A general technique for calculating cascade processes in the framework of narrow resonances' improved approximation is considered. Possible experimental restrictions on the emission angles of final and intermediate particles were taken into account. An important distinguishing feature of the technique is the usage of the Poincare-invariant helicities for the final particles of the cascade reaction, as well as the method of basis spinors. As a result, the section of the cascade process ab → cd → 12 + 34 has a compact form.

Keywords: binary reaction, Feynman diagram, cross section, helicity, partial width, decay.

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

The paper presents the results of developments carried out using computer-aided design programs, additive manufacturing tools such as 3D printers, and numerical control equipment. Samples of metasurfaces were made and experimentally investigated for screening metal or metallized objects of complex shape and reducing the reflection of radiation from them.

Keywords: unit cell, metasurface element, omega element, electrical conductivity, transformation of radiation polarization, anechoic chamber, radiation absorption coefficient.

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

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

S.A. Khakhomov – rector of Francisk Skorina Gomel State University

I.V. Semchenko – vice-rector of Francisk Skorina Gomel State University

J. Wang – Jiangnan University, Wuxi

W. Song – Beijing Institute of Technology

A general theory for a field with spin 2 based on the 30-component system of first-order Fedorov – Regge equations is presented. As a result of the elimination of the additional vector and the third-rank tensor in these equations, the Pauli – Fierz second-order equations for the scalar and symmetric tensor are derived. Transition to the massless limit is analyzed in detail; the gauge symmetry available according to the Pauli – Fierz analysis is investigated. There are explicitly constructed solutions in the form of plane waves for a massive particle, which correspond to five linearly independent states. In the case of a massless field, 6 independent solutions are found, and it is shown that four of them are gauge ones and, therefore, can be excluded as nonphysical. Two independent solutions that do not contain gauge degrees of freedom are found explicitly.

Keywords: spin 2 field, second order equation by Pauli – Fierz, first order equation by Fedorov – Regge, plane waves, linearly independent solutions, massless particle, exclusion of the gauge degrees of freedom.

A.V. Buryy – B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus

A.V. Ivashkevich – B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus

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

V.M. Red’kov – B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus

The dispersion equation is obtained for plane monochromatic electromagnetic waves propagating in a biisotropic medium moving at a constant speed. Exact solutions of the dispersion equation are found in the case of wave propagation along or opposite to the medium motion direction.

Keywords: moving biisotropic medium, material equations, Maxwell equations, refractive index, dispersion equation.

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

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

Results of determination of efficiency of use of vacuum deposition and sol-gel synthesis of study for formation of nanocomposite coatings of ZnO_x : MgO with band gap width greater than 5 eV and with high sensitivity to UV and visible radiation are presented. It has been shown that spectral sensitivity of heterostructure with band gap width ZnO_x : Mg about 6.2 eV to UV radiation (278 nm) was from 0.3 A/W to 3.23 A/W at shift from 10 V to 14 V respectively.

Keywords: ZnO : MgO films, sol-gel method, electron-beam dispersion, band gap, volt-ampere characteristics, spectral sensitivity.

V.V. Malyutina-Bronskayà – SSPA “Optics, optoelectronics and laser technology”, Minsk

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

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

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

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

C.D. Danilchenko – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

S.A. Soroka – SSPA “Optics, optoelectronics and laser technology”, Minsk

E.V. Rusu – D.V. Gitsu Institute of Electronic Engineering and Nanotechnology of the Academy of Sciences of Moldova, Chisinau

The molecular structure, morphology, and radiation resistance of organosilicon coatings doped with copper or tungsten oxides have been determined. The coatings were formed from volatile products of electron beam dispersion of organosilicon resin and metal compounds mechanical mixtures. The copper- and tungsten-containing organosilicon coatings are characterized by a significantly lower content of methyl fragments. The effect of tungsten oxide is manifested in a decrease in the content of crosslinked fragments in the molecular struc-ture of the coating. One-component and metal-doped organosilicon layers are highly dispersed and defect-free. Prolonged exposure to UV radiation does not cause noticeable changes in their morphology and molecular structure.

Keywords: nanocomposite coatings, organosilicon coatings, copper acetate, tungsten oxide, electron beam deposition.

A.À. Rogachev – director of the Institute of Chemistry of New Materials of the National Academy of Sciences of Belarus, Minsk

Yiming Liu – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

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

Lihong Gao – Beijing Institute of Technology

Zhuang Ma – Beijing Institute of Technology

The morphological features and phase composition of carbon coatings reinforced with carbon nanotubes and formed with different pulse repetition rates of a cathode-arc carbon plasma source are considered. It is shown that reinforced carbon coatings are characterized by greater sub-roughness and relief differences (up to 500 nm) as compared to coatings unreinforced with nanotubes, which is due to the presence of MWCNT agglomerates and their random distribution over the surface. It was found that agglomerates of MWCNTs make a significant contribution to the formation of the final Raman spectrum, which is most likely due to an increase in the number of defects and amorphization of the MWCNT walls as a result of the action of a carbon plasma flow.

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

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

D.G. Piliptsou – Faculty of Physics and Information Technology, Francisk Skorina Gomel State University

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

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

The analytical solution of the boundary value problem of axisymmetric bending of a circular three-layer plate under creep conditions is obtained using the experimental-theoretical Ilyushin method. The physical equations of state of the hereditary theory of linear viscoelas-ticity are used. The similarity of the creep cores of the layer materials was assumed. The polyline hypothesis is used to describe the kinematics of a plate package that is not symmetric in thickness. In the bearing layers, the Kirchhoff hypotheses are valid. In a relatively thick placeholder, the Timoshenko hypothesis is accepted. The well-known solution of a similar problem in the theory of elasticity for such a three-layer plate is taken as the initial one. Numerical approbation of the obtained solution is carried out.

Keywords: three-layer plate, bending, similarity of creep nuclei, experimental-theoretical method.

E.I. Starovoitov – Belarusian State University of Transport, Gomel

Yu.M. Pleskatshevsky – Belarusian National Technical University, Minsk

A.V. Yarovaya – Belarusian State University of Transport, Gomel

Within the framework of the theory of thermoelasticity, a comparative analysis of the fields of thermoelastic stresses arising in multilayer glass structures (triplex) in the process of two-beam controlled laser thermal cleavage is carried out. In this case, a series of channels perpendicular to the surface of the material is preliminarily applied along the processing line. It is shown that in the presence of channels, the magnitude of the maximum tensile stresses increases significantly and depends on the radius of the channels, the distance between them and the processing parameters. A series of applied channels along the machining line allows control of crack initiation and propagation along the machining line along curved paths.

Keywords: thermal cleavage, modeling, finite element method, temperature, triplex, thermoelastic stresses.

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

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

A.A. Sereda – 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

The 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} is described.

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

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

The questions of the influence of random forces on oscillatory processes in systems described by differential equations containing both links with lumped parameters and links with distributed parameters and delay, connected by delay connections, are investigated. The step-by-step application of the averaging method of nonlinear mechanics and the method of Fokker – Planck – Kolmogorov equations to the study of such systems is described.

Keywords: stochastic differential equations, delay, stochastic partial differential equations, units with lumped parameters, units with distributed parameters, averaging method of nonlinear mechanics, method of Fokker – Planck – Kolmogorov equations.

S.P. Zhogal – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

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

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

For the functions H_γ=∑^∞_k=1 sin kx / (γ)_k, where (γ)_k =γ (γ +1) … (γ + k-1) and their trigonometric Padé approximations π^t_n,m(x; H_γ) the asymptotics of decreasing difference H_γ (x)- π^t_n,m(x;H_γ) in the case is found, where 0 ≤ m ≤ m(n), m(n) = î(n) as n→∞. Particulary, we determine that, under the same assumption,the trigonometric Padé approximations π^t_n,m(x; H_γ) converge to H_γ uniformly on the R with the asymptotically best rate.

Keywords: Padé approximations, asymptotic equality, best uniform approximation, trigonometric Padé approximations, rational approximations.

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

Throughout this paper, all groups are finite and G always denotes a finite group. Moreover, σ is some partition of the set of all primes P that is, σ={σ_i | i ∈ I} where P = ∪_{i ∈ I} σ_i and σ_i∩σ_j = ∅ for all i≠j. A set of subgroups of G is a complete Hall σ-set of G if every member ≠1 of is a Hall σ_i-subgroup of G for some σ_i∈σ and contains exactly one Hall σ_i-subgroup of G for every σ_i∈σ(G). A subgroup A of G is said to be: &sigma-permutable in G if G possesses a complete Hall σ-set such that AH^x = H^xA for all and all x∈G; σ-subnormal in G if there is a subgroup chain A=A₀≤A₁≤...≤A_i=G such that either or is σ-primary for all i=1,...,t. A subgroup A of G is said to be weakly σ-permutable in G if there is a σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G=AT and A∩T≤S≤A. In this paper it is proved that if in every maximal chain M₃<M₂<M₁<M₀=G of G of length 3 at least one of the subgroups M₃, M₂ or M₁ is either submodular or weakly σ-permutable in G, then G is σ-soluble.

Keywords: finite group, σ-soluble group, σ-subnormal subgroup, σ-permutable subgroup, weakly σ-permutable subgroup, modular subgroup.

V.M. Sel'kin – Faculty of Mathematics and Programming Technologies, Francisk Skorina Gomel State University

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

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

Possible methods are considered and a computational experiment technique for calculating the aerodynamic characteristics of feathered aerodynamic objects is proposed. Its verification was carried out on an object of simple shape (cone + cylinder) with subsequent matching of the obtained values for the object under study with the available experimental data. The shape coefficients and drag coefficients are calculated depending on the Mach number for an aerodynamic object with two forms of the head by the method of a computational experiment. Ballistic coefficients are obtained according to the laws of 1943 and 1958, as well as according to the Siacci law as functions of the Mach number. The obtained drag coefficients will be used for calculation of the initial data for launching the aerodynamic object.

Keywords: aerodynamic object, drag coefficient, body flow simulation, ballistic coefficient.

A.Ph. Ilyushchanka – State Research and Production Powder Metallurgy Association, Minsk

A.K. Kryvanos – State Research and Production Powder Metallurgy Association, Minsk

A.D. Chorny – A.V. Luikov Heat and Mass Transfer Institute of NAS of Belarus, Minsk

The developed algorithms for determining the emotions of interest in students by facial expressions using key points on the face image are presented. These algorithms allow organizing effective real-time control over the quality of the educational process in distance learning.

Keywords: algorithms, emotions, facial images, facial marks, classifier.

S.V. Sadau – Belarusian State University, Minsk

O.M. Demidenko – vice-rector of the Francisk Skorina Gomel State University

I.A. Yerafeyeu – Belarusian State University, Minsk

V.S. Sadau – Belarusian State University, Minsk