Modern classification of carbon nanotubes (CNT) is considered. A comparative analysis of the methods for their synthesis has been carried out; key technological parameters affecting the type and architecture of the formed CNTs have been identified. The most effective catalysts for the growth process of CNT have been established, the main methods of detection, the fields of application, and promising directions of introduction into production are given.

Keywords: carbon nanotubes, synthesis, catalyst, purification, application.

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

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

The article presents the results of studies of the dielectric characteristics of tantalum oxide films deposited by reactive magnetron sput-tering of a Ta target in an Ar/O₂ gas mixture. Dependencies of dielectric constant, dielectric loss tangent, band gap, leakage current density on oxygen content in Ar/O₂ gas mixture during film deposition were established. Films with a dielectric constant of 12–30 units, a dielectric loss tangent of 0.01, a leakage current density of less than 0.1 A/cm2 at an electric field strength of 2.0õ10⁶ V/cm, and band gap of 4.5–4.85 eV, were obtained.

Keywords: tantalum oxide, reactive magnetron sputtering, MOS structure, dielectric properties.

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

D.A. Golosov – Belarusian State University of Informatics and Radioelectronics, Minsk

T.D. Nguyen – Belarusian State University of Informatics and Radioelectronics, Minsk

The features of the formation of the molecular structure of the individual layers that make up the multilayer coatings, the processing of which leads to the formation of hydroxyapatite (HA). The efficiency of using polymethylsilsesquioxane (PMSQ) as a separate layer of a multilayer coating system is estimated. It is shown that the exchange processes between the multilayer system and the buffer solution are responsible for the destruction of the crosslinked organosilicon layer. Established the formation of calcium phosphates and zinc in the in-teraction of etidronate and phosphorus oxide (V) with the corresponding metals.

Keywords: hydroxyapatite, polymethylsilsesquioxane, etidronate, electron-beam dispersion.

Liu Yiming – Faculty of Physics and Information Technology, F. Scorina Gomel State University

A.V. Rogachev – Faculty of Physics and Information Technology, F. Scorina Gomel State University

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

A.A. Rogachev – Faculty of Physics and Information Technology, F. Scorina Gomel State University

Jiang Xiaohong – Nanjing University of Science and Technology

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

The composition of a multilayer coating on the basis of Mg + Ca + P₂O₅ applied to the surface of the intramedullary implant in order to initiate bone tissue growth has been proposed and substantiated. It is shown that the Mg + Ca + Ethidronate + P₂O₅ + Zn + P₂O₅ multi-layer system, which was heat-treated at 200⁰ C, has high resistance to abrasion. In SBF buffer solution the layer system is transformed into HA. Influence of heat treatment is manifested in increase of structural order of separate layers.

Keywords: hydroxyapatite, polymethylsilsesquioxane, etidronate, electron-beam dispersion.

Liu Yiming – Faculty of Physics and Information Technology, F. Scorina Gomel State University

A.V. Rogachev – Faculty of Physics and Information Technology, F. Scorina Gomel State University

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

A.A. Rogachev – Faculty of Physics and Information Technology, F. Scorina Gomel State University

Jiang Xiaohong – Nanjing University of Science and Technology

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

A new principle of physical quantities measurement is described, based on recording changes in the frequency of single optical pulses recirculation at different wavelengths in a closed fiber optic loop with their periodic regeneration in amplitude, shape and duration. It was established that the relative long-term instability of the recirculation frequency does not exceed 2õ10^-6 for fibers longer than 200 m with a measurement time of 1 s and an observation time more than 1 hour. Using numerical simulations, it was shown that the resolution of the temperature sensor is 0.15–0.1⁰ C in the temperature range 0–500⁰ C using quartz fiber with a metallic coating.

Keywords: fiber optic sensor, recirculation frequency, relative long-term instability, temperature, resolution.

A.V. Polyakov – Belarusian State University, Minsk

The canonical linear theory of a massive spinless field is adapted as special-relativistic gauge-invariant model of gravity with a quadratic self-action.

Keywords: scalar gravity, massive graviton, scale invariance, gravitationally dependent mass.

M.A. Serdyukova – Faculty of Physics and Information Technology, F. Scorina Gomel State University

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

Generalized tetraedron groups have a presentation of the form There exists a Rosenberger’s conjecture that the Tits alternative holds for generalized tetrahedron groups. This conjecture is open for groups of the form In this paper, a number of suf-ficient conditions are found for fulfillment the Tits alternative for groups

Keywords: generalized tetraedron group, Tits alternative, free group, almost solvavle group.

V.V. Beniash-Kryvets – Belarusian State University, Minsk

Y.A. Yushkevich – M. Tank Belarusian State Pedagogical University, Minsk

The factorization properties of the -projector introduced by V.A. Vedernikov and M.M. Sorokina in 2016 ( is a non-empty set of primes and is a non-empty class of groups) were investigated. Necessary and sufficient conditions are found for the equality for any -projector H and any normal -subgroups N₁ and N₂ of G, where G is an extension of the -group with the help of an -group.

Keywords: finite group, -projector, -saturated formation, -primitive closed homomorph.

T.I. Vasil'eva – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State Uni-versity; Belarusian State University of Transport

The problems of the continuous in the mean square dependence on the initial data of the solutions of systems of differential equations containing a stochastic differential equation in partial derivatives of a hyperbolic type and ordinary stochastic differential equations with delay, which are connected by delay connections, are investigated.

Keywords: stochastic differential equations, delay, stochastic partial differential equations, units with lumped parameters, units with distributed parameters.

S.P. Zhogal – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

S.I. Zhogal – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

Let be a Fitting set of a group G and Then is called a Fischer set of G, if and H/K is a p-subgroup of L/K for some prime p, then A subgroup F of a group G is said to be Fischer -subgroup of G if the following conditions are hold: (1) (2) if then Let be some nonempty set of prime numbers. A Fitting set of a group G is said to be -saturated if where is the class of all -groups. In this paper it is proved that if is a -saturated Fischer set of a -soluble group G, then a subgroup V of a group G is -injector of G if and only if V is a Fischer -subgroup of G, which contains Hall -subgroup of G.

Keywords: Fitting set, Fischer set, -injector, Fischer -subgroup of G.

T.B. Karaulova – P.M. Masherov Vitebsk State University

We establish new criteria of partial solubility of a finite factored group with restrictions on indices in chains of subgroups from factors to the group. In particular, it is proved that if A and B are soluble subgroups of a group G such that there exist chains from A and B to G in which indices of neighboring non-normal subgroups are either odd or equal to 2 or 4 and G = AB, then G is soluble.

Keywords: finite group, soluble group, factored group, generalized subnormal subgroup.

V.N. Kniahina – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

I.K. Chirik – University of Civil Protection of the Ministry for Emergency Situations of the Republic of Belarus, Minsk

The set of square matrices with positive principal leading minors and the similarity transformation of them using lower triangular matrices with positive diagonal elements are discussed. The criterion of the preservation of the principal corner minors' of a matrix positivity in the similarity transformation using the above-stated transformation is established. Moreover a sufficient condition of the preservation of the positive separation property for square above mentioned matrices in the similarity transformation is obtained. The received results are planned to be further used in the theory of controllability of asymptotic invariants of linear systems of ordinary differential equations.

Keywords: invariance, similarity transformation, principal leading (corner) minors of a matrix, positive separation, law-obedience of two matrices.

A.A. Kozlov – Polotsk State University

K.D. Kalita – Polotsk State University

A finite group G is called -special if where We say that a finite group G is semi--special if the normalizer of every non-normal -special subgroup of G is -special. We prove that if G is not -special but is -special for every subgroup A of G such that A is either a -group or a p-group for some then the following state-ments hold: (i) G/F(G) is -special. Hence G has a Hall -subgroup H and a soluble Hall -subgroup E. (ii) If G is not p-closed for each then: (1) H is normal in G and E is nilpotent. (2) is a maximal -special subgroup of G and every minimal normal subgroup of G is contained in F(G).

Keywords: finite group, -soluble group, -special group, Sylow subgroup, Hall subgroup.

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

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

V.N. Mitsik – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

V.N. Rizhik – Bryansk State Agrarian University

New concepts are introduced in the work. They are quite normal index and a quite perfect system of functions. Using these concepts, a uniqueness criterion was formulated and proved, explicit determinant representations of type II Hermite – Pade polynomials for an arbitrary system of power series were obtained. The results obtained complement and generalize the well-known result in the theory of Hermite – Pade approximations.

Keywords: Hermite – Pad’e polynomials, normal index, perfect system, Hadamard determinant, Hankel determinant.

A.P. Starovoitov – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

N.V. Ryabchenko – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

D.A. Volkov – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

Strict justification and extension to the linear stationary system with delay in the equation of state and in the output of one approach to reception of complete observability conditions for systems with delay is executed. The approach is based on the reduction of the problem of complete observability of the system with delay to the problem of uniqueness of the solution of a special homogeneous boundary value problem for a system of differential equations without delay in the extended state space. The necessary and sufficient conditions for complete observability, complete identifiability are proved. Complete observability, identifiability in the sense of unambiguous reconstruction of an unobservable piece of the trajectory on the time period of the delay length by the known output function are proved. The conditions are of rank type and are expressed in terms of matrices of the original observation system.

Keywords: complete observability, complete identifiability, time delay, output, criterion.

V.B. Tsekhan – Yanka Kupala State University of Grodno

A model of physical events to provide digital monitoring. RFID-objects is described in accordance with the requirements of international standards ISO/IEC 19987: 2017 Information technology - EPC Information Services (EPCIS) Standard and ISO/IEC 19988: 2017 Information technology - Core Business Vocabulary Standard. Both of these standards are designed to be used together. Using standardized vocabulary provided CBV standard, is critical to ensure unambiguous identification of RFID-objects using EPCIS services.

Keywords: EPC Information Services, Core Business Vocabulary, GS1, traceability in supply chains.

V.I. Dravitsa – Centre for Identification Systems, Minsk

A.V. Reshetniak – Centre for Identification Systems, Minsk

Å.À. Yakushkin – Centre for Identification Systems, Minsk