This is a
list of theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
s, by Wikipedia page. See also
- Classification of finite simple groups
The classification of the finite simple groups, also called the enormous theorem, is believed to classify all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural...
- Comparison theorem
A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.-Riemannian geometry:In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.*Rauch...
- Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
- Fixed point theorem
- Homotopy theorem
- list of fundamental theorems
- list of lemmas (Fodor's lemma
In mathematics, particularly in set theory, Fodor's lemma states the following:If is a regular, uncountable cardinal, is a stationary subset of , and is regressive then there is some and some stationary such that for any...
, etc.)
- list of conjectures
- list of inequalities
- list of mathematical proofs
- list of misnamed theorems
- Open mapping theorem
In mathematics, there are two theorems with the name "open mapping theorem". In both cases, they give conditions under which certain maps are open maps, i.e. they map open sets to open sets...
- Toy theorem
In mathematics, a toy theorem is a simplified version of a more general theorem. For instance, by introducing some simplifying assumptions in a theorem, one obtains a toy theorem....
- Vanishing theorem
In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish.*Grauert–Riemenschneider vanishing theorem*Kawamata–Viehweg vanishing theorem*Kollár vanishing theorem*Kodaira vanishing theorem*Miyaoka vanishing theorem...
Most of the results below come from
pure mathematicsBroadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty...
, but some are from
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
,
economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and other
appliedApplied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
fields.
0–9
- 15 and 290 theorems (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- 2π theorem (Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
A
- AF+BG theorem
In algebraic geometry, a field of mathematics, the AF+BG theorem is a result of Max Noether which describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic curves....
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- ATS theorem
In mathematics, the ATS theorem is the theorem on the approximation of atrigonometric sum by a shorter one. The application of the ATS in certain problems of mathematical and theoretical physics can be very helpful.- History of the problem :...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Abel's binomial theorem
Abel's binomial theorem, named after Niels Henrik Abel, states the following:...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Abel's curve theorem
Abel's curve theorem, named after Niels Henrik Abel, is a theorem that relates fixed points to the parameters of rational functions....
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Abelian and tauberian theorems
In mathematics, abelian and tauberian theorems relate to the meaningful assignment of a value as the "sum" of a class of divergent series. A large number of methods have been proposed for the summation of such series, generally taking the form of some linear functional L with domain contained in...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Abel–Jacobi theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Abel–Ruffini theorem
The Abel–Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher.-Misinterpretation:...
(theory of equationsIn mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...
, Galois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
)
- Abhyankar–Moh theorem
In mathematics, the Abhyankar–Moh theorem states that any embedding of the affine line in the affine plane extends to an automorphism of the plane....
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Abouabdillah's theorem
Abouabdillah's theorem refers to two distinct theorems in mathematics, proven by Moroccan mathematician Driss Abouabdillah: one in geometry and one in number theory.-Geometry:In geometry, similarities of an Euclidean space preserve circles and spheres...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
,number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Absolute convergence theorem
In mathematics, especially calculus, the absolute convergence theorem states that if a series of real or complex numbers is absolutely convergent, then it is also convergent.-Proof:Assume is convergent...
(mathematical series)
- Acyclic models theorem
In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish...
(algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
)
- Addition theorem
In mathematics, an addition theorem is a formula such as that for the exponential functionthat expresses, for a particular function f, f in terms of f and f...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Adiabatic theorem
The adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock , can be stated as follows:...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Ado's theorem
In mathematics, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket...
(Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
)
- Akra–Bazzi theorem (computer science
Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
)
- Albert–Brauer–Hasse–Noether theorem
In mathematics, the Albert–Brauer–Hasse–Noether theorem or has been characterized as the most profound result in the theory of central simple algebras. The theorem states that if A is a central simple algebra over an algebraic number field K, then A is a cyclic algebra...
(algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
s)
- Alperin–Brauer–Gorenstein theorem (finite groups)
- Analytic Fredholm theorem
In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and Hilbert-Schmidt theorems...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Anderson's theorem
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin...
(real analysisReal analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Andreotti–Frankel theorem
In mathematics, the Andreotti–Frankel theorem states that if is a smooth affine variety of complex dimension or, more generally, if is any Stein manifold of dimension , then in fact is homotopy equivalent to a CW complex of real dimension at most n. In other words has only half as much...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Angle bisector theorem
In geometry, the angle bisector theorem relates the length of the side opposite one angle of a triangle to the lengths of the other two sides of the triangle....
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Ankeny–Artin–Chowla theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Apéry's theorem
In mathematics, Apéry's theorem is a result in number theory that states the number ζ is irrational. That is, the numbercannot be written as a fraction p/q.-History:...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Apollonius' theorem
In elementary geometry, Apollonius' theorem is a theorem relating several elements in a triangle.It states that given a triangle ABC, if D is any point on BC such that it divides BC in the ratio n:m , then...
(plane geometryIn mathematics, plane geometry may refer to:*geometry of a plane,*geometry of the Euclidean plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;...
)
- Appell–Humbert theorem
In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety.It was proved for 2-dimensional tori by and .-Statement:...
(complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
)
- Area theorem (conformal mapping)
In the mathematical theory of conformal mappings, the area theoremgives an inequality satisfied bythe power series coefficients of certain conformal mappings....
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Arithmetic Riemann–Roch theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Aronszajn–Smith theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Arrival theorem (queueing theory
Queueing theory is the mathematical study of waiting lines . The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served by the server at the front of the queue...
)
- Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
(game theoryGame theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy...
)
- Art gallery theorem (geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Artin approximation theorem
In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k....
(commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
)
- Artin–Schreier theorem (real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
s)
- Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that a semisimple ring R is isomorphic to a product of ni-by-ni matrix rings over division rings Di, for some integers ni, both of...
(abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
)
- Artin–Zorn theorem (algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
)
- Artstein's theorem
Artstein's theorem states that a dynamical system has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback ....
(control theoryControl theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
)
- Arzelà–Ascoli theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
(differential topologyIn mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
)
- Atiyah–Segal completion theorem (homotopy theory)
- Atiyah–Singer index theorem
In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
(elliptic differential operators, harmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
)
- Atkinson's theorem
In operator theory, Atkinson's theorem gives a characterization of Fredholm operators.- The theorem :Let H be a Hilbert space and L the bounded operators on H...
(operator theoryIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
)
- Aumann's agreement theorem
Aumann's agreement theorem, informally stated, says that two people acting rationally and with common knowledge of each other's beliefs cannot agree to disagree...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Autonomous convergence theorem
In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system.-History:...
(dynamical systemThe dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
s)
- Auxiliary polynomial theorem
The construction of auxiliary polynomials is an important concept in diophantine approximation and transcendental number theory.-Statement:Let β equal the cube root of b/a in the equation ax3 + bx3 = c and assume m is and integer that satisfies m + 1 > 2n/3 ≥ m ≥ 3 where n is...
(diophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
)
- Ax–Grothendieck theorem (model theory
In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language...
)
- Ax–Kochen theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for any given natural number d there is a finite set Y of exceptional primes, such that if p is any prime not in Y then every non-constant homogeneous polynomial of degree d over the p-adic numbers in at least...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Aztec diamond theorem (combinatorics
Combinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
B
- BEST theorem (graph theory
In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Babuška-Lax-Milgram theorem
In mathematics, the Babuška-Lax-Milgram theorem is a generalization of the famous Lax-Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem...
(partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s)
- Baily-Borel theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Baire category theorem
The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
(topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
, metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s)
- Balian-Low theorem
In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low.Suppose g is a square-integrable function on the real line, and consider the so-called Gabor systemfor integers m and n...
(Fourier analysis)
- Balinski's theorem
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher-dimensional polytopes...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Banach-Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Banach–Mazur theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Banach fixed point theorem
The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points...
(metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s, differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s)
- Banach-Steinhaus theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Banach–Stone theorem (operator theory
In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
)
- Barbier's theorem
Barbier's theorem is a basic result on curves of constant width first proved by Joseph Emile Barbier.The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. A circle of width w has perimeter πw. A Reuleaux triangle of width w consists of three arcs of...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Bapat-Beg theorem
In probability theory, the Bapat–Beg theorem gives the joint cumulative distribution function of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Baranyai's theorem
In combinatorial mathematics, Baranyai's theorem deals with the decompositions of complete hypergraphs proved by Zsolt Baranyai.-Statement of the theorem:...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Barta's theorem
In mathematics, Barta's theorem involves the estimation of first eigenvalues of precompact domains on manifolds that can be extended to precompact domains of geometrized graphs.-Theorem:...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Bass's theorem (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Basu's theorem
In statistics, Basu's theorem states that any complete sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu....
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Bauer-Fike theorem
In mathematics, the Bauer-Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact...
(spectral theoryIn mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many...
)
- Bayes' theorem
In probability theory, Bayes' theorem shows how one conditional probability depends on its inverse . The theorem expresses the posterior probability In probability theory, Bayes' theorem (often called Bayes' law or Bayes' rule, and named after Rev. Thomas Bayes; IPA:/'beɪz/) shows how one...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Beatty's theorem (diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
)
- Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by .-The theorem:...
(vector bundles)
- Beck's monadicity theorem
In category theory, a branch of mathematics, Beck's monadicity theorem asserts that a functoris monadic if and only if# U has a left adjoint;# U reflects isomorphisms; and...
(category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
)
- Beck's theorem
In incidence geometry, Beck's theorem is a more quantitative form of the more classical Sylvester–Gallai theorem. It says that finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines...
(incidence geometry)
- Beer's theorem (metric geometry)
- Bell's theorem
Bell's theorem is a no-go theorem, loosely stating that:It is the most famous legacy of the late physicist John S. Bell. The theorem has important implications for physics itself and philosophy of science as well.- Overview :...
(quantum theoryQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
- physics)
- Beltrami's theorem
In mathematics — specifically, in Riemannian geometry — Beltrami's theorem is a result named after the Italian mathematician Eugenio Beltrami which states that geodesic maps preserve the property of having constant curvature...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Belyi's theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.It follows that the Riemann surface in...
(algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s)
- Bendixson-Dulac theorem
In mathematics, the Bendixson-Dulac theorem on dynamical systems states that if there exists such thatalmost everywhere in the region of interest, which must be simply connected, then the plane autonomous system...
(dynamical systemThe dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
s)
- Berger–Kazdan comparison theorem (Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Bernstein's theorem
In real analysis, a branch of mathematics, Bernstein's theorem states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Berry-Esséen theorem (probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
)
- Bertini's theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Bertrand–Diquet–Puiseux theorem
In the mathematical study of the differential geometry of surfaces, the Bertrand–Diquet–Puiseux theorem expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle, or the area of a geodesic disc. The theorem is named for Joseph Bertrand, Victor Puiseux, and V...
(differential geometry)
- Bertrand's ballot theorem
In combinatorics, Bertrand's ballot theorem is the solution to the question: "In an election where one candidate receives p votes and the other q votes with p>q, what is the probability that the first candidate will be strictly ahead of the second candidate throughout the count?" The answer isIt...
(probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Bertrand's postulate
Bertrand's postulate states that if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n − 2...
(prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s)
- Betti's theorem
Betti's theorem, discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,m and {Qj}, j=1,2,...,n, the work done by the set P through the displacements produced by the set Q is equal to the work done by the set Q...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Beurling–Lax theorem
In mathematics, the Beurling–Lax theorem is a theorem due to and which characterizes the shift-invariant subspaces of the Hardy space . It states that each such space is of the formfor some inner function ....
(Hardy spaceIn complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
s)
- Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
(algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s)
- Bing metrization theorem
The Bing metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space is metrizable if and only if it is regular and T0 and has a σ-discrete basis...
(general topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
)
- Bing's theorem
In topology, Bing's theorem, named for RH Bing, asserts that a necessary and sufficient condition for a 3-manifold M to be homeomorphic to the 3-sphere is that every Jordan curve in M be contained within a topological ball....
(geometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
)
- Binomial inverse theorem
In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways.If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then...
(matrix theory)
- Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power
n into a sum involving terms of the form ax
by
c, where the coefficient of each term is a positive...
(algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
, combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Birch's theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.-Statement of Birch's theorem:...
(Diophantine equationIn mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
)
- Birkhoff-Grothendieck theorem
In mathematics, the Birkhoff–Grothendieck theorem concerns properties of vector bundles over complex projective space . It reduces every vector bundle over into direct sum of tautological line bundles, which enables one to deal with the bundle in a practical way...
(vector bundles)
- Birkhoff-Von Neumann theorem (matrix theory)
- Birkhoff's representation theorem
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets...
(lattice theory)
- Birkhoff's theorem
Birkhoff's theorem may refer to several theorems named for the American mathematician G. D. Birkhoff:* Birkhoff's theorem * Birkhoff's theorem * Birkhoff's ergodic theorem...
(ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systemswith an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
)
- Birkhoff's theorem (relativity)
In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric....
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Bishop-Cannings theorem
The Bishop–Cannings theorem is a theorem in evolutionary game theory. It states that all members of a mixed evolutionarily stable strategy have the same payoff , and that none of these can also be a pure ESS...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Blaschke selection theorem
The Blaschke selection theorem is a result in topology about sequences of convex sets. Specifically, given a sequence of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence and a convex set such that converges to in the Hausdorff metric...
(geometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
)
- Bloch's theorem
Bloch's theorem, named after André Bloch is as follows.Let ƒ be a holomorphic function on a region that includes as a subset the closed unit disk |z| ≤ 1. Suppose ƒ = 0 and ƒ ′ = 1...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Blondel's theorem
Blondel's theorem, named after André Blondel, states that, in a system of N electrical conductors, N-1 electrical meter or wattmeter elements, when properly connected, will measure the electrical power or energy taken. The connection must be such that all potential coils have a common tie to the...
(electric powerElectric power is defined as the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt.When electric current flows in a circuit, it can transfer energy to do mechanical or thermodynamic work...
) (physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Blum's speedup theorem
In computational complexity theory Blum's speedup theorem, first stated by Manuel Blum in 1967, is an important theorem about the complexity of computable functions....
(computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
)
- Bôcher's theorem
In mathematics, Bôcher's theorem can refer to one of two theorems proved by the American mathematician Maxime Bôcher.-Bôcher's theorem in complex analysis:...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Bogoliubov-Parasyuk theorem
The Bogoliubov-Parasyuk theorem in quantum field theory states that renormalized Green's functions and matrix elements of the scattering matrix are free of ultraviolet divergencies. Green's functions and scattering matrix are the fundamental objects in quantum field theory which determine basic...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, defined for x > 0 by...
(gamma functionIn mathematics, the Gamma function is an extension of the factorial function to real and complex numbers...
)
- Bohr–van Leeuwen theorem
The Bohr–van Leeuwen theorem is a theorem in the field of solid state physics. The theorem posits that when applying classical statistics the magnetization in the thermal balance is zero because the kinetic energy of a pull in the magnetic field does not alter it...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Bolyai-Gerwien theorem (geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Bolzano's theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
, calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
)
- Bolzano-Weierstrass theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
, calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
)
- Bombieri's theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Bombieri–Friedlander–Iwaniec theorem
In analytic number theory, an advanced branch of mathematics, the Friedlander–Iwaniec theorem asserts that there are infinitely many prime numbers of the form...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Bondareva-Shapley theorem
In game theory, the Bondareva-Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva-Shapley theorem implies that market games and convex...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Bondy's theorem
In mathematics, Bondy's theorem is a theorem in combinatorics that appeared in a 1972 article by John Adrian Bondy. The theorem is as follows:In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Bondy-Chvátal theorem (graph theory
In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Bonnet theorem
In the mathematical field of differential geometry, more precisely, the theory of surfaces in Euclidean space, the Bonnet theorem states that the first and second fundamental forms determine a surface in R3 uniquely up to a rigid motion. It was proven by Pierre Ossian Bonnet in about...
(differential geometry)
- Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Borel-Bott-Weil theorem (representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...
)
- Borel–Carathéodory theorem
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory....
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Borel-Weil theorem (representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...
)
- Borel determinacy theorem
In descriptive set theory, the Borel determinacy theorem shows that any Gale-Stewart game whose winning set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. It was proved by Donald A. Martin in 1975...
(set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
)
- Borel fixed-point theorem
In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry. The result was proved by the Swiss mathematician Armand Borel in 1956.-Statement of the theorem:...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Borsuk-Ulam theorem (topology
Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
)
- Bott-Duffin theorem (network theory
Network theory is an area of applied mathematics and network science and part of graph theory. It has application in many disciplines including particle physics, computer science, biology, economics, operations research, and sociology...
)
- Bott periodicity theorem
In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable...
(homotopy theory)
- Bounded convergence theorem (measure theory)
- Bounded inverse theorem
In mathematics, the bounded inverse theorem is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1...
(operator theoryIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
)
- Bourbaki–Witt theorem
In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a chain complete poset, andsuch thatthen f has a fixed point...
(order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
)
- Brahmagupta theorem
Brahmagupta's theorem is a result in geometry. It states that if a cyclic quadrilateral has perpendicular diagonals, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Branching theorem
In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.-Statement of the theorem:...
(complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
)
- Brauer–Nesbitt theorem (representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
)
- Brauer-Siegel theorem
In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Brauer–Suzuki theorem
In mathematics, the Brauer-Suzuki theorem states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2...
(finite groups)
- Brauer's theorem
In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.-Statement of Brauer's theorem:...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Brauer's theorem on induced characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group...
(representation theory of finite groupsIn mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
)
- Brauer's three main theorems
Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group with those of its p-local subgroups, that is to say, the normalizers of its non-trivial p-subgroups....
(finite groups)
- Brauer–Cartan–Hua theorem
The Brauer–Cartan–Hua theorem is a theorem in abstract algebra pertaining to division rings, which says that given two division rings K ≤ D such that xKx−1 is contained in K for every x not equal to 0 in D, then either K is contained in Z, the center of D, or...
(ring theoryIn mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
)
- Brianchon's theorem
In geometry, Brianchon's theorem, named after Charles Julien Brianchon , is as follows. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then lines AD, BE, CF intersect at a single point....
(conics)
- Brooks’ theorem
In graph theory, Brooks' theorem states a relationship between the degree of a graph and its chromatic number. According to the theorem, in a graph in which every vertex has at most Δ neighbors, the vertices may be colored with only Δ colors, except for two cases, complete graphs and cycle graphs...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Brouwer fixed point theorem
In mathematics, Brouwer's fixed point theorem is a theorem in topology, named after Luitzen Brouwer. It is one of many fixed point theorems, which state that for any continuous function f with certain properties there is a point x
0 such that f = x
0.The simplest form...
(topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
)
- Browder-Minty theorem
In mathematics, the Browder-Minty theorem states that a bounded, continuous, coercive and monotone function T from a real, reflexive Banach space X into its continuous dual space X∗ is automatically surjective...
(operator theoryIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
)
- Brown's representability theorem
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor...
(homotopy theory)
- Bruck-Chowla-Ryser theorem (combinatorics
Combinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Brun's theorem
In mathematics, Brun's theorem is a result in number theory proved by Viggo Brun in 1919. It states that the sum of the reciprocals of the twin primes is convergent with a finite value known as Brun's constant. It has historical importance in the introduction of sieve methods.Let P denote the...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Brun-Titchmarsh theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Brunn-Minkowski theorem
In mathematics, the Brunn–Minkowski theorem is an inequality relating the volumes of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem applied to convex sets; the generalization to compact nonconvex sets stated here is due to L.A...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Buckingham π theorem (dimensional analysis
In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature,...
)
- Burke's theorem
In probability theory, Burke's theorem is a theorem in queueing theory by Paul J. Burke while working at Bell Telephone Laboratories that states for an M/M/1, M/M/m or M/M/∞ queue in the steady state with arrivals a Poisson process with rate parameter λ then:# The departure process is a Poisson...
(probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) (queueing theoryQueueing theory is the mathematical study of waiting lines . The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served by the server at the front of the queue...
)
- Burnside theorem
In mathematics, Burnside's theorem in group theory states that if G is a finite group of orderwhere p and q are prime numbers, and a and b are non-negative integers, then G is solvable...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Busemann's theorem
In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces.-Statement of the theorem:...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Butterfly theorem
The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
C
- CPCTC
In geometry, CPCTC is the abbreviation of a theorem involving congruent triangles. CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent...
(triangle geometry)
- Cameron–Martin theorem (measure theory)
- Cantor–Bernstein–Schroeder theorem
In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B...
(Set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, cardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s)
- Cantor's theorem
In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
(Set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, Cantor's diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers...
)
- Carathéodory-Jacobi-Lie theorem
The Carathéodory-Jacobi-Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.-Statement:Let M be a 2n-dimensional symplectic manifold with symplectic form ω...
(symplectic topologySymplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
)
- Carathéodory's existence theorem
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem...
(rdinary differential
- Carathéodory's theorem
In mathematics, Carathéodory's theorem in complex analysis states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map...
(conformal mapping)
- Carathéodory's theorem
In convex geometry Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset P′ of P consisting of d+1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in a r-simplex with vertices in P, where...
(convex hull)
- Carathéodory's theorem (measure theory)
- Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem proves that for a given set Ω, one can always extend a σ-finite measure defined on R to the σ-algebra generated by R, where R is a ring included in the power set of Ω; moreover, the extension is unique...
(measure theory)
- Caristi fixed point theorem
In mathematics, the Caristi fixed point theorem generalizes the Banach fixed point theorem for maps of a complete metric space into itself. Caristi's fixed point theorem is a variation of the ε-variational principle of Ekeland...
(fixed pointsIn mathematics, a fixed point of a function is a point that is mapped to itself by the function...
)
- Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem about a summable expansion of an analytic function. It is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for expansions in other bases of...
(Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Carmichael's theorem
Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F has at least one prime factor that is not a factor of any earlier Fibonacci number. The only exceptions for n up to 12 are:...
(Fibonacci numberIn mathematics, the Fibonacci numbers are the numbers in the following sequence:By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two...
s)
- Carnot's theorem
In Euclidean geometry, Carnot's theorem, named after Lazare Carnot , is as follows. Let ABC be an arbitrary triangle. Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Carnot's theorem (thermodynamics
In physics, thermodynamics is the study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature, volume and pressure...
)
- Cartan–Dieudonné theorem (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Cartan–Hadamard theorem
The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Cartan–Kähler theorem
In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler....
(partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s)
- Cartan–Kuranishi prolongation theorem (partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s)
- Cartan's theorem
In mathematics, there are three results in Lie group theory that go by the name Cartan's theorem. They are both named for Élie Cartan.See also Cartan's theorems A and B, results of Henri Cartan....
(Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
)
- Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.Theorem A states that F is spanned...
(several complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
)
- Casey's theorem
In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.- Formulation of the theorem :...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Castelnuovo theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Castelnuovo–de Franchis theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Castigliano's first and second theorems
Castigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy...
(structural analysisStructural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. The subjects of structural analysis are engineering artifacts whose integrity is judged largely based upon their ability to withstand loads; they commonly include...
)
- Cauchy integral theorem (Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Cauchy-Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard...
(Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Cauchy-Kowalevski theorem (partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s)
- Cayley-Bacharach theorem
In mathematics, the Cayley-Bacharach theorem is a statement in projective geometry which contains as a special case Pascal's theorem. The Cayley-Bacharach theorem pertains to the family of cubic curves passing through eight given pointsthat lie "in general position" in the projective plane over...
(projective geometryIn mathematics, projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry In...
)
- Cayley-Hamilton theorem (Linear algebra
Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
)
- Cayley-Salmon theorem (algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more...
s)
- Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed . The central limit theorem also requires the random variables to...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Cesaro's theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Ceva's theorem
Ceva's theorem is a well-known theorem in elementary geometry.Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several "abstract primes" in the ring of algebraic integers of K. There are...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Chen's theorem
Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime . The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original proof was much simplified...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Cheng's eigenvalue comparison theorem
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that the larger the domain, the smaller the first Dirichlet eigenvalue of the Laplace–Beltrami operator. This general characterization is not precise, in part because the notion of "size" of the domain must also...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Chern-Gauss-Bonnet theorem (differential geometry)
- Chevalley–Shephard–Todd theorem
In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex...
(finite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
)
- Chevalley-Warning theorem
The Chevalley-Warning theorem is a mathematical theorem on solvability of polynomial equations in several variables over a finite field. The theorem was proved by Ewald Warning in 1936. A slightly weaker form of the theorem, known as Chevalley's theorem, was proved by French mathematician Claude...
(field theoryField theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
)
- Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.- Theorem statement :...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Choi's theorem on completely positive maps
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite dimensional C*-algebras...
(operator theoryIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
)
- Chomsky-Schützenberger theorem (linguistics
Linguistics is the scientific study of natural language. Linguistics encompasses a number of sub-fields. An important topical division is between the study of language structure and the study of meaning...
)
- Choquet–Bishop–de Leeuw theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Chow's theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Chowla-Mordell theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Church-Rosser theorem (lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. It was introduced by Alonzo Church in 1932 as part of an investigation into the foundations of mathematics...
)
- Clairaut's theorem
Clairaut's theorem, published in 1743 by Alexis Claude de Clairaut in his Théorie de la figure de la terre, tirée des principes de l'hydrostatique, synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It is a general mathematical law applying to spheroids of...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Clapeyron's theorem
In the linear theory of elasticity Clapeyron's theorem states that the potential energy of deformation of a body, which is in equilibrium under a given load, is equal to half the work done by the external forces computed assuming these forces had remained constant from the initial state to the...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Clark-Ocone theorem
In mathematics, the Clark–Ocone theorem is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itō integral with respect to that path...
(stochastic processes)
- Classification of finite simple groups
The classification of the finite simple groups, also called the enormous theorem, is believed to classify all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Clausius theorem
The Clausius theorem states that in a cyclic processThe equality holds in the reversible case and the '<' is in the irreversible case. The reversible case is used to introduce the function state entropy...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Clifford's theorem on special divisors (algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s)
- Closed graph theorem
In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.- The closed graph theorem :...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Closed range theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Cluster decomposition theorem
In physics, the cluster decomposition theorem guarantees locality in quantum field theory. According to this theorem, the vacuum expectation value of a product of many operators - each of them being either in region A or in region B where A and B are very separated - asymptotically equals the...
(quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
)
- Coase theorem
In law and economics, the Coase theorem, attributed to Ronald Coase, describes the economic efficiency of an economic allocation or outcome in the presence of externalities. The theorem states that when trade in an externality is possible and there are no transaction costs, bargaining will lead to...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Cochran's theorem
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used in the analysis of variance.- Statement :Suppose U1, ..., Un are independent standard normally distributed random variables, and an identity of the formcan be written where each Qi is...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Codd's theorem
Codd's theorem states that relational algebra and the domain-independent relational calculus queries, two well-known foundational query languages for the relational model, are precisely equivalent in expressive power. That is, a database query can be formulated in one language if and only if it can...
(relational modelThe relational model for database management is a database model based on first-order predicate logic, first formulated and proposed in 1969 by E.F. Codd.- Overview :...
)
- Cohn's irreducibility criterion
Arthur Cohn's irreducibility criterion is a test to determine whether a polynomial is irreducible in .The criterion is often stated as follows:The base-10 version of the theorem attributed to Cohn by Pólya and Szegő in one of their books while the generalization to any base, 2 or greater, is due to...
(polynomials)
- Coleman-Mandula theorem
The Coleman–Mandula theorem, named after Sidney Coleman and Jeffrey Mandula, is a no-go theorem in theoretical physics. It states that the only conserved quantities in a "realistic" theory with a mass gap, apart from the generators of the Poincaré group, must be Lorentz scalars.In other words,...
(quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
)
- Commutation theorem
In mathematics, a commutation theorem explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by F.J...
(von Neumann algebraIn mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group...
)
- Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Compression theorem
In computational complexity theory the compression theorem is an important theorem about the complexity of computable functions.The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.- Compression theorem :Given a Gödel...
(computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
) (structural complexity theoryIn computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms...
)
- Conservativity theorem
In mathematical logic, the conservativity theorem states the following: Suppose that a closed formulais a theorem of a first-order theory...
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Constant rank theorem (multivariate calculus)
- Convolution theorem
In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
(Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
s)
- Cook's theorem
In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete...
(computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
)
- Corona theorem
In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by ....
(Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Cox's theorem
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability. As the laws of probability derived by Cox's theorem are applicable to...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
foundations)
- Craig's theorem
In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable. Distinguish this result from the more well-known Craig interpolation theorem....
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Cramér's theorem
In mathematical statistics, Cramér's theorem refers to two theorems of Harald Cramér, a Swedish statistician and probabilist.- Normal random variables :...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Cramér-Wold theorem
In mathematics, the Cramér-Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.Letand...
(measure theory)
- Critical line theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Crooks fluctuation theorem
The Crooks equation is an equation in statistical mechanics that relatesthe work done on a system during a non-equilibrium transformation to thefree energy difference between the final and the initial state of the...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, crystallographyCrystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals...
)
- Curtis–Hedlund–Lyndon theorem
In mathematics, the Curtis–Hedlund–Lyndon theorem, also called Hedlund's theorem, after Gustav A. Hedlund, characterizes the global mappings of cellular automata as the mappings which are continuous and translation-invariant...
(cellular automata)
- Cut-elimination theorem
The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen 1934 in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively...
(proof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
)
- Cybenko theorem
The Cybenko theorem is a theorem proved by George Cybenko in 1989 that says that a single hidden layer, feed forward neural network is capable of approximating any continuous, multivariate function to any desired degree of accuracy and that failure to map a function arises from poor choices for ...
(neural networksNeural Networks is the official journal of the three oldest societies dedicated to research in neural networks: International Neural Network Society, European Neural Network Society and Japanese Neural Network Society, published by Elsevier. A subsription to the journal is part of the membership...
)
D
- Dandelin's theorem (solid geometry
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry...
)
- Danskin's theorem
In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the formThe theorem has applications in optimization, where it sometimes is used to solve minimax problems.- Statement :...
(convex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, a subdomain of optimization theory.- References :...
)
- Darboux's theorem
Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval....
(real analysisReal analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...
(symplectic topologySymplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
)
- Davenport–Schmidt theorem
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, Diophantine approximations)
- Dawson–Gärtner theorem (asymptotic analysis
In computer science and applied mathematics, particularly the analysis of algorithms, real analysis, and engineering, asymptotic analysis is a method of describing limiting behaviour...
)
- De Branges' theorem
In complex analysis, the Bieberbach conjecture or de Branges's theorem, posed by and proven by , states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane....
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- De Bruijn–Erdős theorem
In mathematics, the de Bruijn–Erdős theorem states that an infinite graph G can be colored by k colors if and only if every finite subgraph of G can be colored by k colors...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- De Finetti's theorem
In probability theory, de Finetti's theorem explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- De Franchis theorem
In mathematics, the De Francis theorem is one of a number of closely-related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite...
(Riemann surfaces)
- De Gua's theorem
De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves. If a tetrahedron has a right-angle corner , then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- De Moivre's theorem
In mathematics, De Moivre's formula, named after Abraham de Moivre, states that for any complex number x and any integer n it holds that...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- De Rham's theorem (differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
)
- Deduction theorem
In mathematical logic, the deduction theorem states that if a formula B is deducible from a set then the implication A → B is deducible from In symbols,impliesIn particular, if is the empty set, then as a special case we haveimplies...
(logicLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
)
- Denjoy theorem (dynamical system
The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
s)
- Denjoy–Carleman theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Desargues' theorem
In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:To understand this, denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Descartes' theorem
In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.-History:...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Descartes' theorem on total angular defect (polyhedra)
- Dilworth's theorem
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician Robert P...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
, order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
)
- Dimension theorem for vector spaces
In mathematics, the dimension theorem for vector spaces states that a vector space has a definite, well-defined number of dimensions. This may be finite, or an infinite cardinal number.Formally, the dimension theorem for vector spaces states that...
(vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, linear algebraLinear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
)
- Dini's theorem
In the mathematical field of analysis, Dini's theorem states that if X is a compact topological space, and { fn } is a monotonically increasing sequence of continuous real-valued functions on X which converges pointwise to a continuous function f, then the...
(analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
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- Dirac's theorem
In graph theory, there are two theorems that are commonly referred to as Dirac's theorem, both named after the mathematician Gabriel Andrew Dirac:...
s (graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number α and any positive integer n, there is some positive integer m ≤ n such that the difference between mα and the nearest integer is at most...
(Diophantine approximations)
- Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words: there are infinitely many primes which are congruent to a modulo d...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Dirichlet's unit theorem
In algebraic number theory, Dirichlet's unit theorem determines the rank of the group of units in the ring OK of algebraic integers of a number field K...
(algebraic number theoryIn mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as...
)
- Disintegration theorem
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures...
