of abstract groups. Group characters

**512.547.2** Representations of finite groups

**512.547.4** Representations of infinite groups

**512.548** Generalizations of groups. Groupoids

**512.55** Rings and modules

**512.552** Associative rings and algebras

**512.552.1** Structures of rings

**512.552.2** Rings with chain conditions

**512.552.3** Rings with conditions on ideals or subrings

**512.552.4** Defining and identity relations in rings. Operations on rings

**512.552.5** Embeddings of rings

**512.552.6** Extensions of rings

**512.552.7** Semigroup and group rings

**512.552.8** Representations of rings

**512.553** Modules

**512.554** Nonassociative rings and algebras

**512.554.3** Lie rings. Lie algebras

*References: 512.81*

**512.554.5** Alternative rings and closely related rings

**512.554.7** Jordan rings and algebras

**512.555** Ordered rings and modules

**512.556** General theory of topological rings, modules, fields

*References: 517.986.2*

**512.558** Generalizations of rings

**512.56** Lattices, including Boolean rings and algebras

**512.562** Partially ordered sets

**512.563** Boolean rings and algebras

*References: 510.633.3*

**512.565** Types of lattice

**512.565.2** Distributive lattices

**512.565.3** Dedekind lattices and similar lattices

**512.565.6** Complemented lattices

**512.565.8** Complete lattices

**512.567** Representations of lattices

**512.568** Generalizations of lattices

**512.568.2** Semilattices

**512.568.3** Noncommutative lattices

**512.568.5** Multilattices

**512.57** Universal algebras. Free algebras

**512.58** Categories. Category theory

*References: 517.983.8*

**512.581** General considerations of category theory

**512.582** Functors

*References: 512.664*

**512.583** Representations of categories

**512.585** Abelian categories

**512.587** General theory of species of structures (according to Ehresmann)

**512.6** Special branches of algebra

**512.62** Fields. Polynomials

*References: 511.2*

**512.622** Polynomials

*References: 511.176*

**512.622.2** General topics

**512.622.22** Euclidean algorithm

**512.622.23** Interpolation polynomials

**512.622.24** Diophantine problems for polynomials

**512.622.25** Rational functions

**512.622.26** Resultants, elimination theory

**512.622.27** Irreducible polynomials

**512.622.28** Composition of polynomials

**512.622.4** Polynomials over fields and rings of a special type

**512.622.42** Polynomials over real and complex fields

**512.622.422** Localization and computation of zeros

**512.622.424** Location of zeros

**512.622.424.4** Stable polynomials

**512.622.426** Systems of non-linear algebraic equations

**512.622.428** Fewnomials

**512.622.44** Polynomials over local and number fields

**512.622.442** Irreducibility criteria

**512.622.442.4** Trinomials

**512.622.444** Factorization

**512.622.446** Integer-valued polynomials

**512.622.448** Polynomials over integers

**512.622.448.2** Cyclotomic polynomials

**512.622.46** Polynomials over finite fields

**512.622.462** Factorization

**512.622.464** Period of polynomials

**512.622.464.4** Primitive polynomials

**512.622.466** Permutation polynomials

**512.622.468** Polynomials over commutative finite rings

**512.622.6** Polynomials of a special type

**512.622.62** Reciprocal polynomials

**512.622.63** Chebyshev polynomials and generalizations

**512.622.64** Bernulli and Euler polynomials

**512.622.65** Symmetric polynomials and generalizations

**512.622.66** Schur functions and generalizations

**512.622.68** Macdonald and related polynomials

**512.622.8** Applications and generalizations of polynomials

**512.622.82** Dynamics of polynomial and rational maps

**512.622.83** Recurrent sequences

**512.622.84** Loran polynomials

**512.622.85** Quantum polynomials

**512.622.86** Power series

**512.622.88** Relations with linear algebra

**512.623** General field theory

**512.623.2** Field extensions

**512.623.22** Algebraic extensions

**512.623.221** Finite extensions

**512.623.221.2** Purely inseparable extensions

**512.623.224** Infinite algebraic extensions

**512.623.23** Transcendental extensions

**512.623.28** Finite-dimensional algebras

**512.623.282** Division algebras (finite-dimensional skew-fields)

**512.623.282.2** Quaternion algebras

**512.623.284** Brauer group of a field

**512.623.3** General Galois theory

*References: 512.625.27*

**512.623.31** Galois group

**512.623.312** Group actions on fields. Fields of invariants

**512.623.314** Computation of Galois groups

**512.623.316** Infinite Galois extensions. Profinite groups

**512.623.32** Embedding problem

**512.623.35** Constructing fields (polynomials) with a prescribed Galois group (inverse Galois theory)

**512.623.36** Galois cohomology

**512.623.362** Cohomological dimension of a field

**512.623.38** Generalizations of Galois theory

**512.623.4** Valuations of fields

**512.623.42** Archimedian and non-Archimedian valuations

**512.623.43** Discrete valuations

**512.623.44** Non-discrete valuations

**512.623.45** Multiple-normed fields

**512.623.46** Extensions of valuations

**512.623.48** Generalization of valuations

**512.623.5** Ordered fields

**512.623.52** Formally real fields

**512.623.522** Real-closed fields

**512.623.54** Structure of ordered fields

**512.623.56** Forms over fields

**512.623.562** Quadratic forms

**512.623.562.2** Sums of squares

**512.623.562.4** Witt group of a field

**512.623.564** Ci-fields

**512.623.6** Topological fields

**512.623.62** Normed fields

**512.623.64** Absolute values

**512.623.65** Completion of fields

**512.623.7** Fields of a special type

**512.623.72** Real and complex fields

**512.623.74** Power series fields and generalizations

**512.623.75** Hilbertian fields. Hilbert's irreducibility theorem

**512.623.76** Formally p-adic fields

**512.623.77** Pseudo algebraic closed fields

**512.623.78** Logical aspects of fields theory

**512.624** Finite fields

**512.624.1** Structure of finite fields

**512.624.2** Polynomials over finite fields

**512.624.22** Factorization

**512.624.24** Period of polynomials

**512.624.244** Primitive polynomials

**512.624.26** Permutation polynomials

**512.624.28** Recurrent sequences over finite fields

**512.624.3** Equations and systems of equations over a finite field

**512.624.5** Linear sequences over a finite field

**512.624.6** Exponential sums and character sums over finite fields

**512.624.7** Perfect difference sets

**512.624.8** Generalization of finite fields

**512.624.85** Galois rings

**512.624.9** Applications of finite fields

**512.624.95** Algebraic coding theory and cryptography

**512.625** Local fields (algebraic aspects)

*References: 511.22*

**512.625.1** Structure of local fields

**512.625.2** Extensions of local fields

**512.625.23** Ramification theory

*References: 511.222*

**512.625.27** Galois theory

*References: 512.623.3*

**512.625.3** Local class field theory

*References: 511.223*

**512.625.32** Local symbols. Connections with K-theory

**512.625.34** Connections with formal group theory

**512.625.36** Non-Abelian class field theory

**512.625.4** Zeta functions and L-functions

**512.625.5** Analysis over complete non-Archimedean valued fields

**512.625.52** Analytical functions over non-Archimedean fields

**512.625.53** Special functions over local fields

**512.625.55** Differential equations over local fields

**512.625.57** Non-Archimedean topological vector spaces

**512.625.58** Applications to mathematical physics, probability theory, etc.

**512.625.6** Higher-dimensional local fields

**512.625.7** Forms over local fields

**512.625.8** Algebras and orders over local fields

**512.626** Global fields

*References: 511.23*

**512.626.1** Valuations and completions

**512.626.2** Different and discriminant

**512.626.3** Number fields of a special type

**512.626.32** Quadratic fields

**512.626.33** Cubic fields

**512.626.34** Quartic fields

**512.626.35** Abelian fields

*References: 511.236*

**512.626.352** Cyclic fields

**512.626.354** Cyclotomic fields and generalizations

**512.626.354.5** Iwasawa theory

**512.626.37** Real fields

**512.626.4** Algebraic integers. Units

**512.626.42** Integral bases

**512.626.44** Units of number fields

**512.626.46** Special algebraic number

**512.626.5** Class groups and class numbers

**512.626.6** Idels and adels

**512.626.7** Forms over global fields

**512.626.72** Quadratic forms

**512.626.8** Algebras and orders over global fields

**512.626.82** Division algebras

**512.626.824** Quaternion algebras

**512.626.826** Braur group

**512.626.84** Arithmetic properties of orders

**512.626.9** Function fields (special properties)

**512.627** Class field theory, Galois theory, Zeta-functions and L-functions

*References: 511.236 , 512.623.3*

**512.627.1** Class field theory

*References: 511.236*

**512.627.14** Non-Abelian class field theory

**512.627.2** Zeta-functions and L-functions of number fields

**512.627.4** Galois theory of number fields

*References: 511.238*

**512.627.41** Galois modules

**512.627.42** Inverse Galois theory over global fields

**512.627.43** Ramified coverings of the projective line

**512.627.44** Belyi functions

**512.627.45** Absolute Galois group

**512.627.46** Galois cohomology

*References: 511.238*

**512.627.6** K-theory of global fields

**512.628** Fields with additional structures

**512.628.2** Differential algebra

**512.628.25** Differential Galois theory

**512.628.4** Difference algebra

**512.628.6** Exponential fields

**512.64** Linear algebra

**512.642** Vector spaces, linear spaces, linear vector spaces

**512.642.2** Vector spaces over skew fields

**512.643** Theory of matrices

**512.643.1** General problems of matrix theory

**512.643.12** Matrix operations

**512.643.122** Matrix multiplication

**512.643.125** Decompositions of matrices

**512.643.16** Rank of matrices

**512.643.165** Orderings of matrix sets

**512.643.2** Determinants and their generalizations

**512.643.22** Computation of determinants

**512.643.225** Non-degeneration criteria

**512.643.25** Permanents

**512.643.27** Immanents

**512.643.3** Vector and matrix norm

**512.643.32** Vector norms

**512.643.33** Numerical range

**512.643.35** Matrix norm

**512.643.355** Conditioning of matrices

**512.643.37** Approximation of matrices

**512.643.4** Matrix equations and matrix inversion

**512.643.42** Computation of inverse matrices

**512.643.43** Generalized inverse matrices

**512.643.47** Special types of matrix equations

**512.643.5** Spectral theory

**512.643.53** Eigenvalues

**512.643.532** Eigenvectors and generalized vectors

**512.643.533** Localization of eigenvalues

**512.643.534** Inverse eigenvalue problems

**512.643.535** Spectral radius

**512.643.537** Relations for eigenvalues

**512.643.538** Generalizations of eigenvalues

**512.643.55** Singular values

**512.643.555** Singular value decomposition

**512.643.7** Canonical forms, functions, inequalities of matrices

**512.643.72** Canonical forms of matrices

**512.643.721** Jordan form

**512.643.74** Matrix functions

**512.643.75** Orthogonal and unitary equivalence

**512.643.76** Matrix inequalities

**512.643.762** Inequalities between matrices

**512.643.767** Inequalities between numerical characteristics of matrices

**512.643.77** Matrix pencils

**512.643.8** Basic and special types of matrices

**512.643.81** Symmetric, skew-symmetric, Hermitian, and skew-Hermitian matrices

**512.643.82** Orthogonal, unitary, and normal matrices

**512.643.83** Simplistic matrices

**512.643.84** Matrices with positivist conditions

**512.643.843** Positive definite matrices

**512.643.845** Non-negative matrices

**512.643.845.3** Cones of matrices

**512.643.845.5** Stochastic matrices

**512.643.85** Toeplitz, Hankel, and circulate matrices

**512.643.86** Arithmetical matrices

**512.643.87** Matrix subspaces with special properties

**512.643.875** Commutative matrices

**512.643.88** Partial matrices and patterns

**512.643.89** Sparse matrices

**512.643.9** Matrices over rings

**512.643.92** Matrices over skew-fields

**512.643.93** Boolean matrices

**512.643.94** Matrices of integers

**512.643.95** Polynomial and other function matrices

**512.643.97** Matrices over finite fields and rings

**512.643.98** Quantum matrices

**512.644** Linear equations and inequalities

**512.644.3** Linear equation systems of a special kind

**512.644.5** Overdetermined systems

**512.644.7** Linear equations over commutative rings

**512.644.8** Linear inequalities

**512.645** Linear transformations

**512.645.2** Semi-linear transformations

**512.645.3** Orthogonal and unitary transformations

**512.645.4** Simplistic transformations

**512.645.5** Linear mappings of matrix spaces

**512.645.7** Linear representation of quivers and posits

**512.646** Bilinear, Hermitian, and quadratic forms

*References: 511.55*

**512.646.2** Canonical forms

**512.646.3** Inner product spaces

**512.646.33** Euclidean and pseudo-Euclidean spaces

**512.646.37** Simplistic spaces

**512.646.5** Quadratic forms over rings

**512.646.55** Integer quadratic forms

**512.646.7** Clifford algebras

**512.647** Multilinear algebra

*References: 511.5*

**512.647.3** Tensor product of vector spaces

**512.647.5** Exterior (Grassmann) algebra

**512.647.7** Special types of tensors

**512.66** Homological algebra

**512.661** Foundation of homological algebra

**512.661.1** Projective modules, injective modules, flat modules, and generalizations

**512.661.2** Envelopes and covers

**512.661.3** Chain complexes

**512.661.4** Differential graded algebras and modules

**512.661.5** Resolutions, syzygies

**512.661.6** Spectral sequences

**512.661.7** Relative homological algebra

**512.661.8** Nonabelian homological algebra

**512.663** Categorical foundation and constructions

**512.663.1** Abelian categories. Exact categories

**512.663.2** Localization of categories. Triangulated categories. Derived categories

**512.663.3** Monads. Triples. Cotriples

**512.663.4** Operads

**512.663.5** Monoidal categories

**512.663.6** Grothendieck topologies. Sites. Sheaves and presheaves

**512.663.7** Semisimplicial theory and generalizations

**512.663.8** Homotopical algebra

**512.664** Derived functors

*References: 512.582*

**512.664.5** General theory of derived functors

**512.664.6** Extension (Ext) functors

**512.664.7** Torsion (Tor) functors

**512.664.9** Derived functors of the inverse limit

**512.665** Various (co)homology theories

**512.665.1** Co)homology theory for small categories

**512.665.2** (Co)homology of groups and semigroups

**512.665.3** (Co)homology of Lie algebras, Leibniz algebras, and Hopf algebras

**512.665.32** Leibniz (co)homology

**512.665.34** (Co)homology of Hopf algebras

**512.665.4** (Co)homology of associative rings and algebras

**512.665.43** Hochschild (co)homology and generalizations

**512.665.45** Cyclic (co)homology and generalizations

**512.665.5** (Co)homology of commutative rings

**512.665.51** Harrison homology

**512.665.52** AndrÃ©-Quillen homology

**512.665.53** Hochschild (co)homology, cyclic (co)homology, and generalizations

**512.665.54** Homology of local rings

**512.665.55** Local cohomology

**512.665.56** Homological invariants of local rings (depth, grade etc.)

**512.665.57** Homological conjectures

**512.665.6** Koszul complex and generalizations

**512.665.7** De Rham complex and generalizations

**512.665.75** Non-commutative differential calculus

**512.665.8** Deformations of algebraic structures

**512.665.82** Infinitesimal deformations and related cohomology

**512.665.84** Spaces of algebraic structures

**512.665.86** Quantum deformations

**512.666** Algebraic K-theory

**512.666.1** Grothendieck groups and K0

**512.666.12** Stability for projective modules

**512.666.15** Burnside rings

**512.666.17** Represention rings

**512.666.2** Whitehead groups and K1

**512.666.22** Stable range

**512.666.24** Stability for linear groups

**512.666.26** Congruence subgroupsols

**512.666.3** Steinberg groups and K2

**512.666.32** Central extensions

**512.666.34** Symbols

**512.666.36** Brauer group

**512.666.4** Higher algebraic K-theory

**512.666.42** Q- and plus-construction

**512.666.44** Algebraic K-theory of spaces

**512.666.46** Milnor K-theory

**512.666.48** Relations with homology theory

**512.666.8** Applications of algebraic K-theory

**512.666.82** Quadratic forms

**512.666.83** Applications to algebraic number theory

**512.666.84** Applications to algebraic geometry

**512.666.85** Applications to topology

**512.666.855** Surgery obstructions

**512.666.86** Applications to operator algebras

**512.666.865** Kasporov K-theory

**512.668** Homological and related dimensions

**512.668.2** Projective dimension

**512.668.3** Flat dimension

**512.668.4** Injective dimension

**512.668.5** G-dimension and generalizations

**512.668.6** Finitely presented dimension

**512.668.7** Hilbert functions. Hilbert-Poincare series

**512.668.75** Gelfand-Kirillov dimension

**512.668.8** Global dimension

**512.668.82** Weak global dimension

**512.668.85** Finistic dimension

**512.669** Rings with special homological properties

**512.669.2** Regular rings and generalizations

**512.669.3** Complete intersections and generalizations

**512.669.4** Cohen-Macaulay rings and generalizations

**512.669.42** Cohen-Macaulay modules

**512.669.45** Buchsbaum rings and modules

**512.669.5** Gorenstein rings and generalizations

**512.669.52** Quasi-Frobenius algebras

**512.669.54** Gorenstein modules

**512.669.6** Dualizing and semi-dualizing modules and complexes

**512.669.7** Koszul algebras

**512.669.8** Homological aspects of representation theory

**512.669.81** Representation type

**512.669.82** Representation theory of algebras with special properties

**512.669.83** Auslander-Reiten quiver

**512.669.84** Almost split sequences

**512.669.85** Tilting and cotilting modules

**512.669.86** Morita equivalence and duality

**512.669.87** Derived equivalence

**512.669.88** Homological aspects of groups representation theory

**512.669.89** Homological aspects of Lie algebra representation theory

**512.7** Algebraic geometry

**512.71** Commutative rings and algebras. Local theory. Foundations of algebraic geometry

**512.711** General commutative ring theory

**512.712** Valuations on commutative rings and theory of divisibility

**512.713** Arithmetic rings. Dedekind and PrÃ¼fer rings

**512.714** Polynomial rings

**512.715** Modules over commutative rings

*References: 512.553*

**512.717** Local algebra. Local theory

**512.72** Structure and variations of algebraic varieties. Fibre bundles. Vector bundles. Fibrations

**512.721** Moduli of algebraic varieties

**512.722** Structure of families. Picard varieties

**512.723** Algebraic vector bundles

**512.725** Algebraic fibrations with degenerate fibres

**512.73** Cohomology theory of algebraic varieties and schemes

**512.732** Algebraic sheaves and cohomology with coefficients therein

**512.734** Cycles: intersection theory, equivalence

**512.736** Serre cohomology. K-theory

**512.737** Grothendieck cohomology and topology

**512.74** Algebraic groups

**512.741** Formal groups

**512.742** Abelian varieties and schemes

**512.743** Linear algebraic groups

**512.745** Algebraic groups of transformations

**512.747** Pro-algebraic groups and group schemes

**512.75** Arithmetic problems of algebraic varieties

**512.76** Birational geometry. Mappings etc.

**512.761** Singularities. Singular points of algebraic varieties

**512.763** Linear systems and rational mappings

**512.765** Modifications and the problem of minimal models

**512.77** Algebraic curves. Algebraic surfaces. Three-dimensional algebraic varieties

**512.772** Algebraic curves

**512.774** Algebraic surfaces

**512.776** Three-dimensional algebraic varieties

**512.78** Analytic spaces on arbitrary complete normed fields

*References: 515.17*

**512.81** Lie groups

*References: 512.554.3*

**512.812** General Lie group theory

**512.813** Special classes of Lie groups

**512.814** Continuous subgroups of Lie groups

**512.815** Linear representations of Lie groups

**512.816** Lie transformation groups

**512.817** Discrete subgroups. Discrete transformation groups

**512.818** Theory of continuous pseudogroups (infinite Lie groups)

**514** Geometry

*References: 515.1*

**514.01** Foundations of geometry. Axiomatics

**514.1** General geometry

**514.11** Elementary geometry, trigonometry, polygonometry

**514.112** Plane geometry. Planimetry

**514.112.3** Geometry of the triangle

**514.112.4** Geometry of polygons, rectangles etc.

**514.112.6** Elementary geometry of circles

**514.113** Solid geometry

**514.113.4** Geometry of tetrahedra

**514.113.5** Geometry of polytopes, polyhedra

**514.113.6** Geometry of spheres and cylinders

**514.114** Elementary geometry in multidimensional spaces

**514.115** Theory of geometric constructions

**514.116** Trigonometry. Polygonometry

**514.116.2** Plane trigonometry

**514.116.3** Spherical trigonometry

**514.12** Euclidean and pseudo-Euclidean geometries. Analytic geometry

**514.122** Analytic geometry in the Euclidean plane

**514.123** Analytic geometry in three-dimensional Euclidean space

**514.124** Analytic geometry in multidimensional Euclidean space

**514.125** Pseudo-Euclidean spaces

**514.126** Galilean spaces

**514.127** Semi-Euclidean spaces

**514.128** Flag spaces

**514.13** Non-Euclidean metric geometries

**514.132** Lobachevsky-Bolyai geometry

**514.133** Other hyperbolic geometries

**514.135** Elliptic geometries

**514.14** Affine geometry. Projective geometry

**514.142** Affine geometry

**514.142.1** Synthetic geometry in affine space

**514.142.2** Analytic geometry in affine space

**514.144** Projective geometry

**514.144.1** Synthetic geometry in projective space

**514.144.2** Analytic geometry in projective space

**514.146** Algebraic theory of affine and projective geometries

**514.15** Geometry in spaces with other fundamental groups

**514.152** Conformal geometry and its analogues

**514.152.2** Conformal geometry

**514.152.4** Laguerre geometry

**514.152.6** Lie geometry of spheres

**514.152.8** Other generalizations of conformal geometry

**514.154** Symplectic geometry

**514.157** Biaxial geometry and its generalizations

**514.16** Geometries over algebras

**514.161** Affine and projective spaces over algebras

**514.162** Quadratic Euclidean and non-Euclidean spaces

**514.163** Hermitian Euclidean and non-Euclidean spaces

**514.164** Symplectic spaces

**514.17** Convex sets. Geometric figure arrangements. Geometric inequalities

**514.172** Convex sets

**514.172.2** Convex curves and surfaces

**514.172.4** Convex bodies

**514.174** Arrangements of geometric figures

**514.177** Geometric inequalities

**514.18** Descriptive geometry

**514.181** Theoretical questions of descriptive geometry

**514.181.2** Descriptive geometry of curves and surfaces

**514.181.4** Affine and projective foundations of methods of descriptive geometry

**514.181.6** Theoretical questions of parallel and central axonometry

**514.181.8** Theory of accuracy of graphic constructions

**514.182** Practical methods of descriptive geometry

**514.182.2** Methods of diagram transformations

**514.182.3** Perspective constructions and the theory of shadows

**514.185** Generalizations of descriptive geometry

**514.185.2** Multidimensional descriptive geometry

**514.185.4** Descriptive geometry of non-Euclidean spaces

**514.185.7** Nonlinear descriptive geometry

**514.7** Differential geometry. Algebraic and analytic methods in geometry

**514.74** Algebraic and analytic methods in geometry

**514.742** Vector algebra. Vector analysis

**514.742.2** Vector algebra

**514.742.4** Vector analysis (vector field theory)

**514.743** Tensor algebra. Tensor analysis

**514.743.2** Tensor algebra

**514.743.4** Tensor analysis

**514.744** Spinors

**514.745** Calculus of exterior forms

**514.745.2** Grassman algebra and its generalizations

**514.745.4** Theory of exterior differential forms

**514.745.5** Differential algebras and their geometric applications

**514.745.8** Theory of compatibility of systems of differential equations

*References: 514.763.8*

