The Kolmogorov theorem gives that the representation of continuous and bounded real-valued functions of n variables by the superposition of functions of one variable and addition is always possible. Based on the fact that each proof of the Kolmogorov theorem or its variants was a constructive one so far, there is the principal possibility to attain such a representation. This paper reviews a pr...

In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then   the ergodic properties of relative  semi-dynamical systems are investigated.  Also,  a relative version of Kolmogorov-Sinai theorem  is given. Finally, it is proved  that the relative entropy of a...

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not א0-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an א1-categorical but not א0-categorical sa...

A general theory of summation of divergent series based on the HardyKolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Ko...

We consider the goodness of fit testing problem for stochastic differential equation with small diffusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We dis...

In this article, we extend the theory of multiplicative chaos for positive definite functions in R of the form f(x) = λ ln T |x| + g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent ...

Kolmogorov superpositions and Hecht-Nielsen's neural network based on them are dimension reducing. This dimension reduction can be understood in terms of space-filling curves that characterize Kolmogorov's functions, and the subject of this paper is the construction of such a curve. We construct a space-filling curve with Lebesgue measure 1 in the unit square, [0,1]2, with approximating curves ...

In this article, we extend the theory of multiplicative chaos for positive definite functions in R of the form f(x) = λ ln T |x| + g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent ...