1000 Sentences With "theorem" | Random Sentence Generator

In fact, the relevant theorem in quantum mechanics is called the "no cloning" theorem.

You can't really use theorem six or theorem seven, because those are yet to be proven down the line.

Their result, called the Nagaoka-Thouless theorem (also Nagaoka's theorem), relies on an idealized system of electrons on an atomic lattice.

Typically teachers derive and/or prove a new theorem and then present sample problems and applications showing how this theorem is used.

Some recent work includes a "decoupling theorem" — a sort of very abstract generalization of the Pythagorean theorem applied to oscillating waves like light or radio waves.

And so, it's a graph coloring problem, but Lovász, in 1978, gave a proof that was a technical tour de force, that used a topological theorem, the Borsuk-Ulam theorem.

She proved her most influential theorem after she turned 50.

But physicists have tried generalizing the theorem to multiple-hole systems.

Acclaim for the Stolper-Samuelson theorem was not instant or universal.

Consider the longstanding math problem called the four-color map theorem.

Perhaps the most famous graph coloring question is the four-color theorem.

I defy anybody to show me that theorem in AI. It's just baloney.

The strategy worked for Andrew Wiles when he proved Fermat's Last Theorem in 1994.

The index theorem and K-theory are actually two sides of the same coin.

Stonehenge secret: Did builders use Pythagoras&apos theorem 0007,000 years before the philosopher lived ?

The Bayesians focus on the probabilistic inference and Bayes' theorem to solve the problems.

Fans called her work on these mind-spinning abstractions the "theorem of the decade".

There's even a theorem, he tells me, to calculate the wisdom of a crowd.

The company has also grown through acquisitions, including of Theorem Clinical Research in 2015.

But, surely only thin wizards have the ability to master this wild, fantastical theorem!

He concluded, with a smile and a laugh, that Coase's theorem was completely wrong.

Perhaps Wiles' proof, with a few simplifications, is God's proof for Fermat's Last Theorem.

People think mathematics begins when you write down a theorem followed by a proof.

To understand their thinking, consider that classic political science concept: the median voter theorem.

If the Nagaoka-Thouless theorem really explains ferromagnetism, then it should apply to all lattices.

On the face of it, this wage pattern is consistent with the Stolper-Samuelson theorem.

But I don't know how the experts judge Andrew Wiles' proof of Fermat's Last Theorem.

But his theorem didn't account for the fact that equivalent fractions should only count once.

His 1983 theorem, the Johnson homomorphism, is still studied in the field of geometric topology.

"If there had been no Uhlenbeck theorem, people would not dare to try," he said.

Like black people are the Pythagorean theorem — they don't really come up in real life.

As it turns out, the Minsky-Papert perceptron theorem applies only to single-layer perceptrons.

The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem. The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.

The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

174) the Ryll-Nardzewski theorem See Theorem 7.3.1 Cf. (2.10) in model theory, and the Kuratowski and Ryll-Nardzewski measurable selection theorem. See Theorem 6.9.

A toy theorem of the Brouwer fixed- point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem. Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function value at the endpoints.

Illustration of the squeeze theorem When a sequence lies between two other converging sequences with the same limit, it also converges to this limit. In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, the police theorem and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of carabinieri. The squeeze theorem is used in calculus and mathematical analysis.

The commutant lifting theorem can be used to prove the left Nevanlinna-Pick interpolation theorem, the Sarason interpolation theorem, and the two-sided Nudelman theorem, among others.

The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (PNT) (Chandrasekharan, 1969).

Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, Law of cosines, and others that are linked to here.

Concerning such "best unbiased estimators", see also Cramér–Rao bound, Gauss–Markov theorem, Lehmann–Scheffé theorem, Rao–Blackwell theorem.

An earlier theorem of Ribet's, the Herbrand–Ribet theorem, is the converse to Herbrand's theorem on the divisibility properties of Bernoulli numbers and is also related to Fermat's Last Theorem.

In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.

He has made a number of discoveries in combinatorics and computer science, including Szemerédi's theorem, the Szemerédi regularity lemma, the Erdős–Szemerédi theorem, the Hajnal–Szemerédi theorem and the Szemerédi–Trotter theorem.

Kőnig's theorem is equivalent to numerous other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem.

The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is the dual LP. See Max-flow min-cut theorem#Linear program formulation. Other graph-related theorems can be proved using the strong duality theorem, in particular, Konig's theorem. The Minimax theorem for zero-sum games can be proved using the strong-duality theorem.

A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem.

In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem. In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise. Toy theorems can also have educational value as well.

A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem.

The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem. The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.

In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.

Mohr's theorem can be used to derive the three moment theorem (TMT).

The polar reciprocal and projective dual of this theorem give Pascal's theorem.

In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by .

Chapter nine discusses ways to weaken Ramsey's theorem, and the final chapter discusses stronger theorems in combinatorics including the Dushnik–Miller theorem on self-embedding of infinite linear orderings, Kruskal's tree theorem, Laver's theorem on order embedding of countable linear orders, and Hindman's theorem on IP sets. An appendix provides a proof of a theorem of Jiayi Liu, part of the collection of results showing that the graph Ramsey theorem does not fall into the big five subsystems.

Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied. Rogers uses the term weak recursion theorem for the first recursion theorem and strong recursion theorem for the second recursion theorem. One difference between the first and second recursion theorems is that the fixed points obtained by the first recursion theorem are guaranteed to be least fixed points, while those obtained from the second recursion theorem may not be least fixed points. A second difference is that the first recursion theorem only applies to systems of equations that can be recast as recursive operators.

96; . Threshold graphs are another special kind of comparability graph. Every comparability graph is perfect. The perfection of comparability graphs is Mirsky's theorem, and the perfection of their complements is Dilworth's theorem; these facts, together with the perfect graph theorem can be used to prove Dilworth's theorem from Mirsky's theorem or vice versa.

Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem. He also gives his name to the Pappus chain and to the Pappus configuration and Pappus graph arising from his hexagon theorem.

Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.

The Rendiconti also provided the introduction of normal numbers,. the original publications of the Plancherel theorem. and Carathéodory's theorem,. Hermann Weyl's proof of the equidistribution theorem,.

The converse is often included as part of the theorem. The theorem is very similar to Ceva's theorem in that their equations differ only in sign.

In abstract algebra, Ado's theorem is a theorem characterizing finite- dimensional Lie algebras.

In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by .

Minkowski's theorem can be used to prove Dirichlet's theorem on simultaneous rational approximation.

In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrashov the Lp theorem.

In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.

This theorem is a generalization of Pappus's (hexagon) theorem – Pappus's theorem is the special case of a degenerate conic of two lines. Pascal's theorem is the polar reciprocal and projective dual of Brianchon's theorem. It was formulated by Blaise Pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside titled "Essay pour les coniques. Par B. P.", translation Pascal's theorem is a special case of the Cayley–Bacharach theorem.

Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices..

The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of standard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem.

The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem.

Some authors refer to the "High school exterior angle theorem" as the strong form of the exterior angle theorem and "Euclid's exterior angle theorem" as the weak form.

In 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 problems.

Bloch's theorem was inspired by the following theorem of Georges Valiron: Theorem. If f is a non-constant entire function then there exist discs D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D. Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's Principle.

However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem.

This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

In particular, the theorem applies to doubling strategies. The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing.

There is a deeper preparation theorem for smooth functions, due to Bernard Malgrange, called the Malgrange preparation theorem. It also has an associated division theorem, named after John Mather.

The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.

The theorem is important because of the connection to the fundamental theorem of projective geometry.

The process of thermalisation can be described using the H-theorem or the relaxation theorem.

Foundational results include the Erdős–Wintner theorem and the Erdős–Kac theorem on additive functions.

Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes. Part 2 covers Galois number fields, including in particular Hilbert's theorem 90. Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields. Part 4 covers cyclotomic fields, including the Kronecker–Weber theorem (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) .

The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic.

According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices., Theorem 9.3.9, p. 570; , Section 9.3, "The Four Vertex Theorem", pp.

Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof of Feuerbach's theorem uses the converse theorem.

In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. Theorem 20.10. Lee states and proves this theorem in all generality.

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (i.e. the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.

Eduard Helly (June 1, 1884 in Vienna – 1943 in Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem were named.

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.

In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.

In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem .

In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum- modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion.

In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck.

For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem.

In Euclidean geometry, the trillium theorem – (from , literally 'lemma about trident', , literally 'theorem of trillium' or 'theorem of trefoil') is a statement about properties of inscribed and circumscribed circles and their relations.

He proved the Rademacher–Menchov theorem, the Looman–Menchoff theorem, and the Lusin–Menchoff theorem. Menshov was an Invited Speaker of the ICM in 1928 in Bologna and in 1958 in Edinburgh.

Therefore, if X is a metrizable space with a countable basis, one implication of Bing's metrization theorem holds. In fact, Bing's metrization theorem is almost a corollary of the Nagata-Smirnov theorem.

The density conjecture was finally proved using the tameness theorem and the ending lamination theorem by and .

Desargues' theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane.

In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was named after Siméon Denis Poisson (1781-1840). A generalization of this theorem is Le Cam's theorem.

Kleene's second recursion theorem and Rogers's theorem can both be proved, rather simply, from each other . However, a direct proof of Kleene's theorem does not make use of a universal program, which means that the theorem holds for certain subrecursive programming systems that do not have a universal program.

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the residue theorem and the well-known Mellin inversion theorem.

In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a p-group. The Glauberman replacement theorem is a generalization of it introduced by .

The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine, who first proved it for the Euclidean plane. Sometimes this theorem is also called Kirszbraun–Valentine theorem.

Typically lower hemicontinuous correspondences admit single-valued selections (Michael selection theorem, Bressan–Colombo directionally continuous selection theorem, Fryszkowski decomposable map selection). Likewise, upper hemicontinuous maps admit approximations (e.g. Ancel–Granas–Górniewicz–Kryszewski theorem).

195–196 Since the closure of C_x is also a connected subset containing x,Kelley, Theorem 20, p. 54; Willard, Theorem 26.8, p.193 it follows that C_x is closed.Willard, Theorem 26.12, p.

Another conjecture of Berge, proved in 1972 by László Lovász, is that the complement of every perfect graph is also perfect. This became known as the perfect graph theorem, or (to distinguish it from the strong perfect graph conjecture/theorem) the weak perfect graph theorem. Because Berge's forbidden graph characterization is self-complementary, the weak perfect graph theorem follows immediately from the strong perfect graph theorem.

Given a proof, it can find the theorem, making it a theorem-checker. Given part of a proof and part of a theorem, it will fill in the missing parts of the proof and the theorem, making it a theorem-explorer. There are implementations of miniKanren in Haskell, Racket, Ruby, Clojure, JavaScript, Scala, Swift and Python. The canonical implementation is an embedded language in Scheme.

In computability theory, the theorem, or universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions. It affirms the existence of a computable universal function, which is capable of calculating any other computable function. The universal function is an abstract version of the universal Turing machine, thus the name of the theorem. Roger's equivalence theorem provides a characterization of the Gödel numbering of the computable functions in terms of the smn theorem and the UTM theorem.

In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on a even-dimensional n-sphere having no sources or sinks. According to the Poincare-Hopf theorem, closed trajectories can encircle two centres and one saddle or one centre, but never just the saddle.

At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem. The following theorem of Artin then takes the place of the classical primitive element theorem. ;Theorem Let E/F be a finite degree field extension. Then E=F(\alpha) for some element \alpha\in E if and only if there exist only finitely many intermediate fields K with E\supseteq K\supseteq F. A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed): ;Corollary Let E/F be a finite degree separable extension.

The name "no-ghost theorem" is also a word play on the no-go theorem of quantum mechanics.

An axiomatization is sound when every theorem is a tautology, and complete when every tautology is a theorem.

The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.

The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and ) is a mathematical theorem detailing nonlinear stability of systems.

The hypergraph removal lemma can be used to prove, for instance, Szemerédi's theorem, and multi- dimensional Szemerédi's theorem.

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.

Several proofs of Fermat's little theorem are known. It is frequently proved as a corollary of Euler's theorem.

The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the Carathéodory existence theorem, which proves existence for some discontinuous functions ƒ.

The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem. The following theorem characterizes when any scalar function on (not necessarily linear) has a continuous linear extension to all of .

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.. The factor theorem states that a polynomial f(x) has a factor (x - k) if and only if f(k)=0 (i.e. k is a root)..

The Isabelle automated theorem prover is an interactive theorem prover, a higher order logic (HOL) theorem prover. It is an LCF-style theorem prover (written in Standard ML). It is thus based on small logical core (kernel) to increase the trustworthiness of proofs without requiring (yet supporting) explicit proof objects.

The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology. The theorem states that the number of multiple images produced by a bounded transparent lens must be odd.

In computational complexity theory the compression theorem is an important theorem about the complexity of computable functions. The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

Littlewood's theorem follows from the later Hardy–Littlewood tauberian theorem, which is in turn a special case of Wiener's tauberian theorem, which itself is a special case of various abstract Tauberian theorems about Banach algebras.

The following examples show how Fubini's theorem and Tonelli's theorem can fail if any of their hypotheses are omitted.

The process of equilibration can be described using the H-theorem or the relaxation theorem, see also entropy production.

The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the exponents are required to be rational integers and linear independence is only assured over the rational integers,. a result sometimes referred to as Hermite's theorem.. Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882.

In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just "extinction theorem", is a theorem that underlies the common understanding of scattering (as well as refraction, reflection, and diffraction). It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.

Campbell's theorem, also known as Campbell’s embedding theorem and the Campbell-Magaarrd theorem, is a mathematical theorem that evaluates the asymptotic distribution of random impulses acting with a determined intensity on a damped system. The theorem guarantees that any n-dimensional Riemannian manifold can be locally embedded in an (n + 1)-dimensional Ricci-flat Riemannian manifold.Romero, Carlos, Reza Tavakol, and Roustam Zalaltedinov. The Embedding of General Relativity in Five Dimensions. N.p.

These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by .

A typical example of this is Rochlin's theorem, which follows from the index theorem. The index problem for elliptic differential operators was posed in 1959 by Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem.

The electrokinematics theorem connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector. Since it contains, as a particular application, the Ramo-Shockley theorem, the electrokinematics theorem is also known as Ramo-Shockly-Pellegrini theorem.

In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches in parallel. It is named after Jacob Millman, who proved the theorem.

His most famous publication is the paper The representation of Lie algebras by matrices. Ado's Theorem has attracted the attention of many famous mathematicians who tried to improve its proof. Kenkichi Iwasawa proved the theorem over a field of prime characteristic. For this reason Ado's Theorem is also called Ado-Iwasawa Theorem.

The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.

The Walsh–Lebesgue theorem has been generalized to Riemann surfaces and to . In 1974 Anthony G. O'Farrell gave a generalization of the Walsh–Lebesgue theorem by means of the 1964 Browder–Wermer theorem with related techniques.

The theorem is thus also known by the names Whittaker–Shannon sampling theorem, Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, and Whittaker–Nyquist–Kotelnikov–Shannon, and may also be referred to as the cardinal theorem of interpolation.

In mathematics, the Sarason interpolation theorem, introduced by , is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation.

The shortest known proof of the four color theorem still has over 600 cases.See Four color theorem#Simplification and verification.

Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.

Over the complex numbers, the theorem is (or can be interpreted as) a special case of the equivariant index theorem.

Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.

Nqthm is a theorem prover sometimes referred to as the Boyer–Moore theorem prover. It was a precursor to ACL2.

The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture. It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.

In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful.

The axiom of choice, or some weaker version of it, is needed to prove this theorem in Zermelo–Fraenkel set theory. Conversely, this theorem together with the Boolean prime ideal theorem can prove the axiom of choice.

In the mathematical theory of probability, the Heyde theorem is the characterization theorem concerning the normal distribution (the Gaussian distribution) by the symmetry of one linear form given another. This theorem was proved by C. C. Heyde.

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–Kochen–Specker theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden-variable theories, which try to explain the predictions of quantum mechanics in a context-independent way. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors. The theorem is a complement to Bell's theorem (to be distinguished from the (Bell–)Kochen–Specker theorem of this article).

107–117 Further investigations reveal that if P is not in the interior of the triangle, but rather on the circumcircle, then PA, PB, PC form a degenerate triangle, with the largest being equal to the sum of the others, this observation is also known as Van Schooten's theorem. Pompeiu published the theorem in 1936, however August Ferdinand Möbius had published a more general theorem about four points in the Euclidean plane already in 1852. In this paper Möbius also derived the statement of Pompeiu's theorem explicitly as a special case of his more general theorem. For this reason the theorem is also known as the Möbius-Pompeiu theorem.

Kazimierz Kuratowski published his theorem in 1930.. The theorem was independently proved by Orrin Frink and Paul Smith, also in 1930, but their proof was never published. The special case of cubic planar graphs (for which the only minimal forbidden subgraph is K3,3) was also independently proved by Karl Menger in 1930. Since then, several new proofs of the theorem have been discovered.. In the Soviet Union, Kuratowski's theorem was known as either the Pontryagin–Kuratowski theorem or the Kuratowski–Pontryagin theorem, as the theorem was reportedly proved independently by Lev Pontryagin around 1927. However, as Pontryagin never published his proof, this usage has not spread to other places..

In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem.

There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem – a step in the direction of proof – or a short theorem appearing at an intermediate stage in a proof.

Sir Andrew John Wiles Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge. Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".

The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published.

In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem.

In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε-variational principle of Ekeland (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.

The entanglement-assisted classical capacity theorem is proved in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the quantum mutual information of the channel is an achievable rate, by a random coding strategy that is effectively a noisy version of the super-dense coding protocol. The converse theorem demonstrates that this rate is optimal by making use of the strong subadditivity of quantum entropy.

Thus the sequences considered in Fourier's theorem and in Budan's theorem have the same number of sign variations. This strong relationship between the two theorems may explain the priority controversy that occurred in 19th century, and the use of several names for the same theorem. In modern usage, for computer computation, Budan's theorem is generally preferred since the sequences have much larger coefficients in Fourier's theorem than in Budan's, because of the factorial factor.

The complexity of finding the point guaranteed by Minkowski's theorem, or the closely related Blitchfields theorem, have been studied from the perspective of TFNP search problems. In particular, it is known that a computational analogue of Blichfeldt's theorem, a corollary of the proof of Minkowski's theorem, is PPP- complete. It is also known that the computational analogue of Minkowski's theorem is in the class PPP, and it was conjectured to be PPP complete .

Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy. Norbert Wiener first proved Wiener's 1/f theorem, see Wiener's theorem. It states that if has absolutely convergent Fourier series and is never zero, then its inverse also has an absolutely convergent Fourier series.

On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.

The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem.

In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed.

In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.See .

His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem.

The equivalence of 3. and 1. is known as the primitive element theorem or Artin's theorem on primitive elements. Properties 4.

Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.

The theorem had in fact been previously discovered by Mitio Nagumo in 1942 and is also known as the Nagumo theorem.

Brouwer fixed-point theorem is a fixed-point theorem in topology, named after Dutchman Luitzen Brouwer, who proved it in 1911.

This concept stems from two fundamental theorems of quantum mechanics: the no-cloning theorem and the no-deleting theorem. But the no-hiding theorem is the ultimate proof of the conservation of quantum information. The importance of the no-hiding theorem is that it proves the conservation of wave function in quantum theory. This has never been proved earlier.

The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self- adjoint operators, (P,Q), satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L^2(\R). The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.

In mathematics, the bounded inverse theorem (or inverse mapping 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 graph theorem.

Poncelet discovered the following theorem in 1822: Euclidean compass and straightedge constructions can be carried out using only a straightedge if a single circle and its center is given. Swiss mathematician Jakob Steiner proved this theorem in 1833, leading to the name of the theorem. The constructions that this theorem states are possible are known as Steiner constructions.

In probability theory, Lévy’s continuity theorem, or Lévy's convergence 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. This theorem is the basis for one approach to prove the central limit theorem and it is one of the major theorems concerning characteristic functions.

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston DarbouxDarboux (1882). who established it as the solution of the Pfaff problem.

Van der Waerden's theorem, a precursor of Szemerédi's theorem, was proven in 1927. The cases k = 1 and k = 2 of Szemerédi's theorem are trivial. The case k = 3, known as Roth's theorem, was established in 1953 by Klaus Roth via an adaptation of the Hardy–Littlewood circle method. Endre Szemerédi proved the case k = 4 through combinatorics.