**514.747** Geometric objects

**514.748** Differential geometric methods in the study of immersed manifolds

**514.748.2** Moving frame of a manifold

**514.748.4** Geometric objects on immersed manifolds

**514.75** Differential geometry in spaces with fundamental groups

**514.752** Differential geometry in Euclidean and pseudo-Euclidean spaces

**514.752.2** Theory of curves

**514.752.3** Theory of surface strips

**514.752.4** Theory of surfaces

**514.752.5** Theory of families of straight lines and planes

**514.752.6** Theory of families of curves and surfaces

**514.752.7** Differential geometry of vector fields

**514.752.8** Theory of nonholonomic manifolds

**514.753** Differential geometry in non-Euclidean spaces

**514.753.2** With nondegenerate absolute

**514.753.3** With degenerate absolute

**514.754** Affine differential geometry

**514.755** Projective differential geometry

**514.756** Differential geometry in spaces with other fundamental groups

**514.756.2** Differential geometry in conformal and pseudoconformal spaces

**514.756.24** Conformal theory of surfaces

**514.756.4** Differential geometry in symplectic spaces

**514.756.7** Differential geometry in biaxial, biaffine spaces and their generalizations

**514.757** Differential geometry of pointwise mappings

**514.757.2** Pointwise mappings of affine and projective spaces

**514.757.3** Pointwise mappings of Euclidean, pseudo-Euclidean, conformal and other spaces with a metric

**514.757.4** Mapping of submanifolds under the pointwise mappings of spaces with fundamental group

**514.757.6** Dual mappings

**514.758** Kinematic geometry

**514.76** Geometry of differentiable manifolds and of their submanifolds

**514.762** Geometry of fibre spaces

**514.762.2** General questions of fibre space geometry

**514.762.3** Spaces and fields of geometric objects in fibre spaces

**514.762.5** Connections in fibre spaces

**514.762.7** Holonomy groups of fibre spaces

**514.763** Infinitesimal structures and fields of geometric objects on differentiable manifolds

**514.763.2** Differential geometry of vector and tensor fields on manifolds

**514.763.3** G-structures on differentiable manifolds

**514.763.4** Manifolds with complex or almost-complex structure

**514.763.5** Infinitesimal structures and fields of geometric objects of higher order

**514.763.6** Finsler geometry and its generalizations

**514.763.62** Finsler geometry

**514.763.63** Geometry of the integral

**514.763.64** Geometry of the spaces of linear elements

**514.763.65** Geometry of spaces with other generating elements

**514.763.7** Geometry of webs

**514.763.8** Geometry of differential equations

*References: 514.745.8*

**514.764** Classical spaces with connections and their generalizations

**514.764.2** Riemannian and pseudo-Riemannian spaces

**514.764.21** General theory of Riemannian and pseudo-Riemannian spaces

**514.764.22** Special types of Riemannian space

**514.764.25** Mappings of Riemannian and pseudo-Riemannian spaces

**514.764.27** Submanifolds of Riemannian and pseudo-Riemannian spaces

**514.764.3** Spaces with an affine connection

**514.764.32** Special types of space with an affine connection

**514.764.4** Spaces with a projective connection

**514.764.5** Spaces with a conformal connection

**514.764.6** Spaces with a symplectic connection

**514.764.7** Spaces with a Euclidean, pseudo-Euclidean or generalized Euclidean connection

**514.765** Geometry of homogeneous spaces. Lie group geometry

**514.765.1** Invariant infinitesimal structures in homogeneous spaces

**514.765.2** Vector and tensor fields in homogeneous spaces

**514.765.7** Integral geometry

**514.77** Differential geometry of submanifolds in the large and of metrizable manifolds

**514.772** Differential geometry of submanifolds in the large. Nonregular submanifolds

**514.772.2** Geometry in the large of curves and surfaces in Euclidean space

**514.772.3** Images in the large

**514.774** Geometry of metrizable manifolds. Minkowsky geometry. Hilbert geometry. Geometry of geodesics

**514.774.2** Minkowsky geometry

**514.774.4** Hilbert geometry

**514.774.8** Geometry of geodesics

**514.8** Geometric study of objects of mechanics and physics

**514.82** Geometric questions and methods in relativity theory. Geometric questions of cosmology, of unified field theory etc.

**514.83** Geometric study of fields of physical objects

**514.84** Geometric methods in quantum mechanics and in the theory of elementary particles

**514.85** Geometric methods in general mechanics

**514.86** Geometric methods in mechanics of continuous media

**514.87** Geometric questions and methods in crystallography

**514.88** Geometric questions and methods in optics

**515.1** Topology

**515.12** General topology

**515.122** Topological spaces

**515.122.2** Axiomatic theory of topological spaces

**515.122.3** Nonaxiomatic theory of topological spaces

**515.122.4** Topological properties of spaces with supplementary structure and topological transformation groups

**515.122.5** Construction of topological spaces. Operations on topological spaces

**515.122.8** Generalizations of topological spaces

**515.123** Uniform spaces. Proximity spaces

**515.123.2** Axioms of uniform and proximity spaces

**515.123.4** Uniform spaces. Uniformly continuous mappings

**515.123.5** Proximity spaces

**515.123.7** Comparison of topologies, uniformities and proximities

**515.124** Metric spaces

**515.124.2** Axioms and generalizations of metric spaces

**515.124.3** Topological properties of metric spaces

**515.124.4** Metric properties of metric spaces

**515.124.5** Classes of metric spaces defined by topological properties

**515.124.6** Classes of metric spaces defined by extrinsic conditions (imposed on possible imbedding spaces)

**515.125** Topology of Euclidean spaces

**515.126** Continuous mappings

**515.126.2** Special types of continuous mapping

**515.126.4** Fixed points and coincidences

**515.126.8** Generalizations of continuous mappings

**515.127** Dimension. Other numerical topological invariants

**515.127.1** Dimension theory

**515.127.2** Dimension-type invariants

**515.128** Descriptive set theory of topological spaces

**515.14** Algebraic topology

**515.142** General theorems on fundamental categories and functors

**515.142.2** General topological categories. Categories whose objects are topological spaces subject to various general restrictions, and whose morphisms are either continuous mappings or homotopy classes of such mappings. Other closely related categories

**515.142.21** Homology and cohomology groups of spaces: definitions, basic properties. Axiomatic questions

**515.142.22** Study of topological spaces and continuous mappings by homological methods

**515.142.23** Homology and cohomology with non-Abelian coefficients

**515.142.25** Homotopy and cohomotopy groups: definitions, basic properties. Axioms etc.

**515.142.27** Functors with values in general topological categories (operations on topological spaces)

**515.142.3** Polyhedral categories. Categories whose objects are polyhedra

**515.142.32** CW-complexes

**515.142.33** Simplicial complexes (triangulations). Simplicial schemes

**515.142.4** Categories approximating to general topological and polyhedral categories

**515.142.42** Categories whose morphisms are stable mappings or homotopy classes of such mappings. Categories of spectra

**515.142.44** Categories of semi-exact functors

**515.142.5** Simplicial sets

**515.143** Homotopy theory: basic problems

**515.143.2** Decompositions of spaces and mappings

**515.143.3** Theory of obstructions. General classification and extension theorems for continuous mappings and cross-sections

**515.143.5** Cohomology operations

**515.145** Spaces with various additional general properties or derived by various general constructions

**515.145.2** Fibre spaces. Fibre bundles

**515.145.3** Spaces with operators

**515.145.5** Spaces with multiplication (H-spaces). Loop spaces

**515.145.6** Spaces with comultiplication and suspensions

**515.145.8** Spaces with only a finite number of non-homotopy groups

**515.146** Concrete spaces. Computation of homotopy invariants

**515.146.2** Computation of homotopy groups

**515.146.3** Computation of homology and cohomology groups

**515.146.5** Computation of K-functors, of J-functors

**515.146.6** Computation of bordism and cobordism groups

**515.148** Isotopy theory

**515.16** Topology of manifolds

**515.162** Low-dimensional manifold topology

**515.163** Topological manifolds

**515.164** Topology of smooth and piecewise linear manifolds

**515.164.2** Classification of smooth and piecewise linear manifolds

**515.164.3** Fibrations of smooth manifolds. Fibre bundles with smooth manifolds as bases

**515.164.6** Smooth and piecewise linear imbeddings and immersions of manifolds

**515.164.8** Groups acting on smooth and piecewise linear manifolds

**515.165** Topology of smooth manifolds provided with additional structure

**515.168** Differential and integral operators on manifolds

**515.17** Analytic spaces

*References: 512.78 , 517.547.7*

**515.171** General theory of complex and real analytic spaces

**515.172** Analytic mappings and constructions of complex spaces

**515.172.2** Holomorphic mappings of complex spaces

**515.172.3** Plurisubharmonic functions, pseudoconvex and pseudoconcave domains and their generalizations

**515.172.4** Meromorphic functions and mappings

**515.172.5** Multivalent analytic functions

**515.172.6** Quotient spaces of complex spaces

**515.172.7** Analytic coverings

**515.172.8** Modifications of complex spaces

**515.173** Complex spaces of one, two and three dimensions

**515.173.2** One-dimensional complex manifolds

**515.173.4** Complex surfaces

**515.173.6** Three-dimensional complex spaces

**515.174** Classes of complex spaces defined by global conditions

**515.175** Generalizations of analytic spaces

**515.176** Holomorphic fibre spaces

**515.177** Complex spaces with a group of automorphisms

**515.178** Automorphic functions

*References: 511.384 , 512.817 , 517.547.58*

**515.178.2** Abelian functions

**515.178.3** Modular functions

**515.179** Deformation of structures. Pseudogroups

**517** Analysis

**517.1** Introduction to analysis

**517.13** Theory of real numbers

**517.15** Asymptotic formulae and expressions

**517.16** Analytical means. Inequalities

**517.17** Basic properties of functions

**517.18** Study of concrete functions

**517.2** Differential calculus. Differentiation

**517.22** Differentials and derivatives

**517.23** Derivatives of arbitrary order

**517.26** Taylor's formula in the real domain

**517.27** Study of variation of functions

**517.272** Maxima. Minima

**517.275** Mappings. Implicit functions

**517.28** Other analytical applications of differential calculus

**517.3** Integral calculus. Integration

**517.31** Single indefinite integrals. General method of integration

**517.37** Multiple integrals

**517.38** Simple or multiple definite integrals

**517.382** Improper integrals

**517.383** Mean-value theorem

**517.387** Differentiation and integration under the integral sign

**517.4** Functional determinants. Integral transforms. Operational calculus

**517.41** Functional determinants

**517.43** Operators of vector calculus (symbolism of Hamilton)

**517.44** Integral transforms. Operational calculus

**517.442** Laplace transforms

**517.443** Fourier integral and Fourier transforms

*References: 517.518.5*

**517.444** Other integral transforms and their inversions. Convolutions

*References: 517.968.2*

**517.445** Operational calculus

**517.5** Theory of functions

**517.51** Functions of a real variable. Real functions

**517.517** Descriptive theory of functions

**517.518** Metric theory of functions

**517.518.1** Measure. Integration. Differentiation

**517.518.11** Measure. Capacity

*References: 517.544.5*

**517.518.12** Theory of integration

**517.518.13** Singular integrals

*References: 517.544.73*

**517.518.14** Integrals of potential type

*References: 517.956.224*

**517.518.15** Theory of differentiation

**517.518.17** Mappings

**517.518.18** Curved surfaces

**517.518.2** Classes of functions (sets of functions)

**517.518.22** Compact families of functions

**517.518.23** Imbedding theorems for classes of differentiable functions

**517.518.24** Functions of bounded variation

**517.518.25** Quasi-analytic functions

**517.518.26** Other classes of functions

**517.518.27** Superpositions

**517.518.28** Inequalities

**517.518.3** Systems of functions. Series in systems of functions

**517.518.4** Trigonometric series

**517.518.42** Representation of a function by trigonometric series

**517.518.43** Uniqueness problem

**517.518.45** Fourier series

**517.518.47** Multiple trigonometric series

**517.518.5** Theory of the Fourier integral

**517.518.6** Almost-periodic functions. Almost-periodicity

**517.518.8** Approximation of functions by polynomials and their generalizations

**517.518.82** Approximation by algebraic polynomials

**517.518.83** Approximation by trigonometric polynomial and entire functions of exponential type

**517.518.84** Approximation by rational functions

**517.518.85** Interpolation

**517.518.86** Extremal properties of polynomials and their generalizations

**517.518.87** Theory of quadratures and cubatures

*References: 519.644*

**517.518.88** Theory of moments

**517.52** Series and sequences

*References: 517.962.27*

**517.521** Numerical series and sequences. Functional series and sequences

**517.523** Infinite products

**517.524** Continued fractions

**517.53** Functions of a complex variable

**517.535** Rational functions in the complex domain

**517.537** Sequence and series of analytic functions

**517.537.3** Power series

**517.537.6** Dirichlet series

*References: 511.331*

**517.538** Systems of functions

**517.54** Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations

**517.542** Mappings of special domains

**517.544** Boundary properties of analytic functions. Boundary value problems

**517.544.4** Bounded functions

**517.544.5** Harmonic measure and capacity

**517.544.7** Boundary properties of holomorphic functions

**517.544.72** Cluster values. Cluster sets

**517.544.73** The Cauchy-type integral

*References: 517.518.13 , 517.968.2*

**517.544.8** Boundary value problems of the theory of analytic functions

**517.545** Riemannian surfaces of functions of a complex variable. Uniformization

**517.546** Univalent and multivalent functions

**517.547** Classes and spaces of analytic functions

**517.547.2** Entire and meromorphic functions

**517.547.22** Entire functions of finite order

**517.547.24** Meromorphic functions

**517.547.26** Picard theorem and its generalizations

**517.547.28** Value distribution theory

**517.547.3** Analytic functions in a finite circle and other domains

**517.547.5** Special classes of analytic functions

**517.547.52** Functions of bounded type

**517.547.54** The classes H (H-delta) and others

**517.547.56** Algebraic and algebroid functions

**517.547.58** Automorphic functions (analytic theory)

*References: 515.178*

**517.547.7** Spaces of analytic functions

*References: 515.17*

**517.548** Generalizations of analytic functions

**517.55** Functions of several complex variables

**517.57** Harmonic functions and their generalizations

*References: 517.956.224*

**517.58** Special functions. Hyperbolic functions

**517.581** Euler integrals and their generalizations. Gamma function G(x) and related functions

**517.582** Probability integrals

**517.583** Elliptic functions and integrals

**517.584** Bessel functions and polynomials and other cylindrical functions

**517.585** Mathieu functions

**517.586** Spherical functions. Legendre polynomials and functions. Harmonic polynomials. Ultraspheric polynomials. Gegenbauer functions

**517.587** Orthogonal polynomials and their generalizations (Chebyshev, Hermite, Jacobi, Laguerre etc.)

**517.588** Hypergeometric series and functions. Generalized and confluent hypergeometric functions

**517.9** Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis

**517.91** Ordinary differential equations: general theory

**517.911** General questions. Existence theorems. Uniqueness theorems. Theorems on differential properties of solutions

**517.912** Methods of solution of different types of equation and systems of equations

**517.913** First integrals

**517.922** Equations not solved with respect to the highest order derivative. Singular solutions

**517.923** Special types of ordinary differential equation

**517.925** Systems and analytic theory of ordinary differential equations

**517.925.4** Theory of equations and systems of second order

**517.925.5** Theory of equations and systems of arbitrary order

**517.925.7** Analytic theory of ordinary differential equations

**517.926** Linear ordinary differential equations and systems

**517.927** Boundary value problems for ordinary differential equations

**517.928** Asymptotic methods in ordinary differential equation theory. Averaging methods. Invariant manifolds

**517.929** Differential difference equations (with retarded arguments etc.)

**517.929.2** Linear difference equations

**517.93** Special differential equations. Systems of analytic mechanics, automatic control, operators. Dynamic systems

**517.933** Equations of analytic mechanics

**517.935** Equations of automatic control systems

*References: 681.5*

**517.936** Pfaffian equations and systems

**517.937** Operator differential equations

**517.938** Theory of dynamic systems

**517.95** Partial differential equations

**517.951** General theory of partial differential equations and systems

**517.951.1** Local and global solvability problems (local and global uniqueness theorems)

**517.951.2** Fundamental solutions

**517.951.3** Geometric theory. Characteristics. Transformations

**517.951.4** Qualitative theory

**517.951.5** Analytic methods. Singularities

**517.951.6** Variational methods

**517.951.7** Microlocal methods. Method of sheaf theory, homological algebra in partial differential equations

**517.951.8** Topological methods

**517.951.9** Other aspects of the general theory

**517.952** General first-order equations and systems: properties, types etc.

**517.952.1** General theory of first-order equations

**517.952.3** Cauchy problem. Initial value problems of first-order equations

**517.952.5** Boundary-value problems of first-order equations

**517.953** General higher-order equations and systems: properties, types etc.

**517.953.1** General theory of higher-order equations

**517.953.3** Cauchy problem of higher-order equations

**517.953.5** Boundary-value problems of higher-order equations

**517.954** Boundary value problems

**517.955** Cauchy problem for partial differential equations

**517.955.2** Well-posedness theory

**517.955.4** Semigroups related to the Cauchy problem

**517.955.8** Asymptotic behaviour of solutions

**517.956** Linear and quasilinear equations and systems

**517.956.1** Equations with constant coefficients

**517.956.2** Elliptic equations and systems

**517.956.21** General theory of elliptic second-order equations and systems

**517.956.22** Linear equations of elliptic type

**517.956.222** General properties of elliptic equations

**517.956.223** Boundary value problems of elliptic equations

**517.956.224** Potential theory. Potentials of elliptic equations

*References: 517.518.14 , 517.57 , 517.982.26*

**517.956.225** Laplace and Poisson equations

**517.956.226** Degenerate equations. Equations with a small parameter

**517.956.227** Spectral theory

**517.956.23** Higher-order elliptic equations

**517.956.232** General properties of higher-order elliptic equations

**517.956.233** Boundary-value problems of higher-order elliptic equations

**517.956.25** Quasilinear equations of elliptic type

**517.956.27** Inverse problems

**517.956.28** Nonlocal problems

**517.956.29** Ill-posed problems

**517.956.3** Hyperbolic equations and systems

**517.956.32** Linear hyperbolic equations

**517.956.321** Second-order hyperbolic equations

**517.956.322** Higher-order hyperbolic equations

**517.956.323** Pseudohyperbolic equations

**517.956.326** Degenerate equations. Equations with a small parameter

**517.956.328** Spectral problems

**517.956.35** Quasilinear hyperbolic equations

**517.956.36** Hyperbolic conservation laws

**517.956.37** Inverse problems

**517.956.4** Parabolic equations and systems

**517.956.41** Linear parabolic equations and systems

**517.956.42** Second-order parabolic equations

**517.956.43** Higher-order parabolic equations

**517.956.44** Free-boundary problems

**517.956.45** Nonlinear equations and/or nonlinear problems

**517.956.46** Ultraparabolic, pseudoparabolic, etc. equations

**517.956.47** Inverse problems

**517.956.6** Equations of mixed and composite types

**517.956.65** Nonlinear equations of mixed and composite types

**517.957** Nonlinear equations and systems

**517.957.6** Nonlinear equations of mathematical physics

**517.957.7** Nonlinear equations from other fields of science

**517.958** Differential and integral equations of mathematical physics

**517.96** Finite differences. Functional and integral equations

**517.962** Finite differences

**517.962.1** Theory of finite differences

**517.962.2** Finite difference equations

**517.962.22** Linear difference equations

**517.962.24** Nonlinear difference equations

**517.962.26** Theory of generating functions

**517.962.27** Recurrent sequences and series

*References: 517.52*

**517.962.8** Application of the calculus of finite differences to differential equations

**517.965** Functional equations. Functional inequalities

**517.968** Integral equations

**517.968.2** Linear integral equations

**517.968.21** Fredholm integral equations

**517.968.211** Equations of the first kind

**517.968.212** Equations of the second kind

**517.968.213** Equations with partial integrals

**517.968.22** Volterra integral equations

**517.968.221** Volterra partial integral equations

**517.968.23** Singular integral equations and related boundary value problems

**517.968.231** Equations with kernels of the Cauchy type

**517.968.232** Convolution-type equations (Abel, Wiener-Hopf etc. equations)

**517.968.233** Other singular equations

**517.968.25** Linear integral equations in Banach and function spaces

**517.968.26** Linear integral equations with special kernels

**517.968.28** Systems of linear integral equations

**517.968.281** Systems of linear singular integral equations

**517.968.282** Systems of linear integral equations of mixed type

**517.968.283** Dual, triple and other integral equations

**517.968.4** Nonlinear integral equations

**517.968.43** Nonlinear singular integral equations

**517.968.44** Nonlinear partial integral equations

**517.968.46** Nonlinear integral equations in Banach and function spaces

**517.968.48** Systems of non-linear integral equations

**517.968.7** Integro-differential equations

**517.968.72** Linear integro-differential equations

**517.968.721** Linear integro-ordinary differential equations

**517.968.722** Linear integro-partial differential equations

**517.968.73** Singular integro-differential equations

**517.968.74** Nonlinear integro-differential equations

**517.968.741** Nonlinear ordinary integro-differential equations

**517.968.742** Nonlinear partial integro-differential equations

**517.968.75** Integro-differential equations in Banach and function spaces

**517.968.78** Systems of integro-differential equations

**517.97** Calculus of variations. Mathematical theory of control

**517.972/.974** Calculus of variations

**517.972** Functional analytic methods in calculus of variations

**517.972.1** Functionals of the calculus of variations

**517.972.11** General properties: semicontinuity, differentiability, etc.

**517.972.13** Homogenization. Problems of variational convergence

**517.972.15** Geometric problems of the calculus of variations and certain problems of Finsler and symplectic geometry

**517.972.17** Geometric measure theory and integration. Integral and normal currents

**517.972.2** Necessary conditions based on the theory of first and second variations

**517.972.21** One-dimensional problems

**517.972.22** Higher-dimensional problems

**517.972.24** Isoperimetrical problems

**517.972.26** Periodic and almost periodic problems

**517.972.3** Sufficient conditions

**517.972.31** One-dimensional problems

**517.972.32** Higher-dimensional problems

**517.972.4** Questions of existence of solutions of variational problems. Existence theory

**517.972.42** Relaxations of variational problems. Generalized solutions. Young measures

**517.972.44** Regularity of minimum points

**517.972.46** Nonconvex and/or noncoercitive problems

**517.972.5** Variational methods for solution of differential, integral and other functional equations

*References: 517.988.8*

**517.972.51** Direct variational methods

**517.972.53** Methods based on variational principles

**517.972.55** Minimax methods

**517.972.58** Other methods: concentrated compactness principle, degree-theoretic methods, Morse and Conley indices, etc.

