If F is continuous then under the null hypothesis converges to the Kolmogorov distribution, which does not depend on F. This result may also be known as the Kolmogorov theorem ; see Kolmogorov's theorem for disambiguation.

The Kolmogorov – Arnold – Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

When this condition is expressed in terms of probability densities, the result is called the Chapman – Kolmogorov equation.

As a result, the Kolmogorov microscales were named after him.

In statistics, the Rao – Blackwell theorem, sometimes referred to as the Rao – Blackwell – Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

An axiomatic approach to Kolmogorov complexity based on Blum axioms ( Blum 1967 ) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov ( Burgin 1982 ).

Therefore, given any positive integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed length U. The input to the program GenerateComplexString is an integer n. Here, the size of n is measured by the number of bits required to represent n, which is log < sub > 2 </ sub >( n ).

A topological space is Hausdorff if and only if it is both preregular ( i. e. topologically distinguishable points are separated by neighbourhoods ) and Kolmogorov ( i. e. distinct points are topologically distinguishable ).

Both the form of the Kolmogorov – Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Vasilyevich Smirnov.

The goodness-of-fit test or the Kolmogorov – Smirnov test is constructed by using the critical values of the Kolmogorov distribution.

The original breakthrough to this problem was given by Andrey Kolmogorov in 1954.

The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory.

The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov ( 1931 ).

It has its origins in correspondence discussing the mathematics of games of chance between Blaise Pascal and Pierre de Fermat in the seventeenth century, and was formalized and rendered axiomatic as a distinct branch of mathematics by Andrey Kolmogorov in the twentieth century.

This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov.

Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933.

Not being a mathematician, she included in the book memorial articles about his mathematical works by Pavel Alexandrov, Vadim Efremovich, Andrei Kolmogorov, Lazar Lyusternik, and Mark Krasnosel ' skii.

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates are restricted to lie in measurable subsets of.

This paradigm was championed by A. N. Kolmogorov along with contributions Levin and Gregory Chaitin.

Speaking more theoretically, the conditions of regularity and T < sub > 3 </ sub >- ness are related by Kolmogorov quotients.

Thus a regular space encountered in practice can usually be assumed to be T < sub > 3 </ sub >, by replacing the space with its Kolmogorov quotient.

Kolmogorov soon invited him to join his graduate programme at Moscow University, and, maintaining his post at Ivanovo, Maltsev effectively became Kolmogorov's student.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the viscosity () and the rate of energy dissipation ().

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range are universally and uniquely determined by the scale r and the rate of energy dissipation.

* Kolmogorov – Sinai entropy, the rate of information generation by a measure-preserving dynamical system

Under suitable topological restrictions, a suitably " consistent " collection of finite-dimensional distributions can be used to define a stochastic process ( see Kolmogorov extension in the next section ).

This is today known as the Kolmogorov length scale ( see Kolmogorov microscales ).

* Kolmogorov complexity — absolute complexity ( within a constant, depending on the particular choice of Universal Turing Machine ); MML is typically a computable approximation ( see Wallace and Dowe ( 1999a ) below for elaboration )

In 1920 Lotka extended, via Kolmogorov ( see above ), the model to " organic systems " using a plant species and a herbivorous animal species as an example and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics arriving at the equations that we know today.

Such measures can be used in statistical hypothesis testing, e. g. to test for normality of residuals, to test whether two samples are drawn from identical distributions ( see Kolmogorov – Smirnov test ), or whether outcome frequencies follow a specified distribution ( see Pearson's chi-squared test ).

In probability theory, the probability P of some event E, denoted, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after the famous Russian mathematician Andrey Kolmogorov, which are described below.

For finite random sequences, Kolmogorov defined the " randomness " as the entropy, i. e. Kolmogorov complexity, of a string of length K of zeros and ones as the closeness of its entropy to K, i. e. if the complexity of the string is close to K it is very random and if the complexity is far below K, it is not so random.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

Algorithmic information theory is the area of computer science that studies Kolmogorov complexity and other complexity measures on strings ( or other data structures ).

* Andrey Kolmogorov ( 1903 – 1987 ) co-developed the Wiener – Kolmogorov filter ( 1941 ).

In statistics, the Kolmogorov – Smirnov test ( K – S test ) is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution ( one-sample K – S test ), or to compare two samples ( two-sample K – S test ).

It follows that if a topological group is T < sub > 0 </ sub > ( Kolmogorov ) then it is already T < sub > 2 </ sub > ( Hausdorff ), even T < sub > 3½ </ sub > ( Tychonoff ).

During the same period ( 1921 — 22 ), Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series ( trigonometric series ).

Kolmogorov works on his talk ( Tallinn, 1973 ).

In classical mechanics, he is best known for the Kolmogorov – Arnold – Moser theorem ( first presented in 1954 at the International Congress of Mathematicians ).

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds Numbers, the small scale turbulent motions are statistically isotropic ( i. e. no preferential spatial direction could be discerned ).

These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow ( i. e. ).

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov and Khinchin ( who finally provided a complete proof of the LLN for arbitrary random variables ).

The relation between Strict MML ( SMML ) and Kolmogorov complexity is outlined in Wallace and Dowe ( 1999a ).

In probability theory, the Borel – Kolmogorov paradox ( sometimes known as Borel's paradox ) is a paradox relating to conditional probability with respect to an event of probability zero ( also known as a null set ).

Noted recipients are Pope John XXIII ( 1962 ), Andrey Kolmogorov ( 1962 ), Paul Hindemith ( 1962 ), Jean Piaget ( 1979 ), Jorge Luis Borges ( 1980 ), Edward Shils ( 1983 ), Jan Hendrik Oort ( 1984 ), Otto E. Neugebauer ( 1986 ), Emmanuel Levinas ( 1989 ), Paul Ricoeur ( 1999 ), Abdul Sattar Edhi ( 2000 ), Eric Hobsbawm ( 2003 ), Bruce A. Beutler ( 2007 ), and Carlo Ginzburg ( 2010 ).

A directed complete partial order ( dcpo ) with the Scott topology is always a Kolmogorov space ( i. e., it satisfies the T < sub > 0 </ sub > separation axiom ).