The new solutions of the parabolic equation featuring paraxial light beams are offered. Such beams are featured by Kummer functions of complex argument with two free parameters without a Gaussian. Restrictions on these parameters, at which beams of Kummer transfer terminating power and are physically realized, are discovered.

Keywords: paraxial beams, beams of Kummer, square integrability.

S.S. Girgel – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

The features of the propagation of singular beams with various signs of the topological charge in a cubic optically active photorefractive crystal of cut, to which an external electric field is applied, are investigated. The dependence of the angle of rotation of the sin-gular beam on value of external electric field applied to the crystal and sign of unit topological charge of the beam is analyzed. It is shown how the optical activity affects the distribution of singular beam in the crystal.

Keywords: singular beam, photorefractive crystal, optical activity.

Zh.V. Kolyadko – I.P. Shamyakin Mozyr State Pedagogical University, Mozyr, Belarus

V.V. Shepelevich – I.P. Shamyakin Mozyr State Pedagogical University, Mozyr, Belarus

V.V. Davydovskaya – I.P. Shamyakin Mozyr State Pedagogical University, Mozyr, Belarus

The effect of laser radiation pressure on the metal nanoparticles located near an air bubble in water is experimentally investigated. The reasons of the observed movement of nanoparticles along a bubble surface are discussed.

Keywords: light pressure of laser radiation, metal nanoparticles.

E.V. Matuk – Y. Kupala Grodno State University, Grodno, Belarus

L.S. Gaida – Y. Kupala Grodno State University, Grodno, Belarus

D.V. Guzatov – Y. Kupala Grodno State University, Grodno, Belarus

Exact solutions with spherical symmetry of the Maxwell equations in the oscillating anti de Sitter Universe are constructed. 3-Dimensional complex formalism by Majorana – Oppenheimer, generalized to pseudo-Riemannian space-time models in accordance with the tetrad method, is applied. The established dependence of the constructed solutions on the time variable leads to independence of the Pointing vector on time. This means that the change in time of the anti de Sitter geometry cannot be seen by measuring the energy flow density.

Keywords: Maxwell's equations, tetrad formalism, anti de Sitter Universe, non-static coordinates, complex formalism of the Majorana – Oppenheime, exact solutionsr.

Å.Ì. Ovsiyuk – I.P. Shamyakin Mozyr State Pedagogical University, Mozyr, Belarus

The morphological features, phase composition, microhardness and tribological properties of composite carbon coatings formed by co-precipitation of carbon plasma streams of pulsed cathodic arc discharge and flows of atoms of the metal ions (Cr, Ti, Cu) generated by magnetron sputtering, and their dependence on the power source sputtering are determined. It is shown that such layers are characterized by high density and order of sp²-clusters and low roughness. The hardness increases and the wear rate of counterbody decreases with in-creasing the power of magnetron.

Keywords: carbon coatings, sp²-clusters, alloying, magnetron, phase composition, morphology, roughness, hardness, friction, wear.

A.S. Rudenkov – F. Skorina Gomel State University, Gomel, Belarus

On the example of a two-layer structure of a semiconductor – insulator – semiconductor the possibility of an analytical determination of the optical parameters and thickness of the upper layer on the reflection spectrum envelope at normal incidence of light is considered. It is very difficult to calculate these functions even in the case of absence of the transition zones between the layers. Mismatching enve-lopes calculated and measured reflection coefficients in certain areas of the spectra indicates the presence of inhomogeneous surface and transition zones in real structures pSi – SiO₂ – cSi.

Keywords: bilayer structure, optical spectra, envelopes extrema, transition layers, pSi – SiO₂ – cSi.

N.I. Staskov – À.A. Kuleshov Mogilev State University, Mogilev, Belarus

S.O. Parashkov – À.A. Kuleshov Mogilev State University, Mogilev, Belarus

A.V. Shylov – À.A. Kuleshov Mogilev State University, Mogilev, Belarus

N.A. Krekoten – JSC “INTEGRAL”, Minsk, Belarus

The possibility of temporal modulation of quasi-continuous radiation in the reflection of thin film of dense resonant medium is theoretically defined for the relative weakness of the relaxation processes. Resonant conversion of the light field in the reflection mode coherent interaction can be used to form the optical signals in ultrafast interaction processes of light radiation with thin layers.

Keywords: resonant reflection, thin optical films, dense resonant medium, dipole-dipolar interaction.

