Том 25, № 1 (2023)

This paper is devoted to a class of self-affine sets on the plane determined by six homotheties. Centers of these homotheties are located at the vertices of a regular hexagon P, and the homothetic coefficients belong to the interval (0, 1). One must note that equality of homothetic coefficients is not assumed. A self-affine set on the plane is a non-empty compact subset that is invariant with respect to the considered family of homotheties. The existence and uniqueness of such a set is provided by Hutchinson’s theorem. The goal of present work is to investigate the influence of homothetic coefficients on the properties of a self-affine set. To describe the set, barycentric coordinates on the plane are introduced. The conditions are found under which the self-affine set is: a) the hexagon P; b) a Cantor set in the hexagon P. The Minkowski and the Hausdorff dimensions of the indicated sets are calculated. The conditions providing vanishing Lebesgue measure of self-affine set are obtained. Examples of self-affine sets from the considered class are presented.

In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

The aim of this work is to construct direct and iterative numerical methods for solving functional equations with hereditary components. Such equations are a convenient tool for modeling dynamical systems. In particular, they are used in population models structured by age with a finite life span. Models based on integro-differential and integral equations with various kinds of delay arguments are considered. For nonlinear equations, the operators are linearized according to the modified Newton-Kantorovich scheme. Direct quadrature and simple iteration methods are used to discretize linear equations. These methods are constructed in the paper: an iterative method for solving a nonlinear integro-differential equation on the semiaxis  (-∞,0)(−∞,0]">, a direct method for solving the signal recovery problem, and iterative methods for solving a nonlinear Volterra integral equation with a constant delay. Special quadrature formulas based on orthogonal Lagger polynomials are used to approximate improper integrals on the semiaxis. The results of numerical experiments confirm the convergence of suggested methods. The proposed approaches can also be applied to other classes of nonlinear equations with delays.