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Theorem and Richard
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.
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.
, 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.
* Richard J. Lipton, Savitch ’ s Theorem.
His Ph. D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic " The Bridge Theorem For Minimal Surfaces.

Theorem and Let
* 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 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: 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 and T
In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.
By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.
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.
Since the theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others, it's also known as the Nyquist – Shannon – Kotelnikov, Whittaker – Shannon – Kotelnikov, Whittaker – Nyquist – Kotelnikov – Shannon, WKS, as well as the Cardinal Theorem of Interpolation Theory.
Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:
By the First Fundamental Theorem, 0 ≤ δ ( a, f ) ≤ 1, if T ( r, f ) tends to infinity ( which is always the case for non-constant functions meromorphic in the plane ).
# The Functional Indeterminacy Theorem ( F. I. T.
# The Fundamental Failure-Mode Theorem ( F. F. T.
: Theorem: Assume T is a bounded linear operator from L < sup > p </ sup > to L < sup > p </ sup > and at the same time from L < sup > q </ sup > to L < sup > q </ sup >.
Theorem: Assume T is a bounded linear operator from to and at the same time from to.
Theorem For a bounded operator T, σ < sub > r </ sub >( T ) ⊂ σ < sub > p </ sub >( T *) ⊂ σ < sub > r </ sub >( T ) ∪ σ < sub > p </ sub >( T ).
* T. R. Johansen, The Bochner-Minlos Theorem for nuclear spaces and an abstract white noise space, 2003.

Theorem and M
For example Newton wrote to Oldenberg in 1676 " I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag.
In its general form, the Löwenheim – Skolem Theorem states that for every signature σ, every infinite σ-structure M and every infinite cardinal number κ ≥ | σ | there is a σ-structure N such that | N |
As with the two-dimensional Gauss – Bonnet Theorem, there are generalizations when M is a manifold with boundary.
* Harold M. Edwards, Fermat's Last Theorem.
* The integral of the Gaussian curvature K of a 2-dimensional Riemannian manifold ( M, g ) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.
* Lovell, M., 2008, A Simple Proof of the FWL ( Frisch, Waugh, Lovell ) Theorem, Journal of Economic Education.

Theorem and N
so the First Fundamental Theorem says that the sum N ( r, a, f ) + m ( r, a, f ), tends to infinity at the rate which is independent of a.
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 ).
Forsythe directed the Frankfurt Ballet ( Ballett Frankfurt ) from 1984 until 2004, choreographing such seminal pieces such as Artifact ( 1984 ), Die Befragung des Robert Scott ( 1986 ) Impressing the Czar ( 1988 ), Limb ’ s Theorem ( 1990 ), The Loss of Small Detail ( 1991 ), ALIE /< u > N A ( C ) TION </ u > ( 1992 ), Eidos: Telos ( 1995 ), Endless House ( 1999 ), Kammer / Kammer ( 2000 ), and Decreation ( 2003 ).
: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G / N is simple.
The Cournot Theorem then states that, in absence of fixed costs of production, as the number of firms in the market, N, goes to infinity, market output, Nq, goes to the competitive level and the price converges to marginal cost.

Theorem and be
: 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 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.

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