Exact solutions of relativistic two-particle equations for scattering s-states are obtained in cases of the -function potential and superpo-sition of two -function potentials. Scattering amplitudes and phase shifts are calculated on the basis of wave functions found. The analysis of values obtained was carried out. As a result, the unitarity condition and vanishing conditions of scattering amplitudes are proved. It is shown that the non-relativistic limit of relativistic expressions obtained yields results which coincide with corresponding expressions which were found in the process of solving the Schrodinger equation.

Keywords: relativistic two-particle equation, relativistic configurational representation, delta-function potential, scattering amplitude, S-matrix, phase shift, unitarity condition, Ramsauer – Townsend effect.

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

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

Creation and investigation of -structures with optimal parameters obtained by magnetron sputtering are described. A metallic coating in the form of -elements was deposited on a layer of polyamide or polytetrafluorethylene. The samples were investigated in the anechoic chamber as polarization transformers of microwave radiation. Structures on the polytetrafluorethylene substrate converted incident linearly polarized wave into the reflected elliptically polarized with the ellipticity coefficient K = 0.53–0.57 at a frequency of 3.7 GHz, while on the polyamide substrate – into a wave with the axial ratio K = 0.36–0.47 at the same frequency.

Keywords: magnetron sputtering, ellipticity, polymer substrate, metamaterial, -element.

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

I.V. Semchenko – F. Skorina Gomel State University, Gomel, Belarus

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

The problem of multiple reflections of light in the prism coupler at waveguide spectroscopy of thin films is investigated. With the use of a 2D ray model, algorithms for computing the trajectory of a light beam in the triangular prism coupler and the prism reflectivity are developed. Criteria for optimal choice of the prism angles and rotation axis are proposed and illustrated numerically. The criteria imply stabilization of the light insertion point into the layered medium under investigation and minimization of influence of high-order rays to full reflectivity of the prism coupler. Examples of the waveguide spectroscopy inverse problem solution for a bi-layer structure of silicon oxinitride on a silicon substrate are presented.

Keywords: inverse problem of the waveguide spectroscopy, multiple reflections of light in the prism coupler, optimization of the prism coupler.

A.B. Sotsky – A.A. Kuleshov Mogilev State University, Mogilev, Belarus

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

The nonlinear dynamic effect - splitting of the supershort optical pulse reflected by a dense resonant medium thin layer is simulated. The effect arises as a result of the coherent interaction, developing in a dynamic phase adjustment of the laser pulse field and the polarization of the medium of a thin layer. The decisive factor in the development of the splitting process is the effect of the phase shift due to the influence of neighbor dipole interactions on the absorption line.

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

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

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

E.V. Timoschenko – A.A. Kuleshov Mogilev State University, Mogilev, Belarus

The behavior of -accessible subgroups in generalized Frattini extension is studied.

Keywords: maximal subgroup, formation, -accessible subgroups, group of operators.

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

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

E.N. Borodich – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The concept of (m,n)-ideals in ordered -groupoids is introduced and the (0,2)-ideals and (1,2)-ideals of an ordered -groupoid in terms of left ideals are characterised. It is shown that an ordered -groupoid S is 0-(0,2)-bisimple if and only if S is right 0-simple. The results of this paper extend the concept of an -groupoid without order. Finally, we characterize an intra-regular ordered -groupoid in terms of left and right ideals.

Keywords: ordered -groupoids, left invertive law, left identity, (m,n)-ideals.

Waqar Khan – University of Science and Technology of China, Hefei, China; COMSATS Institute of Information Technology, Abbottabad, Pakistan

Faisal Yousafzai – University of Science and Technology of China, Hefei, China

Asad Khan – University of Science and Technology of China, Hefei, China

The Shemetkov problem about description of hereditary saturated lattice formations and lattice subgroup functors in the class of all finite metanilpotent groups has been solved.

Keywords: finite group, lattice formation, lattice subgroup functor, -subnormal subgroup.

A.F. Vasil’ev – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

I.N. Khalimonchik – F. Skorina Gomel State University, Gomel, Belarus

Subgroup H of G is called s-normal in G, if G has a subnormal subgroup T such that G = HT and New criteria for the p-solvability and solvability of finite groups are obtained on the basis of this definition.

Keywords: finite group, solvable group, maximal subgroup, subnormal subgroup.

