This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

The latter theorem has been generalized by Yamabe and Yujobo, and Cairns to show that in Af there are families of such cubes.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

** The theorem that every Hilbert space has an orthonormal basis.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

The no-hair theorem states that, once it achieves a stable condition after formation, a black hole has only three independent physical properties: mass, charge, and angular momentum.

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as the Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

This implies, by the Bolzano – Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.

# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information.

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.

There were several early failed attempts at proving the theorem.

First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.

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.

Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

In differential geometry, Stokes ' theorem ( also called the generalized Stokes ' theorem ) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

The Principle of Moments, also known as Varignon's theorem ( not to be confused with the geometrical theorem of the same name ) states that the sum of torques due to several forces applied to a single point is equal to the torque due to the sum ( resultant ) of the forces.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

This work described several now famous results, including Condorcet's jury theorem, which states that if each member of a voting group is more likely than not to make a correct decision, the probability that the highest vote of the group is the correct decision increases as the number of members of the group increases, and Condorcet's paradox, which shows that majority preferences become intransitive with three or more options – it is possible for a certain electorate to express a preference for A over B, a preference for B over C, and a preference for C over A, all from the same set of ballots.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

There are several combinatorial analogs of the Gauss – Bonnet theorem.

The most important among them are Zorn's lemma and the well-ordering theorem.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

It was introduced in 1971 by Stephen Cook in his seminal paper " The complexity of theorem proving procedures " and is considered by many to be the most important open problem in the field.

This led in 1828 to an important theorem, the Theorema Egregium ( remarkable theorem ), establishing an important property of the notion of curvature.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

Notably he was the first to use discharging for proving the theorem, which turned out to be important in the unavoidability portion of the subsequent Appel-Haken proof.

His concept " Dyson's transform " led to one of the most important lemmas of Olivier Ramaré's theorem that every even integer can be written as a sum of no more than six primes.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem.

Some of the more important results in the study of monoids are the Krohn-Rhodes theorem and the star height problem.

An important step in the evolution of classical model theory occurred with the birth of stability theory ( through Morley's theorem on uncountably categorical theories and Shelah's classification program ), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories.

Abel had sent most of his work to Berlin to be published in Crelles Journal, but he had saved what he regarded his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials.