See also invariance theorem.

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

Automated theorem proving ( also known as ATP or automated deduction ) is the proving of mathematical theorems by a computer program.

The binomial theorem also holds for two commuting elements of a Banach algebra.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.

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

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

It is also an example of a more generalized version of the central limit theorem that is characteristic of all stable distributions, of which the Cauchy distribution is a special case.

Special cases of the Chinese remainder theorem were also known to Brahmagupta ( 7th century ), and appear in Fibonacci's Liber Abaci ( 1202 ).

Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument ( changing only that the minimal counterexample requires 6 colors ) and use Kempe chains in the degree 5 situation to prove the five color theorem.

The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs ( possibly with an uncountable number of vertices ) for which every finite subgraph is planar.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

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.

" Financial economics ", at least formally, also considers investment under " certainty " ( Fisher separation theorem, " theory of investment value ", Modigliani-Miller theorem ) and hence also contributes to corporate finance theory.

The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel – Ruffini theorem on the algebraic insolubility of quintic equations.

This exclusion of certain notes leads to the statement of the prime number theorem.

The constructive proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem.

and the samples of x ( t ) are denoted by x ( nT ) for integer values of n. The sampling theorem leads to a procedure for reconstructing the original x ( t ) from the samples and states sufficient conditions for such a reconstruction to be exact.

This criterion leads to a proof ( Kelley, 1950 ) of Tychonoff's theorem, which is, word for word, identical to the Cartan / Bourbaki proof using filters, save for the repeated substitution of " universal net " for " ultrafilter base ".

Because of the Mordell – Weil theorem, Faltings ' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell – Lang conjecture, which has been proved.

Combining this equality with König's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.

This leads to the idea of including the deduction theorem as a formal rule in the system, which happens in natural deduction.

This leads up to Brahmagupta's famous theorem,

Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity:

Likewise Isenkrahe's violation of the energy conservation law is unacceptable, and Kelvin's application of Clausius ' theorem leads ( as noted by Kelvin himself ) to some sort of perpetual motion mechanism.

Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed point theorem in quantitative form.

which leads to the generalised least squares version of the Gauss-Markov theorem ( Chiles & Delfiner 1999, p. 159 ):

In signal processing, the Poisson summation formula leads to the Discrete-time Fourier transform and the Nyquist – Shannon sampling theorem.

The Hellinger – Toeplitz theorem leads to some technical difficulties in the mathematical formulation of quantum mechanics.

This leads to the general envelope theorem.

If the poset X is finite then the statement of the theorem has a clear interpretation that leads to the proof.

The cases n ≤ 7 are ruled out by the Brauer-Suzuki theorem, the case n = 8 leads to the McLaughlin group, the case n = 9 was ruled out by Zvonimir Janko, Lyons himself ruled out the case n = 10 and found the Lyons group for n = 11, while the cases n ≥ 12 were ruled out by J. G.

The main idea of the proof of the theorem is to use a counting argument, to show that any failure of local injectivity ( twin patterns ) leads to an orphan pattern, and vice versa.

In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties.