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Abstract. The paper analyzes the history of the formation and development of the famous algebraic school at Gomel State University. The role of Academician of the Academy of Sciences of the BSSR S.A. Chunikhin and his student, Corresponding Member of the National Academy of Sciences of Belarus, L.A. Shemetkov as scientific leaders in the formation and development of the scientific school is shown.

Keywords: finite group, scientific school, S.A. Chunikhin, L.A. Shemetkov.

For citation: On the formation and development of research in Algebra at Gomel State University: history, results and prospects / A.F. Vasil’ev, V.S. Monakhov, V.G. Safonov, A.N. Skiba // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 7–15. – DOI: https://doi.org/10.54341/20778708_2025_3_64_7 (in Russian). – EDN: YLSTHQ

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Abstract. Based on many years of research conducted at Francisk Skorina Gomel State University and joint developments with scien-tific institutions in Belarus and foreign scientific centers, a review of literature on the creation of mathematical models and methods for calculating the stress-strain state of tribological elements of parts made of composite materials has been made.

Keywords: tribology, gears, contact stress, coatings, contact area, functionally graded material, composite.

For citation: Mozharovsky, V.V. Mathematical models for calculating the stress-strain state of composite products and coatings for ma-chine elements and tribological systems / V.V. Mozharovsky // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 16–28. – DOI: https://doi.org/10.54341/20778708_2025_3_64_16 (in Russian). – EDN: XQJHGI

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Abstract. The analysis of the regularities of the main stages of applying composite polymer coatings from the active gas stage generated by electron-beam dispersion is carried out. The most effective applications of this method are the formation of medical coatings, layers with high hydrophobic and hydrophilic properties, as well as antifriction coatings for precision friction units. Based on the advantages of the method of deposition of coatings from the gas phase, a number of other promising applications are proposed for solving a wide range of technological problems.

Keywords: polymer coatings, electron-beam dispersion, medical coatings, hydrophobic coatings, antifriction coatings.

For citation: Main results and prospects of forming functional nanocomposite polymer coatings from the active gas phase / A.V. Rogachev, V.A. Yarmolenko, A.A. Rogachev, Jiang Xiao Hong // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 29–36. – DOI: https://doi.org/10.54341/20778708_2025_3_64_29 (in Russian). – EDN: VGMOLJ

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Abstract. The study focuses on developing neuro-fuzzy and fuzzy models for laser alloying of structural steel 30XΓCH2A with chromium, based on numerical experiment results. Finite element modeling of laser alloying for steel 30XΓCH2A was performed using APDL programming language, accounting for temperature-dependent thermophysical material properties and power densities, and employing a face-centered central composite experimental design. The experimental factors included time intervals corresponding to the durations of three laser pulse fronts and their peak power densities, while the response variables were maximum temperatures in the processing zone. The model quality was evaluated using statistical metrics (RMSE, MAE, MAPE, R²) on a test dataset. The results demonstrate the superior predictive accuracy of neuro-fuzzy models over conventional fuzzy models for laser alloying parameters.

Keywords: laser alloying, Mamdani fuzzy inference.

For citation: Neuro-fuzzy and fuzzy modeling of laser alloying of steel 30XΓCH2A / G.A. Baevich, Yu.V. Nikityuk, L.N. Marchenko, A.V. Maksimenko // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 37–42. – DOI: https://doi.org/10.54341/20778708_2025_3_64_37. – EDN: VTZKRT

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Abstract. New solutions of the parabolic equation describing vector rotating paraxial TM Arnaud beams with complex astigmatism are investigated. Permissible values of free parameters for which Arnaud beams carry finite power are established. Explicit expressions describing the energy and polarization properties of vector Arnaud beams with complex astigmatism are proposed. Graphical modeling and corresponding analysis of polarization ellipses, intensity and transverse energy fluxes of vector paraxial Arnaud beams with homogeneous and inhomogeneous (TM modes) polarization are carried out.

Keywords: polarization, vector beams, rotating beams, Gaussian beams, complex astigmatism, TM-modes, transverse energy fluxes.

For citation: Girgel, S.S. Vector rotating Gaussian Arnaud beams with complex astigmatism / S.S. Girgel // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 43–48. – DOI: https://doi.org/10.54341/20778708_2025_3_ 64_43 (in Russian). – EDN: UPGBEC

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Abstract. In the present paper, we examine the equation for a spin 1/2 particle with two additional characteristics, anomalous magnetic moment and polarizability in presence of external uniform magnetic field. After separating the variables, we derive the system of four differential equations in the polar coordinate. To resolve the system of equations, we apply the method by Fedorov – Gronskiy, which is based on the use of projective operators. According to this approach, four polar components are expressed through only two different functions, the last reduce to the confluent hypergeometric equation; moreover there arises a definite quantization rule. We have constructed two types of the wave functions, the corresponding energy spectra are found in analytical form. The energy spectra are studied numerically as well.

Keywords: spin 1/2 particle, additional electromagnetic characteristics, magnetic field, projective operators, exact solutions, generalized energy spectra.

For citation: Spin 1/2 particle with anomalous magnetic moment and polarizability in the external magnetic field / A.V. Ivashkevich, P.O. Sachenok, E.M. Ovsiyuk, V.V. Kisel // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 49–55. – DOI: https://doi.org/10.54341/20778708_2025_3_64_49. – EDN: TMXGPY

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Abstract. Numerical solutions are obtained for the two-dimensional partial-wave integral quasipotential equations in the relativistic configuration space representation, describing the bound states of a system of two scalar particles of equal mass interacting via a Gaussian potential. A non-trivial spectral property is established: within certain ranges of the potential parameters, the ground state possesses a wave function with one, two or more nodes. The limiting transition is demonstrated as the Gaussian potential degenerates into the singular "delta-circle" potential: the numerical solutions asymptotically approach those for this limiting case.

Keywords: two-dimensional Logunov – Tavkhelidze equation, relativistic configurational representation, momentum representation, two-dimensional Green's function, wave function, two-particle system, bound states, Gaussian potential, delta-circle potential.

For citation: Kapshai, V.N. Numerical solution of the two-dimensional Logunov – Tavkhelidze equation for Gaussian potential in the relativistic configuration representation / V.N. Kapshai, A.V. Paulenka, Yu.A. Grishechkin // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 56–61. – DOI: https://doi.org/10.54341/20778708_2025_3_64_56 (in Russian). – EDN: NGSETJ

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Abstract. The structure and parameters of an experimental sample of a tunable Fabry – Perot resonator operating on the principle of the thermo-optical effect are presented for its use as a narrow-band light filter with a tunable bandwidth. The peculiarity of this device is the integration of a heating element and a Fabry – Perot resonator into a single structure. It has been experimentally established that with an increase in the current passing through the control heating element, the resonator structure heats up, leading to a shift in the maximum transmission coefficient to a shorter-wave region of the light spectrum.

Keywords: tunable resonator, Fabry – Perot, thermo-optic effect, resonator base.

For citation: Tunable Fabry – Perot resonator / V.M. Kravchenko, H.S. Kuzmitskaya, V.V. Malyutina-Bronskaya, A.I. Konoiko // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 62–66. – DOI: https://doi.org/10.54341/20778708_2025_3_64_62 (in Russian). – EDN: KUAOZX

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Abstract. The exact solutions of the two-dimensional partial integral equations of Logunov-Tavkhelidze for the scattering states of a system of two scalar particles of equal mass were obtained. The particle interaction was modeled by a "delta-circle" quasipotential defined in the relativistic configurational representation and by a superposition of two such quasipotentials. The analysis of the partial scattering cross-sections and the full two-dimensional scattering amplitude revealed their resonant behavior. It was established that a peculiarity of two-dimensional scattering, unlike its three-dimensional counterpart, is the unlimited growth of the scattering cross-section corresponding to states with a zero azimuthal quantum number as the rapidity tends to zero (energy tends to the rest mass). This feature is caused by the logarithmic behavior of the partial Green's function at small rapidity values. Using the found exact solutions as an example, the fulfillment of the unitarity condition for the two-dimensional partial scattering amplitudes is demonstrated.

Keywords: two-dimensional Logunov-Tavkhelidze equation, two-dimensional Green's function, wave function, two-particle system, "delta-circle" potential, partial integral equation, exact solution, unitarity condition, two-dimensional scattering amplitude, two-dimensional scattering cross-section.

For citation: Paulenka, A.V. Exact solutions of the two-dimensional Logunov – Tavkhelidze equation with the "delta-circle" potential for scattering states / A.V. Paulenka, V.N. Kapshai, Yu.A. Grishechkin // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 67–72. – DOI: https://doi.org/10.54341/20778708_2025_3_64_67 (in Russian). – EDN: HZYUGC

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Abstract. It has been established that during nitriding of carbon coatings alloyed with silicon and zirconium, the formation of compounds of the CN_x, C – Zr, Zr – N, Si – C, SiO_õ, Zr – O type takes place, and the degree of structural ordering of sp²-clusters increases. At a heating temperature of up to 350^o C, the processes of coating graphitization occur due to nitrogen desorption with simultaneous rupture of C = N bonds, which ensures better thermal stability of hardness and elastic modulus compared to non-nitrided coatings. At the same time, the values of friction coefficients and volumetric wear coefficients of the counterbody during heat treatment up to 350 oC increase due to the abrasive effect of solid carbide phases, a further increase in the heat treatment temperature up to 500^o C leads to an improvement in the tribotechnical characteristics of the coatings due to a more intensive formation of graphite clusters, which act as a solid lubricant.

Keywords: heat treatment, carbon coatings, nitrogen, silicon, zirconium, hardness, elastic modulus, friction coefficient.

For citation: Rudenkov, A.S. Influence of heat treatment on the structure and mechanical properties of carbon coatings doped with zirco-nium and silicon / A.S. Rudenkov, Xiahong Jiang, D.G. Piliptsou // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 73–83. – DOI: https://doi.org/10.54341/20778708_2025_3_64_73 (in Russian). – EDN: IBPEYG

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Abstract. The article studies of l-ary quotient groups of polyadic groups of special form, that is polyadic groups with l-ary operation η_s,σ,k, that is called polyadic operation of special form and is defined on Cartesian power A^k of n-ary group <A,η> by substitution σ∈S_k which order divides l – 1 and n-ary operation η.

Keywords: polyadic operation, semiinvariant l-ary subgroup, n-semiinvariant l-ary subgroup, quotient group, congruence, coset.

For citation: Gal'mak, A.M. Polyadic quotient groups of polyadic groups of special form. I / A.M. Gal'mak // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 84–89. – DOI: https://doi.org/10.54341/20778708_ 2025_3_64_84 (in Russian). – EDN: IJQIOV

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Abstract. The values of the operation that assigns to each class of finite groups X the class of finite groups NZX whose Sylow subgroups lie in the X-hypercenters of their normalizers are formations. Fixed points of this operation are studied. R. Baer's supersolvability criterion is improved.

Keywords: finite group, Sylow subgroup, formation, F-hypercenter, F-residual.

For citation: Koranchuk, A.G. Classes of finite groups with formationally hypercentral embedding of Sylow subgroups into their normalizers / A.G. Koranchuk // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 90–95. – DOI: https://doi.org/10.54341/20778708_2025_3_64_90 (in Russian). – EDN: HEUGQT

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Abstract. Sufficient conditions are established for the presence of a center for the differential equation, the right-hand side of which is the ratio of two power series in the polar coordinate ρ with 2π-periodic coefficients by polar angle φ.

Keywords: differential equation, center, focus, center-focus problem, reflecting function.

For citation: Mayorovskaia, S.V. Sufficient center conditions for some differential equations / S.V. Mayorovskaia, V.V. Mironenko // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 96–98. – DOI: https://doi.org/10.54341/20778708_2025_3_64_96 (in Russian). – EDN: ARETIN

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Abstract. We study -critical formations or, in other words, minimal τ-closed σ-local non--formations of finite groups, where is some class of groups, σ is a partition of the set of all prime numbers P, and τ is a subgroup functor. A criterion for minimal τ-closed σ-local formations is obtained. A description of σ-local formations of this type is given for the formations of all Ï-groups, all σ-soluble and all σ-nilpotent Ï-groups, where ∅≠Ï⊆σ. In particular, a description of minimal τ-closed σ-local non-σ-soluble and non-σ-nilpotent formations of finite groups is obtained.

Keywords: finite group, formation σ-function, σ-local formation, subgroup functor, critical functor-closed σ-local formation, σ-soluble group.

For citation: Safonova, I.N. On -critical formations of finite groups / I.N. Safonova, V.V. Skrundz // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 99–111. – DOI: https://doi.org/10.54341/20778708_2025_3_64_99 (in Russian). – EDN: BFOPLO

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Abstract. Multilayer TiN – TiCuN – TiN – TiCuN and TiN – TiCuN – TiÑuCO coatings were obtained by vacuum-arc deposition from separated plasma flows. It has been established that the adhesive strength of the formed multilayer coatings depends both on their architecture and on the difference in chemical composition. The use of TiÑuCO as a surface layer increases the corrosion resistance of the multilayer coating, but reduces the load indicators before the first crack appears. The electrochemical behavior of multilayer TiN – TiCuN – TiN – TiCuN and TiN – TiCuN – TiÑuCO coatings was studied depending on their thickness and the number of layers in them. It is shown that for all coatings the greatest influence on the corrosion resistance of the coatings is exerted by the thickness of the TiCuN layer, an increase in which leads to an improvement in the corrosion resistance of the coatings regardless of the ratio of layers in them, which is due to a decrease in the porosity of the coatings due to the formation of microstructural barriers in them, as well as the ability of the elements of the coating material to form passive surface layers in aggressive environments.

Keywords: vacuum arc deposition, multilayer coatings, corrosion resistance, electrochemical testing, adhesion resistance.

For citation: Formation of corrosion-resistant multilayer coatings based on the Ti – Cu system by vacuum-arc deposition / A.V. Belyi, S.D. Latushkina, O.I. Posylkina, I.A. Sechko, V.M. Komarovskaya // Problems of Physics, Mathematics and Technics. – 2025. – № 3 (64). – P. 112–118. – DOI: https://doi.org/10.54341/20778708_2025_3_64_112 (in Russian). – EDN: CAJESQ

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