## Vector circular paraxial Kummer – Gauss beams. Polarization and power properties

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Abstract. Analytical expressions in the closed form for vector circular 3D light of Kummer – Gauss beams from a uniform and non-uniform polarizations are offered and are investigated. Restrictions on free parameters that such of Kummer – Gauss beams transferred final power are formulated. Polarizing properties, longitudinal and crossflows of energy of such beams are calculated and are graphically investigated.

Keywords: paraxial beams, vector circular beams, Kummer – Gauss beams, streams of energy.

For citation: Girgel, S.S. Vector circular paraxial Kummer – Gauss beams. Polarization and power properties / S.S. Girgel // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 7–11. – DOI: https://doi.org/10.54341/20778708_2023_3_56_7 (in Russian). – EDN: AEFCIV

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##
^{2}+ρ_{0}^{2})^{-1} in the relativistic configurational representation

Approximate analytical solution of the
one-dimensional quasipotential equation with the potential
(ρ## QR code

Abstract. The approximate analytical solutions of the one-dimensional Logunov – Tavkhelidze equation in integral form, that describes bound states, with a model potential of the form
(ρ^{2}+ρ_{0}^{2})^{-1} in the relativistic configuration representation are found. To solve the problem an approximate transformation of the relativistic integral equation in the momentum representation to the Sturm – Liouville problem for the Schrodinger equation with a potential in the form of the modified Poschl – Teller potential well is performed.

Keywords: Logunov – Tavkhelidze equation, model potential, relativistic configurational representation, momentum representation, Sturm – Liouville problem, approximate analytical solution, Schrodinger equation, modified Poschl – Teller potential, hyper-geometric series, energy quantization condition.

For citation: Grishechkin, Yu.A. Approximate analytical solution of the one-dimensional quasipotential equation with the potential
(ρ^{2}+ρ_{0}^{2})^{-1} in the relativistic configurational representation / Yu.A. Grishechkin, A.V. Buzhan, V.N. Kapshai // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 12–15. – DOI: https://doi.org/10.54341/20778708_2023_3_56_12 (in Russian). – EDN: ANAMGY

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## Acousto-optical modulation of Bessel light beams in paratellurite crystals

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Abstract. Acousto-optical modulation of Bessel light beams on ultrasound in uniaxial gyrotropic crystals of paratellurite has been theo-retically investigated. It is established that the efficiency of anisotropic Bragg diffraction of zero-order Bessel light beams is higher than that of first-order asimutally inhomogeneous beams. It is shown that the highest diffraction efficiency for a first-order Bessel beam is achieved for the acousto-optical transformation of a left-elliptically polarized light wave into a right-elliptically polarized one. The bandwidth of the device remains unchanged under optimal conditions of one hundred percent diffraction efficiency for Bessel beams of various orders and decreases with increasing acousto-optic interaction length.

Keywords: acousto-optical interaction, gyrotropic uniaxial crystal, Bessel light beam, light modulation.

For citation: Acousto-optical modulation of Bessel light beams in paratellurite crystals / G.V. Kulak, T.V. Nikolaenko, P.I. Ropot, O.V. Shakin // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 16–20. – DOI: https://doi.org/10.54341/20778708_2023_3_56_16 (in Russian). – EDN: ASYHAV

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## Influence of composite pipe reinforcement schemes on the wave speed in case of water hammer

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Abstract. The studies of the wave velocity during water hammer for single-layer pipes made of a composite with a liquid flowing inside with various reinforcement schemes are presented. To develop mathematical models for calculating the wave speed, the mathematical theory of elasticity of an anisotropic medium is used, based on the concepts of macromechanics. Using the example of calculating a polyethylene pipe reinforced with a winding of steel fibers, an analysis was made of the results on the effect of the percentage of fiber in the matrix of a composite material on the wave velocity during hydraulic shock. It is shown that the results obtained are in good agreement with the other experimental and theoretical data.

Keywords: composite pipe, orthotropy, fibers, matrix, wave velocity, water hammer.

For citation: Mozharovsky, V.V. Influence of composite pipe reinforcement schemes on the wave speed in case of water hammer / V.V. Mozharovsky, S.V. Kirhintsava // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 21–25. – DOI: https://doi.org/10.54341/20778708_2023_3_56_21 (in Russian). – EDN: BTLCZN

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## Multi-criteria optimization of quartz glass laser cutting parameters using neural network simulation and genetic algorithm

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Abstract. Using neural network modeling and a genetic algorithm, the values of technological parameters were determined that ensure efficient laser cutting of quartz glass when the workpiece is exposed to a laser beam with a wavelength of 10,6 μm and a coolant. The multi-criteria optimization of laser cutting of quartz plates was performed according to the criteria of maximum ten-sile stresses and maximum processing speed. The algorithms for choosing the optimal architecture of neural networks are described.

Keywords: neural network modeling, laser cutting, genetic algorithm, parameter optimization.

For citation: Nikitjuk, Y.V. Multi-criteria optimization of quartz glass laser cutting parameters using neural network simulation and genetic algorithm / Y.V. Nikitjuk, V.A. Prokhorenko, A.I. Kulyba // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 26–31. – DOI: https://doi.org/10.54341/20778708_2023_3_56_26 (in Russian). – EDN: BXVAMJ

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## The thermally controlled terahertz hyperlens

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Abstract. The presented article proposes an improved design of a controllable cylindrical hyperlens based on alternating layers of indium antimonide and silicon. This design is intended for obtaining sub-wavelength resolution images in the terahertz range. Using numerical simulation, the ability of dynamic tuning of the hyperlens over a wide frequency range by varying the temperature is demonstrated. The key characteristics of this structure include small dimensions, low losses in the dielectric, and the ability to form images with super-high resolution. This research can contribute to enhancing the resolution of visualization systems in the terahertz range, as well as promoting the development of ultra-high resolution imaging and probing systems operating in this range.

Keywords: hyperlens, temperature, numerical simulation, THz range.

For citation: The thermally controlled terahertz hyperlens / I.A. Fanyaev, D.V. Slepiankou, A.Y. Kravchenko, I.V. Semchenko, J. Li, S.A. Khakhomov // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 32–37. – DOI: https://doi.org/10.54341/20778708_2023_3_56_32 (in Russian). – EDN: CCFYWF

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##
*l*-ary groups of special form. II

Powers in ## QR code

Abstract. The study on the powers in polyadic groups of special form is carried on.

Keywords: polyadic operation, *n*-ary group, power.

For citation: Gal’mak, A.M. Powers in *l*-ary groups of special form. II / A.M. Gal’mak // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 38–43. – DOI: https://doi.org/10.54341/20778708_2023_3_56_38 (in Russian). – EDN: EQNIHG

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##
*p*-length of a finite factorizable group with given permutability conditions for subgroups of factors

On the ## QR code

Abstract. A subgroup *A* of a group *G* is called *tcc*-subgroup in *G*, if there is a subgroup *T* of *G* such that *G=AT* and for any *X*≤*A* and for any *Y*≤*T* there exists an element *u*∈<*X*,*Y*> such that
*XY ^{n}*≤

*G*. Suppose that

*G=AB*is a product of two

*p*-soluble

*tcc*-subgroups

*A*and

*B*. We give a bound of the

*p*-length of

*G*from the nilpotent class and the number of generators of

*A*and

_{p}*B*where

_{p}*A*and

_{p}*B*are the Sylow subgroups of

_{p}*A*and

*B*respectively.

Keywords: finite group, *p*-solvable group, *tcc*-subgroup, *p*-length.

For citation: Zubei, E.V. On the p-length of a finite factorizable group with given permutability conditions for subgroups of factors / E.V. Zubei, A.A. Trofimuk // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 44–47. – DOI: https://doi.org/10.54341/20778708_2023_3_56_44 (in Russian). – EDN: GKWXXU

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##
*G*-network with many-lines systems with control and quarantine queues

Asymptotic analysis of ## QR code

Abstract. A method for finding average characteristics and their variances for a *G*-network in systems with control and quarantine queues in case of a big but limited number of customers in network is proposed. This network is a mathematical model of informational systems and networks, which consists of multiprocessor devices with antivirus software installed on each of them in case of heavy load in the network.

Keywords: *G*-network, control and quarantine queues, asymptotic analysis, antivirus software

For citation: Kopats, D.Y. Asymptotic analysis of *G*-network with many-lines systems with control and quarantine queues / D.Y. Kopats // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 48–55. – DOI: https://doi.org/10.54341/20778708_2023_3_56_48 (in Russian). – EDN: JQKVQE

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## Portable device for quantitative tremor parameters

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Abstract. The paper presents a developed portable device for detecting hand tremors from several accelerometer sensors integrated on an Arduino board with an ATmega32u4 microcontroller. A variant of using this portable device as part of a hardware-software complex for assessing tremor parameters in the early diagnosis and treatment of diseases of the central nervous system is also proposed.

Keywords: tremor, tremor diagnostics, accelerometer sensor, portable device, hardware and software complex.

For citation: Babrova, T.S. Portable device for quantitative tremor parameters / T.S. Babrova, M.V. Davydov // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 56–59. – DOI: https://doi.org/10.54341/20778708_2023_3_56_56 (in Russian). – EDN: MXDOUE

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## Modeling of plasma-chemical etching of silicon nitride functional layer on silicon dioxide sublayer in microelectronics technologies

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Abstract. Mathematical modeling of plasma-chemical etching of a silicon nitride film in the plasma of a gas mixture containing sulfur hexafluoride as a fluorine-containing gas in an amount of 70-91 vol. % and oxygen in the amount of 30-9 vol. %, at plasma power density *I*=0,2-0,4 W/cm^{2} and operating pressure *P*=4-8 Pa.

Keywords: plasma-chemical etching, silicon nitride, mathematical modeling.

For citation: Emelyanov, V.V. Modeling of plasma-chemical etching of silicon nitride functional layer on silicon dioxide sublayer in microelectronics technologies / V.V. Emelyanov, A.N. Kupo, V.A. Emelyanov // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 60–63. – DOI: https://doi.org/10.54341/20778708_2023_3_56_60 (in Russian). – EDN: PTZREH

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## Modeling of IR radiation influence on the phase composition of silicon dioxide coatings

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Abstract. Mathematical modeling of the interaction of radiation in the near infrared range with a coating of silicon dioxide was carried out at a pulse duration τ=0,05-0,5 seconds and exposure *E*=0,1 to 1,0 J/cm^{2}. An estimate is made of the decrease in the rate of plasma-chemical etching of a SiO_{2} coating due to an increase in the average binding energy in the crystal lattice due to thermal modification of the phase composition of the specified coating.

Keywords: plasma-chemical etching, silicon dioxide, infrared radiation, mathematical modeling.

For citation: Kupo, A.N. Modeling of IR radiation influence on the phase composition of silicon dioxide coatings / A.N. Kupo, V.V. Emelyanov, V.A. Emelyanov // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 64–68. – DOI: https://doi.org/10.54341/20778708_2023_3_56_64 (in Russian). – EDN: RUIKVY

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## Investigation of the cross-correlation of spectral components of cavitation noise

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Abstract. The results of the study of the cross-correlation of the spectral components of cavitation noise (CN), in particular, the fundamental frequency, i.e. frequency f0 of the ultrasonic field, subharmonics, harmonics of the fundamental frequency, the continuous component of the CN, and a number of others are proposed. It has been established that the fundamental frequency correlates weakly with other NC components, i.e. the relationship f0 with the other frequencies is characterized by a low correlation coefficient. A linear relationship has been found between the harmonics of the fundamental frequency, the continuous component of the spectrum, and the integral power of cavitation noise.

Keywords: cavitation, cavitation noise, spectral components, correlation.

For citation: Investigation of the cross-correlation of spectral components of cavitation noise / V.S. Minchuk, A.Yu. Perkhunova, V.S. Gavrilyuk, N.V. Dezhkunov // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 69–74. – DOI: https://doi.org/10.54341/20778708_2023_3_56_69 (in Russian). – EDN: TIQLKT

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## Nonlinear mathematical model of the heat and mass transfer process in diamond thermochemical processing technologies

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Abstract. A non-stationary nonlinear axisymmetric model of the process of laser thermochemical processing of diamond has been developed. The values of temperature and diffusion coefficients in the three-phase system "hydrogen – metal – diamond" are calculated. The diffusion coefficients are determined and the diamond removal rate is estimated in the range of heat source power densities *q*=104-107 W/m^{2} and metal coating thicknesses *h*=10 nm to 1 μm.

Keywords: laser processing, diamond, diffusion, carbon.

For citation: Shershnev, E.B. Nonlinear mathematical model of the heat and mass transfer process in diamond thermochemical processing technologies / E.B. Shershnev // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 75–80. – DOI: https://doi.org/10.54341/20778708_2023_3_56_75 (in Russian). – EDN: XCTKQN

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## Determination of the properties of the anisotropic medium over hydrocarbon reserves in the mode of amplitude-modulated signals

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Abstract. The analysis of the effect of amplitude-modulated signals on the characteristics of an anisotropic medium over oil and gas deposits based on a quasi-hydrodynamic approach is considered. The components of the surface impedance of the dielectric constant of the medium formed over the hydrocarbon deposits are investigated. The modeling of the characteristics of the medium above the accumulations of hydrocarbons for electromagnetic waves from the frequency of the carrier vibration, the amplitude modulation coefficient, the modulation frequency and the conductivity of the dielectric filler has been carried out. The modes of sounding of anisotropic media over hydrocarbons based on the changes in the absolute and phase components of the surface impedance of the underlying surface are recommended. The introduction of various methods and equipment is proposed to improve the accuracy of determining the boundaries of hydrocarbon deposits. The research results can be applied in exploration geophysics.

Keywords: hydrocarbon reservoir, electromagnetic wave, amplitude-modulated signal.

For citation: Determination of the properties of the anisotropic medium over hydrocarbon reserves in the mode of amplitude-modulated signals / V.F. Yanushkevich, S.V. Kalintsev, I.V. Sudzko, V.A. Bogush // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 81–87. – DOI: https://doi.org/10.54341/20778708_2023_3_56_81 (in Russian). – EDN: XIXGXE

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## Blended learning technology in the study of descriptive characteristics of a statistical sample

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Abstract. A substantive description of blended learning is presented, the main elements of the blended learning model and methodology are noted: rotation, flipped class, individual plan, flexible model. The analysis of information and communication technologies and Internet resources with data sets for solving the problem of statistical data processing is performed. On an example of solving the problem of determining the descriptive characteristics of the statistical sample there is used programming language R with a set of libraries: WDI, xtable, openxlsx, psych, sm, ggplot2, tidyverse, dunn.test, rstatix, ggpubr. The implementation of parsing data from the World Bank's open data web resource (data.worldbank.org) is shown. The script in the programming language R for finding descriptive characteristics of the statistical sample is developed. The technology of using information and communication technology for the educational process in the study of statistical methods of data processing is proposed.

Keywords: blended learning, information and communication technology, parsing, R programming language, descriptive data sampling characteristics.

For citation: Blended learning technology in the study of descriptive characteristics of a statistical sample / O.M. Demidenko, A.I. Yakimov, Y.A. Yakimov, K.G. Tishchenko // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 88–94. – DOI: https://doi.org/10.54341/20778708_2023_3_56_88 (in Russian). – EDN: XKUAHU

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## Method for calculating the adequacy parameters of image mathematical model

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Abstract. A structure of an artificial neural network for assessing the similarity of a pair of images, comparing the quality of its work with other criteria for image similarity are presented. Based on the proposed artificial neural network, a methodology for estimating the adequacy of mathematical image model by the estimation of its similarity to real images has been developed. A comparison of the estimation of the adequacy of the mathematical image model by the classical method and using the proposed methodology on the example of a Gaussian mathematical image model has ben conducted.

Keywords: image modeling, model adequacy estimate, artificial neural networks, similarity estimate.

For citation: Sergeyenko, A.V. Method for calculating the adequacy parameters of image mathematical model / A.V. Sergeyenko, A.Y. Liplyanin, A.V. Khijnyak // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 95–99. – DOI: https://doi.org/10.54341/20778708_2023_3_56_95 (in Russian). – EDN: YPKQTQ

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## Subjective aspects of digital images perceptions

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Abstract. The subjective perception of digital images were investigated. A digital image model at the macro, micro and information levels was proposed, and a description of the questionnaire for the study of the psychophysiological effects of digital images was given. In addition, other factors that affect subjective perceptions were also introduced in this paper.

Keywords: digital images, psychophysiological impact, questionnaire.

For citation: Zhang, Y. Subjective aspects of digital images perceptions / Y. Zhang // Problems of Physics, Mathematics and Technics. – 2023. – № 3 (56). – P. 100–104. – DOI: https://doi.org/10.54341/20778708_2023_3_56_100. – EDN: YVTACB

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