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

Let G be a finite group and H a subgroup of G. We say that H is -quasinormal in G if H permutes with all Sylow subgroups Q of G such that and . The main result here is the following: Let G = AT, where A is a Hall -subgroup of G and T is p-nilpotent for some prime , let P denote a Sylow p-subgroup of T and assume that A is -quasinormal in G. Suppose that there is a number such that and A permutes with every subgroup of P of order and with every cyclic subgroup of P of order 4 (if and P is non-abelian). Then G is p-supersoluble.

Keywords: -quasinormal subgroup, Sylow subgroup, Hall subgroup, p-soluble group, p-supersoluble group.

V.O. Lukyanenko – P.O. Sukhoi Gomel State Technical University, Gomel, Belarus

T.V. Tihonenko – P.O. Sukhoi Gomel State Technical University, Gomel, Belarus

On the example of the Riccati equation the method of construction of differential equations with a given partial solution and the same reflecting function as the given equation is described.

Keywords: differential equation, reflecting function, periodic solution.

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

Three classes of completely solvable linear nonautonomous multidimensional differential systems (diagonalizable systems, triangular systems, the Lappo-Danilevsky systems) are considered. The regular methods of building first integrals for these differential systems are developed. In addition, some examples are given to illustrate the obtained results.

Keywords: system of total differential equations, first integral.

A.F. Pranevich – Y. Kupala Grodno State University, Grodno, Belarus

P.B. Pauliuchyk – Y. Kupala Grodno State University, Grodno, Belarus

The structure of finite groups in which every proper subgroup is either -subnormal or -abnormal, where is a saturated hereditary formation with the Shemetkov property containing all nilpotent groups is studied. In particular, descriptions of these groups in the cases when is either the formation of all p-nilpotent groups or all p-decomposable groups were obtained.

Keywords: finite group, -subnormal subgroup, -abnormal subgroup, saturated formation, formation with the Shemetkov property.

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

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

The question under consideration is reduction of a weighted composition operator to an operator with invariant coefficient by Liapunov transformation. Topological obstruction to be reducible is described in the case of periodic mapping generating operator. Explicit form of the corresponding Liapunov transformation is given.

Keywords: weighted composition operator, Liapunov transformation, Cauchy index, reducibility, homological equation.

Teube Cyrille Mbainaissem – University Cheikh Anta Diop de Dakar, Dakar, Senegal

Serine Alou Lo – University Cheikh Anta Diop de Dakar, Dakar, Senegal

Moussa Ould Ahmed Salem – University Cheikh Anta Diop de Dakar, Dakar, Senegal

A new sharp result for a trace operator in analytic Bergman spaces in products of Siegel domains of the second type extending previously known assertions in less general domains is provided. These are the first results of this type for such a type productdomains.

Keywords: Traces, Siegel domains, analytic functions, Bergman-type spaces.

R.F. Shamoyan – Bryansk State University, Bryansk, Russia

S.M. Kurilenko – Bryansk State University, Bryansk, Russia

The application area of hardware models of the type studied is described. The purpose of this study is to optimize performance and time characteristics of flow systems based on the integration of the results of the computational core flow model systems with web-access.

Keywords: fluid outflow at variable pressure, strain gauge automated system flow control, proportional-digital control, callback function, IP-protocol PC-compatible controller.

Yu.R. Beytuk – Y. Kupala Grodno State University, Grodno, Belarus

V.M. Ramazanov – Y. Kupala Grodno State University, Grodno, Belarus

G.P. Sebrovskaya – Y. Kupala Grodno State University, Grodno, Belarus

O.I. Sadovskaya – Y. Kupala Grodno State University, Grodno, Belarus

Design philosophy of conical detector is considered. It is intended for investigation of multicomponent liquid-dispersed systems using the complex electrophysical method. Utilization of the detector allows studing the constitution of such systems in laboratory and indus-trial conditions as well as to control the composition of bi- and multicomponent liquid systems, for example biological liquids, drilling fluids and polymer solutions. The detector is characterized by high structural perceptibility, measurement accuracy of depolarization current and its derivatives such as physico-chemical parameters.

Keywords: structural perceptibility, measuring electrode, depolarization current, multicomponent liquid-dispersed systems.

V.A. Goldade – F. Skorina Gomel State University, Gomel, Belarus; V.A. Belyi Metal-Polymer Research Institute of NAS of Belarus, Gomel, Belarus

I.V. Shalamov – Limited liability company «VINEX», Minsk, Belarus

E.A. Tsvetkova – F. Skorina Gomel State University, Gomel, Belarus

T.V. Rjabchenko – V.A. Belyi Metal-Polymer Research Institute of NAS of Belarus, Gomel, Belarus