Yu.V. Yurevich – A.A. Kuleshov Mogilev State University, Mogilev, Belarus

V.A. Yurevich – Mogilev State University of Food Technologies, Mogilev, Belarus

Let be a nonempty formation, some set of prime numbers. The theorems reflecting the general regularities on intersections of maximal subgroups of a finite group mutually simple with numbers from indexes including statements on the intersections and are received.

Keywords: formations of finite groups, subgroup m-functor, intersections of maximal -subgroups in a finite group.

L.M. Belokon – Mogilev State University of Food Technologies, Mogilev, Belarus

The necessary and sufficient conditions under which linear periodic differential system has strongly irregular periodic solutions were obtained.

Keywords: strongly irregular periodic solution, linear differential system, periodic coefficient.

M.S. Belokursky – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The behavior of normal subgroups in generalized Frattini extension is studied.

Keywords: formation, normal subgroups, Sylow p-subgroup, group of operators.

R.V. Borodich – F. Skorina Gomel State University, Gomel, Belarus

S.N. Bykov – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The problem of characterization of all hereditary regular transitive subgroup functors is solved. It is proved that any natural transitive subgroup functor is -subnormal for some primitive class .

Keywords: finite solvable group, primitive group, subgroup functor, natural transitive subgroup functor (NT-functor), formation.

S.F. Kamornikov – Gomel Branch of International University «MITSO», Gomel, Belarus

The conditions of existence of periodic solutions of differential equation with linear-fractional right-hand side are given.

Keywords: differential equation, reflecting function, Poincare mapping, linear-fractional function, periodic solution.

V.V. Mironenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The system describing the motion of four bodies in a plane is considered. The method of the small parameter of a simplified system consisting of nonlinear differential equations, each of which has a second order is determined. For these systems, a set of interparticle interaction constants in which all its solutions of the system are meromorphic functions is specified.

Keywords: motion of four bodies in a plane, interaction constant, simple system, meromorphic function.

A.T. Sazonava – Y. Kupala Grodno State University, Grodno, Belarus

Let G be a finite group, some partition of the set of all primes and Ï a subset of the set . A set of subgroups of G is said to be a complete Hall Ï-set of G if contains exact one Hall -subgroup of G for every such that We say also that G is: Ï-full if G possess a complete Hall Ï-set; a Ï-full group of Sylow type if for each every subgroup E of G is a -group, that is, E has a Hall -subgroup H and every -subgroup of E is contained in some conjugate of In this paper we study properties of finite Ï-full groups. The work continues the research of the paper [1].

Keywords: finite group, Ï-full group, -soluble group, -nilpotent group, -quasinilpotent group.

Skiba A.N. – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The upper estimates are determined for modules of the zeros of Hermite polynomials (Hermite – Pade approximation type I) of the exponential system where are arbitrary complex numbers. The proven statements complement and generalize known results about the behavior of the zeros of Hermite polynomials for exponential system by E. Saff and R. Varga, H. Stahl, F. Wielonsky.

Keywords: exponential system, Hermite polynomials, Hermite – Pad’e approximation, zeros of Pad’e approximation.

A.P. Starovoitov – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

E.P. Kechko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The limiting bearing ability of a pile-cit in the nonlinear-deformed soil basis, taking into account ground consolidation in zone running-impulse processing is considered.

Keywords: pile-cit, mathematical model, limiting bearing ability.

V.E. Bykhautsau – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

D.V. Prokopenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

L.A. Curganova – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

This paper considers the construction of cryptographic S-boxes of the length which are optimal from the point of view of the ab-sence of correlation between the output and input vectors. Constructed sets of S-boxes can be recommended for upgrading of the existing block symmetric ciphers, as well as for the synthesis of new high-speed encryption algorithms based on the principles of multi-valued logic.

Keywords: S-box, matrix of correlation coefficients, Kim scheme.

O.N. Zhdanov – M.F. Reshetnev Siberian State Aerospace University, Krasnoyarsk, Russia

A.V. Sokolov – Odessa National Polytechnic University, Odessa, Ukraine

The idea of creating software is considered. With this software, one can automate the calculations for determining the rational schemes and designs of electrical networks, determine the electrical load, loss of power and loss of supply, reactive power compensation, maintain the required voltage quality, quantity and variety of power transformers.

Keywords: software tools, web-application, power supply, power consumption, circuitry.

S.F. Maslovich – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.B. Gorbach – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

V.N. Halushko – Belarusian State University of Transport, Gomel, Belarus

A.V. Drobov – Gomel State Technical College, Gomel, Belarus