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Abstract. The data on the electrical properties of SiO₂:Cu^o films are presented. The dependence of voltage on the solar irradiation intensity is presented. It was revealed that at increase in copper ions concentration in the thin films of SiO₂:Cu^o their electrical conductivity is decreasing due to the increase of the distance between the nanoparticles of copper distributed in the dielectric matrix of SiO₂. The films have the non-linear VA characteristic with the parts of voltage limits at the range of 7...10 V. At the same time, the photoelectric response was caused by generation of voltage of the order of temperature potential with the evident dependence on the solar irradiation. That means the possibility of the films application as the irradiation sensors.

Keywords: thin films, copper ions, silica, pulsed laser evaporation, volt-ampere characteristics, photoelectric, solar radiation.

For citation: Electrical properties of SiO₂:Cu^o thin films produced by pulse laser deposition / M.F.S.H. Al-Kamali, D.I. Zalizny, A.A. Boika, N.N. Fedosenko // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 7–11. – DOI: https://doi.org/10.54341/20778708_2022_2_51_7 (in Russian). – EDN: AWNUID

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Abstract. The compositions have been determined and a series of film-forming solutions has been prepared to study the effect of the methyltriethoxysilane concentration in tetraethylorthosilicate-based solutions on the electrical properties. The optimal modes of deposition and heat treatment of film-forming solutions for the synthesis of dielectric sol-gel coatings have been selected experimentally. The viscosity of the resulting solutions has been studied using a capillary viscometer. The studies of cleavage were carried out using a scanning electron microscope to analyze the thickness and homogeneity of the structure of the obtained materials. The capacitance-voltage and current-voltage dependences have been established. The calculations of the dielectric constant, the value of the voltage shift of the flat zones and the capacitance of the flat zones have been performed.

Keywords: film-forming solution, sol-gel, heat treatment, coating thickness according to SEM cleavage, capacitance-voltage characteristics, leakage current, dielectric constant.

For citation: Electrophysical properties of dielectric sol-gel coatings based on silica dioxide / V.V. Vaskevich, D.L. Kovalenko, V.E. Gaishun, V.V. Sidski, Y.A. Kosenok, D.V. Zhyhulin // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 12–17. – DOI: https://doi.org/10.54341/20778708_2022_2_51_12 (in Russian). – EDN: DVGXHB

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Abstract. The analytical expressions for Kummer circular 3D light beams without the Gaussian apodization with the continuous complex coefficient ν are obtained. The restrictions on possible values of the free parametres of such beams provided that they possess transferred power, have been discovered and physically realised. The pictorial modelling of such Kummer 3D beams has confirmed, that, indeed, there are certain continuous values of the free complex parametre ν for which Kummer beams trans-fer terminating power.

Keywords: paraxial beams, circular beams, Kummer beams.

For citation: Girgel, S.S. Kummer 3D light beams without the Gaussian apodization with transferable terminating power / S.S. Girgel // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 18–21. – DOI: https://doi.org/10.54341/20778708_2022_2_51_18 (in Russian). – EDN: GMTCSS

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Abstract. The approximate analytical solutions of the Logunov – Tavkhelidze equation in integral form with a linear potential in the relativistic configurational representation are found. The obtained wave functions are expressed in terms of the Macdonald function while the energy quantization condition is a transcendental equation.

Keywords: Logunov – Tavkhelidze equation, relativistic configurational representation, linear potential, Laplace transformation, approximate analytical solution, Macdonald function, energy quantization condition.

For citation: Grishechkin, Yu.A. Approximate analytical solution of the Logunov – Tavkhelidze equation with a linear potential in the relativistic configurational representation / Yu.A. Grishechkin, V.N. Kapshai // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 22–25. – DOI: https://doi.org/10.54341/20778708_2022_2_51_22 (in Russian). – EDN: IMQFYS

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Abstract. In this work, using the method of full factorial experiment and the method of enumeration of parameters, the process of laser polishing of quartz glass is optimized. As a result of finite element modeling, the modes of laser polishing of quartz glass using a CO2 la-ser were determined. These modes provide the specified temperatures in the treatment zone at the minimum values of thermoelastic stresses in quartz plates.

Keywords: laser polishing, quartz plate, ANSYS.

For citation: Optimization of laser polishing of quartz glass using the method of planning computational experiments / V.A. Emelyanov, E.B. Shershnev, Y.V. Nikitjuk, S.I. Sokolov, I.Y. Aushev // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 26–30. – DOI: https://doi.org/10.54341/20778708_2022_2_51_26 (in Russian). – EDN: NRYSQT

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Abstract. An axisymmetric bending of an elastic three-layer circular plate connected to the Pasternak foundation is considered. The effect of the temperature field is taken into account. To describe the kinematics of a plate package asymmetric in thickness, the hypothesis of a broken normal is accepted. The statement of the problem and its solution are carried out in a cylindrical coordinate system. The system of equilibrium equations was obtained using the Lagrange variational method. Its general analytical solution is written out in displacements. A numerical parametric analysis of the stress-strain state of a three-layer metal-polymer plate under a uniformly distributed load has been carried out. The significant influence of temperature on the stresses in the layers is shown.

Keywords: thermoelasticity, Pasternak foundation, three-layer plate.

For citation: Kozel, A.G. Thermal elastic bending of a circular three-layer plate, associated with the Pasternak foundation / A.G. Kozel // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 31–37. – DOI: https://doi.org/10.54341/20778708_2022_2_51_31 (in Russian). – EDN: QSYVXB

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Abstract. The energy coefficients of reflection and transmission of zero- and first-order diffracted waves with anisotropic Bragg diffraction by holographic phase gratings in cubic non-centrosymmetric crystals under Fresnel reflection conditions are theoretically investigated. It is established that at significant values of the amplitude of the perturbation of the refractive index of the medium, there is a high amplitude modulation of light in the transmitted and reflected diffraction orders. It is shown, that significant periodic changes in the transmission coefficients of diffracted waves of the zero and first orders with a change in the azimuth of polarization of incident light are determined by a significant difference in the Fresnel reflection coefficients at the layer boundaries for the s- and p-components of incident light.

Keywords: holographic grating, anisotropic Bragg diffraction, reflection and transmission coefficients, light modulation, polarization azimuth.

For citation: Kulak, G.V. Anisotropic bragg diffraction of light on holographic phase gratings under fresnel reflection conditions / G.V. Kulak, V.N. Naunyka, T.V. Nikolaenko // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 38-43. – DOI: https://doi.org/10.54341/20778708_2022_2_51_38 (in Russian). – EDN: RIXPHL

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Abstract. The problem of calculating the stress-strain state of orthotropic double-layer pipes made of composites and determining the wave velocity during a water hammer is solved, in relation to the modern CIPP “Cured-in-place pipe” technology. The dependences are derived that determine the wave velocity during a water hammer for various combinations of elastic orthotropic properties of the pipe and lining. The numerical results are given on the effect of cylindrical anisotropy of elastic properties of the lining (or pipe) on the wave velocity during a water hammer.

Keywords: wave velocity, water hammer, lining, stresses, layered pipe, orthotropic material.

For citation: Mozharovsky, V.V. The wave velocity during a water hammer and the stress-strain state of layered lined pipes made of orthotropic materials / V.V. Mozharovsky, S.V. Kirhintsava // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 44-51. – DOI: https://doi.org/10.54341/20778708_2022_2_51_44 (in Russian). – EDN: SGFXWP

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Abstract. The morphological features, phase composition, and mechanical properties of chromium-doped silicon-carbon coatings formed by ion-beam sputtering of a composite target based on silicon and chromium carbide are determined. It has been estab-lished that the introduction of chromium into the coating leads to a decrease in the oxygen concentration and contributes to the structural ordering of sp² clusters. Alloying with chromium Si/C coatings have a smoother surface and are character-ized by increased plasticity, lower values of friction coefficients and volumetric wear of the counterbody.

Keywords: carbon coatings, silicon, chromium, morphology, phase composition, hardness, plasticity, friction coefficient.

For citation: Structure and mechanical properties silicon-carbon coatings alloyed with chromium / A.S. Rudenkov, A.V. Rogachev, S.M. Zavadski, A.N. Kupo // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 52-57. – DOI: https://doi.org/10.54341/20778708_2022_2_51_52 (in Russian). – EDN: STABSW

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Abstract. Let π be some set of primes, G be a π-soluble group and . It is proved that if for any prime p∈π∩π(G) and Sylow p-subgroup P from G the normalizer N_G(P) is π-supersoluble and its nilpotent residual is semisubnormal in G, then G is π-supersoluble.

Keywords: finite group, Sylow normalizer, semisubnormal subgroup, nilpotent residual, π-supersoluble group.