Lord Rayleigh published a generalization of the virial theorem in 1903. Henri Poincaré applied a form of the virial theorem in 1911 to the problem of determining cosmological stability. A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of the virial theorem was developed by Parker, Chandrasekhar and Fermi.

Newton derived an early theorem which attempted to explain apsidal precession. This theorem is historically notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995.Chandrasekhar, p. 183.

A topological characterization of Cantor spaces is given by Brouwer's theorem:. The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as: This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic.

In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields .

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.

In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces. This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to multivalued functions. One also meets the theorem and its variants outside topology.

The classical PCP theorem states that simulating the ground states of classical systems is hard. The quantum analog of the PCP theorem concerns simulations of quantum systems. Proving the quantum analog of the PCP theorem is an open problem.

The Theorem Proving System (TPS) is an automated theorem proving system for first-order and higher-order logic. TPS has been developed at Carnegie Mellon University. An educational version of it is known as ETPS (Educational Theorem Proving System).

The theorem was proved by Nikolay Bogoliubov and Ostap Parasyuk in 1955. The proof of the Bogoliubov–Parasyuk theorem was simplified later.

Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space ℓ1. An elementary proof can also be given.

As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder–Schreier theorem in Goursat varieties.

While the theorem is true, Kempe's proof is incorrect. Percy John Heawood illustrated it in 1890P. J. Heawood, "Map colour theorem", Quart.

Many earlier results such as the Riemann–Roch theorem and the Hodge theorem have been generalized or understood better using sheaf cohomology.

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Brianchon's theorem In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864).

In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem.

The Gibbard–Satterthwaite theorem is a similar theorem that deals with voting systems that elect a single winner. Likewise, Arrow's impossibility theorem deals with voting systems that yield a complete preference order of the candidates, rather than choosing only winners.

In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators.

In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz.

The proof of Lusin's theorem can be found in many classical books. Intuitively, one expects it as a consequence of Egorov's theorem and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.

It can be verified with elementary algebra. The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows.

Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.

George Rushing Kempf (Globe, Arizona, August 12, 1944 – Lawrence, Kansas, July 16, 2002) was a mathematician who worked on algebraic geometry, who proved the Riemann–Kempf singularity theorem, the Kempf–Ness theorem, the Kempf vanishing theorem, and who introduced Kempf varieties.

In reverse mathematics, Brouwer's theorem can be proved in the system WKL0, and conversely over the base system RCA0 Brouwer's theorem for a square implies the weak König's lemma, so this gives a precise description of the strength of Brouwer's theorem.

The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation. In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin.

In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911. Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.

In probability theory, the reversed compound agent theorem (RCAT) is a set of sufficient conditions for a stochastic process expressed in any formalism to have a product form stationary distribution (assuming that the process is stationary). The theorem shows that product form solutions in Jackson's theorem, the BCMP theorem and G-networks are based on the same fundamental mechanisms. The theorem identifies a reversed process using Kelly's lemma, from which the stationary distribution can be computed.

Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem). Today he is mostly remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered.

Jörg Siebeck discovered this theorem 81 years before Marden wrote about it. However, Dan Kalman titled his American Mathematical Monthly paper "Marden's theorem" because, as he writes, "I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book". attributes what is now known as Marden's theorem to and cites nine papers that included a version of the theorem. Dan Kalman won the 2009 Lester R. Ford Award of the Mathematical Association of America for his 2008 paper in the American Mathematical Monthly describing the theorem. A short and elementary proof of Marden’s theorem is explained in the solution of an exercise in Fritz Carlson’s book “Geometri” (in Swedish, 1943).

This theorem can be used to prove Lagrange's four- square theorem, which states that all natural numbers can be written as a sum of four squares. Gauss pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct proof of the four-square theorem that does not use the three-square theorem. Indeed, the four-square theorem was proved earlier, in 1770.

In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro.

In fact, the superposition theorem establishes the relation between the values of the resistances, the uniqueness theorem guarantees the uniqueness of such solution.

In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.

In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results.

An introduction to the theory of numbers, Oxford 1938. This generalization of Fermat's theorem is known as the sum of two squares theorem.

The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal.

Just as van der Waerden's theorem has a stronger density version in Szemerédi's theorem, the Hales–Jewett theorem also has a density version. In this strengthened version of the Hales–Jewett theorem, instead of coloring the entire hypercube WnH into c colors, one is given an arbitrary subset A of the hypercube WnH with some given density 0 < δ < 1\. The theorem states that if H is sufficiently large depending on n and δ, then the set A must necessarily contain an entire combinatorial line. The density Hales–Jewett theorem was originally proved by Furstenberg and Katznelson using ergodic theory.

All proofs of the De Bruijn–Erdős theorem use some form of the axiom of choice. Some form of this assumption is necessary, as there exist models of mathematics in which both the axiom of choice and the De Bruijn–Erdős theorem are false. More precisely, showed that the theorem is a consequence of the Boolean prime ideal theorem, a property that is implied by the axiom of choice but weaker than the full axiom of choice, and showed that the De Bruijn–Erdős theorem and the Boolean prime ideal theorem are equivalent in axiomatic power.For this history, see .

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

The First Fundamental Welfare Theorem asserts that market equilibria are Pareto efficient. In a pure exchange economy, a sufficient condition for the first welfare theorem to hold is that preferences be locally nonsatiated. The first welfare theorem also holds for economies with production regardless of the properties of the production function. Implicitly, the theorem assumes complete markets and perfect information.

The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of f at 0 in case f is injective; that is, univalent. The Koebe 1/4 theorem provides a related estimate in the case that f is univalent.

The lower bound 1/72 in Bloch's theorem is not the best possible. The number B defined as the supremum of all b for which this theorem holds, is called the Bloch's constant. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown. The similarly defined optimal constant L in Landau's theorem is called the Landau's constant.

The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these are further generalized by Schanuel's conjecture.

The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890Poincaré, Œuvres VII, 262–490 (theorem 1 section 8) and proved by Constantin Carathéodory using measure theory in 1919.Carathéodory, Ges. math. Schr.

A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation S is often used to indicate that S is a theorem. Formal theorems consist of formulas of a formal language and the transformation rules of a formal system.

In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that constrain the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states.

To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance. The geometrization theorem has been called Thurston's Monster Theorem, due to the length and difficulty of the proof.

With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus integration by parts, and Taylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign.

Schlickerei's theorem implies the Thue-Siegel-Roth theorem, whose p-adic analogue was already proved in 1958 by David Ridout. In 1998 Schlickewei was an invited speaker with talk The Subspace Theorem and Applications at the International Congress of Mathematicians in Berlin.

The advisor of many PhD. His main research areas are measure theory, functional analysis, foundations of mathematics and probability theory. Several theorems bear his name: the Ryll-Nardzewski fixed point theorem, “9. Theorem of Ryll-Nardzewski” (p. 171), “(9.6) Theorem (Ryll- Nardzewski)” (p.

Then there exists a color c and an infinite set D of natural numbers, all colored with c, such that every finite sum over D also has color c. The Milliken–Taylor theorem is a common generalisation of Hindman's theorem and Ramsey's theorem.

In 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. The theorem is named for Henri Lebesgue.

In 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. The Kawamata–Viehweg vanishing theorem generalizes it.

In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by , and the full result by .

In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold.

The Wold decomposition and the related Wold's theorem inspired Beurling's factorization theorem in harmonic analysis and related work on invariant subspaces of linear operators.

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by .

In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.

48 & 65. Rothe was also the first to formulate the q-binomial theorem, a q-analog of the binomial theorem, in an 1811 publication...

Important theoretical results include the Banach–Steinhaus theorem, spectral theorem (central to operator theory), Hahn–Banach theorem, open mapping theorem, and the closed graph theorem. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The proof of the theorem may be most easily understood as an application of the Perron-Frobenius theorem. This latter theorem comes from a branch of linear algebra known as the theory of nonnegative matrices. A good source text for the basic theory is Seneta (1973). The statement of Okishio's theorem, and the controversies surrounding it, may however be understood intuitively without reference to, or in-depth knowledge of, the Perron-Frobenius theorem or the general theory of nonnegative matrices.

In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups.

In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941.

In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere. proved the theorem for continuous functions, extended it to measurable functions, and extended it to arbitrary functions. and give historical accounts of the theorem.

The Łoś-Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formulas preserved under taking substructures is exactly the set of universal formulas (Hodges 1997). The theorem was discovered by Jerzy Łoś and Alfred Tarski.

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: every finite simple alternative ring is a field.

For Budan's theorem, it is the sequence of the coefficients. For the Budan–Fourier theorem, it is the sequence of values of the successive derivatives at a point. For Sturm's theorem it is the sequence of values at a point of the Sturm sequence.

Part of Wilson's theorem states that :(p-1)!\ \equiv\ -1 \pmod p for every prime p. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number np of Sylow's p-subgroups in the symmetric group Sp is (p − 2)!.

Rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its nullity (the dimension of its kernel) .

Khayyam was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyam had a general binomial theorem is based on his ability to extract roots.J. L. Coolidge, The Story of the Binomial Theorem, Amer. Math. Monthly, Vol.

The binomial theorem is valid more generally for any elements and of a semiring satisfying . The theorem is true even more generally: alternativity suffices in place of associativity. The binomial theorem can be stated by saying that the polynomial sequence is of binomial type.

At the end of this section, a short alternate proof of the Kelvin-Stokes theorem is given, as a corollary of the generalized Stokes' Theorem.

In 1826, Ostrogradsky gave the first general proof of the divergence theorem, which was discovered by Lagrange in 1762.For references, see Divergence theorem#History.

Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem.

We mention a few of the results which can be viewed as consequences of Stinespring's theorem. Historically, some of the results below preceded Stinespring's theorem.

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.

An axiomatic system is complete if every tautology is a theorem (derivable from axioms). An axiomatic system is sound if every theorem is a tautology.

Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms . By the perfect graph theorem of , the complement of any perfect graph is also perfect. Therefore, the complement of any comparability graph is perfect; this is essentially just Dilworth's theorem itself, restated in graph-theoretic terms . Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.

In political science, Arrow's impossibility theorem states that it is impossible to devise a voting system that satisfies a set of five specific axioms. This theorem is proved by showing that four of the axioms together imply the opposite of the fifth. In economics, Holmström's theorem is an impossibility theorem proving that no incentive system for a team of agents can satisfy all of three desirable criteria.

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set- valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem.

The Delta-compactness theorem of T. C. Lim states that if (X,d) is an asymptotically complete metric space, then every bounded sequence in X has a Delta-convergent subsequence. The Delta- compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable case) its proof does not depend on the Axiom of Choice.

In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows. If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points. It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930... cites pp. 26–31 of this 68-page pamphlet for the theorem.

With the definitions of multiple integration and partial derivatives, key theorems can be formulated, including the fundamental theorem of calculus in several real variables (namely Stokes' theorem), integration by parts in several real variables, the symmetry of higher partial derivatives and Taylor's theorem for multivariable functions. Evaluating a mixture of integrals and partial derivatives can be done by using theorem differentiation under the integral sign.

In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.

If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , and , then there exists at least one in the open interval such that :f'(c) = 0. This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

Berge conjectured the converse, that every Berge graph is perfect. This was finally proven as the strong perfect graph theorem of Chudnovsky, Robertson, Seymour, and Thomas (2006). It trivially implies the perfect graph theorem, hence the name. The perfect graph theorem has a short proof, but the proof of the strong perfect graph theorem is long and technical, based on a deep structural decomposition of Berge graphs.

However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the Cantor–Bernstein–Schroeder theorem, is non-constructive. It has recently been shown that the Cantor–Bernstein–Schroeder theorem implies the law of the excluded middle, hence there can be no constructive proof of the theorem.

Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm does not exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson which spans 21 years, with Matiyasevich completing the theorem in 1970.S. Barry Cooper, Computability theory, p. 98 The theorem is now known as Matiyasevich's theorem or the MRDP theorem.

In mathematics, the Freidlin–Wentzell theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path. This statement is made precise using rate functions. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.

In 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 (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0). This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions.

The second recursion theorem is a generalization of Rogers's theorem with a second input in the function. One informal interpretation of the second recursion theorem is that it is possible to construct self-referential programs; see "Application to quines" below. :The second recursion theorem. For any partial recursive function Q(x,y) there is an index p such that \varphi_p \simeq \lambda y.Q(p,y).

A quite different proof given by David Gale is based on the game of Hex. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed-point theorem for dimension 2. By considering n-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.

In the comparability graph of a partially ordered set, a clique represents a chain and a coloring represents a partition into antichains, and induced subgraphs of comparability graphs are themselves comparability graphs, so Mirsky's theorem states that comparability graphs are perfect. Analogously, Dilworth's theorem states that every complement graph of a comparability graph is perfect. The perfect graph theorem of states that the complements of perfect graphs are always perfect, and can be used to deduce Dilworth's theorem from Mirsky's theorem and vice versa .

The polynomial remainder theorem may be used to evaluate f(r) by calculating the remainder, R. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem. The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.

In mathematics, Fermat's theorem (also known as interior extremum 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 (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. By using Fermat's theorem, the potential extrema of a function \displaystyle f, with derivative \displaystyle f', are found by solving an equation in \displaystyle f'. Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum).

In algebra, the Mori–Nagata theorem introduced by and , states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral closure of A in K is a direct product of r Krull domains, where r is the number of minimal prime ideals of A. The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if R is a Nagata ring, then every R-subalgebra of finite type is again a Nagata ring . The Mori–Nagata theorem follows from Matijevic's theorem.

Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis.

In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.

Jouko Väänänen, Lindström's Theorem Lindström's theorem has been extended to various other systems of logic in particular modal logics by Johan van Benthem and Sebastian Enqvist.

A series of simplifications of the proof culminated in the proofs by and . In modern textbooks Petersen's theorem is covered as an application of Tutte's theorem.

The statement does not generalize to higher degree forms, although there are a number of partial results such as Darboux's theorem and the Cartan-Kähler theorem.

The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155–234. that was motivated by the classic Riemann mapping theorem in complex analysis.

In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

In 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 .

In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets.

In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to a larger field.

Their result would be valid for the general case if the Nash embedding theorem can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian manifolds in 1956. In 1943 Allendoerfer and Weil published their proof for the general case, in which they first used an approximation theorem of H. Whitney to reduce the case to analytic Riemannian manifolds, then they embedded "small" neighborhoods of the manifold isometrically into a Euclidean space with the help of the Cartan-Janet local embedding theorem, so that they can patch these embedded neighborhoods together and apply the above theorem of Allendoerfer and Fenchel to establish the global result. This is, of course, unsatisfactory for the reason that the theorem only involves intrinsic invariants of the manifold, then the validity of the theorem should not rely on its embedding into a Euclidean space.

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems.

4-37 gave a Nielsen equivalence version of Grushko's theorem (stated above) and provided some generalizations of Grushko's theorem for amalgamated free products. Scott (1974) gave another topological proof of Grushko's theorem, inspired by the methods of 3-manifold topologyScott, Peter. An introduction to 3-manifolds.

It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash- Kuiper Isometric C^1 embedding theorem and the Smale-Hirsch Immersion theorem.

Nearly all of the important theorems in the traditional theory of the Lebesgue integral, such as Lebesgue's dominated convergence theorem, the Riesz–Fischer theorem, Fatou's lemma, and Fubini's theorem may also readily be proved using this construction. Its properties are identical to the traditional Lebesgue integral.

The theorem is due to Julius Petersen, a Danish mathematician. It can be considered as one of the first results in graph theory. The theorem appears first in the 1891 article "Die Theorie der regulären graphs". By today's standards Petersen's proof of the theorem is complicated.

In game theory, Aumann's agreement theorem is a theorem which demonstrates that rational agents with common knowledge of each other's beliefs cannot agree to disagree. It was first formulated in the 1976 paper titled "Agreeing to Disagree" by Robert Aumann, after whom the theorem is named.

In 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: every finite alternative division ring is a field.

In 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 and elliptic modular functions.

Among other problems, it assumed implicitly a theorem (now known as Puiseux's theorem), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795).

Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers. It was first proven by Klaus Roth in 1953. Roth's Theorem is a special case of Szemerédi's Theorem for the case k = 3.

Bartlett's bisection theorem is an electrical theorem in network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical two-port network can be transformed into a lattice network.Bartlett, AC, "An extension of a property of artificial lines", Phil. Mag., vol 4, p902, November 1927.

4 and Corollary 2.1.5; , Theorem 11, p. 83. However, this method requires a specific finite obstruction set to work, and the theorem does not provide one. The theorem proves that such a finite obstruction set exists, and therefore the problem is polynomial because of the above algorithm.

The 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".; Jeffrey E. Mandula (2015). "Coleman-Mandula theorem" Scholarpedia 10(2):7476.

The assertion that quadrilateral A2 is a square is equivalent to the assertion that the diagonals of A1 are equal and perpendicular to each other. The latter assertion is the content of van Aubel's theorem. Thus van Aubel's theorem is a special case of the PDN-theorem.

In symplectic topology and dynamical systems, Poincaré–Birkhoff theorem (also known as Poincaré–Birkhoff fixed point theorem and Poincaré's last geometric theorem) states that every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points.

A statement and proof for the theorem was given by J.E. Littlewood in 1912, but he attributes it to no one in particular, stating it as a known theorem. Harald Bohr and Edmund Landau attribute the theorem to Jacques Hadamard, writing in 1896; Hadamard published no proof.

However, sequent calculus and cut- elimination were not known at the time of Herbrand's theorem, and Herbrand had to prove his theorem in a more complicated way.

The 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.

Many other important generalizations of Picard's theorem can be obtained from Ahlfors theory. One especially striking result (conjectured earlier by André Bloch) is the Five Island theorem.

The proof uses a fixed-point theorem by Eilenberg and Montgomery. Note: Every convex set is contractible, so Diamantaras' theorem is more general than the previous three.

It provides clues with which to identify the eclipse, and says that Hipparchus used a formula "as in Theorem 12," a theorem of Ptolemy's which is extant.

A related theorem is CPCFC, in which "triangles" is replaced with "figures" so that the theorem applies to any pair of polygons or polyhedrons that are congruent.

Siegel's theorem on integral points occurs in Chapter 28. Mordell's theorem on the finite generation of the group of rational points on an elliptic curve is in Chapter 16, and integer points on the Mordell curve in Chapter 26. In a hostile review of Lang's book, Mordell wrote He notes that the content of the book is largely versions of the Mordell–Weil theorem, Thue–Siegel–Roth theorem, Siegel's theorem, with a treatment of Hilbert's irreducibility theorem and applications (in the style of Siegel). Leaving aside issues of generality, and a completely different style, the major mathematical difference between the two books is that Lang used abelian varieties and offered a proof of Siegel's theorem, while Mordell noted that the proof "is of a very advanced character" (p. 263).

Thus, in this case, the perfect graph theorem implies Kőnig's theorem that the size of a maximum independent set in a bipartite graph is also n − M,, later rediscovered by . a result that was a major inspiration for Berge's formulation of the theory of perfect graphs. Mirsky's theorem characterizing the height of a partially ordered set in terms of partitions into antichains can be formulated as the perfection of the comparability graph of the partially ordered set, and Dilworth's theorem characterizing the width of a partially ordered set in terms of partitions into chains can be formulated as the perfection of the complements of these graphs. Thus, the perfect graph theorem can be used to prove Dilworth's theorem from the (much easier) proof of Mirsky's theorem, or vice versa.

In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm.

In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning "proposition" or "theorem"). Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal over I has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.

Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although, there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.

Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least , the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least . In turn Ore's theorem is generalized by the Bondy–Chvátal theorem. One may define a closure operation on a graph in which, whenever two nonadjacent vertices have degrees adding to at least , one adds an edge connecting them; if a graph meets the conditions of Ore's theorem, its closure is a complete graph.

An Introduction Kluwer Academic Publishers (new edition 2001) . Brouwer's theorem is probably the most important."... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu Fixed Point Theory an Introduction Kluwer Academic Publishers (new edition 2001) . It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.E.g.: S. Greenwood J. Cao Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem University of Auckland, New Zealand.

