Theorem

Cauchy's mean value theorem Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem.

It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3. Wilson's theorem Part of Wilson's theorem states that : for every prime p. One may easily prove this theorem by Sylow's third theorem.

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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 furthermore the proof assumed the fundamental theorem of algebra.

A simplified formulation of the central limit theorem under strong mixing is: Billingsley (1995, Theorem 27.4) Theorem.

Dynkin's π-λ theorem This theorem (or the related monotone class theorem ) is an essential tool for proving many results about properties of specific σ-algebras.

Proof of Cauchy's mean value theorem The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.

Relationship to the compactness theorem The completeness theorem and the compactness theorem are two cornerstones of first-order logic.

The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the Nash–Moser theorem and Newton's method with postconditioning.

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.

A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the central limit theorem of statistics.

A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus in a coordinate neighborhood of the manifold.

Although 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.

An analogue of Rice's theorem for recursive sets One can regard Rice's theorem as asserting the impossibility of effectively deciding for any recursively enumerable set whether it has a certain nontrivial property.

Another theorem of this type is the Four color theorem whose computer generated proof is too long for a human to read.

Any of the several well-known axiomatisations will do; we assume without proof all the basic well-known results about our formalism (such as the normal form theorem or the soundness theorem ) that we need.

As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory.

Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras: Theorem.

Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem ), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.