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Abstract. In the literature in physics, devoted to the section "Electricity and magnetism", various forms of conductors with currents that create a magnetic field are considered. One of them is the ring. A ring or a single-turn circuit is an element of many electrical devices. One of the main parameters of the ring is inductance. In this work, we first consider the magnetic flux created by a current-carrying ring. Then, based on a certain magnetic flux, the inductance of the ring is found. The resulting formula is in good agreement with the other expressions for the inductance of the ring.

Keywords: current ring, magnetic field induction, magnetic flux, inductance.

For citation: Akhramenko, N.A. Magnetic flux of ring with current. Ring inductance / N.A. Akhramenko // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 7–10. – DOI: https://doi.org/10.54341/20778708_2024_2_59_7. – EDN: YSFOSG

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Abstract. Bragg diffraction of two-parameter ring light beams on a slow shear ultrasonic wave in paratellurite crystals was investigated. It has been shown that the energy of the ring beam is efficiently pumped from the zero to the first diffraction order, and the more efficient energy exchange of the diffracted light beams is realized for circular polarized light waves. It has been found that the in-teracting waves have the shape of circular beams, wherein the parameters of the diffracted beam are determined by the order parameter and the radius of the incident beam, as well as the power of ultrasound, the length of acoustooptical interaction and the Bragg angle.

Keywords: ring two-parameter beam, Bragg diffraction of light, diffraction efficiency, uniaxial gyrotropic crystal, paratellurite crystal.

For citation: Diffraction of two-parameter ring light beams by ultrasound in paratellurite crystals / V.N. Bely, G.V. Kulak, L.A. Tozik, O.V. Shakin // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 11–15. – DOI: https://doi.org/10.54341/20778708_2024_2_59_11 (in Russian). – EDN: YYBWHX

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Abstract. A technology for manufacturing low-cost flexible multilayer microwave absorbers is proposed. The low cost of the absorbers is due to the fact that they contain aluminum shavings, which are a production waste. The second layer of the absorbers is made from these shavings. This is realized by forming from it the elements characterized by the same shape and size, and fixing these elements with the same step between the first and third layers made from synthetic non-woven fiber material. It has been established that electromagnetic radiation absorption coefficient values of the absorbers manufactured in accordance with the presented technology reach a value of 0.9 in the frequency range of 2.0–17.0 GHz, and the width of their effective absorption band is 140.0%.

Keywords: aluminum shavings, absorption coefficient, electromagnetic radiation absorber, effective absorption band.

For citation: Flexible multilayer electromagnetic radiation absorbers with an ordered structured intermediate layer based on aluminum shawings / O.V. Boiprav, V.S. Chelyadinsky, L.M. Lynkou, M.V. Tumilovich // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 16–21. – DOI: https://doi.org/10.54341/20778708_2024_2_59_16 (in Russian). – EDN: ZEBMHD

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Abstract. New vector solutions of the parabolic equation describing Cartesian vector astigmatic TM Kummer – Gauss (K-G) light beams are proposed. The polarization and energy properties of such beams are investigated. The graphical modeling of ellipses of polarization, intensity and transverse energy fluxes of vector astigmatic TM of K-G light beams is carried out.

Keywords: TM-modes, astigmatic beams, Kummer-Gaussian beams, polarization, transverse energy fluxes.

For citation: Girgel, S.S. Polarization and energy properties of the cartesian TM-modes of Kummer – Gauss beams / S.S. Girgel // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 22–26. – DOI: https://doi.org/10.54341/ 20778708_2024_2_59_22 (in Russian). – EDN: VDUCZP

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Abstract. The results of studying the structure and properties of electrical steels modified by laser radiation are presented. The steels with silicon content of 6.67% and 3.32% were used as the object of the study. The steels were irradiated with laser radiation generated by a GOR-100M ruby laser with a flux density of 10⁴ W/cm² and 10⁵ W/cm². By the method of XRD analysis, it was found that the laser irradiation with selected flux densities of the silicon electrical steels did not lead to a change in the structural-phase state. The coherent crystalline domain sizes and the dislocation density of steels were calculated. It is shown that the coherent crystalline domain sizes increase after laser irradiation and dislocation density decreases. The data obtained by XRD analysis correlate with decreasing in microhardness.

Keywords: electrical steel, laser radiation, structure, hardness, X-ray phase analysis, corrosion resistance.

For citation: Moroz, N.I. Modification of the structure and properties of electrical steel by laser radiation / N.I. Moroz, N.G. Valko, S.V. Vasilyev // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 27–31. – DOI: https://doi.org/10.54341/20778708_2024_2_59_27 (in Russian). – EDN: UVGFQC

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Abstract. This study proposes a solution to the applied research problem of predicting the characteristics of laser cleaving of silicate glasses. The results of a numerical experiment conducted in APDL (Ansys Parametric Design Language) were used to build regression, neural network and fuzzy models for the controlled laser cleaving of silicate glasses. The processing speed, radius, and power of the laser beam were considered as variable factors, whereas the maximum temperature and thermoelastic tensile stresses in the laser-treated area were regarded as responses. The regression model for the responses of laser cutting of glass plates at a specified significance level was estimated using the findings from the face-centered version of the central composite design experiment. Artificial neural networks that exhibit response dependence on input factors were created and trained. The most effective neural network models of the maximum temperature and thermoelastic tensile stresses in the laser-treated area were de-termined using MAPE (mean absolute percentage error) heat maps. Fuzzy modeling of controlled laser cleaving of silicate glasses was conducted according to the developed linguistic variables of input and output parameters. An evaluation was performed to compare the results of regression, neural network, and fuzzy modelling based on accuracy criteria, ultimately identifying the most effective model. The research findings can be suggested for practical application in approximating the maximum temperature and thermoelastic tensile stress in the laser-treated area.

Keywords: laser cutting, fuzzy logic, artificial neural network, ANSYS – Universal Finite Element Analysis Software System.

For citation: Determination of parameters for controlled laser cleaving of silicate glasses using regression, neural network and fuzzy models / Yu.V. Nikityuk, A.F. Vasilyev, L.N. Marchenko, J. Ma, L. Wang, Y. Qin, I.Yu. Aushev // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 32–38. – DOI: https://doi.org/10.54341/20778708_ 2024_2_59_32. – EDN: UZDLHO

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Abstract. The paper provides a quantitative assessment of the effectiveness of the Rayleigh – Jeans approximation for the thermal radiation of stars in the infrared region of the spectrum. For a number of fixed temperatures, the relative deviations of the Rayleigh – Jeans spectral density from the Planck one at the edges of the NIR, MIR and FIR ranges are calculated. For the same ranges, the relative deviations of emissivities, luminosities and fluxes are also determined. The efficiency of the approximation for some stars is assessed as well.

Keywords: infrared radiation, spectral density, Planck curve, Rayleigh – Jeans formula, relative deviation, luminosity.

For citation: Novikova, O.V. Rayleigh – Jeans approximation in the infrared range of thermal radiation of stars / O.V. Novikova, G.Yu. Tyumenkov // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 39–42. – DOI: https://doi.org/10.54341/20778708_2024_2_59_39 (in Russian). – EDN: RRKPNB

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Abstract. This article studies the issues of recording and processing of the signals that excite plasma, as well as the control of plasma interaction with the surface of objects on the basis of the analysis of its energy characteristics. The article offers assessment of influence/interaction of cold plasma on / with objects on the basis of the measurement and analysis of dynamics of changes of energy characteristics of the signals that excite plasma.

Keywords: diagnostics of cold low-temperature plasma, lasing signals, released power on plasma.

For citation: Changing the parameters of signals exciting cold plasma when interacting with objects / A.N. Osipov, Y.N. Kalenkovich, V.A. Rokach, T. Ma // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 43–47. – DOI: https://doi.org/10.54341/20778708_2024_2_59_43 (in Russian). – EDN: TFXMZH

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Abstract. The influence of heat treatment modes on the chemical composition, morphological features, and mechanical properties of carbon coatings alloyed with zirconium and silicon was determined. The deposition of coatings was carried out from a plasma flow of complex composition generated by pulsed electric arc and electric spark evaporation. It has been established that heat treatment leads to graphiti-zation of coatings, the formation of ordered sp2-clusters and, as a consequence, an increase in the elastic modulus, a decrease in nano-hardness values with their subsequent stabilization during heat treatment above 350° Ñ.

Keywords: carbon coatings, silicon, zirconium, carbide, cluster, chemical composition, morphology, phase composition, nanohardness, elastic modulus.

For citation: Rudenkov, A.S. Influence of heat treatment on the structure and mechanical properties of carbon coatings doped with zirconium and silicon / A.S. Rudenkov, D.G. Piliptsou // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 48–56. – DOI: https://doi.org/10.54341/20778708_2024_2_59_48 (in Russian). – EDN: OBHLQQ

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Abstract. The kinetic regularities of electron beam dispersion of polyoxymethylene (POM) have been established and the molecular structure of coatings formed by volatile polymer destruction products has been studied. The effect on the coating growth rate, the pressure in the vacuum chamber and the molecular structure of the deposited layers of laser assisted influence and the copper particles introduction into the target were determined. It has been shown that the reactivity of POM dispersion products can be significantly increased by co-dispersing of POM and polyethylene (PE). The deposition of composite coatings occurs at a higher rate. For the POM component of the composite coating, the proportion of folded structures in the volume of the thin layer is significantly higher than the value typical for a single-component POM-based layer.

Keywords: electron beam dispersion, laser assistance, coating, polyoxymethylene, molecular structure.

For citation: Kinetic regularities of deposition and molecular structure of polyoxymethylene coatings formed from the active gas phase / M.A. Yarmolenko, A.V. Rogachev, Jiang Xiao Hong, S.A. Frolov // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 57–63. – DOI: https://doi.org/10.54341/20778708_2024_2_59_57 (in Russian). – EDN: PEFHRO

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Abstract. In this work, for a maximal subgroup of a group G, the concept of an n-modularly embedded subgroup (n is some natural number) is introduced. A criterion is established under which every maximal subgroup in G is n-modularly embedded, as well as necessary and sufficient conditions under which in every subgroup A of G any maximal subgroup is n-modularly embedded in A for some natural number n, n≤k (k – fixed natural number).

Keywords: supersoluble group, maximal subgroup, n-modularly embedded subgroup, Schunck class.

For citation: Vasilyeva, T.I. On some groups from the formation of supersoluble finite groups / T.I. Vasilyeva // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 64–69. – DOI: https://doi.org/10.54341/20778708_2024_2_59_64 (in Russian). – EDN: RJFGYO

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Abstract. Steady-state random oscillations in a self-oscillatory system with two and a half degrees of freedom, such as a molecular generator, have been studied using the method of canonical representations. The relationships have been found for finding the most probable values of the characteristics of the stationary mode of oscillations of a molecular oscillator, taking into account the thermal noise of the resonator and the spontaneous emission of a beam of active molecules.

Keywords: self-oscillatory system, stochastic oscillations, method of canonical representations, amplitude of stationary stochastic oscillations, molecular generator.

For citation: Zhogal, S.P. Investigation of fluctuations in molecular generator as self-oscillatory system by method of canonical representations / S.P. Zhogal, S.I. Zhogal, V.V. Orlov // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 70–72. – DOI: https://doi.org/10.54341/20778708_2024_2_59_70 (in Russian). – EDN: NIDZVS

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Abstract. For the telegraph equation with a nonlinear potential given in the first quadrant, we consider the first and the second mixed problem, for which we study issues related to the absence, non-uniqueness, and blow-up of classical solutions.

Keywords: semilinear wave equation, mixed problem, classical solution, absence of solution, non-uniqueness of solution, blow-up of solution, method of characteristics, energy methods, matching conditions.

For citation: Korzyuk, V.I. On the absence, non-uniqueness, and blow-up of classical solutions of mixed problems for the telegraph equation with a nonlinear potential / V.I. Korzyuk, J.V. Rudzko // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 73–78. – DOI: https://doi.org/10.54341/20778708_2024_2_59_73 (in Russian). – EDN: LGNECC

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Abstract. Throughout the article, all groups are finite and G always denotes a finite group. A subgroup A of a group G is called quasinormal in G if AH=HA for all subgroups H of G. If A is a subgroup of G, then A_qG is the subgroup of A generated by all those subgroups of A that are quasinormal in G. We say that the subgroup A is N-quasinormal in G (N≤G), if for some quasinormal subgroup of T of G, containing A, N avoids the pair (T, A_qG), i.e. N∩T=N∩A_qG. Using these concepts, we give new characterizations of soluble and supersoluble finite groups.

Keywords: finite group, soluble group, supersoluble group, subgroup lattice, quasinormal subgroup, modular lattice.

For citation: Finite groups with systems of N-quasinormal subgroups / N.S. Kosenok, I.V. Blisnetz, I.A. Sobol, Ya.A. Kuptsova // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 79–83. – DOI: https://doi.org/ 10.54341/20778708_2024_2_59_79 (in Russian). – EDN: MRRNFM

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Abstract. The paper considers one two-stage optimal control problem, described at different time intervals by various integro-differential equations of the Volterra type. Under different assumptions two types of formulas for the increment of the quality functional are constructed for these problems. Studying them on various control variations, the analogues of the maximum principle of L.S. Pontryagin and the linearized maximum condition are proved.

Keywords: two-stage control problem, integro-differential equation, increment formula, admissible control, necessary optimality condition, optimal control.

For citation: Mansimov, K.B. To the necessary optimality conditions in one two-stage control problem for integro-differential equations of Volterra type / K.B. Mansimov, A.F. Mansimzade // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 84–89. – DOI: https://doi.org/10.54341/20778708_2024_2_59_84 (in Russian). – EDN: BMLYHS

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Abstract. The article considers the solution technique for inverse modelling problems of magnetic-abrasive finishing of planar surfaces. The solution of the problem enables the determination of control law of the pressure exerted onto the workpiece needed for realization of material removal according to the prescribed function. The problem is posed in matrix form on the basis of Preston’s equation, and the corresponding linear system of equations is solved using iterative Kaczmarz method. On the basis of numerical examples it is shown that pressure control during machining enables sufficiently precise (with deviation no more than 2.2%) realization of the prescribed material removal function, particularly, a nearly uniform removal function as well as the removal function providing compensation of initial shape errors of the workpiece.

Keywords: magnetic-abrasive finishing, uniformity of material removal, Preston’s equation, inverse problems, Kaczmarz method.

For citation: Stepanenko, D.A. Application of Kaczmarz method for solving inverse modelling problems of magnetic-abrasive finishing process of planar surfaces / D.A. Stepanenko, E.S. Eromin // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. 90–98. – DOI: https://doi.org/10.54341/20778708_2024_2_59_90 (in Russian). – EDN: KOFBHW

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Abstract. The scientific biography of Doctor of Physical and Mathematical Sciences, Professor S.S. Girgel, a well-known specialist in the field of physical optics and crystal physics, whose scientific interests are related to the development of the theory of optical properties of crystals with an ordered magnetic structure, the study of phase transitions in crystals, as well as the study of dif-fraction-free propagation of light beams with a special energy distribution in their cross section, is presented.

Keywords: Girgel Sergej Sergeevich, physical optics, crystal physics, diffraction-free propagation of light beams.

For citation: Girgel Sergej Sergeevich (to the 80th Anniversary) // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. – DOI: https://doi.org/10.54341/20778708_2024_2_59_99 (in Russian). – EDN: FDMRYP

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Abstract. The scientific biography of Corresponding Member of the National Academy of Sciences of Belarus, Doctor of Physical and Mathematical Sciences, Professor A.N. Serdyukov, a famous Belarusian scientist in the field of phenomenological electrodynamics, nonlinear crystal optics, acousto-optics, quantum electronics, theory of gravity, is presented.

Keywords: Serdyukov Anatoly Nikolaevich, phenomenological linear and nonlinear electrodynamics, gyrotropic media, crystal optics, acousto-optics, relativistic gauge-invariant model of a gravitational field with a scalar potential.

For citation: Serdyukov Anatoly Nikolaevich (to the 80th Anniversary) // Problems of Physics, Mathematics and Technics. – 2024. – № 2 (59). – P. – DOI: https://doi.org/10.54341/20778708_2024_2_59_100 (in Russian). – EDN: PLBHRZ