In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e. the smallest total weight of the edges which if removed would disconnect the source from the sink. The max-flow min-cut theorem is a special case of the duality theorem for linear programs and can be used to derive Menger's theorem and the Kőnig–Egerváry theorem.

In differential calculus, the domain-straightening theorem states that, given a vector field X on a manifold, there exist local coordinates y_1, \dots, y_n such that X = \partial / \partial y_1 in a neighborhood of a point where X is nonzero. The theorem is also known as straightening out of a vector field. The Frobenius theorem in differential geometry can be considered as a higher- dimensional generalization of this theorem.

As of July 2012, the MML included 1150 articles written by 241 authors.The MML Query search engine In aggregate, these contain more than 10,000 formal definitions of mathematical objects and about 52,000 theorems proved on these objects. More than 180 named mathematical facts have so benefited from formal codification. Some examples are the Hahn–Banach theorem, Kőnig's lemma, Brouwer fixed point theorem, Gödel's completeness theorem and Jordan curve theorem.

If Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.

Shankar initially served as a research associate at Stanford University, from 1986 to 1988. In 1989, he joined SRI International's Computer Science Laboratory. While at SRI, he has used the Boyer–Moore theorem prover to prove metatheorems such as the tautology theorem, Godel's incompleteness theorem and the Church-Rosser theorem. He has contributed to the development of automated reasoning technology, deductive systems and computational engines, including the Prototype Verification System.

The Vogt–Russell theorem states that the structure of a star, in hydrostatic and thermal equilibrium with all energy derived from nuclear reactions, is uniquely determined by its mass and the distribution of chemical elements throughout its interior. Although referred to as a theorem, the Vogt–Russell theorem has never been formally proved. The theorem is named after astronomers Heinrich Vogt and Henry Norris Russell, who devised it independently.

It is an extension of Thévenin's theorem stating that any collection of voltage sources and resistors with two terminals is electrically equivalent to an ideal current source. Edward Lawry Norton likewise described this in 1926 in an internal report for Bell Labs. The theorem is well-known under the name Norton's theorem or Mayer-Norton theorem. Hans Ferdinand Mayer published some 25 technical articles and held more than 80 patents.

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others.

The theorem was proven by Bing in 1951 and was an independent discovery with the Nagata–Smirnov metrization theorem that was proved independently by both Nagata (1950) and Smirnov (1951). Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem. It is a common tool to prove other metrization theorems, e.g. the Moore metrization theorem – a collectionwise normal, Moore space is metrizable – is a direct consequence.

In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element. If the element is neutral, then the theorem reduces to the usual index theorem. The classical formula such as the Atiyah–Bott formula is a special case of the theorem.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz.

Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving. The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. The theorem has also been generalized to triangles on other surfaces of constant curvature.

Illustration of the hyperplane separation theorem. In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.

For example, if = 2 and = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem..

Cederbaum's theorem defines hypothetical analog electrical networks which will automatically produce a solution to the minimum s–t cut problem. Alternatively, simulation of such a network will also produce a solution to the minimum s–t cut problem. This article gives basic definitions, a statement of the theorem and a proof of the theorem. The presentation in this article closely follows the presentation of the theorem in the original publication.

The inscribed angle theorem is used in many proofs of elementary Euclidean geometry of the plane. A special case of the theorem is Thales' theorem, which states that the angle subtended by a diameter is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle.

14 August 2012.Serafino, Jason (2012). "Christoph Waltz Signs On for Terry Gilliam’s 'Zero Theorem'". Complex. 14 August 2012.Schafer, Sandy (2012). "Terry Gilliam Is Making ‘The Zero Theorem’ with Christoph Waltz", screenrant.com. 14 August 2012.Brown, Todd (2012)."Terry Gilliam Resurrects the Zero Theorem with Christoph Waltz" . Twitch Film. 14 August 2012. The Zero Theorem premiered at the 70th Venice International Film Festival on 2 September 2013.

Holland's schema theorem, also called the fundamental theorem of genetic algorithms, is an inequality that results from coarse-graining an equation for evolutionary dynamics. The Schema Theorem says that short, low-order schemata with above-average fitness increase exponentially in frequency in successive generations. The theorem was proposed by John Holland in the 1970s. It was initially widely taken to be the foundation for explanations of the power of genetic algorithms.

In fact, this is another way to state the Lie–Kolchin theorem. Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace. The result for Lie algebras was proved by and for algebraic groups was proved by . The Borel fixed point theorem generalizes the Lie–Kolchin theorem.

The history of this important theorem is described by Collingwood and Lohwater. It was published by Weierstrass in 1876 (in German) and by Sokhotski in 1868 in his Master thesis (in Russian). So it was called Sokhotski's theorem in the Russian literature and Weierstrass's theorem in the Western literature. The same theorem was published by Casorati in 1868, and by Briot and Bouquet in the first edition of their book (1859).

The three circles theorem follows from the fact that for any real a, the function Re log(zaf(z)) is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant a so that this harmonic function has the same maximum value on both circles. The theorem can also be deduced directly from Hadamard's three-lines theorem.

The left action given by turns into a homogeneous -space. The closed subgroup theorem now simplifies the hypotheses considerably, a priori widening the class of homogeneous spaces. Every closed subgroup yields a homogeneous space. In a similar way, the closed subgroup theorem simplifies the hypothesis in the following theorem.

In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by , where it was used to prove the Thompson uniqueness theorem.

Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres \pi^S_3 is cyclic of order 24; this is Rokhlin's original approach. It can also be deduced from the Atiyah–Singer index theorem. See Â genus and Rochlin's theorem. gives a geometric proof.

Example of a theorem painting (c.1850) from the Metropolitan Museum of Art Theorem stencil, sometimes also called theorem painting or velvet painting, is the art of making stencils and using them to make drawings or paintings on fabric or paper.Chotner, Deborah (1992). American Naive Paintings, pp. 370-71.

In modern language, Holmgren's uniquess theorem states that any distributional solution of such a system of equations must be analytic and therefore unique, by the Cauchy–Kowalevski theorem.

Kakutani's fixed-point theorem is used in proving the existence of cake allocations that are both envy-free and Pareto efficient. This result is known as Weller's theorem.

In the proof, both the Japanese theorem for cyclic quadrilaterals and the quadrilateral case of the cyclic polygon theorem are proven as a consequence of Thébault's problem III.

Unlike the Urysohn's metrization theorem which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable.

The open mapping theorem, also known as Banach's homomorphism theorem, gives a sufficient condition for a continuous linear operator between complete metrizable TVSs to be a topological homomorphism.

A generalization of the Kac–Bernstein theorem is the Darmois–Skitovich theorem, in which instead of sum and difference linear forms from n independent random variables are considered.

In the theory of causal structure on Lorentzian manifolds, Geroch's theorem or Geroch's splitting theorem (first proved by Robert Geroch) gives a topological characterization of globally hyperbolic spacetimes.

In statistics, the Fisher–Tippett–Gnedenko theorem (also the Fisher–Tippett theorem or the extreme value theorem) is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem and its convergence details are given to Fréchet (1927), Ronald Fisher and Leonard Henry Caleb Tippett (1928), Mises (1936) and Gnedenko (1943). The role of the extremal types theorem for maxima is similar to that of central limit theorem for averages, except that the central limit theorem applies to the average of a sample from any distribution with finite variance, while the Fisher–Tippet–Gnedenko theorem only states that if the distribution of a normalized maximum converges, then the limit has to be one of a particular class of distributions.

In a 2010 interview, David X. Cohen revealed that the episode writer Ken Keeler, a PhD mathematician, penned and proved a theorem based on group theory, and then used it to explain the plot twist in this episode. However, Keeler does not feel it carries enough importance to be designated a theorem, and prefers to call it a proof. Cut-the-Knot, an educational math website created by Alexander Bogomolny, refers to Keeler's result as the "Futurama Theorem",The Futurama Theorem and Puzzle at Cut-the-Knot while mathematician James Grime of the University of CambridgeThe Guardian-James Grime profile calls it "Keeler's Theorem".Grime, J. (May 2, 2012) Futurama and Keeler's Theorem Twitter.

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives.

The Nielsen–Ninomiya theorem is a no-go theorem in physics, in particular in lattice gauge theory, concerning the possibility of defining a theory of chiral fermions on a lattice in even dimensions. The theorem can be stated as follows: let S[\psi] be the (Euclidean) action describing fermions \psi on a regular lattice of even dimensions with periodic boundary conditions, and suppose that S is local, hermitian and translation invariant; then the theory describes as many left-handed as right-handed states. Equivalently, the theorem implies that there are as many states of chirality +1 as of chirality −1\. The proof of the theorem relies on the Poincaré–Hopf theorem or on similar results in algebraic topology.

The Pythagorean theorem has at least 370 known proofs Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics. In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement.

3-tangents degeneration of Brianchon's theorem As for Pascal's theorem there exist degenerations for Brianchon's theorem, too: Let coincide two neighbored tangents. Their point of intersection becomes a point of the conic. In the diagram three pairs of neighbored tangents coincide. This procedure results in a statement on inellipses of triangles.

F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

The signature theorem is a special case of the Atiyah–Singer index theorem for the signature operator. The analytic index of the signature operator equals the signature of the manifold, and its topological index is the L-genus of the manifold. By the Atiyah–Singer index theorem these are equal.

This theorem has many important consequences. Since Büchi automata and ω-regular languages are equally expressive, the theorem implies that Büchi automata and deterministic Muller automata are equally expressive. Since complementation of deterministic Muller automata is trivial, the theorem implies that Büchi automata/ω-regular languages are closed under complementation.

In 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. They showed that a certain finitistic special case of a theorem in Ramsey theory due to Erdős and Rado is not provable in Peano arithmetic.

In formal language theory, the Chomsky-Schützenberger enumeration theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger 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.

He conducted research in the field of theory of functions. His most famous student was Leonida Tonelli. In 1889 he generalized the Ascoli theorem to Arzelà–Ascoli theorem, an important theorem in the theory of functions. He was a member of the Accademia Nazionale dei Lincei, and of several other academies.

The Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic number of the subgraph.

This is the fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.

Unrestricted domain is one of the conditions for Arrow's impossibility theorem. Under that theorem, it is impossible to have a social choice function that satisfies unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non- dictatorship. However, the conditions of the theorem can be satisfied if unrestricted domain is removed.

Figure 3. Marginal value theorem shown graphically. The marginal value theorem is a type of optimality model that is often applied to optimal foraging. This theorem is used to describe a situation in which an organism searching for food in a patch must decide when it is economically favorable to leave.

In the mathematical discipline of algebraic geometry, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem due to Jean-Pierre Serre which gives sufficient conditions for a scheme to be affine.. The theorem was first published by Serre in 1957..

Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion. Lévy's modulus of continuity theorem is named after the French mathematician Paul Lévy.

In mathematics, the Jacobson-Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , .

This approach is essential for classifying algebraic varieties. The Riemann–Roch theorem also holds for holomorphic vector bundles on a compact complex manifold, by the Atiyah–Singer index theorem.

The proof also relies on the following theorem proven in p. 185: :Theorem. Let be a simple right -module, , and a finite set. Write for the annihilator of in .

In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.

5, pp. 385-387 gave a version of Grushko's theorem for free products with infinitely many factors. A 1976 paper of ChiswellI. M. Chiswell, The Grushko-Neumann theorem. Proc.

Birkhoff's theorem states that in fact all finite distributive lattices can be obtained this way, and later generalizations of Birkhoff's theorem state a similar thing for infinite distributive lattices.

The theorem is named after Henri Poincaré, who conjectured it in 1883, and Carlo Miranda, who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem..

This is in stark contrast to the situation in Banach spaces. The inverse function theorem is not true in Fréchet spaces; a partial substitute is the Nash–Moser theorem.

Clote & Kranakis (2002) p.50 Barrington's theorem says that BWBP is exactly nonuniform NC1. The proof uses the nonsolvability of the symmetric group S5. The theorem is rather surprising.

Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastly more complicated. The final paper is 255 pages long.

The Sipser–Lautemann theorem or Sipser–Gács–Lautemann theorem states that Bounded-error Probabilistic Polynomial (BPP) time, is contained in the polynomial time hierarchy, and more specifically Σ2 ∩ Π2.

World Scientific 2011, p. 314 With Lagrange's four-square theorem and the two-square theorem of Girard, Fermat and Euler, the Waring's problem for k = 2 is entirely solved.

The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890).D. Hilbert, Über die Theorie der algebraischen Formen, Mathematische Annalen 36, 473–530. The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem (Theorem III), which is used in part IV to discuss the Hilbert polynomial.

In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob., see The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.

In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of . Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise.

Repeated application of the Finsler–Hadwiger theorem can be used to prove Van Aubel's theorem, on the congruence and perpendicularity of segments through centers of four squares constructed on the sides of an arbitrary quadrilateral. Each pair of consecutive squares forms an instance of the theorem, and the two pairs of opposite Finsler–Hadwiger squares of those instances form another two instances of the theorem, having the same derived square., problem 15, pp. 25–26.

For example, Gauss's Theorema Egregium is a deep theorem which relates a local phenomenon (curvature) to a global phenomenon (area) in a surprising way. In particular, the area of a triangle on a curved surface is proportional to the excess of the triangle and the proportionality is curvature. Another example is the fundamental theorem of calculus (and its vector versions including Green's theorem and Stokes' theorem). The opposite of deep is trivial.

Shimura's formulation of the Taniyama–Shimura conjecture (later known as the modularity theorem) in the 1950s played a key role in the proof of Fermat's Last Theorem by Andrew Wiles in 1995. In 1990, Kenneth Ribet proved Ribet's theorem which demonstrated that Fermat's Last Theorem followed from the semistable case of this conjecture. Shimura dryly commented that his first reaction on hearing of Andrew Wiles's proof of the semistable case was 'I told you so'.

In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.

Together, these effects are called the inflationary "no- hair theorem"Kolb and Turner (1988). by analogy with the no hair theorem for black holes. The "no-hair" theorem works essentially because the cosmological horizon is no different from a black-hole horizon, except for philosophical disagreements about what is on the other side. The interpretation of the no- hair theorem is that the Universe (observable and unobservable) expands by an enormous factor during inflation.

The deduction theorem described above holds in some versions of paraconsistent logic. Usually the classical deduction theorem does not hold in paraconsistent logic. However, the following "two-way deduction theorem" does hold in one form of paraconsistent logic:Hewitt 2008 ::\vdash E \rightarrow F if and only if (E \vdash F and eg F \vdash eg E) that requires the contrapositive inference to hold in addition to the requirement of the classical deduction theorem.

Shortly after Slepian–Wolf theorem on lossless distributed compression was published, the extension to lossy compression with decoder side information was proposed as Wyner–Ziv theorem. Similarly to lossless case, two statistically dependent i.i.d. sources X and Y are given, where Y is available at the decoder side but not accessible at the encoder side. Instead of lossless compression in Slepian–Wolf theorem, Wyner–Ziv theorem looked into the lossy compression case.

Any finite group that has a Hall π-subgroup for every set of primes π is solvable. This is a generalization of Burnside's theorem that any group whose order is of the form p aq b for primes p and q is solvable, because Sylow's theorem implies that all Hall subgroups exist. This does not (at present) give another proof of Burnside's theorem, because Burnside's theorem is used to prove this converse.

This is a special case of Szemerédi's theorem on the density of sets of integers that avoid longer arithmetic progressions. To distinguish this result from Roth's theorem on Diophantine approximation of algebraic numbers, this result has been called Roth's theorem on arithmetic progressions. After several additional improvements to Roth's theorem, the size of a Salem–Spencer set has been proven to be O\bigl(n(\log\log n)^4/\log n\bigr).

In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field.

Indian mathematician Bhāskara II (1114–1185) is credited with knowledge of Rolle's theorem. Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem.

Instead of describing the Feit–Thompson theorem directly, it is easier to describe Suzuki's CA theorem and then comment on some of the extensions needed for the CN-theorem and the odd order theorem. The proof can be broken up into three steps. We let G be a non-abelian (minimal) simple group of odd order satisfying the CA condition. For a more detailed exposition of the odd order paper see or or .

The theorem is actually a collection of related theorems. The first theorem states that if two different Bernoulli shifts have the same Kolmogorov entropy, then they are isomorphic as dynamical systems. The third theorem extends this result to flows: namely, that there exists a flow T_t such that T_1 is a Bernoulli shift. The fourth theorem states that, for a given fixed entropy, this flow is unique, up to a constant rescaling of time.

In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by and , though essentially the same theorem was used by Stieltjes in the theory of continued fractions many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.

160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case s=1, and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that \zeta(1)=\log\infty, and makes use of this latter result in his Theorem 19, in order to show that the sum of the inverses of the prime numbers is \log\log\infty.

This follows from putting D=K in the theorem. In particular, as long as D has degree at least 2g-1, the correction term is 0, so that :\ell(D) = \deg(D) - g + 1. The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

In 2012 Webster was elected a Fellow of the American Mathematical Society. In 1977 he proved a significant theorem on biholomorphic mappings between algebraic real hypersurfaces. Using his expertise on Chern-Moser invariants, he developed a theory that provides a complete set of invariants for nondegenerate real hypersurfaces under volume-preserving biholomorphic transformations. He used the edge-of-the-wedge theorem to prove an extension theorem that generalized a 1974 theorem of Charles Fefferman.

The PBR theorem is a no-go theorem in quantum foundations due to Matthew Pusey, Jonathan Barrett, and Terry Rudolph (for whom the theorem is named). It has particular significance for how one may interpret the nature of the quantum state. With respect to certain realist hidden variable theories that attempt to explain the predictions of quantum mechanics, the theorem rules that pure quantum states must be "ontic" in the sense that they correspond directly to states of reality, rather than "epistemic" in the sense that they represent probabilistic or incomplete states of knowledge about reality. The PBR theorem may also be compared with other no-go theorems like Bell's theorem and the Bell–Kochen–Specker theorem, which, respectively, rule out the possibility of explaining the predictions of quantum mechanics with local hidden variable theories and noncontextual hidden variable theories.

300px The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others.

Feuerbach's theorem has also been used as a test case for automated theorem proving.. The three points of tangency with the excircles form the Feuerbach triangle of the given triangle.

Clapeyron also worked on the characterisation of perfect gases, the equilibrium of homogeneous solids, and calculations of the statics of continuous beams, notably the theorem of three moments (Clapeyron's theorem).

Minkowski's theorem is also useful to prove Lagrange's four-square theorem, which states that every natural number can be written as the sum of the squares of four natural numbers.

The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.Bombieri & Gubler (2006) pp. 176–230.

The theorem can also be extended to nonmeagre sets with the Baire property. The proof of these extensions, sometimes also called Steinhaus theorem, is almost identical to the one below.

In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every vector space has a basis. In the case of a single indeterminate, Hilbert's syzygy theorem is an instance of the theorem asserting that over a principal ideal ring, every submodule of a free module is itself free.

Otter is an automated theorem prover developed by William McCune at Argonne National Laboratory in Illinois. Otter was the first widely distributed, high- performance theorem prover for first-order logic, and it pioneered a number of important implementation techniques. Otter is an acronym for Organized Techniques for Theorem-proving and Effective Research.

Turing's thesis that every function which would naturally be regarded as computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX." Indeed immediately before this statement, Kleene states the Theorem XXX: ::"Theorem XXX (= Theorems XXVIII + XXIX). The following classes of partial functions are coextensive, i.e.

There are many known proofs of the circle packing theorem. Paul Koebe's original proof is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain. There are several different topological proofs that are known. Thurston's proof is based on Brouwer's fixed point theorem.

Because Ronald Coase did not originally intend to set forth any one particular theorem, it has largely been the effort of others who have developed the loose formulation of the Coase theorem. What Coase initially provided was fuel in the form of “counterintuitive insight”Andrew Halpin, "Disproving the Coase Theorem?", 23 Econ. & Phil.

In physics, in the context of electromagnetism, Birkhoff's theorem concerns spherically symmetric static solutions of Maxwell's field equations of electromagnetism. The theorem is due to George D. Birkhoff. It states that any spherically symmetric solution of the source-free Maxwell equations is necessarily static. Pappas (1984) gives two proofs of this theorem.

Similarly, the PBR theorem could be said to rule out preparation independent hidden variable theories, in which quantum states that are prepared independently have independent hidden variable descriptions. This result was cited by theoretical physicist Antony Valentini as "the most important general theorem relating to the foundations of quantum mechanics since Bell's theorem".

In 2009, the Polymath Project developed a new proof of the density Hales–Jewett theorem based ideas on from the proof of the corners theorem. Dodos, Kanellopoulos, and Tyros gave a simplified version of the Polymath proof. The Hales–Jewett is generalized by the Graham–Rothschild theorem, on higher-dimensional combinatorial cubes.

In microeconomics, quasiconcave utility functions imply that consumers have convex preferences. Quasiconvex functions are important also in game theory, industrial organization, and general equilibrium theory, particularly for applications of Sion's minimax theorem. Generalizing a minimax theorem of John von Neumann, Sion's theorem is also used in the theory of partial differential equations.

In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered Fermat's little theorem. He invented a factorization method—Fermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic. and found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.

In some occasions, the number of cases is quite large, in which case a brute-force proof may require as a practical matter the use of a computer algorithm to check all the cases. For example, the validity of the 1976 and 1997 brute-force proofs of the four color theorem by computer was initially doubted, but was eventually confirmed in 2005 by theorem-proving software. When a conjecture has been proven, it is no longer a conjecture but a theorem. Many important theorems were once conjectures, such as the Geometrization theorem (which resolved the Poincaré conjecture), Fermat's Last Theorem, and others.

In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold are connected by a horizontal path in the manifold. It is named after Wei- Liang Chow who proved it in 1939, and Petr Konstanovich Rashevskii, who proved it independently in 1938. The theorem has a number of equivalent statements, one of which is that the topology induced by the Carnot–Carathéodory metric is equivalent to the intrinsic (locally Euclidean) topology of the manifold. A stronger statement that implies the theorem is the ball-box theorem.

The proof of the theorem consists of 4 steps. We assume Green's theorem, so what is of concern is how to boil down the three-dimensional complicated problem (Kelvin–Stokes theorem) to a two-dimensional rudimentary problem (Green's theorem). When proving this theorem, mathematicians normally deduce it as a special case of a more general result, which is stated in terms of differential forms, and proved using more sophisticated machinery. While powerful, these techniques require substantial background, so the proof below avoids them, and does not presuppose any knowledge beyond a familiarity with basic vector calculus.

See also (Tilson 1989) The Krohn–Rhodes theorem for semigroups/monoids is an analogue of the Jordan–Hölder theorem for finite groups (for semigroups/monoids rather than groups). As such, the theorem is a deep and important result in semigroup/monoid theory. The theorem was also surprising to many mathematicians and computer scientistsC.L. Nehaniv, Preface to (Rhodes, 2009) since it had previously been widely believed that the semigroup/monoid axioms were too weak to admit a structure theorem of any strength, and prior work (Hartmanis & Stearns) was only able to show much more rigid and less general decomposition results for finite automata.

In mathematics, the Banach–Caccioppoli fixed-point theorem (also known as the contraction mapping theorem or contractive mapping 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. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) and Renato Caccioppoli (1904–1959), and was first stated by Banach in 1922. Caccioppoli independently proved the theorem in 1931.

The source of Green and Tao's arithmetic progressions is Endre Szemerédi's seminal 1975 theorem on existence of arithmetic progressions in certain sets of integers. Green and Tao showed that one can use a "transference principle" to extend the validity of Szemerédi's theorem to further sets of integers. The Green-Tao theorem then arises as a special case, although it is not trivial to show that the prime numbers satisfy the conditions of Green and Tao's extension of the Szemerédi theorem. In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions.

A year later Dmitri Egorov published his independently proved results,In the note and the theorem became widely known under his name: however, it is not uncommon to find references to this theorem as the Severini–Egoroff theorem or Severini–Egorov Theorem. The first mathematicians to prove independently the theorem in the nowadays common abstract measure space setting were , and in :According to and . an earlier generalization is due to Nikolai Luzin, who succeeded in slightly relaxing the requirement of finiteness of measure of the domain of convergence of the pointwise converging functions in the ample paper .According to .

Hogg independently discovered a special case of "Basu's theorem", a few years before the publication by Deb Basu.Basu's theorem appears in For Hogg's independent and early (1953) discovery of special cases of Basu's theorem, see page 511 in Hogg's second paper on the topic of Basu's theorem was never published, because of a negative report by an anonymous referee in 1953.Randles and Calvin, page 2476. (Later, in his interview with Randles, Hogg speculates that the anonymous referee was Erich Leo Lehmann.) Later, Basu refers "to Hogg and Craig (1956) for several interesting uses [of Basu's theorem] in proving results in distribution theory".

Although named for Edgar Buckingham, the theorem was first proved by French mathematician Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh. The first application of the theorem in the general caseWhen in applying the pi–theorem there arises an arbitrary function of dimensionless numbers.

In doing so, he discovered a connection between Riemann zeta function and prime numbers, known as the Euler product formula for the Riemann zeta function. Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n. Using properties of this function he was able to generalize Fermat's little theorem to what would become known as Euler's theorem.

David Alan Plaisted is a computer science professor at the University of North Carolina at Chapel Hill. Plaisted's research interests include term rewriting systems, automated theorem proving, logic programming, and algorithms. His research accomplishments in theorem proving include work on the recursive path ordering, the associative path ordering, abstraction, the simplified and modified problem reduction formats, ground reducibility, nonstandard clause form translations, rigid E-unification, Knuth–Bendix completion, replacement rules in theorem proving, instance-based theorem proving strategies, and semantics in theorem proving. He received his B.S. from the University of Chicago in 1970 and his Ph.D. from Stanford University in 1976.

Boltzmann in his original publication writes the symbol E (as in entropy) for its statistical function. Years later, Samuel Hawksley Burbury, one of the critics of the theorem, wrote the function with the symbol H, a notation that was subsequently adopted by Boltzmann when referring to his "H-theorem". The notation has led to some confusion regarding the name of the theorem. Even though the statement is usually referred to as the "Aitch theorem", sometimes it is instead called the "Eta theorem", as the capital Greek letter Eta (Η) is undistinguishable from the capital version of Latin letter h (H).

The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927. In 1937 it was independently rediscovered by George Pólya, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of chemical compounds. The Pólya enumeration theorem has been incorporated into symbolic combinatorics and the theory of combinatorial species.

The Borde–Guth–Vilenkin theorem, or the BGV theorem, is a theorem in physical cosmology which deduces that any universe that has, on average, been expanding throughout its history cannot be infinite in the past but must have a past spacetime boundary. It is named after the authors Arvind Borde, Alan Guth and Alexander Vilenkin, who developed its mathematical formulation in 2003. The BGV theorem is also popular outside physics, especially in religious and philosophical debates. The theorem does not assume any specific mass content of the universe and it does not require gravity to be described by Einstein field equations.

Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively.

252, Theorem 10.1. Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.

151; , Theorem 3.2.3, p. 41. Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of star graphs.; ; , Theorem 4.4.

In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.

In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x^p + y^p = z^p of Fermat's Last Theorem for odd prime p.

The justification for the existence of point D is the often unstated crossbar theorem. For this particular result, other proofs exist which do not require the use of the crossbar theorem.

This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory. The second recursion theorem can be applied to any total recursive function.

This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski.

The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.See, for example, proofs of quadratic reciprocity for more.

He is also known for his work on the Morse-Sard theorem and the Federer–Morse theorem. Anthony Morse should not be confused with Marston Morse, known for developing Morse Theory.

In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring R of subsets of a given set Ω can be extended to a measure on the σ-algebra generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure. The theorem is also sometimes known as the Carathéodory-Fréchet extension theorem, the Carathéodory-Hopf extension theorem, the Hopf extension theorem and the Hahn-Kolmogorov extension theorem.

Every finite planar graph can be colored with four colors, by the four-color theorem. The De Bruijn–Erdős theorem then shows that every graph that can be drawn without crossings in the plane, finite or infinite, can be colored with four colors. More generally, every infinite graph for which all finite subgraphs are planar can again be four-colored.. states the same result for the five-color theorem for countable planar graphs, as the four-color theorem had not yet been proven when he published his survey, and as the proof of the De Bruijn–Erdős theorem that he gives only applies to countable graphs. For the generalization to graphs in which every finite subgraph is planar (proved directly via Gödel's compactness theorem), see .

Using Solèr's theorem, the field K over which the vector space is defined can be proven, with additional hypotheses, to be either the real numbers, complex numbers, or the quaternions, as is needed for Gleason's theorem to hold. By invoking Gleason's theorem, the form of a probability function on lattice elements can be restricted. Assuming that the mapping from lattice elements to probabilities is noncontextual, Gleason's theorem establishes that it must be expressible with the Born rule.

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval. When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by and extended to finite volume manifolds by in 3 dimensions, and by in all dimensions at least 3. gave an alternate proof using the Gromov norm. gave the simplest available proof.

Another, preceding to the diamond theorem, sufficient permanence condition was given by Haran-Jarden: Theorem. Let K be a Hilbertian field and N, M two Galois extensions of K. Assume that neither contains the other. Then their compositum NM is Hilbertian. This theorem has a very nice consequence: Since the field of rational numbers, Q is Hilbertian (Hilbert's irreducibility theorem), we get that the algebraic closure of Q is not the compositum of two proper Galois extensions.

In 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. The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.

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. The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.

Rouché's theorem, named after Eugène Rouché, states that for any two complex- valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour \partial K is simple, that is, without self- intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.

In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) 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. The theorem is named after Pierre Fatou and Henri Léon Lebesgue. If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to Lebesgue's dominated convergence theorem.

In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.

Thomson went on to prove Stokes' theorem, which earned that name after Stokes asked students to prove in on a test in 1854. Stokes learned it from Thomson in a letter in 1850. Stokes' theorem generalises Green's theorem, which itself is a higher-dimensional version of the Fundamental Theorem of Calculus. Arthur Cayley is credited with the creation of the theory of matrices—rectangular arrays of numbers—as distinct objects from determinants, studied since the mid-eighteenth century.

It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. The theorem can also be found in existence proofs for the solutions of certain partial differential equations.These examples are taken from: F. Boyer Théorèmes de point fixe et applications CMI Université Paul Cézanne (2008–2009) Archived copy at WebCite (August 1, 2010).

Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.

It gives a beautiful solution of an important problem. # His theorem that a compact analytic variety in a projective space is algebraic is justly famous. The theorem shows the close analogy between algebraic geometry and algebraic number theory. # Generalizing a result of Caratheodory on thermodynamics, he formulated a theorem on accessibility of differential spaces.

In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of the Gabriel–Popescu theorem, given by (for an AB5 category with a set of generators), , (for triangulated categories).

In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a multifunction to have a measurable selection function. Theorem (12.13) on page 76. Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem". It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.

In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth compact oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem.

Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.

In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.

Automated reasoning has been most commonly used to build automated theorem provers. Oftentimes, however, theorem provers require some human guidance to be effective and so more generally qualify as proof assistants. In some cases such provers have come up with new approaches to proving a theorem. Logic Theorist is a good example of this.

The Jacobson density theorem is a theorem concerning simple modules over a ring .Isaacs, p. 184 The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.Such rings of linear transformations are also known as full linear rings.

This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led Haran to his diamond theorem. ;Weissauer's theorem Let K be a Hilbertian field, N a Galois extension of K, and L a finite proper extension of N. Then L is Hilbertian.

The invariance of domain theorem states that a continuous and locally injective function between two -dimensional topological manifolds must be open. In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map. This theorem has been generalized to topological vector spaces beyond just Banach spaces.

In mathematics, Glaeser's theorem, introduced by , is a theorem giving conditions for a smooth function to be a composition of F and θ for some given smooth function θ. One consequence is a generalization of Newton's theorem that every symmetric polynomial is a polynomial in the elementary symmetric polynomials, from polynomials to smooth functions.

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.Google books 1800–1900 search for "freshman's dream": Bentley's miscellany, Volume 26, p. 176, 1849 Since the expansion of is correctly given by the binomial theorem, the freshman's dream is also known as the "child's binomial theorem" or "schoolboy binomial theorem".

However, this interpretation of its implications has been criticized in several publications reviewed in Altenberg, L. (1995). The Schema Theorem and Price’s Theorem. Foundations of genetic algorithms, 3, 23-49., where the Schema Theorem is shown to be a special case of the Price equation with the schema indicator function as the macroscopic measurement.

He also shows that such a calculus satisfies two natural criteria. First, a calculus defines an evaluation function that maps closed terms (representations of programs) to answers (representations of outputs). This theorem relies on a conventional Church–Rosser theorem for the modified calculus. The evaluation function is defined via the traditional Curry–Feys standardization theorem.

The theorem leaves open the question of how well (uniformly) the multiples mP of P fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.

In 2010, academics at The University of Edinburgh offered people the chance to "buy their own theorem" created through a computer-assisted proof. This new theorem would be named after the purchaser.

Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, Studien zur Kombinatorik.

John Stewart Bell FRS (28 June 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum physics regarding hidden variable theories.

In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to .

Later, László Székely discovered a much simpler proof using the crossing number inequality for graphs. (See below.) The Szemerédi–Trotter theorem has a number of consequences, including Beck's theorem in incidence geometry.

Vitold Lvovich Shmulyan (, August 29, 1914 – August 27 1944), was a Soviet mathematician known for his work in functional analysis. The Eberlein–Šmulian theorem and Krein–Smulian theorem are named after him.

Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.

In 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. It is named after Paul Erdős and Arthur Stone, who proved it in 1946, and it has been described as the “fundamental theorem of extremal graph theory”.

The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after Euclid and Leonhard Euler. It has been conjectured that there are infinitely many Mersenne primes.

Bernard Malgrange (born 6 July 1928) is a French mathematician who works on differential equations and singularity theory. He proved the Ehrenpreis–Malgrange theorem and the Malgrange preparation theorem, essential for the classification theorem of the elementary catastrophes of René Thom. He received his Ph.D. from Université Henri Poincaré (Nancy 1) in 1955. His advisor was Laurent Schwartz.

We point out that Theorem 2 is an exact structure theorem since the precise structure of K5-free graphs is determined. Such results are rare within graph theory. The graph structure theorem is not precise in this sense because, for most graphs H, the structural description of H-free graphs includes some graphs that are not H-free.

We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate I(X;B) for communicating classical data over a quantum channel). We first review the minimal amount of quantum mechanics needed for the theorem. We then cover quantum typicality, and finally we prove the theorem using a recent sequential decoding technique.

Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by one-forms. The most famous of these are the Cartan–Kähler theorem, which only works for real analytic differential systems, and the Cartan–Kuranishi prolongation theorem. See Further reading for details. The Newlander-Nirenberg theorem gives integrability conditions for an almost- complex structure.

In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.

Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and Feodor Deahna. Deahna was the first to establish the sufficient conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to Pfaffian systems, thus paving the way for its usage in differential topology.

Stated by Claude Shannon in 1948, the theorem describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption. Shannon's theorem has wide-ranging applications in both communications and data storage. This theorem is of foundational importance to the modern field of information theory. Shannon only gave an outline of the proof.

The Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian)Semisimple rings are necessarily Artinian rings. Some authors use "semisimple" to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.

Many of the extensions of Szemerédi's theorem hold for the primes as well. Independently, Tao and Ziegler and Cook, Magyar, and Titichetrakun derived a multidimensional generalization of the Green–Tao theorem. The Tao–Ziegler proof was also simplified by Fox and Zhao. In 2006, Tao and Ziegler extended the Green–Tao theorem to cover polynomial progressions.

The Knaster-Tarski theorem states that any order-preserving function on a complete lattice has a fixed point, and indeed a smallest fixed point. See also Bourbaki-Witt theorem. The theorem has applications in abstract interpretation, a form of static program analysis. A common theme in lambda calculus is to find fixed points of given lambda expressions.

In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems are a corollary of the other.

Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics. Stephen Smale made significant advances as well.

The Bishop-Cannings theorem is a theorem in evolutionary game theory. It states that (i) all members of a mixed evolutionarily stable strategy (ESS) have the same payoff (Theorem 2), and (ii) that none of these can also be a pure ESSBishop, D.T. and Cannings, C. (1978). A generalized war of attrition. Journal of Theoretical Biology 70:85-124.

Khoa Lu Nguyen (2005), A synthetic proof of Goormaghtigh's generalization of Musselman's theorem. Forum Geometricorum, volume 5, pages 17–20 Ion Pătrașcu and Cătălin Barbu (2012), Two new proofs of Goormaghtigh theorem. International Journal of Geometry, volume 1, pages=10–19, Joseph Neuberg (1884), Mémoir sur le Tetraèdre. According to Nguyen, Neuberg also states Goormaghtigh's theorem, but incorrectly.

This application of the dimension theorem is sometimes itself called the dimension theorem. Let :T: U → V be a linear transformation. Then :dim(range(T)) + dim(kernel(T)) = dim(U), that is, the dimension of U is equal to the dimension of the transformation's range plus the dimension of the kernel. See rank–nullity theorem for a fuller discussion.

In algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) Semisimple rings are necessarily Artinian rings. Some authors use "semisimple" to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.

Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, say > The method of proof here is truly amazing. If the generalized Riemann > hypothesis is true, then the theorem is true. If the generalized Riemann > hypothesis is false, then the theorem is true.

In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard Helly. The theorem has applications throughout mathematical analysis.

Epsteen shows that what he calls the "Jordan–Beke theorem" implies the "Vessiot theorem" in Picard–Vessiot theory. determinants, and mathematical physics. He is known for reforming the teaching of mathematics in Hungary.

In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the D-operators.

In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous are Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al.

The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem.

Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.

In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces.

They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.

This result is now known as the Quillen–Suslin theorem.

This follows directly from the identity theorem for holomorphic functions.

Stallings' proof of Grushko Theorem follows from the following lemma.

The theorem is named after Élie Cartan and Masatake Kuranishi.

The UTM theorem proves the existence of such a function.

The theorem may be generalized in a variety of ways.

Cantor's theorem has been generalized to any category with products.

Taylor's theorem also generalizes to multivariate and vector valued functions.

We use \vdash A to mean A is a theorem.

The theorem was first conjectured by . It was proved by .

This theorem can be proved directly using summation by parts.

The Sylvester–Gallai theorem can be proven within ordered geometry.

The converse result is known as the Five circles theorem.

Bernoulli's theorem is a direct consequence of the Euler equations.

The following statements are equivalent to the Borsuk–Ulam theorem.

In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by Mark Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri and A. I. Vinogradov, Corrigendum. ibid. 30 (1966), pages 719-720.

The converse of the hinge theorem is also true: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle. In some textbooks, the theorem and its converse are written as the SAS Inequality Theorem and the SSS Inequality Theorem respectively.

Cauchy's theorem implies that for any prime divisor of the order of , there is a subgroup of whose order is —the cyclic group generated by the element in Cauchy's theorem. Cauchy's theorem is generalised by Sylow's first theorem, which implies that if is the maximal power of dividing the order of , then has a subgroup of order (and using the fact that a -group is solvable, one can show that has subgroups of order for any less than or equal to ).

A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.

Note: if all agents' preferences are convex (as in theorem 1), then A(u) is obviously convex too. Moreover, if A(u) is singleton (as in theorem 2) then it is obviously convex too. Hence, Svensson's theorem is more general than both Varian's theorems. Theorem 4 (Diamantaras): If all agents' preferences are strongly monotone, and for every PE utility-profile u, the set A(u) is a contractible space (can be continuously shrunk to a point within that space), then PEEF allocations exist.

Crease patterns with more than one vertex do not obey such a simple criterion, and are NP-hard to fold. The theorem is named after one of its discoverers, Toshikazu Kawasaki. However, several others also contributed to its discovery, and it is sometimes called the Kawasaki–Justin theorem or Husimi's theorem after other contributors, Jacques Justin and Kôdi Husimi.The name "Yasuji Husimi" appearing in and sometimes associated with this theorem is a mistranslation of the kanji "康治" in Kôdi Husimi's name.

Reciprocity in electrical networks is a theorem that relates voltages and currents at two different points in a circuit. The reciprocity theorem states that current at one point in a circuit due to a voltage at a second is the same as the current at the first point due to the same voltage at the second. The reciprocity theorem is valid for almost all passive networks. The reciprocity theorem is a feature of a more general principle of reciprocity in electromagnetism.

Paul Guldin (original name Habakkuk Guldin; 12 June 1577 (Mels) – 3 November 1643 (Graz)) was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. (This theorem is also known as the Pappus–Guldinus theorem and Pappus's centroid theorem, attributed to Pappus of Alexandria.) Guldin was noted for his association with the German mathematician and astronomer Johannes Kepler. Guldin composed a critique of Cavalieri's method of Indivisibles.

Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small number of low-degree vertices. It is named after Lajos Pósa, a protégé of Paul Erdős born in 1947, who discovered this theorem in 1962.

A.J. Wilkie, A theorem of the complement and some new o-minimal structures, Sel. Math. 5 (1999), pp.397-421. Wilkie's approach for this latter result is somewhat different from his proof of Wilkie's theorem, and the result that allowed him to show that the Pfaffian structure is model complete is sometimes known as Wilkie's theorem of the complement. See also M. Karpinski and A. Macintyre, A generalization of Wilkie's theorem of the complement, and an application to Pfaffian closure, Sel. math.

In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains. It is named for and is closely related to Dilworth's theorem on the widths of partial orders, to the perfection of comparability graphs, to the Gallai–Hasse–Roy–Vitaver theorem relating longest paths and colorings in graphs, and to the Erdős–Szekeres theorem on monotonic subsequences.

To establish a mathematical statement as a theorem, a proof is required. That is, a valid line of reasoning from the axioms and other already-established theorems to the given statement must be demonstrated. In general, the proof is considered to be separate from the theorem statement itself. This is in part because while more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem.

This type of situation is sometimes referred to as implicit collusion. Morton's theorem contrasts with the fundamental theorem of poker, which states that a player wants their opponents to make decisions which minimize their own expectation. The two theorems differ in the presence of more than one opponent: whereas the fundamental theorem always applies heads-up (one opponent), it does not always apply in multiway pots. The scope of Morton's theorem in multi-way situations is a subject of controversy.

Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows: : If the sequence :: \phi_1, \ \phi_2, \ ... , \ \phi_n, \ \chi \vdash \psi : has been demonstrated, then it is also possible to demonstrate the sequence :: \phi_1, \ \phi_2, \ ..., \ \phi_n \vdash \chi \to \psi . This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a meta-theorem, comparable to theorems about the soundness or completeness of propositional calculus.

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearisation near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearisation has real part equal to zero.

In computability theory the '' theorem', (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the computable functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name ' comes from the occurrence of an S with subscript n and superscript m in the original formulation of the theorem (see below). In practical terms, the theorem says that for a given programming language and positive integers m and n, there exists a particular algorithm that accepts as input the source code of a program with m + n free variables, together with m values.

By definition, two triangles are perspective if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be similar. Under the standard duality of plane projective geometry (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual:This is due to the modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the converse of Desargues's theorem and was always referred to by that name.

In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem, Picard's existence theorem, Cauchy–Lipschitz theorem, or existence and uniqueness theorem gives a set of conditions under which an initial value problem has a unique solution. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Consider the initial value problem :y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. Suppose is uniformly Lipschitz continuous in (meaning the Lipschitz constant can be taken independent of ) and continuous in , then for some value , there exists a unique solution to the initial value problem on the interval [t_0-\varepsilon, t_0+\varepsilon].

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression :a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers.

In 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. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities.

The Miller theorem establishes that in a linear circuit, if there exists a branch with impedance Z, connecting two nodes with nodal voltages V1 and V2, we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively Z/(1 − K) and KZ/(K − 1), where K = V2/V1. The Miller theorem may be proved by using the equivalent two-port network technique to replace the two-port to its equivalent and by applying the source absorption theorem. This version of the Miller theorem is based on Kirchhoff's voltage law; for that reason, it is named also Miller theorem for voltages.

In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds.

Exactly how, when, or why Harry Nyquist had his name attached to the sampling theorem remains obscure. The term Nyquist Sampling Theorem (capitalized thus) appeared as early as 1959 in a book from his former employer, Bell Labs, and appeared again in 1963, and not capitalized in 1965. It had been called the Shannon Sampling Theorem as early as 1954, but also just the sampling theorem by several other books in the early 1950s. In 1958, Blackman and Tukey cited Nyquist's 1928 article as a reference for the sampling theorem of information theory, even though that article does not treat sampling and reconstruction of continuous signals as others did.

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The Lasker–Noether theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components.

From examples given in the Arithmetica it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange.. Adrien-Marie Legendre extended the theorem in 1797–8 with his three-square theorem, by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form 4^k(8m+7) for integers k and m. Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four- square theorem.

The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries.

No purely Hamiltonian description is capable of treating the experiments carried out to verify the Crooks fluctuation theorem, Jarzynski equality and the Fluctuation theorem. These experiments involve thermostatted systems in contact with heat baths.

Similar theorems describe the degree sequences of simple graphs and simple directed graphs. The first problem is characterized by the Erdős–Gallai theorem. The latter case is characterized by the Fulkerson–Chen–Anstee theorem.

The following wonderful theorem was proved by Pogorelov in 1973 Theorem. Any two- dimensional continuous complete flat metric is a \sigma-metric. Thus Hilbert's fourth problem for the two-dimensional case was completely solved.

109 The result is named for Constantin Carathéodory, who proved the theorem in 1907 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets P in Rd.

The constant is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that is algebraic. By the theorem, is transcendental, but , which is a contradiction.

The Weierstrass preparation theorem can be used to show that the ring of germs of analytic functions in n variables is a Noetherian ring, which is also referred to as the Rückert basis theorem.

This was essentially done by the Trichotomy theorem. # Classification of simple groups of characteristic 2 type. This was handled by the Gilman–Griess theorem, with 3-elements replaced by p-elements for odd primes.

Dana Scott first proved the theorem in 1963. The theorem, in a slightly less general form, was independently proven by Haskell Curry. It was published in Curry's 1969 paper "The undecidability of λK-conversion".

In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics. The theorem is named after Andrey Nikolayevich Tikhonov.

Any n-vertex forest has pathwidth O(log n)., Theorem 5, p. 99; , Theorem 66, p. 30. gives a tighter upper bound of log3(2n + 1) on the pathwidth of an n-vertex forest.

3 on p. 36 and the historical comment on p. 441 See Theorem (12.13) at the bottom of p. 76 See Theorem A He became a member of the Polish Academy of Sciences in 1967.

He is often called the Father of Russian Aviation. The Joukowsky transform is named after him, while the fundamental aerodynamical theorem, the Kutta–Joukowski theorem, is named after both him and German mathematician Martin Kutta.

From the definition, split graphs are clearly closed under complementation., Theorem 6.1, p. 150. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal.; , Theorem 6.3, p.

One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions. The synopsis below is primarily based on Spivak (1965) and Tu (2011).

The Stolper–Samuelson theorem is closely linked to the factor price equalization theorem, which states that, regardless of international factor mobility, factor prices will tend to equalize across countries that do not differ in technology.

This theorem has applications to Ramsey theory, specifically graph Ramsey theory. Using this theorem, a relationship between the graph Ramsey numbers and the extremal numbers can be shown (see Graham-Rothschild-Spencer for the details).

In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below.

Friedman's theorem of 1971 showed that there is a model of Zermelo set theory (with the axiom of choice) in which Borel determinacy fails, and thus Zermelo set theory cannot prove the Borel determinacy theorem.

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.

Kneser's theorem on differential equations is named after him, and provides criteria to decide whether a differential equation is oscillating. He is also one of the namesakes of the Tait–Kneser theorem on osculating circles.

Menger's theorem asserts that for distinct vertices u,v, equals , and if u is also not adjacent to v then equals . This fact is actually a special case of the max-flow min-cut theorem.

Marion Walter (born July 30, 1928 in Berlin, Germany) is a mathematician who retired as Professor of Mathematics at University of Oregon in 1994. There is a theorem named after her, called Marion Walter's theorem.

In quantum mechanics, the Landau–Yang theorem is a selection rule for particles that decay into two on-shell photons. The theorem states that a massive particle with spin 1 cannot decay into two photons.

These formal statements are also known as Lagrange's Mean Value Theorem.

For semismall maps, the decomposition theorem also applies to Chow motives.

The same applies to definability; see for example Tarski's undefinability theorem.

He also derived a short elementary proof of Stone–Weierstrass theorem .

For a proof of this theorem, see analyticity of holomorphic functions.

In 1870 he introduced the virial theorem which applied to heat.

Schoen, Richard; Yau, Shing Tung. Proof of the positive mass theorem.

R or theorem,J. F. C. Kingman. Poisson processes, volume 3.

They can be used to formulate a very general Stokes' theorem.

Do Carmo–Dajczer theorem is named after his teacher and him.

Helly's theorem gave rise to the notion of a Helly family.

The answer to this question is called the Mackey–Arens theorem.

This coordinate system can be used to prove the theorem directly.

The long exact sequence of relative homology then gives the theorem.

However, there are many planes in which Desargues's theorem is false.

The Balian–Low theorem has been extended to exact Gabor frames.

For that reason it is sometimes called the Cauchy–Goursat theorem.

The answer to this question is called the Mackey–Arens theorem.

Also many generalizations and refinements of Ado's Theorem have been considered.

In other words, the theorem is valid for every possible weight.

Gentzen's theorem spurred the development of ordinal analysis in proof theory.

He developed the Prototype Verification System, which is a theorem prover.

The theorem was proved in 1834 by Carl Gustav Jakob Jacobi.

Using trigonometry and the Pythagorean Theorem to construct a regular pentagon.

Meanwhile, Denis Sargan refers to it as the general transformation theorem.

The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.

The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

In his work with his coauthors Navin Aswal and Shurojit Chatterji, he provides a comprehensive description of environments where GS theorem holds. In his works and with coauthors Shurojit Chatterji, Huaxia Zeng, and Remzi Sanver, he identifies environments where GS theorem does not hold, i.e., well- behaved voting rules exist. In his work with coauthors Shurojit Chatterji and Huaxia Zeng, he has identified environments where the GS theorem type result continues to hold even if the voting rule allows for randomization (which generalizes Gibbard's theorem).

The theorem was first announced by , who showed this result to Antoni Zygmund shortly before he died in World War II. The theorem was almost forgotten by Zygmund, and was absent from his original works on the theory of singular integral operators. Later realized that Marcinkiewicz's result could greatly simplify his work, at which time he published his former student's theorem together with a generalization of his own. In 1964 Richard A. Hunt and Guido Weiss published a new proof of the Marcinkiewicz interpolation theorem.

A related result is the Weierstrass division theorem, which states that if f and g are analytic functions, and g is a Weierstrass polynomial of degree N, then there exists a unique pair h and j such that f = gh + j, where j is a polynomial of degree less than N. In fact, many authors prove the Weierstrass preparation as a corollary of the division theorem. It is also possible to prove the division theorem from the preparation theorem so that the two theorems are actually equivalent.

By contrast, neither a maximum nor a minimum exists for S. Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem, the Kirszbraun theorem, Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.

A closely related theorem by states that an n-vertex strongly connected digraph with the property that, for every two nonadjacent vertices u and v, the total number of edges incident to u or v is at least 2n − 1 must be Hamiltonian. Ore's theorem may also be strengthened to give a stronger conclusion than Hamiltonicity as a consequence of the degree condition in the theorem. Specifically, every graph satisfying the conditions of Ore's theorem is either a regular complete bipartite graph or is pancyclic .

Front page of Arkiv för Matematik, Astronomi och Fysik where Jebsen's work was published 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 (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric. The theorem was proven in 1923 by George David Birkhoff (author of another famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory).

Theorem 1, Section 2.7 Rossmann states the theorem for linear groups. The statement is that there is an open subset such that is an analytic bijection onto an open neighborhood of in . For linear groups, Hall proves a similar result in Corollary 3.45. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, See § 26.

In mathematics, the Denjoy–Luzin–Saks theorem states that a function of generalized bounded variation in the restricted sense has a derivative almost everywhere, and gives further conditions of the set of values of the function where the derivative does not exist. N. N. Luzin and A. Denjoy proved a weaker form of the theorem, and later strengthened their theorem.

The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary representations. The theorem was first published in 1943.И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943), 301–316, (I.

Journal of Differential Geometry, Volume 35 (1992), pp. 85–101 The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory and has had many applications and generalizations (e.g.EMINA ALIBEGOVIC, A COMBINATION THEOREM FOR RELATIVELY HYPERBOLIC GROUPS.

In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by as a consequence of a theorem about the compactness of sets of discrete subgroups of semisimple Lie groups generalizing Mahler's compactness theorem.

In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2\. proved a version of this theorem for polynomial rings, and proved a more general version. Several other authors later rediscovered and published variations of this theorem. gives a statement and proof.

Earnshaw's theorem has no exceptions for non-moving permanent ferromagnets. However, Earnshaw's theorem does not necessarily apply to moving ferromagnets, certain electromagnetic systems, pseudo-levitation and diamagnetic materials. These can thus seem to be exceptions, though in fact they exploit the constraints of the theorem. Spinning ferromagnets (such as the Levitron) can—while spinning—magnetically levitate using only permanent ferromagnets.

These theorems are for Banach spaces with the Radon–Nikodym property. A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded). Edgar's theorem implies Lindenstrauss's theorem.

Frederick Soddy rediscovered the equation in 1936. The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature (June 20, 1936). Soddy also extended the theorem to spheres; Thorold Gosset extended the theorem to arbitrary dimensions.

The circle packing theorem was first proved by Paul Koebe. William Thurston, Chap. 13. rediscovered the circle packing theorem, and noted that it followed from the work of E. M. Andreev. Thurston also proposed a scheme for using the circle packing theorem to obtain a homeomorphism of a simply connected proper subset of the plane onto the interior of the unit disk.

The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective module over a polynomial ring is free.

In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff.

In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by . A consequence of the Keel–Mori theorem is the existence of a coarse moduli space of a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space.

Elitzur'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 Shmuel Elitzur, who proved it for Abelian gauge fields on a lattice. It is nonetheless possible to spontaneously break a global symmetry within a theory that has a local gauge symmetry, as in the Higgs mechanism.

Proof of Apollonius's theorem The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines. Let the triangle have sides with a median drawn to side . Let be the length of the segments of formed by the median, so is half of .

In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of , which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields. :Hilbert–Speiser Theorem. A finite abelian extension has a normal integral basis if and only if it is tamely ramified over .

In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.

These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters. In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape.. This theorem was first published by Joseph-Émile Barbier in 1860.. See in particular pp. 283–285.

In mathematics, more specifically in multivariable calculus, the implicit function theoremAlso called Dini's theorem by the Pisan school in Italy. In the English-language literature, Dini's theorem is a different theorem in mathematical analysis. is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function.

The basic idea of RSA cryptosystem is thus: if a message is encrypted as y=x^e\pmod n, using public values of and , then, with the current knowledge, it cannot be decrypted without finding the (secret) factors and of . Fermat's little theorem is also related to the Carmichael function and Carmichael's theorem, as well as to Lagrange's theorem in group theory.

Hamnet, "Geometrical theorem", The Quarterly Journal of Pure and Applied Mathematics 2, 1858, p. 38. While not mentioned by Holditch, the proof of the theorem requires an assumption that the chord be short enough that the traced locus is a simple closed curve.Broman, Arne, "A fresh look at a long-forgotten theorem", Mathematics Magazine 54(3), May 1981, 99–108.

The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish book. In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder–Tychonoff fixed point theorem.

The inequalities satisfied by power series coefficients of conformal mappings were of considerable interest to mathematicians prior to the solution of the Bieberbach conjecture. The area theorem is a central tool in this context. Moreover, the area theorem is often used in order to prove the Koebe 1/4 theorem, which is very useful in the study of the geometry of conformal mappings.

Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.

In the mathematical field of differential geometry, more precisely, the theory of surfaces in Euclidean space, the Bonnet theorem states that the first and second fundamental forms determine a surface in R3 uniquely up to a rigid motion.. It was proven by Pierre Ossian Bonnet in about 1860. This is not to be confused with the Bonnet–Myers theorem or Gauss–Bonnet theorem.

A screw axis. Mozzi–Chasles' theorem says that every Euclidean motion is a screw displacement along some screw axis.In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a translation along a line (called its screw axis or Mozzi axis) followed (or preceded) by a rotation about an axis parallel to that line.

The Bolzano–Weierstrass theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass. It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem. Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of analysis.

In Proposition 45 Newton extended his theorem to arbitrary central forces by assuming that the particle moved in nearly circular orbit. As noted by astrophysicist Subrahmanyan Chandrasekhar in his 1995 commentary on Newton's Principia, this theorem remained largely unknown and undeveloped for over three centuries.Chandrasekhar, p. 183. Since 1997, the theorem has been studied by Donald Lynden-Bell and collaborators.

Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces. Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis.

A heuristic principle known as Bloch's Principle (made precise by Zalcman's lemma) states that properties that imply that an entire function is constant correspond to properties that ensure that a family of holomorphic functions is normal. For example, the first version of Montel's theorem stated above is the analog of Liouville's theorem, while the second version corresponds to Picard's theorem.

Kawamata was involved in the development of the minimal model program in the 1980s. The program aims to show that every algebraic variety is birational to one of an especially simple type: either a minimal model or a Fano fiber space. The Kawamata-Viehweg vanishing theorem, strengthening the Kodaira vanishing theorem, is a method. Building on that, Kawamata proved the basepoint-free theorem.

The theorem is clearly true for three non-collinear points. We proceed by induction. Assume n > 3 and the theorem is true for n − 1\. Let P be a set of n points not all collinear.

Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group.

Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching. The Tutte theorem provides a characterization for arbitrary graphs. A perfect matching is a spanning 1-regular subgraph, a.k.a. a 1-factor.

The Hardy–Littlewood maximal operator appears in many places but some of its most notable uses are in the proofs of the Lebesgue differentiation theorem and Fatou's theorem and in the theory of singular integral operators.

In mathematics, the multinomial theorem describes 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 from binomials to multinomials.

List and Pettit argue that the discursive dilemma can be likewise generalized to a sort of "List–Pettit theorem". Their theorem states that the inconsistencies remain for any aggregation method which meets a few natural conditions.

In mathematics, the Fraňková-Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.

The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).

In mathematics, the Dawson-Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson-Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one.

The same results are true of Ω(n), the number of prime factors of n counted with multiplicity. This theorem is generalized by the Erdős–Kac theorem, which shows that ω(n) is essentially normally distributed.

In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map...

With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.

Above we showed how to prove the Borsuk–Ulam theorem from Tucker's lemma. The converse is also true: it is possible to prove Tucker's lemma from the Borsuk–Ulam theorem. Therefore, these two theorems are equivalent.

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate. Thus, right triangles in a non-Euclidean geometry do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to /2, and all its angles are right angles, which violates the Pythagorean theorem because a^2 + b^2 = 2 c^2 > c^2 .

From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting edge developments. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995.

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators. The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli.

Ronald Coase's work itself emphasized a problem in applying the Coase theorem: transactions are "often extremely costly, sufficiently costly at any rate to prevent many transactions that would be carried out in a world in which the pricing system worked without cost." (Coase, 1960—first paragraph of section VI.) This isn't a criticism of the theorem itself, since the theorem considers only those situations in which there are no transaction costs. Instead, it is an objection to applications of the theorem that neglect this crucial assumption. So, a key criticism is that the theorem is almost always inapplicable in economic reality, because real- world transaction costs are rarely low enough to allow for efficient bargaining.

Atiyah and Segal combined this fixed point theorem with the index theorem as follows. If there is a compact group action of a group G on the compact manifold X, commuting with the elliptic operator, then one can replace ordinary K theory in the index theorem with equivariant K-theory. For trivial groups G this gives the index theorem, and for a finite group G acting with isolated fixed points it gives the Atiyah–Bott fixed point theorem. In general it gives the index as a sum over fixed point submanifolds of the group G. Atiyah solved a problem asked independently by Hörmander and Gel'fand, about whether complex powers of analytic functions define distributions.

In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. The theorem was proved by Donald A. Martin in 1975, and is applied in descriptive set theory to show that Borel sets in Polish spaces have regularity properties such as the perfect set property and the property of Baire. The theorem is also known for its metamathematical properties. In 1971, before the theorem was proved, Harvey Friedman showed that any proof of the theorem in Zermelo–Fraenkel set theory must make repeated use of the axiom of replacement.

