# Theoretical and applied problems of Physics and Mathematics

# On the Approximate Solution of One Infinite-Dimensional Problem of Optimal Stabilization with Nonautonomous Perturbations in the Coefficients

This paper considers the optimal stabilization problem for solutions of parabolic inclusion in which nonautonomous perturbations act on the differential operator coefficients and multivalue interaction function. Such objects naturally occur in applied problems where medium characteristics change over time, and the interaction functions are discontinuous on a phase variable. Under general conditions on nonautonomous coefficients the solvability of the initial problem was proved.

# Fractal Recursive Function Posses-sing Hyperexponential Growth

This article considers the research methods of deterministic fractal sets. Specifically, it develops constructive methods for studying recursive functions. The obtained results allow consider deterministic fractal functions from the novel theoretical standpoint.

# Convergence of Series Whose Terms are Elements of Weakly Dependent and Strongly Dependent Gaussian Markov Sequences

This paper is devoted to the finding of necessary and sufficient conditions for the almost sure convergence of the random series whose terms are elements of one-dimensional zero-mean Gaussian Markov sequences. Mainly in the paper the series whose terms are elements of weakly dependent and strongly dependent Gaussian Markov sequences are considered. Criterions which provide almost sure convergence of the series whose terms are elements of weakly dependent and strongly dependent Gaussian Markov sequences respectively are the main results of this work.

# An Analysis of the Anticipatory Logistic Equation with the Strong Anticipation

This article investigates the logistic equation with the first order strong anticipation, studies the stability scopes of its fixed points in parameter space and the sufficient condition for existing of the accumulating hyperincursion behaviors. We consider the anticipatory system constructed by the multivalue evolution operator with two selectors. We use the dynamic system iterating tools with the multivalue operators, Lamerey diagrams. On the basis of Lamerey diagrams we describe main types of hyperincursion. We employ basic concepts of the discrete anticipatory system theory of such type.

# Integration of the System of Linear Partial Differential Equations of the First Order

In this paper the method for determining the general solution of the system of first-order linear homogeneous partial differential equations is proposed. This method is the generalization of Euler method for defining general solution of linear homogeneous partial differential equation.

# r-Confluent Hypergeometric Functions and Their Applications

The new properties of the r-generalized confluent hypergeometric functions are investigated. The integral representations, the representation by series are constructed. Applications of these functions to the calculation of integrals absent in cash scientific and reference mathematical literature are given.

# The Modification of the Method of Resolving Functions for the Difference-Differential Pursuit’s Games

The subject of investigation is game problems for the object control under the conditions of counteractions. The paper suggests that the object dynamics is described by the system of difference-differential equations. We consider the approach problem with fixed time. In the course of the game the information on the initial function and the prehistory of the evader’s control is used. We suggest the way to solution of problems with fixed time. The game is completed when an integral of some numeral function describing the game course turns into a unit.

# An Anticommutation of Locally Measurable Operators Affiliated with a von Neumann Algebra

The purpose of this paper is to complete the investigation of a q-commutation of two self adjoint locally measurable operators affiliated with an arbitrary von Neumann algebra. Since possible values of the q parameter are q є 1 і -1, the problem reduces to consideration of conditions of commutation (q = 1) and anticommutation (q = −1) of locally measurable operators. The first case has been researched earlier. In this paper, we consider the case An intersection of any two domains of locally measurable operators is a locally measurable subspace.

# Clark–Ocone Type Formulas in the Meixner White Noise Analysis for Non-Dif¬feren¬tiable by Hida Chance Quantities

Clark–Ocone type formulas allow presenting square integrable and differentiable by Hida chance quantities as stochastic integrals of some random processes as well as reconstructing a chance quantity by its Hida derivative. These formulas can be used in the stochastic analysis and in the financial mathematics. This paper aims to significantly extend a class of chance quantities, for which the Clark–Ocone type formulas in the Meixner white noise analysis can be applied.

# The Structure of a Set of Continuous Solutions of Systems of Linear Functional Difference Equations

This paper considers the structure of a set of systems of continuous solutions in a number of cases depending on the hypotheses for the matrices *А*, *В*, number *q* and their properties. Using the methods of the theory of differential and difference equations, we define new conditions for the existence of continuous solutions of these systems of equations. Specifically, we develop the method of their construction and examine their properties.