List of theorems
Encyclopedia
This is a list of theorem
s, by Wikipedia page. See also
Most of the results below come from pure mathematics
, but some are from theoretical physics
, economics
, and other applied
fields.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s, by Wikipedia page. See also
- Classification of finite simple groupsClassification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. 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...
- Comparison theorem
- Existence theoremExistence theoremIn 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 lemmaFodor's lemmaIn mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a regular, uncountable cardinal, S is a stationary subset of \kappa, and f:S\rightarrow\kappa is regressive In mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a...
, etc.) - list of conjectures
- list of inequalities
- list of mathematical proofs
- list of misnamed theorems
- Open mapping theorem
- Toy theoremToy theoremIn 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
Most of the results below come from pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, but some are from theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, economics
Economics
Economics is the social science that analyzes 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 applied
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
fields.
0–9
- 15 and 290 theorems (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - 2π theorem (Riemannian geometryRiemannian 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...
)
A
- AF+BG theoremAF+BG theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - ATS theoremATS theoremIn mathematics, the ATS theorem is the theorem on the approximation of atrigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Abel's binomial theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Abel's curve theoremAbel's curve theoremAbel's curve theorem, named after Niels Henrik Abel, is a theorem that relates fixed points to the parameters of rational functions....
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Abel's theoremAbel's theoremIn 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 analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Abelian and tauberian theoremsAbelian and tauberian theoremsIn mathematics, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Abel–Jacobi theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Abel–Ruffini theoremAbel–Ruffini theoremIn algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...
(theory of equationsTheory 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 theoryGalois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
) - Abhyankar–Moh theoremAbhyankar–Moh theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Abouabdillah's theoremAbouabdillah's theoremAbouabdillah'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...
(geometryGeometryGeometry 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Absolute convergence theorem (mathematical series)
- Acyclic models theoremAcyclic models theoremIn 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 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Addition theoremAddition theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Adiabatic theoremAdiabatic theoremThe adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock , can be stated as follows:...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Ado's theoremAdo's theoremIn abstract algebra, 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 algebraLie 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...
) - Ahiezer's theoremAhiezer's theoremIn the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.-Statement:Let be an entire function of exponential type , with for real...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Akra–Bazzi theorem (computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Albert–Brauer–Hasse–Noether theoremAlbert–Brauer–Hasse–Noether theoremIn algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits over every completion Kv is a matrix algebra over K...
(algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
s) - Alchian–Allen theorem (economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Alperin–Brauer–Gorenstein theorem (finite groups)
- Amitsur–Levitzki theoremAmitsur–Levitzki theoremIn algebra, the Amitsur–Levitzki theorem states that the algebra of n by n matrices satisfies a certain identity of degree 2n. It was proved by...
(linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Analytic Fredholm theoremAnalytic Fredholm theoremIn 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 the Hilbert–Schmidt theorem...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Anderson's theoremAnderson's theoremIn 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 analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Andreotti–Frankel theoremAndreotti–Frankel theoremIn mathematics, the Andreotti–Frankel theorem from 1959 states that if V is a smooth affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n, then in fact V is homotopy equivalent to a CW complex of real dimension at most n...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Angle bisector theoremAngle bisector theoremIn geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.Consider a triangle...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Ankeny–Artin–Chowla theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Apéry's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Apollonius' theoremApollonius' theoremIn geometry, Apollonius' theorem is a theorem relating the length of a median of a triangle to the lengths of its side.Specifically, in any triangle ABC, if AD is a median, thenAB^2 + AC^2 = 2\,...
(plane geometry) - Appell–Humbert theoremAppell–Humbert theoremIn 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 manifoldComplex 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)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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Arithmetic Riemann–Roch theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Aronszajn–Smith theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Arrival theorem (queueing theoryQueueing theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
) - Arrow's impossibility theoremArrow's impossibility theoremIn social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
(game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Art gallery theorem (geometryGeometryGeometry 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 theoremArtin approximation theoremIn 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 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 fieldReal closed fieldIn 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 theoremArtin–Wedderburn theoremIn abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
(abstract algebraAbstract 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 (algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
) - Artstein's theoremArtstein's theoremArtstein's theorem states that a dynamical system has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback u....
(control theoryControl 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 analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Atiyah–Bott fixed-point theoremAtiyah–Bott fixed-point theoremIn 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 topologyDifferential 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 theoremAtiyah–Singer index theoremIn differential geometry, the Atiyah–Singer index theorem, proved by , 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 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 theoremAtkinson's theoremIn operator theory, Atkinson's theorem gives a characterization of Fredholm operators.- The theorem :Let H be a Hilbert space and L the set of bounded operators on H...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Aumann's agreement theoremAumann's agreement theoremAumann's agreement theorem says that two people acting rationally and with common knowledge of each other's beliefs cannot agree to disagree...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Autonomous convergence theoremAutonomous convergence theoremIn 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 systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Auxiliary polynomial theorem (diophantine approximationDiophantine 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 theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) - Ax–Kochen theoremAx–Kochen theoremThe Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables has a nontrivial...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Aztec diamond theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
)
B
- BEST theoremBEST theoremIn graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed graphs. The name is an acronym of the names of people who discovered it: de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte.- Precise statement :Let...
(graph theoryGraph 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...
) - Babuška–Lax–Milgram theorem (partial differential equationPartial 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 their partial derivatives with respect to those variables...
s) - Baily–Borel theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Baire category theoremBaire category theoremThe 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....
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, metric spaceMetric 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 (Fourier analysis)
- Balinski's theoremBalinski's theoremIn 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...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Banach–Alaoglu theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Banach–Mazur theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Banach fixed point theoremBanach fixed point theoremIn mathematics, 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 spaceMetric 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 equationDifferential 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 analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Banach–Stone theorem (operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Barbier's theoremBarbier's theoremBarbier's theorem, a basic result on curves of constant width first proved by Joseph Emile Barbier, states that the perimeter of any curve of constant width w is πw....
(geometryGeometryGeometry 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 (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Baranyai's theoremBaranyai's theoremIn combinatorial mathematics, Baranyai's theorem deals with the decompositions of complete hypergraphs.-Statement of the theorem:The statement of the result is that if 2\le r...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Barta's theorem (Riemannian geometryRiemannian 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...
) - Barwise compactness theoremBarwise compactness theoremIn mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967....
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Bass's theorem (group theoryGroup 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...
) - Basu's theoremBasu's theoremIn statistics, Basu's theorem states that any boundedly complete sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu....
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Bauer–Fike theorem (spectral theorySpectral theoryIn mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
) - Bayes' theoremBayes' theoremIn probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Beatty's theorem (diophantine approximationDiophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
) - Beauville–Laszlo theoremBeauville–Laszlo theoremIn 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 theoremBeck's monadicity theoremIn category theory, a branch of mathematics, Beck's monadicity theorem asserts that a functorU: C \to Dis monadic if and only if# U has a left adjoint;# U reflects isomorphisms; and...
(category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
) - Beck's theoremBeck's theorem (geometry)In the context of discrete geometry, Beck's theorem may refer to several different results, two of which are given below. Both appeared, alongside several other important theorems, in a well-known paper by József Beck. The two results described below primarily concern lower bounds on the number of...
(incidence geometry) - Beckman–Quarles theoremBeckman–Quarles theoremIn geometry, the Beckman–Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher dimensional Euclidean space preserves unit distances, then it preserves all distances...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Beer's theorem (metric geometry)
- Behnke–Stein theoremBehnke–Stein theoremIn mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k of domains of holomorphy is again a domain of holomorphy....
(several complex variablesSeveral 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...
) - Bell's theoremBell's theoremIn theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
(quantum theoryQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
– physics) - Beltrami's theoremBeltrami's theoremIn 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 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 theoremBelyi's theoremIn 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 curveAlgebraic 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 (dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Berger–Kazdan comparison theorem (Riemannian geometryRiemannian 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...
) - Bernstein's theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Berry–Esséen theoremBerry–Esséen theoremThe central limit theorem in probability theory and statistics states that under certain circumstances the sample mean, considered as a random quantity, becomes more normally distributed as the sample size is increased...
(probability theoryProbability 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...
) - Bertini's theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Bertrand–Diquet–Puiseux theoremBertrand–Diquet–Puiseux theoremIn the mathematical study of the differential geometry of surfaces, the Bertrand–Diquet–Puiseux theoremexpresses 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...
(differential geometry) - Bertrand's ballot theorem (probability theoryProbability 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...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Bertrand's postulate (prime numberPrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s) - Besicovitch covering theoremBesicovitch covering theoremIn mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover....
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Betti's theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Beurling–Lax theoremBeurling–Lax theoremIn mathematics, the Beurling–Lax theorem is a theorem due to and which characterizes the shift-invariant subspaces of the Hardy space H^2. It states that each such space is of the formfor some inner function \theta.-References:...
(Hardy spaceHardy 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 theoremBézout's theoremBézout'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 curveAlgebraic 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 theoremBing metrization theoremIn topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis...
(general topologyGeneral 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 recognition theorem (geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Binomial inverse theoremBinomial inverse theoremIn 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 theoremBinomial theoremIn 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 axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
(algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Birch's theoremBirch's theoremIn 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 equationDiophantine 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 (vector bundles)
- Birkhoff–Von Neumann theorem (matrix theory)
- Birkhoff's representation theoremBirkhoff's representation theoremIn 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 (ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Birkhoff's theorem (relativity)Birkhoff's theorem (relativity)In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric....
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Bishop–Cannings theorem (economicsEconomicsEconomics is the social science that analyzes 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 (geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Bloch's theoremBloch's theorem (complex variables)In complex analysis, a field within mathematics, Bloch's theorem is a result that gives a lower bound on the size of the image of a certain class of holomorphic functions. It is named after André Bloch.-Statement:...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Blondel's theoremBlondel's theoremBlondel's theorem, named after its discoverer, French electrical engineer 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...
(electric powerElectric powerElectric power is the rate at which electric energy is transferred by an electric circuit. The SI unit of power is the watt.-Circuits:Electric power, like mechanical power, is represented by the letter P in electrical equations...
) (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Blum's speedup theoremBlum's speedup theoremIn computational complexity theory Blum's speedup theorem, first stated by Manuel Blum in 1967, is a fundamental theorem about the complexity of computable functions....
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Bôcher's theoremBôcher's theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Bogoliubov–Parasyuk theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Bohr–Mollerup theorem (gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
) - Bohr–van Leeuwen theoremBohr–van Leeuwen theoremThe Bohr–van Leeuwen theorem is a theorem in the field of statistical mechanics. The theorem shows that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Bolyai–Gerwien theoremBolyai–Gerwien theoremIn geometry, the Bolyai–Gerwien theorem, named after Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e...
(discrete geometryDiscrete geometryDiscrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
) - Bolzano's theorem (real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
, calculusCalculusCalculus is a branch of 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 theorem...
) - Bolzano–Weierstrass theoremBolzano–Weierstrass theoremIn real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
, calculusCalculusCalculus is a branch of 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 theorem...
) - Bombieri's theoremSchneider–Lang theoremIn mathematics, the Schneider–Lang theorem is a refinement by of a theorem of about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Bombieri–Friedlander–Iwaniec theoremBombieri–Friedlander–Iwaniec theoremIn analytic number theory, an advanced branch of mathematics, the Friedlander–Iwaniec theorem asserts that there are infinitely many prime numbers of the form a^2 + b^4...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Bondareva–Shapley theorem (economicsEconomicsEconomics is the social science that analyzes 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 theoremBondy's theoremIn 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 theoryGraph 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...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Bondy–Chvátal theorem (graph theoryGraph 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...
) - Bonnet theoremBonnet theoremIn 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 1860.This is...
(differential geometry) - Boolean prime ideal theoremBoolean prime ideal theoremIn 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 logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 theoremBorel–Bott–Weil theoremIn mathematics, the Borel–Weil-Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Borel–Carathéodory theoremBorel–Carathéodory theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Borel–Weil theoremBorel–Weil theoremIn mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Borel determinacy theoremBorel determinacy theoremIn descriptive set theory, the Borel determinacy theorem states 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 theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Borel fixed-point theoremBorel fixed-point theoremIn mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie-Kolchin theorem. The result was proved by .-Statement of the theorem:...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Borsuk–Ulam theoremBorsuk–Ulam theoremIn mathematics, the Borsuk–Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point....
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Bott periodicity theoremBott periodicity theoremIn mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , 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...
(homotopy theory) - Bounded convergence theorem (measure theory)
- Bounded inverse theoremBounded inverse theoremIn 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. It is equivalent to both the open mapping theorem and the closed...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Bourbaki–Witt theoremBourbaki–Witt theoremIn 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 non-empty chain complete poset, andsuch that...
(order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Brahmagupta theoremBrahmagupta theoremBrahmagupta's theorem is a result in geometry. It states that if a cyclic quadrilateral is orthodiagonal , then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Branching theoremBranching theoremIn 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 manifoldComplex 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 groupsRepresentation 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–Siegel theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Brauer–Suzuki theoremBrauer–Suzuki theoremIn mathematics, the Brauer–Suzuki theorem, proved by , , , 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. In particular, such a group cannot be simple.A generalization of the...
(finite groups) - Brauer's theoremBrauer's theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Brauer's theorem on induced charactersBrauer's theorem on induced charactersBrauer'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 groupsRepresentation 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 theoremsBrauer's three main theoremsBrauer'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 theoremBrauer–Cartan–Hua theoremThe 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 theoryRing theoryIn abstract algebra, 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 theoremBrianchon's theoremIn 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...
(conics) - British flag theoremBritish flag theoremIn Euclidean geometry, the British flag theorem says that if a point P is chosen inside rectangle ABCD then the sum of the squared Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners....
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Brooks’ theoremBrooks’ theoremIn graph theory, Brooks' theorem states a relationship between the maximum 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...