The Bondy–Chvátal theorem states that a graph is Hamiltonian if and only if its closure is Hamiltonian; since the complete graph is Hamiltonian, Ore's theorem is an immediate consequence. found a version of Ore's theorem that applies to directed graphs. Suppose a digraph G has the property that, for every two vertices u and v, either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G. Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every edge in a given undirected graph by a pair of directed edges.

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if K is a nonempty convex closed subset of a Hausdorff topological vector space V and T is a continuous mapping of K into itself such that T(K) is contained in a compact subset of K, then T has a fixed point. A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was proved earlier by Juliusz Schauder and Jean Leray.

In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually.

Lindström's theorem states that first-order logic is the strongest (subject to certain constraints) logic satisfying both compactness and completeness. A completeness theorem can be proved for modal logic or intuitionistic logic with respect to Kripke semantics.

The 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.

This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.

The multi-dimensional case of the Fourth Hilbert problem was studied by Szabo.Z. I. Szabo, Hilbert's fourth problem I, Adv. Math. 59 (1986), 185—301. In 1986, he proved, as he wrote, the generalized Pogorelov theorem. Theorem.

In mathematical analysis, Haar's tauberian theorem named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood tauberian theorem.

The theorem is named after R. Leonard Brooks, who published a proof of it in 1941. A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring or a Δ-coloring.

One application of the low basis theorem is to construct completions of effective theories so that the completions have low Turing degree. For example, the low basis theorem implies the existence of PA degrees strictly below \emptyset'.

Tennenbaum'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 first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).

For example, the intermediate value theorem for functions from the reals to the reals is provable in RCA0 (Simpson 2009, p. 87), while the Bolzano–Weierstrass theorem is equivalent to ACA0 over RCA0 (Simpson 2009, p. 34).

The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (for example, using the Lax–Milgram theorem, a consequence of the Riesz representation theorem).

One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox.

Converse of this theorem is an consequence of cellular homology, and the fact that every free module is projective. Theorem: Let X be an aspherical n-dimensional CW complex with π1(X) = G, then cdZ(G) ≤ n.

This result may be proven using Serre's theorem on regular local rings.

Examples of automated reasoning engines include inference engines, theorem provers, and classifiers.

From here the proof is the same as the Erdos-Selfridge theorem.

Sabidussi wrote foundational work on Cayley graphs, graph products and Frucht's theorem.

The "four functions theorem" was independently generalized to 2k functions in and .

The PCP theorem states that : NP = PCP[O(log n), O(1)].

The theorems in this section simultaneously imply Euclid's theorem and other results.

Later Pop proved the Theorem for arbitrary characteristic by developing "rigid patching".

This was the first full published proof of the second incompleteness theorem.

The Myhill–Nerode theorem can be generalized to trees. See tree automaton.

1865 – 1883 This theorem of Shelah answers a question of H. Friedman.

There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982).

However, Briot and Bouquet removed this theorem from the second edition (1875).

The concept was introduced by to study the consequences of Arrow's theorem.

The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras. The theorem has been given numerous proofspossibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.

The distinction between elementary and non-elementary proofs has been considered especially important in regard to the prime number theorem. This theorem was first proved in 1896 by Jacques Hadamard and Charles Jean de la Vallée-Poussin using complex analysis. Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs: However, in 1948, Atle Selberg produced new methods which led him and Paul Erdős to find elementary proofs of the prime number theorem.

In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of GrushkoI. A. Grushko, On the bases of a free product of groups, Matematicheskii Sbornik, vol 8 (1940), pp. 169-182. and then, independently, in a 1943 article of Neumann.

Mantel's Theorem (1907) and Turán's Theorem (1941) were some of the first milestones in the study of Extremal graph theory. In particular, Turán's theorem would later on become a motivation for the finding of results such as the Erdős-Stone-Simonovits Theorem (1946). This result is surprising because it connects the chromatic number with the maximal number of edges in an H-free graph. An alternative proof of Erdős-Stone-Simonovits was given in 1975, and utilised the Szemerédi regularity lemma, an essential technique in the resolution of extremal graph theory problems.

In 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 (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan. The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Grigory Margulis, Anders Karlsson, and François Ledrappier.

Goodstein's theorem is a statement about the Ramsey theory of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic. Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.

Fuss's theorem, which is the analog of Euler's theorem for triangles for bicentric quadrilaterals, says that if a quadrilateral is bicentric, then its two associated circles are related according to the above equations. In fact the converse also holds: given two circles (one within the other) with radii R and r and distance x between their centers satisfying the condition in Fuss' theorem, there exists a convex quadrilateral inscribed in one of them and tangent to the other. (and then by Poncelet's closure theorem, there exist infinitely many of them).

The Second Fundamental Theorem allows to give an upper bound for the characteristic function in terms of N(r,a). For example, if f is a transcendental entire function, using the Second Fundamental theorem with k = 3 and a3 = ∞, we obtain that f takes every value infinitely often, with at most two exceptions, proving Picard's Theorem. Nevanlinna's original proof of the Second Fundamental Theorem was based on the so-called Lemma on the logarithmic derivative, which says that m(r,f'/f) = S(r,f). A similar proof also applies to many multi-dimensional generalizations.

The Cut-insertion theorem, also known as Pellegrini's theorem,Bruno Pellegrini has been the first Electronic Engineering graduate at the University of Pisa where is currently Professor Emeritus. He is also author of the Electrokinematics theorem, that connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector. is a linear network theorem that allows transformation of a generic network N into another network N' that makes analysis simpler and for which the main properties are more apparent.

Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951. It states the following: Let K be a compact subset of the complex plane C such that C∖K is connected. Then, every continuous function f : K\to C, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K. Mergelyan's theorem is the ultimate development and generalization of the Weierstrass approximation theorem and Runge's theorem.

Because of Justin's contribution to the problem, Kawasaki's theorem has also been called the Kawasaki–Justin theorem. The fact that this condition is sufficient—that is, that crease patterns with evenly many angles, alternatingly summing to can always be flat-folded—may have been first stated by . Kawasaki himself has called the result Husimi's theorem, after Kôdi Husimi, and some other authors have followed this terminology as well. The name "Kawasaki's theorem" was first given to this result in Origami for the Connoisseur by Kunihiko Kasahara and Toshie Takahama (Japan Publications, 1987).

In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair.

In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them. Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem.

Since the Banach–Alaoglu theorem is usually proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, and in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC. However, the theorem does not rely upon the axiom of choice in the separable case (see below): in this case one actually has a constructive proof. In the non-separable case, the Ultrafilter Lemma, which is strictly weaker than the axiom of choice, suffices for the proof of the Banach-Alaoglu theorem, and is in fact equivalent to it.

The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization: :252 = 2 × 3 × 7 Euclid's Elements first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss. The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number.

In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem. The theorem is named after Stephen Cook and Leonid Levin. An important consequence of this theorem is that if there exists a deterministic polynomial time algorithm for solving Boolean satisfiability, then every NP problem can be solved by a deterministic polynomial time algorithm.

In mathematical logic, the compactness theorem states that a set of first- order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces,See Truss (1997). hence the theorem's name.

In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games .In mathematics, the term folk theorem refers generally to any theorem that is believed and discussed, but has not been published. Roger Myerson has recommended the more descriptive term "general feasibility theorem" for the game theory theorems discussed here. See Myerson, Roger B. Game Theory, Analysis of conflict, Cambridge, Harvard University Press (1991) The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game.

In the 60s he developed an axiomatic complexity theory which was independent of concrete machine models. The theory is based on Gödel numberings and the Blum axioms. Even though the theory is not based on any machine model it yields concrete results like the compression theorem, the gap theorem, the honesty theorem and the Blum speedup theorem. Some of his other work includes a protocol for flipping a coin over a telephone, median of medians (a linear time selection algorithm), the Blum Blum Shub pseudorandom number generator, the Blum- Goldwasser cryptosystem, and more recently CAPTCHAs.

For instance, the subspace theorem proved by demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation. Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by and respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.

Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite dimensional complex vector space has a nontrivial invariant subspace, the fundamental theorem of noncommutative algebra asserts that Lat(Σ) contains nontrivial elements for certain Σ. Theorem (Burnside) Assume V is a complex vector space of finite dimension. For every proper subalgebra Σ of L(V), Lat(Σ) contains a nontrivial element. Burnside's theorem is of fundamental importance in linear algebra. One consequence is that every commuting family in L(V) can be simultaneously upper-triangularized.

Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem. Because the sum of the reciprocals of the primes diverges, the Green-Tao theorem on arithmetic progressions is a special case of the conjecture. The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem, which appears as the main result in a 2020 preprint by Bloom and Sisask. The former strongest bound in Roth's theorem is due to Bloom.

The Riemann–Stieltjes integral appears in the original formulation of F. Riesz's theorem which represents the dual space of the Banach space C[a,b] of continuous functions in an interval [a,b] as Riemann–Stieltjes integrals against functions of bounded variation. Later, that theorem was reformulated in terms of measures. The Riemann–Stieltjes integral also appears in the formulation of the spectral theorem for (non-compact) self-adjoint (or more generally, normal) operators in a Hilbert space. In this theorem, the integral is considered with respect to a spectral family of projections.

There is also a dual version of Miller theorem that is based on Kirchhoff's current law (Miller theorem for currents): if there is a branch in a circuit with impedance Z connecting a node, where two currents I1 and I2 converge to ground, we can replace this branch by two conducting the referred currents, with imperespectively equal to (1 + α)Z and (1 + α)Z/α, where α = I2/I1. The dual theorem may be proved by replacing the two-port network by its equivalent and by applying the source absorption theorem.

Uzawa's theorem, also known as the steady state growth theorem, is a theorem in economic growth theory concerning the form that technological change can take in the Solow–Swan and Ramsey–Cass–Koopmans growth models. It was first proved by Japanese economist Hirofumi Uzawa. One general version of the theorem consists of two parts. The first states that, under the normal assumptions of the Solow and Neoclassical models, if (after some time T) capital, investment, consumption, and output are increasing at constant exponential rates, these rates must be equivalent.

For pseudovarieties, there is no general finitary counterpart to Birkhoff's theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived.E.g. Banaschewski, B. (1983), "The Birkhoff Theorem for varieties of finite algebras", Algebra Universalis, Volume 17(1): 360-368, DOI 10.1007/BF01194543 Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory. Eilenberg's theorem, often referred to as the variety theorem, describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.

The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem. Fig. 8b – The triangle (pink), an auxiliary circle (light blue) and two auxiliary right triangles (yellow) Case of acute angle , where . Drop the perpendicular from onto = , creating a line segment of length . Duplicate the right triangle to form the isosceles triangle . Construct the circle with center and radius , and a chord through perpendicular to half of which is Apply the Pythagorean theorem to obtain :b^2 = c^2 + h^2.

Isabelle has been used to formalize numerous theorems from mathematics and computer science, like Gödel's completeness theorem, Gödel's theorem about the consistency of the axiom of choice, the prime number theorem, correctness of security protocols, and properties of programming language semantics. Many of the formal proofs are maintained in the Archive of Formal Proofs, which contains (as of 2019) at least 500 articles with over 2 million lines of proof in total. The Isabelle theorem prover is free software, released under the revised BSD license. Isabelle was named by Lawrence Paulson after Gérard Huet's daughter.

Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem.

In mathematics, the well-ordering theorem, also known as Zermelo's 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. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.

The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry.. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The importance of this theorem is as follows: very little was known about the geometry behind symplectic transformations. One easy consequence of a transformation being symplectic is that it preserves volume.

The exception of Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs. The Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs.Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs.

Bulatov proved a dichotomy theorem for domains of three elements. Another dichotomy theorem for constraint languages is the Hell- Nesetril theorem, which shows a dichotomy for problems on binary constraints with a single fixed symmetric relation. In terms of the homomorphism problem, every such problem is equivalent to the existence of a homomorphism from a relational structure to a given fixed undirected graph (an undirected graph can be regarded as a relational structure with a single binary symmetric relation). The Hell-Nesetril theorem proves that every such problem is either polynomial-time or NP-complete.

This is a graph with a vertex for each element of the order and an edge for each pair of incomparable elements. Using this coloring interpretation, together with a separate proof of Dilworth's theorem for finite partially ordered sets, it is possible to prove that an infinite partially ordered set has finite width if and only if it has a partition into chains., Theorem 5.6, p. 60. In the same way, the De Bruijn–Erdős theorem extends the four-color theorem from finite planar graphs to infinite planar graphs.

The Hahn–Banach theorem is a central tool in functional analysis (a field of mathematics). 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 vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

Among all closed manifolds with non-positive sectional curvature, flat manifolds are characterized as precisely those with an amenable fundamental group. This is a consequence of the Adams-Ballmann theorem (1998), which establishes this characterization in the much more general setting of discrete cocompact groups of isometries of Hadamard spaces. This provides a far-reaching generalisation of Bieberbach's theorem. The discreteness assumption is essential in the Adams-Ballmann theorem: otherwise, the classification must include symmetric spaces, Bruhat-Tits buildings and Bass-Serre trees in view of the "indiscrete" Bieberbach theorem of Caprace-Monod.

In modern sources, the Adian–Rabin theorem is usually stated as follows:Roger Lyndon and Paul Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ; Ch. IV, Theorem 4.1, p. 192G. Baumslag.

The 1-skeleton of any k-dimensional convex polytope forms a k-vertex-connected graph (Balinski's theorem, ). As a partial converse, Steinitz's theorem states that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

11 in . (James' theorem) Since norm-closed convex subsets in a Banach space are weakly closed,Theorem 2.5.16 in . it follows from the third property that closed bounded convex subsets of a reflexive space X are weakly compact.

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

CP violation implied nonconservation of T, provided that the long-held CPT theorem were valid. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together.

The van Cittert–Zernike theorem rests on a number of assumptions, all of which are approximately true for nearly all astronomical sources. The most important assumptions of the theorem and their relevance to astronomical sources are discussed here.

The Droz-Farny line theorem concerns a property of two perpendicular lines intersecting at a triangle's orthocenter. Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle.

Many texts prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.

This theorem, which is an existence theorem for measures on infinite product spaces, says that if any finite- dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite- dimensional distributions.

Reformulated in the language of semi-simple categories, Maschke's theorem states :Maschke's theorem. If is a group and is a field with characteristic not dividing the order of , then the category of representations of over is semi-simple.

Nevertheless, it is possible to formulate a version of the Sylvester–Gallai theorem that is valid within the axioms of constructive analysis, and to adapt Kelly's proof of the theorem to be a valid proof under these axioms.

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non- zero complex number in the spectrum of a compact operator is an eigenvalue.

In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015 Alexandre Eremenko gave a simplified proof of Stahl's theorem.

Here Petersen proves his first major result, viz. that any such graph has a 2-factorization (2-factor theorem). :(iii) Criteria for the existence of edge-separating factorizations of 4-regular graphs. :(iv) The factorization of regular graphs of odd degree, in particular, the theorem that any bridgeless 3-regular graph can be decomposed into a l-factor and a 2-factor (Petersen's theorem).

In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name. The first theorem exhibited the basis of an algebra through infinitesimal transformations. The second theorem exhibited structure constants of the algebra as the result of commutator products in the algebra. The third theorem showed these constants are anti-symmetric and satisfy the Jacobi identity.

An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of level sets of harmonic functions. There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions.

The motivation behind Artin and Grothendieck's proof for constructible sheaves was to give a proof that could be adapted to the setting of étale and \ell-adic cohomology. Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic. The theorem can also be generalized to intersection homology. In this setting, the theorem holds for highly singular spaces.

The envelope theorem is a result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models.

Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

Along with Samuel L. Braunstein, he proved the quantum no-deleting theorem. Similar to the no-cloning theorem, the no-deleting theorem is a fundamental consequence of the linearity of quantum mechanics. This proves that given two copies of an unknown quantum state we cannot delete one copy. The no-cloning and the no-deleting theorems suggest that we can neither create nor destroy quantum information.

In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the importance of and being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.

The Brylinski–Kostant filtration of weight spaces is named after her. She originally developed this filtration in 1989, motivated by earlier work of Bertram Kostant. She is also known for Brylinski's theorem, a theorem from her dissertation on the closures of orbits of algebraic groups. Another result, also called "Brylinski's theorem", comes from a paper written jointly by Brylinski and her husband, characterizing universal quantum logic gates.

Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson. This can be viewed as a kind of generalization of the Artin- Wedderburn theorem's conclusion about the structure of simple Artinian rings. More formally, the theorem can be stated as follows: :The Jacobson Density Theorem.

Attending Littlewood's lectures, she solved one of the open problems which he posed. Her mathematical theorem, now known as Cartwright's theorem, gives an estimate for the maximum modulus of an analytic function that takes the same value no more than p times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Lars Ahlfors for conformal mappings.

The Baire category theorem (BCT) is an important result 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 (a topological space such that the intersection of countably many dense open sets is still dense). The theorem was proved by French mathematician René-Louis Baire in his 1899 doctoral thesis.

1 (1957), pp. 55–67. showing that Gödelian incompleteness held for formal systems considerably more elementary than that of Kurt Gödel's 1931 landmark paper. The contemporary understanding of Gödel's theorem dates from this 1931 paper. Smullyan later made a compelling case that much of the fascination with Gödel's theorem should be directed at Tarski's theorem, which is much easier to prove and equally disturbing philosophically.

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.

Kőnig's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.

In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called the Gromov-Hausdorff convergence, is currently widely used in geometry. A relatively simple proof of the theorem was found by Bruce Kleiner. Later, Terence Tao and Yehuda Shalom modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces from Lebesgue integration theory are complete.

He also proved several theorems concerning convergence of sequences of measurable and holomorphic functions. The Vitali convergence theorem generalizes Lebesgue's dominated convergence theorem. Another theorem bearing his name gives a sufficient condition for the uniform convergence of a sequence of holomorphic functions on an open domain. This result has been generalized to normal families of meromorphic functions, holomorphic functions of several complex variables, and so on.

The parallel axis theorem, also known as Huygens-Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

In fluid dynamics, Prandtl–Batchelor theorem states that if in a two- dimensional laminar flow at high Reynolds number closed streamlines occur, then the vorticity in the closed streamline region must be a constant. The theorem is named after Ludwig Prandtl and George Batchelor. Prandtl in his celebrated 1904 paper stated this theorem in arguments,Prandtl, L. (1904). Über Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandl.

The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style. It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof.

In 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 of A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney.

The work culminated in what Arrow called the "General Possibility Theorem," better known thereafter as Arrow's (impossibility) theorem. The theorem states that, absent restrictions on either individual preferences or neutrality of the constitution to feasible alternatives, there exists no social choice rule that satisfies a set of plausible requirements. The result generalizes the voting paradox, which shows that majority voting may fail to yield a stable outcome.

Among hundreds of fixed-point theorems,E.g. F & V Bayart Théorèmes du point fixe on [email protected] Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.

Diagram illustrating the fact that Napoleon's theorem is a special case of Petr–Douglas–Neumann theorem. In the case of triangles, the value of n is 3 and that of n − 2 is 1. Hence there is only one possible value for k, namely 1. The specialisation of the theorem to triangles asserts that the triangle A1 is a regular 3-gon, that is, an equilateral triangle.

There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kovalevskaya–Kashiwara theorem, due to . This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor Ext^1.

For u = 0 above, the statement is also known as Bolzano's theorem. (Since there is nothing special about u = 0, this is obviously equivalent to the intermediate value theorem itself.) This theorem was first proved by Bernard Bolzano in 1817. Augustin- Louis Cauchy provided a proof in 1821. Both were inspired by the goal of formalizing the analysis of functions and the work of Joseph-Louis Lagrange.

Then the fiber f^{-1}(y) has at most n connected components. In particular, if f is birational, then the fibers of unibranch points are connected. In EGA, the theorem is obtained as a corollary of Zariski's main theorem.

Other generalizations include considering subgroups of free pro- finite products and a version of the Kurosh subgroup theorem for topological groups.Peter Nickolas, A Kurosh subgroup theorem for topological groups. Proceedings of the London Mathematical Society (3), 42 (1981), no.

