The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|^s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|^s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|^s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|^s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|^s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.

The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

We study the Lipschitz-like properties of multivalued mappings defined by functional parametric equalities and inequalities. Sufficient conditions of pseudo-Lipschitzian continuity are obtained on the basе of the regularity condition of the relaxed constant positive linear dependence (RCPLD) by Andreani et al. The results of the article generalize some known sufficient conditions for pseudo-Lipschitzian continuity of the systems of parametric equalities and inequalities.

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.

Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

We consider a special class of optimization problems where the objective function is linear w.r.t. decision variable х and the constraints are linear w.r.t. х and quadratic w.r.t. index t defined in a given cone. The problems of this class can be considered as a generalization of semi-definite and copositive programming problems. For these problems, we formulate an equivalent semi-infinite problem and define a set of immobile indices that is either empty or a union of a finite number of convex bounded polyhedra. We have studied properties of the feasible sets of the problems under consideration and use them to obtain new efficient optimality conditions for generalized copositive problems. These conditions are CQ-free and have the form of criteria.

This article is devoted to the Error Bound property (also named R-regularity) in mathematical programming problems. This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems. In the article, new sufficient conditions for the error bound property are described, and it’s also shown that several known necessary conditions are insufficient. The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints.

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

In the framework of the relativistic quark model based on the point form of the Poincaré-invariant quantum mechanics, the parameters were fixed using the integral representations of the lepton decay constants of pseudoscalar and vector mesons containing u-, d- and s-quarks. As a result of numerical calculations using the oscillator wave function, the basic parameters of the model are obtained using the pseudoscalar density constant and current quark masses. The analysis showed that the obtained calculation results in the framework of the model and the experimental data on the lepton decays of hadrons agree well with each other. As a result, the calculation method is generalized to the case of hadronic transitions with γ-quantum emission and a subsequent calculation of the integral representations of radiative decay constants of pseudoscalar and vector mesons. The obtained values of the anomalous magnetic moments are compared with the baryon data. As a test of the model, the authors studied the behavior of the form-factors of radiative decays of vector mesons with a subsequent comparison to the modern experimental data in the q < 0.5 GeV range where the resonance effects are insignificant. As a result, self-consistent descriptions of lepton and radiative transitions were obtained within the framework of the model proposed by the authors.

The wave equation for the vector bispinor Ψ_a(x), which describes a zero mass spin 3/2 particle in the Rarita – schwinger form, is transformed into a new basis of Ψ_a(x), in which the gauge symmetry in the theory becomes evident: there exist solutions in the form of the 4-gradient of an arbitrary bispinor Ψ_a⁰(x) = ∂_аΨ(x), For 16-component equation in this new basis, two independent solutions are constructed in explicit form, which do not contain any gauge constituents. Zero mass solutions are transformed into linear combinations of helicity states, the derived formulas contain the terms with all helicities σ = ±1/2, ±3/2.

An analytical and numerical modeling of the process of obtaining hydroxyl radicals OH⁰ and atomic hydrogen H⁰ from water molecules on a square lattice based on electrical neutralization of ions OH⁻ on an anode and ions H⁺ on a cathode is conducted. The numerical solution of a system of equations describing a stationary migration of ions H⁺ and OH⁻ over the interstitial sites of a square lattice located in an external electric field is considered. The ions H⁺ and OH⁻ in the interstitial sites of a square lattice are generated as a result of dissociation of a water molecule under the action of external electromagnetic radiation and external constant (stationary) electric field. It is assumed that anode and cathode are unlimited ion sinks. The problem is solved using the finite difference approximation for the initial system of differential equations with the construction of an iterative process due to the nonlinearity of the constituent equations. It is shown by using calculation that the dependence of the ion current on a difference of electric potentials between anode and cathode is sublinear.

With the development of nuclear industry, the requirements for materials capable of operating under the conditions of ionizing radiation have increased. Such materials are nitride coatings based on titanium and chromium. In the work, using X-ray diffraction, X-ray microanalysis, nanoindentation method of Oliver and Farr, scratch testing, the structural phase state and the mechanical properties of nanostructured Cr–N and Ti–Cr–N coatings formed by vacuum-arc deposition from filtered plasma on substrates of steel 12X18H10T and alloy Zr2.5%Nb are investigated. It is established that the coating based on titanium and chromium has a single-phase structure (Ti,Cr)N with a face-centered cubic crystal lattice (FCC), and the coating based on chromium consists of chromium nitride CrN (FCC). It is shown that the Ti–Cr–N coating has greater hardness and toughness than the Cr–N coating. The Ti–Cr–N coating, due to its alloying with Ti atoms, has a higher adhesive strength as compared to the Cr–N coating. At the same time, in the Ti–Cr–N coating, the adhesive strength for a substrate made of Zr2.5%Nb alloy is ≈2 times greater than for a substrate based on steel 12X18H10, which may be associated with the formation of solid solutions between Ti and Zr elements. It is shown that on the contrary, the Cr–N coating can withstand heavy loads before tearing from a substrate based on steel 12X18H10T than from that based on a Zr2.5%Nb alloy. On the basis of the obtained data, one can say about the positive effect on the mechanical properties of titanium additive in the chromium-based coating composition.

The results of study of the characteristics of the proposed method [1] for correction of errors arising during information transmission via communication lines are presented. The estimates of the efficiency of search for errors and the performance of an algorithm developed to realize the proposed method using the parity values of binary matrix coordinates are obtained; among these errors are rows, columns, main and auxiliary diagonals, are obtained. We have determined the dependence of algorithm characteristics on the intensity (density) of bit errors in the message obtained after transmission via communication lines and on the size of matrices, into which a transmitted message is divided.

The time spent for calculating the parity values of matrix coordinates and for the algorithm used to find transmitted information errors are given. Recommendations on an optimal choice of sizes of binary matrices are presented. It is shown that, when the bit error rate is 10^–2 and less, the algorithm detects all the available errors.