If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

Wiles discovered Fermat's Last Theorem on his way home from school when he was 10 years old.

* Lawrence C. Paulson of the University of Cambridge, work on higher-order logic system, co-developer of the Isabelle Theorem Prover

George Boolos ( 1989 ) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way.

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

Fine, Do Correlations need to be explained ?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin ( University of Notre Dame Press, 1986 ).

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

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.

He was appointed a lecturer in mathematics at Cambridge in 1927, where his 1935 lectures on the Foundations of Mathematics and Gödel's Theorem inspired Alan Turing to embark on his pioneering work on the Entscheidungsproblem ( decision problem ) using a hypothetical computing machine.

While in Copenhagen, Abel did some work on Fermat's Last Theorem.

Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.

Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it très ingénieuse ( See Best Work on Fermat's Last Theorem ).

The first letter, dated 21 November 1804, discussed Gauss ' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem.

She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case ( see Best Work on Fermat's Last Theorem ).

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

On 13 August, 2012, this project was officially announced to be The Zero Theorem, set to start shooting in Bucharest on October 22, produced by Dean Zanuck ( son to the late Richard D. Zanuck who was to originally produce in 2009 ), worldwide sales handled by Voltage Pictures, Toronto and starring Academy Award winner Christoph Waltz in the lead, replacing Billy Bob Thornton who had been attached to the project in 2009.

His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-Completeness.

* PixelMonkeys. org-Artist, Matt Kane's application of the Infinite Monkey Theorem on pixels to create images.

* Theorem Let X be a normed space.

Theorem: Let V be a topological vector space

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

Theorem: Let R be a Dedekind domain.

Theorem: Let A ∈ C < sup > n × n </ sup > be a complex-valued matrix and ρ ( A ) its spectral radius ; then

Theorem ( Fuglede ) Let T and N be bounded operators on a complex Hilbert space with N being normal.

Theorem ( Calvin Richard Putnam ) Let T, M, N be linear operators on a complex Hilbert space, and suppose that M and N are normal and MT

Theorem: Let T be a bounded linear operator from to and at the same time from to.

Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V ( viewed as an F-module with the action of x given by A and extending by linearity ) satisfies the F-module isomorphism

Theorem of Oka: Let M be a complex manifold,

Theorem: K is not a computable function.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

The Second Main Theorem, more difficult than the first one tells that there are relatively few values which the function assumes less often than average.

" The Implicit Function Theorem states that if is defined on an open disk containing, where,, and and are continuous on the disk, then the equation defines as a function of near the point and the derivative of this function is given by ..."

Tautology -- Temporal logic -- Term -- Term logic -- Ternary logic -- Theorem -- Tolerance -- Trilemma -- Truth -- Truth condition -- Truth function -- Truth value -- Type theory

This theorem is called the Second Fundamental Theorem of Nevanlinna Theory, and it allows to give an upper bound for the characteristic function in terms of N ( r, a ).

The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

Theorem For ƒ in L < sup > 1 </ sup >( R < sup > d </ sup >), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L < sup > 1 </ sup >( Λ ) with || Pƒ ||< sub > 1 </ sub > ≤ || ƒ ||< sub > 1 </ sub >.

Theorem: A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

Theorem ( Dini's test ): Assume a function f satisfies at a point t that

However, unlike the value of most other assets, the value of land is largely a function of government spending on services and infrastructure ( a relationship demonstrated by economists in the Henry George Theorem ).

Schwarz's Theorem then provides us with a necessary criterion for the existence of a potential function.

The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin in 1975 and is sometimes known as Voronin's Universality Theorem.

Theorem For any normalized continuous positive definite function f on G ( normalization here means f is 1 at the unit of G ), there exists a unique probability measure on such that