(graph theoryGraph 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 theoremBrouwer fixed point theoremBrouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Browder–Minty theorem (operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Brown's representability theoremBrown's representability theoremIn 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 theoremBruck–Chowla–Ryser theoremThe Bruck–Ryser–Chowla theorem is a result on the combinatorics of block designs. It states that if a -design exists with v = b , then:* if v is even, then k − λ is a square;...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Brun's theoremBrun's theoremIn number theory, Brun's theorem was 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, usually denoted by B2...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Brun–Titchmarsh theoremBrun–Titchmarsh theoremIn analytic number theory, the Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Brunn–Minkowski theorem (Riemannian geometryRiemannian 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 analysisDimensional analysisIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
) - Burke's theoremBurke's theoremIn 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 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 theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
) - Burnside theoremBurnside theoremIn mathematics, Burnside's theorem in group theory states that if G is a finite group of orderp^a q^b\ where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each...
(group theoryGroup 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 theoremBusemann's theoremIn 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 geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Butterfly theoremButterfly theoremThe 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 geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
)
C
- CPCTCCPCTCIn geometry, CPCTC is the abbreviation of a theorem involving congruent triangles. CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. CPCTC states that if two or more triangles are proven congruent by: ASA, AAS, SSS, HL, or SAS, then all of their corresponding parts are...
(triangle geometry) - Cameron–Martin theorem (measure theory)
- Cantor–Bernstein–Schroeder theoremCantor–Bernstein–Schroeder theoremIn set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function...
(Set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, cardinal numberCardinal 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 intersection theoremCantor's intersection theoremIn real analysis, a branch of mathematics, Cantor's intersection theorem, named after Georg Cantor, is a theorem related to compact sets in R, the set of real numbers. It states that a decreasing nested sequence of non-empty, closed and bounded subsets of R has nonempty intersection...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Cantor's theoremCantor's theoremIn 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 theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, Cantor's diagonal argumentCantor'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 mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
) - Carathéodory–Jacobi–Lie theorem (symplectic topologySymplectic 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 theoremCarathéodory's existence theoremIn 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...
(ordinary differential equations) - Carathéodory's theoremCarathéodory's theorem (conformal mapping)In mathematical complex analysis, Carathéodory's theorem, proved by , 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 theoremCarathéodory's theorem (convex hull)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 an r-simplex with vertices in P, where r \leq d...
(convex hull) - Carathéodory's theorem (measure theory)
- Carathéodory's extension theoremCarathéodory's extension theoremIn measure theory, Carathéodory's extension theorem states that any σ-finite measure defined on a given ring R of subsets of a given set Ω can be uniquely extended to the σ-algebra generated by R...
(measure theory) - Caristi fixed point theoremCaristi fixed point theoremIn 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 pointsFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
) - Carlson's theoremCarlson's theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Carmichael's theoremCarmichael's theoremCarmichael'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:Carmichael's theorem, named after...
(Fibonacci numberFibonacci numberIn mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s) - Carnot's theoremCarnot's theoremIn 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...
(geometryGeometryGeometry 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 (thermodynamicsThermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
) - Cartan–Dieudonné theorem (group theoryGroup 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...
) - Cartan–Hadamard theoremCartan–Hadamard theoremThe 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 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 theoremCartan–Kähler theoremIn 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 equationPartial 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 their partial derivatives with respect to those variables...
s) - Cartan–Kuranishi prolongation theorem (partial differential equationPartial 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 their partial derivatives with respect to those variables...
s) - Cartan's theoremCartan's theoremIn mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....
(Lie groupLie 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 BCartan's theorems A and BIn 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 variablesSeveral 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 theoremCasey's theoremIn 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 geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Castelnuovo theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Castelnuovo–de Franchis theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Castigliano's first and second theoremsCastigliano's methodCastigliano'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 analysisStructural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
) - Cauchy integral theorem (Complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Cauchy–Hadamard theorem (Complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Cauchy–Kowalevski theorem (partial differential equationPartial 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 their partial derivatives with respect to those variables...
s) - Cauchy's theoremCauchy's theorem (geometry)Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...
(geometryGeometryGeometry 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 ....
) - Cauchy's theoremCauchy's theorem (group theory)Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the order of G , then G contains an element of order p...
(finite groups) - Cayley–Bacharach theorem (projective geometryProjective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
) - Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
(Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Cayley-Salmon theorem (algebraic surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s) - Cayley's theoremCayley's theoremIn 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 acting on G...
(group theoryGroup 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 theoremCentral limit theoremIn 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 has a number of variants. In its common...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Cesàro's theorem (real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Ceva's theoremCeva's theoremCeva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively...
(geometryGeometryGeometry 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 ....
) - Chasles' theorems
- Chebotarev's density theoremChebotarev's density theoremChebotarev'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 ideal primes in the ring of algebraic integers of K. There are only...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Chen's theoremChen's theoremright|thumb|Chen JingrunChen'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...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Cheng's eigenvalue comparison theoremCheng's eigenvalue comparison theoremIn Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also...
(Riemannian geometryRiemannian 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 theoremChevalley–Shephard–Todd theoremIn 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 groupFinite 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 (field theoryField theory (mathematics)Field 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 theoremChinese remainder theoremThe Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Choi's theorem on completely positive mapsChoi's theorem on completely positive mapsIn mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional C*-algebras...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Chomsky–Schützenberger theoremChomsky–Schützenberger theoremIn formal language theory, the Chomsky–Schützenberger theorem is a statement about the number of words of a given length generated by an unambiguous context-free grammar. The theorem provides an unexpected link between the theory of formal languages and abstract algebra...
(formal language theory) - Choquet–Bishop–de Leeuw theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Chow's theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Chowla-Mordell theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Church-Rosser theorem (lambda calculusLambda calculusIn mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
) - Clairaut's theoremClairaut's theoremClairaut's theorem, published in 1743 by Alexis Claude 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...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Clapeyron's theoremClapeyron's theorem (elasticity)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...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Clark-Ocone theoremClark-Ocone theoremIn 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 groupsClassification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. 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...
(group theoryGroup 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 (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Clifford's circle theoremsClifford's circle theoremsIn geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.The first theorem considers any four circles passing through a common point M...
(circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) - Clifford's theorem on special divisors (algebraic curveAlgebraic 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) - Closed graph theoremClosed graph theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Closed range theoremClosed range theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Cluster decomposition theoremCluster decomposition theoremIn 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 theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) - Coase theoremCoase theoremIn 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 if trade in an externality is possible and there are no transaction costs, bargaining will lead to...
(economicsEconomicsEconomics is the social science that analyzes 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 theoremCochran's theoremIn statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used in to justify results relating to the probability distributions of statistics that are used in the analysis of variance.- Statement :...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Codd's theoremCodd's theoremCodd'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 modelRelational modelThe relational model for database management is a database model based on first-order predicate logic, first formulated and proposed in 1969 by Edgar F...
) - Cohn's irreducibility criterion (polynomials)
- Coleman-Mandula theoremColeman-Mandula theoremThe Coleman–Mandula theorem, named after Sidney Coleman and Jeffrey Mandula, is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a trivial way"...
(quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) - Commutation theoremCommutation theoremIn 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 algebraVon 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 his study of single operators, group...
) - Compactness theoremCompactness theoremIn 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 logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 theoremCompression theoremIn 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 theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) (structural complexity theoryStructural 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 theoremConservativity theoremIn mathematical logic, the conservativity theorem states the following: Suppose that a closed formula\exists x_1\ldots\exists x_m\,\varphiis a theorem of a first-order theory T...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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)
- Continuous mapping theoremContinuous mapping theoremIn probability theory, the continuous mapping theorem states that continuous functions are limit-preserving even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if xn → x...
(probability theoryProbability 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...
) - Convolution theoremConvolution theoremIn 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 transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s) - Cook's theoremCook's theoremIn 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 theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Corners theoremCorners theoremIn mathematics, the corners theorem is an important result, proved by Miklós Ajtai and Endre Szemerédi, of a statement in arithmetic combinatorics...
(arithmetic combinatoricsArithmetic combinatoricsArithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations...
) - Corona theoremCorona theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Cox's theoremCox's theoremCox'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...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
foundations) - Craig's theoremCraig's theoremIn mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem....
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Craig's interpolation theoremCraig interpolationIn mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ then there is a third formula ρ, called an interpolant, such that every nonlogical symbol...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 theoremCramér's theoremIn mathematical statistics, Cramér's theorem is one of several theorems of Harald Cramér, a Swedish statistician and probabilist.- Normal random variables :...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Cramér–Wold theorem (measure theory)
- Critical line theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Crooks fluctuation theoremCrooks Fluctuation TheoremThe 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...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Crossbar theoremCrossbar theoremIn geometry, the crossbar theorem states: if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC....
(Euclidean plane geometry) - Crystallographic restriction theoremCrystallographic restriction theoremThe 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 theoryGroup 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...
, crystallographyCrystallographyCrystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
) - Curtis–Hedlund–Lyndon theoremCurtis–Hedlund–Lyndon theoremThe Curtis–Hedlund–Lyndon theorem is a mathematical characterization of cellular automata in terms of their symbolic dynamics. It is named after Morton L. Curtis, Gustav A...
(cellular automata) - Cut-elimination theoremCut-elimination theoremThe 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 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 (neural networksNeural 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...
)
D
- Dandelin's theorem (solid geometrySolid geometryIn 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 (convex analysisConvex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
) - Darboux's theoremDarboux's theorem (analysis)Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval....
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Darboux's theoremDarboux's theoremDarboux'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 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 theoremDavenport–Schmidt theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Diophantine approximations) - Dawson–Gärtner theorem (asymptotic analysisAsymptotic analysisIn mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
) - De Branges' theoremDe Branges' theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - de Bruijn's theoremDe Bruijn's theoremDe Bruijn's theorem is a theorem concerning computational geometry and packing problems attributed to Nicolaas Govert de Bruijn.-Theorem:A box can be packed with a brick a×ab×abc if and only if the box has dimensions ap×abq×abcr for some natural numbers p, q, r....
(discrete geometryDiscrete geometryDiscrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
) - De Bruijn–Erdős theorem (incidence geometry)De Bruijn–Erdős theorem (incidence geometry)In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős, states a lower bound on the number of lines determined by n points in a projective plane...
- De Bruijn–Erdős theorem (graph theory)
- De Finetti's theoremDe Finetti's theoremIn 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...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - De Franchis theorem (Riemann surfaces)
- De Gua's theorem (geometryGeometryGeometry 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 theoremDe Moivre's formulaIn mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - De Rham's theorem (differential topologyDifferential 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 :...
) - Deduction theoremDeduction theoremIn mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...
(logicLogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
) - Denjoy theorem (dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Denjoy–Carleman theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Desargues' theoremDesargues' theoremIn projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
(projective geometryProjective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
) - Descartes' theoremDescartes' theoremIn 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:...
(plane geometry) - Descartes' theorem on total angular defect (polyhedra)
- Dilworth's theoremDilworth's theoremIn 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...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Dimension theorem for vector spacesDimension theorem for vector spacesIn mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite, or given by an infinite cardinal number, and defines the dimension of the space....
(vector spaceVector 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 algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Dini's theorem (analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Dirac's theorems (graph theoryGraph 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 theoremDirichlet's approximation theoremIn 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 exists integers p and q such that 1 ≤ q ≤ N and...
(Diophantine approximations) - Dirichlet's theorem on arithmetic progressionsDirichlet's theorem on arithmetic progressionsIn 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...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Dirichlet's unit theoremDirichlet's unit theoremIn mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K...
(algebraic number theoryAlgebraic number theoryAlgebraic 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 factorization,...
) - Disintegration theoremDisintegration theoremIn 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 theoremDivergence theoremIn 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 calculus) - Dominated convergence theoremDominated convergence theoremIn measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
(Lebesgue integrationLebesgue integrationIn mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , 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 subset of the real line or a higher...
) - Donaldson's theoremDonaldson's theoremIn 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 topologyDifferential 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 theoremDonsker's theoremIn 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 the functional central limit theorem....
(probability theoryProbability 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 theoremDoob decomposition theoremIn 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 theoremsDoob's martingale convergence theoremsIn 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 theoremDudley's theoremIn 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...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Duggan–Schwartz theorem (voting theory)
- Dunford–Pettis theorem (probability theoryProbability 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...
) - Dunford-Schwartz theoremDunford-Schwartz theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
)
E
- Earnshaw's theoremEarnshaw's theoremEarnshaw'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 British mathematician Samuel Earnshaw in 1842. It is usually referenced to magnetic...
(electrostaticsElectrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
) - Easton's theoremEaston's theoremIn set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. showed via forcing that...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Eberlein–Šmulian theoremEberlein–Šmulian theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Edge-of-the-wedge theoremEdge-of-the-wedge theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Edgeworth's limit theoremEdgeworth's limit theoremEdgeworth'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...
(economicsEconomicsEconomics is the social science that analyzes 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 theoremEgorov's theoremIn 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 theoremEhresmann's theoremIn 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 topologyDifferential 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 theoremEilenberg–Zilber theoremIn 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 X \times Y and those of the spaces X and Y. The theorem first appeared in a 1953 paper in the American Journal of...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Elitzur's theoremElitzur's theoremElitzur's theorem is a theorem in quantum and statistical field theory stating that local gauge symmetries cannot be spontaneously broken. The theorem was proposed in 1975 by S. Elitzur, who proved it for Abelian gauge fields on a lattice...
(physics) - Envelope theoremEnvelope theoremThe envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
(calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
) - Equal incircles theoremEqual incircles theoremIn 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 geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Equidistribution theoremEquidistribution theoremIn mathematics, the equidistribution theorem is the statement that the sequenceis uniformly distributed on the unit interval, when a is an irrational number...
(ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Equipartition theoremEquipartition theoremIn 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 theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Erdős–Anning theoremErdos–Anning theoremThe 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 geometryDiscrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
) - Erdos–Dushnik–Miller theorem (set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Erdős–Gallai theoremErdős–Gallai theoremThe Erdős–Gallai theorem is a result in graph theory, a branch of combinatorics, that gives a necessary and sufficient condition for a finite sequence to be the degree sequence of a simple graph...
(graph theoryGraph 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...
) - Erdos–Ginzburg–Ziv theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Erdős–Kac theoremErdos–Kac theoremIn number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω is the number of distinct prime factors of n, then, loosely speaking, the probability distribution ofis the standard normal...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Erdős–Ko–Rado theoremErdos–Ko–Rado theoremIn combinatorics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem on hypergraphs, specifically, on uniform hypergraphs of rank r.The theorem is as follows...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Erdős–Nagy theorem (discrete geometryDiscrete geometryDiscrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
) - Erdős–Pósa theoremErdős–Pósa theoremIn the mathematical discipline of graph theory, the Erdős–Pósa theorem, named after Paul Erdős and Lajos Pósa, states that there is a function such for each positive integer , every graph either contains vertex-disjoint circuits or it has a feedback vertex set of vertices that intersects every...
(graph theoryGraph 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 theoremErdős–Rado theoremIn partition calculus, part of combinatorial set theory, which is a branch of mathematics, the Erdős–Rado theorem is a basic result, extending Ramsey's theorem to uncountable sets.-Statement of the theorem:If r≥2 is finite, κ is an infinite cardinal, then...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Erdős–Stone theoremErdos–Stone theoremIn extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a non-complete graph H...
(graph theoryGraph 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...
) - Euclid's theoremEuclid's theoremEuclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Euclid–Euler theoremPerfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Euler's polyhedron theorem (polyhedra)
- Euler's rotation theoremEuler's rotation theoremIn geometry, 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 single rotation about some axis that runs through the fixed point. It also means that the composition of two...
(geometryGeometryGeometry 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 theoremEuler's theorem (differential geometry)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 (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Euler's theorem in geometryEuler's theorem in geometryIn 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 d^2=R \,...
(triangle geometry) - Euler's theorem on homogeneous functions (multivariate calculus)
- Exchange theorem (linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Excision theoremExcision theoremIn 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 theoryHomology 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.-The general idea:...
) - Exterior angle theoremExterior angle theoremThe exterior angle theorem can mean one of two things: Postulate 1.16 in Euclid's Elements which states that the exterior angle of a triangle is bigger than either of the remote interior angles, or a theorem in elementary geometry which states that the exterior angle of a triangle is equal to the...
(triangle geometry) - Extreme value theoremExtreme value theoremIn 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 theoremF. and M. Riesz theoremIn 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 theoremFWL theoremIn econometrics, the Frisch–Waugh–Lovell theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell.The Frisch–Waugh–Lovell theorem states that if the regression we are concerned with is:...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Faltings' theoremFaltings' theoremIn number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension...
(diophantine geometryDiophantine geometryIn mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
) - Fáry's theoremFáry's theoremIn mathematics, 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 theoryGraph 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 (knot theoryKnot theoryIn topology, knot theory is the study of 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 so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
) - Fatou's theoremFatou's theoremIn 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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Fatou–Lebesgue theoremFatou–Lebesgue theoremIn mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Faustman–Ohlin theorem (economicsEconomicsEconomics is the social science that analyzes 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 theoremFeit–Thompson theoremIn mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
(finite groupFinite 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...
s) - Fenchel's duality theoremFenchel's duality theoremIn 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 Rn and let g be a proper concave function on Rn...
(convex analysisConvex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
) - Fenchel's theorem (differential geometry)
- Fermat's Last TheoremFermat's Last TheoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fermat's little theoremFermat's little theoremFermat'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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fermat's theorem on sums of two squares (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fermat's theorem (stationary points)Fermat's theorem (stationary points)In mathematics, Fermat's theorem is 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 analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Fermat polygonal number theoremFermat polygonal number theoremIn additive number theory, 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...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fernique's theoremFernique's theoremIn 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 theoremFieller's theoremIn statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.-Approximate confidence interval:...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Final value theoremFinal value theoremIn 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 analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Fisher separation theoremFisher separation theoremIn economics, the Fisher separation theorem asserts that the objective of a corporation will be the maximization of its present value, regardless of the preferences of its shareholders. The theorem therefore separates management's "productive opportunities" from the entrepreneur's "market...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Fisher–Tippett–Gnedenko theorem (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Fitting's theoremFitting's theoremFitting'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 theoryGroup 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 circles theoremFive circles theoremIn geometry, the five circles theorem states that, given five circles centered on a common sixth circle and intersecting each other chainwise on the same circle, the lines joining the their second intersection points forms a pentagram whose points lie on the circles themselves....
(circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) - Five color theoremFive color theoremThe 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 theoryGraph 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 spacesFixed point theorems in infinite-dimensional spacesIn 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....
- Floquet's theorem (differential equations)
- Fluctuation dissipation theoremFluctuation dissipation theoremThe fluctuation-dissipation theorem is a powerful tool in statistical physics for predicting the behavior of non-equilibrium thermodynamical systems. These systems involve the irreversible dissipation of energy into heat from their reversible thermal fluctuations at thermodynamic equilibrium...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Fluctuation theoremFluctuation theoremThe 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 mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
) - Ford's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Focal subgroup theoremFocal subgroup theoremIn abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and...
(abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Foster's theoremFoster's theoremIn 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...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Four color theoremFour color theoremIn mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...
(graph theoryGraph 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 theoremFour-vertex theoremThe four-vertex theorem states that the curvature function of a simple, closed, smooth plane curve has at least four local extrema...
(differential geometry) - Fourier inversion theoremFourier inversion theoremIn mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.Sometimes the following expression is used as the definition of the Fourier transform:...
(harmonic analysisHarmonic 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 analysisHarmonic 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...
) - Franel–Landau theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fraňková–Helly selection theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Fredholm's theoremFredholm's theoremIn mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on...
(Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Freidlin–Wentzell theorem (stochastic processes)
- Freiman's theoremFreiman's theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Freudenthal suspension theoremFreudenthal suspension theoremIn 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 theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
) - Frobenius determinant theoremFrobenius determinant theoremIn mathematics, the Frobenius determinant theorem is a discovery made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it ....
(group theoryGroup 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...
) - Frobenius reciprocity theorem (group representationGroup representationIn 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 theoremFrobenius theorem (differential topology)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...
(foliationFoliationIn 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 theoremFrobenius theorem (real division algebras)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 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 theoremFroda's theoremIn mathematics, Froda's theorem, named after Alexandru Froda, describes the set of discontinuities 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 analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Frucht's theoremFrucht's theoremFrucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph...
(graph theoryGraph 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...
) - Fubini's theoremFubini's theoremIn mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
(integrationIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
) - Fubini's theorem on differentiationFubini's theorem on differentiationIn mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.- Statement :...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Fuchs's theoremFuchs's theoremIn mathematics, the Fuchs' theorem, named after Lazarus Fuchs, states that a second order differential equation of the formy + py'+qy=g\;...
(differential equations) - Fuglede's theoremFuglede's theoremIn 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, where N* denotes the adjoint of N.Normality of N is necessary, as is seen by...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Full employment theoremFull employment theoremIn computer science and mathematics, the term full employment theorem has been used to refer to a theorem showing that no algorithm can optimally perform a particular task done by some class of professionals. The name arises because such a theorem ensures that there is endless scope to keep...
(theoretical computer scienceTheoretical computer scienceTheoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
) - Fulton–Hansen connectedness theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Fundamental theorem of algebraFundamental theorem of algebraThe fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Fundamental theorem of arbitrage-free pricingFundamental theorem of arbitrage-free pricingThe fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
(financial mathematics) - Fundamental theorem of arithmeticFundamental theorem of arithmeticIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Fundamental theorem of calculusFundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
(calculusCalculusCalculus is a branch of 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 theorem...
) - Fundamental theorem on homomorphismsFundamental theorem on homomorphismsIn 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 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 theoremDinitz conjectureIn 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....
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Gauss theorem (vector calculus)
- Gauss's Theorema EgregiumTheorema EgregiumGauss'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 theoremGauss–Bonnet theoremThe Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
(differential geometry) - Gauss–Lucas theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Gauss–Markov theoremGauss–Markov theoremIn statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator of the coefficients is given by the...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Gauss–Wantzel theorem (geometryGeometryGeometry 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 algebraBanach algebraIn 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 theoremGelfand–Naimark theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Gelfond–Schneider theoremGelfond–Schneider theoremIn mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
(transcendence theoryTranscendence theoryTranscendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
) - Gershgorin circle theoremGershgorin circle theoremIn 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 (voting methods)
- Girsanov's theorem (stochastic processStochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es) - Glaisher's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Gleason's theoremGleason's theoremGleason'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 spaceHilbert 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 theoremGlivenko's theoremGlivenko'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 logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 (probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Goddard–Thorn theorem (vertex algebras)
- Gödel's completeness theoremGödel's completeness theoremGödel'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 logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 theoremGodunov's theoremIn 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 analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Going-up and going-down theoremsGoing up and going downIn 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 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 (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Goldie's theoremGoldie's theoremIn mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension as a right module over itself, and satisfies the ascending chain condition on right...
(ring theoryRing theoryIn abstract algebra, 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...
) - Goldstine theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Goldstone theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Golod–Shafarevich theoremGolod–Shafarevich theoremIn mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.-The inequality:...
(group theoryGroup 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...
) - Gomory's theorem (mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Goodstein's theoremGoodstein's theoremIn mathematical logic, Goodstein's theorem is a statement about the natural numbers, made by Reuben Goodstein, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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 theoremGordon–Newell theoremIn 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 theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
) - Gottesman–Knill theorem (quantum computation)
- Gradient theoremGradient theoremThe 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: \phi\left-\phi\left = \int_L...
(vector calculus) - Graph structure theoremGraph structure theoremIn 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 theoryGraph 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 theoremGrauert–Riemenschneider vanishing theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Great orthogonality theorem (group theoryGroup 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...
) - Green–Tao theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Green's theoremGreen's theoremIn 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 calculus) - Grinberg's theoremGrinberg's theoremIn 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 give new counterexamples to Tait's conjecture...
(graph theoryGraph 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 theoremGromov's compactness theorem (geometry)In Riemannian geometry, Gromov's compactness theorem states thatthe set of Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov-Hausdorff metric. It was proved by Mikhail Gromov....
(Riemannian geometryRiemannian 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 compactness theoremGromov's compactness theorem (topology)In symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or "bubbles". A bubble is a holomorphic sphere...
(Symplectic topologySymplectic 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...
) - Gromov's theorem on groups of polynomial growthGromov's theorem on groups of polynomial growthIn geometric group theory, Gromov's theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index....
(geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
) - Gromov–Ruh theorem (differential geometry)
- Gross–Zagier theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Grothendieck–Hirzebruch–Riemann–Roch theoremGrothendieck–Hirzebruch–Riemann–Roch theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Grothendieck's connectedness theoremGrothendieck's connectedness theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Grötzsch's theoremGrötzsch's theoremIn the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors...
(graph theoryGraph 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...
) - Grunwald–Wang theorem (algebraic number theoryAlgebraic number theoryAlgebraic 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 factorization,...
) - Grushko theoremGrushko theoremIn 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 theoryGroup 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 topologyDifferential 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 :...
) - H-theoremH-theoremIn Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...
(thermodynamicsThermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
) - Haag's theoremHaag's theoremRudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory , something now commonly known as Haag's Theorem...
(quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) - Haag–Lopuszanski–Sohnius theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Haboush's theorem (algebraic groupAlgebraic groupIn 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 theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, invariant theoryInvariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
) - Hadamard three-circle theoremHadamard three-circle theoremIn complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions.Let f be a holomorphic function on the annulusr_1\leq\left| z\right| \leq r_3....
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hadamard three-lines theoremHadamard three-lines theoremIn complex analysis, a branch of mathematics, theHadamard three-lines theorem, named after the French mathematician Jacques Hadamard, is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane....
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hadwiger's theoremHadwiger's theoremIn integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...
(geometryGeometryGeometry 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 theoremHahn decomposition theoremIn 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:...
(measure theory) - Hahn embedding theoremHahn embedding theoremIn 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 linearly ordered abelian groups....
(ordered groupOrdered 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 theoremHairy ball theoremThe hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Hahn–Banach theoremHahn–Banach theoremIn mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Hahn–Kolmogorov theorem (measure theory)
- Hahn–Mazurkiewicz theorem (continuum theory)
- Hajnal–Szemerédi theorem (graph theoryGraph 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...
) - Hales–Jewett theoremHales–Jewett theoremIn mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".An informal geometric statement...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Halpern–Lauchli theorem (Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Ham sandwich theoremHam sandwich theoremIn 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 measurable "objects" in -dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Hammersley–Clifford theoremHammersley–Clifford theoremThe 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 can be represented as a Markov network...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Hardy's theoremHardy's theoremIn mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a constant function...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hardy–Littlewood maximal theorem (real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Hardy–Littlewood tauberian theoremHardy–Littlewood tauberian theoremIn 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 analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Hardy–Ramanujan theoremHardy–Ramanujan theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Harish-Chandra theoremHarish-Chandra theoremIn matheamtics, Harish-Chandra theorem may refer to one of several theorems due to Harish-Chandra,including:*Harish-Chandra's theorem on the Harish-Chandra isomorphism*Harish-Chandra's classification of discrete series representations...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Harish-Chandra's regularity theoremHarish-Chandra's regularity theoremIn mathematics, Harish-Chandra's regularity theorem, introduced by , states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Harnack's curve theoremHarnack's curve theoremIn real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve...
(real algebraic geometryReal algebraic geometryIn mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them ....
) - Harnack's theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hartman-Grobman theoremHartman-Grobman theoremIn mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is an important theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point....
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Hartogs–Rosenthal theoremHartogs–Rosenthal theoremIn mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hartogs' theoremHartogs' theoremIn mathematics, Hartogs' theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hartogs' extension theoremHartogs' extension theoremIn mathematics, precisely in the theory of functions of several complex variables, Hartogs' extension theorem is a statement about the singularities of holomorphic functions of several variables...