**517.972.59** Inverse problem of the calculus of variations

**517.972.8** Extremal problems in linear topological spaces and related concepts

*References: 519.85*

**517.972.81** Semi-infinite mathematical programming problems

**517.972.82** Infinite-dimensional mathematical programming problems

**517.972.83** Abnormal problems

**517.972.84** Minimax problems

**517.972.85** Selected topics of convex analysis

**517.972.86** Selected topics of smooth analysis

**517.972.87** Selected topics of nonsmooth analysis

**517.972.88** Selected topics of multivalued mapping theory

**517.972.89** Approximate methods of the calculus of variations

**517.972.9** Various particular problems of the calculus of variations

**517.972.91** Harmonic mappings and their generalizations

**517.972.92** Extremal problems of Riemannian geometry

**517.972.93** Problems of sub-Riemannian geometry

**517.972.94** Extremal problems of function theory

**517.972.96** Vector optimization problems. Minimal surfaces

**517.972.99** Other problems of the calculus of variations

**517.974** Topological methods in the calculus of variations

**517.974.2** Morse theory

**517.974.3** Lyusternik-Shnirelman theory

**517.974.4** Other smooth critical point theories

**517.974.5** Nonsmooth critical point theory

**517.974.6** Selected topics of infinite-dimensional manifold theory

**517.974.8** Variatinal theory of geodesics

**517.977** Mathematical control theory. Optimal control.

**517.977.1** General theory of control systems described by ordinary continuous- and discrete-time differential and differential-functional equations and inclusions

**517.977.11** Controllability and reachability. Reachable set of nonlinear control systems

**517.977.12** Observability and identification

**517.977.13** Input-output mapping. Minimal realization theory

**517.977.14** Uncertain control systems

**517.977.15** Variable structure systems. Sliding-mode controls

**517.977.16** Feedback nonlinear control systems

**517.977.163** Stabilizability

**517.977.166** Linearization and partial linearization

**517.977.169** Other structural theory problems

**517.977.17** Particular classes of nonlinear control systems

**517.977.18** Differential-geometric, topological, and algebraic methods for studying nonlinear control systems

**517.977.2** Linear control systems with continuous and discrete time

**517.977.22** Reachability, controllability, and observability. Reachable set

**517.977.24** Linear feedback systems

**517.977.26** Algebraic theory of linear control systems

**517.977.28** H?-control

**517.977.3** General theory of systems described by abstract equations

**517.977.32** Controllability and approximate controllability

**517.977.34** Observability and approximate observability

**517.977.36** Feedback control systems

**517.977.4** Viability theory

**517.977.5** Optimal control

*References: 519.85*

**517.977.51** Optimal control of systems described by continuous- and discrete-time ordinary differential and functional-differential equations and inclusions

**517.977.52** Maximum principle

**517.977.53** Higher-order necessary optimality conditions. Higher-order maximum principle

**517.977.54** Sufficient optimality conditions. Dynamic programming. Optimal synthesis

**517.977.55** Linear and linear-quadratic optimal control problems

**517.977.56** Optimal control problems for distributed parameter systems

**517.977.57** Problems of existence of optimal controls

**517.977.58** Approximate methods for solutions of optimal control problems

**517.977.59** Bang-bang theorems and related topics

**517.978** Differential games

*References: 519.837*

**517.978.2** Two-player differential games

**517.978.4** Many-player differential games

**517.98** Functional analysis and operator theory

**517.981** Axiomatics of functional analysis

*References: 510.2*

**517.982** Linear spaces with topology and order or other structures

**517.982.1** Ordered and semiordered spaces

**517.982.2** Topological linear spaces

**517.982.22** Normed spaces. Banach spaces. Hilbert spaces

**517.982.23** Countably normed spaces. FrÃ©chet spaces

**517.982.24** Montel and kernel spaces

**517.982.25** Geometry of topological linear spaces

**517.982.26** Absolute potential theory

*References: 517.956.224 , 519.217.5*

**517.982.3** Linear functionals and conjugate spaces

**517.982.4** Theory of generalized functions (distributions)

**517.983** Linear operators. Linear operator equations

**517.983.2** Linear operators in infinite-dimensional linear spaces: general properties

**517.983.22** Linear operators in locally convex spaces

**517.983.23** Linear operators in Banach spaces

**517.983.24** Linear operators in Hilbert spaces

**517.983.26** Completely continuous and kernel operators

**517.983.27** Linear operators in semiordered spaces

**517.983.28** Perturbation theory of linear operators

**517.983.5** Linear equations in infinite-dimensional linear spaces

**517.983.6** Vector functions. Operator functions

**517.983.8** Families of linear spaces. Categories of linear operators

*References: 512.58*

**517.984** Spectral theory of linear operators

**517.984.2** Spectral theory in general topological linear spaces and spaces with quasitopology

**517.984.3** Spectral theory in Banach spaces

**517.984.4** Spectral theory in Hilbert spaces

**517.984.5** Study of the spectrum of concrete operators, including differential and integral operators

**517.984.7** Operator extensions

**517.986** Topological algebras. Theory of infinite-dimensional representations

*References: 511.384*

**517.986.2** Topological algebras (rings) and their continuous representations

*References: 512.556*

**517.986.3** Algebras (rings) of linear operators

**517.986.4** Infinite-dimensional representations of groups

**517.986.5** Infinite-dimensional representations of Lie algebras

**517.986.6** Harmonic analysis of functions of groups and homogeneous spaces

**517.986.7** Semigroups of linear or nonlinear operators. Evolution equations

**517.986.9** Other algebraic structures in functional analysis

**517.987** Measures. Representations of Boolean algebras. Metric theory of dynamic systems

**517.987.1** Theory of measure and integral

*References: 517.518.1*

**517.987.3** Representations of Boolean algebras

*References: 512.563*

**517.987.4** Functional integrals and their applications to evaluation equations

*References: 517.518.12*

**517.987.5** Metric theory of dynamic systems

*References: 517.938*

**517.988** Nonlinear functional analysis and approximation methods

**517.988.3** Nonlinear functionals

**517.988.5** Nonlinear operators

**517.988.6** Nonlinear equations in functional spaces

**517.988.8** Application of functional analysis to theory of approximation methods

*References: 517.972.5*

**519.1** Combinatorial analysis. Graph theory

**519.101** General theory of combinatorial analysis

**519.11** Enumeration theory

**519.111** General methods of enumeration

**519.111.1** Recurrences

**519.111.2** Incidence algebras. Inversion formulae. The method of inclusion and exclusion

**519.111.3** Combinatorics of formal power series. Generating functions

**519.114** Combinatorial sequences of numbers and polynomials

**519.115** Enumeration problems in combinatorial analysis

**519.115.1** Combinatorial problems on substitutions and permutations

*References: 512.542.7*

**519.115.2** Enumeration of mappings and transformations

**519.115.3** Enumeration of integral matrices and their equivalence classes

**519.115.4** Enumeration of Boolean functions and disjunctive normal forms

**519.115.5** Problems on set partitions and set coverings. Finite topologies

**519.115.6** Problems on paths in integral lattices

**519.115.8** Combinatorics of sequences

**519.116** Combinatorial theory of number partitions

**519.117** Combinatorial identities

**519.118** Combinatorial inequalities

**519.119** Combinatorial problems in probability theory

*References: 519.21*

**519.14** Combinatorial configurations and arrays

**519.142** Matrix combinatorial problems

**519.143** Latin squares and their generalizations. Magic squares and cubes

**519.144** Block designs

**519.145** Finite affine and projective geometries as block designs

*References: 514.146*

**519.145.4** Generalized polygons

**519.146** Combinatorics of arrangements of geometric objects

*References: 514.174*

**519.147** Packing and covering

**519.148** Problems on plane mosaics and tesselations

**519.15** Combinatorial theory of choice

**519.151** Matroids

**519.152** Transversals

**519.154** Ramsey theory

**519.156** Combinatorics of ordered sets

*References: 512.562*

**519.157** Extremal combinatorial problems

**519.16** Algorithmic problems of combinatorial analysis

*References: 519.178*

**519.161** General questions of combinatorial algorithms. Polynomial and NP-complete problems

**519.163** Constructions of efficient combinatorial algorithms. Search trees

**519.165** Combinatorial algorithms in algebraic problems

**519.168** Problems of optimum choice. Algorithms for extremal problems

**519.17** Graph theory

**519.171** General graph theory and representations of graphs

**519.172** Particular classes of graphs. Trees

**519.172.1** Trees

**519.172.2** Planar graphs

**519.172.3** Digraphs. Tournaments

**519.172.4** Regular graphs

**519.172.5** Bipartite graphs

**519.173** Topological and metric problems of graph theory

*References: 515.14*

**519.174** Structural properties of graphs. Colourings

**519.174.1** Graphical partitions

**519.174.2** Walks in graphs

**519.174.3** Coverings and packings in graphs

**519.174.5** Factorization of graphs

**519.174.7** Colourings of graphs

**519.175** Classification and enumeration of graphs and digraphs

**519.176** Extremal problems of graph theory

**519.177** Matrix methods in graph theory. Spectral graph theory

**519.178** Algorithmic questions of graph theory

*References: 519.16*

**519.179** Generalizations of graphs

**519.179.1** Hypergraphs

**519.179.2** Nets

**519.2** Probability. Mathematical statistics

**519.21** Probability theory. Stochastic processes

**519.211** Foundations and axioms of probability theory

**519.212** Abstract probability theory

**519.212.2** Combinatorial probabilities

**519.212.3** Geometrical probabilities

**519.213** Probability distributions and densities

**519.213.1** Normal distribution

**519.213.2** Characteristic functions, moments, semi-invariants and other characteristics

*References: 517.5*

**519.213.3** Measures of dependence

**519.213.5** Infinitely divisible laws

**519.213.7** Stable laws

**519.214** Limit theorems

**519.216** Stochastic processes in general

**519.216.3** Prediction theory

**519.216.5** Stopping times

**519.216.8** Martingales

**519.217** Markov processes

**519.217.1** General theory of Markov processes

**519.217.2** Markov chains. Processes with finite or countable set of states

**519.217.3** Processes with independent increments

**519.217.4** Diffusion processes and their applications

**519.217.5** Additive functionals. Probabilistic potential theory

*References: 517.982.26*

**519.217.6** Transformation of Markov processes

**519.217.7** Boundary theory for Markov processes

**519.217.8** Controlled Markov processes

**519.218** Special stochastic processes

**519.218.2** Branching and epidemic processes

**519.218.4** Renewal processes

**519.218.5** Point random processes

**519.218.7** Gaussian processes and measures

**519.218.8** Stationary and harmonizable sequences and processes

**519.218.82** Extrapolation. Interpolation. Filtration

**519.218.84** Ergodic theorems

**519.219** Random functions of several variables

**519.22** Statistical theory. Statistical models. Mathematical statistics in general

*References: 311*

**519.221** Foundations of statistical theory

**519.222** Statistical dispersion and relationships. Statistical mean, deviations etc.

**519.223** Sufficiency. Sufficient statistics

**519.224** Distribution theory

**519.226** Inference and decision theory

**519.23** Statistical analysis. Inference methods

**519.233** Parametric methods

**519.233.2** Estimation of parameters and functionals

**519.233.22** Point estimation

**519.233.24** Confidence regions, tolerance, limits

**519.233.3** Hypothesis testing

**519.233.4** Variance analysis. Covariance analysis

**519.233.5** Correlation analysis. Regression analysis

**519.233.6** Ranking and selection

**519.233.8** Paired and multiple comparisons

**519.234** Nonparametric methods

**519.234.7** Order statistics

**519.234.8** Paired comparison methods

**519.235** Statistics of dependent variables. Contingency tables

**519.237** Multivariate statistical methods

**519.237.7** Factor analysis

**519.237.8** Cluster analysis. Classification

**519.24** Special statistical applications and models

**519.242** Experimental design

**519.242.5** Optimal designs

**519.242.6** Block designs

*References: 519.144*

**519.242.7** Factorial designs

**519.243** Sampling. Sampling theory

**519.244** Sequential methods

**519.245** Stochastic approximation. Monte Carlo methods

*References: 536.911*

**519.246** Statistics of stochastic processes

**519.246.8** Time series analysis

**519.248** Engineering statistics. Operational research statistics

**519.25** Statistical data handling

**519.252** Collection of data and planning of surveys

**519.253** Sample surveys

*References: 519.243*

**519.254** Computation of data. Algorithms

**519.255** Presentation of data

**519.256** Preservation of original data. Data banks

**519.257** Use of statistical data

**519.258** Types of statistical data

**519.6** Computational mathematics. Numerical analysis

**519.61** Numerical methods of algebra

**519.612** Numerical methods for the solution of systems of linear alphabetic equations

**519.613** Numerical methods for matrix inversion

**519.614** Numerical methods for computing eigenvalues and eigenvectors of matrices

**519.615** Numerical methods for the solution of transcendental equations and systems of equations

**519.62** Numerical methods for solution of ordinary differential equations

**519.622** Cauchy problems for ordinary differential equations

**519.624** Boundary value problems for ordinary differential equations

**519.63** Numerical methods for solution of partial differential equations

**519.632** Second-order equations of elliptic type

**519.633** Second-order equations of parabolic and hyperbolic type

**519.635** Partial differential equations of higher order (and systems of such)

**519.64** Numerical methods for solution of integral equations. Quadrature formulae

**519.642** Numerical methods for solution of integral equations

**519.644** Quadrature formulae

*References: 517.518.87*

**519.65** Approximation. Interpolation

*References: 511.4*

**519.651** Approximation of functions

**519.652** Interpolation. Extrapolation

**519.653** Numerical differentiation

**519.654** Least squares method

**519.657** Empirical formulae

**519.66** Mathematical tables and their compilation

*References: 51-3*

**519.66.09** Tables according to degree of accuracy

**519.66.094** Four-figure tables

**519.66.095** Five-figure tables

**519.66.099** Nine-figure tables. Tables to more than nine figures

**519.661** General tables. Ready reckoners etc.

**519.662** Logarithm tables. Tables of elementary functions

**519.665** Tables of higher transcendental functions

**519.67** Machine, graphical and other methods of computational mathematics

*References: 51-3*

**519.672** Mechanical methods of calculation

**519.673** Solution of mathematical problems by use of modelling systems

**519.674** Graphical methods of computation

**519.675** Nomography. Nomographic devices

**519.676** Probabilistic methods of computation

**519.677** Solutions of problems of mathematical analysis and applied problems

**519.7** Mathematical cybernetics

**519.71** Control systems theory: mathematical aspects

*References: 303.725.36 , 330.46*

**519.711** General questions of control theory

**519.712** Cybernetic problems of theory of algorithms

*References: 510.5*

**519.713** Automata

*References: 510.586*

**519.714** Synthesis questions in control theory

**519.715** Analysis problems in control theory

**519.716** Functional systems

**519.717** Identity transformations. Transformation rules

**519.718** Stability. Reliability. Checking

**519.72** Information theory: mathematical aspects

*References: 303.725.37 , 330.47*

**519.722** Entropy. Amount of information and its properties

**519.723** Asymptotic theorems of optimal coding. Shannon theory. Channels

**519.724** Computation of information-theory characteristics for specific channels and messages

**519.725** Algebraic theory of error-correcting codes

**519.726** Nonhomogeneous codes for messages

**519.727** Methods of sequential coding

**519.728** Methods of coding in continuous channels

**519.728.2** Quantification of communications

**519.728.4** Gaussian channels

**519.728.6** Channels with fading

**519.76** Semiotics. Mathematical theory of systems of symbols. Mathematical linguistics

*References: 003 , 510.287 , 510.62 , 81`32*

**519.762** Syntactic study of systems of symbols

**519.763** Meaningful interpretation of sets of symbols

**519.764** Deciphering of systems of symbols

**519.765** Mathematical linguistics in general. General mathematical investigations of languages

*References: 510.567*

**519.766** Models for languages and language structures

**519.766.2** Algebrological models. Set-theory models

**519.766.4** Probabilistic-statistical models

**519.767** Semantics of a language: mathematical aspects

*References: 510.27*

**519.768** Mathematics of machine translation

**519.8** Operational research

*References: 005.31 , 303.725.35 , 330.45*

**519.81** Theory of utility and acceptability of solutions

**519.812** General theory of utility

**519.813** Games of chance. Experimental games

**519.814** Theory of statistical solutions

**519.816** Theory of decision-making

**519.83** Theory of games

**519.832** Antagonistic games. Matrix games

**519.833** Noncoalition games

**519.834** Coalition games

**519.837** Dynamic games

*References: 517.978*

**519.85** Mathematical programming

*References: 004.42 , 517.972.8 , 517.977.5 , 519.7*

**519.852** Linear programming

**519.853** Nonlinear programming

**519.854** Discrete programming

**519.855** Parametric programming

**519.856** Stochastic programming

**519.857** Dynamic programming

**519.86** Theory of economic-mathematical models

**519.862** Descriptive models

*References: 51-7*

**519.863** Optimization models

**519.865** Theory of economic behaviour

**519.866** Modelling of specific aspects of the economy

**519.868** Branching models

**519.87** Mathematical models for operational research

**519.872** Queuing theory

**519.873** Theory of reliability and reservation. Quality control

**519.874** Theory of supplies and stock control. Inventory theory

**519.876** Theory of large systems

**519.876.2** Modelling of control processes

**519.876.3** Network planning

**519.876.5** Numerical imitation of systems. Simulation

**519.878** Search theory

**52** Astronomy. Astrophysics. Space research. Geodesy

**52-1/-8** Properties, processes, parts etc.

**52-1** Mode of treatment

**52-12** Initial investigation

**52-121** Unidentified objects

**52-123** Initial hypotheses

**52-125** Predictions

**52-126** Searches

**52-128** Discoveries

**52-13** Observations and measurements. Reduced data

**52-14** Results of analysis. Evaluation of observations and measurements

**52-16** Theoretical treatment and development

**52-17** Numerical treatment. Simulation. Mathematical techniques

**52-3** Properties and phenomena, especially geometrical

**52-32** Position and motion

**52-323** Position

**52-323.2** Angular coordinates

**52-323.3** Distance

**52-323.33** Parallax

**52-323.35** Light-time

**52-323.4** Doppler shift. Radial velocity

**52-323.6** Position in space

**52-323.7** Distribution in space

**52-323.8** Orbital motion. Elements of orbit

**52-325** Motion

**52-325.2** Proper motion

**52-325.4** Radial velocities

**52-327** Rotation

**52-33** Physical properties (except radiation)

**52-332** Dimensions. Shape parameters

**52-332.2** Apparent angular dimensions

**52-332.4** Linear dimensions

**52-332.5** Oblateness parameters

**52-332.6** Irregularities in shape

**52-333** Mass

**52-334** Physical bulk properties

**52-334.2** Mechanical properties

**52-334.4** Viscosity

**52-334.5** Opacity. Refractive index

**52-334.6** Thermal conductivity

**52-334.7** Electrical and magnetic properties

**52-335** Physical state variables

**52-335.3** Density

**52-335.5** Pressure

**52-335.7** Temperature

**52-336** Gravity field

**52-337** Magnetic field

**52-338** Age

**52-34** Distribution of bodies in a system or systems

**52-35** Radiation from a body or system

**52-352** Apparent intensity (intensity as observed). Apparent magnitude. Flux density

**52-353** Absolute intensity (intensity as emitted). Absolute magnitude

**52-355** Spectrum. Spectral type

**52-357** Polarization

**52-36** Chemical composition. Abundances

**52-37** Biological properties

**52-38** Geometrical phenomena

**52-383** Seasonal phenomena

**52-384** Diurnal phenomena

**52-4** Processes of bodies and systems

**52-42** Interactions between bodies within systems

**52-423** Gravitational interactions

**52-423.3** Tidal interactions

*References: 521.16*

**52-423.4** Mass-flow. Exchange of mass

**52-424** Collisions

**52-425** Electromagnetic forces

**52-43** Condensation. Accretion

**52-44** Disintegration. Separation. Mass-loss

**52-46** Bulk-matter processes

**52-462** Conduction

**52-464** Convection

**52-466** Turbulence

**52-468** Shock processes

**52-47** Atomic processes

**52-472** Excitation

**52-473** Ionization

**52-48** High-energy processes

**52-5** Stages in development of bodies and systems

**52-52** Origin. Formation. Cosmogony

*References: 524.85 , 550.2 , 551.12*

**52-54** Evolution. Change of state or structure

**52-55** Stability. Equilibrium

**52-56** Variability

**52-563** Regular variability. Pulsation

**52-565** Irregular variability. Explosion

**52-567** Isolated change

**52-59** Catastrophic change of state or structure. Final stages

**52-6** Processes of radiation

**52-62** Emission

**52-622** Thermal emission

**52-624** Synchrotron radiation

**52-626** Stimulated emission. Laser and maser emission

**52-628** Luminescence

**52-64** Radiative transfer

**52-65** Propagation

**52-652** Reflection

**52-653** Refraction

**52-655** Polarization

**52-656** Interference. Diffraction. Coupling

**52-657** Absorption. Reddening

**52-658** Scintillation

**52-67** Frequency and wavelength shifts. Doppler effect. Aberration

**52-68** Velocity. Speed

**52-683** Acceleration

**52-687** Retardation. Deceleration

**52-7** Character of radiation

**52-72** Corpuscular radiation

**52-724** Protons and atomic nuclei

**52-726** Plasma

**52-728** Neutrinos

**52-73** Very short-wave photons

**52-732** Gamma-rays

**52-735** X-rays (lambda less than 1 nm)

**52-739** Extreme ultraviolet (lambda 1 nm to 50 nm)

**52-74** Ultraviolet radiation (lambda 50 nm to 300 nm)

**52-75** Visible radiation (lambda 0.3 Âµm to 0.7 Âµm)

**52-76** Infrared radiation (lambda 0.7 Âµm to 1 mm)

**52-77** Radio radiation (lambda more than 1 mm)

**52-78** Waves in plasmas

**52-782** Shock waves

**52-8** Parts and features of individual systems

**52-82** Interior. Central region. Head (of comet)

**52-823** Core. Nucleus (of comet)

**52-825** Mantle

**52-827** Crust. Subsurface region

**52-83** Surface. Surface features

**52-834** Solid surface. Lithosphere

**52-836** Oceans. Hydrosphere

**52-837** Photosphere

**52-84** Regional features

**52-842** Polar features

**52-843** Mid-latitude features

**52-844** Equatorial features

**52-846** Apparent disc

**52-847** Apparent edge

**52-848** Terminator

**52-85** Exterior regions

**52-852** Atmosphere

**52-853** Ionosphere

**52-854** Magnetosphere

**52-856** Spiral arms

**52-857** Halo. Extended atmosphere

**52-86** Exterior features. Surroundings

**52-862** Rings and disc features

**52-864** Tails

**52-866** Filaments. Jets. Links

**52-87** Satellites. Companions

**52-88** Peculiar features

**520** Astronomical instruments and techniques

*References: 681.7*

**520.1** Observatories

*References: 727*

**520.2** Astronomical telescopes

*References: 681.751.2*

**520.2.01** Theory and general properties

**520.2.02** Construction

**520.2.03** Adjustment and testing. Instrumental errors

**520.2.06** Optical components

**520.2.07** Mechanical components

**520.22** General-purpose optical telescopes

**520.222** Refractors. Long-focus telescopes

**520.224** Reflectors. Reflecting telescopes in general. Foci

*References: 681.7.062.42*

**520.224.062.42** Conic section mirrors. E.g. parabolic mirrors

*References: 520.224*

**520.24** Solar instruments

**520.242** Coelostats. Heliostats. Siderostats

**520.243** Solar towers

**520.244** Spectroheliographs

**520.245** Monochromators

**520.246** Coronagraphs

**520.248** Solar magnetographs

**520.25** Astrometric instruments

*References: 521*

**520.252** Transit instruments

**520.253** Meridian circles

**520.254** Vertical circles

**520.255** Zenith tubes (PZT, VZT, LZT etc.)

**520.256** Modern astrolabes. Impersonal astrolabes

**520.257** Astrographs. Astrographic cameras

**520.259** Other astrometric instruments

**520.27** Radio telescopes

**520.272** Simple telescopes

**520.272.2** Antennas

**520.272.4** Transmission systems

**520.272.5** Receivers

**520.274** Interferometer-type radio telescopes

**520.3** Auxiliary instruments. Recording instruments

*References: 621.397 , 681.782.4*

**520.32** Visual equipment

**520.322** Oculars. Eyepieces

**520.328** Micrometers

**520.33** Photographic instruments and equipment

**520.33:771.319.5** Astronomical cameras

*References: 520.33*

**520.34** Photoelectric instruments and equipment

**520.35** Spectroscopic instruments

**520.36** Interferometers

**520.37** Recording and timing instruments used when making observations

**520.373** Analogue recorders

**520.374** Digital recorders

**520.378** Timing equipment

**520.4** Measuring instruments, normally separate from telescopes

**520.42** Coordinate measuring instruments

**520.43** Densitometers. Microphotometers

**520.44** Spectrophotometers

**520.47** Blink comparators

**520.48** Automatic measuring instruments

**520.6** Instruments for specific environments

**520.6.02** Aeroplane-borne instruments

**520.6.03** Balloon-borne instruments

**520.6.04** Rocket-borne instruments

**520.6.05** Satellite-borne instruments

**520.6.07** Space-probe instruments

**520.6.08** Instruments for observations from the Moon and other celestial bodies

**520.62** Electromagnetic wave detectors

**520.64** Plasma analysers (in situ)

**520.66** Suprathermal particle detectors

**520.68** Dust detectors

**520.8** Techniques of observation, measurement, analysis

**520.8.07** Calibration

**520.82** Photometry

*References: 535.24 , 681.782.43*

**520.84** Spectrometry

*References: 543.42 , 681.782.46*

**520.849** Classification systems

**520.849.07** Standard stars

*References: 520.849*

**520.85** Polarimetry

*References: 543.454 , 681.785.3*

**520.86** Observation techniques for eclipses, transits, occultations

**520.87** Special techniques

**520.872** Interferometry

**520.874** Ranging techniques

**520.874:621.375.82** Ranging by laser/maser

*References: 520.874*

**520.874:621.396.96** Ranging by radar

*References: 520.874*

**520.876** Techniques used for non-ground-based observations

**520.9** Other instruments or techniques

**520.98** Planetariums

**520.99** Other simulators

**521** Theoretical astronomy. Celestial mechanics

**521.1** Celestial mechanics. General principles of dynamical astronomy

**521.13** General problems

**521.131** General n-body problem

**521.135** Restricted 3-body problem

**521.14** Theories of figures, and gravity fields and rotation of extended bodies

**521.16** Theories of tidal interaction and other dissipative forces

**521.17** Special problems

**521.172** Planetary theory (mutual perturbations important)

**521.174** Lunar theory (perturbations by distant third body important)

**521.176** Close-satellite theory (oblateness of primary important)

**521.18** Special techniques

*References: 521.9*

**521.181** Coordinate reference systems

**521.182** Numerical integration of orbits

**521.184** Integration of variations of orbit elements or parameters

**521.186** Averaging methods for long-term orbits

**521.19** Special techniques for development of analytical theories

**521.19:517.952** Using first-order partial differential equations

*References: 521.19*

**521.3** Orbit determination

**521.31** Specification of orbits. Orbital elements and parameters

**521.32** Initial orbit determination

**521.35** Orbit improvement

**521.8** Astrometric aspects of eclipses, transits, occultations

**521.81** Eclipses of the Sun and Moon

**521.83** Satellite and ring phenomena

**521.832** Eclipses of a satellite by its planet

*References: 52-87*

**521.832:523.45-87*1** Eclipses of Io by Jupiter

*References: 521.832*

**521.834** Transits and occultations of satellites over and by planets

**521.835** Mutual phenomena of satellites

**521.837** Ring phenomena

**521.84** Transits of planets across the Sun

**521.85** Occultations by the Moon

**521.86** Occultations by planets and their satellites, and by minor planets

**521.87** Occultations by comets

**521.89** Computation of rising, setting and local transit phenomena

**521.9** Astrometry. Spherical astronomy

**521.91** Coordinate and time systems in general

**521.92** Theory, determination and constants of precession and nutation

**521.93** Theory and determination of Earth rotation and polar motion

**521.932** Determination of time by astronomical observations

**521.932:521.84** Determination by transit

*References: 521.932*

**521.932:528.37** Determination by altitudes

*References: 521.932*

**521.932:681.786.3** Determination by sextant

*References: 521.932*

**521.933** Irregularity of the Earth's rotation

**521.934** Timescales. Systems of time reckoning

**521.936** Latitude determination from astronomical observations

**521.937** Movement of terrestrial poles. Variations of latitude and longitude

**521.94** System of astronomical constants

**521.95** Reduction of astrometric observations

**521.96** Fundamental position and proper-motion systems

**521.97** Observational (instrumental) corrections

**521.98** Calculation of ephemerides

**523** The Solar System

**523-52** Origin, formation of the Solar System

*References: 523*

**523.2** General properties of the Solar System

**523.21** Laws of planetary distribution

**523.23** Conjunctions and oppositions

**523.24** Motion of the Solar System in space

**523.3** Earth-Moon system

**523.31** The Earth as an astronomical body

**523.31-121** Unidentified objects in the vicinity of the Earth. Near-Earth objects (NEOs)

*References: 523.31*

**523.34** Moon. Selenology

**523.4** Planets and their satellites. Planetology

**523.4-87** Satellites (of the Solar System planets)

*References: 523.4*

**523.41/.43** Terrestrial, interior planets

**523.41** Mercury

**523.42** Venus

**523.43** Mars

**523.44** Minor planets (Planetoids. Asteroids)

**523.45/.48** Outer planets

**523.45** Jupiter

**523.46** Saturn

**523.47** Uranus

**523.481** Neptune

**523.482** Pluto

**523.489** Hypothetical planets

**523.6** Interplanetary medium. Comets. Meteors. Meteorites

**523.62** Interplanetary medium

**523.62-65** Zodiacal light

*References: 523.62*

**523.62-726** Interplanetary plasma. Solar wind

*References: 523.62*

**523.64** Comets

**523.68** Meteors. Meteoroids. Meteorites

**523.681** Meteorites. Meteor remnants on Earth's surface

*References: 552.6*

**523.682** Meteors. Meteoroids. Shooting stars

**523.683** Meteor showers

**523.684** Tektites. Australites. Moldavites

*References: 549.691.1 , 552.64*

**523.9** The Sun. Solar physics

**523.9:520.876** Observations of the Sun from space

*References: 523.9*

**523.92** Solar interior

**523.94** Quiet Sun

**523.942** Photosphere

**523.942.3** Granulation

**523.945** Chromosphere

**523.947** Corona

**523.947:520.86** Observations of the corona during eclipses

*References: 523.947*

**523.947-335.7** Temperature of the corona

*References: 523.947*

**523.947-355** Spectrum of the corona

*References: 523.947*

**523.98** Active Sun. Solar activity

**523.982** Sunspots

**523.983** Faculae. Plages

**523.985** Flares. Bursts

**523.987** Prominences

**523.988** Streamers

**524** Stars. Stellar systems. The Universe

**524-86** Circumstellar features. Including: planets (other than those of the Solar System). Extrasolar planets

*References: 524*

**524.1** Cosmic rays. Primary cosmic rays

**524.3** Stars

**524.31** Simple stars. Stars according to luminosity and spectral type

**524.31*1/*5** Stars according to luminosity class

*References: 524.31*

**524.31*1** Supergiant stars. Class I stars

*References: 524.31*

**524.31*2/*3** Giant stars. Class II and III stars

*References: 524.31*

**524.31*4** Subgiant stars. Class IV stars

*References: 524.31*

**524.31*5** Dwarf and subdwarf stars. Class V stars

*References: 524.31*

**524.312/.318** Stars according to spectral type

**524.312** Early-type stars

**524.314** A stars. F stars

**524.316** Late-type stars

**524.318** Other (non-main-sequence) classes of star

**524.33** Variable stars

**524.331** Pulsating variables in general

**524.332** RR Lyrae stars

**524.333** Cepheids

**524.334** Semiregular variables

**524.335** Miscellaneous variables

**524.336** Long-period variables

**524.337** Novae. Explosive and eruptive variables

**524.338** Special variable stars

**524.338.2** Shell stars. Envelope stars

*References: 524.37*

**524.338.3** Symbiotic variables. Symbiotic stars

**524.338.5** Nebular variables. T Tau stars

**524.338.6** Flare stars

**524.338.7** R Coronae Borealis stars

**524.35** Supernovae and related objects. Peculiar stars

*References: 52-121*

**524.35-76-121** Unidentified infrared objects believed to lie in the Galaxy

*References: 524.35*

**524.352** Supernovae

**524.354** Objects believed to be supernovae remnants

**524.354-423** Gravitational collapse of supernovae remnants

*References: 524.354*

**524.354.2** Crab nebula

**524.354.2-62** Emission mechanism of Crab nebula

*References: 524.354.2*

**524.354.2-735** X-ray radiation from Crab nebula

*References: 524.354.2*

**524.354.4** Pulsars

**524.354.4-327** Rotation of pulsars

*References: 524.354.4*

**524.354.4-563** Pulsation processes of pulsars

*References: 524.354.4*

**524.354.6** Neutron stars

**524.354.6-423** Gravitational collapse of neutron stars

*References: 524.354.6*

**524.354.7** White dwarf stars (as supernovae remnants)

**524.355** Other objects believed to be disintegrating or collapsing

**524.358** Wolf-Rayet stars

**524.37** Planetary nebulae

**524.38** Double and multiple stars. Binary systems

**524.382** Double stars (optical or visual doubles)

**524.383** Visual binary systems

**524.384** Spectroscopic binary systems

**524.386** Eclipsing binary systems

**524.387** Close binary systems. Very close binary systems

**524.388** Multiple stars. Multiple systems

**524.4** Star clusters. Associations of stars

**524.42** Associations of stars

**524.45** Open clusters. Galactic clusters

**524.47** Globular clusters

**524.5** Interstellar medium. Galactic nebulae

**524.52** Interstellar gas

**524.522** H I regions

**524.523** H II regions

**524.527** Interstellar molecules

**524.54** Emission nebulae

**524.57** Interstellar dust

**524.6** The Galaxy (Milky Way)

**524.62** Structural parameters of the Galaxy

**524.63** Local region of the Galaxy. Solar neighbourhood

**524.64** Galactic centre. Nucleus of the Galaxy

**524.66** Spiral arms of the Galaxy

**524.68** Halo of the Galaxy

**524.7** Extragalactic systems

**524.7-323.3** Distance of system

*References: 524.7*

**524.7-327** Rotation of system

*References: 524.7*

**524.7-52** Origin, formation of systems

*References: 524.7*

**524.72** Local Group of galaxies

**524.722** Magellanic Clouds

**524.724** Irregular galaxies in Local Group

**524.726** Spiral galaxies in Local Group

**524.728** Elliptical galaxies in Local Group

**524.74** Types of galaxy

**524.77** Clusters and systems of galaxies

**524.78** Intergalactic matter

**524.8** The Universe. Metagalaxy. Cosmology

**524.82** Observational cosmology

**524.822** Red-shift relation. Hubble constant

**524.827** Sky-background radiation

**524.83** Fundamental theories of cosmology

**524.85** Origin and formation of the Universe

*References: 52-52*

**524.852** Big-bang theories

**524.854** Steady-state theories

**524.86** Origin and formation of the elements

**524.88** Special hypotheses

**524.882** Black holes

**524.883** Quasi-stellar objects (QSOs). Quasars

**524.884** Antimatter

**528** Geodesy. Surveying. Photogrammetry. Remote sensing. Cartography

**528-1/-7** Details of instruments, devices etc.

**528.01** Preparatory work

**528.02** Geodetic measuring methods

*References: 53.08*

**528.021** Methods of distance measurement

**528.021.1** Mechanical methods of distance measurement

**528.021.4** Optical and trigonometric methods of distance measurement

**528.021.6/.7** Methods of distance measurement by measuring the transit time of electromagnetic waves

**528.021.6** Electric methods of distance measurement

*References: 621.396.96*

**528.021.7** Electro-optical methods of distance measurement

**528.022** Methods of angular and direction measurement

**528.022.6** Measurement of oriented directions

**528.023** Methods for plotting position

**528.024** Levelling methods

**528.026** Gravity measurement

**528.029.4/.6** Frequency ranges and wavelengths of electromagnetic oscillations

**528.029.67/.69** Light rays. Gamma-rays. Cosmic rays

**528.029.67** Light rays

**528.029.68** Gamma-rays

**528.029.69** Cosmic rays

**528.03** Auxiliary measurement

**528.031** Determination of points

**528.033** Simple auxiliary figures

**528.034** Plumbing down

**528.037** Centring measurements

**528.06** Evaluation of measuring results

**528.061** Correction and reduction of measuring results

**528.063** Geodetic computations

**528.065** Textual representation of survey results

**528.066** Representation of survey results in numerical tables. Tables of coordinates. Trig lists. Station summaries

**528.067** Graphical representation of survey results

**528.08** General principles and theories of measurement and of construction of measuring instruments

**528.1** Theory of errors and adjustment in geodesy and photogrammetry

*References: 519.233*

**528.11** Theory of errors in observation

**528.13** Preparation of observation programmes respecting theory of errors

**528.14** Adjustment by least squares method

**528.16** Other methods of adjustment

**528.162** Adjustment by successive approximation

**528.165** Graphical adjustment

**528.181** Treatment of adjustment problems by means of matrix and tensor calculus

**528.2** Figure of the Earth. Earth measurement. Mathematical geodesy. Physical geodesy. Astronomical geodesy

*References: 52-3*

**528.21** Fundamentals derived from potential theory. Level surfaces. Geoids

**528.22** Methods of determining the figure of the Earth

**528.221** Geometric (static) methods. Use of longitudinal and latitudinal measurements

**528.223** Physical methods. Dynamic methods

**528.225** Astronomical methods

**528.23** Mathematical surfaces approaching the shape of the Earth. Geodetic projections. Coordinate systems

**528.231** Spheres

**528.232** Ellipsoids of revolution

**528.232.1** Reference ellipsoids and their basic values

**528.232.2** Computations on the ellipsoid

**528.232.22** Normal sections. Geodetic lines

**528.232.23** The geodetic triangle. Excess (spherical excess). Area. Transformation into the plane

**528.232.24** The main problem of geodesy. Computation of geographical coordinates when azimuth and side are given. Inversion of the problem

**528.233** Triaxial ellipsoids

**528.234** Level spheroid

**528.235** Representations of approximate surfaces. Geodetic projections. Map projections. Graticule design. Grids

**528.236** Coordinate systems. Conversion of coordinates

**528.24** Relations between the shape of the Earth and its approximate surfaces

**528.241** Deflections of the vertical

**528.242** Geoid undulations

**528.247** Gravitational anomalies

**528.26** Arc measurement

**528.27** Gravity measurement

*References: 530.122*

**528.28** Astro-geodetic determination of position. Geographical coordinates

**528.28.087.2** Astronomic-optical observation

*References: 528.28*

**528.28.087.5** Astronomic-photographic position recording

*References: 528.28*

**528.281** Determination of latitude. Observation of polar altitudes

**528.282** Determination of longitude

**528.283** Determination of azimuth

**528.288** Astronomical orientation. Astronomical connection

**528.3** Geodetic surveying

**528.31/.35** Planimetric determination of stations

**528.31** Outline and structure of the geodetic survey

**528.32** Base-line measurement

**528.33** Trigonometric networks

**528.331** Primary triangulation. First-order triangulation. First-order astro-geodetic networks

**528.335** Filling-in of network. Triangulation of second and lower orders

**528.338** Planimetric stations. Trigonometric stations. Traverse stations

**528.34** Special methods of trigonometric determination of stations

**528.341** Flare triangulation

**528.342** Solar eclipse method

**528.343** Star occultation method

**528.344** Stellar triangulation

**528.35** Trilateration

**528.37/.38** Vertical determination of stations. Levelling

**528.37** Kinds of elevation and principles of levelling

**528.371** Datum planes. Reference points. Gauges

**528.38** Figuration and observation of level network

**528.381** Primary level network. First-order level network

**528.385** Filling-in of level networks. Level networks of second and lower orders

**528.388** Bench marks. Levelling stations

**528.389** Special applications of levelling and depth measurement

**528.4** Field surveying. Land surveying. Cadastral survey. Topography. Engineering survey. Special fields of surveying

**528.41** Local geodetic networks

**528.411** Local triangulation networks. Detail triangulation

**528.412** Intercalation of single points. Determination of points by intersection

**528.414** Traversing. Supplementary points

*References: 528.335*

**528.42** Topographic surveying

**528.421** Planimetric surveying

**528.422** Levelling

*References: 625.725*

**528.425** Tacheometry. Topography

**528.44** Cadastral survey

**528.45** Urban surveying

**528.46** Surveying for soil improvement

**528.47** Hydrographic surveying. Coast surveying

**528.48** Engineering survey. Special fields of surveying

**528.48:624.19** Surveying for tunnelling

*References: 528.48*

**528.481** Observations of local ground displacements

**528.482** Observations of structures. Geodetic operations during erection and operation of structures. Observations of settling. Deformation measurement

*References: 622.83 , 624.131.5*

**528.484** Connection in underground survey

**528.486** Staking-out operations. Transfer of construction plan into terrain. Tracing

**528.5** Geodetic instruments and equipment

*References: 531.7 , 532.29 , 681.2 , 681.783*

**528.5-187** Accuracy

*References: 528.5*

**528.5-217** Ground plates. Base plates

*References: 528.5*

**528.5-539** Precision regulation

*References: 528.5*

**528.5-752.3** Stabilizing devices

*References: 528.5*

**528.51** Instruments and equipment for distance measurement

*References: 531.71*

**528.511** Base apparatus

**528.512** Geodetic equipment for length measurement, profiles etc.

**528.514** Optical instruments for distance measurement

**528.516** Electric instruments for distance measurement

**528.517** Electro-optical instruments for distance measurement

**528.519** Accessories for distance measurement

**528.52** Instruments and equipment for angular measurement, direction measurement, staking-out

**528.521** Theodolites and accessories

**528.526** Instruments for measurement of oriented directions

**528.526.1** Magnetic instruments

**528.526.6** Gyroscopic instruments. Surveying gyroscopes

**528.528** Devices for staking-out angles and directions

**528.53** Instruments and equipment for simultaneous measurement of angles and distances

**528.531** Tacheometer (tachymeter) theodolites and accessories

**528.533** Plane tables

**528.534** Alidades

**528.535** Topometers

**528.54** Instruments and equipment for measurement of elevations, depths, inclinations

**528.541** Instruments and equipment for geometric levelling and inclination measurement

**528.541.2** Automatic levels

**528.541.4** Simple inclinometers, clinometers, gradient meters

**528.543** Automatic height indicators

**528.546** Instruments and devices for hydrostatic levelling

**528.548** Direction-finding devices. Plumbing devices

**528.56** Instruments and equipment for gravity measurement

**528.561** Pendulum instruments

**528.563** Gravimeters

**528.565** Torsion balances. Horizontal pendulums

**528.58** Devices for marking-out. Signalling equipment. Devices for finding of marked points

**528.59** Plotting instruments and equipment

*References: 744.3*

**528.7** Photogrammetry: aerial, terrestrial

*References: 355.404.4 , 550.81 , 681.783.3 , 778.35*

**528.71** Production of photograms

**528.711** Methods of production of photograms

**528.711.1** Photographic production of photograms in central projection

*References: 528.835 , 771.319.5*

**528.714** Exterior influences on image formation

**528.715** Photogrammetric survey flight

*References: 528.837*

**528.715:629.735** Photogrammetric survey aircraft

*References: 528.715*

**528.716** Determination of camera orientation elements at the exposure station

**528.716.1** Planimetry. Determination of position

**528.716.2** Altitude and altitude differences. Altimetry

**528.716.3** Deviation of camera axis from the vertical (verticality). Tilt

**528.716.4** Swing. Azimuth curvature of the flight line

**528.718** Practical procedures in production of terrestrial photograms

**528.72** Restitution of photograms

*References: 514.14*

**528.72:528.236** Photogrammetric coordinate systems

*References: 528.72*

**528.721** Orientation and restitution of photograms in photographic central projection

**528.721.1** Orientation and restitution of single photograms

**528.721.15** Compilation of mosaics

**528.721.16** Compilation of maps by tracing of photograms

**528.721.2** Orientation and restitution of pairs of photograms

**528.721.28** Stereophotograms and stereomosaics

**528.722** Instruments for restitution of photograms in photographic central projection

**528.722.2** Transformation printers

**528.722.3** Restitution instruments for single photograms

**528.722.4** Orthophoto instruments

**528.722.6** Stereorestitution instruments

**528.722.8** Instruments for measurement of plate coordinates and angles

**528.73** Photogrammetric aerial triangulation. Determination of control points and pass points

**528.731** Radial triangulation

**528.732** Other methods of planimetric control from photograms

**528.735** Spatial aerial triangulation

**528.737** Control points. Transfer points

**528.738** Terrestrial methods of ground control determination

**528.74** Applications of photogrammetry

**528.74:52** Applications in astronomy

*References: 528.74*

**528.74:528.93** Application to topographic cartography

*References: 528.74*

**528.77** Interpretation of photograms

*References: 528.873*

**528.8** Remote sensing

**528.8.04** Remote sensing techniques

**528.8.041** Photographic central projection techniques

*References: 528.711.1*

**528.8.041.3** Infrared photography

**528.8.041.5** Multispectral photography

**528.8.041.8** Television techniques

**528.8.042** Scanning techniques

**528.8.044** Active techniques

**528.8.044.1** Side looking real aperture radar (SLAR)

**528.8.044.2** Side looking synthetic aperture radar (SAR)

**528.8.044.4** Scatterometer techniques

**528.8.044.6** Light detection and ranging (LIDAR)

**528.8.044.8** Airborne profile recording (APR)

**528.81** Fundamental and physical principles

**528.811** Nature of electromagnetic radiation

**528.813** Interaction mechanisms

*References: 537.87*

**528.83** Data acquisition

**528.831** Resolution

**528.831.1** Spatial resolution

**528.831.2** Spectral resolution

**528.831.3** Intensity resolution

**528.831.4** Temporal resolution

**528.831.5** Temperature resolution

**528.835** Remote sensors and their use

*References: 528.711.1 , 778.35 , 53.089.6*

**528.835.042.3.089.6** Calibration of multispectral scanners

*References: 528.835*

**528.837** Remote sensing platforms

*References: 528.715*

**528.837:629.73** Aircraft used for remote sensing

*References: 528.837*

**528.837:629.783** Artificial satellites used for remote sensing

*References: 528.837*

**528.85** Data processing

**528.851** On-board processing. Data transmission

**528.852** Image processing

**528.854** Image analysis. Pattern recognition

*References: 519.237.8*

**528.855** Ground investigations. Test sites

**528.856** Geometric transformations. Mapping

**528.87** Interpretation

**528.871** Elements of image interpretation

**528.873** Interpretation of imagery

**528.873.044.1** Interpretation of SLAR imagery

*References: 528.873*

**528.875** Instruments for image interpretation

**528.88** Applications

*References: 632.2*

**528.88:551.4** Application of remote sensing to geomorphology

*References: 528.88*

**528.88.041.3:632.2** Detection of plant injuries by means of infrared photography

*References: 528.88*

**528.9** Cartography. Mapping (textual documents)

*References: 912*

**528.91** Theoretical cartography. Methodology of cartography

**528.912** Science and study of mapping generally

**528.913** Principles of map design. Map editing

*References: 528.235*

**528.914** Generalization

**528.915** Scales. Series of scales. Choice of scale

**528.916** Neat-line systems. Designation of map sheets. Sheet size

**528.918** Marginal data. Key to symbols. Legend

**528.92** Practical cartography: processes, operations

**528.921** Execution of map design

**528.923** Cartographic drawing techniques

*References: 744.43*

**528.924** Cartographic engraving with and without coating

*References: 681.94*

**528.925** Mechanical auxiliary methods for map plotting

**528.926** Cartographic colour separation methods

**528.927** Cartographic reproduction

*References: 655.3.026*

**528.93** Topographic cartography. Chorographic cartography (landscape cartography). Representation of place and space

*References: 003.62*

**528.931** Representation of planimetric detail

**528.932** Representation of elevation. Relief mapping

**528.933** Colouring

**528.935** Map lettering

**528.936** Map correction, revision, updating

**528.936:528.4** Topographic report service

*References: 528.936*

**528.94** Thematic cartography. Topical cartography

**528.94:551.4** Geomorphological cartography

*References: 528.94*

**528.951** Preparation of globes

**528.952** Preparation of reliefs (three-dimensional)

**528.97** Style-sheets. Mapping specifications

**53** PhysicsPhysik

**53.01** Theory and nature of phenomena

**53.02** General laws of phenomena

**53.023** Empirical formulae

**53.03** Production and causes of phenomena

**53.04** Effects of phenomena

**53.043** Physical effects

**53.044** Chemical effects

**53.047** Biological effects

**53.05** Observation and recording of phenomena. Visual indication of phenomena

**53.06** Application, utilization of phenomena

**53.07** Apparatus for production and study of phenomena

**53.072** Models

**53.08** General principles and theory of measurement and the design of measuring apparatus. Measurement methods

**53.081** Units. Constants

**53.081.3** Absolute units

**53.081.4** Relative and practical units

**53.081.5** Dimensions of units

**53.081.6** Physical constants

**53.081.7** Constants for specific substances

**53.082** Principles of measurement and of construction of instruments

**53.082.1** Use of mechanical phenomena

**53.082.12** Use of weight, static forces

**53.082.13** Use of dynamic phenomena

**53.082.14** Use of pendulums, balance wheels

**53.082.15** Use of centrifugal force

**53.082.16** Use of gyroscopic devices

**53.082.17** Use of elasticity, torsion

**53.082.18** Use of friction

**53.082.2** Use of hydrostatic and hydrodynamic phenomena

**53.082.22** Use of hydrostatic phenomena

**53.082.25** Use of hydrodynamic phenomena

**53.082.3** Use of aerostatic and aerodynamic phenomena

**53.082.32** Use of aerostatic phenomena

**53.082.36** Use of aerodynamic phenomena

**53.082.4** Use of acoustic phenomena

**53.082.5** Use of optical phenomena

**53.082.52** Use of photoelectric phenomena. Optoelectronic measurements

**53.082.53** Use of reflection, refraction, absorption

**53.082.54** Use of interference

**53.082.55** Use of polarization, double refraction (birefringence), dispersion in anisotropic substances

**53.082.56** Use of colour

**53.082.6** Use of thermal phenomena

**53.082.7** Use of electrical, magnetic, nuclear phenomena

**53.082.72** Use of electrostatic phenomena, electrostatic force, capacitance

**53.082.73** Use of electrokinetic phenomena

**53.082.74** Use of electromagnetic, electrodynamic phenomena

**53.082.75** Use of electrolytic, electrochemical phenomena

**53.082.76** Use of perturbation phenomena

**53.082.77** Use of discharge phenomena

**53.082.78** Use of magnetic phenomena

**53.082.79** Use of nuclear phenomena

**53.082.8** Use of chemical phenomena

**53.082.9** Use of biological phenomena

**53.083** Methods of measurement. Instruments according to measurement method

**53.083.1** Direct measurements

**53.083.2** Resonance methods

**53.083.3** Ballistic methods

**53.083.4** Null methods. Bridge methods

**53.083.5** Compensation methods

**53.083.6** Differential methods

**53.083.62** Using devices for measuring and indicating small differences

**53.083.64** Using devices for measuring and indicating maxima and minima

**53.083.7** Remote measurement. Telemetering

**53.083.8** Summation methods. Integral measurement

**53.083.98** Micromeasurement methods

**53.084** Construction of instruments. Primary components

**53.084.2** Receivers. Parts in contact with phenomena measured

**53.084.4** Taring. Tare pans

**53.084.6** Devices for magnification, amplification, of measured quantity

**53.084.8** Supply, storage, transmission, discharge of energy or matter

**53.084.81** Containers for powders, granular substances

**53.084.82** Liquids: supply, reservoirs, conveyance, discharge

**53.084.83** Gases: supply, reservoirs, conveyance, discharge

**53.084.84** Sound waves: sources, distribution, reflection, damping

**53.084.85** Light: sources, distribution, reflection, extinction

**53.084.86** Heat: production, distribution, reflection, cooling

**53.084.87** Electricity: supply, distribution, regulation, discharge

**53.084.88** Magnetism

**53.084.89** Radioactive substances. Ionized substances

**53.085** Indicating apparatus

**53.085.1** Braking. Damping

**53.085.2** Zero point. Zero adjustment. Adjusting devices

**53.085.3** Indicating devices

**53.085.31** Mechanical indicating devices

**53.085.32** Indicators

**53.085.33** Optical devices for reading scales. Magnifiers

**53.085.34** Indicating devices using light beam

**53.085.35** Optical devices for reading liquid levels, menisci

**53.085.39** Accessories

**53.085.4** Scales

**53.085.41** Rectilinear scales

**53.085.42** Circular scales. Curved scales. Cylindrical scales

**53.085.49** Illumination of scales

**53.085.5** Springs

*References: 62-272*

**53.085.6** Casings. Housings

*References: 62-213*

**53.086** Microscopic investigation

**53.087** Observation. Recording

**53.087.2** Subjective observation

**53.087.21** Observation at normal distance

**53.087.22** Observation at small distances. Microscopic observation

**53.087.23** Observation at great distances. Telescopic observation

**53.087.24** Stereoscopic observation

**53.087.25** Observation of variable phenomena

**53.087.251** Cinematographic observation

**53.087.252** Stroboscopic observation

**53.087.3** Apparatus for subjective observation

**53.087.4** Objective observation. Automatic recording

**53.087.42** Recording of maxima and minima

**53.087.47** Intermittent recording

**53.087.5** Photographic recording

**53.087.6** Graphic recording

**53.087.7** Coded recording

**53.087.9** Devices for converting, calculating, counting, printing out

**53.088** Errors. Error correction. Evaluation, interpretation of measurements

**53.088.2** Errors

**53.088.21** Errors due to observer

**53.088.22** Errors due to instrument

**53.088.23** Errors due to object of measurement

**53.088.24** Errors due to transmission medium

**53.088.3** Mean errors. Accuracy of measurement. Sensitivity. Correction of nonlinear inaccuracies by calibration curves

**53.088.4** Limits of use of measuring instruments

**53.088.6** Error correction

**53.088.7** Automatic error correction. Automatic compensation

**53.089** Various questions of measurement

**53.089.2** Coupling of instruments with each other and with other apparatus

**53.089.5** Characteristic accuracy of instruments

**53.089.6** Calibration and testing of instruments

**53.089.62** Calibration equipment, accessories

**53.089.68** Standard test objects, masters, gauges for testing and calibration of instruments

**53.09** Dependence of phenomena on certain fundamental physical effects

**53.091** Mechanical effects (except pressure)

**53.092** Pressure

**53.093** Humidity

**53.096** Temperature

**53.097** Electrical effects

**53.098** Magnetic effects

**530** Theoretical foundations of physics

**530.12** Relativity theory

**530.121** Special relativity

**530.122** General relativity. Gravitation. Space-time

**530.122.1** Gravitational waves

**530.122.2** Space-time. Curved space-time

*References: 530.2*

**530.122.3** Quantum gravity

**530.122.4** Kaluza-Klein type theories

*References: 539.120.4*

**530.122.5** Supergravity

**530.13** Symmetry and conservation law

**530.145** Quantum theory. Quantum mechanics

*References: 535.14*

**530.145.6** Wave mechanics. Corpuscular waves. Matrices

**530.145.61** Wave mechanics

**530.145.63** Matrices. q-numbers

**530.145.65** Corpuscular waves

**530.145.7** Perturbation calculation

**530.145.81** Mathematical methods of quantum mechanics

**530.145.811** Path integral

**530.145.82** Quantum theory of measurements

**530.145.83** Many-particle quantum systems

**530.145.84** Quantum theory of scattering

**530.145.85** Relativistic quantum mechanics

**530.145.86** Coherent states of quantum systems. Squeezed states. Correlated states

**530.182** Nonlinear phenomena

*References: 535.18 , 532.517.4 , 533.951 , 534.182*

**530.182.1** Solitons

**530.182.2** Chaos

**530.191** Fractals

**530.192** Synergetics

**530.2** Field theory

*References: 539.120*

**530.21** Classical field theories

*References: 539.120*

**530.22** Quantum field theories

*References: 539.120*

**530.225** Vacuum

*References: 539.120*

**530.23** Renormalization theory

*References: 539.120*

**530.24** Gauge field theories

*References: 539.120*

**531/534** Mechanics

**531** General mechanics. Mechanics of solid and rigid bodies

**531-1** One-dimensional. Lines

**531-2** Two-dimensional. Planes. Surfaces

**531-3** Three-dimensional. Bodies. Spaces

**531-4** Hyperspace (space of more than three dimensions)

**531-9** Non-Euclidean spaces

**531.01** Mathematical theory of mechanics

**531.1** Kinematics. Mathematical-mechanical geometry of motion

**531.11** Principles. Definitions

**531.111** Dimensions. Space and time

**531.111.4** Continuum of space and time

**531.111.5** Symmetry of space and time

**531.111.9** Kinematics in continuous and discontinuous space

**531.112** Velocity

**531.113** Acceleration

**531.114** Accelerations of high degree

**531.115** Initial conditions

**531.12** Motion of points

**531.13** Motion of extended figures

**531.131** Motion of one-dimensional figures

**531.132** Motion of two-dimensional figures. Moving planes

**531.132.1** Motion of two-dimensional rigid figures

**531.132.2** Motion of two-dimensional deformable figures

**531.133** Motion of three-dimensional figures

**531.133.1** Motion of three-dimensional rigid figures

**531.133.2** Motion of three-dimensional deformable figures

**531.133.3** Motion of three-dimensional deformable systems composed of rigid elements. Articulated systems

**531.134** Motion of multidimensional figures (more than three dimensions)

**531.14** Translation. Sliding

**531.15** Rotation. Angular velocity. Angular acceleration

**531.16** Free motion. Composition of translations and rotations

**531.17** Guided motion. Constrained motion

**531.2** Statics. Forces. Equilibrium. Attraction

**531.21** Statics in general

**531.211** Forces

**531.213** Principle of virtual displacement. Laws of Clapeyron, Maxwell, Bett, Castigliano

**531.215** Geometry of masses in general

**531.216** Geometry of segments in general

**531.22** Composition and resolution of forces. Stresses

**531.221** Equilibrium. Parallelogram of forces. Equilibrium of external and internal forces. Parallel and nonparallel forces. Funicular polygons. Force polygons. Catenaries

**531.222** Tensions

**531.223** Forces of compression

**531.224** Bending stresses

**531.225** Torsional stresses

**531.226** Shear stresses

**531.228** Secondary stresses

**531.23** Moments

**531.231** Determination of moments of inertia. Axes of inertia. Centrifugal moments. Property of beam section

**531.232** Torque. Turning moments

**531.233** Moments of various orders with reference to a plane

**531.234** Static moment. Moment of a force

**531.235** Properties of the product of inertia Ã¤mxy and analogous expressions

**531.24** Centre of gravity

**531.24-1** Centre of gravity of lines

*References: 531.24*

**531.24-2** Centre of gravity of planes

*References: 531.24*

**531.24-3** Centre of gravity of volumes

*References: 531.24*

**531.25** Graphical statics. Analytical statics

**531.252** Specific stresses

**531.255** One-dimensional bodies

**531.258** Plane bodies. Two-dimensional bodies

**531.259** Three-dimensional bodies

**531.259.1** Solid bodies

**531.259.2** Hollow bodies

**531.26** Attraction. Laws and applications of potential. Specific solutions of integration problems

*References: 530.122*

**531.261** Newtonian attraction. Potential. Equipotential. Equipotential lines and surfaces

**531.261.1** Attraction of lines

**531.261.2** Attraction of surfaces, especially single shells

**531.261.3** Attraction of homogeneous bodies, volumes

**531.261.4** Attraction of nonhomogeneous bodies

**531.261.5** Attraction of bodies with irregular boundaries

**531.262** Solution of Neumann's problem, Dirichlet's problem

**531.266** Various generalizations of attraction and potential

**531.3** Dynamics. Kinetics

**531.31** Fundamental principles of dynamics and kinetics

**531.311** Relationships between force, mass and acceleration

**531.312** D'Alembert's principle and derived principles

**531.313** Principle of virtual velocities

**531.314** Principle of least action. Hamilton's principle

**531.315** Principle of least constraint

**531.32** Generalities on the motion of a point mass

**531.33** Problems of tautochronism. Brachistochronism. Isochronism

**531.35** Motion under the action of central forces. Central forces

**531.351** Centripetal forces. Centrifugal forces

**531.352** Motion about a centre of gravity

**531.353** Motion under particular laws of force. Investigation of forces causing motion satisfying certain given conditions

**531.355** Motion with reference to resistance of the medium

**531.36** Generalities on motion of a solid body or of a system of rigid solid bodies. Small motions. Stability

**531.37** Motion of a solid body unaffected by force. Free motion

**531.38** Other motions of a solid body

**531.381** Bodies movable about a fixed point

**531.382** Bodies movable about an axis. Solids of revolution

**531.383** Application to the gyroscope, the spinning top

**531.384** Bodies on a fixed plane

**531.391** Dynamics of material systems

**531.392** Integrals common to various problems of dynamics

**531.394** Methods of transformation in mechanics

**531.395** Dynamics of systems subject to conditions varying in time

**531.396** Relative motion of systems of solid bodies

**531.4** Work. Weight. Mass. Friction. Passive resistance

**531.41** Work. Efficiency

**531.42** Weight. Mass. Relative density (specific gravity). Density

*References: 531.75*

**531.421** Weight as a measure of quantity of material

**531.422** Mass as a measure of quantity of material. Heavy masses

**531.423** Relative density (specific gravity). Mass of unit volume

**531.424** Density

**531.43** Friction in general

*References: 539.92*

**531.44** Sliding friction

**531.45** Rolling friction

**531.46** Combination of gliding and rolling friction

**531.47** Resistance of an axis of rotation

**531.49** Other passive resistances

**531.5** Gravity. Gravitation. Pendulums. Ballistics

**531.52** Laws of falling bodies

**531.53** Pendulum theory. Various kinds of pendulum

**531.55** External ballistics. Path and motion of projectiles. Trajectory

*References: 623.54*

**531.55.011** Abscissa. Range

**531.55.012** Ordinate. Rise

**531.55.013** Fall. Descent phase

**531.55.014** Flight duration

**531.55.015** Inclination of tangent

**531.55.016** Final velocity

**531.55.018** Radius of curvature

**531.551** Motion in a vacuum

**531.552** Motion in a resisting medium

*References: 532.582*

**531.553** Differential equations of movement. Transformations. Hodographs

**531.554** Inverse ballistic problem

**531.555** Elements of the trajectory. Laws of air resistance

**531.557** Study of rectilinear horizontal and vertical motions

**531.558** Firing tables. Various other tables

**531.56** Secondary motions in the air. Effect of spin of projectile and of variation of ballistic elements

**531.561** Laws of spin of projectiles

**531.562** Influence of density variations in resisting medium

**531.564** Influence of atmospheric air movement. Deviations due to wind

**531.565** Influence of projectile characteristics, e.g. weight, centre of gravity, stability

**531.566** Experimental study of misfirings, failures, stray shots

**531.567** Probability of hitting target. Strike probability. Distribution of projectiles

**531.57** Internal ballistics

**531.58** Terminal ballistics. Target penetration

**531.6** Mechanical energy. Conservation of mechanical energy

**531.61** Energy. Momentum

**531.611** Mechanical energy in the strict sense

**531.612** Momentum

**531.62** Law of conservation of energy in mechanics

**531.64** Potential energy

**531.65** Kinetic energy. Principle of kinetic forces

**531.66** Collision of bodies. Impact

**531.7** Measurement of geometric and mechanical quantities: instruments, methods and units

**531.71** Linear measurement

**531.71.081** Units of linear measurement. Units of length

*References: 531.71*

**531.711** Reference lengths. Standard lengths, measures

**531.712** Measurement of datum lines

**531.713** Comparators. Cathetometers

**531.714** Measuring instruments and machines

**531.715** Optical precision methods and apparatus

**531.716** Everyday measuring devices. Industrial measurement

**531.717** Measurement of thickness, diameter, height, depth, inclination, flatness, parallelism

**531.717.1** Methods and apparatus for measurement of thickness, diameter, separation (perforation pitch etc.)

**531.717.2** Measurement of shapes, profiles etc.

**531.717.3** Height measurement. Depth measurement. Pitch measurement

**531.717.5** Indicators. Pointers. Minimeters. Dial gauges

**531.717.7** Measurement of spheres. Spherometers

**531.717.8** Testing of flatness, straightness, parallelism

**531.718** Gauges. Templates. Limit gauges

*References: 621.713.3*

**531.719** Other instruments for length measurement

**531.719.1** Curvimeters

**531.719.2** Distance measurement, ranging. Ranging instruments

**531.719.3** Depth measurement and instruments

**531.719.4** Height measurement and instruments, e.g. for aircraft. Altimeters

**531.72** Measurement of surfaces, sections

**531.72.081** Units of surface measurement. Units of area. Square units

*References: 531.72*

**531.721** Planimeters. Integrators

**531.722** Special compasses for area measurement

**531.724** Measurement of specific surface of granular materials

**531.728** Surface measurement using microscope

**531.729** Calipers. Gap gauges

**531.73** Measurement of volumes

**531.73.081** Units of volume. Cubic units

*References: 531.73*

**531.731** Volume measurement of solid bodies, hollow bodies, loose matter

**531.732** Measurement of liquid volumes

**531.733** Measurement of gas volumes

**531.74** Measurement of angles. Goniometry

**531.741** Protractors. Contact goniometers

**531.742** Other direct measurement of angles

**531.743** Measurement by reflection. Reflection goniometry

*References: 681.786*

**531.744** Graduated scales for mirror instruments

**531.745** Indirect measurement. Sine-bars. Surface and angle plates

**531.746** Gradient measurement. Clinometers

**531.747** Standard angles. Set squares

**531.748** Photographic goniometry. Photogoniometry

**531.749.1** Solid angle measurement

**531.75** Measurement of mass and density

*References: 681.26*

**531.751** Standards of mass. Measurement of mass

**531.753** Methods of weighing

**531.754** Density measurement generally. Density of solids. Determination of relative density (specific gravity)

**531.755** Measurement of density or relative density of lumpy, granular, pulverous materials. Density of powders

**531.756** Measurement of density of liquids

**531.758** Measurement of density of gases

**531.76** Measurement of time intervals, velocity, acceleration

**531.761** Measurement of time intervals. Time standards

*References: 681.11*

**531.761.081** Units of time measurement

*References: 531.761*

**531.767** Velocity measurement. Velocimeters

**531.768** Acceleration measurement. Accelerometers

**531.77** Measurement of angular velocity

**531.771** Revolution counters

**531.775** Mechanical velocity indicators. Tachometers

**531.776** Optical velocity indicators

**531.776.087.252** Stroboscopes

*References: 531.776*

**531.78** Measurement of force, work, pressure

*References: 620.1*

**531.781** Measurement of force. Measurement of torque, twisting moment. Tension measurement

**531.783** Measurement of work in general

**531.787** Pressure measurement. Manometers. Pressure cells. Barometers

**531.787.087.4** With automatic recording. Recording manometers. Barographs

*References: 531.787*

**531.787.1** With flexible tubes. Bourdon tubes

**531.787.2** With diaphragms. Pressure cells. Metal barometers. Aneroid barometers. Hypsometers. Hypsometric barometers

**531.787.3** With piston. Piston manometers

**531.787.4** With liquid, air, float. Mercury barometers. Ring-balance manometers

**531.787.5** Optical manometers

**531.787.6** Micromanometers

**531.787.7** Differential pressure gauges

**531.787.91** Electrical pressure measurement

**531.788** Measurement of vacuums. Vacuum meters

**531.788.1** Mechanical vacuum gauges

**531.788.5** Optical vacuum gauges. Discharge tubes as vacuoscopes

**531.788.6** Thermal vacuum gauges. Thermal conductivity vacuum gauges (Pirani gauges)

**531.788.7** Electrical vacuum gauges. Ionization gauges. Discharge tubes as vacuum gauges

**531.79** Other measuring apparatus

**531.791** Counting methods. Methods of estimating, totalizing, a number of units

**531.792** Methods of division. Dividing apparatus. Parallel and angular division

**531.793** Measurement of degree of curvature

**531.8** Theory of machines. Technical mechanics in general

*References: 621.01*

**532** Fluid mechanics in general. Mechanics of liquids (hydromechanics)

**532.1** Hydrostatics in general

**532.1-1/-9** In lines, surfaces, spaces

**532.11** Pressure

*References: 531.787*

**532.12** Compressibility. Elasticity

**532.13** Internal friction. Viscosity

**532.132** Fluidity. Superfluidity

**532.133** Viscosity coefficients

**532.135** Rheology

**532.137** Viscometers. Consistometers

**532.14** Density

**532.2** Equilibrium of liquids

**532.21** Surface level. Surfaces

**532.217** Measurement of surface level. Depth gauges

**532.22** Free surfaces

**532.23** Equilibrium of several superposed liquids

**532.24** Relative equilibrium

**532.25** Equilibrium of a heavy liquid rotating about an axis

**532.29** Levels. Spirit levels. Bubble levels. Communicating vessels

**532.3** Immersed bodies (submerged bodies). Floating bodies

**532.31** Conditions of equilibrium

**532.32** Stability. Theory of floating vessels

**532.33** Equilibrium of bodies of particular shapes, e.g. right prisms

**532.5** Liquid motion. Hydrodynamics

**532.5-1/-9** In lines, surfaces, spaces

**532.5.01/.08** Principles, characteristics, measurement

**532.51** Liquid motion in general. Hydrodynamics in the strict sense

**532.511** Dynamic equations of motion

**532.513** Bernoulli's equation

**532.516** Classical theory of viscous liquids. Friction in lubricants

**532.516.5** Theory of viscous lubrication. Fluid wedge theories

**532.517** Liquid motion according to type of flow

**532.517.2** Laminar flow

**532.517.3** Transition region between laminar and turbulent flow

**532.517.4** Turbulent flow

**532.517.6** Pulsating flow

**532.518** Surfaces of discontinuity. Helmholtz free streamlines

**532.52** Conditions of fluid motion

**532.521** Theory of liquid filaments

**532.522** Flow through orifices

**532.525** Flow through nozzles

**532.526** Boundary layers. Surface friction (skin friction)

**532.527** Rotational motion. Vortices

**532.528** Cavitation. Formation of hollow spaces

**532.529** Mixing of gases and liquids. Two-phase fluids. Motion of gas-liquid mixtures

**532.53** Open-channel phenomena

**532.531** Falls. Waterfalls. Profiles of nappes in general

**532.532** Weirs. Notches

**532.533** Overflows. Spillways

**532.535** Hydraulic jumps

**532.537** Inlet, outlet channels. Backwaters. Drop-down curves

**532.54** Liquid motion in various systems

**532.542** In tubes, pipes. Closed full conduits

**532.543** In open conduits

**532.543.1** Flow as function of periphery and gradient

**532.543.2** With low gradient

**532.543.3** With high gradient

**532.543.4** Flow as function of discharge

**532.543.5** Flow as function of suspension

**532.543.6** Obstructions to flow

**532.543.7** Variations in flow

**532.546** In porous media

**532.55** Energy loss. Pressure loss

**532.551** Flow in straight pipes

**532.552** Flow round bends

**532.553** Flow in sluices

**532.555** Flow at constrictions

**532.556** Flow at widenings. Diffusers

**532.559.2** Energy losses at places where the flows unite

**532.559.3** Energy losses at places where the flows divide

**532.559.5** Entry and exit losses

**532.559.8** Energy losses caused by screens, gauzes, honeycombs etc.

**532.57** Flow measurement. Velocity, rate of flow. Quantity, total flow

**532.572** Gauging with weirs, spillways

**532.573** Measurement of surface velocity (with floats, rods etc.)

**532.574** Measurement of internal velocity (with Pitot tubes, with tracers etc.)

**532.575** Direct measurement of rate of flow (with Venturi meters etc.)

**532.576** Measurement of loss of head

*References: 531.787*

**532.58** Pressures induced by relative motion of bodies and fluids. Resistance to motion

**532.581** Fundamental equations

**532.582** Immersed bodies (submerged bodies)

**532.583** Floating bodies. Floating vessels

**532.584** Suspensions as liquids

**532.584.1** Light suspensions

**532.584.2** Heavy suspensions

**532.585** Influence of cross-sectional shape of stream. Wall effects. Interference effects

**532.59** Wave motion

**532.591** Translational waves. Velocity of wave propagation

**532.592** Single waves. Propagation of waves in a moving liquid

*References: 556.536*

**532.593** Billows. Shallow waves. Large waves. Theory of sea waves. Deep waves. Ground swell

**532.594** Capillary waves. Effect of capillarity on other wave phenomena. Ripple tanks

**532.595** Oscillating columns of liquid. Oscillation of water in tubes. Propagation of liquid waves in elastic pipes

**532.596** Classical theory of tides

*References: 52-423.3 , 521.16 , 551.466*

**532.6** Surface phenomena. Surface tension. Capillarity

*References: 539.61*

**532.61** Surface tension. Interfacial tension

*References: 544.722.2*

**532.61.041** Static tension

**532.61.042** Dynamic tension

**532.612** Surface tension

**532.613** Interfacial tension (tension of contact surfaces, boundary layers)

**532.614** Surface energy. Interfacial energy (energy of contact surfaces, boundary layers)

**532.615** Interfacial activity of dissolved substances

**532.62** Properties of thin films of liquid

**532.63** Capillarity

**532.64** Wetting angle. Contact angle

**532.65** Free surface under action of capillary forces. Rayleigh phenomena of break-up of jets by surface-tension effects and droplet formation

**532.66** Capillary tubes

**532.68** Motion due to capillary forces

**532.69** Various properties depending on surface tension

**532.691** Surface activity in general

**532.692** Power of forming thin films. Unctuosity (oiliness)

**532.694** Formation of gas-liquid systems

**532.695** Formation of emulsions. Emulsifiability

**532.696** Behaviour relating to solids

**532.696.1** Wetting capacity

**532.696.2** Washing capacity

**532.696.5** Retention of liquids in solid-liquid systems

**532.7** Kinetic theory of liquids. Osmosis. Solution and solutions

**532.71** Osmosis

**532.72** Diffusion

**532.73** Solution. Solution phenomena. Physical solution. Solution kinetics

**532.74** Molecular state. Association. Structure

**532.77** Physical properties of solutions

**532.771** Dilute solutions

**532.772** Concentration with reference to amount of particular constituents

**532.773** Concentrated solutions

**532.78** Crystallization of liquids. Crystallization from solutions

**532.781** Crystallization of pure liquids

**532.782** Metastable, supercooled liquids. Solidified liquids

*References: 536.421.48*

**532.783** Anisotropic liquids. Liquid crystals

**532.785** Crystallization from solutions

**532.787** Supersaturated solutions

**533** Mechanics of gases. Aeromechanics. Plasma physics

**533.1** Properties of gases

**533.11** Specific volume

**533.12** Density

*References: 531.758*

**533.13** Archimedes' principle applied to gases. Correction of weighings. Upthrust

*References: 533.61*

**533.15** Diffusion and penetration of gases. Inflow

**533.16** Internal friction. Viscosity

**533.17** Outflow

**533.2** Elasticity. Compressibility. Liquefaction. Gas mixing. Gaseous mixtures

**533.21** Compressibility

**533.22** Approximate laws

**533.23** Limits of Boyle's law. Experiments of Pouillet, Dulong, Arago, Regnault

**533.24** Liquefaction. Solidification of gases. Liquid air

**533.27** Gaseous mixtures. Properties of gas mixtures

**533.273** Partial pressure

**533.275** Humidity. Hygrometry

*References: 543.275.1*

**533.276** Katathermometers. Katathermometry

**533.5** Rarefied gases. Vacuum physics

**533.51/.57** Rarefaction by mechanical means. Air pumps

**533.51** Mechanical air pumps. Oil vacuum pumps

**533.52** Mercurial air pumps

**533.54** Water jet pumps. Fluid jet pumps

**533.55** Steam jet pumps. Mercury vapour jet pumps

**533.56** Molecular pumps. Diffusion pumps

**533.57** Batteries of air pumps. Combinations of kinds of pump listed above. Pump controls

**533.58** Rarefaction by other physical and chemical means

**533.581** Use of low temperatures

**533.582** Ion pumps

**533.583** Use of absorbent and adsorbent substances. Charcoal. Silica gel. Getters

*References: 544.723*

**533.585** Use of purely chemical means. Reducing agents

**533.587** Gas consumption ('clean-up')

**533.59** Experiments in a vacuum. Vacuum plant

**533.6** Aerodynamics

**533.6.01** Aerodynamic principles and theory

**533.6.011** Motion of and in gases. Gas flow

**533.6.011.1** Dimensional analysis of gas and air flow

**533.6.011.12** Reynolds number. Scale effect

**533.6.011.3** Subsonic flow. Transonic flow

**533.6.011.32** Subsonic flow without compressibility effect (below M 0.8)

**533.6.011.34** Subsonic flow with compressibility effect (M 0.8 to 1.0)

**533.6.011.35** Transonic flow. Flow at sonic velocities (M 0.8 to 1.3)

**533.6.011.5** Flow at supersonic velocities (M 1.0 to 4.99)

**533.6.011.55** Flow at hypersonic velocities (M 5+)

**533.6.011.6** Heat-transfer phenomena of moving bodies and gases. Aerothermodynamics

**533.6.011.7** Compressibility. Compressibility effects

**533.6.011.8** Superaerodynamics (flow in rarefied gases)

**533.6.013** Dynamics of aeronautic machines

*References: 629.7.015 , 629.7.017*

**533.6.013.1** Dynamic theory applied to dynamics of aeronautic machines

**533.6.013.12** Air resistance. Aerodynamic drag

**533.6.013.13** Lift. Normal force

**533.6.013.14** Lateral force (side force)

**533.6.013.15** Moments

**533.6.013.16** Thrust

**533.6.013.2** Unsteady motion

**533.6.013.3** Steady motion

**533.6.013.4** Stability

**533.6.013.42** Flutter. Aeroelasticity

**533.6.013.43** Buffeting

**533.6.013.6** Flight position

**533.6.013.61** Level flight

**533.6.013.63** Turning flight

**533.6.013.64** Diving and climbing

**533.6.013.642** Diving. Nose-dive

**533.6.013.644** Climbing

**533.6.013.66** Stalling

**533.6.013.67** Hovering. Low-speed flight

**533.6.013.68** Nonpowered flight

**533.6.013.7** Spiral dive. Spin

**533.6.013.8** Acceleration stresses. Vg tests (vertical gust tests). Gust loads

**533.6.015** Performance

**533.6.015.1** Take-off

**533.6.015.2** Landing

**533.6.015.3** Speed. Rate of ascent and descent

**533.6.04** Aerofoil characteristics and variables

**533.6.05** Operational observations, testing, recording, in service

**533.6.052** Ground testing

**533.6.054** Flight testing. Test flights (manned, piloted)

**533.6.055** Model testing (unmanned, pilotless)

**533.6.07** Wind tunnels. Shock tubes. Tanks. Other ground testing techniques and apparatus

**533.6.071** Wind tunnels: design, construction, fitting out

**533.6.072** Models. Simulators. Analogies

**533.6.074** Water tanks. Water tunnels

**533.6.075** Apparatus with rotating devices

**533.6.076** Electrolytic tanks for solving potential flow problems

**533.6.078** Ballistic and other special testing apparatus and techniques

**533.6.08** Flow and flight measurement. Anemometry etc.

**533.61** Theory of lighter-than-air aircraft (aerostats)

**533.62** Theory of free balloons

**533.63** Theory of captive balloons

**533.64** Theory of dirigibles (airships)

**533.65** Theory of heavier-than-air aircraft (Aerodynes. Aeroplanes)

**533.652** Landplanes

**533.655** Seaplanes

**533.657** Amphibious aircraft

**533.66** Theory of other flying machines

**533.661** Rotorcraft. Helicopters

**533.662** Airscrews

**533.662.2** Driving screws. Propellers

**533.662.3** Fans

**533.662.5** Tilting propellers. Tilting rotors

**533.662.6** Lifting airscrews. Helicopter rotors

**533.662.7** Windmill sails

**533.664** Aircraft with moving wings (except helicopters). Ornithopters etc.

**533.665** Missiles, rockets (aerodynamic aspects)

*References: 531.5 , 623.46 , 629.76*

**533.666** Devices for nonpowered flight

**533.68** Aerodynamic studies, including interference problems, of bodies moving on or near surfaces (on ground, decks, water). Ground effect

**533.69** Aerodynamic studies of aircraft elements

**533.69.04** Aerofoil design and dimensional characteristics. Air-flow variables

**533.69.041** Aspect ratio. Span-chord ratio

**533.69.043** Chord line and related characteristics

**533.69.044** Thickness distribution

**533.69.045** Dihedral design

**533.69.046** Surface conditions

**533.69.047** Other aerofoil characteristics

**533.69.048** Air-flow variables

**533.69.048.1** Aerodynamic loading

**533.69.048.2** Pressure distribution

**533.69.048.3** Wake. Slipstream. Downwash

**533.692** Wing sections

**533.693** Complete wings. Planforms

**533.694** High-lift devices. Controls

**533.694.2** Flaps. Slots. Slats

**533.694.5** Control surfaces

**533.694.51** Lateral controls. Ailerons

**533.694.52** Elevons

**533.694.53** Longitudinal controls

**533.694.54** Directional controls

**533.694.58** Tabs

**533.694.6** Jets as high-lift devices or controls

**533.694.7** Boundary layer control

**533.694.71** Suction

**533.694.72** Blowing

**533.694.73** Fences. Vortex generators. End plates

**533.695** Components in combination. Interference problems

**533.695.1** Wing

**533.695.1`4** Interference between wing [533.695.1] and tailplane [533.695.4]

*References: 533.695*

**533.695.2** Body

**533.695.4** Tail: vertical, horizontal

**533.695.5** Cascades. Multiplanes

**533.695.7** Jet. Exhaust

**533.695.8** Propeller. Airscrew. Rotor

**533.695.9** Other components

**533.696** Bodies

**533.696.2** Spheres

**533.696.3** Cylinders

**533.696.4** Cones. Nose cones

**533.696.5** Other bodies of circular cross-section. Bodies of revolution

**533.696.6** Bodies of noncircular cross-section

**533.696.7** Forebodies in general

**533.696.8** Afterbodies in general

**533.697** Internal aerodynamics

**533.697.2** Inlets. Intakes

**533.697.3** Ducts. Passages. Bends. Diffusers. Vortex tubes

**533.697.4** Nozzles. Jets

**533.697.5** Ejectors. Jet pumps. Outlets. Bleeds

**533.697.6** Bodies in ducts

**533.7** Kinetic theory of gases. Continuity of state

*References: 533.15 , 536.97 , 539.1 , 539.2*

**533.72** Kinetic phenomena

**533.723** Brownian movement

**533.73** Kinetic energy. Mean velocity of molecules. Mean energy of molecules

**533.735** Thermodiffusion

**533.74** Avogadro's number

**533.75** Continuity of state. Van der Waals' equation and corrections of it

**533.77** Kinetic theory of equilibrium of phases

**533.9** Plasma physics

*References: 538.9 , 621.039.61*

**533.9-11** Plasma (and plasma waves) according to state

**533.9-112** Electron state

**533.9-113** Ion state

**533.9-115** Low-temperature plasmas (10 000K to 100 000K)

**533.9-116** High-temperature plasmas (over 100 000K)

**533.92** Plasma reactions, including atomic cross-sections, reaction rates

**533.922** Plasma interactions with charged particles

**533.924** Plasma interactions with solids

**533.93** Statistical behaviour of plasmas

**533.95** Plasma dynamics

**533.951** Plasma waves, oscillations. Plasma resonances. Shock waves

**533.952** Pinch effect

**533.98** Plasmoids

**534** Vibrations. Waves. Acoustics

**534-13** Vibrations in gases

**534-14** Vibrations in liquids

**534-16** Vibrations in solids

**534-18** Vibrations in heterogeneous, multiphase systems

**534-6** Low-frequency vibrations (long wavelength)

*References: 534.321.8*

**534-7** Audible-frequency vibrations. Sound waves

*References: 534.321.1*

**534-8** High-frequency vibrations (short wavelength)

*References: 534.321.9*

**534.01** Elementary theory of vibrations

**534.011** Transverse waves

**534.012** Longitudinal waves

**534.013** Torsion waves

**534.014** Oscillations of a mass-point

**534.014.1** Free oscillations. Natural vibrations

**534.014.2** Forced oscillations

**534.015** Coupling of systems of oscillating points

**534.1** Vibration of bodies. Excitation of vibrations. Vibratory formations with distributed mass and elasticity

**534.11** Vibration of linear bodies

**534.112** Vibration of strings

**534.113** Vibration of straight rods. Vibration of hollow cylinders

**534.114** Vibration of curved rods

*References: 534.321.7*

**534.115** Vibration of liquid columns. Vibration of gas columns

**534.12** Vibration of two-dimensional bodies

**534.121** Theory of vibrations in two dimensions. Solid vibrating bodies

**534.122** Vibration of liquid layers. Vibration of gas layers

**534.13** Vibration of three-dimensional bodies

**534.131** Theory of vibrations in three-dimensional bodies

**534.132** Vibration in closed cylinders

**534.133** Vibrations in quartz

**534.138** Vibrations in small particles dispersed in a liquid

**534.14** Methods of exciting vibrations

**534.141** Mechanical excitation

**534.142** Thermal excitation

**534.143** Electric and magnetic excitation

**534.17** Various influences on vibrations

*References: 537.871.7*

**534.18** Waves

**534.182** Nonlinear waves

**534.2** Propagation of vibrations. Processes in the acoustic field

**534.21** Propagation of vibrations

**534.22** Velocity of propagation

**534.221** Uniform propagation velocity

**534.222** Non-uniform propagation velocity

**534.222.2** Nonlinear acoustics. Shock waves. Detonation

**534.23** Transmission of vibrations through the acoustic field

**534.231** Vibration field. Acoustic field

**534.232** Vibration sources. Acoustic radiators. Vibrators

**534.24** Reflection of waves

**534.241** Echo

**534.242** Resonance and associated phenomena

**534.25** Refraction of waves

**534.26** Diffraction of waves

**534.27** Polarization of waves

**534.28** Physical effects on waves

**534.29** Physical effects produced by sound waves

**534.3** Musical sounds and perception

*References: 681.8 , 78.01 , 781.1*

**534.31** Objective causes of sensation of sound

**534.32** Sound quality. Nature of sound sensation

**534.321** Pitch

**534.321.1** Limits of audibility

*References: 534-7*

**534.321.2** Intervals

**534.321.3** Musical scales

**534.321.4** Harmonic overtones. Subharmonic tones. Undertones

**534.321.5** Harmonies, concords. Dissonances, discords

**534.321.6** Chords from the acoustic point of view

**534.321.7** Absolute pitch

**534.321.8** Subsonic vibrations

*References: 534-6*

**534.321.9** Ultrasonic vibrations

*References: 534-8*

**534.322** Timbre, tone-colour. Amplitude

**534.322.3** Noise

*References: 534.77 , 534.83 , 612.85 , 628.517 , 699.844*

**534.323** Subjectively perceived sound intensity. Loudness

**534.324** Comparisons and contrasts of sounds and colours

**534.4** Analysis and synthesis of sounds

**534.41** Qualitative analysis of sounds

**534.42** Synthesis of timbre

**534.44** Quantitative analysis of sounds

**534.5** Composition of vibrations

**534.51** Composition of parallel vibrations

**534.511** Propagation of vibrations in the same direction

**534.512** Interaction of vibrations in opposite directions

**534.52** Interaction of nonparallel vibrations

**534.6** Acoustic measurements

**534.61** Measurement of intensity

**534.612** Measurement of sound pressure

**534.614** Measurement of the velocity of sound

**534.615** Measurement of the displacement, density variation and temperature variation in a sound wave

**534.62** Sound-absorbent chambers. Anechoic chambers (dead rooms). Echo chambers. Resonance chambers. Measurement of total energy of sound sources

**534.63** Measurement of frequency

**534.64** Measurement of acoustic-mechanical impedance

*References: 681.883*

**534.647** Measurement of shocks. Seismographs

*References: 550.34*

**534.7** Physiological acoustics. Medical acoustics

**534.75** Hearing (audition)

*References: 612.85*

**534.76** Binaural, directional hearing (stereoaudition)

**534.77** Hearing defects. Hearing aids

**534.771** Measurement of hearing acuity

**534.773** Apparatus for the improvement of hearing. Hearing aids

*References: 621.395.92*

**534.78** Physics of speech sounds. Human voice. Phonation. Speech production

**534.79** Loudness. Concept of loudness, volume

**534.8** Applications of acoustics (theory)

*References: 681.8*

**534.83** Vibrations and their control. Control of disturbing noise. Noise abatement

*References: 351.759.4 , 628.517 , 699.84*

**534.831** Unpleasantness of vibrations. Degree of irritation. Noise. Shocks

**534.832** Prevention of vibrations at source

**534.833** Attenuation of vibrations

**534.833.5** Attenuation of acoustic vibrations

**534.833.52** Attenuation of sound during transmission

*References: 534.23*

**534.833.522** Attenuation of airborne sound

**534.833.524** Attenuation of structure-borne sound

**534.833.53** Attenuation by reflection. Sound absorption

**534.833.6** Attenuation of subsonic and ultrasonic vibrations. Vibration damping

**534.835** Noise in dwellings. Noise in buildings

**534.836** Noise outside buildings

**534.84** Architectural acoustics. Sound level. Tone quality

**534.843** Acoustic phenomena in rooms, and related problems

**534.843.2** Reflections (in room acoustics)

**534.843.5** Intelligibility (in room acoustics). Syllabic clarity, distinctness

**534.843.6** Shapes of auditoriums with reference to distribution of directly radiated sound

**534.843.7** Problems related to distribution of total sound energy

**534.85** Sound recording and reproduction

**534.851** Mechanical recording. Gramophone records

**534.852** Magnetic recording

**534.853** Optical recording

**534.86** Sound transmission (radio, television, film). Electroacoustics

**534.861** Studio techniques

**534.863** Telephone acoustics

**534.864** Public address systems

**534.87** Signalling acoustics. Sound signals

*References: 654.92*

**534.873** Range of acoustic signals

**534.874** Directional characteristics

**534.88** Sound location. Acoustic direction finding. Sound ranging

*References: 681.88*

**535** Optics

*References: 159.937.51 , 537.53 , 537.86 , 537.87 , 539.1*

**535-1/-3** Radiation according to wavelength. Hertzian waves. Light colours

**535-1** Long waves. Infrared

**535-11** Very long waves, above radio wavelengths (about 10 000 m)

**535-12** Radio wavelengths

**535-14** Wavelengths between Hertzian and infrared

**535-15** True infrared waves

**535-2** Visible light. Colours

**535-2-026.613.01** Colourless (in clear transparent bodies)

**535-2-026.613.02** White

**535-2-026.613.03** Grey

**535-2-026.613.04** Black

**535-2-026.613.1** Red colour group

**535-2-026.613.11** Bluish red. Magenta. Crimson. Carmine

**535-2-026.613.114** Rose. Pink

**535-2-026.613.13** Bright red. Primary red. Scarlet

**535-2-026.613.15** Vermilion. Cinnabar

**535-2-026.613.17** Yellowish red

**535-2-026.613.174** Yellowish pink

**535-2-026.613.2** Orange colour group

**535-2-026.613.21** Reddish orange

**535-2-026.613.215** Reddish brown

**535-2-026.613.24** Orange

**535-2-026.613.245** Brown

**535-2-026.613.27** Yellowish orange

**535-2-026.613.3** Yellow colour group

**535-2-026.613.31** Reddish yellow. Golden yellow

**535-2-026.613.33** Pure yellow. Primary yellow

**535-2-026.613.334** Cream

**535-2-026.613.335** Yellowish brown

**535-2-026.613.36** Greenish yellow

**535-2-026.613.366** Olive

**535-2-026.613.38** Yellow-green

**535-2-026.613.4** Green colour group

**535-2-026.613.41** Yellowish green. Leaf green

**535-2-026.613.43** Pure green. Primary green

**535-2-026.613.47** Sea green. Bluish green

**535-2-026.613.55** Blue-green

**535-2-026.613.6** Blue colour group

**535-2-026.613.61** Greenish blue. Cyan. Ice blue

**535-2-026.613.612** Turquoise

**535-2-026.613.63** Pure blue. Primary blue

**535-2-026.613.634** Sky blue. Azure

**535-2-026.613.75** Indigo blue

**535-2-026.613.8** Violet colour group

**535-2-026.613.81** Bluish violet

**535-2-026.613.814** Lilac

**535-2-026.613.82** Violet

**535-2-026.613.83** Bluish purple. Red-violet

**535-2-026.613.85** Purple

**535-2-026.613.87** Reddish purple

**535-3** Short waves

**535-31** Ultraviolet

**535-32** Region between ultraviolet rays and X-rays

**535-34** X-rays

*References: 621.386*

**535-36** Gamma-rays

**535-4** Polarized light

**535-5** Natural light

**535-6** Partially polarized light

**535-7** Totally reflected light

**535-92** Reflected light

**535-94** Refracted light

**535.012** Optical isotropy and anisotropy

**535.012.1** Isotropy

**535.012.2** Anisotropy

**535.1** Theory of light

**535.11** Emission theory of light (Newton)

**535.12** Wave theory of light (Huygens, Fresnel)

**535.13** Electromagnetic theory of light (Maxwell)

*References: 537.8*

**535.14** Quantum optics

*References: 530.145 , 539.122 , 539.184*

**535.18** Nonlinear optics

**535.181** Bistability. Multistability

**535.182** Phase conjugation

**535.2** Propagation and energetics of radiation. Photometry

**535.21** Effects of light radiation

**535.211** Thermal effects of light

**535.212** Luminous effects of light

**535.214** Mechanical effects of light. Radiation pressure

**535.215** Electrical effects of light. Photoelectric phenomena. Photoelectric effect

*References: 621.383*

**535.217** Chemical effects of light. Actinism. Actinology

*References: 544.52*

**535.218** Magnetic effects of light

**535.22** Velocity of propagation of light

**535.23** Energetics of radiation

**535.231** Total radiation

**535.231.1** Fundamental quantities

**535.231.2** Relationship between absorption and emission

**535.231.6** Measurement of total radiation (with nonselective receivers)

**535.232** Spectral distribution of energy in general

**535.232.1** Spectral density. Relative spectral distribution

**535.232.5** Equal energy spectrum

**535.232.6** Properties of radiation receiver. Radiation measurement (except photometry and colour measurement)

**535.233** Thermactinic radiation

**535.233.2** Black-body radiation

**535.233.4** Non-black-body radiation

**535.233.5** Concepts of radiation temperature

**535.233.52** Partition temperature

**535.233.55** Black temperature

**535.233.56** Colour temperature

**535.24** Photometry

*References: 520.82 , 681.785.4 , 771.534.5*

**535.241** General principles of measurement. Standards. Units

**535.241.1** Attenuation techniques

**535.241.3** Standards representing the unit of light intensity

**535.241.4** Units. Fundamental quantities. Definitions

**535.241.41** Luminous flux

**535.241.42** Luminous intensity

**535.241.43** Radiance. Light emission related to the unit solid angle

**535.241.44** Luminance (brightness)

**535.241.45** Quantity of light

**535.241.46** Illumination. Illuminance

**535.241.5** Proportions between photometric and other quantities

**535.241.6** Auxiliary photometric apparatus

**535.242** Subjective photometry

*References: 612.843.36*

**535.242.1** Principles of subjective photometry

**535.242.3** Problems of field of view

**535.242.5** Isochromatic photometry

**535.242.6** Heterochromatic photometry

**535.242.61** Methods using equalization of brilliance

**535.242.62** Methods based on free flicker. Flicker methods

**535.242.64** Trichromatic methods

**535.242.65** Mathematical methods

**535.242.66** Colour extinction methods

**535.242.67** Visual acuity methods

**535.242.7** Photometry with dark-adapted eyes

**535.243** Spectrophotometry

*References: 535.651*

**535.243.2** Spectrophotometry using photocells, photoelements, thermoelements, bolometers etc.

**535.243.3** Photographic spectrophotometry

**535.247** Objective photometry (using chemical, electrochemical, thermoelectric, photoelectric and similar methods)

**535.249.2** Determination of luminosity (luminous power)

**535.3** Propagation. Reflection. Refraction. Absorption. Emission

*References: 681.7*

**535.31** Geometrical optics (theory and calculations)

**535.311** Rectilinear propagation of light. Shadows

**535.312** Reflection

**535.313** Spherical mirrors. Curved mirrors. Aberration. Focal lines. Focal planes

**535.314** Refraction

**535.315** Prisms

**535.316** Thin lenses. Astigmatic lenses. Cylindrical lenses

**535.317** Thick lenses. Optical systems. Mathematical optics

**535.317.2** Optical systems

**535.317.22** Focal point. Focal length. Refractivity of optical systems

**535.317.23** Apertures. Relative aperture. Restriction of rays

**535.317.24** Visual field. Image field

**535.317.25** Resolving power

**535.317.6** Mathematical and graphical determination of image errors in spherical lenses and lens systems. Theory of image errors

**535.317.61** Aperture errors

**535.317.62** Coma

**535.317.63** Image field curvature

**535.317.64** (Lens) aberration. Distortion

**535.317.68** Chromatic aberration

**535.317.7** Experimental determination of image errors in spherical lenses and lens systems

**535.317.9** Aspherical lenses

**535.318** Optical systems containing both refracting and reflecting surfaces. Catadioptric systems

**535.319** Testing for optical homogeneity. Testing of plane parallelism of plates

**535.32** Refractive indices. Dispersion

*References: 535.58 , 681.785.2*

**535.321** Methods of measurement of refractive index

**535.321.2** Minimum deviation method

**535.321.4** Determination from angle of total reflection

**535.321.5** Immersion methods

**535.323** Refractive indices of solids

**535.324** Refractive indices of liquids

**535.325** Refractive indices of gases

**535.326** Bodies with spatial variation of refractive index

**535.327** Various influences on refractive index

**535.328** Measuring indices for dispersion

**535.329** Anomalous dispersion

**535.33** Spectra in general. Emission spectra

*References: 52-355*

**535.33(083.4)** Tables of spectra

*References: 535.33*

**535.331** Spectra of metals

**535.332** Spectra of metalloids

**535.333** Spectra of compounds

**535.337** Various influences on spectra

**535.338** Composition of spectra

**535.338.1** Continuous emission spectrum

**535.338.3** Lines. Line spectra

**535.338.4** Bands. Band spectra

**535.339** Monochromatic light and its production

*References: 535.622 , 681.785.5*

**535.34** Absorption. Absorption spectra

*References: 543.42*

**535.341** Absorption coefficient. Extinction coefficient. Similar measures of absorption

**535.342** Absorption spectra in general

**535.343** Particular absorption spectra

**535.343.1** Of flames and incandescent gases

**535.343.2** Of transparent solid bodies

**535.343.3** Of liquids

**535.343.4** Of gases

**535.345** Direct transmission of light through bodies

**535.345.1** Transparency in general. Transmittance. Transmission coefficient

**535.345.3** Nontransparency in general. Opacity

**535.345.6** Light filters

**535.345.61** Spectral transmission curves

**535.345.62** Absorption filters

**535.345.65** Utilization of the chromatic aberration of lenses

**535.345.66** Filters employing rotation dispersion

**535.345.67** Interference filters

**535.346** Reflection of absorbing surfaces. Reflectivity. Reflection coefficient

**535.347** Dichroism. Pleochroism

**535.348** Pleochroic haloes

**535.349** Special absorption effects

**535.35** Excitation. Extinction

**535.352** Electron collisions

**535.353** Collisions between atoms and molecules

**535.354** Duration of the excited state

**535.36** Light scattering. Diffusion

**535.361** Scattered reflection and transmission through turbid media

**535.361.1** Diffuse transmission

**535.361.2** Diffuse reflection. Scattered reflection. Opalescence generally

**535.362** Dispersion-absorption complex

**535.37** Luminescence. Fluorescence. Phosphorescence. Their spectra. Excitation and amplification of radiation

**535.371** Fluorescence in general

**535.372** Fluorescence spectra

**535.373** Phosphorescence in general

**535.373.1** Production of phosphors

*References: 661.14*

**535.373.2** Excitation. Theory of luminous mechanisms. Energy transfer

**535.373.3** Duration of phosphorescence

**535.374** Amplification of radiation by stimulated emission

**535.374:621.375.82** Amplification by maser, laser

*References: 535.374*

**535.375** Molecular scattering as quantum effect

**535.375.5** Raman effect

**535.376** Luminescence excited by charged particles. Electroluminescence

**535.377** Thermoluminescence

**535.378** Triboluminescence (luminescence generated by friction)

**535.379** Chemiluminescence (chemically produced luminescence)

**535.39** Relationships between reflection, refraction, absorption

**535.391** Behaviour of transparent bodies. Scattered reflection

*References: 535.361.2 , 535.44*

**535.391.2** Surface reflection of transparent bodies. Vitreous lustre (glass lustre) up to adamantine lustre (diamond lustre)

*References: 535.361.2*

**535.391.4** Fresnel reflection formula

**535.391.5** Decrease of reflection at surfaces

**535.391.6** Increase of reflection at surfaces

**535.392** Bodies with surface colours. Selective reflection. Coloured lustre

**535.393** Metal bodies. Metallic lustre

**535.394** Total reflection

**535.399** Relationships of refraction, dispersion, absorption, to chemical composition

*References: 544.164*

**535.4** Interference. Diffraction. Scattering by diffraction

*References: 535.36*

**535.41** Interference

*References: 681.785.5*

**535.412** Visibility and origin of interference fringes

**535.413** Achromatism of interference phenomena

**535.415** Colours of isotropic thin plates. Newton's rings

*References: 535.662.1*

**535.416** Standing waves

**535.417** Applications

*References: 531.715 , 681.785.57 , 681.787*

**535.417.2** Resonators

**535.42** Diffraction

**535.421** Plane waves. Diffraction gratings. Normal spectrum

**535.422** Spherical waves

**535.423** Cylindrical waves. Graphical methods

**535.428** Polarization by diffraction

*References: 535.514*

**535.43** Scattering by diffraction. Tyndall effect

*References: 535.361.2*

**535.44** Special phenomena produced by interference, diffraction and reflection in regular structures, especially crystals

**535.441** Produced by parallel needle-shaped inclusions

**535.442** Produced by parallel plate-shaped inclusions

**535.443** Pearl lustre. Pearliness

**535.5** Polarization. Double refraction. Dispersion in anisotropic bodies

*References: 681.785.3*

**535.51** Polarization

**535.512** Interference of polarized light

**535.513** Direction of vibrations. Plane of polarity

**535.514** Production of polarized light

**535.515** Birefringence (double refraction) in general. Polarization by double refraction

**535.516** Double reflection

**535.518** Degree of polarization

**535.52** Optical datum planes and axes

**535.521** Wave surfaces. Single-sheet surfaces of refraction

**535.522** Double-sheet surfaces of refraction

**535.524** Optic axes and ray axes in general

**535.525** Measurement of axial angle

**535.526** Ray axes

**535.527** Conic refraction

**535.54** Chromatic polarization

**535.542** Measurement of phase difference, of birefringence

**535.543** Interference phenomena in crystals. Colours of thin crystalline plates. Newton's interference colours. Normal interference colours

