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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

References: 59 , 598.2

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

References: 551.46 , 574.6

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

References: 582 , 59

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

References: 519.76 , 81`32

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

References: 515.128 , 517.517

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

References: 81`32 , 519.765

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)

References: 511.344 , 519.1

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

References: 512.7 , 517

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

References: 511.178 , 519.116

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

References: 512.75 , 512.774

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

References: 511.2 , 512.74

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 André-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

References: 512.55 , 512.62

512.712 Valuations on commutative rings and theory of divisibility

512.713 Arithmetic rings. Dedekind and Prüfer 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

References: 511.384 , 515.178

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

References: 510.35 , 517.517

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

References: 510.35 , 515.128

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. Fréchet 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

References: 510.75 , 510.825

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

References: 004.3 , 51-3

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

References: 007 , 621.391

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

References: 330.4 , 510.8

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

References: 52-6 , 52-7

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 µm to 0.7 µm)

52-76 Infrared radiation (lambda 0.7 µm 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

References: 681.782.7 , 727

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

References: 520.25 , 520.4

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

References: 626.8 , 711

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

References: 551.24 , 624.131

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

References: 535 , 537.86

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

References: 528.918 , 744.43

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

References: 531.7 , 681.2

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

References: 621.398 , 654.94

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

References: 549.086 , 57.086

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

References: (15) , 115.4

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

References: 531.35 , 531.42

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 ämxy 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

References: 532.14 , 533.12

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

References: 542.3 , 681.12

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

References: 534 , 551.466

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

References: 536.97 , 544.35

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

References: 536.421.4 , 548.5

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

References: 621.592 , 66.078

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

References: 531.788 , 621.52

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

References: 539.91 , 532.593

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

References: 615.837 , 66.084

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

References: 534.61 , 534.79

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

References: 534.323 , 534.79

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

References: 534.323 , 534.61

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

References: 681.84 , 681.85

534.851 Mechanical recording. Gramophone records

534.852 Magnetic recording

534.853 Optical recording

534.86 Sound transmission (radio, television, film). Electroacoustics

References: 621.39 , 654.191

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

References: 537.53 , 539.184

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)

References: 577.34 , 591.148

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 (Schrödinger)

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 Réaumur 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

References: 621.362 , 621.363

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

References: 532.7 , 533.7

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

References: 537.533 , 539.124

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

References: 536.532 , 621.363

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

References: 533.9 , 621.387

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