The generalized interpolation formula of Hermite – Birkhoff type with respect to the arbitrary Chebyshev system of functions is constructed. Theorem on the satisfaction of the interpolation conditions is proved. A class of polynomials, for which the interpolation formula is exact, is determined. The error estimate of the obtained formula is constructed. The particular cases of the interpolation formula for the systems of trigonometric and exponential functions are considered.

Introduction of the present article indicates the object of investigation: the fourth-order differential equation. The purpose of the research is to study the analytic properties of solutions of the equation considered. In the main part of the article the solution is constructed in the form of the Laurent series. The solutions in the form of Dirichlet series and exponential series with respect to fractional-linear functions have been formulated. The issues of the series convergence, which represent the solution of this fourth-order differential equation, have been explored. The existence of three-parameter solutions with a movable singular line has been established. The obtained results can be used in the analytical theory of ordinary differential equations.

For vector autoregressive model with heterogeneous endogenous-exogenous structure and Markov switching states we propose the EM-algorithm for mixture decomposition, as well as discriminant analysis algorithm for classification of new observations. Accuracy of the algorithms is examined by means of computer simulation experiments.

A relation between the best polynomial approximations of the function assigned on a segment and the function conjugated to it is found. In the periodic case, a similar relation is obtained by S. B. Stechkin. The inequality of G. Szegö type is also proved for the derivatives of an algebraic polynomial.

Summary Character varieties of Baumslag – Soliter groups are investigated. Irreducible components of these varieties are found, their dimensions are calculated, and their rationality is proved.

In this article, the theorems of existence of solutions of nonlocal Cauchy problems for differential equations of fractional order in weighted spaces with Caputo derivatives are proved. Sufficient conditions for compactness of integral operators operating in the space of real-valued continuous functions on a segment are also obtained.

The system of third-order nonlinear differential equations for components of two-soliton solution of the Korteweg-de Vries equation at t → ± ∞ is constructed. The equation describing the relation between these components is derived, and a general solution of this equation is obtained for one special case.

The known result that electromagnetic solutions in the form of the gradient of an arbitrary scalar function make a zero contribution to the energy-momentum tensor is extended to a massless spin 2 field. Within the theory of 30-component first-order equations, it is shown that in Minkowski’s space, the generalized gradient- type solutions for a spin 2 field make a contribution to the energy-momentum tensor in the form of a 4-divergence of a third-rank tensor; its contribution therefore vanishes during integration.

It has been shown by a spectral method that the efficiency of solubilization of a borondifluoride dipyrromethene derivative by Pluronic F-127 inmixed systems depends on the proportions of mass concentrations of the dye and Pluronic in an initial liquid solution. It has been established that the values of the Pluronic F-127 concentration, at which the micelles solubilizing the dye are efficiently formed, exceed 10 % (mass). The analysis of spectral characteristics of the solution of the “dye – Pluronic” adducts has shown that the dye under study exists in the Pluronic micelles in the highest degree of disaggregation manifesting the spectral characteristics of monomeric form.

The method based on suppressing speckle noise by a rotating quasi-spiral 2D diffractive optical element (DEO) created using binary pseudo-random sequences is presented. The unique DOE structure for the case of M-sequence code15 inlength was fabricated. Speckle suppression for green, purple, and red lasers by a factor more than 11.6 and 4, respectively, was obtained. Numerical simulations have shown that strong dispersion of photoresist (DOE structure medium) results in a significant decrease of the efficiency of the method on the borders of the visible range of electromagnetic radiation, and the use of material with anomalous dispersion can significantly improve the parameters of suppression of speckles on the borders of the visible range of wavelengths.

This article proposes a method for high-amplitude modulation of non-polarized radiation on the basis of the transverse electrooptic effect. The essence of the method is to minimize the effect of temperature changes in the electro-optical elements due to their parallel location in the insulating housing having high thermal conductivity. An amplitude electro-optic modulator with a cross application of control field KTP crystals made according to the ring scheme is described.

The sequence of phase transformations during Sr₂FeMoO_6–δ crystallization by the solid phase method was studied for powders synthesized from the stoichiometric mixture of SrCO₃ Fe₂O₃, MoO₃ oxides. It is found that the synthesis of a strontium ferromolybdate solid solution proceeds through a series of parallel chemical reactions. It is revealed that at the beginning stage of interaction, the initially formed Sr₂FeMoO_6–δ powder is enriched with iron. During the annealing process, the composition of strontium ferromolybdate changes and the molybdenum content increases upon further heating. It is shown that in the process of crystallization of double perovskite in the temperature range T = (300–1420) K, there is the following sequence of phase transformations Fe₂O₃, MoO₃, SrCO₃} (300 К) → {SrMoO₄, Fe₂O₃, SrCO₃} (770 К) → {SrMoO₄, SrFeO_3–х (traces), Sr₂FeMoO_6–δ (traces)} (970 K) → {SrMoO₄, Sr₂FeMoO_6–δ } (1170 К) → { Sr₂FeMoO_6–δ } (1420 К)..

The solid solutions with the rhombohedral distorted perovskite structure of the (1–x)(NaBi)_1/2TiO₃–xBiCoO₃ system (x < 0.2) have been prepared by solid state reactions. The unit cell parameters a_H and c_H for these perovscites were found to increase linearly with increasing x. The dielectric properties of the solid solutions undergo a diffuse ferroelectric phase transition at 590–610 K. For compositions with x < 0.05, the transition to relaxor state at a temperature below 500 K was observed. At x > 0.05 such a transition was not revealed.

The influence of high energy (6 MeV, fluencies 10¹⁵–10¹⁸ cm^–2) electron irradiation on the shift of a fundamental absorption edge and on the luminescence properties of thin InN films, which were grown on sapphire substrates by the molecular-beam epitaxy method, has been studied. It is found that electron irradiation increases the electron concentration and optical band gap energy E_g of InN. The increase in the optical band gap energy E_g of irradiated thinInN films is caused by the formation of radiation defects of donor type and by a manifestation of the Burshtein – Moss effect.

Electrical conductivity and dielectric properties of TlGaS₂, TlGa_0,999Yb_0,001S₂ and TlGa_0,99 Co_0,01S₂  single crystals are investigated in the temperature range 150–320 K at the measuring field frequencies of 10³ –10⁶ Hz. The values of generalized activation energy of charge carriers in these crystals are determined. It is shown that the absolute values of the characteristics studied increase with temperature. The temperature dependences of the dielectric constant of these crystals have revealed the anomalies in the form of wide peaks, indicating the presence of their structural changes at temperatures ~ 170–250 K. The dispersion of the dielectric properties of the single crystals under study is seen: with a frequency growth the dielectric constant values decrease, and electrical conductivity values increase. It is found that the cobalt and ytterbium doping of TlGaS₂ crystals decrease permittivity values and increase electrical conductivity values.

The problem of optimization of multi-level representations of logic circuits taking into account two main characteristics of CMOS circuits (area and average dissipated power value) is developed. The results of a comparative study of two approaches to the construction of multi-level logic circuits design on the base of gates are presented. Such multi-level logic circuits are intended to cover a CMOS library with elements.

The aspects of parameterized tiling in application to algorithms with index domain represented by a convex polyhedron are investigated. The structure of the set of partial tiles is proposed and the formulas to determine this set are constructed. The formula to define the boundaries of local loops in partial tiles is obtained as well. These formulas enable one to minimize the calculation time of local loop boundaries in the implementation of the tiling in sequential and parallel programs.