**535.544** Measurement of constants of crystals

**535.545** Extinction angles and their measurement. Stauroscopes

**535.547** Irregular crystals. Clusters. Radiative polarization. Polarization by aggregates

**535.548** Applications. Phase contrast methods

**535.55** Stress birefringence. Double refraction in rods and plates

**535.56** Rotation of the plane of polarization. Optical activity

**535.561** Properties of quartz

**535.562** Circular and elliptic birefringence

**535.563** Rotatory power (specific rotation). Rotatory dispersion colours

**535.565** Crystals with rotatory power (except quartz)

**535.566** Amorphous bodies with rotatory power: solids, liquids, gases

**535.568** Applications of rotation of the plane of polarization

**535.568:681.785.3** Polarimeters. Polariscopes

*References: 535.568*

**535.58** Dispersion in anisotropic bodies

*References: 535.32*

**535.581** Dispersion of double refraction (birefringence). Abnormal interference colours. Brewster-Herschel colours

**535.582** Axial dispersion

**535.583** Dispersion of principal optic axis

**535.6** Colours and their properties. Colour theory

*References: 159.937.51 , 612.84 , 62-777.6 , 666.24 , 667 , 75 , 778.6*

**535.61** Generalities on colour theory

**535.611** Hue

**535.612** Brightness. Luminosity

**535.613** Saturation. Purity. Whiteness

**535.616** Equivalent colours. Colours of equal stimulus

**535.617** Two-dimensional colour representation in general

**535.618** Three-dimensional colour effects

**535.62** Composition of colours. Colour stimulus function. Relationship between colour and spectrum

**535.621** Complementary colours. Compensatory colours. Contrast colours

**535.622** Optimal colours

**535.623** Additive colour mixing

**535.624** Subtractive colour mixing

**535.625** Colour stimulus function. Illumination and object colour. Effect of light source

**535.64** Colour systems. Colour theories. Theory of colour measurement, chromaticity measurement

**535.641** Monochromatic systems

**535.642** Dichromatic systems

**535.643** Trichromatic systems

**535.643.1** Physiological chromaticity systems

**535.643.2** Standard chromaticity system (CIE 1931)

**535.643.6** Relationships between various trichromatic systems. Transformations. Transformation aids

**535.644** Tetrachromatic systems. Yellow-blue. Red-green

**535.645** Chromaticity systems adapted to subjective measurements

**535.646** Measurement of colour perception. Higher, non-Euclidean colour measurement (SchrÃ¶dinger)

**535.646.1** Relationship of stimulus to sensation. Measurement formulae. Line elements

**535.646.3** Interval dimensions in colour series. Question of interval equality. Equidistant (uniform) colour space

**535.646.5** Equivalence. Very similar colours. Specific comparison

**535.646.7** Colour tolerances. Small colour differences

**535.647** Colour systems based on subtractive mixing

**535.648** Colour theories of Goethe and Schopenhauer

**535.649** Other colour theories and colour classification systems

**535.65** Colour measurement (colorimetric) methods and apparatus

**535.651** Spectral methods. Evaluation of spectrophotometric data

**535.652** Compensation methods

**535.653** Matching methods

**535.653.1** Methods using a coloured primary stimulus

**535.653.2** Methods using two primary stimuli (Guild's vector method)

**535.653.3** Methods using three primary stimuli. Three-colour colorimeters in general

**535.653.4** Methods using more than three primary stimuli

**535.653.6** Polarization colour methods

**535.653.7** Colour circle methods with control colours

**535.653.8** Matching methods using standards from colour collections. Colour charts and colour atlases. Colour samples

**535.653.8:635.9** Tables of flower colours

*References: 535.653.8*

**535.654** Luminosity methods

**535.654.4** Subjective luminosity methods

**535.654.5** Objective luminosity methods

**535.658** Empirical methods

**535.658.1** Three-filter methods: subjective, objective

**535.658.2** Ostwald's suppression filter and pass filter experiments

**535.66** Colours of objects. Physical conditions of their production

**535.661** Absorption colours (except metallic absorption) in transmitted and reflected light. Intrinsic colours of bodies

**535.661.1** Idiochromatic colours

**535.661.2** Allochromatic colours

**535.662** Colours depending on structure of particles

**535.662.1** Diffraction colours. Interference colours

**535.662.2** Diffusion colours. Colloidal colours. Colours of turbid media

**535.663** Metallic absorption colours

**535.664** Dispersion colours. Imbibition colours

**535.665** Colours of self-luminous bodies. Coloured light

*References: 535.233*

**535.665.2** Colour of daylight

*References: 551.521.16*

**535.665.7** Luminescence colours

*References: 535.37*

**535.668** Technical properties of colours and pigments

**535.668.1** Covering power

**535.668.2** Colouring power. Brightening power

**535.668.6** Whites. Degree of whiteness

**535.67** Colour rendering

**535.672** Colour rendering with light sources and viewing through spectacles

**535.674** Colour rendering by reproduction

**535.68** Mutability of colours of bodies. Fastness of dyes. Colour fastness

**535.682** Influence of heat. Thermal effects on colour

**535.683** Influence of radiation on colour. Fastness to light

**535.684** Influence of chemicals on colours

**535.684.5** Fastness to perspiration

**535.685** Fastness to water. Fastness to washing, boiling etc. Similar kinds of fastness

**535.686** Fastness to mechanical and climatic influences

**535.8** Applications of optics in general

*References: 520 , 543.4 , 621.32 , 628.9 , 681.7*

**535.92** Wave optics

**535.93** Integrated optics

**536** Thermodynamics. Statistical physics

**536-11** Reversible

**536-12** Irreversible

**536-3** Under constant external influences

**536-33** At constant temperature. Isothermal

**536-34** With constant energy. Adiabatic. Isentropic

**536-35** Polytropic

**536-36** At constant pressure. Isobaric. Isopiesic

**536-37** Of constant volume. Isochoric. Isometric

**536.08** Measurement methods and apparatus

**536.1** General theory of heat

**536.11** Fundamentals of heat theory

**536.12** Temperature distribution in bodies

**536.13** Heat potential

**536.14** Quantity of heat

**536.2** Heat conduction. Heat transfer

**536.2.01** Theory and nature of conduction

*References: 536.2*

**536.2.02** General laws of conduction

*References: 536.2*

**536.2.022** Coefficient of thermal conductivity. Heat insulating capacity

**536.21** Conduction in solids

**536.22** Conduction in liquids

**536.23** Conduction in gases

**536.24** Heat transfer. Heat exchange. Heat transmission

**536.241** Solid-solid

**536.242** Solid-liquid. Solid-gas

**536.243** Liquid-solid-liquid

**536.244** Gas-solid

**536.245** Gas-solid-gas

**536.246** Gas-solid-liquid

**536.247** Liquid-liquid

**536.25** Convection without material boundary action

**536.251** In solids

**536.252** In liquids

**536.253** In gases

**536.255** In heterogeneous systems

**536.27** Direct flow and counterflow in heat exchangers

**536.3** Effect of bodies on heat radiation

**536.3:535.312** Reflection of heat

*References: 536.3*

**536.3:535.34** Absorption of heat

*References: 536.3*

**536.33** Heating and cooling by radiation

**536.331** Heating by radiation

**536.332** Cooling by radiation. Cooling curves

**536.35** Diathermancy

**536.37** Other thermal effects due to physical processes

**536.4** Effect of heat input and temperature on volume and structure of bodies

**536.4.032** Lowering and raising of transition points, e.g. melting point

**536.4.033** Influence of heterogeneities on thermal change of state

**536.41** Influence of temperature on volume

**536.411** Expansion of gases

**536.412** Expansion of liquids

**536.413** Expansion of solids

**536.414** Effect of sudden change of state

**536.416** Combined effects. Expansion. Contraction. Shrivelling

**536.42** Effect of heat gain or heat loss on volume and structure. Phase change

**536.421** Change of solid to liquid, liquid to solid

**536.421.1** Change of solid to liquid. Melting. Melting point

**536.421.2** Plastic-liquid phase. Flow point. Softening point

**536.421.3** Droplets. Dropping point

**536.421.4** Change of liquid to solid. Solidification. Freezing. Freezing point

*References: 532.78*

**536.421.48** Supercooling

*References: 532.782*

**536.421.5** Sintering

**536.422** Direct conversion of solid to gas, gas to solid. Sublimation

**536.422.1** Change of solid to gas. Vaporization. Gasification. Vapour pressure

**536.422.4** Change of gas to solid. Solidification of gas

**536.423** Change of liquid to gas, gas to liquid

**536.423.1** Change of liquid to gas. Evaporation. Boiling point

**536.423.15** Vapour pressure of liquids. Saturation pressure

**536.423.16** Volatility

**536.423.18** Superheating

**536.423.4** Change of gas to liquid. Condensation. Condensation point (dew point)

**536.423.8** Prevention of evaporation

**536.424** Allotropic transformations

**536.424.1** Solid to solid

**536.424.5** Liquid to liquid

**536.425** Phase changes in heterogeneous bodies

**536.44** Critical temperatures

**536.45** High-temperature phenomena. Production of high temperatures

**536.48** Low-temperature phenomena. Production of low temperatures

**536.483** Very low temperatures. Cryogenics

**536.485** Resistance to cold. Resistance to freezing. Frost resistance

**536.49** Other effects of heat input and temperature on volume and structure

**536.491** Dependence of the effect of heat on the shape of bodies. Accumulation of heat

**536.495** Heat resistance

**536.5** Temperature. Temperature scales. Temperature measurement. Thermometers. Temperature control

**536.5.081.3** Absolute temperature scales

*References: 536.5*

**536.5.081.3K** Kelvin scale

*References: 536.5*

**536.5.081.4** Relative and practical temperature scales

*References: 536.5*

**536.5.081.4C** Celsius scale

*References: 536.5*

**536.5.081.4R** RÃ©aumur scale

*References: 536.5*

**536.51** Thermometry and thermometers in general

**536.511** Gas thermometers

**536.512** Mercury thermometers

**536.512.083.62** Mercury thermometers for measuring small temperature differences, e.g. Beckmann thermometer

*References: 536.512*

**536.513** Other thermometers based on expansion of liquids

**536.513.083.64** Six's thermometer

*References: 536.513*

**536.514** Vapour pressure thermometers

**536.516** Thermometers based on expansion of solids

**536.52** Pyrometry

**536.521** Optical pyrometers

**536.521.2** Heat radiation pyrometers

**536.521.3** Light radiation pyrometers

**536.522** Pyrometry based on change of state

**536.522.2** Fusible bodies

**536.522.3** Colours as temperature indicators. Thermocolour methods

**536.53** Electrical thermometry

**536.531** Resistance thermometers

**536.532** Thermometers with thermocouple elements

**536.54** Calorimetric thermometry

*References: 536.6*

**536.55** Other methods

**536.551** Thermometry based on outflow of gases

**536.58** Temperature control

*References: 681.536.5*

**536.581** Thermostats

**536.582** Control by expansion

**536.584** Control with freezing mixtures or boiling media

**536.587** Electrical temperature control

**536.6** Measurement of heat quantity. Calorimetry

**536.6.081** Calorimetric units

*References: 536.6*

**536.62** Calorimetric methods and apparatus in general

**536.621** Mixing calorimeters

**536.622** Ice calorimeters

**536.623** Vaporization calorimeters

**536.624** Condensation calorimeters

**536.626** Combustion calorimeters

**536.627** Calorimeters with electric energizing of test substance

**536.628** Experimental methods of overcoming heat losses

**536.629.7** Heat flow meters

**536.63** Specific heat. Heat capacity

**536.631** Heat capacity of solids

**536.632** Heat capacity of liquids

**536.633** Heat capacity of gases and vapours

**536.65** Latent heat. Other physical thermal effects

**536.651** Heat of expansion

**536.652** Heat of fusion

**536.653** Heat of solution. Heat of dilution

**536.654** Heat of sublimation

**536.655** Heat of vaporization

**536.656** Heat of allotropic transformation

**536.657** Heat of dissociation

**536.658** Heat of adsorption

**536.659** Heat of surface energy

**536.66** Heat of reaction. Chemical heat effects. Heat of formation

*References: 544.332*

**536.662** Heat of combustion. Calorific value

**536.664** Heat of hydration

**536.665** Heat of neutralization

**536.666** Heat generated during other chemical reactions

**536.68** Determination of heat requirements for room heating, heating of solid masses etc.

*References: 697.1*

**536.7** Thermodynamics. Energetics

**536.71** Changes of state. Thermal equations of state

**536.711** Adiabatic changes of state

**536.712** Isothermal changes of state

**536.713** Changes of state at constant pressure

**536.714** Changes of state with constant volume

**536.715** Changes of state under other conditions

**536.717** Cycles. Change-of-state diagrams. Heat diagrams

**536.72** First law of thermodynamics. Mechanical equivalent of heat. Law of conservation of energy

**536.721** Mechanical equivalent of heat

**536.722** Forms of energy. Internal and external energy. Enthalpy

**536.725** Perpetual motion of the first kind

**536.73** Second law of thermodynamics

**536.735** Perpetual motion of the second kind

**536.74** Thermodynamic temperature

**536.75** Entropy. Statistical thermodynamics. Irreversible processes

**536.76** Relationships deduced from the two laws of thermodynamics. Phase transformations

**536.763** Clausius-Clapeyron equation. Phase transformations of the first order

**536.764** Phase transformations of a higher order

**536.765** Maxwell's laws

**536.77** Free energy. Thermodynamic potential

**536.773** Van't Hoff-Le Chatelier principle

**536.775** Gibbs-Helmholtz principle. Phase rule. Degrees of freedom

**536.78** Joule's law of intrinsic energy of gases

**536.791** Third law of thermodynamics. Nernst's heat theorem.

**536.791.2** Degeneration of gases

**536.791.5** Perpetual motion of the third kind. Theorem of the unattainability of absolute zero

**536.9** Statistical physics

**536.91** Computational statistical physics

**536.911** Monte Carlo methods

*References: 519.245*

**536.912** Molecular dynamics methods

**536.92** Classical statistical mechanics. Phase transformation

**536.93** Quantum statistical mechanics

**536.931** Fermions

**536.932** Bosons

**536.933** Boltzmann distributions, Boltzmann factor. Fermi-Dirac distributions, Bose-Einstein distributions. Condensations. Ideal Bose gas. Ideal Fermi gas. Marginal cases

**536.934** Fractional quantum statistics

**536.935** Applied quantum statistical mechanics. Superconductivity Superfluidity. Econophysics

**536.94** Fluctuations. Noise. Random processes

**536.95** Brownian motion

**536.96** Lattice models

**536.97** Physical kinetics

**536.971** Transport phenomena

**537** Electricity. Magnetism. Electromagnetism

*References: 621.3*

**537-7** Potential

**537-72** Low potential

**537-73** Medium potential

**537-77** High potential. Very high potential

**537-9** Frequency

**537-94** Low frequency

**537-95** Medium frequency

**537-96** High frequency. Very high frequency

**537-961** Short waves

**537-962** Very short waves. Microwaves

**537.1** Theory of electricity

**537.11** Fundamental concepts

**537.12** Negative electricity

**537.13** Positive electricity

*References: 537.534 , 539.124.6 , 539.125.4*

**537.2** Static electricity. Electrostatics

*References: 621.319*

**537.21** Generalities

**537.211** Law of electric attraction

**537.212** Electric fields

**537.213** Potential. Potential fields

**537.214** Quantity of electricity. Electrical energy

**537.215** Identity of static and dynamic electricity

**537.217** Attraction. Repulsion

**537.218** Electric intensity

**537.22** Generation, distribution and collection of static electricity

**537.221** Triboelectricity (frictional electricity). Sources of static electricity

*References: 537.226.86*

**537.222** Electric charge

**537.226** Electrical properties of dielectrics

**537.226.1** Permittivity in general

**537.226.2** Permittivity in materials. Relative permittivity, *@br@B (dielectric constant)

**537.226.3** Dielectric losses, tan*

**537.226.4** Ferroelectricity

**537.226.5** Isotropy, anisotropy of dielectrics

**537.226.7** Dielectric breakdown strength

**537.226.8** Effects of external factors on electrical properties of dielectrics

**537.226.82** Temperature effect. Pyroelectricity

**537.226.83** Electrical field effects

**537.226.86** Mechanical effect. Effect of mechanical stress (deformation). Piezoelectricity

*References: 537.221*

**537.228** Effects of electric field

**537.228.3** Electrical double refraction. Kerr effect

**537.228.4** Johnsen-Rahbek effect

**537.228.5** Stark effect

**537.24** Static charge and discharge

**537.241** Charge

**537.242** Discharge. Electroprecipitation

**537.243** Change of sign of charge

**537.246** Permanent charge

**537.27** Effects on electric fields

**537.291** Trajectories of charged particles in electric fields

**537.3** Current electricity. Electric current. Electrokinetics

*References: 544.6*

**537.31** Transmission, conduction of electricity. Electrical conduction

**537.311** Generalities

**537.311.1** Nature of conduction. Theories of motion of electrons in conductors

**537.311.2** Ohm's law

**537.311.3** Resistivity and conductivity. Specific resistance. Specific conductance

**537.311.31** In conductors: metals, electrolytes

*References: 544.623 , 621.315.5*

**537.311.32** In dielectrics

**537.311.322** In semiconducting materials

**537.311.324** In insulating materials

**537.311.33** In a vacuum

**537.311.35** In nonelectrolytic liquids

**537.311.37** In gases

**537.311.4** Contact resistance. Intermediate resistance

**537.311.5** Distribution of current in conductors

**537.311.6** Resistance to alternating currents

**537.312** Effects of external factors on conductivity and resistivity

**537.312.5** Effects of radiation

**537.312.6** Effects of temperature

*References: 538.945*

**537.312.7** Effects of electric field

**537.312.8** Effects of magnetic field

**537.312.9** Effects of mechanical stresses

**537.313** Branching of currents. Branch circuits (shunt circuits)

**537.314** Velocity of propagation of current

**537.315** Problems of voltage, emf (electromotive force)

**537.32** Thermoelectricity

*References: 621.362*

**537.321** Thermal effects of electric current

*References: 537.226.82*

**537.322** Thermoelectric effects

**537.323** Thermoelectric constants of materials

**537.36** Electrokinetics

*References: 621.359*

**537.361** Electrocapillarity

**537.362** Electroendosmosis

**537.363** Electrophoresis

*References: 543.55*

**537.364** Excitation of potential difference by the motion of a liquid over a surface

**537.365** Excitation of potential difference by the motion of particles through liquids

**537.37** Effects on electric currents other than electrodynamic and electromagnetic effects

**537.39** Effects of electric currents other than electrodynamic and electromagnetic effects

**537.5** Electron and ion phenomena

**537.52** Electric discharges

**537.521** Generalities

*References: 537.56*

**537.523** Discharges in gases at normal pressure

**537.523.2** Discharges from points and edges at normal pressure

**537.523.3** Corona discharges at normal pressure

**537.523.4** Spark discharges at normal pressure

**537.523.5** Arc discharges at normal pressure

**537.525** Discharges in gases at low pressures. Glow discharges

**537.525.2** Emissions, point discharges at low pressures

**537.525.3** Corona discharges at low pressures

**537.525.4** Spark discharges at low pressures

**537.525.5** Arc discharges at low pressures

**537.525.6** Electron discharges at low pressures

**537.525.7** Annular discharges. Discharges without electrodes

**537.527** Discharges in gases at high pressures

**537.527.2** Emissions, point discharges at high pressures

**537.527.3** Corona discharges at high pressures

**537.527.4** Spark discharges at high pressures

**537.527.5** Arc discharges at high pressures

**537.528** Discharges in liquids

**537.529** Discharges in solids

**537.53** Emission of radiation or particles, from discharges

*References: 539.16*

**537.531** Electromagnetic radiation

*References: 621.386*

**537.531.2** Bremsstrahlung (free-free and free-bound transitions)

**537.531.3** Transition radiation (bound-bound transitions)

**537.531.7** Effects on X-rays

**537.531.8** Secondary radiations due to bombardment by X-rays

**537.533** Electron rays (cathode rays) and associated geometrical optics

*References: 539.124*

**537.533.1** Velocity of particles

**537.533.2** Electron emissions. Work function for electron emission

*References: 537.58*

**537.533.3** Electron optics

**537.533.35** Electron microscopy

*References: 621.385.833*

**537.533.7** Effects on electron rays. Passage of electrons through matter

**537.533.71** Reflection

**537.533.72** Refraction

**537.533.73** Diffraction

**537.533.74** Scattering. Ramsauer effect. Effective radius

**537.533.75** Absorption

**537.533.8** Secondary radiation due to bombardment by electrons. Secondary emission

**537.534** Canal rays

**537.534.1** Particle charge. Velocity

**537.534.2** Positive ion sources (positive ray sources)

**537.534.3** Ion optics

**537.534.6** Change of charge in positive rays

**537.534.7** Effects on anode rays

**537.534.8** Secondary radiations due to bombardment by positive ions. Secondary emission

**537.56** Ionization of gases

**537.561** Ionization potential. Resonance potential

**537.562** Properties of ionized gases

**537.563** Kinds of ionization of gases

**537.563.2** Ionization by means of electron impacts

**537.563.3** Ionization by means of impact of atoms and ions

**537.563.5** Ionization by light. Photoionization

**537.563.7** Surface ionization

**537.565** Ion movement in gases

**537.566** Electrical conductivity of flames

**537.567** Effects on ionization

**537.568** Effects of ions

**537.569** Deionization

**537.57** Ionization of nonelectrolytic liquids and solids

**537.571** Ionization of nonelectrolytic liquids

**537.572** Ionization phenomena in solids

**537.58** Thermoionization. Thermoemission of electrons, ions, other particles

**537.6/.8** Magnetism. Electromagnetism

**537.6** Magnetism

*References: 621.318 , 669.018.58*

**537.61** Theory of magnetism

**537.611** Fundamental laws of magnetism

**537.611.2** Theory of elementary magnets, magnetic poles, dipoles and magnetic moments (magnetons)

**537.611.3** Theory of magnetic domains

**537.611.4** Theories of particular types of magnetism

**537.612** Magnetic field. Magnetic potential

**537.613** Distribution of magnetism

**537.62** Magnetization processes

**537.621** Magnetic properties in general

**537.621.2** Magnetic resistance (reluctance)

**537.621.3** Magnetic permeability. Influences on magnetic permeability

**537.621.4** Magnetic susceptibility

**537.621.5** Magnetic moment

**537.622** Magnetic properties of materials

*References: 621.318.1*

**537.622.2** Diamagnetic materials

**537.622.3** Paramagnetic materials

**537.622.4** Ferromagnetic materials

**537.622.5** Antiferromagnetic materials

**537.622.6** Ferrimagnetic materials

**537.623** Magnetic hysteresis. Hysteresis loss. Other energy losses in a magnetic field

*References: 539.98*

**537.624** Magnetic polarity reversal. Magnetization. Demagnetization

**537.626** Magnetic circuit. Flux-bar physics

**537.63** Effects of and on magnetic fields

**537.632/.636** Effects of magnetic fields

**537.632** Optical effects of magnetic fields. Magneto-optics

**537.632.2** Double refraction

**537.632.3** Effects on polarized light

**537.632.4** Rotatory polarization in a magnetic field

**537.632.5** Effects on light emission and absorption

**537.633** Electrical effects of magnetic fields

**537.634** Mechanical effects of magnetic fields

**537.635** Magnetic resonances

**537.636** Other effects of magnetic fields

**537.638/.639** Effects on magnetic fields

**537.638** Effects of temperature on magnetization and thermal changes accompanying magnetization

**537.638.2** Effects of temperature on magnetic fields

**537.638.3** Pyromagnetism

**537.638.5** Magnetothermal effects

**537.639** Other effects on magnetic fields

**537.67** Terres