* Theorem Let X be a normed space.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

Theorem: Let V be a topological vector space

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,

* Brown, R., Higgins, P. J. and Sivera, R .. 2011, EMS Tracts in Mathematics Vol. 15 ( 2011 ) Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids ; ( The first of three Parts discusses the applications of the 1-and 2-dimensional versions of the Seifert-van Kampen Theorem.

Theorem: If R is a filtered ring whose associated graded ring gr ( R ) is a domain, then R itself is a domain.

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 >.

Gallager's January 1965 paper in the IEEE Transactions on Information Theory, " A Simple Derivation of the Coding Theorem and some Applications, won the 1966 IEEE W. R. G.

* Kahneman, D., Knetsch, J. L., Thaler, R. H. " Experimental Tests of the Endowment Effect and the Coase Theorem " ( 1990 ) Journal of Political Economy, 98 ( 6 ), 1325-1348.

* Although Sard's Theorem does not hold in general, every continuous map f: X → R < sup > n </ sup > from a Hilbert manifold can be arbitrary closely approximated by a smooth map g: X → R < sup > n </ sup > which has no critical points

* T. R. Johansen, The Bochner-Minlos Theorem for nuclear spaces and an abstract white noise space, 2003.

: 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.

This result is now known as Church's Theorem or the Church – Turing Theorem ( not to be confused with the Church – Turing thesis ).

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 ).

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

It was then simplified in 1947, when Leon Henkin observed in his Ph. D. thesis that the hard part of the proof can be presented as the Model Existence Theorem ( published in 1949 ).

The Model Existence Theorem and its proof can somehow be formalized in the framework of PA.

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.

In the General Possibility Theorem, Kenneth Arrow argues that if a legislative consensus can be reached through a simple majority, then minimum conditions must be satisfied, and these conditions must provide a superior ranking to any subset of alternative votes ( Arrow 1963 ).

~ p ∨ p. Since p → p is true ( this is Theorem 2. 08, which is proved separately ), then ~ p ∨ p must be true.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

This theorem was established by John von Neumann, who is quoted as saying " As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved ".

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.

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.

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.

On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.

Where the angle is a right angle, also known as the Hypotenuse-Leg ( HL ) postulate or the Right-angle-Hypotenuse-Side ( RHS ) condition, the third side can be calculated using the Pythagoras ' Theorem thus allowing the SSS postulate to be applied.

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 König-Egerváry Theorem.

Theorem: The angle may be trisected if and only if is reducible over the field extension Q.

At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents using what we now recognize as the fact that the ring is a Dedekind domain.

Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the Fundamental Theorem of Arithmetic: every Euclidean domain is a unique factorization domain.

Theorem ( Modified Schwarz inequality for 2-positive maps ) For a 2-positive map Φ between C *- algebras, for all a, b in its domain,

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.

Problem II. 8 in the Arithmetica ( edition of 1670 ), annotated with Fermat's comment which became Fermat's Last Theorem.

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.

Germain's best work was in number theory, and her most significant contribution to number theory dealt with 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 ).

* A space elevator is also constructed in the course of Clarke's final novel ( co-written with Frederik Pohl ), The Last Theorem.

:: Boone-Rogers Theorem: There is no uniform partial algorithm which solves the word problem in all finitely presented groups with solvable word problem.

Strangely, this approach is often used for cases where Theorem I applies, which creates problems with the basic model assumptions.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

Remembrance of the story of the bull's hide and the foundation of Carthage is preserved in mathematics in connection with the Isoperimetric problem which is sometimes called Dido's Problem ( and similarly the Isoperimetric theorem is sometimes called Dido's Theorem ).

, with some help from Richard Taylor, proved the Taniyama – Shimura – Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

* Zenkov, DV, AM Bloch, and JE Marsden The Lyapunov-Malkin Theorem and Stabilization of the Unicycle with Rider.

Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization.

The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7. 4 in the preprint ' Ricci Flow with surgery on three-manifolds '.

Under Desargues ' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically.

Gentzen's so-called " Main Theorem " ( Hauptsatz ) about LK and LJ was the cut-elimination theorem, a result with far-reaching meta-theoretic consequences, including consistency.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble ( i. e., the total or convective time derivative is zero ).

Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:

Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.