External evaluations of trajectories for a stationary two-parameter discrete system with interval coefficients were obtained, with the use of which a constructive method of management, resulting in a minimum set of solutions in a neighborhood of a given polyhedron, was proposed.

Numerical methods for solving singular integral first-kind equations with a special form of the right-hand side are developed. The proposed schemes are based on the decomposition of singular integrals with power-logarithmic singularity in Chebyshev’s polynomials of first and second kind. Accuracy estimates of the considered methods are presented.

Gauss-type quadrature formulas for rational functions with simple poles have been investigated.

We consider a linear control periodic system with reverse communication. It is supposed that a right block of coefficint matrix has an imperfect column rank. The control problem of an asynchronous spectrum is solved.

Let ƒ(X) and g(Y) be nonsingular quadratic forms over a field K having dimensions m and n  , charK≠ 2. The following problem of a birational compositions ƒ(X) and g(Y)is considered: under which conditions is the product ƒ(X) and g(Y)birationally equivalent over K to a quadratic form h(Z) of dimension m+n over K?
The main result of the paper is a complete solution of the birational composition problem for quadratic forms ƒ(X) and g(Y)over the function field F, char F ≠ 2.

In this paper we studied the question of constructing rational interpolation operators of Hermite Fejer. Uniform convergence of considered interpolation process for function on condition of completeness of system of rational functions is thus proved. Also, we proved uniform convergence for considered interpolation process for functions ƒ (x) є C[-1,1] of complete system of rational functions The rational interpolating functions construction with nodes in the zeros of Chebyshev Markov polynomials of the second kinds.

In the article some properties of Frattini-like subgroups of finite groups are investigated.

The multicriteria investment boolean Markowitz problem with extreme optimism criteria is considered. Upper and lower bounds of the radius of the stability of this problem are given in the case of the arbitrary Holder metric lp, 1 <р < ∞ in the portfolio space and the Chebyshev metric lx in the space of financial market states and in the space of investment project profitability.

The problem of statistical assignment of arbitrarily distributed multivariate samples with the fixed probability distribution is considered. The decision rule based on the maximum likelihood method is proposed and its efficiency is analytically investigated. The case of two samples and the Fisher model is investigated.

In the paper an algorithm is suggested for quick estimate of sieving time and the number of relations for the number field sieve algorithm.

Taking into account the star light pressure it is proved that for a star-dark body-test body system in the bounded planar circular problem of three bodies there exist a countless number of photolibtation points filling up three segments of the straight line passing through the star and the dark body (collinear case) or those points filling up the two circular arcs passing through the Lagrangian libration points L₄, L₅ (triangular case).

The introduction contains the object of investigation the system consisting of N ordinary differential equations, which is a mathematical model of the motion of N bodies in the plane. The basic concepts are: motion of four bodies, interparticle interaction constant, Painleve property, simplified system.
The purpose of this study is to establish the analytic properties of solution of simplified systems for a system of nonlinear differential equations describing the motion of the four bodies.
The main part deals with the study of simplified systems of a system describing the planar motion of the four bodies. These systems consist of nonlinear differential equations, each of which is of second order.
A set of interparticle interaction constants in the two cases of the problem under investigation in the plane is found. The general solution at these constants can be written in closed (rather simple) form.
Obtained are 15 nonlinear autonomous differential equations of first order with respect to one of the components of the system, whose general solution is an integer, i.e., these ordinary differential equations possess the Painleve property, as well as 23 autonomous nonlinear differential equations of first order, whose general solution contains a logarithm. Therefore, these equations are not the equations of Painleve type.
Necessary and sufficient conditions are established for the existence of the Painleve property of the studied system that reveal 56 cases in the problem of four bodies in the plane when trajectories of the given bodies in the plane can be described.
The results can be applied in the analytical theory of differential equations, as well as in celestial mechanics theory.

Two eleven-component tensor field systems of Maxwell type are considered. It has been shown that an additional scalar function in one of the system is a gauge function. Therefore, this system describes а massless vector field. The second system provides а simultaneous description of massless fields with helicities 0 and ±1 as a united physical object.

The properties of changeable optical solitons in anisotropic media are considered. It is shown that variable solitions also keep the basic specific features of light propagation in crystals.

Participants of shape transformation of two-dimensional nearly absent-diffracted (cosine-supergaussian) light beams under Braggs diffraction by ultrasonic waves in iniaxial gyrotropic crystals are investigated. It is shown that at diffraction of such light beams travelling in the vicinity of the crystal optical axis on a slow shear ultrasonic wave under the conditions of a strong acoustooptical interaction, light beams experience a significant deformation of the intensity distribution in the beam cross section. At large widths of a supergaussian component of an incident beam, the shape of a diffracted beam remains underchanged. It is stated that the diffraction efficiency weakly depends on the width of the supergaussian component of the cosine-supergaussian beam.

Hopping migration of blpolarons (electron pairs) via immobile defects of one kind in three charge states (-1, 0, +1) in partly disordered semiconductors is considered theoretically. They are considered to have negative correlation energy and to “stabilize” the Fermi level in the vicinity of the middle of the band gap of a semiconductor. The expression is written for the drift and diffusion components of the dc current density of blpolarons hopping from defects in the charge states (-1) to those in the charge states (+1). The analytical expression for screening length of an external stationary electric field is found. It is shown that the concentration of mobile electric charges responsible for electric field screening is equal to the blpolaron concentration.

The high-frequency (ƒ= 1 MHz) negative capacitance of avalanche diodes irradiated with fast electrons in the breakdown mode has been studied for the first time. It is found that the appearance of a negative capacitance area on the curve for temperature-dependent capacitance is accompanied by a conductivity maximum on a similar curve for temperature-dependent conductivity. It is suggested that this effect is due to the influence of trapping centers on the process of recharging deep levels in the entire space-charge region.

The influence of differently focused grains of a copper substrate and the condition of their surface on the growth of thin electrolytic nickel films is investigated.

The treatment of dental enamel by UV (308 nm) radiation is made and the dependence of the ablation value of the peak power density of laser radiation at a fixed number of pulses is determined. It is established that the ablation threshold of dental enamel is about 0.3 J/cm². It is found that at an energy density of more than 0.15 J/cm² dental calcium is removed without dental enamel ablation.

Ultraviolet, visible and near infra-red spectroscopy is used for identification of the authenticity of matured wine distillates. The use of principal component analysis, the discriminant analysis, the classification trees and projection latent structures to broad-band transmission spectra of wine distillates allowed determining manufacturer and aging of wine distillates.

It is proved that the Dirichlet’s theorem on the approximation of real numbers by rational can not be improved.

To the 85th Anniversary of Birthday