(several complex variablesSeveral 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 theoremHasse norm theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Hasse's theorem on elliptic curves (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Hasse–Arf theoremHasse–Arf theoremIn 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 theoremHasse–Minkowski theoremThe 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Heckscher-Ohlin theoremHeckscher-Ohlin theoremThe Heckscher–Ohlin theorem is one of the four critical theorems of the Heckscher–Ohlin model. It states that a country will export goods that use its abundant factors intensively, and import goods that use its scarce factors intensively...
(economicsEconomicsEconomics is the social science that analyzes 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 analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Heine–Cantor theoremHeine–Cantor theoremIn mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M and N are metric spaces and M is compact then every continuous functionis uniformly continuous....
(metric geometry) - Hellinger–Toeplitz theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Hellmann–Feynman theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Helly–Bray theoremHelly–Bray theoremIn probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray....
(probability theoryProbability 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 theoremHelly's selection theoremIn 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. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Helly's theoremHelly's theoremHelly's theorem is a basic result in discrete geometry describing the ways that convex sets may intersect each other. It was discovered by Eduard Helly in 1913, but not published by him until 1923, by which time alternative proofs by and had already appeared...
(convex setConvex 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...
s) - Helmholtz theorem (classical mechanics) (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Helmholtz's theoremsHelmholtz's theoremsIn fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex filaments...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Herbrand's theorem (logicLogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
) - Herbrand–Ribet theoremHerbrand–Ribet theoremIn 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 fieldCyclotomic 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 theoremHigman's embedding theoremIn 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 theoryGroup 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 theoremHilbert's basis theoremIn mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
(commutative algebraCommutative 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 theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
) - Hilbert's NullstellensatzHilbert's NullstellensatzHilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis 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 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Hilbert-Schmidt theoremHilbert-Schmidt theoremIn 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 analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Hilbert-Speiser theoremHilbert-Speiser theoremIn mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension K of the rational field Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic...
(cyclotomic fieldCyclotomic 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 theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Hilbert's irreducibility theoremHilbert's irreducibility theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Hilbert's syzygy theoremHilbert's syzygy theoremIn 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 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 (differential geometry)
- Hilbert's theorem 90Hilbert's Theorem 90In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Hilbert projection theoremHilbert projection theoremIn mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every point x in a Hilbert space H and every closed convex C \subset H, there exists a unique point y \in C for which \lVert x - y \rVert is minimized over C....
(convex analysisConvex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
) - Hille–Yosida theoremHille–Yosida theoremIn functional analysis, the Hille–Yosida theorem characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It is sometimes stated for the special case of contraction semigroups, with the general case being called the Feller–Miyadera–Phillips...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Hindman's theorem (Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Hinge theoremHinge theoremIn geometry, the hinge theorem 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...
(geometryGeometryGeometry 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 geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Hirzebruch signature theorem (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Hirzebruch–Riemann–Roch theorem (complex manifolds)
- Hjelmslev's theoremHjelmslev's theoremIn geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... on a 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 a line.The proof is easy if one...
(geometryGeometryGeometry 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 theoremHobby–Rice theoremIn 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 Charles R. Hobby and John R. Rice; a simplified proof was given in 1976 by A...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Hodge index theoremHodge index theoremIn 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 surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s) - Hölder's theoremHölder's theoremIn 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 analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Holditch's theoremHolditch's theoremIn plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate around a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by...
(plane geometry) - Holland's schema theoremHolland's Schema TheoremHolland's schema theorem is widely taken to be the foundation for explanations of the power of genetic algorithms. It was proposed by John Holland in the 1970s....
(genetic algorithmGenetic algorithmA genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...
) - Holmström's theoremHolmström's theoremIn 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 ,...
(economicsEconomicsEconomics is the social science that analyzes 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 theoremHurewicz theoremIn 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 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Hurwitz's automorphisms theorem (algebraic curveAlgebraic 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 theoremHurwitz's theorem (complex analysis)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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Hurwitz's theorem (normed division algebraNormed division algebraIn mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
s) - Hurwitz's theoremHurwitz's theorem (number theory)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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
)
I
- Identity theoremIdentity theoremIn complex analysis, a branch of mathematics, 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...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Identity theorem for Riemann surfacesIdentity theorem for Riemann surfacesIn 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 (Computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Implicit function theoremImplicit function theoremIn 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 there may be such a function on...
(vector calculus) - Increment theoremIncrement theoremIn non-standard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f is differentiable at x and that Δx is infinitesimal...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Infinite monkey theoremInfinite monkey theoremThe 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....
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Integral root theorem (algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, polynomials) - Initial value theoremInitial value theoremIn mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let...
(integral transform) - Integral representation theorem for classical Wiener spaceIntegral representation theorem for classical Wiener spaceIn 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 theoremIntermediate value theoremIn 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 at least one point in its domain that the function maps to that value....
(calculusCalculusCalculus is a branch of 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 theorem...
) - Intercept theoremIntercept theoremThe intercept theorem, also known as Thales' theorem , is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Intersection theoremIntersection theoremIn 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 geometryProjective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
) - Inverse eigenvalues theorem (Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Inverse function theoremInverse function theoremIn 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 calculus) - Isomorphism extension theoremIsomorphism extension theoremIn 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 algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Isomorphism theoremIsomorphism theoremIn 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 algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Isoperimetric theorem (curveCurveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s, calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
)
J
- Jackson's theorem (queueing theoryQueueing theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
) - Jacobi's four-square theoremJacobi's four-square theoremIn 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 theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Jacobson density theoremJacobson density theoremIn 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....
(ring theoryRing theoryIn abstract algebra, 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 algebraLie 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...
) - Japanese theorem for concyclic polygons (Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Japanese theorem for concyclic quadrilateralsJapanese theorem for concyclic quadrilateralsIn geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle....
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - John’s theorem (geometryGeometryGeometry 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 theoremJordan curve theoremIn topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Jordan–Hölder theorem (group theoryGroup 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...
) - Jordan–Schönflies theorem (geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Jordan–Schur theoremJordan–Schur theoremIn mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan...
(group theoryGroup 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...
) - Jordan's theorem (multiply transitive groups) (group theoryGroup 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...
) - Jung's theoremJung's theoremIn 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...
(geometryGeometryGeometry 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 theoremJurkat–Richert theoremThe 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 theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. 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 theoremKachurovskii's theoremIn 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 analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
) - Kanamori–McAloon theoremKanamori–McAloon theoremIn mathematical logic, the Kanamori–McAloon theorem, due to , gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem....
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Kantorovich theoremKantorovich theoremThe Kantorovich theorem is a mathematical statement on the convergence of Newton's method. It was first stated by Leonid Kantorovich in 1940.Newton's method constructs a sequence of points that—with good luck—will converge to a solution x of an equation f = 0 or a vector solution of a...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Kaplansky density theoremKaplansky density theoremIn 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 algebraVon 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 his study of single operators, group...
) - Kaplansky's theorem on quadratic formsKaplansky's theorem on quadratic formsIn 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 formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s) - Karhunen–Loève theorem (stochastic processes)
- Karp–Lipton theorem (computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Kawamata–Viehweg vanishing theoremKawamata–Viehweg vanishing theoremIn 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 geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Kawasaki's theoremKawasaki's theoremKawasaki's theorem is a theorem in the mathematics of paper folding, named after Toshikazu Kawasaki, that gives a criterion for determining whether a crease pattern with a single vertex may be folded to form a flat figure.- Statement of the theorem :...
(paper foldingPaper foldingPaper craft is the collection of art forms employing paper or card as the primary artistic medium for the creation of three-dimensional objects. It is the most widely used material in arts and crafts. It lends itself to a wide range of techniques, it can for instance be folded, cut, glued,...
) - Kelvin's circulation theoremKelvin's circulation theoremIn 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...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Kharitonov's theoremKharitonov's theoremKharitonov'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 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 theoremKhinchin's theoremKhinchin'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...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Kirby–Paris theorem (proof theoryProof 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...
) - Kirchhoff's theoremKirchhoff's theoremIn 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 theoryGraph 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 theoremKirszbraun theoremIn 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 continuityLipschitz continuityIn mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
) - Kleene's recursion theoremKleene's recursion theoremIn computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions...
(recursion theoryRecursion theoryComputability theory, also called recursion 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 theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Knaster–Tarski theorem (order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Kneser theoremKneser theoremIn 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 (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Kodaira embedding theoremKodaira embedding theoremIn mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Kodaira vanishing theorem (complex manifoldComplex 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 theoremKoebe 1/4 theoremIn complex analysis, a branch of mathematics, the Koebe 1/4 theorem states that the image of an injective analytic function f:\mathbb D\to\mathbb C from the unit disk \mathbb D onto a subset of the complex plane contains the disk whose center is f\, and whose radius is |f\,'|/4. The theorem is...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Kolmogorov–Arnold–Moser theoremKolmogorov–Arnold–Moser theoremThe Kolmogorov–Arnold–Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics....
(dynamical systems) - Kolmogorov extension theoremKolmogorov extension theoremIn 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) - Kolmogorov's three-series theoremKolmogorov's three-series theoremIn probability theory, Kolmogorov's three-series theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions.-...
(mathematical series) - Kōmura's theorem (measure theory)
- König's theoremKönig'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 m_i In set theory, König's theorem In set theory, König's theorem (named after the Hungarian mathematician Gyula Kőnig, who published under the...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - König's theorem (graph theory)König's theorem (graph theory)In the mathematical area of graph theory, König's theorem, proved by Dénes Kőnig in 1931, describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs...
(bipartite graphBipartite 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)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 to the movement of the particles relative to the center...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - König's theorem (set theory)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 m_i In set theory, König's theorem In set theory, König's theorem (named after the Hungarian mathematician Gyula Kőnig, who published under the...
(cardinal numbers) - Kövari–Sós–Turán theorem (graph theoryGraph 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...
) - Kraft–McMillan theorem (coding theoryCoding theoryCoding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
) - Kramers theoremKramers theoremThe 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 . It was first discovered in 1930 by H. A...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Krein–Milman theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, discrete geometryDiscrete geometryDiscrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
) - Krener's theoremKrener's theoremIn mathematics, 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...
(control theoryControl 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 theoremKronecker's theoremIn mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.- The existence of extension fields :This is a theorem stating that a polynomial in a field, p ∈ F[x], has a root in an extension field E \supset F.For example, a polynomial in the reals such...
(diophantine approximationDiophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
) - Kronecker–Weber theoremKronecker–Weber theoremIn algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Krull's principal ideal theoremKrull's principal ideal theoremIn 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 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...
) - Krull–Schmidt theoremKrull–Schmidt theoremIn mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.-Definitions:...
(group theoryGroup 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...
) - Kruskal–Katona theoremKruskal–Katona theoremIn algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes. It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs. The theorem is named after Joseph Kruskal...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Kruskal's tree theoremKruskal's tree theoremIn 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...
(order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Krylov–Bogolyubov theorem (dynamical systems)
- Kuiper's theoremKuiper's theoremIn mathematics, Kuiper's theorem is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Künneth theoremKünneth theoremIn 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 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Kurosh subgroup theoremKurosh subgroup theoremIn 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 Alexander Kurosh, a Russian mathematician, in 1934...
(group theoryGroup 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 theoremKutta–Joukowski theoremThe 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...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
)
L
- L-balance theoremL-balance theoremIn mathematical finite group theory, the L-balance theorem was proved by .The letter L stands for the layer of a group, and "balance" refers to the property discussed below.-Statement:...
(finite groups) - Labelled enumeration theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Ladner's theoremLadner's theoremIn 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. Ladner's theorem, shown in 1975 by Richard Ladner, is a result asserting that, if P ≠ NP, then NPI is...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Lagrange's theoremLagrange's theorem (group theory)Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
(group theoryGroup 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...
) - Lagrange's theoremLagrange's theorem (number theory)In number theory, Lagrange's theorem states that:If the modulus is not prime, then it is possible for there to be more than n solutions. The exact number of solutions can be determined by finding the prime factorization of n...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Lagrange's four-square theoremLagrange's four-square theoremLagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Lagrange inversion theoremLagrange inversion theoremIn mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.-Theorem statement:...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Lagrange reversion theoremLagrange reversion theoremIn mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions....
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Lambek–Moser theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Lami's theoremLami's theoremIn statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear forces, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding forces...
(staticsStaticsStatics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
) - Landau prime ideal theoremLandau prime ideal theoremIn algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Larmor's theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Lasker–Noether theoremLasker–Noether theoremIn mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection of finitely many primary ideals...
(commutative algebraCommutative 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...
) - Lattice theoremLattice theoremIn mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups...
(abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Laurent expansion theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Lauricella's theoremLauricella's theoremIn the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely:...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Lax–Milgram theorem (partial differential equations)
- Lax–Richtmyer theorem (numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Lax–Wendroff theoremLax–Wendroff theoremIn computational mathematics, the Lax–Wendroff theorem, named after Peter Lax and Burton Wendroff, states that if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution....
(numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Lebesgue covering dimensionLebesgue covering dimensionLebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.-Definition:...
(dimension theoryDimension theoryIn mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
) - Lebesgue's decomposition theoremLebesgue's decomposition theoremIn mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given \mu and \nu two σ-finite signed measures on a measurable space , there exist two σ-finite signed measures \nu_0 and \nu_1 such that:* \nu=\nu_0+\nu_1\, * \nu_0\ll\mu *...
(dimension theoryDimension theoryIn mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
) - Lebesgue's density theoremLebesgue's density theoremIn mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A, the "density" of A is 1 at almost every point in A...
(dimension theoryDimension theoryIn mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
) - Lee Hwa Chung theoremLee Hwa Chung theoremThe Lee Hwa Chung theorem is a theorem in symplectic topology.The statement is as follows. Let M be a symplectic manifold with symplectic form ω. Let \alpha be a differential k-form on M which is invariant for all Hamiltonian vector fields...
(symplectic topologySymplectic 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...
) - Lebesgue differentiation theoremLebesgue differentiation theoremIn mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Le Cam's theoremLe Cam's theoremIn probability theory, Le Cam's theorem, named after Lucien le Cam , is as follows.Suppose:* X1, ..., Xn are independent random variables, each with a Bernoulli distribution , not necessarily identically distributed.* Pr = pi for i = 1, 2, 3, ...* \lambda_n = p_1 + \cdots + p_n.\,* S_n = X_1...
(probability theoryProbability 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...
) - Lee–Yang theoremLee–Yang theoremIn statistical mechanics, the Lee–Yang theorem states that if partition functions of certain models in statistical field theory with ferromagnetic interactions are considered as functions of an external field, then all zeros...
(statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
) - Lefschetz fixed-point theoremLefschetz fixed-point theoremIn mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
(fixed pointsFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
, algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Lefschetz–Hopf theorem (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Lefschetz hyperplane theoremLefschetz hyperplane theoremIn mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Lefschetz theorem on (1,1)-classesLefschetz theorem on (1,1)-classesIn algebraic geometry, a branch of mathematics, the Lefschetz theorem on -classes, named after Solomon Lefschetz, is a classical statement relating divisors on a compact Kähler manifold to classes in its integral cohomology. It is the only case of the Hodge conjecture which has been proved for all...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Lehmann–Scheffé theoremLehmann–Scheffé theoremIn statistics, the Lehmann–Scheffé theorem is prominent in mathematical statistics, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Leray's theoremLeray's theoremIn algebraic geometry, Leray's theorem relates abstract sheaf cohomology with Čech cohomology.Let \mathcal F be a sheaf on a topological space X and \mathcal U an open cover of X...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Leray–Hirsch theorem (algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Lerner symmetry theoremLerner symmetry theoremThe Lerner symmetry theorem is a result used in trade theory, which states that, based on an assumption of a zero balance of trade , an ad valorem import tariff will have the same effects as an export tax...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Lester's theoremLester's theoremIn Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter are concyclic.- References :...
(Euclidean plane geometry) - Levi's theorem (Lie groups)
- Levitzky's theoremLevitzky's theoremIn mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe...
(ring theoryRing theoryIn abstract algebra, 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...
) - Lévy continuity theoremLévy continuity theoremIn probability theory, the Lévy’s continuity theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Lévy's modulus of continuity theorem (probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Lickorish twist theorem (geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Lickorish–Wallace theoremLickorish–Wallace theoremIn mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to...
(3-manifolds) - Lie's third theoremLie's third theoremIn mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G. The relationship to the history has though become confused....
(Lie algebraLie 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...
) - Lindemann–Weierstrass theoremLindemann–Weierstrass theoremIn mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ..., are algebraic numbers which are linearly independent over the rational numbers ', then 1, ..., are algebraically...
(transcendence theoryTranscendence theoryTranscendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
) - Lie–Kolchin theoremLie–Kolchin theoremIn mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras....
(algebraic groupAlgebraic groupIn 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 theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Liénard's theoremLiénard's theoremIn mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard....
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Lindelöf's theoremLindelöf's theoremIn mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Lindström's theoremLindström's theoremIn mathematical logic, Lindström's theorem states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström) states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Linear congruence theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, modular arithmeticModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
) - Linear speedup theorem (computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Linnik's theoremLinnik's theoremLinnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that, if we denote p the least prime in the arithmetic progressiona + nd,\...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Lions–Lax–Milgram theoremLions–Lax–Milgram theoremIn mathematics, the Lions–Lax–Milgram theorem is a result in functional analysis with applications in the study of partial differential equations...
(partial differential equations) - Liouville's theorem (complex analysis)Liouville's theorem (complex analysis)In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
(entire functionEntire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
s) - Liouville's theorem (conformal mappings) (conformal mappings)
- Liouville's theorem (Hamiltonian)Liouville's theorem (Hamiltonian)In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...
(Hamiltonian mechanicsHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
) - Löb's theoremLöb's theoremIn mathematical logic, Löb's theorem states that in a theory with Peano arithmetic, for any formula P, if it is provable that "if P is provable then P", then P is provable...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Lochs' theoremLochs' theoremIn number theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number. The theorem was proved by Gustav Lochs in 1964....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Looman–Menchoff theoremLooman–Menchoff theoremIn the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Łoś' theorem (model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) - Löwenheim–Skolem theoremLöwenheim–Skolem theoremIn mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Lucas' theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Lukacs's proportion-sum independence theoremLukacs's proportion-sum independence theoremIn statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Lumer–Phillips theorem (semigroup theory)
- Luzin's theoremLuzin's theoremIn mathematics, Lusin's theorem in real analysis is a form of Littlewood's second principle.It states that every measurable function is a continuous function on nearly all its domain:...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Lyapunov–Malkin theorem (stability theoryStability theoryIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
) - Lyapunov's central limit theorem (probability theoryProbability 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...
)
M
- M. Riesz extension theoremM. Riesz extension theoremThe M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.-Formulation:Let E be a real vector space, F ⊂ E a vector subspace, and let K ⊂ E be a convex cone.A linear functional ...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - MacMahon Master theoremMacMahon Master theoremThe MacMahon Master theorem is a result in enumerative combinatorics and linear algebra, both branches of mathematics. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis...
(enumerative combinatoricsEnumerative combinatoricsEnumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations...
) - Mahler's compactness theoremMahler's compactness theoremIn mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate in a sequence of...
(geometry of numbersGeometry of numbersIn number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
) - Mahler's theoremMahler's theoremIn mathematics, Mahler's theorem, introduced by , expresses continuous p-adic functions in terms of polynomials.In any field, one has the following result. Let=f-f\,be the forward difference operator...
(p-adic analysisP-adic analysisIn mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
) - Maier's theoremMaier's theoremIn number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which the Cramér's probabilistic model of primes gives the wrong answer.It states that if π is the prime counting function and λ is greater than 1 then...
(analytic number theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. 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...
) - Malgrange preparation theoremMalgrange preparation theoremIn mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by .-Statement of Malgrange preparation theorem:...
(singularity theorySingularity theory-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
) - Malgrange–Ehrenpreis theoremMalgrange–Ehrenpreis theoremIn mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function...
(differential equations) - Mann's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Marcinkiewicz theoremMarcinkiewicz theoremIn mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces....
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Marden's theoremMarden's theoremIn mathematics, Marden's theorem, named after Morris Marden, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative....
(polynomials) - Mergelyan's theoremMergelyan's theoremMergelyan's theorem is a famous result from complex analysis, a branch of mathematics, named after the Armenian mathematician Sergei Nikitovich Mergelyan and was put forward by him in 1951. It states the following:...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Marginal value theoremMarginal value theoremIn behavioral ecology, the marginal value theorem considers an optimally foraging animal exploiting resources distributed in patches and that must decide when to leave a patch to start searching for a fresh one. The animal is assumed to have evolved to optimize a cost/benefit ratio: searching for...
(biologyBiologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
) - Markus−Yamabe theorem (2D stability theoryStability theoryIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
) - Marriage theoremMarriage theoremIn mathematics, Hall's marriage theorem is a combinatorial result that gives the condition allowing the selection of a distinct element from each of a collection of finite sets...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Martingale representation theoremMartingale representation theoremIn probability theory, the martingale representation theorem states that a random variable which is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion....
(probability theoryProbability 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...
) - Mason–Stothers theoremMason–Stothers theoremThe Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after W. Wilson Stothers, who published it in 1981, and R. C...
(polynomials) - Master theoremMaster theoremIn the analysis of algorithms, the master theorem provides a cookbook solution in asymptotic terms for recurrence relations of types that occur in the analysis of many divide and conquer algorithms...
(recurrence relationRecurrence relationIn mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
s, asymptotic analysisAsymptotic analysisIn mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
) - Maschke's theoremMaschke's theoremIn mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces...
(group representationGroup representationIn 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) - Matiyasevich's theorem (mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Max flow min cut theorem (graph theoryGraph 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...
) - Max Noether's theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Maximal ergodic theorem (ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Maximum power theoremMaximum power theoremIn electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must be equal to the resistance of the source as viewed from the output terminals...
(electrical circuits) - Maxwell's theoremMaxwell's theoremIn probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = T is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then the components...
(probability theoryProbability 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...
) - May's theoremMay's theoremIn social choice theory, May's theorem states that simple majority voting is the only anonymous, neutral, and monotone choice function between two alternatives. Further, this procedure is resolute when there are an odd number of voters and ties are not allowed. Kenneth May first published this...
(game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Mazur–Ulam theorem (normed spaces)
- Mazur's torsion theoremMazur's torsion theoremIn algebraic geometry Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers....
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Mean value theoremMean value theoremIn calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
(calculusCalculusCalculus is a branch of 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 theorem...
) - Measurable Riemann mapping theorem (conformal mapping)
- Median theorem (triangle geometry)
- Mellin inversion theoremMellin inversion theoremIn mathematics, the Mellin inversion formula tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function....
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Menelaus' theoremMenelaus' theoremMenelaus' theorem, named for Menelaus of Alexandria, is a theorem about triangles in plane geometry. Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, thenThis equation uses signed lengths of...
(geometryGeometryGeometry 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 ....
) - Menger's theoremMenger's theoremIn the mathematical discipline of graph theory and related areas, Menger's theorem is a basic result about connectivity in finite undirected graphs. It was proved for edge-connectivity and vertex-connectivity by Karl Menger in 1927...
(graph theoryGraph 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...
) - Mercer's theoremMercer's theoremIn mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most notable results of the work of James Mercer...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Mermin–Wagner theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Mertens' theoremsMertens' theoremsIn number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Metrization theorems (topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s) - Meusnier's theoremMeusnier's theoremIn differential geometry, Meusnier's theorem states that all curves on a surface passing through a given point p and having the same tangent line at p also have the same normal curvature at p and their osculating circles form a sphere. The theorem was first announced by Jean Baptiste Meusnier in...
(differential geometry) - Midy's theoremMidy's theoremIn mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Mihăilescu's theoremMihailescu's theoremCatalan's conjecture is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Milliken–Taylor theorem (Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Milliken's tree theoremMilliken's tree theoremIn mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets....
(Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Milman–Pettis theoremMilman–Pettis theoremIn mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.The theorem was proved independently by D. Milman and B. J. Pettis . S. Kakutani gave a different proof in , and John R. Ringrose published a shorter proof in 1959.- References :* S. Kakutani,...
(Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
) - Min-max theoremMin-max theoremIn linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Minimax theorem (game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Minkowski's theoremMinkowski's theoremIn mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n d contains a non-zero lattice point...
(geometry of numbersGeometry of numbersIn number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
) - Minkowski-Hlawka theorem (geometry of numbersGeometry of numbersIn number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
) - Minlos' theoremMinlos' theoremIn mathematics, Minlos' theorem states that a cylindrical measure on the dual of a nuclear space is a Radon measure if its Fourier transform is continuous. It can be proved using Sazonov's theorem....
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Miquel's theoremMiquel's theoremMiquel's theorem is a theorem in geometry, named after Auguste Miquel, about the intersection pattern of circles defined from six points on a triangle...
(circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) - Mitchell's embedding theoremMitchell's embedding theoremMitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating that every abelian category admits a full and exact embedding into the category of R-modules...
(category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
) - Mittag-Leffler's theoremMittag-Leffler's theoremIn complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.-Theorem:Let D...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Modigliani-Miller theoremModigliani-Miller theoremThe Modigliani–Miller theorem forms the basis for modern thinking on capital structure. The basic theorem states that, under a certain market price process , in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is...
(finance theory) - Modularity theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Mohr–Mascheroni theoremMohr–Mascheroni theoremIn mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. The result was originally published by Georg Mohr in 1672, but his proof languished in obscurity until 1928. The theorem was...
(geometryGeometryGeometry 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 ....
) - Monge's theoremMonge's theoremIn geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the three intersection points of the three pairs of external tangent lines are in fact collinear....
(geometryGeometryGeometry 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 ....
) - Monodromy theoremMonodromy theoremIn complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function along curves starting in the original domain of the function and ending in the larger set...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Monotone class theorem (measure theory)
- Monotone convergence theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Montel's theoremMontel's theoremIn complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal....
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Moore-Aronszajn theorem (Hilbert spaceHilbert 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...
) - Mordell–Weil theoremMordell–Weil theoremIn mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Moreau's theoremMoreau's theoremIn mathematics, Moreau's theorem is a result in convex analysis. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent...
(convex analysisConvex analysisConvex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
) - Morera's theoremMorera's theoremIn complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic....
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Morley's categoricity theoremMorley's categoricity theoremIn model theory, a branch of mathematical logic, a theory is κ-categorical if it has exactly one model of cardinality κ up to isomorphism....
(model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) - Morley's trisector theoremMorley's trisector theoremIn plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley...
(geometryGeometryGeometry 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 ....
) - Morton's theoremMorton's theoremMorton's theorem is a poker principle articulated by Andy Morton in a Usenet poker newsgroup. It states that in multi-way pots, a player's expectation may be maximized by an opponent making a correct decision....
(game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Mostow rigidity theoremMostow rigidity theoremIn mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...
(differential geometry) - Mountain pass theoremMountain pass theoremThe mountain pass theorem is an existence theorem from the calculus of variations. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point...
(calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
) - Moving equilibrium theorem (economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Multinomial theoremMultinomial theoremIn mathematics, the multinomial theorem says how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials.-Theorem:...
(algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Multiplication theoremMultiplication theoremIn mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...
(special functions) - Multiplicity-one theoremMultiplicity-one theoremIn the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group...
(group representationGroup representationIn 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) - Mumford vanishing theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Müntz–Szász theoremMüntz–Szász theoremThe Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass theorem on polynomial approximation can have holes dug into it, by restricting certain coefficients in the...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Mycielski's theorem (graph theoryGraph 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...
) - Myers theorem (differential geometry)
- Myhill–Nerode theoremMyhill–Nerode theoremIn the theory of formal languages, the Myhill–Nerode theorem provides a necessary and sufficient condition for a language to be regular. The theorem is named for John Myhill and Anil Nerode, who proved it at the University of Chicago in 1958 ....
(formal languageFormal languageA formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s)
N
- Nachbin's theoremNachbin's theoremIn mathematics, in the area of complex analysis, Nachbin's theorem is commonly used to establish a bound on the growth rates for an analytic function. This article will provide a brief review of growth rates, including the idea of a function of exponential type...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Nagata–Smirnov metrization theoremNagata–Smirnov metrization theoremThe Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular and Hausdorff and has a countably locally finite basis.Unlike Urysohn's metrization theorem, which...
(general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
) - Nagell–Lutz theoremNagell–Lutz theoremIn mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.-Definition of the terms:Suppose that the equationy^2 = x^3 + ax^2 + bx + c \...
(elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s) - Napoleon's theoremNapoleon's theoremIn mathematics, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centres of those equilateral triangles themselves form an equilateral triangle....
(triangle geometry) - Nash embedding theoremNash embedding theoremThe Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...
(differential geometry) - Nash–Moser theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Newlander–Niremberg theorem (differential geometry)
- Newton's theorem about ovalsNewton's theorem about ovalsIn mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant....
(curveCurveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s) - Nicomachus's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Nielsen fixed-point theorem (fixed pointsFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
) - Nielsen realization problemNielsen realization problemThe Nielsen realization problem is a question asked by about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by .-Statement:...
(geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Nielsen–Schreier theoremNielsen–Schreier theoremIn group theory, a branch of mathematics, the Nielsen–Schreier theorem is the statement that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.-Statement of the theorem:...
(free groupFree groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s) - No cloning theoremNo cloning theoremThe no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.The state of one system can be...
(quantum computation) - No free lunch theoremNo free lunch theoremIn mathematical folklore, the "no free lunch" theorem of Wolpert and Macready appears in the 1997 "No Free Lunch Theorems for Optimization." Wolpert had previously derived no free lunch theorems for machine learning...
(philosophy of mathematicsPhilosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
) - No hair theoremNo hair theoremThe no-hair theorem postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - No wandering domain theoremNo wandering domain theoremIn mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985.The theorem states that a rational map f : Ĉ → Ĉ with deg ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere...
(ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - No-broadcast theoremNo-broadcast theoremThe no-broadcast theorem is a result in quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem: since quantum states cannot be copied in general, they cannot be broadcast...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - No-communication theoremNo-communication theoremIn quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Noether's theoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
(Lie groupLie 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...
s, calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, differential invariantDifferential invariantIn mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view...
s, physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Noether's second theoremNoether's second theoremIn mathematics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least...
(calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Noether's theorem on rationality for surfacesNoether's theorem on rationality for surfacesIn mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective...
(algebraic surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s) - No-ghost theorem (vertex algebras)
- No-trade theoremNo-trade theoremIn financial economics, the no-trade theorem states that if markets are in a state of efficient equilibrium, if there are no noise traders or other non-rational interferences with prices, and if the structure by which traders or potential traders acquire information is itself common knowledge, then...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Nonsqueezing theoremNonsqueezing theoremThe nonsqueezing theorem is one of the most important theorems in symplectic geometry. It was first proven by Mikhail Gromov in 1985. The theorem states that one cannot embed a sphere into a cylinder via a symplectic map unless the radius of the sphere is less than or equal to the radius of the...
(symplectic geometry) - Norton's theoremNorton's theoremNorton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R...
(electrical networkElectrical networkAn electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
s) - Novikov's compact leaf theoremNovikov's compact leaf theoremIn mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that- Novikov's compact leaf theorem for S3 :Theorem: A smooth codimension-one foliation of the 3-sphere S3 has a compact leaf. The leaf is a torus T2 bounding a solid torus with the Reeb foliation.The theorem was...
(foliationFoliationIn 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) - Nyquist–Shannon sampling theoremNyquist–Shannon sampling theoremThe Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
(information theoryInformation theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
)
O
- Odd number theoremOdd number theoremThe odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology. It says that the number of multiple images produced by a bounded transparent lens must be odd....
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Open mapping theoremOpen mapping theorem (complex analysis)In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map .The open mapping theorem points to the sharp difference between holomorphy and real-differentiability...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Open mapping theoremOpen mapping theorem (functional analysis)In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Optical equivalence theorem (quantum opticsQuantum opticsQuantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...
) (quantum physics) - Optional stopping theoremOptional stopping theoremIn probability theory, the optional stopping theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value...
(probability theoryProbability 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...
) - Orbit theorem (control theoryControl 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...
) - Orbit-stabilizer theorem (group theoryGroup 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...
) - Ore's theoremOre's theoremOre's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore. It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with "sufficiently many edges" must contain a Hamilton cycle...
(graph theoryGraph 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...
) - Ornstein theorem (ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Ortsbogen theoremOrtsbogen theoremIn geometry, for any line segment AB and angle θ < π, the locus of points C on one side of line AB such that angle ACB equals θ is an arc of a circle...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Oseledec theoremOseledec theoremIn mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets in 1965 and reported at the International Mathematical Congress in Moscow in 1966...
(ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Osterwalder-Schrader theorem (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Ostrowski's theoremOstrowski's theoremOstrowski's theorem, due to Alexander Ostrowski , states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.- Definitions :...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Ostrowski–Hadamard gap theoremOstrowski–Hadamard gap theoremIn mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
)
P
- PCP theoremPCP theoremIn computational complexity theory, the PCP theorem states that every decision problem in the NP complexity class has probabilistically checkable proofs of constant query complexity and logarithmic randomness complexity .The PCP theorem says that for some universal constant K, for every...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Paley's theoremPaley's theoremIn mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley....
(algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
) - Paley–Wiener theorem (Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s) - Pandya theoremPandya theoremPandya theorem provides a theoretical framework for connecting the energy levels in jj coupling of a nucleon-nucleon and nucleon-hole system. It is also referred to as Pandya Transformation or Pandya Relation in literature...
(nuclear physicsNuclear physicsNuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
) - Pappus's centroid theoremPappus's centroid theoremIn mathematics, Pappus' centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution....
(geometryGeometryGeometry 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 ....
) - Pappus's hexagon theoremPappus's hexagon theoremIn mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
(geometryGeometryGeometry 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 ....
) - Paris–Harrington theoremParis–Harrington theoremIn mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Parovicenko's theorem (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Parallel axis theoremParallel axis theoremIn physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Parseval's theoremParseval's theoremIn mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
(Fourier analysis) - Parthasarathy's theoremParthasarathy's theoremIn mathematics and in particular the study of games on the unit square, Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem...
(game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Pascal's theoremPascal's theoremIn projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This theorem...
(conics) - Pasch's theoremPasch's theoremIn geometry, Pasch's theorem, stated in 1882 by a German mathematician Moritz Pasch, is a result of plane geometry which cannot be derived from Euclid's postulates. It would now be considered as order theory, but the point it makes is in relation to the axiomatic method.The statement is as follows...
(order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Peano existence theoremPeano existence theoremIn mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy-Peano theorem, named after Giuseppe Peano and Augustin Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value...
(ordinary differential equations) - Peetre theoremPeetre theoremIn mathematics, the Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Peixoto's theoremPeixoto's theoremIn the theory of dynamical systems, Peixoto theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties:...
(dynamical systems) - Penrose–Hawking singularity theorems (physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Pentagonal number theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Perfect graph theorem (graph theoryGraph 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...
) - Perlis theorem (graph theoryGraph 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...
) - Perpendicular axis theoremPerpendicular axis theoremIn physics, the perpendicular axis theorem can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Perron–Frobenius theoremPerron–Frobenius theoremIn linear algebra, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components, and also asserts a similar statement for certain classes of...
(matrix theory) - Peter–Weyl theoremPeter–Weyl theoremIn mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Phragmen–Lindelöf theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Picard theoremPicard theoremIn complex analysis, the term Picard theorem refers to either of two distinct yet related theorems, both of which pertain to the range of an analytic function.-Little Picard:...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Picard–Lindelöf theorem (ordinary differential equations)
- Pick's theoremPick's theoremGiven a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
(geometryGeometryGeometry 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 ....
) - Pickands–Balkema–de Haan theoremPickands–Balkema–de Haan theoremThe Pickands–Balkema–de Haan theorem is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown...
(extreme value theoryExtreme value theoryExtreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. The general theory sets out to assess the type of probability distributions generated by processes...
) - Pitman–Koopman–Darmois theorem (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Pitot theoremPitot theoremIn geometry, the Pitot theorem, named after the French engineer Henri Pitot, states that in a tangential quadrilateral the two sums of lengths of opposite sides are the same. The theorem is a consequence of the fact that two tangent line segments from a point outside the circle to the circle have...
(plane geometry) - Pivot theoremPivot theoremIn geometry, the pivot theorem states that, given any three points P, Q, and R on each respective side of a triangle ABC, the three circles through the points AQR, BPR and CPQ share a common point M...
(circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) - Planar separator theoremPlanar separator theoremIn graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices...
(graph theoryGraph 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...
) - Plancherel theoremPlancherel theoremIn mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
(Fourier analysis) - Plancherel theorem for spherical functionsPlancherel theorem for spherical functionsIn mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Poincaré–Bendixson theoremPoincaré–Bendixson theoremIn mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane.-Theorem:...
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Poincaré–Birkhoff–Witt theoremPoincaré–Birkhoff–Witt theoremIn the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem is a result giving an explicit description of the universal enveloping algebra of a Lie algebra...
(universal enveloping algebraUniversal enveloping algebraIn mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
s) - Poincaré–Hopf theoremPoincaré–Hopf theoremIn mathematics, the Poincaré–Hopf theorem is an important theorem that is still used today in differential topology...
(differential topologyDifferential 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 :...
) - Poincaré duality theoremPoincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
of manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s) - Poincaré recurrence theoremPoincaré recurrence theoremIn mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Poisson limit theoremPoisson limit theoremThe Poisson theorem gives a Poisson approximation to the binomial distribution, under certain conditions. The theorem was named after Siméon-Denis Poisson .- The theorem :If...
(probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
) - Pólya enumeration theoremPólya enumeration theoremThe Pólya enumeration theorem , also known as the Redfield–Pólya Theorem, is a theorem in combinatorics that both follows and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by John Howard Redfield in 1927...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Pompeiu's theoremPompeiu's theoremPompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is quite simple, but not classical. It states the following:...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Poncelet–Steiner theorem (geometryGeometryGeometry 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 ....
) - Positive energy theoremPositive energy theoremIn general relativity, the positive energy theorem states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Post's theoremPost's theoremIn computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.- Background :The statement of Post's theorem uses several concepts relating to definability and recursion theory...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Poynting's theoremPoynting's theoremPoynting's theorem is a statement due to British physicist John Henry Poynting about the conservation of energy for the electromagnetic field. Poynting's theorem takes into account the case when the electric and magnetic fields are coupled – static or stationary electric and magnetic fields are not...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Preimage theoremPreimage theoremIn mathematics, particularly in differential topology, the preimage theorem is a theorem concerning the preimage of particular points in a manifold under the action of a smooth map.-Statement of Theorem:...
(differential topologyDifferential 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 :...
) - Price's theoremPrice's theoremIn theoretical physics, particularly general relativity, Price's theorem can be informally stated as the principle that any inhomogeneities in the spacetime geometry outside a black hole will be radiated away as gravitational radiation. The theorem was formulated in 1972 by the American physicist...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Prime number theoremPrime number theoremIn number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Primitive element theoremPrimitive element theoremIn mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
(field theoryField theory (mathematics)Field 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....
) - Principal axis theoremPrincipal axis theoremIn the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated to an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse...
(linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Principal ideal theoremPrincipal ideal theoremIn mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, is the statement that for any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, thenIO_L\ is a principal ideal αOL, for OL the ring of...
(algebraic number theoryAlgebraic number theoryAlgebraic 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 factorization,...
) - Prokhorov's theoremProkhorov's theoremIn measure theory Prokhorov’s theorem relates tightness of measures to weak compactness in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov, who considered probability measures on complete separable metric spaces...
(measure theory) - Proth's theoremProth's theoremIn number theory, Proth's theorem is a primality test for Proth numbers.It states that if p is a Proth number, of the form k2n + 1 with k odd and k In number theory, Proth's theorem is a primality test for Proth numbers....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Pseudorandom generator theoremPseudorandom generator theoremIn computational complexity theory and cryptography, the existence of pseudorandom generators is related to the existence of one-way functions through a number of theorems, collectively referred to as the pseudorandom generator theorem.- Pseudorandomness :...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Ptolemaios' theorem (geometryGeometryGeometry 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 ....
) - Pythagorean theoremPythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
(geometryGeometryGeometry 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 ....
)
Q
- Quillen–Suslin theoremQuillen–Suslin theoremThe Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings...
(abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Quadratic reciprocity theoremQuadratic reciprocityIn number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
- Quantum threshold theoremQuantum threshold theoremIn quantum computing, the Threshold Theorem states that a quantum computer with noise can quickly and accurately simulate an ideal quantum computer, provided the level of noise is below a certain threshold...
(computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) (theoretical computer scienceTheoretical computer scienceTheoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
) - Quotient of subspace theoremQuotient of subspace theoremThe quotient of subspace theorem is an important property of finite dimensional normed spaces, discovered by Vitali Milman.Let be an N-dimensional normed space...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
)
R
- Rademacher's theoremRademacher's theoremIn mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn andis Lipschitz continuous, then f is Fréchet-differentiable almost everywhere in U In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Rado's theorem (harmonic analysisHarmonic 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...
) - Radon's theoremRadon's theoremIn geometry, Radon's theorem on convex sets, named after Johann Radon, states that any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect...
(convex setConvex 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...
s) - Radon–Nikodym theoremRadon–Nikodym theoremIn mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
(measure theory) - Ramanujam vanishing theoremRamanujam vanishing theoremIn algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to , that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Ramanujan–Skolem's theorem (diophantine equations)
- Ramsey's theoremRamsey's theoremIn combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs...
(graph theoryGraph 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...
, combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Rank-nullity theoremRank-nullity theoremIn mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...
(linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Rao–Blackwell theoremRao–Blackwell theoremIn statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Rashevsky–Chow theorem (control theoryControl 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...
) - Rational root theorem (algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, polynomials) - Rationality theoremRationality theoremThe rationality theorem is a theory introduced by political scientist Graham Allison in his book Essence of Decision: Explaining the Cuban Missile Crisis.Allison defined the rationality theorem like this:...
(politicsPoliticsPolitics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...
) - Ratner's theoremsRatner's theoremsIn mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive...
(ergodic theoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
) - Rauch comparison theoremRauch comparison theoremIn Riemannian geometry, the Rauch comparison theorem is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for large curvature, geodesics tend to converge, while for small curvature,...
(Riemannian geometryRiemannian 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...
) - Rédei's theorem (group theoryGroup 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...
) - Reeb sphere theoremReeb sphere theoremIn mathematics, Reeb sphere theorem, named after Georges Reeb, states that-Morse foliation:A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function...
(foliationFoliationIn 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) - Reeh–Schlieder theoremReeh–Schlieder theoremThe Reeh–Schlieder theorem is a result of relativistic local quantum field theory, stating that the vacuum is a cyclic vector for the field algebra of any open set in Minkowski space...
(local quantum field theoryLocal quantum field theoryThe Haag-Kastler axiomatic framework for quantum field theory, named after Rudolf Haag and Daniel Kastler, is an application to local quantum physics of C*-algebra theory. It is therefore also known as Algebraic Quantum Field Theory...
) - Reflection theoremReflection theoremIn algebraic number theory, a reflection theorem or Spiegelungssatz is one of a collection of theorems linking the sizes of different ideal class groups , or the sizes of different isotypic components of a class group...
(algebraic number theoryAlgebraic number theoryAlgebraic 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 factorization,...
) - Regev's theoremRegev's theoremIn algebra, Regev's theorem, proved by , states that the tensor product of two PI algebras is a PI algebra....
(ring theoryRing theoryIn abstract algebra, 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...
) - Reidemeister–Singer TheoremHeegaard splittingIn the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...
(geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Reider's theoremReider's theoremIn algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.-Statement:Suppose that L is a line bundle on a smooth projective surface with canonical bundle K...
(algebraic surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s) - Residue theoremResidue theoremThe residue theorem, sometimes called Cauchy's Residue Theorem, in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Reynolds transport theoremReynolds transport theoremReynolds' transport theorem , or in short Reynolds theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign....
(fluid dynamicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
) - Ribet's theorem (elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s) - Rice's theoremRice's theoremIn computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property...
(recursion theoryRecursion theoryComputability theory, also called recursion 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...
, computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Rice–Shapiro theorem (computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Richardson's theoremRichardson's theoremIn mathematics, Richardson's theorem establishes a limit on the extent to which an algorithm can decide whether certain mathematical expressions are equal...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Riemann mapping theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Riemann series theorem (mathematical series)
- Riemann's existence theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Riemann's theorem on removable singularities (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Riemann–Roch theoremRiemann–Roch theoremThe Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
(Riemann surfaceRiemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s, algebraic curveAlgebraic 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) - Riemann–Roch theorem for smooth manifolds (differential topologyDifferential 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 :...
) - Riemann–Roch theorem for surfacesRiemann–Roch theorem for surfacesIn mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by , after preliminary versions of it were found by and...
(algebraic surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s) - Riemann singularity theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Riesz representation theoremRiesz representation theoremThere are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, Hilbert spaceHilbert 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...
) - Riesz–Fischer theoremRiesz–Fischer theoremIn mathematics, the Riesz–Fischer theorem in real analysis refers to a number of closely related results concerning the properties of the space L2 of square integrable functions...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Riesz–Thorin theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Ringel–Youngs theorem (graph theoryGraph 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...
) - Robbins theoremRobbins theoremIn graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge that makes it into a strongly connected graph. Robbins' theorem, named after , states that the graphs that have strong orientations are exactly the 2-edge-connected graphs...
(graph theoryGraph 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...
) - Robertson–Seymour theoremRobertson–Seymour theoremIn graph theory, the Robertson–Seymour theorem states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering...
(graph theoryGraph 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...
) - Robin's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Robinson's joint consistency theoremRobinson's joint consistency theoremRobinson's joint consistency theorem is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability.The classical formulation of Robinson's joint consistency theorem is as follows:...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Rokhlin's theoremRokhlin's theoremIn 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4-manifold M has a spin structure , then the signature of its intersection form, a quadratic form on the second cohomology group H2, is divisible by 16...
(geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Rolle's theoremRolle's theoremIn calculus, Rolle's theorem essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero.-Standard version of the theorem:If a real-valued function ƒ is continuous on a closed...
(calculusCalculusCalculus is a branch of 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 theorem...
) - Rosser's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Rouché's theoremRouché's theoremRouché's theorem, named after , states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g| ...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Rouché–Capelli theoremRouché–Capelli theorem–Capelli theorem is the theorem in linear algebra that allows computing the number of solutions in a system of linear equations given the ranks of its augmented matrix and coefficient matrix...
(Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Routh's theoremRouth's theoremIn geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the intersection of three cevians...
(triangle geometry) - Routh–Hurwitz theoremRouth–Hurwitz theoremIn mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz-stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.-Notations:...
(polynomials) - Runge's theoremRunge's theoremIn complex analysis, Runge's theorem, also known as Runge's approximation theorem, named after the German Mathematician Carl Runge, and put forward by him in the year 1885, states the following: If K is a compact subset of C , A is a set containing at least one complex number from every bounded...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Rybczynski theoremRybczynski theoremThe Rybczynski theorem was developed in 1955 by the Polish-born English economist Tadeusz Rybczynski . The theorem states: At constant relative goods prices, a rise in the endowment of one factor will lead to a more than proportional expansion of the output in the sector which uses that factor...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Ryll-Nardzewski fixed point theoremRyll-Nardzewski fixed point theoremIn functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E is a normed vector space and K is a nonempty convex subset of E which is compact under the weak topology, then every group of affine isometries of K has at least one fixed point...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
)
S
- S–cobordism theorem (differential topologyDifferential 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 :...
) - Saccheri–Legendre theoremSaccheri–Legendre theoremIn absolute geometry, the Saccheri–Legendre theorem asserts that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate...
(absolute geometryAbsolute geometryAbsolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...
) - Sahlqvist correspondence theorem (modal logicModal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
) - Saint-Venant's theoremSaint-Venant's theoremIn solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Sard's theorem (differential geometry)
- Sarkovskii's theoremSarkovskii's theoremIn mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky, is a result about discrete dynamical systems. One of the implications of the theorem is that if a continuous discrete dynamical system on the real line has a periodic point of period 3, then it must have...
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Savitch's theoremSavitch's theoremIn computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, states that for any function ƒ ≥ log,...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Sazonov's theoremSazonov's theoremIn mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov , is a theorem in functional analysis.It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is Hilbert–Schmidt...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Schaefer's dichotomy theorem (computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Schauder fixed point theoremSchauder fixed point theoremThe Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Schilder's theoremSchilder's theoremIn mathematics, Schilder's theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, Schilder's theorem gives an estimate for the probability that a sample path of Brownian motion will stray far from the mean path . This statement is made precise using rate...
(stochastic processes) - Schnyder's theoremSchnyder's theoremIn graph theory, Schnyder's theorem is a characterization of planar graphs in termsof the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989....
(graph theoryGraph 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...
) - Schreier refinement theoremSchreier refinement theoremIn mathematics, the Schreier refinement theorem of group theory states that any two normal series of subgroups of a given group have equivalent refinements....
(group theoryGroup 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...
) - Schröder–Bernstein theorems for operator algebrasSchröder–Bernstein theorems for operator algebrasThe Schröder–Bernstein theorem, from set theory, has analogs in the context operator algebras. This article discusses such operator-algebraic results.- For von Neumann algebras :...
(operator algebraOperator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s) - Schroeder–Bernstein theorem for measurable spaces (measure theory)
- Schur's lemmaSchur's lemmaIn mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Schur's theoremSchur's theoremIn discrete mathematics, Schur's theorem is either of two different theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of A. Schur...
(Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Schur–Zassenhaus theoremSchur–Zassenhaus theoremThe Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product of N and G/N....
(group theoryGroup 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...
) - Schwartz kernel theoremSchwartz kernel theoremIn mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz himself have a two-variable theory that includes all reasonable...
(generalized functionGeneralized functionIn mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
s) - Schwartz–Zippel theorem (polynomials)
- Schwarz–Ahlfors–Pick theoremSchwarz–Ahlfors–Pick theoremIn mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model...
(differential geometry) - Schwenk's theorem (graph theoryGraph 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...
) - Scott core theoremScott core theoremIn mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott,...
(3-manifolds) - Seifert–van Kampen theoremSeifert–van Kampen theoremIn mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Separating axis theoremSeparating axis theoremFor objects lying in a plane , the separating axis theorem states that, given two convex shapes, there exists a line onto which their projections will be separate if and only if they are not intersecting. A line for which the objects have disjoint projections is called a separating axis...
(convex geometryConvex geometryConvex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...
) - Shannon–Hartley theoremShannon–Hartley theoremIn information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy channel coding theorem to the archetypal case of a continuous-time...
(information theoryInformation theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
) - Shannon's expansion theorem (Boolean algebra)
- Shannon's source coding theorem (information theoryInformation theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
) - Shell theoremShell theoremIn classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Shirshov–Witt theorem (Lie algebraLie 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...
) - Shannon's theoremNoisy channel coding theoremIn information theory, the noisy-channel coding theorem , establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data nearly error-free up to a computable maximum rate through the channel...
(information theoryInformation theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
) - Shift theoremShift theoremIn mathematics, the shift theorem is a theorem about polynomial differential operators and exponential functions...
(differential operatorDifferential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s) - Siegel–Walfisz theorem (analytic number theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. 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...
) - Silverman–Toeplitz theoremSilverman–Toeplitz theoremIn mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Simplicial approximation theoremSimplicial approximation theoremIn mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices — that is,...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Sinkhorn's theoremSinkhorn's theorem-Theorem:If A is an n × n matrix with strictly positive elements, then there exist diagonal matrices D1 and D2 with strictly positive diagonal elements such that D1AD2 is doubly stochastic....
(matrix theory) - Sion's minimax theoremSion's minimax theoremIn mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem.It states:Let X be a compact convex subset of a linear topological space and Y a convex subset of a linear topological space. If f is a real-valued function on X\times Y...
(game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
) - Sipser–Lautemann theorem (probabilistic complexity theory) (structural complexity theoryStructural 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...
) - Six circles theoremSix circles theoremIn geometry, the six circles theorem relates to a chain of six circles together with a triangle, such that each circle is tangent to two sides of the triangle and also to the preceding circle in the chain...
(circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) - Six exponentials theoremSix exponentials theoremIn mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials.-Statement:...
(transcendence theoryTranscendence theoryTranscendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
) - Sklar's theorem (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Skoda–El Mir theorem (complex geometryComplex geometryIn mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
) - Skolem–Mahler–Lech theoremSkolem–Mahler–Lech theoremIn additive number theory, the Skolem–Mahler–Lech theorem, named after Thoralf Skolem, Kurt Mahler, and Christer Lech, states that the indices of the null elements of a linear recurrence sequence are the union of a finite set and finitely many arithmetic progressions. The proofs of this result use...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Skolem–Noether theoremSkolem–Noether theoremIn mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in ring theory which characterizes the automorphisms of simple rings....
(simple algebras) - Skorokhod's embedding theoremSkorokhod's embedding theoremIn mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A.V...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Skorokhod's representation theoremSkorokhod's representation theoremIn mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables...
(statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Śleszyński–Pringsheim theoremŚleszyński–Pringsheim theoremIn mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century....
(continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
) - Slutsky's theoremSlutsky's theoremIn probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.The theorem was named after Eugen Slutsky. Slutsky’s theorem is also attributed to Harald Cramér....
(probability theoryProbability 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...
) - Smn theorem (recursion theoryRecursion theoryComputability theory, also called recursion 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...
, computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Sobolev embedding theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Sokhatsky–Weierstrass theorem (complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Sonnenschein–Mantel–Debreu Theorem (economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Sophie Germain's theoremSophie Germain's theoremIn number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation xp + yp = zp of Fermat's Last Theorem...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Soul theoremSoul theoremIn mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Meyer...
(Riemannian geometryRiemannian 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...
) - Soundness theorem (mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Space hierarchy theoremSpace hierarchy theoremIn computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in more space, subject to certain conditions. For example, a deterministic Turing machine can solve more decision problems...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Specht's theoremSpecht's theoremIn mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940....
(matrix theory) - Spectral theoremSpectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Speedup theoremSpeedup theoremIn computational complexity theory, a speedup theorem is a theorem that considers some algorithm solving a problem and demonstrates the existence of a more efficient algorithm solving the same problem....
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Sperner's theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Sphere theoremSphere theoremIn Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows...
(Riemannian geometryRiemannian 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...
) - Spin-statistics theoremSpin-statistics theoremIn quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Sprague–Grundy theoremSprague–Grundy theoremIn combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber. The Grundy value or nim-value of an impartial game is then defined as the unique nimber that the game is equivalent to...
(combinatorial game theoryCombinatorial game theoryCombinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...
) - Squeeze theoremSqueeze theoremIn calculus, the squeeze theorem is a theorem regarding the limit of a function....
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Stallings theorem about ends of groupsStallings theorem about ends of groupsIn the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
(group theoryGroup 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...
) - Stallings–Zeeman theorem (algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Stanley's reciprocity theoremStanley's reciprocity theoremIn combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone and the generating function of the cone's interior.A rational cone is the set of all...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Star of David theoremStar of David theoremThe Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by H.W. Gould in 1972.- Statement :...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Stark–Heegner theoremStark–Heegner theoremIn number theory, a branch of mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Stein–Strömberg theorem (measure theory)
- Steiner–Lehmus theorem (triangle geometry)
- Steinhaus theorem (measure theory)
- Steinitz theorem (graph theoryGraph 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...
) - Stewart's theoremStewart's theoremIn geometry, Stewart's theorem yields a relation between a lengths of the sides of the triangle and the length of a cevian of the triangle. Its name is in honor of the Scottish mathematician Matthew Stewart who published the theorem in 1746.- Theorem :...
(plane geometry) - Stinespring factorization theoremStinespring factorization theoremIn mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra as a composition of two completely positive maps each of which has a...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Stirling's theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Stokes' theoremStokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
(vector calculus, differential topologyDifferential 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 :...
) - Stolper–Samuelson theorem (economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Stolz–Cesàro theorem (calculusCalculusCalculus is a branch of 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 theorem...
) - Stone's representation theorem for Boolean algebrasStone's representation theorem for Boolean algebrasIn mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Stone...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Stone's theorem on one-parameter unitary groupsStone's theorem on one-parameter unitary groupsIn mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Stone–Tukey theorem (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Stone–von Neumann theoremStone–von Neumann theoremIn mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the Heisenberg group, quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
) - Stone–Weierstrass theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Strassman's theorem (field theoryField theory (mathematics)Field 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....
) - Structure theorem for finitely generated modules over a principal ideal domainStructure theorem for finitely generated modules over a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...
(abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Structure theorem for Gaussian measuresStructure theorem for Gaussian measuresIn mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space...
(measure theory) - Structured program theoremStructured program theoremThe structured program theorem is a result in programming language theory. It states that every computable function can be implemented in a programming language that combines subprograms in only three specific ways...
(computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Sturm's theoremSturm's theoremIn mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
(theory of equationsTheory 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...
) - Sturm–Picone comparison theorem (differential equations)
- Subspace theorem (Diophantine approximationDiophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
) - Supporting hyperplane theorem (convex geometryConvex geometryConvex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...
) - Swan's theoremSwan's theoremIn the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative...
(module theory) - Sylow theorems (group theoryGroup 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...
) - Sylvester's determinant theoremSylvester's determinant theoremIn matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named after James Joseph Sylvester....
(determinants) - Sylvester's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Sylvester's law of inertiaSylvester's law of inertiaSylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...
(quadratic forms) - Sylvester–Gallai theoremSylvester–Gallai theoremThe Sylvester–Gallai theorem asserts that given a finite number of points in the Euclidean plane, either# all the points are collinear; or# there is a line which contains exactly two of the points.This claim was posed as a problem by...
(plane geometry) - Symmetric hypergraph theoremSymmetric hypergraph theoremThe Symmetric hypergraph theorem is a theorem in combinatorics that puts an upper bound on the chromatic number of a graph...
(graph theoryGraph 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...
) - Symphonic theorem Altitude (triangle)In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
(triangle geometry) - Synge's theoremSynge's theoremIn mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936. Let M be a compact Riemannian manifold with positive sectional curvature...
(Riemannian geometryRiemannian 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...
) - Sz.-Nagy's dilation theoremSz.-Nagy's dilation theoremThe Sz.-Nagy dilation theorem states that every contraction T on a Hilbert space H has a unitary dilation U to a Hilbert space K, containing H, withT^n = P_H U^n \vert_H,\quad n\ge 0....
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Szegő limit theoremsSzegő limit theoremsIn mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.-Notation:...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Szemerédi's theoremSzemerédi's theoremIn number theory, Szemerédi's theorem is a result that was formerly the Erdős–Turán conjecture...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Szemerédi–Trotter theoremSzemerédi–Trotter theoremThe Szemerédi–Trotter theorem is a mathematical result in the field of combinatorial geometry. It asserts that given n points and m lines in the plane,the number of incidences The Szemerédi–Trotter theorem is a mathematical result in the field of combinatorial geometry. It asserts that given n...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Szpilrajn extension theoremSzpilrajn Extension TheoremIn mathematics, the Szpilrajn extension theorem, due to , is one of many examples of the use of the axiom of choice to find a maximal of a set with certain given properties....
(axiom of choice)
T
- Takagi existence theoremTakagi existence theoremIn class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K and the generalized ideal class groups defined via a modulus of K.It is called an existence theorem because a...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Takens' theoremTakens' theoremIn mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system...
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Tameness theoremTameness conjectureIn mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold....
(3-manifolds) - Tarski's indefinability theoremTarski's indefinability theoremTarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Taylor's theoremTaylor's theoremIn calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...
(calculusCalculusCalculus is a branch of 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 theorem...
) - Taylor–Proudman theoremTaylor–Proudman theoremIn fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high \Omega, the fluid velocity will be uniform along any line parallel to the axis of rotation...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Tennenbaum's theoremTennenbaum's theoremTennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of Peano arithmetic can be recursive.-Recursive structures for PA:...
(model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) - Thabit ibn Qurra's theoremThabit numberIn number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n...
(amicable numberAmicable numberAmicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s) - Thales' theoremThales' theoremIn geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem...
(geometryGeometryGeometry 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 ....
) - The duality theorem (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Thébault's theoremThébault's theoremThébault's theorem is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.-Thébault's problem I:...
(geometryGeometryGeometry 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 ....
) - Theorem of de Moivre–Laplace (probability theoryProbability 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...
) - Theorem of the cubeTheorem of the cubeIn mathematics, the theorem of the cube is a condition for a line bundle of a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry...
(algebraic varieties) - Theorem of three momentsTheorem of three momentsIn civil engineering and structural analysis Clapeyron's theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam....
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Theorem on friends and strangersTheorem on friends and strangersThe theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.-Statement:Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met...
(Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
) - Thévenin's theoremThévenin's theoremIn circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...
(electrical circuits) - Thue's theoremThue equationIn mathematics, a Thue equation is a Diophantine equation of the formwhere ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number...
(Diophantine equationDiophantine 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...
) - Thue–Siegel–Roth theoremThue–Siegel–Roth theoremIn mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very...
(diophantine approximationDiophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
) - Tietze extension theorem (general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
) - Tijdeman's theoremTijdeman's theoremIn number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equationy^m = x^n + 1,\...
(diophantine equations) - Tikhonov fixed point theorem (functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Time hierarchy theoremTime hierarchy theoremIn computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Titchmarsh theorem (integral transform)
- Titchmarsh convolution theoremTitchmarsh convolution theoremThe Titchmarsh convolution theorem is named after Edward Charles Titchmarsh,a British mathematician. The theorem describes the properties of the support of the convolution of two functions.- Titchmarsh convolution theorem :...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Tits alternativeTits alternativeIn mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
(geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
) - Toda's theoremToda's theoremToda's theorem was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy" and was given the 1998 Gödel Prize. The theorem states that the entire polynomial hierarchy PH is contained in PPP; this implies a closely related statement, that PH is contained in P#P...
(computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Tomita's theorem (operator algebraOperator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s) - Tonelli's theoremTonelli's theorem (functional analysis)In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for functional analysis and the calculus of variations. Roughly, it shows that weak lower semicontinuity for...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Topkis's theoremTopkis's TheoremIn mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes...
(economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
) - Toponogov's theoremToponogov's theoremIn the mathematical field of Riemannian geometry, Toponogov's theorem is a triangle comparison theorem. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region...
(Riemannian geometryRiemannian 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...
) - Torelli theoremTorelli theoremIn mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve C is determined by its Jacobian variety J, when the latter is given in the form of a principally polarized abelian variety...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Trichotomy theoremTrichotomy theoremIn mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes...
(finite groups) - Trudinger's theoremTrudinger's theoremIn mathematical analysis, Trudinger's theorem or the Trudinger inequality is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger ....
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Tsen's theoremTsen's theoremIn mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed. This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H i vanish for...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Tunnell's theoremTunnell's theoremIn number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. The congruent number problem asks which rational numbers can be the area of a right triangle with all three sides rational...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Tutte theoremTutte theoremIn the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings...
(graph theoryGraph 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...
) - Turán's theoremTurán's theoremIn graph theory, Turán's theorem is a result on the number of edges in a Kr+1-free graph.An n-vertex graph that does not contain any -vertex clique may be formed by partitioning the set of vertices into r parts of equal or nearly-equal size, and connecting two vertices by an edge whenever they...
(graph theoryGraph 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...
) - Turán–Kubilius theoremTurán–Kubilius inequalityThe Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a special case in 1934 by Paul Turán and generalized in 1956 and 1964 by Jonas Kubilius.-Statement of...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Tychonoff's theoremTychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
(general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
)
U
- Ugly duckling theoremUgly duckling theoremThe Ugly Duckling theorem is an argument asserting that classification is impossible without some sort of bias. It is named for Hans Christian Andersen's famous story of "The Ugly Duckling." It gets its name because it shows that, all things being equal, an ugly duckling is just as similar to a...
(computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
) - Uniformization theoremUniformization theoremIn mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, differential geometry) - Universal approximation theoremUniversal approximation theoremIn the mathematical theory of neural networks, the universal approximation theorem states that the standard multilayer feed-forward network with a single hidden layer that contains finite number of hidden neurons, and with arbitrary activation function are universal approximators on a compact...
(neural networksNeural 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...
) - Universal coefficient theoremUniversal coefficient theoremIn mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Unmixedness theorem (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
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- Valiant–Vazirani theorem (computational complexity theoryComputational complexity theoryComputational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
) - Van Aubel's theoremVan Aubel's theoremIn geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given quadrilateral , construct a square on each side. Van Aubel's theorem holds that the two line segments connecting the centers of opposite squares are of equal...
(quadrilaterals) - Van der Waerden's theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Van Vleck's theorem (mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Vantieghems theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Varignon's theoremVarignon's theoremVarignon's theorem is a statement in Euclidean geometry by Pierre Varignon that was first published in 1731. It deals with the construction of a particular parallelogram from an arbitrary quadrangle....
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Vietoris–Begle mapping theoremVietoris–Begle mapping theoremThe Vietoris–Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale.-Theorem:...
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Vinogradov's theoremVinogradov's theoremIn number theory, Vinogradov's theorem implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Virial theorem (classical mechanicsClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
) - Vitali convergence theoremVitali convergence theoremIn mathematics, the Vitali convergence theorem is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem...
(measure theory) - Vitali covering theorem (measure theory)
- Vitali theoremVitali setIn mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
(measure theory) - Vitali–Hahn–Saks theorem (measure theory)
- Viviani's theoremViviani's theoremViviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from a point to the sides of an equilateral triangle equals the length of the triangle's altitude....
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Von Neumann bicommutant theoremVon Neumann bicommutant theoremIn mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Von Neumann's theoremVon Neumann's theoremIn mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.-Statement of the theorem:Let G and H be Hilbert spaces, and let T : dom ⊆ G → H be a densely-defined operator from G into H...
(operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
) - Von Staudt–Clausen theoremVon Staudt–Clausen theoremIn number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by and ....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
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- Wagner's theorem (graph theoryGraph 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...
) - Waldhausen's theoremHeegaard splittingIn the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...
(geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Walter theoremWalter theoremIn mathematics, the Walter theorem, proved by , describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method to give a simpler proof.-Statement:...
(finite groups) - Weber's theoremWeber's theoremIn mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following....
(algebraic curveAlgebraic 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) - Wedderburn's little theoremWedderburn's little theoremIn mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
(ring theoryRing theoryIn abstract algebra, 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...
) - Wedderburn's theorem (abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Weierstrass–Casorati theoremWeierstrass–Casorati theoremIn complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Weierstrass factorization theoremWeierstrass factorization theoremIn mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Weierstrass preparation theoremWeierstrass preparation theoremIn mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P...
(several complex variablesSeveral 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...
,commutative algebraCommutative 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...
) - Weinberg–Witten theorem (quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
) - Well-ordering theoremWell-ordering theoremIn mathematics, the well-ordering theorem states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
) - Whitehead theoremWhitehead theoremIn homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....
(homotopy theory) - Whitney embedding theorem (differential manifolds)
- Whitney extension theoremWhitney extension theoremIn mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function off A in such a way as to have...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Whitney immersion theorem (differential topologyDifferential 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 :...
) - Whitney–Graustein TheoremRegular homotopyIn the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions....
(algebraic topologyAlgebraic 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, though usually most classify up to homotopy equivalence.Although algebraic topology...
) - Wick's theoremWick's theoremWick's theorem is a method of reducing high-order derivatives to a combinatorics problem . It is named after Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators...
(physicsPhysicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
) - Wiener's tauberian theoremWiener's tauberian theoremIn mathematical analysis, Wiener's tauberian theorem refers to one of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations of translations of a given...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Wiener–Ikehara theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Wigner–Eckart theorem (Clebsch–Gordan coefficients)
- Wilkie's theoremWilkie's theoremIn mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.-Formulations:...
(model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
) - Wilson's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Witt's theoremWitt's theoremWitt theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian...
(quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s) - Wold's theorem (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Wolstenholme's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
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- Z* theorem (finite groups)
- ZJ theoremZJ theoremIn mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′Z is a normal subgroup of G, for any Sylow p-subgroup S.-Notation and definitions:*J is the Thompson subgroup of a p-group S:...
(finite groups) - Zahorski theorem (real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. 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 the real...
) - Zariski's main theoremZariski's main theoremIn algebraic geometry, a field in mathematics, Zariski's main theorem is a theorem proved by that states that there is only one branch at any point of a normal variety...
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which 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 diverse fields as complex...
) - Zeckendorf's theoremZeckendorf's theoremZeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers....
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Zeilberger–Bressoud theorem (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Zsigmondy's theorem (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
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