Kleene's Church – Turing Thesis: A few years later ( 1952 ) Kleene would overtly name, defend, and express the two " theses " and then " identify " them ( show equivalence ) by use of his Theorem XXX:

: 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.

* James ` s Theorem For a Banach space the following two properties are equivalent:

Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called “ Sophie Germain's Theorem ”:

In 1870, Rudolf Clausius delivered the lecture " On a Mechanical Theorem Applicable to Heat " to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20 year study of thermodynamics.

General Impossibility Theorem: It is impossible to formulate a social preference ordering that satisfies all of the following conditions:

The Myhill-Nerode Theorem for tree automaton states that the following three statements are equivalent:

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 ).

A. Karatsuba proved the following two theorems, which completely solved Moore's problem on the improvement of the bounds of the experiment length of his Theorem 8.

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:

h-cobordism Theorem: Suppose the following holds:

The following Theorem is due to Wassily Hoeffding and hence is called Chernoff-Hoeffding theorem.

The term " Helmholtz Theorem " can also refer to the following.

In addition to the above characterizations of Pappus's Theorem and its dual, the following are equivalent statements:

In a September 1904 lecture in St. Louis named The Principles of Mathematical Physics, Poincaré draw some consequences from Lorentz's theory and defined ( in modification of Galileo's Relativity Principle and Lorentz's Theorem of Corresponding States ) the following principle: " The Principle of Relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.

A generalized form of Zariski Main Theorem is the following :< ref > EGA IV < sub > 3 </ sub >, Théorème 8. 12. 6 .</ ref > Suppose Y is quasi-compact and quasi-separated.

The following Theorem is a corollary of the above Proposition.

An example is the following result ( Fröhlich and Taylor, Chapter V, Theorem 1. 25 ).

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.

The Gauss – Bonnet Theorem can be seen as a special instance in the theory of characteristic classes.

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

Kummer referred to his own partial proof of Fermat's Last Theorem for regular primes as " a curiosity of number theory rather than a major item " and to the higher reciprocity law ( which he stated as a conjecture ) as " the principal subject and the pinnacle of contemporary number theory.

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

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 principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the ' Gaussian Product Theorem ', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them.

Theorem 2 establishes a similar, but " more subtle ," NFL result for time-varying objective functions.

The stronger version of Montel's Theorem ( occasionally referred to as the Fundamental Normality Test ) states that a family of holomorphic functions, all of which omit the same two values, is normal.

The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

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.

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.

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

Therefore both and are inverses of By Theorem 1. 5,

They are expressed using the Reynolds Transport Theorem.

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.

Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that ( q − 1 )< sup >( p − 1 )</ sup > ≡ 1 ( mod p ).

Pavel Samuilovich Urysohn, Pavel Uryson () ( February 3, 1898, Odessa – August 17, 1924, Batz-sur-Mer ) was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

: Theorem: Armstrong's axioms are sound and complete ; given a header and a set of FDs that only contain subsets of, if and only if holds in all relation universes over in which all FDs in hold.

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.

This classical statement, as well as the classical Divergence theorem and Green's Theorem, are simply special cases of the general formulation stated above.

An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.

The Coase Theorem states that assigning property rights will lead to an optimal solution, regardless of who receives them, if transaction costs are trivial and the number of parties negotiating is limited.

For Theorem I the data are generated in full range, while in Theorem II data is only generated when it surpasses a certain threshold, called Peak Over Threshold models ( POT ).

The Modigliani-Miller Theorem describes conditions under which corporate financing decisions are irrelevant for value, and acts as a benchmark for evaluating the effects of factors outside the model that do affect value.

Other Merton alumni are Bodleian Library founder Thomas Bodley, the Oxford Calculators, Director-General of the BBC Mark Thompson and Sir Andrew Wiles who proved Fermat's Last Theorem.

: Task Completion Theorem: Nevertheless, some tasks are completed, since the intervening presence is itself attempting a task and is, of course, subject to interference.