# Theoretical and applied problems of Physics and Mathematics

# The Calculation of Droplet’s Vertical Motion that Evaporates under the Sreznevsky Law

By employing the Airy functions, we find the solution of the nonlinear Cauchy problem describing the vertical motion of the droplet evaporating under the Sreznevsky law. Using a special variable transformation in the differential equation of motion, we find the first integral with the cylindrical functions. This transformation has increased the order of the differential equation, but it became linear with variable coefficients. We also propose an approximate solution for calculating the displacement.

# o the Theory of the Generalized Integral Transform

Using the generalized confluent hypergeometric function, we describe a new generalization of integral transforms (Laplace, Stieltjes, the potential theory). Specifically, we study main properties of these new integral transforms (linearity, similarity). We find representations of generalized integral Laplace transforms of the unit and power functions. Some composition relations are proved. Relying on the tables of the classical integral transforms, they permit finding the representations of more composite functions. Parseval-type equalities are proved.

# Problem of Finiteness Conjecture and Joint Spectral Radius

This paper studies the constraint of a vector norm under periodic and aperiodic action of matrices from a finite set of matrices with rational elements as well as the presence of AZR and PZR. We consider a complex case when every matrix from has its own numbers that are both bigger and smaller units.

# Primitive Program Algebra of the Calculated Functions for Graphs

The problem of finding algebraic characteristics of representative classes of functions and predicates is closely connected with issues of the programming theory and practice. In this paper, we study a class of calculated functions and predicates for finite graphs. We choose graph structures because of their importance and popularity in the theoretical and applied programming. We also use the primitive program algebra as a research tool.

# Parameters of Surface Bifocal Spin-Wave Lens

The paper investigates the process of spin waves refraction when passing through the inhomogeneous structure, a biaxial ferromagnet in the form of the biconvex lens placed in the medium of an uniaxial ferromagnet. We theoretically calculate the dependencies of “optical” parameters (refractive index, focal length) on such spin-wave lens. The paper uses geometrical optics formalism to describe the behavior of surface spin waves propagating in the ferromagnetic medium with nonuniform distribution of magnetic parameters.

# Optimal Control for Singular Perturbed Periodic Parabolic Equations with Nonlocal Boundary Value Conditions

This paper considers the issues of optimal control for singular perturbed by the spatial value linear parabolic equations with nonlocal boundary value conditions and the quadratic performance criterion. We construct the complete asymptotes of optimal solutions for the initial problem under optimal conditions by boundary functions method. Unlike similar problems for parabolic equations with local boundary value conditions, the iterative problems for boundary functions do not “decompose”.

# Single-Electron Optical Properties of Nanoeggs

The aim of this paper is to investigate single-electron optical properties of a spherical nanoegg comprising a dielectric core and a thin metal shell with a slight shift of the core center relative to the geometric center of the nanoparticle. Furthermore, we propose a model for this type of the composite nanoparticle. It allows calculating the wave functions and the electron wavenumber spectrum of the electron in the metal shell of the nanoegg. Specifically, the contribution of the latter dominates in the optical properties of the whole particle.

# Correlation Analysis of Characteristics Gloss and Smoothness of Paper Tape with Purpose of Their Control in Technological Process

Through experiments conducted, we study the characteristics of gloss and smoothness of paper tape to develop the method of their technological control. By utilizing the method of statistical analysis, we prove that there is nonlinear correlation dependence between them. We calculate the parameters of regression equation, which has a parabolic form. Moreover, we find that for certain angles of incidence of light ray 70–80, the parameter of correlation communication is the correlation relation signifying 0,80–0,85.

# The Study of the Structure of a Set of Continuously Differentiable on <strong>R<sup>+</sup></strong> Solutions for Systems of Differential-Functional Equations with the Linearly Transformed Argument

In this paper, we consider the structure of a set of continuously differentiable on **R ^{+}** solutions for systems of linear inhomogeneous differential-functional equations with the linearly transformed argument. Crucially, we focus on equations with the delay. We utilize basic methods of the theory of ordinary differential and differential-functional equations as well as the method of successive approximations.

# The Study of the Structure of a Set of Continuously Differentiable on <strong>R<sup>+</sup></strong> Solutions for Systems of Differential-Functional Equations with the Linearly Transformed Argument

In this paper, we consider the structure of a set of continuously differentiable on **R ^{+}** solutions for systems of linear inhomogeneous differential-functional equations with the linearly transformed argument. Crucially, we focus on equations with the delay. We utilize basic methods of the theory of ordinary differential and differential-functional equations as well as the method of successive approximations.