Abstract. A new version of the modified regularized long-wave equation is considered. The equations of such a type are used as an alternative to the Korteweg-de Vries equation. A modification of the equation consists in an accounting the term which describes an interaction of dispersion and dissipation. Using the direct Hirota method for nonlinear equations in partial derivatives a kink-type (antikink-type) solution for modified equation is constructed. A possibility to construct a coupled solution of kink and antikink is analysed.

Keywords: regularized long-wave equation, kink, anti-kink, Hirota direct method.

For citation: Knyazev, M.A. Kink for modificated regularized long-wave equation / M.A. Knyazev // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 7–10. – DOI: https://doi.org/10.54341/20778708_2021_4_49_7 (in Russian)

Information about the authors:

M.A. Knyazev – Belarusian National Technical University, Minsk

Abstract. The conditions for the formation of nanostructured materials based on ZnO nanorods and TiO₂ nanotubes formed on their basis, the optimal modes of application and the required thickness of the seeding ZnO layers have been determined. The time and temperature regimes for hydrothermal synthesis of columnar nanostructures based on zinc oxide were selected experimen-tally. The method of atomic force microscopy was used to study the morphology and sub-roughness of the surface. Using a scanning electron microscope, a detailed study of the structure and morphology of the obtained nanostructured materials was carried out at each stage of synthesis. The main characteristics influencing the size and shape of the obtained ZnO / TiO₂ nanostructured materials have been es-tablished.

Keywords: hydrothermal method, sol-gel method, heat treatment, ZnO nano-rods, TiO₂ nanotubes, surface morphology, structural properties.

For citation: Development of a combined formation method of ZnO / TiO₂ nanotube / D.L. Kovalenko, M. Dobromir, V.V. Vaskevich, A.V. Semchenko, V.V. Sidski, O.I. Tyulenkova, Y.A. Kosenok, G.Y. Ayvazyan, D. Luca // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 11–16. – DOI: https://doi.org/10.54341/20778708_2021_4_49_11 (in Russian)

Information about the authors:

D.L. Kovalenko – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

M. Dobromir – Alexandru Ioan Cuza University, Iasi

V.V. Vaskevich – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

À.V. Semchenko – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

V.V. Sidski – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

O.I. Tyulenkova – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

Y.A. Kosenok – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

G.Y. Ayvazyan – National Polytechnic University of Armenia, Yerevan

D. Luca – Alexandru Ioan Cuza University, Iasi

Abstract. The problem of approximate calculation of the stress-strain state of an isotropic base (pavement) is considered using the example of the contact between a tire and a road surface. The main dependences for calculating the stress-strain isotropic base are presented. Based on the presented formulas, an algorithm for solving the problem under study has been developed and a program has been written to calculate the stress-strain state of a massive tire in contact with a road surface.

Keywords: coating, wheel tire, stress-strain state, method of final elements.

For citation: Mozharovsky, V.V. Calculation of the stress-deformed state of a tire on the example of its contact with a road surface / V.V. Mozharovsky, D.S. Kuzmenkov, I.I. Kolyaskin // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 17–20. – DOI: https://doi.org/10.54341/20778708_2021_4_49_17 (in Russian)

Information about the authors:

V.V. Mozharovsky – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

D.S. Kuzmenkov – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

I.I. Kolyaskin – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

Abstract. The parameters of laser splitting of quartz plates are optimized. Optimization calculations were carried out using the genetic algorithm MOGA, implemented in the ANSYS Workbench program.

Keywords: laser splitting, quartz plate, genetic algorithm MOGA, software system of finite element analysis ANSYS.

For citation: Nikitjuk, Y.V. Optimization of the parameters of laser splitting of quartz glass / Y.V. Nikitjuk, A.N. Serdyukov, I.Y. Aushev / Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 21–28. – DOI: https://doi.org/ 10.54341/20778708_2021_4_49_21 (in Russian)

Information about the authors:

Y.V. Nikitjuk – Francisk Skorina Gomel State University

A.N. Serdyukov – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

I.Y. Aushev – University of Civil Protection of the Ministry for Emergency Situations of the Republic of Belarus, Minsk

Abstract. The influence of heat treatment modes on the mechanical properties of coatings based on carbon and composite nitride layers has been determined by the nanoindentation. An increase in the hardness of the (Al-TiN) / a-C coating to 19,7 GPa after annealing in vacuum at 400^o C was established. The ratios H / E, H³ / E² were determined, which characterize the elastic properties of the coatings. An increase in the resistance to plastic deformation for coatings containing aluminum after heat treatment in vacuum has been established. The analysis of the elastic recovery coefficients η_IT obtained at different indentation depths indicates the formation of a more homogeneous structure and a decrease in the gradient of mechanical properties along the thickness in comparison with unannealed coatings.

Keywords: carbon coatings, nitride sublayers, hardness, elastic modulus, nanoindentation.

For citation: Piliptsou, D.G. Influence of heat treatment on the mechanical properties of layered carbon coatings / D.G. Piliptsou // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 29–37. – DOI: https://doi.org/ 10.54341/20778708_2021_4_49_29 (in Russian)

Information about the authors:

D.G. Piliptsou – Francisk Skorina Gomel State University

Abstract. The features of the morphology and phase composition of carbon coatings deposited on sublayers based on polytetrafluoroethylene and lead formate of various compositions have been determined. The effect of the mass ratio of the components of the initial target used for the deposition of the sublayer and the heat treatment of carbon coatings on their morphology and phase composition has been established. The efficiency of using metal oxides and carbonized polymer layers for a given change in the phase composition and relief of carbon coatings is estimated.

Keywords: carbon coatings, lead, polytetrafluoroethylene, carbonation, morphology, phase composition.

For citation: Rudenkov, A.S. Morphology and phase composition of nanostructured carbon coatings, formed on the sublayer based on polytetrafluoroethylene and lead formate / A.S. Rudenkov // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 38–44. – DOI: https://doi.org/10.54341/20778708_2021_4_49_38 (in Russian)

Information about the authors:

A.S. Rudenkov – Francisk Skorina Gomel State University

Abstract. The results of study of the influence of the features of sol-gel synthesis, composition and thermal treatment on the microstructure, surface topography and ferroelectric properties of Ba_xSr_1-xTiO₃ (x = 0,8; 0,9; 1) thin sol-gel films are presented.

Keywords: BST films, sol-gel method, ferroelectrics, hydrolysis, polycondensation, annealing, ferroelectric properties.

For citation: Influence of the composition and conditions of the sol-gel process on the properties of barium-strontium titanate ferroelectric thin films / S.A. Khakhomov, A.V. Semchenko, A.V. Sidsky, V.V. Vaskevich, A.A. Maevsky, O.I. Tyulenkova, V.E. Gaishun, D.L. Kovalenko, O.V. Pakhomov, A.V. Es’kov, A.A. Starkov, A.L. Kholkin, V.A. Pilipenko // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 45–50. – DOI: https://doi.org/10.54341/20778708_2021_4_49_45 (in Russian)

Information about the authors:

S.A. Khakhomov – Francisk Skorina Gomel State University

A.V. Semchenko – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

V.V. Sidsky – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

V.V. Vaskevich – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

A.A. Maevsky – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

O.I. Tyulenkova – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

V.E. Gaishun – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

D.L. Kovalenko – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

O.V. Pakhomov – ITMO University, St. Petersburg

A.V. Es’kov – ITMO University, St. Petersburg

A.A. Starkov – ITMO University, St. Petersburg

A.L. Kholkin – ITMO University, St. Petersburg

V.A. Pilipenko – JSC «INTEGRAL», Minsk

Abstract. The conditions for obtaining pronounced structural coloring of a photonic crystal fiber with hexagonal symmetry are defined. It is shown that the parameters of the fiber should be selected in accordance with the configuration of the bandgaps of a two-dimensional photonic crystal formed by air channels filling the fiber. The conclusions of the band theory are confirmed by calculating the scattering cross sections of photonic crystal fibers made of polyethylene terephthalate with a circular outer boundary.

Keywords: structural coloring, photonic-crystal fiber, band theory, scattering cross section.

For citation: Shilov, A.V. To attainment of pronounced structural coloring of a photonic crystal fiber / A.V. Shilov, A.B. Sotsky // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 51–56. – DOI: https://doi.org/10.54341/ 20778708_2021_4_49_51 (in Russian)

Information about the authors:

A.V. Shilov – A.A. Kuleshov Mogilev State University

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

Abstract. The dynamics of energy exchange of a low-dimensional planar lattice formed by quantum dots with a resonant light field of a pulse probing from the outside has been investigated. In the average field approximation, the problem of describing the collective effect of superradiance in the considered quasi-two-dimensional super-crystal (SC) is solved. In the estimation of SC response to a resonant field, the dipole-dipole interaction is taken into account, which also occurs under conditions of absorption in quasi-resonant transitions. Both mechanisms of phase nonlinearity are typical for semiconductor SC and cause a self-modulation shift of the resonance frequency of the absorption (gain) spectral line.

Keywords: super-crystals of quantum dots, dipole-dipole interaction, optical collective effects, Bloch vector formalism.

For citation: Yurevich, V.A. Supershot optical pulse nonlinear reflection by low-dimensional super-crystal / V.A. Yurevich // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 57–63. – DOI: https://doi.org/10.54341/ 20778708_2021_4_49_57 (in Russian)

Information about the authors:

V.A. Yurevich – Belarussian State University of Food and Chemical Technologies, Mogilev

Abstract. The structure of a subgroup associated with the intersection of kernels of maximal A-admissible subgroups that do not contain the -residual and do not belong to the formation whose indices have restrictions are considered. The properties of the corresponding generalized Frattini subgroup are established.

Keywords: finite group, formation, -residual.

For citation: Borodich, R.V. On the intersection of non--subgroups, distinguished by a subgroup function, in groups with operators / R.V. Borodich // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 64–68. – DOI: https://doi.org/10.54341/20778708_2021_4_49_64 (in Russian)

Information about the authors:

R.V. Borodich – Francisk Skorina Gomel State University

Abstract. Let be a saturated S-closed formation of finite soluble groups containing the class of all groups whose nilpotent length does not exceed k, where k ≥ 3. In this paper, we obtain a constructive description of all Fitting formations in such that for any -group G = AB, there is

Keywords: finite group, nilpotent length, di--group, -radical, Fitting formation, radical formation with the Monakhov condition in the class .

For citation: Vasil'ev, A.F. On radicals of factorized finite -groups / A.F. Vasil'ev // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 69–75. – DOI: https://doi.org/10.54341/20778708_2021_4_49_69 (in Russian)

Information about the authors:

A.F. Vasil'ev – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

Abstract. The article goes on with the studies on the earlier described relationship between sets of generators in group A and sets of generators in polyadic group < A^k, [ ]_{l, σ, k} > with l-ary operation [ ]_{l, σ, k}, that is defined on Cartesian power A^k of group A for arbitrary integer l ≥ 2 and arbitrary substitution σ from the set S_k of all substitutions of the set {1, 2, …, k}.

Keywords: group, l-ary group, set of generators.

For citation: Gal'mak, A.M. On sets of generators of l-ary group < A^k, [ ]_{l, σ, k} >. III / A.M. Gal'mak // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 76–80. – DOI: https://doi.org/10.54341/20778708_2021_4_49_76 (in Russian)

Information about the authors:

A.M. Gal'mak – Belarusian State University of Food and Chemical Technologies, Mogilev

Abstract. Throughout this paper all groups are finite and G always denotes a finite group. The group G is said to be -separable if every chief factor of G is either a -group or a -group. A subgroup A of G is said to be , -subnormal in G if there is a subgroup chain such that either or is a -separable group for all i = 1, …, n. In this paper we study the influence of , -subnormal subgroups on the structure of the group.

Keywords: finite group, -separable group, , -subnormal subgroup, Hall subgroup.

For citation: Dergacheva, I.M. Criteria for -separability of a finite group / I.M. Dergacheva, I.P. Shabalina, E.A. Zadorozhnyuk // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 81–84. – DOI: https://doi.org/10.54341/ 20778708_2021_4_49_81 (in Russian)

Information about the authors:

I.M. Dergacheva – Belarusian State University of Transport, Gomel

I.P. Shabalina – Belarusian State University of Transport, Gomel

E.A. Zadorozhnyuk – Belarusian State University of Transport, Gomel

Abstract. A polling system with two queues with a limited number of places in buffers are considered. Each queue receives markovian flow of requests. The times of servicing the requests and switching between the queues have phase type distribution. The queuing discipline is gated. The formulas for finding the stationary probabilities of system states at an arbitrary moment in time, and the formulas for computing the main performance measures for the system are obtained. The expressions for the Laplace – Stieltjes transforms of waiting time distributions in buffers are found.

Keywords: polling system, markovian arrival process, phase type service, stationary distribution, waiting time.

For citation: Dudin, A.N. Polling system with two markovian arrival proñesses, finite buffers and phase type distribution of service and switching times / A.N. Dudin, Y.V. Sinyugina // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 85–91. – DOI: https://doi.org/10.54341/20778708_2021_4_49_85 (in Russian)

Information about the authors:

A.N. Dudin – Belarusian State University, Minsk

Y.V. Sinyugina – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

Abstract. This article proves the coincidence of generalized moduli of smoothness of the k-s order, defined with the help of Gegenbauers generalized shift operator, with different and identical shifts and, as a consequence, direct and inverse theorems of approximation theory by algebraic polynomials are obtained.

Keywords: the best approximation by algebraic polynomials, Gegenbauers generalized shift operator, generalized modulus of smoothness.

For citation: Kazimirov, G.N. Direct and inverse theorems of approximation theory for the generalized modulus of smoothness of some class of functions / G.N. Kazimirov // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 92–94. – DOI: https://doi.org/10.54341/20778708_2021_4_49_92 (in Russian)

Information about the authors:

G.N. Kazimirov – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

Abstract. We give examples of finite soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. We prove that if in a group G there is a strictly 2-maximal subgroup of order 2, then G is a supersoluble group of order 2pq, where p and q are primes, not necessarily distinct, or G is isomorphic to the alternating group A₄. We establish the structure of a finite group in which every 2-maximal subgroup is a Hall subgroup. We prove that the requirement of -subnormality of all strictly 2-maximal subgroups coincides with the requirement of subnormality of all 2-maximal subgroups of a group G for a subgroup-closed saturated lattice formation - containing all nilpotent groups and G ∉.

Keywords: finite group, 2-maximal subgroup, strictly 2-maximal subgroup, Hall subgroup, lattice formation.

For citation: Konovalova, M.N. On strictly 2-maximal subgroups of finite groups / M.N. Konovalova, V.S. Monakhov, I.L. Sokhor // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 95–100. – DOI: https://doi.org/ 10.54341/20778708_2021_4_49_95 (in Russian)

Information about the authors:

M.N. Konovalova – Russian Presidential Academy of National Economy and Public Administration, Bryansk

V.S. Monakhov – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

I.L. Sokhor – Brest State A.S. Pushkin University

Abstract. All considered groups are finite. Let G be a group. Then denotes the intersection of all totally ω-composition formations containing G. The formation is called a totally ω-composition formation generated by G or a one-generated totally ω-composition formation. A totally ω-composition formation is called a bounded, if is a subformation of some one-generated totally ω-composition formation, that is, for some group G. In this paper, criteria for the one-generation (boundedness) of a totally ω-composition formation are obtained.

Keywords: formation of finite groups, ω-composition formation, one-generated formation, bounded formation, totally ω-composition formation.

For citation: Los, I.P. On one-generated and bounded totally ω-composition formations of finite groups / I.P. Los, V.G. Safonov // Problems of Physics, Mathematics and Technics. – 2021. – N 4 (49). – P. 101–107. – DOI: https://doi.org/ 10.54341/20778708_2021_4_49_101 (in Russian)

Information about the authors:

I.P. Los – Belarusian State University, Minsk

V.G. Safonov – Belarusian State University, Minsk