# Theoretical and applied problems of Physics and Mathematics

# Asymptotic Properties of Continuous Solutions of Linear Functional Difference Equations

The paper studies asymptotic properties of continuous solutions of linear functional difference equations

# Spectral Properties of Singularly Perturbed qs-Normal Operator

Using described singularly perturbed rank one qs-normal operator, we study their spectral properties. We construct the singularly perturbed qs-normal operator with the prescribed set of eigenvectors and eigenvalues. When constructing this operator, we use the previously proven theorem on the structure of singularly perturbed self-adjoint operators with prescribed set of eigenvalues and corresponding eigenvectors. In such case the eigenvalues are located on the real axis. Its construction was a step-by-step process.

# A Method of Linear Summation of Trigonometric Fourier Serie

The paper considers the method of linear summation of trigonometric Fourier series of functions f(t) with limited variation due to σ(k,α)introducing into this series the factors, depending on the number of series coefficient and the parameter α.

# Asymptotic Properties of Solutions for Systems of Differential-Functional Equations with Linearly Transformed Argument

The differential-functional equations with linear deviations of the argument have been the main object of the research by many mathematicians. Currently many directions are quite well studied. However, in the modern theory of such equations there are a number of vague issues. Specifically, among them are issues on asymptotic properties of continuously differentiable solutions for systems of linear differential-functional equations with linear transformed argument. Here we focus on equations in cases when deviation of the argument Δi(t) =(1−λi)t, i =1,2,..., can be either positive or negative.

# Strong Law of Large Numbers for Random Variables with Superadditive Moment Function

In this paper, we study random variables with moment function of superadditive structure. We do not impose any assumptions on the structure of dependence of these random variables. We prove the strong law of large numbers for such random variables under regularly varying normalization by the method developed by Fazekas and Klesov. In this proof we use different properties of superadditive and ragularly varying functions. The key role in the proof is played by the possibility of approximating the nondifferentiable slowly varying function by differentiable slowly varying function.

# Convergence of Generalized Spitzer Series

The proposed method for proving the convergence of Spitzer series can be applied for other classes of functions are studied in the theory of pseudo-regularly varying (PRV) functions.

# Asymptotic Properties of Estimator of Linear Regression Parameter in Case of Long-Range Dependent Regressors

The paper considers linear regression model with long-range/weak dependent random noise and time dependent regressors which are observed with long-range dependent errors. Parameter estimation of these models is an important problem of statistics of random processes. We choose widely used least squares estimator for the estimation. The aim of this work is to prove consistency and asymptotic normality of least squares estimator of the regression model.

# Calculation of Energy Chemical Communication Phases Containing Boron in Alloy Fе–B–C

This paper deals with establishing the type of a solid solutio (penetration or substitution) for Fe–B–C system alloys. We investigate the alloys of boron content of 0,0001-0,1% (w.) and of carbon content of 0,005-0,5% (w.).To study the properties of obtained alloys, we use X-ray and durametric analyses. Under boron content of 0,0003-0,003 % (w.) in alloy the parameter of ferrite crystal lattice and microhardness decreases. Under alloy boron content increase α 0,003% (w.) the above parameters are increasing.

# Quasi-Equilibrium Heterogeneous States of Electrolyte at Steel Ball Etching in a Magnetic Field

In this paper, research into the formation of interface separating phases (regions) with different magnetic susceptibilities of paramagnetic etching products is carried out in an inhomogeneous magnetic field of a magnetized steel ball at its corrosion. The theoretical model is proposed describing shape, size of this area and distribution of paramagnetic etching product concentration within the region. The characteristic times of formation, existence and destruction of this interface boundary was experimentally identified.

# Signal Stabilization for Combined Gaussian-Vortex Beam Propagating Through Atmospheric Optical Link

The paper numerically investigates the spatial evolution of the structure of coherent and partially coherent laser beams, including the optical vortices, propagating in turbulent atmospheres. The influence of beam fragmentation and wandering relative to the axis of propagation (z-axis) on the value of the scintillation index of the signal at the detector is analyzed.