(measure theory)
- Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
(vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Dominated convergence theorem
In measure theory, a branch of mathematical analysis, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and pointwise convergence for a sequence of functions...
(Lebesgue integrationIn mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue...
)
- Donaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
(differential topologyIn mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
)
- Donsker's theorem
In probability theory, Donsker's theorem, named after M. D. Donsker, identifies a certain stochastic process as a limit of empirical processes. It is sometimes called functional central limit theorem....
(probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
)
- Doob decomposition theorem
In the theory of discrete time stochastic processes, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of any submartingale as the sum of a martingale and an increasing predictable process. The theorem was proved by and is named for J. L....
(stochastic processes)
- Doob's martingale convergence theorems
In mathematics — specifically, in stochastic analysis — Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph Leo Doob....
(stochastic processes)
- Doob–Meyer decomposition theorem (stochastic processes)
- Dudley's theorem
In probability theory, Dudley’s theorem is a result relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure. The result was proved in a landmark 1967 paper of Richard M...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
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- Duggan–Schwartz theorem (voting theory)
- Dunford–Pettis theorem (probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
)
- Dunford-Schwartz theorem
In mathematics, particularly functional analysis, the Dunford-Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz states that the averages of powers of certain norm bounded operators on L1 converge in a suitable sense....
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
E
- Earnshaw's theorem
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but...
(electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
)
- Easton's theorem
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. W. B. showed via forcing thatand, for , that...
(set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
)
- Eberlein–Šmulian theorem
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result relating three different kinds of weak compactness in a Banach space...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Edge-of-the-wedge theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Edgeworth's limit theorem
Edgeworth's limit theorem is an economic theorem created by Francis Ysidro Edgeworth that examines a range of possible outcomes which may result from free market exchange or barter between groups of people...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Egorov's theorem
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions...
(measure theory)
- Ehresmann's theorem
In mathematics, Ehresmann's fibration theorem states that a smooth mappingwhere M and N are smooth manifolds, such that#f is a surjective submersion, and#f is a proper map,...
(differential topologyIn mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
)
- Eilenberg–Zilber theorem
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space and those of the spaces and . The theorem first appeared in a 1953 paper in the American Journal of...
(algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
)
- Envelope theorem
The envelope theorem is a theorem used in optimisation problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
(calculus of variationsCalculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives...
)
- Equal incircles theorem
In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequenceis uniformly distributed on the unit interval, when a is an irrational number...
(ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systemswith an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
)
- Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...
(ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systemswith an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
)
- Erdős–Anning theorem
The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H...
(discrete geometryDiscrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity.Parts of its domain of...
)
- Erdos-Dushnik-Miller theorem (set theory
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
)
- Erdos-Ginzburg-Ziv theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Erdős-Kac theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Erdős-Ko-Rado theorem (combinatorics
Combinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Erdős–Nagy theorem (discrete geometry
Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity.Parts of its domain of...
)
- Erdős–Pósa theorem
In graph theory, a mathematical discipline, the Erdős–Pósa theorem, named after Paul Erdős and Lajos Pósa, states that if, in a graph, there are no k vertex-disjoint circuits, then all circuits can be covered by f vertices, for some value f.More generally, we can define Erdős–Pósa property, see...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Erdős-Rado theorem (set theory
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
)
- Erdős-Stone theorem (graph theory
In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Euclid's theorem
Euclid's theorem is a fundamental statement in number theory which asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Euclid-Euler Theorem
In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Euler's rotation theorem
In kinematics, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a rotation about a fixed axis through that point. The theorem is named after Leonhard Euler, who proved this in 1775...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Euler's theorem
In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least...
(differential geometry)
- Euler's theorem
In number theory, Euler's theorem states that if n is a positive integer and a is a positive integer coprime to n, then...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Euler's theorem in geometry
In geometry, Euler's theorem, named after Leonhard Euler, states that the distance d between the circumcentre and incentre of a triangle can be expressed as...
(triangle geometry)
- Euler's theorem on homogeneous functions (multivariate calculus)
- Exchange theorem (linear algebra
Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
)
- Excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology—given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out U from both spaces such...
(homology theoryIn mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.- Simple explanation :...
)
- Exterior angle theorem
The exterior angle theorem is a theorem in elementary geometry which states that the exterior angle of a triangle is equal to the sum of the two remote interior angles....
(triangle geometry)
- Extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
F
- F. and M. Riesz theorem
In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure μ on the circle, any part of μ that is not absolutely continuous with respect to the Lebesgue measure dθ can be detected by means of Fourier...
(measure theory)
- FWL theorem
In econometrics, the FWL theorem is named after the econometricians Ragnar Anton Kittil Frisch, Frederick V. Waugh, and Michael C...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Faltings' theorem
In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made; it is now known as Faltings' theorem.-Background:Suppose we are given...
(diophantine geometry)
- Fáry's theorem
Fáry's theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Fary-Milnor theorem
In mathematics, the Fary–Milnor theorem in knot theory states that if K is any closed curve K in Euclidean space that is sufficiently smooth to define the curvature κ at each of its points, and if the total curvature is at most 4π, then K is an unknot...
(knot theoryIn mathematics, knot theory is the area of topology that studies mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together to prevent it from becoming undone. In precise mathematical language, a...
)
- Fatou's theorem
In complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.-Motivation and statement of theorem:...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Fatou-Lebesgue theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Faustman–Ohlin theorem (economics
Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Feit-Thompson theorem (finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s)
- Fenchel's duality theorem
In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.Let ƒ be a proper convex function on R
n and let g be a proper concave function on R
n...
(convex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, a subdomain of optimization theory.- References :...
)
- Fenchel's theorem
In differential geometry, Fenchel's theorem states that the average curvature of any closed convex plane curve iswhere P is the perimeter...
(differential geometry)
- Fermat's last theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fermat's theorem on sums of two squares
In number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible aswith x and y integers, if and only if...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fermat's theorem (stationary points)
In mathematics, Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
(real analysisReal analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Fermat polygonal number theorem
The Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fernique's theorem
In mathematics — specifically, in measure theory — Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has exponential tails. The result was proved in 1970 by the mathematician Xavier...
(measure theory)
- Fieller's theorem
In statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.Variables a and b may be measured in different units, so there is no way to directly combine the standard errors as they may also be in different units...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Final value theorem
In mathematical analysis, the final value theorem is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Fisher separation theorem
In economics, the Fisher separation theorem asserts that the objective of a firm will be the maximization of its present value, regardless of the preferences of its owners. The theorem therefore separates management's "productive opportunities" from the entrepreneur's "market opportunities"...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Fisher–Tippet–Gnedenko theorem
In statistics, the Fisher–Tippet–Gnedenko theorem is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Fitting's theorem
Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups is nilpotent. This can be used to show that the Fitting subgroup of certain types of groups ...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Five color theorem
The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color.The five color theorem is...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Fixed point theorems in infinite-dimensional spaces
In mathematics, a number of fixed point theorems in infinite-dimensional spaces generalise the Brouwer fixed point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations....
- Fluctuation dissipation theorem
In statistical physics, the fluctuation dissipation theorem is a powerful tool for predicting the non-equilibrium behavior of a system — such as the irreversible dissipation of energy into heat — from its reversible fluctuations in thermal equilibrium...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Fluctuation theorem
The fluctuation theorem which originated from statistical mechanics deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time...
(statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
)
- Ford's theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Foster's theorem
In probability theory, Foster's theorem, named after F. G. Foster, is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces...
(statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Four color theorem
In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Four-vertex theorem
The Four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema . The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex.The Four-vertex theorem was first proved for convex...
(differential geometry)
- Fourier inversion theorem
In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.Sometimes the following identity is used as the definition of the Fourier transform:Then it is asserted that...
(harmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
)
- Fourier theorem (harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
)
- Franel-Landau theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fraňková–Helly selection theorem (mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Freidlin-Wentzell theorem
In mathematics, the Freidlin-Wentzell theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin-Wentzell theorem gives an estimate for the probability that a sample path of an Itō diffusion will stray far from the mean path. This statement is made...
(stochastic processes)
- Freiman's theorem
In mathematics, Freiman's theorem is a combinatorial result in number theory. In a sense it accounts for the approximate structure of sets of integers that contain a high proportion of their internal sums, taken two at a time.The formal statement is:...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Freudenthal suspension theorem
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and...