For citation: Vasil'ev, A.F. On finite groups with semisubnormal residuals of Sylow normalizers / A.F. Vasil'ev // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 58-62. – DOI: https://doi.org/10.54341/20778708_2022_2_51_58 (in Russian). – EDN: UEJBDP

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Abstract. The article focuses on idempotents in polyadic groups of a special form. The main result was obtained for l-ary group of a special form, i. e. for polyadic group with l-ary operation η_s,σ,k, that is called polyadic operation of a special form and is defined on Car-tesian power A^k of n-ary group <A, η> by substitution σ∈S_k, satisfying the condition σ^l=σ, and n-ary operation η. As corollaries there were obtained the results for polyadic groups of a special form with (2s+1)-ary operation η_s,σ,k, which is defined on Cartesian power A^k of ternary group <A, η> by substitution σ∈S_k, which satisfies the condition σ^{2s + 1}=σ, and ternary operation η.

Keywords: polyadic operation, n-ary group, idempotent, substitution.

For citation: Gal'mak, A.M. Idempotents in polyadic groupoids of special form / A.M. Gal'mak // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 63-67. – DOI: https://doi.org/10.54341/20778708_2022_2_51_63 (in Russian). – EDN: UJHSDS

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Abstract. A nonlinear differential equation of the second order of the Riccati hierarchy is considered. The concept of a generalized solution for such an equation cannot be introduced in the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, an approach is considered in which the construction of a generalized solution of the second equation of the Riccati hierarchy is carried out using approximation by solutions of the Cauchy problem with complex initial conditions. The total number of generalized solutions depends on the initial conditions of the Cauchy problem, and their form depends on the location of the poles of the approximating solutions.

Keywords: generalized function, generalized solution of a nonlinear differential equation, Painleve property.

For citation: Kuzmina, E.V. Generalized solutions of the second equation of the Rickati hierarchy / E.V. Kuzmina // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 68-75. – DOI: https://doi.org/10.54341/20778708_2022_2_51_68 (in Russian). – EDN: UQMBEZ

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Abstract. For each nonempty Fitting class , Lockett defined the smallest Fitting class , containing such that for all groups G and H and the Fitting class as the intersection of all nonempty Fitting classes for which . Lockett pair of nonempty Fitting classes and is an ordered pair such that . If and is a Lockett class, then is said to satisfy Lockett conjecture in . In the present paper, in the universe of all finite soluble groups, the methods for constructing Lockett pairs are described for the case when is a generalized local Fitting class, and, in particular, for confirmed Lockett conjecture.

Keywords: σ-local Fitting class, Lockett pair, Lockett conjecture.

For citation: Lantsetova, E.D. On Lockett pairs and Lockett conjecture for &sugma;-local Fitting classes / E.D. Lantsetova // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 76-82. – DOI: https://doi.org/10.54341/20778708_2022_2_51_76 (in Russian). – EDN: WQAZZC

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Abstract. For time-invariant singularly perturbed control systems with state delay the method of separation of movements is evolved on the basis of Chang-type non-degenerate transformation. Asymptotic approximations for completely separated subsystems of the considered singularly perturbed system with three-time scales are introduced, boundaries of values of small singularity parameters are constructed and proved, which guarantee the validity of asymptotic representations and estimates of solutions underlying matrix operator equations, asymptotic approximations for the decoupling transformation and matrix operators of the split system. An illustrative example is given.

Keywords: singularly perturbed system, three-time-scale system, time delay, decoupling transformation, decomposition, asymptotic approximation, parameter estimate.

For citation: Naligama, C.A. Asymptotic approximations validity boundaries for decoupling transformation of three-time-scale linear time-invariant singularly perturbed systems with delay / C.A. Naligama, O.B. Tsekhan // Problems of Physics, Mathema-tics and Technics. – 2022. – N 2 (51). – P. 83-93. – DOI: https://doi.org/10.54341/20778708_2022_2_51_83 – EDN: ZLGNYA

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Abstract. In pre-Hilbert function spaces generated by the measures μ₁,...,μ_k, the process of polyorthogonalization of an arbitrary linearly independent system of functions {φ₀(x),φ₁(x),...,φ_m(x)} is described, which allows us to introduce the concept of the nth polyorthogonal function for an arbitrary multi-index n. Necessary and sufficient conditions are found under which this polyorthogonal function is uniquely determined, and its explicit form is described. The main theorem is a multiple analogue of the Gram–Schmidt orthogonalization theorem.

Keywords: linearly independent system, Pre-Hilbert spaces, polyorthogonal polynomials, perfect system, Gram – Schmidt orthogonalization.

For citation: Starovoitov, A.P. On polyorthogonal functions of the first type / A.P. Starovoitov, A.D. Kovalkova // Problems of Physics, Mathematics and Technics. – 2022. – N 2 (51). – P. 94-98. – DOI: https://doi.org/10.54341/20778708_2022_2_51_94 (in Russian). – EDN: ZVSWTZ

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