Interaction of the quasi-nondiffraction beam with semi-infinite medium has been investigated. It has been shown that in case when we deal with absorbing medium, new features of energy flux behavior and the spatial profile of heat generation occurre. Such behavior of energy flux isn’t typicall for plane wave or Gaussian beam. The combination of the detected features, sup-pressed diffraction divergence of beam, self-reconstruction phenomenon is of interest of laser technologies and diagnostics.

Keywords: quasi-nondiffraction beam, semi-infinite medium, energy flux, heat generation region.

V.N. Belyi – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

L.I. Kramoreva – Gomel State Medical University, Gomel, Belarus

E.S. Petrova – P.O. Sukhoi Gomel State Technical University, Gomel, Belarus

N.A. Khilo – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

A.I. Savitski – Gomel State Medical University, Gomel, Belarus

Basic laws governing of the forming of the electroplated films Co-Ni under the influence of X-ray irradiation are presented. The dependences of the electrical properties from electrolysis conditions are discovered. The reasons for the change of the resistivity in the films Co-Ni obtained under irradiation, caused by the change in the morphology of surface, element composition, by the decrease of porous in the coating and by the increase in the speed of electrodeposition under the effect of X ray-irradiation are considered.

Keywords: X-ray irradiation, electroplated films, Co- Ni alloys, resistivity.

N.G. Valko – Y. Kupala Grodno State University, Grodno, Belarus

V.G. Hurtavy – Sctantific and Practical Materials Research Center of NAS of Belarus, Minsk, Belarus

Polarization and energy flux density of an electromagnetic field for the vector paraxial Gaussian light beams with the homogeneous and nonhomogeneous polarization of various types are discovered and explored. Some new types of the vector paraxial Gaussian light beams, for example, beams with spiral elliptic polarization are featured.

Keywords: paraxial beams, vector beams, light beams, Gaussian beams, polarizable properties, energy properties, nonhomogeneous polarization.

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

Absorption of electromagnetic radiation in the frequency range of 35 to 55 THz by the multilayer coating, in which the top layer contains an array of pyramids arranged in a strictly geometric pattern in two mutually perpendicular directions is investigated. It is shown that the proposed structure provides a return loss better than 39.5 dB at the resonance frequency of 42.3 THz and a change of temperature of the sensitive layer of 3 mK at the input power density of radiation of 0.1 W/m².

Keywords: absorbing structure, multi-layer coating, the array of pyramids, thermal sensitivity.

A.K. Esman – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

V.K. Kuleshov – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

G.L. Zykov – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

V.B. Zalesski – B.I. Stepanov Institute of Physics of NAS Belarus, Minsk, Belarus

Luminescence and absorbance spectra of uranyl nitrate and perchlorate solutions in water and some organic solvents such as acetone and acetonitrile have been studied. Distinction of the emerging structures is defined depending on the position of the short-wave band which is assigned to purely electronic transition. The possible explanation of the adherent processes is given. In addition, acetone polymerization resulted from uranyl UO₂²⁺ excited ions, serving as a catalyst, is found.

Keywords: luminescence and absorption spectra, structure, uranyl nitrate, uranyl perchlorate, acetone, photochemistry.

A.P. Zazhogin – Belarusian State University, Minsk, Belarus

D.S. Umreiko – A.N. Sevchenko Institute of Applied Physics Problems, Minsk, Belarus

A.A. Zazhogin – Belarusian State University, Minsk, Belarus

A.I. Komyak – Belarusian State University, Minsk, Belarus

S.D. Umreiko – Belarusian State University, Minsk, Belarus

The PVD method of formation of carbon coatings from plasma of pulse cathode-arc discharge was proposed. The character of the influence of the parameters of discharge on the growth rate and morphology of carbon coatings was states.

Keywords: carbon coating, pulse cathode-arc discharge, morphology, growth rate.

D.G. Piliptsov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.V. Rogachev – Rector of F. Skorina Gomel State University, Gomel, Belarus

N.N. Fedosenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

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

Results of laser thermosplitting process modelling of single-crystalline silicon are presented. Calculation of the thermoelastic fields formed in a silicon plate as a result of consecutive laser heating and coolant influence was carried out for a section (110) in three different variants of laser beam movement, namely in directions [1–10], [001], [1–11].

Keywords: crack, laser splitting, silicon wafer.

A.N. Serdykov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

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

Y.V. Nikitjuk – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.A. Sereda – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

In 1986 V.A. Vedernikov proved that if M is a non-normal maximal subgroup of a finite soluble group G, then M contains a normalizer of some Sylow subgroup of G. In the paper the following generalization of Vedernikov’s result is proved.

Theorem. Let G be a -soluble finite group. Let M be a non-normal maximal subgroup of G such that | G : M | is a power of a prime p in . Let H be a Hall subgroup in M such that p does not divide | H |, and either or | M : H | is a -number. If the core of in is not equal to 1, then is contained in M.
Here is the core of M in G, i. e., the largest normal subgroup in G contained in M ; is the set of prime divisors of | H |.

Keywords: -soluble group, maximal subgroup.

N.M. Adarchenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

Weighted shift operators B in the space generated by a linear mapping are considered. A description of properties of for belonging to spectrum is given. In particular, the necessary and sufficient condition for to be one-sided invertible is obtained.

Keywords: weighted shift operators, spectrum, one-sided invertibility, invariant measure, decomposition of oriented graph.

A.A. Akhmatova – Belarusian State University, Minsk, Belarus

Let be a conjugate -normally embedded -soluble Fitting functor and let be the smallest element of the Lockett section of with respect to strong containment. A generalized version of an open question of Beidleman-Brewster-Hauck is to give a description of . In this paper such a description is presented.

Keywords: Lockett's operation, Lockett section, Fitting -functor.

E.A. Vitko – P.M. Masherov Vitebsk State University, Vitebsk, Belarus

Let A be an RG-module, where R is a commutative noetherian ring with the unit, G is a locally soluble group, and each proper subgroup H of a group G for which is not a noetherian R-module, is finitely generated. It is proved that a locally soluble group G with these conditions is hyperabelian. It is described the structure of a group G under consideration if is a finitely generated soluble group and the quotient module is not a noetherian R-module.

Keywords: group ring, locally soluble group, noetherian R-module.

O.Yu. Dashkova – O. Honchar Dnepropetrovsk National University, Dnepropetrovsk, Ukraine

This article discusses the convergence of the Fourier series for functions of the multidimensional p-adic argument. For this purpose we define the multidimensional Mahler function and partial sums of Fourier series for the functions of multidimen-sional p-adic argument. We calculate the norm of the m -th derivatives of multidimensional Mahler functions and prove the criterion of m times continuously differentiability in terms of Mahler coefficients. We represent coefficients and partial sums of multidimensional Fourier series in terms of coefficients and partial sums of one-dimensional Fourier series. The main result states that for positive integers the Fourier series for function converges uniformly. An example of with divergent Fourier series is given.

Keywords: function of multidimensional p-adic argument, Fourier series, Fourier coefficients, Mahler function.

Zarenok Maxim – Department of Mechanics and Mathematics, Belarussian State University, Minsk, Belarus

Let G be a finite group. A subgroup A of G is said to be quasipermutable in G if A either covers or avoids every maximal pair (K,H) of G. We study the finite groups with given systems of quasipermutable subgroups.

Keywords: finite group, maximal pair, (weakly) quasipermutable subgroup, generalized Fitting subgroup, p-nilpotent group, -hypercentre.

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

Zhao Xiaoyu – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A new criterion for supersolubility of finite groups has been found.

Keywords: primitive subgroups, finite group, soluble group, supersoluble group, subnormally primitive subgroup, normally primitive subgroup.

N.S. Kosenok – Belarusian Trade and Economics University of Consumer Cooperatives, Gomel

The optimal control problem of a thermal process in the rod with the heat exchange on one of its ends is considered in the article. It’s reduced to the optimal control problem of the special large-scale differential equations system. This problem is solved by a method of quasidecomposition. The algorithm of the operation of the optimal regulator which forms current values of the optimal feedback in real time is described. The numerical example illustrating the use of the quasidecomposition method is brought.

Keywords: thermal process, optimal control problem, method of quasidecomposition, optimal feedback, optimal regulator.

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

The algorithm for numerical modeling of equilibrium capillary surfaces with irregular contact conditions is presented. It is a combination of an iteration-difference method to solve the boundary-value problem and a Runge-Kutta method to solve the initial-value problem for modeling of equilibrium shapes of a capillary surface in contact with the fracture line of a solid wall. The algorithm is approved on the well-known problem of capillary hydrostatics about fluid flowing out of the capillary.

Keywords: capillary surface, parametric differential equations, irregular boundary conditions, numerical modeling, computational al-gorithm, numerical results.

V.K. Polevikov – Belarusian State University, Minsk

Y.N. Volotovskaya – Belarusian State University, Minsk

In the study of factorizable groups the authors consider some natural restrictions on the factors. Ito Theorem about two-step solubility of a finite group which is factorized by abelian subgroups, and Kegel-Wielandt Theorem about solubility of a finite group which is a product of two nilpotent subgroups are the classical examples in this trend. We should also note that L.S. Kazarin have obtained validity of the hypothesis of S.A. Chunikhin about non-simplicity of a finite group which is factorized by subgroups with nontrivial center.

Keywords: finite group, simple nonabelian group, factorized group.

N.V. Tyutyanov – International Institute of Labor and Social Relations, Gomel

V.A. Tyutyanova – International Institute of Labor and Social Relations, Gomel

The technique analysis of reliability of probabilistic production systems allowing to define the parameters of the technological operations providing a given level of reliability of the technological cycle. The implementation of the procedure for the assessment of probability values of the parameters of reliability of the components production system to ensure optimal variants of the structure of the technological cycle within specified criterion of the quality evaluation of the technological cycle is proved to be of exceptional practical significance.

Keywords: reliability analysis, state space, estimation of parameter values, structure of the technological cycle.

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

The study deals with the experimental results of the peculiarities of the anodic growth of the porous aluminium oxide films formed in aqueous solutions of oxalic and malonic acids. The results obtained show that during galvanostatic aluminum anodizing in 0.3 M and 0.8 M solutions of oxalic and malonic acids, the films with regular porous structure are formed. Aluminum anodizing in the solutions of malonic and tartaric acids leads to the forming of the porous films with inter-porous spacing in the range of 90–270 nm.

Keywords: aluminium oxide, oxalic acid, malonic acid, porous film.

I.A. Vrublevsky – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

S.K. Dick – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

A.S. Tereh – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

A.V. Smirnov – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

K.V. Chernyakova – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus