Authors:Yuliya S. Mishura, Olha M. Hopkalo, Hanna S. Zhelezniak Pages: 11 - 19 Abstract: The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided. Pages of the article in the issue: 11 - 19 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.1

Authors:O. Yu. Masyutka, I. I. Golichenko, M. P. Moklyachuk Pages: 20 - 33 Abstract: The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary process from observations of the process with missings is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the process is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the process is not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities. Pages of the article in the issue: 20 - 33 Language of the article: English PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.2

Authors:Victoria O. Voloshyna Pages: 34 - 37 Abstract: Bernstein inequality made it possible to obtain a constructive characterization of the approximation of periodic functions by trigonometric polynomials T_n of degree n. Instead, the corollary of this inequality for algebraic polynomials P_n of degree n, namely, the inequality $ φ P_n' ⩽ n P_n $, where $∥ · ∥ := ∥ · ∥_[−1,1]$ and $φ(x) := \sqrt{1-x^2}$, does not solve the problem obtaining a constructive characterization of the approximation of continuous functions on a segment by algebraic polynomials. Markov inequality $ P_n' ⩽ n^2 P_n $ does not solve this problem as well. Moreover, even the corollary $ φ_n P_n' ⩽ 2n P_n $, where $φ_n(x) := \sqrt{1-x^2+1/n^2}$ of Bernstein and Markov inequalities is not enough. This problem, like a number of other theoretical and practical problems, is solved by Dzyadyk inequality $ P_n' φ_n^{1-k} ⩽ c(s) n P_n φ_n^{-s} ,$ valid for each s ∈ R. In contrast to the Bernstein and Markov inequalities, the exact constant in the Dzyadyk inequality is unknown for all s ∈ R, whereas the asymptotically exact constant for natural s is known: c(s) = 1 + s + s^2; and for n ⩾ 2s, s ∈ N, even the exact constant is known. In our note, this result is extended to the case s ⩽ n < 2s. Pages of the article in the issue: 34 - 37 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.3

Authors:L. V. Batyuk, Natalya M. Kizilova Pages: 40 - 43 Abstract: Rheological properties of the red blood cells (RBC) determine their movement in the larger and smaller blood vessels, oxygen and carbon dioxide delivery to/from the cells. Those properties vary significantly with age and health state of an organism. In this paper a new rheological model of RBC as a thin multilayer shell, which includes the cytoskeleton, lipid bilayer, glycocalyx, and hydrate shell as Maxwell's viscoelastic bodies is proposed. Mechanical properties of the rheological model in isotonic, isometric and dynamic experiments are studied. The oscillations of the surfaces of erythrocytes or other cells in the approximation of multilayer viscoelastic shell filled with a viscous fluid are investigated. The expressions for the dynamic Young’s modules and viscosity/fluidity coefficients as functions of the viscoelastic and geometric parameters of the layers are obtained. The problem of propagation of small perturbations along the cell surface is considered. The solutions of the problem in the form of Young and Lamé waves are obtained. The method of identification of the erythrocyte parameters from the experimental measurements of the wave propagation on the basis of the developed mathematical model for the purposes of clinical diagnostics of diseases with use of a microdrop of blood of the patient is proposed. Pages of the article in the issue: 40 - 43 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.4

Authors:I. V. Vovk, Volodymyr S. Malyuga, V. Yu. Duhnovsky Pages: 44 - 49 Abstract: The problem of generation of self-sustained oscillations in the flow past a circular cylinder with a splitter plate is solved numerically. We investigate both the transient process and the steady periodic vortex shedding behind the cylinder. The evolution of the vorticity field is shown for various length of the splitter plate. It is demonstrated that the splitter oriented along the flow direction significantly reduces the forces applied to the cylinder. With increasing splitter length the average drag decreases monotonically but the amplitudes of oscillation of the forces applied to the body change nonmonotonically. In this paper we offer our explanation of this phenomenon. It is shown that when turning the splitter plate at some angle from the flow direction the process of vortex formation and shedding behind the cylinder is no longer strictly regular and periodic. Pages of the article in the issue: 44 - 49 Language of the article: English PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.5

Authors:K. V. Savelieva, O. G. Dashko Pages: 50 - 53 Abstract: The interaction of elastic plane harmonic waves in the material, the nonlinear properties of which are described by the elastic potential of Murnaghan, is investigated theoretically. The displacement vector is depended of only one spatial variable and time, a record of the complete system of equations for plane waves moves along the abscissa axis is recorded and used. The interaction of longitudinal waves with a separate considering cubic nonlinearity is investigated. On the basis of the cubic equation of motion, the interaction of four harmonic waves is studied. The method of slowly variable amplitudes is used. Firstly the two-wave interaction is investigated, then the interaction of four waves is described. Shorten and evolutionary equations are obtained, the first integrals of these equations and the record of the law of conservation for a set of four interacting waves are obtained. An analogy is made between the triplets studied when taking into account the interaction of three waves and the triplets investigated in the case under consideration, taking into account the four-wave interaction, quadruplets. Pages of the article in the issue: 50 - 53 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.6

Authors:I. M. Askerov Pages: 56 - 60 Abstract: In the paper the problem of determination of the boundary function is studied in the initial boundary value problem described by the second order hyperbolic equation. With the help of the additional condition, the functional is constructed, and the problem under consideration is reduced to the optimal control problem. The differential of the function is calculated, a necessary and sufficient condition for optimality is proved. Pages of the article in the issue: 56 - 60 Language of the article: English PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.7

Authors:Petro R. Vavryk Pages: 61 - 66 Abstract: This article examines one of the approaches to the formalization of information dissemination processes based on the diffusion-limited aggregation model, using elements of cellular automata and their analogs. The model describes the dynamics of the information dissemination process without the influence of the mass media by taking into account the facts of information exchange that occurs during communication between participants of an arbitrary target audience. It is believed that the process is characterized by the property of self-similarity. An approach is proposed that makes it possible to study the dynamics of information dissemination processes, taking into account the attitude of the group members to each other and the attitude of the participants to the input information. As a result, an assessment of the effectiveness of the information dissemination process was obtained, which allows drawing conclusions regarding the success of information promotion measures. To demonstrate the processes of information dissemination modeled on the basis of the approach, the results of numerical experiments are presented, in which the implementation of the information exchange procedure for each person is limited to three members of the target group. Pages of the article in the issue: 61 - 66 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.8

Authors:D. Ya. Khusainov, T. I. Shakotko Pages: 67 - 71 Abstract: Quite a lot of works have been devoted to problems of stability theory and, in particular, to the use of the second Lyapunov method for this. The main ones are the following [1-7]. The main attention in these works is paid to obtaining stability conditions. At the same time, when solving practical problems, it is important to obtain quantitative characteristics of the convergence of solutions to an equilibrium position. In this paper, we consider nonlinear scalar differential equations with nonlinearity of a special form (weakly nonlinear equations). Differential equations of this type are encountered in the study of processes in neurodynamics [8,9]. In this paper, we obtain stability conditions for a stationary solution of scalar equations of this type. And also the characteristics of the convergence of the process are calculated. It is shown that the solution of stability problems is closely related to optimization problems [10-12]. Pages of the article in the issue: 67 - 71 Language of the article: Ukrainian PubDate: 2022-04-26 DOI: 10.17721/1812-5409.2022/1.9