In 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédi's theorem. The Furstenberg boundary and Furstenberg compactification of a locally symmetric space are named after him, as is the Furstenberg–Sárközy theorem in additive number theory.

In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.

The Droz-Farny line theorem says that the midpoints of the three segments A_1A_2, B_1B_2, and C_1C_2 are collinear. The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

In 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.

The asymptotic equipartition property for non-stationary discrete-time independent process leads us to (among other results) the source coding theorem for non-stationary source (with independent output symbols) and noisy-channel coding theorem for non- stationary memoryless channels.

This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally: Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

Wiener's Tauberian theorem, a 1932 result of Wiener, developed Tauberian theorems in summability theory, on the face of it a chapter of real analysis, by showing that most of the known results could be encapsulated in a principle taken from harmonic analysis. In its present formulation, the theorem of Wiener does not have any obvious association with Tauberian theorems, which deal with infinite series; the translation from results formulated for integrals, or using the language of functional analysis and Banach algebras, is however a relatively routine process. The Paley–Wiener theorem relates growth properties of entire functions on Cn and Fourier transformation of Schwartz distributions of compact support. The Wiener–Khinchin theorem, (also known as the Wiener – Khintchine theorem and the Khinchin – Kolmogorov theorem), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

The main contribution of Moshe Jarden in algebra, and in mathematics in general, is his research on families of large algebraic extensions of Hilbertian fields (in particular global fields), parametrized by the automorphisms of the absolute Galois group of the base field. Notable results in this domain are the zero theorem, the transfer theorem, the free generators theorem, the Frey-Jarden theorem about the rank of algebraic varieties over large algebraic fields, Geyer-Jarden theorem about torsion points on elliptic curves over large algebraic fields, and the strong approximation theorem over such fields. The remarkable development of Galois theory over a class of large fields, called ample fields,These fields, previously called large fields by Florian Pop, were so named by M. Jarden because the term large fields was already used in another context. is described in the second book of Jarden: Algebraic Patching.

Using the above properties of graphs, one can prove the Nielsen–Schreier theorem.

It can be deduced from the hairy ball theorem illustrated at the right.

The Trotter–Kato theorem can be used for approximation of linear C0-semigroups.

Brianchon's theorem can be proved by the idea of radical axis or reciprocation.

As the arrivals are determined by a Poisson process, the arrival theorem holds.

In Greg Egan's novel Diaspora, two characters discuss the derivation of this theorem.

Siacci's theorem is particularly useful in motions where the angular momentum is constant.

The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925.

Riemann-Roch and Atiyah-Singer are other generalizations of the Gauss-Bonnet theorem.

The independence from A is an analogue of the Excision theorem in homology.

Nikolai Luzin's generalization of the Severini–Egorov theorem is presented here according to .

The excision theorem is taken to be one of the Eilenberg-Steenrod Axioms.

Rainwater's theorem is an important result in summability theory and Banach-space theory.

"Waltz Figures Out the Zero Theorem: Terry Gilliam's Latest!". Empire. 13 August 2012.

Fleming, Mike (2012). "Update: Toronto: Terry Gilliam Confirms Christoph Waltz for ‘Zero Theorem’".

Ekeland was associated with the University of Paris when he proposed this theorem.

Proof: Follows immediately from Ptolemy's theorem: : qs=ps+rs \Rightarrow q=p+r.

Nevertheless, analogues to Whitney's isomorphism theorem can still be derived in this case.

This paper, his doctoral dissertation, introduced Morley rank and proved Morley's categoricity theorem.

The theorem can be generalized to the midpoint polygon of an arbitrary polygon.

The localization theorem is one of the most powerful tools in equivariant cohomology.

Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Another theorem commonly referred to as Krull's theorem: :::Let R be a Noetherian ring and a an element of R which is neither a zero divisor nor a unit. Then every minimal prime ideal P containing a has height 1.

Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Star of David theorem (the rows of the Pascal triangle are shown as columns here). The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972.

It pertains to the classification of finite simple groups, namely the classification of finite primitive permutation groups. The paper contains a complete self-contained proof of the theorem. Praeger later went on to generalise the O'Nan–Scott Theorem to quasiprimitive groups.

Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations.Leonida Tonelli.

A partition of a complete graph on 8 vertices into 7 colors (perfect matchings), the case r = 2 of Baranyai's theorem In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.

In 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.

Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.) Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

The theorem then follows from the lemma. Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of the asymptote index.

The PCP theorem is the culmination of a long line of work on interactive proofs and probabilistically checkable proofs. The first theorem relating standard proofs and probabilistically checkable proofs is the statement that NEXP ⊆ PCP[poly(n), poly(n)], proved by .

In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.

The risk inclination formula (RIF) construct is based upon Varignon's theorem and quantifies feelings of rightness toward knowledge certainty.,Coxeter, H. S. M. and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp.

In the theory of hyperfunctions there is an extension of the edge-of- the-wedge theorem to the case when there are several wedges instead of two, called Martineau's edge-of-the-wedge theorem. See the book by Hörmander for details.

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

The other interior angle (at the North Pole) can be made larger than 90°, further emphasizing the failure of this statement. However, since the Euclid's exterior angle theorem is a theorem in absolute geometry it is automatically valid in hyperbolic geometry.

His work on the theory of real functions was also important in the development of the concept of the measure on a set.See . Lezioni di analisi infinitesimale, 1878 The implicit function theorem is known in Italy as the Dini's theorem.

But this homomorphism is not necessarily onto. If such a parameterization exists, the variety is said unirational. Lüroth's theorem (see below) implies that unirational curves are rational. Castelnuovo's theorem implies also that, in characteristic zero, every unirational surface is rational.

In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.

The main theorem links the maximum flow through a network with the minimum cut of the network. : Max-flow min-cut theorem. The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts.

Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle. Theorem. Let F be a countably (topologically) generated profinite group. Then # F is projective if and only if any finite embedding problem for F is solvable.

The cone theorem and contraction theorem, central results in the theory, are the result of a joint effort by Kawamata, Kollár, Mori, Reid, and Shokurov.Y. Kawamata, K. Matsuda, and K. Matsuki. Introduction to the minimal model program. Algebraic Geometry, Sendai 1985.

The completeness theorem and the compactness theorem are two cornerstones of first-order logic. While neither of these theorems can be proven in a completely effective manner, each one can be effectively obtained from the other. The compactness theorem says that if a formula φ is a logical consequence of a (possibly infinite) set of formulas Γ then it is a logical consequence of a finite subset of Γ. This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deductive system then implies φ is a logical consequence of this finite set. This proof of the compactness theorem is originally due to Gödel.

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds.

He shifted attention from the study of individual varieties to the relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space. In 1956, he applied the same thinking to the Riemann–Roch theorem, which had already recently been generalized to any dimension by Hirzebruch. The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption. The Chinese remainder theorem can also be used in secret sharing, which consists of distributing a set of shares among a group of people who, all together (but no one alone), can recover a certain secret from the given set of shares. Each of the shares is represented in a congruence, and the solution of the system of congruences using the Chinese remainder theorem is the secret to be recovered. Secret sharing using the Chinese remainder theorem uses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret from a set of shares with less than a certain cardinality.

In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations. The theorem states that: :Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate. Professor Sergei K. Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields.

Observe that the above argument also gives the following corollary: if we let A be the set of all eight-digit numbers whose digits are all either 1, 2, 3 (thus A contains numbers such as 11333233), and we color A with two colors, then A contains at least one arithmetic progression of length three, all of whose elements are the same color. This is simply because all of the combinatorial lines appearing in the above proof of the Hales–Jewett theorem, also form arithmetic progressions in decimal notation. A more general formulation of this argument can be used to show that the Hales–Jewett theorem generalizes van der Waerden's theorem. Indeed the Hales–Jewett theorem is substantially a stronger theorem.

In general, a -form is an object that may be integrated over a -dimensional oriented manifold, and is homogeneous of degree in the coordinate differentials. The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation on differential forms known as the exterior derivative that, when given a -form as input, produces a -form as output. This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.

In theoretical computer science, the PACELC theorem is an extension to the CAP theorem. It states that in case of network partitioning (P) in a distributed computer system, one has to choose between availability (A) and consistency (C) (as per the CAP theorem), but else (E), even when the system is running normally in the absence of partitions, one has to choose between latency (L) and consistency (C).

The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics.

In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ringIn this article, rings have a 1. has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice.

Finsler-Hadwiger theorem The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths and area of a triangle.. See in particular p. 324.

In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem (also referred to as Barratt–Priddy–Quillen theorem) expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem (named after Michael Barratt, Stewart Priddy, and Daniel Quillen) is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.

Burke first published this theorem along with a proof in 1956. The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955). A second proof of the theorem follows from a more general result published by Reich. The proof offered by Burke shows that the time intervals between successive departures are independently and exponentially distributed with parameter equal to the arrival rate parameter, from which the result follows.

In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.

Nonuniform sampling is a branch of sampling theory involving results related to the Nyquist–Shannon sampling theorem. Nonuniform sampling is based on Lagrange interpolation and the relationship between itself and the (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker–Shannon–Kotelnikov (WSK) sampling theorem. The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time.

Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to- one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity.

In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring .Isaacs, p. 184 The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.Such rings of linear transformations are also known as full linear rings.

The Borel–Weil theorem provides a concrete model for irreducible representations of compact Lie groups and irreducible holomorphic representations of complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold of the group. The Borel–Weil–Bott theorem is its generalization to higher cohomology spaces. The theorem dates back to the early 1950s and can be found in and .

In automata theory, McNaughton's theorem refers to a theorem that asserts that the set of ω-regular languages is identical to the set of languages recognizable by deterministic Muller automata. McNaughton, R.: "Testing and generating infinite sequences by a finite automaton", Information and control 9, pp. 521–530, 1966. This theorem is proven by supplying an algorithm to construct a deterministic Muller automaton for any ω-regular language and vice versa.

In algebraic geometry, the Gabriel–Rosenberg reconstruction theorem, introduced in , states that a quasi-separated scheme can be recovered from the category of quasi-coherent sheaves on it. The theorem is taken as a starting point for noncommutative algebraic geometry as the theorem says (in a sense) working with stuff on a space is equivalent to working with the space itself. It is named after Pierre Gabriel and Alexander L. Rosenberg.

When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything "behaves nicely". In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed. However, there is no "best" class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain 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 Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space.

The system involved both custom software and custom hardware. In the late 1960s the company developed a system called SAM (Semi-Automated Mathematics) for proving mathematical theories without human intervention. A theorem proved by the system, "SAM's lemma", was "widely hailed as the first contribution of automated reasoning systems to mathematics." The SAM series was one of the first interactive theorem provers and had an influence on subsequent theorem provers.

A generalization of the tennis ball theorem applies to any simple smooth curve on the sphere that is not contained in a closed hemisphere. As in the original tennis ball theorem, such curves must have at least four inflection points. If a curve on the sphere is centrally symmetric, it must have at least six inflection points. A closely related theorem of also concerns simple closed spherical curves.

In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem.

In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the crosscorrelation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Institute of Technology.J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec.

The circle packing theorem is a useful tool to study various problems in planar geometry, conformal mappings and planar graphs. An elegant proof of the planar separator theorem, originally due to Lipton and Tarjan, has been obtained in this way. Another application of the circle packing theorem is that unbiased limits of bounded-degree planar graphs are almost surely recurrent. Other applications include implications for the cover time.

An exotic \R^4 is called small if it can be smoothly embedded as an open subset of the standard \R^4. Small exotic \R^4 can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.

All section meetings were cancelled so that everyone could hear his contribution. Kőnig applied a theorem proved in the thesis of Hilbert's student Felix Bernstein; this theorem, however, was not as generally valid as Bernstein had claimed. Ernst Zermelo, the later editor of Cantor's collected works, found the error already the next day. In 1905 there appeared short notes by Bernstein, correcting his theorem, and Kőnig, withdrawing his claim.

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor power Ln is very ample for n ≥ m. The theorem was proved by Teruhisa Matsusaka in 1972 and named by Lieberman and Mumford in 1975. The theorem has an application to the theory of Hilbert schemes.

A theorem by Gallai and Milgram shows that the number of paths in a smallest path cover cannot be larger than the number of vertices in the largest independent set., Theorem 2.5.1. In particular, for any graph G, there is a path cover P and an independent set I such that I contains exactly one vertex from each path in P. Dilworth's theorem follows as a corollary of this result.

Another easy proof uses Menelaus' theorem, since the ratios can be calculated with the diameters of each circle, which will be eliminated by cyclic forms when using Menelaus' theorem. Desargues' theorem also asserts that 3 points lie on a line, and has a similar proof using the same idea of considering it in 3 rather than 2 dimensions and writing the line as an intersection of 2 planes.

This corollary is also some called "the Krein-Milman theorem". A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners.

Egorov studied potential surfaces and triply orthogonal systems, and made significant contributions to the broader areas of differential geometry and integral equations. His work influenced that of Jean Gaston Darboux on differential geometry and mathematical analysis. A theorem in real analysis and integration theory, Egorov's Theorem, is named after him.He published a proof of this theorem in the short paper , and the result become widely acknowledged under his name.

A path of four upward-sloping edges in a set of 17 points. By the Erdős–Szekeres theorem, every set of 17 points has a path of this length that slopes either upward or downward. The 16-point subset with the central point removed has no such path. In mathematics, the Erdős–Szekeres theorem is a finitary result that makes precise one of the corollaries of Ramsey's theorem.

The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas Grothendieck saw it as a theorem about a morphism between varieties. By finding the right generalization, the proof became simpler while the conclusion became more general. In short, Grothendieck applied a strong categorical approach to a hard piece of analysis.

In combinatorial number theory, the Lambek–Moser theorem is a generalization of Beatty's theorem that defines a partition of the positive integers into two subsets from any monotonic integer-valued function. Conversely, any partition of the positive integers into two subsets may be defined from a monotonic function in this way. The theorem was discovered by Leo Moser and Joachim Lambek. provides a visual proof of the result.

In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

Alexander's Lemma: Up to isotopy, there is a unique (piecewise linear) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the Schoenflies theorem. In dimension two this is the Jordan curve theorem.) This may be restated as follows: the genus zero splitting of S^3 is unique. Waldhausen's Theorem: Every splitting of S^3 is obtained by stabilizing the unique splitting of genus zero.

In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X \times Y and those of the spaces X and Y. The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.

In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an Lp space. It can be thought of as an Lp version of the Arzelà–Ascoli theorem, from which it can be deduced. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov.

In algebraic geometry, the seesaw theorem, or seesaw principle, says roughly that a limit of trivial line bundles over complete varieties is a trivial line bundle. It was introduced by André Weil in a course at the University of Chicago in 1954–1955, and is related to Severi's theory of correspondences. The seesaw theorem is proved using proper base change. It can be used to prove the theorem of the cube.

Kaluznin's research spread wide most notably in group theory and abstract groups. He worked on the Sylow p-subgroups of symmetric groups and even mathematical linguistics. Despite not being able to go to many conferences he contributed to the application of computers in algebra. The universal embedding theorem is sometimes called the "Krasner-Kaloujnine universal embedding theorem" due to his joint proof of the theorem with Marc Krasner.

R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of ACM STOC'98, pp. 203–208. 1998. While Toda's theorem shows that PPP contains PH, P⊕P is not known to even contain NP. However, the first part of the proof of Toda's theorem shows that BPP⊕P contains PH. Lance Fortnow has written a concise proof of this theorem.

The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann–Hilbert problem in 1908.

Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only ifEngel and Nagel Theorem II.3.15, Arent et al. Theorem 3.4.5, Staffans Theorem 3.4.8 # D(A) is dense in X, # A is closed, # A is dissipative, and # A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.

Robinson'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: Let T_1 and T_2 be first-order theories. If T_1 and T_2 are consistent and the intersection T_1\cap T_2 is complete (in the common language of T_1 and T_2), then the union T_1\cup T_2 is consistent.

The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.

A more formal relationship is described via the fundamental theorem of arbitrage-free pricing.

In that area he proved a fundamental result which is now called Tsen's theorem.

Combining the second and third steps, and then applying Green's theorem completes the proof.

One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem.

The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups.

Ignatov's theorem states that the sequences Y1, Y2, Y3, ... are independent and identically distributed.

In the case m = 2, this statement reduces to that of the binomial theorem.

Jürgen Braun, On Kolmogorov's Superposition Theorem and Its Applications, SVH Verlag, 2010, 192 pp.

Then all the assumptions of the diamond theorem are satisfied, hence L is Hilbertian.

The proof of the β function lemma makes use of the Chinese remainder theorem.

The moduli space of instantons was used by Simon Donaldson to prove Donaldson's theorem.

In the parlance of Eric Brewer's CAP Theorem, HBase is a CP type system.

Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear.

Other significant contributions include being the first to prove the Fermat polygonal number theorem.

For this reason, keeping the full name of "Rayleigh Theorem for Eigenvalues" avoids confusions.

Chevalley's structure theorem is used in the proof of the Néron–Ogg–Shafarevich criterion.

Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede.

This fact can be regarded as an instance of the no free lunch theorem.

In this section we recall the objects of interest in the nonabelian Hodge theorem.

The implicit function theorem provides a uniform way of handling these sorts of pathologies.

A special case of this theorem was first described by Parameshvara (1370–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II.J. J. O'Connor and E. F. Robertson (2000). Paramesvara, MacTutor History of Mathematics archive. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823.

For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

In 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 Wolfgang Meyer. The related soul conjecture was formulated by Gromoll and Cheeger in 1972 and proved by Grigori Perelman in 1994 with an astonishingly concise proof. The soul theorem states: :If is a complete connected Riemannian manifold with sectional curvature , then there exists a compact totally convex, totally geodesic submanifold whose normal bundle is diffeomorphic to .

The completeness theorem can also be understood in terms of consistency, as a consequence of Henkin's model existence theorem. We say that a theory T is syntactically consistent if there is no sentence s such that both s and its negation ¬s are provable from T in our deductive system. The model existence theorem says that for any first-order theory T with a well-orderable language, :if T is syntactically consistent, then T has a model. Another version, with connections to the Löwenheim–Skolem theorem, says: :Every syntactically consistent, countable first-order theory has a finite or countable model.

Anatoly Shirshov was a pioneer in several directions of associative, Lie, Jordan, and alternative algebras, as well as groups and projective planes. His name is associated with notions and results on Gröbner-Shirshov bases, the Composition-Diamond Lemma, the Shirshov-Witt Theorem, the Lazard-Shirshov elimination process, Shirshov's Height Theorem, Lyndon-Shirshov words, Hall- Shirshov bases, Shirshov's Theorem on the Kurosh problem for alternative and Jodan algebras, and Shirshov's Theorem on the speciality of Jordan algebras with two generators. Shirshov's ideas were used by his student Efim Zelmanov for the solution of the Restricted Burnside problem.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech. Several texts identify Tychonoff's theorem as the single most important result in general topology [e.g.

Dirichlet's theorem on primes in arithmetic progressions shows that there are an infinity of primes in each co-prime residue class, and the prime number theorem for arithmetic progressions shows that the primes are asymptotically equidistributed among the residue classes. The Bombieri–Vinogradov theorem gives a more precise measure of how evenly they are distributed. There is also much interest in the size of the smallest prime in an arithmetic progression; Linnik's theorem gives an estimate. The twin prime conjecture, namely that there are an infinity of primes p such that p+2 is also prime, is the subject of active research.

According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at y(0)=0, either y(x)=0 or y(x)=x^2/4. The transition between y=0 and y=(x-C)^2/4 can happen at any C. The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.

While it seems elementary enough, at the time the modern definitions didn't exist, and when Cayley introduced what are now called groups it wasn't immediately clear that this was equivalent to the previously known groups, which are now called permutation groups. Cayley's theorem unifies the two. Although Burnside attributes the theorem to Jordan, Eric Nummela nonetheless argues that the standard name--"Cayley's Theorem"--is in fact appropriate. Cayley, in his original 1854 paper, showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding).

The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"Fermat's Last Theorem, Simon Singh, 1997, Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now the modularity theorem.

In functional analysis, the Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz. The Borel graph theorem shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space. The weak dual of a separable Fréchet space and the strong dual of a separable Fréchet–Montel space are Souslin spaces.

