# Theoretical and applied problems of Physics and Mathematics

# Convergence of Baum–Katz Series with OSV-Functions

In this paper conditions for the convergence of series

# Sufficient Conditions of Ergodicity of Solutions of Second Order Abstract Linear Differential Equations

This paper is devoted to second order abstract linear differential equations in a Banach space. For such equations the Cauchy problem is stated, and the behavior of its solutions as is examined. The aim of the paper is to study ergodicity and asymptotic behavior of the solutions of the strongly correct Cauchy problem. For this purpose the theory of complete second order linear differential equations in Banach spaces, developed by Fattorini, is used. As shown in the paper, for a wide class of equations the solutions are either ergodic or unbounded, depending on the initial values.

# Integral Transforms with the r-Hypergeometric Functions

In the paper the r-hypergeometric function is considered in the form

# Asymptotic Unbiasedness and Consistency of Cross-Correlogram Estimators of Response Functions in Linear Continuous Systems

The estimation problem of an unknown real-valued response function of a linear continuous system is considered. We suppose that a family of zero-mean stationary Gaussian processes, which are close, in some sense, to a white noise, disturbs the system. Integral-type sample input-output cross-correlograms are taken as estimators of the response function from . The corresponding cross-correlogram estimator depends on two parameters (a parameter of a scheme of series and a length of an averaging interval) and is biased.

# Research into Equilibrium States of a Spherical Pendulum with Non-Ideal Excitation

The present paper studies the equilibrium states of a dynamic system, formed by a spherical pendulum, whose point of suspension is excited in a vertical plane by the power-constrained electromotor. We identify that in the phase space of the system there is a singular surface, whose all points are the equilibrium states. Furthermore, sufficient conditions of asymptotic stability are obtained for one of equilibrium states and relevant domains of stability in the space of parameters’ system are constructed

# Recurrent Relations with the Generalized Legendre Functions

This paper considers -generalized (by Wright) Legendre functions. By using the known properties of -generalized (by Wright) Gauss hypergeometric function and the relation between and functions, the theorem of recurrent relations, as well as the theorem of differentiation formulas for these functions are formulated and proved

# Inverse Spectral Problem for a Block Matrix of Jacobi Type Corresponds to the Real Two Dimensional Moment Problem

The purpose of this paper is to find matrices that correspond to some finite measure with compact support on the real plane, in other words, to solve the inverse spectral problem for the present two dimensional moment problem (on the real plane).

# Critical Phenomena in Dynamical Visibility Graph

We investigate the time series by mapping them to the complex network. We show that the technique of mapping proposed – the dynamic visibility graph has an intrinsic parameter behaving similarly to the order parameter in the theory of the second order phase transitions. The behavior of the relative number of clusters near the critical angle of view was thoroughly analyzed.

# Direct and Inverse Problems of Computer-Based Materials Design

The paper analyzes tasks associated with new materials design recently applied in material science in light of the principle stating that new knowledge design should be based on the previously accumulated knowledge, as well as wide application of computer technologies and operating with material science databases. We show that the general problem of computer design of materials should be divided into 3 problems. The aim of the direct problem is to construct the interpolation polynomial based on available materials science discrete databases available.

# Didion Inversion Solutions for the Problem of Point Ballistics

The purpose of this study is to develop a convenient way to calculate distance (horizontal projection of the trajectory) flight of a particle in a gas medium with quadratic resistance movement to flatter trajectory. Didion solution converted to a form suitable for calculating the flight range of a particle in a gaseous environment is known in the theory of ballistics. The Lambert function table is proposed by calculating the flight range of a point on the flat trajectory.