(homotopy theory)
- Freyd's adjoint functor theorem (category theory
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
)
- Frobenius reciprocity theorem (group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s)
- Frobenius theorem
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
(foliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s)
- Frobenius theorem
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers...
(abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
s)
- Froda's theorem
In mathematics, Froda's theorem, named after Alexandru Froda, describes the set of of a real-valued function of a real variable. Usually, this theorem appears in literature without Froda's name being mentioned. However, this result was first proven by A...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Fubini's theorem
In mathematical analysis, Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to change the order of integration.-Theorem statement:Suppose A and B are complete measure spaces...
(integrationIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
)
- Fuchs's theorem
In mathematics, the Fuchs's theorem states that a second order differential equation of the formhas a solution expressible by a generalised Frobenius series when , and are analytical at or is a regular singular point...
(differential equations)
- Fuglede's theorem
In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede.- The result :Theorem Let T and N be bounded operators on a complex Hilbert space with N being normal. If TN = NT, then TN* = N*T.Normality of N is necessary, as is seen by taking T=N...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Fulton-Hansen connectedness theorem
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1....
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Fundamental theorem of algebra
In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Fundamental theorem of arbitrage-free pricing
In a general sense, the fundamental theorem of arbitrage/finance is a way to relate arbitrage opportunities with risk neutral measures that are equivalent to the original probability measure.-In a finite state market:...
(financial mathematics)
- Fundamental theorem of arithmetic
In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Fundamental theorem of calculus
The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration....
(calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
)
- Fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
(abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
)
G
- Galvin's theorem
In combinatorics, the Dinitz conjecture is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by Fred Galvin....
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Gauss theorem (vector calculus
Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Gauss's Theorema Egregium
Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
(differential geometry)
- Gauss-Bonnet theorem (differential geometry)
- Gauss-Lucas theorem
In complex analysis, the Gauss–Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Gauss-Markov theorem (statistics
Statistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
)
- Gauss-Wantzel theorem (geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Gelfand–Mazur theorem (Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
)
- Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem is a result which establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
(transcendence theoryIn mathematics, transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
)
- Gershgorin circle theorem
In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Belarusian mathematician Semyon Aranovich Gershgorin in 1931. The spelling of S. A...
(matrix theory)
- Gibbard-Satterthwaite theorem
The Gibbard–Satterthwaite theorem is a result about voting systems designed to choose a single winner from the preferences of certain individuals, where each individual ranks all candidates in order of preference...
(voting methods)
- Girsanov's theorem (stochastic process
In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...
es)
- Glaisher's theorem
In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher.It states that the number of partitions of an integer into parts not divisible by is equal to the number of partitions of the formwhere and that is,...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Gleason's theorem
Gleason's theorem, named after Andrew Gleason, is a mathematical result of particular importance for quantum logic. It proves that the Born rule for the probability of obtaining specific results to a given measurement, follows naturally from the structure formed by the lattice of events in a real...
(Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
)
- Glivenko's theorem
Glivenko's theorem is a basic result showing a close connection between classical and intuitionistic propositional logic. It was proven by Valery Glivenko in 1929, with the aim of showing that intuitionistic logic is consistent and coherent...
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Glivenko–Cantelli theorem (probability
Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Goddard-Thorn theorem
In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem is a theorem about certain vector spaces...
(vertex algebras)
- Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Gödel's incompleteness theorem (mathematical logic
Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Godunov's theorem
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations.The theorem states...
(numerical analysisNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
)
- Going-up and going-down theorems
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions....
(commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
)
- Goldberg–Sachs theorem (physics
Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Goldie's theorem
In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. It gives a result on the noetherian rings that have a classical ring of quotients, that is a semisimple artinian ring, and so of known structure by the Artin–Wedderburn theorem.It...
(ring theoryIn mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
)
- Goldstone theorem (physics
Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Goodstein's theorem
In mathematical logic, Goodstein's theorem is a statement about thenatural numbers, made by Reuben Goodstein, which states that every Goodstein sequence eventually terminates at 0. Kirby & Paris showed that it is unprovable in Peano arithmetic...
(mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
)
- Gordon–Newell theorem
In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers. We cannot apply Jackson's theorem to closed networks because the queue...
(queueing theoryQueueing theory is the mathematical study of waiting lines . The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served by the server at the front of the queue...
)
- Gottesman–Knill theorem (quantum computation)
- Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve:It is a generalisation of the fundamental theorem of calculus to any...
(vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Graph structure theorem
In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Grauert–Riemenschneider vanishing theorem
In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to ....
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Great orthogonality theorem (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Green-Tao theorem
In mathematics, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for any natural number k, there exist k-term arithmetic progressions of primes...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Green's theorem
In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
(vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Grinberg's theorem
In graph theory, Grinberg's theorem is a necessary condition on the planar graph to contain a Hamiltonian cycle. The result has been widely used to construct non-Hamiltonian planar graphs with further properties, such as to disprove Tait's conjecture...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Gromov's compactness theorem
Gromov's compactness theorem can refer to either of two mathematical theorems:* Gromov's compactness theorem in Riemannian geometry* Gromov's compactness theorem in symplectic topology...
(Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length...
)
- Gromov's theorem
Gromov's theorem may mean one of a number of results of Mikhail Gromov:*One of Gromov's compactness theorems:** Gromov's compactness theorem in Riemannian geometry** Gromov's compactness theorem in symplectic topology...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Gromov-Ruh theorem (differential geometry)
- Gross-Zagier theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Grothendieck–Hirzebruch–Riemann–Roch theorem
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Grothendieck's connectedness theorem
In mathematics, Grothendieck's connectedness theorem states thatif A is a complete local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec is -connected...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Grunwald-Wang theorem (algebraic number theory
In mathematics, algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as...
)
- Grushko theorem
In the mathematical subject of group theory, the Grushko theorem or the Grushko-Neumann theorem is a theorem stating that the rank of a free product of two groups is equal to the sum of the ranks of the two free factors...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
H
- H-cobordism theorem (differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
)
- H-theorem
In thermodynamics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process, by considering the Boltzmann equation....
(thermodynamicsIn physics, thermodynamics is the study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature, volume and pressure...
)
- Haag's theorem
Rudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory, something now commonly known as Haag's Theorem. The theorem was subsequently proved by a number of different authors...
(quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
)
- Haboush's theorem (algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...
, invariant theoryInvariant theory is a branch of abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect on functions...
)
- Hadamard three-circle theorem
In complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions.Let be a holomorphic function on the annulus...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Hadwiger's theorem
In integral geometry , Hadwiger's theorem states that the space of "measures" defined on finite unions of compact convex sets in Rn consists of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures".Here "measure" means a...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, measure theory)
- Hahn decomposition theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that:#P ∪ N = X and ...
(measure theory)
- Hahn embedding theorem
In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all totally ordered abelian groups....
(ordered groupIn abstract algebra, a partially-ordered group is a group equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.An element x of G is called positive element if 0 ≤ x...
s)
- Hairy ball theorem
The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on the sphere. If f is a continuous function that assigns a vector in R
3 to every point p on a sphere such that f is always tangent to the sphere at p, then there is at least...
(algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
)
- Hahn-Banach theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Hahn–Kolmogorov theorem (measure theory)
- Hahn-Mazurkiewicz theorem (continuum theory)
- Hales-Jewett theorem (combinatorics
Combinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Halpern-Lauchli theorem
In mathematics, the Halpern-Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false...
(Ramsey theoryRamsey theory, named after Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: how many elements of some structure must there be to guarantee that a particular property will...
)
- Ham sandwich theorem
In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given n measurable "objects" in n-dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane...
(topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
)
- Hammersley–Clifford theorem
The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics, that gives necessary and sufficient conditions under which a positive probability distribution is a Markov network...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Hardy's theorem
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.Let be a holomorphic function on the open ball centered at zero and radius in the complex plane, and assume that is not a constant function. If one defines for then this function is...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Hardy–Littlewood maximal theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Hardy–Littlewood tauberian theorem
In mathematical analysis, the Hardy–Littlewood tauberian theorem is a tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Hardy–Ramanujan theorem
In mathematics, the Hardy–Ramanujan theorem, proved by , states that the normal order of the number ω of distinct prime factors of a number n is log...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Harish-Chandra's regularity theorem (representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...
)
- Harnack's curve theorem
In real algebraic geometry, Harnack's curve theorem states when a curve of degree m can have c components. For any real plane algebraic curve of degree m, the number of components c is bounded by...
(real algebraic geometryIn mathematics, real algebraic geometry is the study of real number solutions to algebraic equations with real number coefficients.-Real plane curves:...
)
- Harnack's theorem (complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Hartogs' theorem
In mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. It states that for complex-valued functions F on Cn, with n > 1, being an analytic function in each variable zi, 1 ≤ i ≤ n, while the others are held...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Hartogs' extension theorem
In mathematics, especially several complex variables, Hartogs' extension theorem states:The theorem does not hold when . It thus constitutes one of elementary phenomena that is unique to several complex variables....
(several complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
)
- Hasse norm theorem
In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm....
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Hasse's theorem on elliptic curves
In mathematics, Hasse's theorem on elliptic curves bounds the number of points on an elliptic curve over a finite field, above and below.If N is the number of points on the elliptic curve E over a finite field with q elements, then Helmut Hasse's result states thatThis had been a conjecture of...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Hasse–Arf theorem
In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of a filtration of the Galois group of a finite Galois extension...
(local class field theory)
- Hasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Heckscher-Ohlin theorem
The Heckscher-Ohlin theorem is one of the four critical theorems of the Heckscher-Ohlin model. It states: "A capital-abundant country will export the capital-intensive good, while the labor-abundant country will export the labor-intensive good."...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Heine-Borel theorem (real analysis
Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
)
- Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M is a compact metric space, then every continuous functionwhere N is a metric space, is uniformly continuous....
(metric geometry)
- Hellinger-Toeplitz theorem (functional analysis
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Hellmann–Feynman theorem (physics
Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Helly–Bray theorem
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. The first eponym is Eduard Helly....
(probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
)
- Helly's selection theorem
In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Helly's theorem
Helly's theorem is a basic result in discrete geometry describing the ways that convex sets may intersect each other. It was proved by Eduard Helly in 1923, and gave rise to the notion of Helly family.-Statement of Helly's theorem:Suppose that...
(convex setIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not...
s)
- Helmholtz theorem (classical mechanics)
The Helmholtz theorem of classical mechanics reads as follows:Letbe the Hamiltonian of a one-dimensional system, whereis the kinetic energy andis a "U-shaped" potential energy profile which depends on a parameter .Let denote the time average. Let...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems describe the three-dimensional motion of fluid in the vicinity of vortex filaments. These theorems apply to inviscid flows and flows where the influence of viscous forces is small and can be ignored....
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Herbrand's theorem (logic
Logic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
)
- Herbrand–Ribet theorem
In mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the...
(cyclotomic fieldIn number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s)
- Higman's embedding theorem
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Hilbert's basis theorem
In mathematics, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a field is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many...
(commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
,invariant theoryInvariant theory is a branch of abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect on functions...
)
- Hilbert's Nullstellensatz
Hilbert's Nullstellensatz is a theorem which makes precise a fundamental relationship between the geometric and algebraic sides of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
(theorem of zeroes) (commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Hilbert-Schmidt theorem
In mathematical analysis, the Hilbert-Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Hilbert-Speiser theorem
In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any abelian extension K of the rational field Q...
(cyclotomic fieldIn number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s)
- Hilbert–Waring theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Hilbert's syzygy theorem
In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it...
(commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
)
- Hilbert's theorem
In differential geometry, Hilbert's theorem states that there exists no complete regular surface of constant negative Gaussian curvature immersed in . This theorem answers the question for the negative case of which surfaces in can be obtained by isometrically immersing complete manifolds with...
(differential geometry)
- Hilbert's theorem 90
In number theory, Hilbert's Theorem 90 refers to an important result on cyclic extensions of number fields that leads to Kummer theory...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Hilbert projection theorem
The Hilbert Projection Theorem is a famous result of convex analysis that says that for every point in a Hilbert space and every closed subspace , there exists a unique point for which is minimized over . A necessary and sufficient condition for is that the vector be orthogonal to ....
(convex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, a subdomain of optimization theory.- References :...
)
- Hille–Yosida theorem
In functional analysis, the Hille–Yosida theorem characterizes the generators of one-parameter semigroups of linear operators on Banach spaces. The theorem is mainly of theoretical interest, the Lumer-Phillips theorem is more useful in determining whether a given operator generates a strongly...
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Hindman's theorem (Ramsey theory
Ramsey theory, named after Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: how many elements of some structure must there be to guarantee that a particular property will...
)
- Hinge theorem
The hinge theorem in geometry states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Hironaka theorem (algebraic geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Hirzebruch signature theorem (topology
Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
, algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Hirzebruch–Riemann–Roch theorem (complex manifolds)
- Hjelmslev's theorem
In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... of one and the same line are isometrically mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP`, QQ´, RR´... also lie on one and the same...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Hobby–Rice theorem
In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by C. R. Hobby and J. R. Rice; a simplified proof was given in 1976 by A...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Hodge index theorem
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V...
(algebraic surfaceIn mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more...
s)
- Hölder's theorem
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. The result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.The theorem also generalizes to...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Holland's schema theorem
Holland's schema theorem is widely taken to be the foundation for explanations of the power of genetic algorithms.A schema is a template that identifies a subset of strings with similarities at certain string positions. Schemata are a special case of cylinder sets; and so form a topological space.-...
(genetic algorithmA genetic algorithm is a search technique used in computing to find exact or approximate solutions to optimization and search problems. Genetic algorithms are categorized as global search heuristics...
)
- Holmström's theorem
In economics, Holmström's theorem is an impossibility theorem attributed to Bengt R. Holmström proving that no incentive system for a team of agents can make all of the following true:# Income equals outflow ,...
(economicsEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
)
- Hopf-Rinow theorem (differential geometry)
- Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
(algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
)
- Hurwitz's automorphisms theorem
In mathematics, Hurwitz's automorphisms theorem bounds the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, telling us that the number of such automorphisms cannot exceed...
(algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s)
- Hurwitz's theorem
In complex analysis, a field within mathematics, Hurwitz's theorem, named after Adolf Hurwitz, roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while those functions and the limit...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Hurwitz's theorem
In algebra, Hurwitz's theorem, named after Adolf Hurwitz, states that the only composition algebras over are , , and , that is the real numbers, the complex numbers, the quaternions and the octonions.- References :}}...
(composition algebraIn mathematics, a composition algebra A over a field K is a unital algebra over K together with a nondegenerate quadratic form N which satisfies...
s)
- Hurwitz's theorem
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many rationals m/n such that...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
I
- Identity theorem
In complex analysis, the identity theorem for holomorphic functions states: given functions f and g holomorphic on a connected open set D, if f = g on some neighborhood of z that is in D, then f = g on D. Thus a holomorphic function is completely determined by its values on a neighborhood in D...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
)
- Identity theorem for Riemann surfaces
In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point.-Statement of the theorem:...
(Riemann surfaces)
- Immerman–Szelepcsényi theorem (complexity theory
Complexity theory may refer to:*The study of complex systems.*Computational complexity theory, a field in theoretical computer science and mathematics dealing with the resources required during computation to solve a given problem....
)
- Implicit function theorem
In the branch of mathematics called multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but...
(vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Increment theorem
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function f is differentiable at x and that Δx is infinitesimal. Thenfor some infinitesimal ε, whereIf then we may write...
(mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
)
- Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare....
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Integral root theorem
In algebra, the rational root theorem states a constraint on rational solutions of the polynomial equationwith integer coefficients.If a0 and an are nonzero,...
(algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
, polynomials)
- Initial value theorem
In mathematics, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Letbe the Laplace transform of ƒ. The initial value theorem then says...
(integral transformIn mathematics, an integral transform is any transform T of the following form:The input of this transform is a function f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator....
)
- Integral representation theorem for classical Wiener space
In mathematics, the integral representation theorem for classical Wiener space is a result in the fields of measure theory and stochastic analysis...
(measure theory)
- Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value....
(calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
)
- Intercept theorem
The intercept theorem is an important theorem in elementary geometry about the ratios of various line segments, that are created if 2 intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Intersection theorem
In projective geometry, an intersection theorem or incidence theorem is an incidence structure consisting of points, lines, and possibly higher-dimensional objects and their incidences, together with a pair of nonincident objects A and B...
(projective geometryIn mathematics, projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry In...
)
- Inverse eigenvalues theorem
In numerical analysis and linear algebra, the Inverse eigenvalues theorem states that, given a matrix A that is nonsingular, with eigenvalue , is an eigenvalue of if and only if is an eigenvalue of .-Proof of the Inverse Eigenvalues Theorem:...
(Linear algebraLinear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
)
- Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
(vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
)
- Isomorphism extension theorem
In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to a larger field.- Isomorphism extension theorem :...
(abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
)
- Isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
(abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
)
- Isoperimetric theorem (curve
In mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave...
s, calculus of variationsCalculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives...
)
J
- Jackson's theorem (queueing theory
Queueing theory is the mathematical study of waiting lines . The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served by the server at the front of the queue...
)
- Jacobi's four-square theorem
In 1834, Carl Gustav Jakob Jacobi found an exact formula for the total number of ways a given positive integer n can be represented as the sum of four squares...
(number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
)
- Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. A special case of the theorem asserts that any primitive ring can be viewed as a dense set of linear operators on some...
(ring theoryIn mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
)
- Jacobson–Morozov theorem (Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
)
- Japanese theorem for concyclic polygons (Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- Japanese theorem for concyclic quadrilaterals
The Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.Triangulate an arbitrary concyclic quadrilateral by its diagonals, this yields four overlapping triangles...
(Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
)
- John’s theorem (geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Jordan curve theorem
In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside" region, and any path connecting a point of one region to a point of the other intersects that loop somewhere.The first to give a proof was...
(topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
)
- Jordan–Hölder theorem (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Jordan–Schönflies theorem
In mathematics, the Jordan–Schönflies theorem, or simply the Schönflies theorem, of geometric topology is a sharpening of the Jordan curve theorem byArthur Schönflies.-Formulation:...
(geometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
)
- Jordan's theorem (multiply transitive groups) (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Jung's theorem
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901.- Statement :...
(geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
)
- Jurkat–Richert theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.It was proved in 1965 by Wolfgang B...
(analytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
)
K
- Kachurovskii's theorem
In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.-Statement of the theorem:...
(convex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, a subdomain of optimization theory.- References :...
)
- Kantorovich theorem
The Kantorovich theorem is a mathematical statement on the convergence of the Newton's method. It was first stated by Leonid Kantorovich in 1940....
(functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
)
- Kaplansky density theorem
In the theory of von Neumann algebras, the Kaplansky density theorem states thatif A is a *-subalgebra of the algebra B of bounded operators on a Hilbert space H, then the strong closure of the unit ball of A in B is the unit ball of the strong closure of A in B...
(von Neumann algebraIn mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group...
)
- Kaplansky's theorem on quadratic forms
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Canadian mathematician Irving Kaplansky .-Statement of the theorem:...
(quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,is a quadratic form in the variables x and y....
s)
- Karhunen-Loève theorem
In the theory of stochastic processes, the Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The distinction is that here, the Fourier...
(stochastic processes)
- Kawamata–Viehweg vanishing theorem
In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by and ....
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Kawasaki's theorem
Kawasaki's theorem is a theorem in the mathematics of paper folding that gives a criterion for whether a given crease pattern is locally flat-foldable.- Statement of the theorem :...
(paper foldingPaper folding is the art of folding paper; it is known in many societies that use paper. In much of the West, the term origami is used synonymously with paper folding, though the term properly only refers to the art of paper folding in Japan....
)
- Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem states "In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time". The theorem was developed by William Thomson, 1st Baron Kelvin...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Kharitonov's theorem
Kharitonov's theorem is a result used in control theory to assess the stability of a dynamical system when the physical parameters of the system are not known precisely. When the coefficients of the characteristic polynomial are known, the Routh-Hurwitz stability criterion can be used to check if...
(control theoryControl theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
)
- Khinchin's theorem
Khinchin's theorem may refer to any of several different results by Aleksandr Khinchin:*Wiener–Khinchin theorem*Khinchin's constant*Khinchin's theorem on the factorization of distributions*Khinchin's theorem on Diophantine approximations...
(probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
)
- Kirby-Paris theorem (proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
)
- Kirchhoff's theorem
In the mathematical field of graph theory Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph...
(graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
)
- Kirszbraun theorem
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and...
(Lipschitz continuityIn mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity...
)
- Kleene's recursion theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions...
(recursion theoryRecursion theory, also called computability theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
)
- Kleene fixed-point theorem (order theory
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
)
- Knaster-Tarski theorem (order theory
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
)
- Kneser theorem
In mathematics, in the field of ordinary differential equations, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.- Statement of the theorem :...
(differential equations)
- Kochen–Specker theorem (physics
Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Kodaira embedding theorem
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds...
(algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
)
- Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero...
(complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
)
- Koebe 1/4 theorem
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states that the image of an injective analytic function from the unit disk onto a subset of the complex plane contains the disk whose center is and whose radius is . The theorem is named after Paul Koebe, who conjectured the...
(complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
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- Kolmogorov-Arnold-Moser theorem (dynamical systems)
- Kolmogorov extension theorem
In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process...
(stochastic processes)
- Kōmura's theorem
In mathematics, Kōmura's theorem is a result on the differentiability of absolutely continuous Banach space-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the indefinite integral, which is that Φ : [0, T] → R given...
(measure theory)
- König's theorem (mathematical logic
Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
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- König's theorem (graph theory)
In the mathematical area of graph theory, König's theorem describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.- Setting :...
(bipartite graphIn the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets...
s)
- König's theorem (kinetics)
Konig's theorem is related to kinetics of a system of particles.- The theorem :It states that the kinetic energy of a system of particles is the kinetic energy associated to the movement of the center of mass and the kinetic energy associated of the movement of the particles relative to the center...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- König's theorem (set theory)
In set theory, König's theorem colloquially states that if the axiom of choice holds, I is a set, mi and ni are cardinal numbers for every i in I, and for every i in I thenThe sum here is the cardinality of the disjoint union of...
(cardinal numbers)
- Kövari–Sós–Turán theorem (graph theory
In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
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- Kraft-McMillan theorem (coding theory
Coding theory is an approach to various science disciplines—such as information theory, electrical engineering, digital communication, mathematics, and computer science -- which helps design efficient and reliable data transmission methods so that redundancy can be removed and errors corrected.It...
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- Kramers theorem
The Kramers degeneracy theorem states that the energy levels of systems with an odd number of electrons remain at least doubly degenerate in the presence of purely electric fields ....
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
- Krein–Milman theorem (mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
, discrete geometryDiscrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity.Parts of its domain of...
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- Krener's theorem
Krener's theorem is a result in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior...
(control theoryControl theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
)
- Kronecker's theorem
In mathematics, Kronecker's theorem refers to one of two theorems named after Leopold Kronecker.- The existence of extension fields :This is a theorem stating that a polynomial in a field, , has a root in an extension field ....
(diophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
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- Kronecker-Weber theorem (number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
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- Krull's principal ideal theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull , gives a bound on the height of a principal ideal in a Noetherian ring...
(commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
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- Krull-Schmidt theorem (group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
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- Kruskal–Katona theorem
The Kruskal–Katona theorem is a combinatorial theorem about uniform hypergraphs. It can be used to derive facts about abstract simplicial complexes. It is named after Joseph Kruskal and Gyula O. H...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
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- Kruskal's tree theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered . The theorem was proved by, and a short proof was given by ....
(order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
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- Krylov-Bogolyubov theorem
In mathematics, the Krylov–Bogolyubov theorem may refer to any of the two related fundamental theorems within the theory of dynamical systems...
(dynamical systems)
- Kuiper's theorem
In mathematics, Kuiper's theorem is a result on the topology of operators on an infinite-dimensional, complex Hilbert space . It states that the topological space of all linear operators from to itself, which are bounded operators and invertible, is such that for any finite complex , there is...
(operator theoryIn mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them...
, topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
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- Künneth theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their...
(algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
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- Kurosh subgroup theorem
In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by a Russian mathematician Alexander Kurosh in 1934...
(group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
)
- Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...
(physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
)
L
- Labelled enumeration theorem
The labelled enumeration theorem is the counterpart of the Pólya enumeration theorem for the labelled case, where we have a set of labelled objects given by an exponential generating function g which are being distributed into n slots and a permutation group G which permutes the slots, thus...
(combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
)
- Ladner's theorem
In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI...
(computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
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- Lagrange's theorem
Lagrange's theorem, in the mathematics of group theory, states that for a