Approximations of the Fejér sums of the Fourier – Chebyshev rational integral operators with restrictions on numerical geometrically different poles are herein studied. The object of research is the class of functions defined by Poisson integrals on the segment [–1, 1]. Integral representations of approximations and upper estimates of uniform approximations are established. In the case when the boundary function has a power singularity on the segment [–1, 1], upper estimates of pointwise and uniform approximations are found, and the asymptotic representation of the majorant of uniform approximations is found. As a separate problem, approximations of Poisson integrals for two geometrically different poles of the approximating rational function are considered. In this case, the optimal values of the parameters at which the highest rate of uniform approximations by the studied method is achieved are found. If the function |x|s, s ∈ (0, 1], is approximated, then this rate is higher than the corresponding polynomial analogues. Consequently, asymptotic expressions of the exact upper bounds of the deviations of Fejer sums of polynomial Fourier – Chebyshev series on classes of Poisson integrals on a segment are obtained. Estimates of uniform approximations by Fejer sums of polynomial Fourier – Chebyshev series of functions given by Poisson integrals on a segment with a boundary function having a power singularity are also obtained.

   This article is devoted to the precise and approximate calculation of the mathematical expectation of non-linear functionals from the solution of the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In [1–4], approximate methods for calculating the mathematical expectation of functionals from solutions of the linear Skorohod stochastic differential equation with a random initial condition and deterministic coefficient functions were proposed and investigated. This paper considers the calculation of the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In this case, the solution is obtained in an analytical form [5]; however, it contains an unknown random parameter, determined as the solution of an auxiliary integral stochastic equation. In this paper we investigate the cases when the solution of this integral equation is found in an explicit form and then evaluate the moments and the mathematical expectations of some types of functional from the solution of the initial Skorohod equation. The construction of approximate formulas for calculating more general nonlinear functionals from the solution is considered. Numerical examples are given to illustrate the accuracy of the obtained formulas.

   In this article, we study discrete nonstationary linear dynamic systems of Volterra type. An essential feature of such kind of systems is that their current states depend on the previous states of this system. The formula Cauchy, which gives us the solution of linear Volterra systems with the control inputs, is obtained. The necessary and sufficient conditions of the pointwise controllability, pointwise output controllability, and observability are proven. Also the linear-quadratic optimization problem for the nonstationary Volterra control systems is studied.

   This paper investigates the influence of the gravitational fields of dark matter and dark energy, the existence of which is currently firmly established, on electromagnetic radiation in space. In the post-Newtonian approximation of the general theory of relativity, a regularity is derived that generalizes the well-known Shapiro time delay (Shapiro effect) to estimate the delay of a light beam during the Mercury location. The generalization consists of the fact that in addition to the gravitational field of the central mass, the influence of the gravitational fields of the visible (observed) medium and dark substance on the processes in space is taken into account. The cases of location of the planet Mercury and the star near the center of our Galaxy in gravitational fields created by a spherically symmetrically distributed medium are considered. Estimates of the delays of location signals are calculated, which can exceed the time delays of signals in a space not filled with a medium by several orders of magnitude. A method for estimating the density of a dark substance is indicated if the experimental estimate of the location signal delay is known. This method is illustrated by the location of Mercury as an example.

   We have herein developed a glass-forming composition and the related sol-gel technology for bonding monocrystalline silicon wafers to produce «silicon–insulator–silicon» structures. A possibility to fabricate defect-free glass-like bonding layers at the annealing temperature decreased to 1000–1100 °C is demonstrated. The composites obtained by the sol-gel method can be used in technological processes of formation of the solid compound of silicon wafers.

   Nanostructured nitride TiAlSiN and carbonitride TiAlSiCN coatings are herein formed by reactive magnetron sputtering on various types of substrates: single-crystal silicon (100) and Titanium Grade2. To control and manage the coating process, the developed modular gas flow control complex (MGFCC) is used. The elemental composition is studied byenergy dispersive X-ray spectroscopy (EDX), the structure by X-ray diffraction (XRD), the morphology by scanning electron microscopy (SEM), whereas the micromechanical properties by nanoindentation. It is discovered that the formed coatings over the entire range of parameters α = 0.421–0.605 have a single-phase structure (Ti,Al)N, which is a disordered solid solution with a face-centered cubic (fcc) lattice. The average crystallite size of the (Ti,Al)N phase varies in the range (20–30) ± 5 nm. It is found that a decrease in the degree of reactivity α from α = 0.605 to α = 0.421 leads to an increase in the rate of deposition of nitride TiAlSiN and carbonitride TiAlSiСN coatings on silicon substrates by 200–300 %. The hardness of the formed coatings varies in the range H = 28.74–48.99 GPa, Young’s modulus E = 324.97–506.12 GPa. TiAlSiN, TiAlSiCN coatings demonstrate high values of impact strength indices H/E* = 0.07–0.12 and plastic deformation resistance indices H3/E*² = 0.13–0.72. It is detected that the degree of reactivity α has a significant effect on the micromechanical properties of the formed coatings. The structure and micromechanical properties of the formed nanostructured nitride and carbonitride TiAlSiN, TiAlSiCN coatings are suitable for use in space technology applications.

   This study focuses on the development of mathematical modelling methods for transport and urban planning. The current research focuses on an innovative mathematical model for simultaneous calculation of passenger and transport flow intensity in urban agglomerations using a common transport graph. An assessment of the model limitation impact on the parameters of individual and public transportation systems in a city (urban agglomeration) is conducted. To verify the model, several experimental calculations are performed for the transport system of the St. Petersburg agglomeration. The model presented by the authors can be used in scientific research in urban and transport planning and in the development of design solutions in cities.