In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers where customers cannot leave the network. Jackson's theorem cannot be applied to closed networks because the queue length at a node in the closed network is limited by the population of the network. The Gordon-Newell theorem calculates the open network solution and then eliminates the infeasible states by renormalizing the probabilities. Calculation of the normalizing constant makes the treatment more awkward as the whole state space must be enumerated.

In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. However, already in 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, using a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods.

Bust of Fourier in Grenoble Fourier left an unfinished work on determining and locating real roots of polynomials, which was edited by Claude-Louis Navier and published in 1831. This work contains much original matter—in particular, Fourier's theorem on polynomial real roots, published in 1820. François Budan, in 1807 and 1811, had published independently his theorem (also known by the name of Fourier), which is very close to Fourier's theorem (each theorem is a corollary of the other). Fourier's proof is the one that was usually given, during 19th century, in textbooks on the theory of equations.

The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable (like voltage, temperature difference, etc.), and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems. The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951 and expanded by Ryogo Kubo.

For example, # Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem # Every vector space has a basis, a result from linear algebra (to which it is equivalent). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis. # Every commutative unital ring has a maximal ideal, a result from ring theory # Tychonoff's theorem in topology (to which it is also equivalent) # Every proper filter is contained in an ultrafilter, a result that yields completeness theorem of first-order logicJ.L. Bell & A.B. Slomson (1969).

Several deep theorems, such as the Hirzebruch–Riemann–Roch theorem, are special cases of the Atiyah–Singer index theorem. In fact the index theorem gave a more powerful result, because its proof applied to all compact complex manifolds, while Hirzebruch's proof only worked for projective manifolds. There were also many new applications: a typical one is calculating the dimensions of the moduli spaces of instantons. The index theorem can also be run "in reverse": the index is obviously an integer, so the formula for it must also give an integer, which sometimes gives subtle integrality conditions on invariants of manifolds.

He is also known for the Szemerédi–Trotter theorem in incidence geometry and the Hajnal–Szemerédi theorem and Ruzsa–Szemerédi problem in graph theory. Miklós Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem. With Ajtai and János Komlós he proved the ct2/log t upper bound for the Ramsey number R(3,t), and constructed a sorting network of optimal depth. With Ajtai, Václav Chvátal, and Monroe M. Newborn, Szemerédi proved the famous Crossing Lemma, that a graph with n vertices and m edges, where has at least crossings.

The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem. The no-cloning theorem has a time-reversed dual, the no-deleting theorem. Together, these underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger compact category. This formulation, known as categorical quantum mechanics, allows, in turn, a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in the same sense that intuitionistic logic arises from Cartesian closed categories).

The most important result is that all points can be constructed by the method of tangents and secants starting with a finite number of points. More precisely the Mordell–Weil theorem states that the group E(Q) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups. The proof of that theorem rests on two ingredients: first, one shows that for any integer m > 1, the quotient group E(Q)/mE(Q) is finite (weak Mordell–Weil theorem).

In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, and Niels Henrik Abel, who provided a proof in 1824. Abel–Ruffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals.

This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria. The Folk Theorem suggests that if the players are patient enough and far-sighted (i.e. if the discount factor \delta \to 1 ), then repeated interaction can result in virtually any average payoff in an SPE equilibrium.

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system itself. Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be proved from within the system itself. The demonstration above shows that if the system is consistent, then p is not provable.

Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930 (Dawson 1996, p. 70). Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by Monatshefte für Mathematik on November 17, 1930.

The Eberlein–Šmulian theorem is important in the theory of PDEs, and particularly in Sobolev spaces. Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact by Alaoglu's theorem. Thus the theorem implies that bounded subsets are weakly sequentially precompact, and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space. Since many PDEs only have solutions in the weak sense, this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE.

In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk. This theorem was thought to be proven by , but found a gap in the proof. The status of Dehn's lemma remained in doubt until using work by Johansson (1938) proved it using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem.

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.

The theorem is named after German mathematician Herbert Grötzsch, who published its proof in 1959. Grötzsch's original proof was complex. attempted to simplify it but his proof was erroneous.. In 2003, Carsten Thomassen derived an alternative proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free planar graphs that are not 3-list-colorable.. In 1989, Richard Steinberg and Dan Younger gave the first correct proof, in English, of the dual version of this theorem.

A tennis ball In geometry, the tennis ball theorem states that any smooth curve on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four inflection points, points at which the curve does not consistently bend to only one side of its tangent line. The tennis ball theorem was first published under this name by Vladimir Arnold in 1994, and is often attributed to Arnold, but a closely related result appears earlier in a 1968 paper by Beniamino Segre, and the tennis ball theorem itself is a special case of a theorem in a 1977 paper by Joel L. Weiner. The name of the theorem comes from the standard shape of a tennis ball, whose seam forms a curve that meets the conditions of the theorem; the same kind of curve is also used for the seams on baseballs.

However, since there are loops such that \gamma([0,1])=S^2 (constructed from the Peano curve, for example), a complete proof requires more careful analysis with tools from algebraic topology, such as the Seifert–van Kampen theorem or the cellular approximation theorem.

While somewhat eclipsed by attention to and progress in Resolution theorem provers, Model Elimination has continued to attract the attention of researchers and software developers. Today there are several theorem provers under active development that are based on the Model Elimination procedure.

In mathematical finite group theory, Thompson's original uniqueness theorem states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3\. gave a shorter proof of the uniqueness theorem.

Most important works of Bloch belong to complex analysis. His early contribution is known as Bloch's theorem. This theorem asserts the existence of certain absolute constant which is called the Bloch constant. The exact value of the Bloch constant is still unknown .

In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.

Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198. Theorem: Let M be a compact complex manifold.

The duality theorem has an economic interpretation. If we interpret the primal LP as a classical "resource allocation" problem, its dual LP can be interpreted as a "resource valuation" problem. See also Shadow price. The duality theorem has a physical interpretation too.

Amongst other fields, financial mathematics uses the theorem extensively, in particular via the Girsanov theorem. Such changes of probability measure are the cornerstone of the rational pricing of derivatives and are used for converting actual probabilities into those of the risk neutral probabilities.

Ernst Zermelo has a theorem (which he calls "Cantor's Theorem") that is identical to the form above in the paper that became the foundation of modern set theory ("Untersuchungen über die Grundlagen der Mengenlehre I"), published in 1908. See Zermelo set theory.

Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem. The main technique involved in making calculations with a characteristic function is recognizing the function as the characteristic function of a particular distribution.

A quote from Ellis: "The core of the reciprocity theorem is the fact that many geometric properties are invariant when the roles of the source and observer in astronomical observations are transposed". This statement is fundamental in the derivation of the reciprocity theorem.

Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables.

In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple groups of characteristic 2 type with e(G) ≥ 4 that have a "standard component", which covers one of the three cases of the trichotomy theorem.

In 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 .

In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane.S. Eilenberg and S. Mac Lane (1953), "Acyclic Models." Amer.

Zernike's work helped awaken interest in coherence theory, the study of partially coherent light sources. In 1938 he published a simpler derivation of Van Cittert's 1934 theorem on the coherence of radiation from distant sources, now known as the Van Cittert–Zernike theorem.

For further details see constructive set theory. Brouwer disavowed his original proof of the fixed- point theorem. The first algorithm to approximate a fixed point was proposed by Herbert Scarf.H. Scarf found the first algorithmic proof: M.I. Voitsekhovskii Brouwer theorem Encyclopaedia of Mathematics .

In differential geometry, Vermeil's theorem essentially states that the scalar curvature is the only (non-trivial) absolute invariant among those of prescribed type suitable for Albert Einstein’s theory of General Relativity. The theorem was proved by the German mathematician Hermann Vermeil in 1917.

In complex analysis, a branch of mathematics, the Hadamard three-lines theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.

In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.

The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.

The Atkinson–Stiglitz theorem is a theorem of public economics which states "that, where the utility function is separable between labor and all commodities, no indirect taxes need be employed" if non-linear income taxation can be used by the government and was developed in a seminal article by Joseph Stiglitz and Anthony Atkinson in 1976. The Atkinson–Stiglitz theorem is generally considered to be one of the most important theoretical results in public economics and spawned a broad literature which delimited the conditions under which the theorem holds, e.g. Saez (2002) which showed that the Atkinson–Stiglitz theorem does not hold if households have heterogeneous rather than homogeneous preferences. In practice the Atkinson–Stiglitz theorem has often been invoked in the debate on optimal capital income taxation: Because capital income taxation can be interpreted as the taxation of future consumption in excess of the taxation of present consumption, the theorem implies that governments should abstain from capital income taxation if non- linear income taxation is an option since capital income taxation would not improve equity by comparison to the non-linear income tax, while additionally distorting savings.

879–898 which is a no-go theorem that relates classical mechanics to quantum mechanics.

He is also one of the namesakes of the Tait-Kneser theorem on osculating circles.

Both orders divide 8, as predicted by Lagrange's theorem. The groups Fp× above have order .

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.

Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.

Theorem 2 establishes a similar, but "more subtle", NFL result for time-varying objective functions.

The theorem was published in 1957 by H. J. Ryser and also by David Gale.

In this section, we will discuss the lamellar vector field based on Kelvin–Stokes theorem.

The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

Claiborne Green Latimer (1893–1960) was an American mathematician, known for the Latimer–MacDuffee theorem.

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.

The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.

It is sometimes called the Lax–Richtmyer theorem, after Peter Lax and Robert D. Richtmyer.

The Rauch comparison theorem is also named after Harry Rauch. He proved it in 1951.

Theorem 5 (Svensson): if all Pareto-optimal allocations are sigma-optimal, then PEEF allocations exist.

This is an example of the use of Budan's theorem as a zero-root test.

II. A Kolchin type theorem. Annals of Mathematics (2), vol. 161 (2005), no. 1, pp.

The theorem can be used to generalise the Stolarsky mean to more than two variables.

So the index theorem can be proved by checking it on these particularly simple cases.

A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al.

For example, the Cayley–Hamilton theorem says that every matrix satisfies its own characteristic polynomial.

The continuous function in this theorem is not required to be bijective or even surjective.

These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function.

A blue neon sign showing the simple statement of Bayes's theorem In probability theory and statistics, Bayes's theorem (alternatively Bayes's law or Bayes's rule), named after Reverend Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes's theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on his age) than simply assuming that the individual is typical of the population as a whole. One of the many applications of Bayes's theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes's theorem may have different probability interpretations.

In quantum mechanics, specifically time-dependent density functional theory, the Runge-Gross theorem (RG theorem) shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential (or potentials) in which the system evolves and the density (or densities) of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the electronic density, ρ(r,t) changes in response to an external scalar potential, v(r,t), such as a time-varying electric field. The Runge-Gross theorem provides the formal foundation of time-dependent density functional theory.

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region. The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.

Logically, many theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not assert B—only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the theorem. Alternatively, A and B can be also termed the antecedent and the consequent, respectively. The theorem "If n is an even natural number, then n/2 is a natural number" is a typical example in which the hypothesis is "n is an even natural number", and the conclusion is "n/2 is also a natural number". In order for a theorem be proved, it must be in principle expressible as a precise, formal statement.

In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in quantum mechanics, quantum entanglement is an effect by which certain widely separated events can be correlated in ways that suggest the possibility of instantaneous communication. The no-communication theorem gives conditions under which such 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 violations of local realism obtained in tests of Bell's theorem.

In theoretical physics, the Vafa–Witten theorem, named after Cumrun Vafa and Edward Witten, is a theorem that shows that vector-like global symmetries (those that transform as expected under reflections) such as isospin and baryon number in vector-like gauge theories like quantum chromodynamics cannot be spontaneously broken as long as the theta angle is zero. This theorem can be proved by showing the exponential fall off of the propagator of fermions.

The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed for that any formal system S powerful enough to represent arithmetic, there is a theorem G which is true but the system is unable to prove. G could be added as an additional axiom to the system in place of a proof. However this would create a new system S' with its own unprovable true theorem G', and so on.

The Bondareva–Shapley theorem, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva–Shapley theorem implies that market games and convex games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.

Given Henkin's theorem, the completeness theorem can be proved as follows: If T \models s, then T\cup\lnot s does not have models. By the contrapositive of Henkin's theorem, then T\cup\lnot s is syntactically inconsistent. So a contradiction (\bot) is provable from T\cup\lnot s in the deductive system. Hence (T\cup\lnot s) \vdash \bot, and then by the properties of the deductive system, T\vdash s.

E is a high performance theorem prover for full first-order logic with equality. It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated into other theorem provers and it has been among the best-placed systems in several theorem proving competitions. E is developed by Stephan Schulz, originally in the Automated Reasoning Group at TU Munich, now at Baden-Württemberg Cooperative State University Stuttgart.

In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano-Weierstrass theorem cannot be applied. Ekeland's principle relies on the completeness of the metric space. Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.

In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. The Beck–Fiala theorem is concerned with the case where each element doesn't appear many times across all sets.

Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p. In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect.Dickson, p. 545.

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field,, in the form of a partial differential equation developed by British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution (i.e. an electrically charged object), through energy flux.

In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem deals with the existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the individual events may be independent or dependent with respect to each other. Thus, the statement goes beyond the mere existence of countable product measures. The theorem was proved by Cassius Ionescu-Tulcea in 1949.

Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed- point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued functions). See Competitive equilibrium#Existence of a competitive equilibrium. The proof was first due to Lionel McKenzie, and Kenneth Arrow and Gérard Debreu. In fact, the converse also holds, according to Uzawa's derivation of Brouwer's fixed point theorem from Walras's law.

The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/f noise, based its central limit-like effect. Much as the central limit theorem requires certain kinds of random processes to have as a focus of their convergence the Gaussian distribution and thus express white noise, the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/f noise.

The previous theorem has further interesting consequences other than the aforementioned generalization of Commandino's theorem. It can be used to prove the following theorem about the centroid of a tetrahedron, first described in the Mathematische Unterhaltungen by the German physicist Friedrich Eduard Reusch:Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen. Zweites Heft. 1973, S. 100, 128In den Mathematische Unterhaltungen (Zweites Heft, S. 128) wird auf die S. 36 von Reuschs Abhandlung Der Spitzbogen verwiesen.

In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy excision theorem.

Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case (Kawasaki's Riemann–Roch). The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for quotient stacks by finite groups. One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class.

Friedgut's sharp threshold theorem states, roughly speaking, that a monotone graph property (a graph property is a property which doesn't depend on the names of the vertices) has a sharp threshold unless it is correlated with the appearance of small subgraphs. This theorem has been widely applied to analyze random graphs and percolation. On a related note, the KKL theorem implies that the width of threshold window is always at most O(1/\log n).

The Grothendieck–Riemann–Roch theorem says that these are equal. When Y is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem. The group K(X) is now known as K0(X). Upon replacing vector bundles by projective modules, K0 also became defined for non-commutative rings, where it had applications to group representations.

Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency. Moreover, Weller's theorem says that there exists a price such that the allocation and the price are a competitive equilibrium (CE) with equal incomes (EI).

Green and Tao's proof has three main components: # Szemerédi's theorem, which asserts that subsets of the integers with positive upper density have arbitrarily long arithmetic progressions. It does not a priori apply to the primes because the primes have density zero in the integers. # A transference principle that extends Szemerédi's theorem to subsets of the integers which are pseudorandom in a suitable sense. Such a result is now called a relative Szemerédi theorem.

Teorema means theorem in Italian. Its Greek root is theorema (θεώρημα), meaning simultaneously "spectacle", "intuition", and "theorem". Viano suggests that the film should be considered as "spectatorship" because each family member gazes at the guest and his loins , although this seems unlikely: the Greek word denotes the object of spectatorship, rather than the actual act of spectatorship, which would be theoresis (θεώρησις). As a term, theorem is also often considered as mathematical or formulaic.

The Erdős–Szekeres theorem is an easy consequence of this statement. Kőnig's theorem in graph theory states that a minimum vertex cover in a bipartite graph corresponds to a maximum matching, and vice versa; it can be interpreted as the perfection of the complements of bipartite graphs. Another theorem about bipartite graphs, that their chromatic index equals their maximum degree, is equivalent to the perfection of the line graphs of bipartite graphs.

In these works, Wigner laid the foundation for the theory of symmetries in quantum mechanics. Wigner's theorem proved by Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT symmetry are represented on the Hilbert space of states. According to the theorem, any symmetry transformation is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space.

In mathematics, what is proven is not the truth of a particular theorem, but that the axioms of the system imply the theorem. In other words, it is impossible for the axioms to be true and the theorem to be false. The strength of deductive systems is that they are sure of their results. The weakness is that they are abstract constructs which are, unfortunately, one step removed from the physical world.

The theorem provides conditions on the lattice under which perfect reconstruction is possible. As with the Nyquist–Shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world functions and sampling techniques, albeit in practice often a very good one.

A 70 (1965), 428--439. His research in functional analysis introduced new concepts, such as F-ordered rings (now known as Ghika rings), which have the property that in any module over a ring in this class, the analogue of the Hahn–Banach theorem holds. He is also remembered for a representation theorem for reflexive Banach spaces, now known as the Ghika-James representation theorem. Among his doctoral students were Silviu Teleman and Petru Mocanu.

In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a mixed state, it can be purified. Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem.

Szemerédi's theorem resolved the original conjecture and generalized Roth's theorem to arithmetic progressions of arbitrary length. Since then it has been extended in multiple fashions to create new and interesting results. Furstenberg and Katznelson used ergodic theory to prove a multidimensional version and Leibman and Bergelson extended it to polynomial progressions as well. Most recently Green and Tao proved the Green-Tao Theorem which says that the prime numbers contain arbitrarily long arithmetic progressions.

Kenneth Alan "Ken" Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's Last Theorem, as well as for his service as President of the American Mathematical Society from 2017 to 2019. He is currently a professor of mathematics at the University of California, Berkeley.

In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The theorem may be deduced from the first main theorem of Turán's method.

In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982. The theorem states that if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.

The result can be considered as a variation of Gabrielov's theorem. This earlier theorem, by Andrei Gabrielov, dealt with sub-analytic sets, or the language Lan of ordered rings with a function symbol for each proper analytic function on Rm restricted to the closed unit cube [0,1]m. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above.A. Gabrielov, Projections of semi-analytic sets, Functional Anal. Appl.

The Edmonds-Gallai decomposition theorem, which was proved independently by Gallai and Jack Edmonds, describes finite graphs from the point of view of matchings. Gallai also proved, with Milgram, Dilworth's theorem in 1947, but as they hesitated to publish the result, Dilworth independently discovered and published it.P. Erdős: In memory of Tibor Gallai, Combinatorica, 12(1992), 373-374. Gallai was the first to prove the higher-dimensional version of van der Waerden's theorem.

Pascal's original note has no proof, but there are various modern proofs of the theorem. It is sufficient to prove the theorem when the conic is a circle, because any (non-degenerate) conic can be reduced to a circle by a projective transformation. This was realised by Pascal, whose first lemma states the theorem for a circle. His second lemma states that what is true in one plane remains true upon projection to another plane.

This therefore means that , where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio. Therefore, are collinear. Another proof for Pascal's theorem for a circle uses Menelaus' theorem repeatedly. Dandelin, the geometer who discovered the celebrated Dandelin spheres, came up with a beautiful proof using "3D lifting" technique that is analogous to the 3D proof of Desargues' theorem.

Hart's most important contribution was contained in his paper Extension of Terquem's theorem respecting the circle which bisects three sides of a triangle (1861). Hart wrote this paper after an carrying out an investigation suggested by William Rowan Hamilton in a letter to Hart. In addition, Hart corresponded with George Salmon on the same topic. This paper contains the result which became known as Hart's Theorem, which is a generalisation of Feuerbach's Theorem.

In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.

Supersymmetry may be considered a possible "loophole" of the theorem because it contains additional generators (supercharges) that are not scalars but rather spinors. This loophole is possible because supersymmetry is a Lie superalgebra, not a Lie algebra. The corresponding theorem for supersymmetric theories with a mass gap is the Haag–Łopuszański–Sohnius theorem. Quantum group symmetry, present in some two- dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.

In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with c colors, the same is true for the whole graph. The theorem was proved by , after whom it is named. The De Bruijn–Erdős theorem has several different proofs, all depending in some way on the axiom of choice.

So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so pq = qp. Equivalently, X, Y, Z are collinear. The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem.