The conditions for the formation of sol-gel coatings for surface planarization were investigated depending on the composition of the film-forming solution. The optimal modes of applying a film-forming solution by spin-coating method were determined. The modes of heat treatment of the obtained coatings have been selected experimentally, and the effect of heat treatment on the thickness and continuity of the formed coatings has been established. By the methods of profilometry and scanning probe microscopy the roughness and planarization of the surface of the aluminum metallization of an integrated circuit using the obtained sol-gel coatings have been studied. To analyze the homogeneity of the layer structures of the deposited sol-gel coatings, studies of planarity and thickness were carried out us-ing a scanning electron microscope.

Keywords: film-forming solution, sol-gel, heat treatment, coating thickness, profilogram, planarization.

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

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

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

Sungwook Mhin – Kyonggi University, Seoul

M.I. Moskvichyov – Faculty of Physics and Information Technology, F. Skorina Gomel State University

A.N. Petlitskiy – JSC “INTEGRAL”, Minsk

D.V. Zhyhulin – JSC “INTEGRAL”, Minsk

The technologies for obtaining and modifying heat-insulating materials using nanosized silica are presented. The effect of a hydrophobizing emulsion upon the addition to the binder of mineral wool boards on their characteristics has been determined. A technology for the production of porous heat-insulating materials based on nanosized aerosils has been developed and their structural and mechanical properties have been studied.

Keywords: pyrogenic silica, hydrophobizing emulsion, mineral wool insulation boards, density, water absorption, thermal conductivity.

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

Ya.À. Kosenok – Faculty of Physics and Information Technology, F. Skorina Gomel State University

M.I. Moskvichev – Faculty of Physics and Information Technology, F. Skorina Gomel State University

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

T.A. Savitskaya – Belarusian State University, Minsk

I.M. Kimlenko – Belarusian State University, Minsk

I.A. Starostenko – JSC “GomelStroyMaterialy”

Analytical expressions in the closed form for nonparaxial and paraxial 3D Tricomi – Kummer (Ò – Ê) beams and other beams with continuous angular index m in parabolic rotary coordinates are offered and analyzed. Physical restrictions on possible values of free parameters of such beams are formulated. It is shown, that paraxial beams without Gaussian can transfer finite power.

Keywords: nonparaxial beams, paraxial beams, parabolic beams, Tricomi – Kummer beams.

S.S. Girgel – Faculty of Physics and Information Technology, F. Skorina Gomel State University

On the basis of the principle of least action, invariants are proposed for describing the processes of formation and evolution of the structure of the interface between metals under external influences. These invariants can be used, among other things, in the creation and destruction of nanomaterials.

Keywords: nano, submicro, micro, meso and macrolevels of deformation, equilibrium and nonequilibrium deformation, invariants.

S.V. Korotkevich – RUE “Gomelenergo”

V.V. Sviridova – Faculty of Physics and Information Technology, F. Skorina Gomel State University

A study of the relationship between the spectral characteristics of cavitation noise and the intensity of sound luminescence (SL) in the field of a focusing emitter was carried out. It was found that the highest degree of correlation with the SL intensity of the parameters studied is characteristic of the broadband component of cavitation noise, i.e., the intensity of the total output signal of the sensor minus all harmonics and subharmonics. This result allows us to conclude that the wide-band component of cavitation noise is generated by col-lapsing cavities and, therefore, can be used as an indicator of the level of transient cavitation activity.

Keywords: cavitation, sound luminescence, collapse of bubbles, cavitation noise spectrum.

À.V. Kotukhov – Belarusian State University of Informatics and Radioelectronics, Minsk

V.S. Gavriluk – Belarusian State University of Informatics and Radioelectronics, Minsk

N.A. Zharko – Belarusian State University of Informatics and Radioelectronics, Minsk

V.S. Minchuk – Belarusian State University of Informatics and Radioelectronics, Minsk

N.V. Dezhkunov – Belarusian State University of Informatics and Radioelectronics, Minsk

The results of the study of the structure, microhardness, and data from tribological tests of AISI 321 steel samples treated with nitrogen ions at various temperatures are presented. It is shown that the ion nitriding of steel forms hardened layers with a thickness of 3 to 18 microns, characterized by increased hardness (up to 1400 HV 0.05) and containing nitrogenous austenite as well as nitride phases and CrN. It is established that during the friction of steel samples with a hardened layer, the transition from the stage of gradual slow wear of the azotated layer to its accelerated wear occurs when the layer thickness decreases to 3 – 5 microns. It is concluded that accelerated wear of thin modified layers is caused by plastic deformation of the substrate and phase transformation in it, which leads to cracking of the surface layer.

Keywords: austenitic steels, nitriding, microhardness, phase composition, wear resistance, contact interaction.

V.A. Kukareko – The Joint Institute of Mechanical Engineering of NAS of Belarus, Minsk

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

A.V. Kushnerou – The Joint Institute of Mechanical Engineering of NAS of Belarus, Minsk

S.A. Marjin – F. Skorina Gomel State University

Using the reoxan holographic medium as an example, the energy reflection and transmission coefficients of diffracted zero and first order waves are theoretically investigated with an increase in the angle of incidence of light (decrease in the phase grating period) incident on the layer and photoinduced change in the refractive index of the medium. It is shown that with an increase in the amplitude of the perturbation of the refractive index of the medium, there is an amplitude modulation of the transmitted and reflected diffracted waves under conditions of Fresnel reflection from the layer boundaries. It has been established that significant periodic changes in the reflection coefficient of diffracted zero-order waves occur with an increase in the angle of incidence of light.

Keywords: holographic lattice, Bragg diffraction of light, reflection and transmission coefficients.

G.V. Kulak – I.P. Shamyakin Mozyr State Pedagogical University

V.N. Naunyka – I.P. Shamyakin Mozyr State Pedagogical University

T.V. Nikolaenko – I.P. Shamyakin Mozyr State Pedagogical University

The phenomenon of the appearance of a photoacoustic signal in a layer of carbon nanotubes under the action of irradiation with Bessel light beams (BLB) is studied for the case of piezoelectric recording. It was found that an increase in the taper angle of BLB affects the frequency of occurrence of resonance peaks depending on the radial coordinate

Keywords: photoacoustic transformation, piezoelectric detection, dense layer of carbon nanotubes, Bessel light beams, energy dissipation rate, amplitude of photoacoustic signal, Bessel function, heat equation.

G.S. Mityurich – F. Sñorina Gomel State University; Belarusian Trade and Economics University of Consumer Cooperatives, Gomel

E.V. Lebedeva – Belarusian Trade and Economics University of Consumer Cooperatives, Gomel

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

The problem of vector particle with electric quadrupole moment in external Coulomb field is investigated. After separation of the vari-ables, two systems of 4 and 6 equations respectively were derived. The terms due to the electric quadrupole moment are present in both systems. The system of 4 equations is reduced to a second order equation which contains two irregular points of the rank 3 and 2, and four regular points. There are constructed its Frobenius type solutions, convergence of involved series is studied. The transcendence condition for such solutions gives a physically interpretable formula for the energy levels. The system of 6 equations turns out to be complicated, in the nonrelativistic approximation it is reduced to two coupled equations of the second order, whence the equation of the fourth order follows. Frobenius solutions of this equation are constructed. Two types of solutions are identified that could correspond to the bound states of a particle.

Keywords: vector particle, quadrupole moment, Coulomb field, nonrelativistic approximation, Frobenius solutions, transcendence condition.

E.M. Ovsiyuk – I.P. Shamyakin Mozyr State Pedagogical University

A.D. Koral’kov – I.P. Shamyakin Mozyr State Pedagogical University

Ya.A. Voynova – I.P. Shamyakin Mozyr State Pedagogical University

The features of the structure of coatings formed by the method of electron-beam dispersion of a mechanical mixture based on cellulose and multilayer carbon nanotubes (MWCNTs) are considered. The effect of low-energy ion-plasma nitriding and the content of nanotubes in the initial target on the morphological features of cellulose-based composite coatings have been established. It has been shown by means of Raman spectroscopy that ion-plasma treatment can lead to partial restoration of the structure of MWCNTs, relaxation of defects in the inner and outer walls, and a change in the degree of crystallinity of the system as a whole.

Keywords: polyacrylamide, carbon coatings, morphology, phase composition.

A.S. Rudenkov – Faculty of Physics and Information Technology, F. Skorina Gomel State University

M.A. Yarmolenko – Faculty of Physics and Information Technology, F. Skorina Gomel State University

A.N. Kupo – Faculty of Physics and Information Technology, F. Skorina Gomel State University

In the framework of the thermodynamic approach to the study of macrosystems the two-parameter semi-empiric Ishikawa – Chung – Lu equation of state is considered. Using the Cardano method for the reduced form of equation of state the explicit form of its representations of the form is found.

Keywords: Ishikawa – Chung – Lu equation of state, cubicity by volume, Cardano method, reduced variables, representation of the form .

G.Yu. Tyumenkov – Faculty of Physics and Information Technology, F. Skorina Gomel State University

The effect of the treatment temperature on the structural properties of sol-gel films of samarium-doped BiFeO₃ was established. Optimal compositions of BiFeO₃:Sm films and their heat treatment modes have been determined. The effect of samarium (Sm) doping on the crystal structure and topography of the surface of thin films BiFeO₃ (BFO) and BiFeO₃:Sm (BFO:Sm) was investigated. X-ray diffraction (XRD) results for BFO and BFO: Sm films calcined at 600^o C revealed the single-phase state of BFO films (Sm:BFO) with the rhombohedral crystal lattice (space group R3c). The presence of samarium was confirmed by EDS. The small distortion of the crystal lattice is explained by the smaller ionic radius of the samarium ion replacing the europium ion.

Keywords: sol-gel method, topography, crystal structure, X-ray diffraction, films, bismuth ferrite.

S.A. Khakhomov – Rector of F. Skorina Gomel State University

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

V.V. Sydsky – Faculty of Physics and Information Technology, F. Skorina Gomel State University

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

A.N. Morozovska – Institute of Physics of National Academy of Sciences of Ukraine, Kyiv

The results of a qualitative analysis of a semiclassical model of radiation generation in solid-state lasers and, based on it, a numerical simulation of the regime of regular pulsations arising under conditions of nonlinear drift and broadening of the resonance gain line due to the influence of near fields of dipoles and absorption in quasi-resonant transitions on the dielectric susceptibility of an active medium are presented. Modeling was carried out for the parameters of semiconductor quantum-dimensional structures.

Keywords: lasing self-sustaining pulsations, nonlinear amplification, quasicrystal of quantum dots, dipole-dipole interaction.

Yu.V. Yurevich – Mogilev State University of Food Technologies

Specific features of electron-beam deposition, structure and properties of ZnO and Zn_xMg_1-xO coatings with the use of mechanical mix-tures of powders of zinc acetate, metallic zinc, and magnesium as a target are established. It is shown that the use of laser radiation, which assists the electron-beam evaporation of zinc acetate, and the subsequent heat treatment make it possible to deposit highly dis-persed thin coatings based on ZnO with stable values of the band gap. In this case, the use of assisting radiation with wavelength 266 nm due to the photochemical decomposition of the salt initiates the formation of less dispersed oxide phases. The introduction of magnesium into the composition of coatings does not noticeably change their molecular structure and leads to an insignificant increase in the band gap. Based on the data obtained, it was concluded that the formation of oxide phases was predominantly independent under the studied synthesis conditions and that the content of solid solutions in the coating composition was insignificant.

Keywords: zinc oxide, magnesium, morphology, phase composition, structure, electron beam dispersion.

M.A. Yarmolenko – Faculty of Physics and Information Technology, F. Skorina Gomel State University

A.S. Rudenkov – Faculty of Physics and Information Technology, F. Skorina Gomel State University

A.V. Rogachev – F. Skorina Gomel State University

E. Rusu – Gitsu Institute of Electronic Engineering and Nanotechnologies

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

V.V. Sydsky – Faculty of Physics and Information Technology, F. Skorina Gomel State University

The sufficient conditions under which almost periodic Abel and Riccati equations have almost periodic solutions were obtained.

Keywords: Abel equation, linear reflecting function, almost periodic solution, Riccati equation.

M.S. Belokursky – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

Throughout this paper, all groups are finite and G always denotes a finite group. Moreover, is some partition of the set of all primes , that is, , where and for all The group G is said to be: -primary if G is a -group for some i; -nilpotent if every chief factor of G is -central in G, that is, is -primary. The symbol denotes the -nilpotent residual of G, that is, the the intersection of all normal subgroups N of G such that is -nilpotent; is the -nilpotent hypercentre of G, that is, the product of all normal subgroups N of G such that either N=1 of and every chief factor of G below N is -central in G. A subgroup A of G is said to be -subnormal in G if there is a subgroup chain such that either or is -primary for all i=1,...,n. In this paper, we prove that if S be a -subnormal subgroup of G and for every subgroup E of G such that then

Keywords: finite group, -nilpotent group, -subnormal subgroup, -nilpotent residual of a finite group, -nilpotent hypercentre.

I.M. Dergacheva – Belarusian State University of Transport

I.P. Shabalina – Belarusian State University of Transport

E.A. Zadorozhnyuk – Belarusian State University of Transport

Let be some partition of the set of all primes and G a finite group. G is said to be -soluble if every chief factor H / K G of G is a -group for some i=i (H / K). We prove the following
Theorem. (i) If G is -separable, H is a nilpotent Hall -subgroup and E a -complement of G such that EX=XE for some subgroup X of H such that then
(ii) If G is -soluble and is a Wielandt -basis of G such that H_i permutes with H_j' for all i, j, then for all i.
(iii) If G is -soluble and is a Wielandt -basis of G such that of H_i permutes with for all i, j, then for all i.
(iv) If then QX=XQ each characteristic subgroup X of H and any Sylow subgroup Q of G such that HQ=QH.
(v) If G is -soluble with for all i and is a -basis of G, then each characteristic subgroup of permutes with each characteristic subgroup of H_j.

Keywords: finite group, -soluble group, -separable group, -length, Hall subgroup.

N.S. Kosenok – Belarusian Trade and Economic University of Consumer Cooperatives

V.M. Selkin – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

The number of points on a curve is calculated. The concept of m / k-power residue (rational power residue) is introduced. Let n be a natural number. The number of rational power residues modulo n is calculated. As a corollary the classic result on the number of quadratic residues is obtained.

Keywords: algebraic curve, number of points on an algebraic curve, power residue, primitive root, indices modulo

V.I. Murashka – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

A.A. Piachonkin – Moscow Institute of Physics and Technology

The critical -local formations of finite groups are studied, where is some partition of the set of all primes . A criterion is obtained for critical -local formations. A des140 cription of critical -local formations is given for formations of all Ï-groups, all -soluble Ï-groups, where and a description of minimal -local non--soluble formations of finite groups is obtained.

Keywords: finite group, formation -function, -local formation, local formation, critical -local formation, -soluble group.

I.N. Safonova – Belarusian State University, Minsk

The problem of computer modeling of two-dimensional surface forest fire spread is considered. The initial-boundary value problem in the form of a system of partial differential equations is described. The results of numerical experiments investigating scenarios how fire zone spreads near fuelbreaks in different directions are provided. The qualitative differences in the evolution of the temperature front at different stages of the combustion process at several characteristic sizes of forest fire breaks (glades), different representative values of wind speed in the forest canopy are revealed and shown.

Keywords: surface forest fire, wildland fire, wildfires, mathematical model, fire front dynamics, fuelbreak size, wind velocity.

D.V. Barovik – Belarusian State University, Minsk

V.B. Taranchuk – Belarusian State University, Minsk

In 2019, a new approach to optimizing network traffic was proposed as an alternative to the current IP system. It involves the segmentation of global network structures with a unique identification of services and a variable-length IP header. Ways of organizing global routing are considered in the article. The analysis of the current situation on the distribution of IP-spaces is carried out.

Keywords: aggregated addressing systems NewIP, IPv4, IPv6, IPv4-IPv6 tunneling, NAT, segment routing.

A.V. Varuyeu – Faculty of Physics and Information Technology, F. Skorina Gomel State University

V.D. Liauchuk – Faculty of Physics and Information Technology, F. Skorina Gomel State University

S.M. Kolaib – F. Skorina Gomel State University

Modern methods of biomechanical analysis of complex coordinated movements of an athlete are analyzed, including a full-scale experiment and the development of mechanical and mathematical models of purposeful movement. It is shown that one of the most difficult and time-consuming stages of the study is to obtain the trajectory positions of the athlete's body when performing the exercise. The features of using “motion capture” technology based on “computer vision” for these purposes are studied. The features of the development of mechanical and mathematical models of motion are shown, taking into account the method of obtaining the coordinates of the biomechanical system (BMS). Another method for obtaining spatial coordinates is proposed: using video shooting with cameras from 1 to 3 or more, with mathematical modeling of motion in spherical and Cartesian coordinates.

Keywords: biomechanical analysis, biomechanical system, computer vision, quaternions, locomotion, modeling, spatial motion, spherical coor-dinate system.

M.A. Kirkor – Mogilev State University of Food Technologies

A.E. Pokatilov – Mogilev State University of Food Technologies

A.M. Gal'mak –Mogilev State University of Food Technologies

Y.V. Voronovich – Mogilev State University of Food Technologies