* Kolmogorov extension theorem

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 ).

For this purpose one traditionally uses a method called Kolmogorov extension.

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions exists, then it can be used to specify the joint probability distribution of finite-dimensional random variables.

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.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions.

* Kolmogorov extension theorem

A far reaching extension of the Gold ’ s approach is developed by Schmidhuber's theory of generalized Kolmogorov complexities, which are kinds of super-recursive algorithms.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

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.

Andrey Kolmogorov later independently published this theorem in Problems Inform.

Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information.

The Kolmogorov – Arnold – Moser ( KAM ) theorem gives the behavior near an elliptic point.

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.

* Kolmogorov – Arnold – Moser theorem

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.

The Kolmogorov continuity theorem guarantees that processes that satisfy certain constraints on the moments of their increments have continuous modifications and are therefore separable.

The Wiener – Khinchin theorem, ( or Wiener – Khintchine theorem or Khinchin – Kolmogorov theorem ), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

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

* Kolmogorov continuity theorem

* Hahn – Kolmogorov theorem

# REDIRECT Kolmogorov – Arnold – Moser theorem

Developed by Andrey Kolmogorov, Vladimir Arnold and Jürgen Moser, this theorem stated the conditions under which a system of partial differential equations will have only mildly chaotic behaviour under small perturbations.

In a 1938 paper, Kolmogorov " established the basic theorems for smoothing and predicting stationary stochastic processes " — a paper that would have major military applications during the Cold War.

In his study of stochastic processes ( random processes ), especially Markov processes, Kolmogorov and the British mathematician Sydney Chapman independently developed the pivotal set of equations in the field, which have been give the name of the Chapman – Kolmogorov equations.

In his study of Markovian stochastic processes and their generalizations, Chapman and the Russian Andrey Kolmogorov independently developed the pivotal set of equations in the field, the Chapman – Kolmogorov equations.

* Chapman – Kolmogorov equation, a mathematical identity relating the joint probability distributions of different sets of coordinates on a stochastic process

" Cramér mentions later work by Andrey Kolmogorov and William Feller but it was Cramér himself who developed Lundberg's ideas on risk and linked them to the emerging theory of stochastic processes.

** Kolmogorov continuity theorem on stochastic processes

In the statistical community the same technique is also known as Gaussian process regression, Kolmogorov Wiener prediction, or best linear unbiased prediction.

In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman – Kolmogorov equation.

Kolmogorov complexity is also known as " descriptive complexity " ( not to be confused with descriptive complexity theory ), Kolmogorov – Chaitin complexity, algorithmic entropy, or program-size complexity.

The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence, while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal a priori probability distribution.

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 ).

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 ).

The " trouble " with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space.

While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby partially solving Hilbert's thirteenth problem.

* Lenin Prize ( 1965, with Andrey Kolmogorov )

The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s.

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

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 ( together with Aleksandr Khinchin ) became interested in probability theory.

* Interview with Kolmogorov ( 1958, Moscow ), Eugene Dynkin Collection of Mathematics Interviews, Cornell University Library.

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

with independent of the scale r. From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the velocity increments ( known as structure functions in turbulence ) should scale as

For low orders the discrepancy with the Kolmogorov n / 3 value is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments.

However, for high order structure functions the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear.

Kolmogorov supports most of Brouwer's results but disputes a few ; he discusses the ramifications of intuitionism with respect to " transfinite judgements ", e. g. transfinite induction.