The Korteweg-de Vries equation is considered in the two systems of coordinates: in a rest system and in a system moving with some velocity. One- and two-soliton solutions for the both equations are presented and analyzed.

Keywords: Korteweg-de Vries equation, one-soliton solution, two-soliton solution, Hirota direct method

M.A. Knyazev – Belarusian National Technical University, Minsk

The modification of the Anisimov – Lukyanchuk model to define the processes of dynamic condensation of Zel'dovich – Reiser for vapor-plasma jets formed during laser ablation of a silver target by intensive nanosecond pulses of optical radiation in a normal atmosphere is described. The initial stage of development of ablative silver jet with a high degree of accuracy corresponds to the theory of dynamic condensation of the Zeldovich – Raiser, and at the stage of disintegration of the ablation torch, it is necessary to take into account an additional mechanism of droplet formation – adhesion of the “nuclear”.

Keywords: ablative laser jet, adiabatic vapor expansion, dynamic condensation, silver nanoparticles.

K.V. Kozadaev – Belarusian State University, Minsk

The mechanisms of destruction of the surface layer of nickel during tribo-loading in a lubricating medium TsIATIM-201 are investigated. The structural-scale levels of deformation of the surface nickel layer were revealed. The evolution of structural-scale levels of deformation of the surface layer of nickel, leading to its phase instability, is shown. The dominant role and the initial role of the nanostructured scale level of deformation are established. The mechanisms of formation of defect structures of various scales are determined.

Keywords: dislocations, mechanisms of plastic deformation, stress fields, pores, slip bands, micro and main cracks.

S.V. Korotkevich – RUE “Gomelenergo”

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

The morphological features of two-layer coatings based on polyacrylamide and carbon, formed in a single technological cycle by electron beam and cathode-pulse methods, respectively, are considered. The effect of heat treatment in air on the morphology, phase composition, and structural transformations in such two-layer systems is established. It was shown that heat treatment of a polyacrylamide layer in air (300^o C, 60 minutes) leads to the complete disappearance of nitrile groups, which may be a consequence of cyclization.

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

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

Problems of reflection of a plane circularly polarized electromagnetic wave incident normally on a biisotropic layer on a reflecting substrate and reflection from a planar periodic biisotropic structure on a substrate are solved. Numerical and graphical analysis of dependences of the reflection coefficients on the thicknesses of layers is performed. The obtained dependences have been found to be periodic; these periods are found analytically. Combinations of parameters of the problem are determined at which reflection coefficients for electromagnetic waves of left and right polarizations coincide.

Keywords: biisotropic medium, biisotropic layer, substrate, planar periodic structure, reflection coefficient, matrix method, recurrence method, equivalent coefficient method.

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

V.N. Kapshai – Faculty of Physics and Information Technology, F. Scorina Gomel State University

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

The 7-layer polarization coating design based on SiO₂ and ZrO₂ layers deposited by electron beam evaporation onto substrates of various types has been developed. Determination of the effect of the nature of the substrate on the degree of polarization of the coating showed that the maximum degree of polarization of 0,75 transmitted radiation was achieved at an angle of incidence of 75^o when spraying the coating on a substrate of polyethylene terephthalate. It was found that the polarization coating is characterized by low roughness values of 5,41 nm.

Keywords: polarizer, refractory oxides, multilayer coating, electron beam evaporation, morphology, degree of polarization.

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

V.A. Emelyanov – JSC «Integral», Minsk

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

The results of the formation CdS thin films by chemical deposition using various thechnics of stirring an aqueous solution and the position of the substrate are presented. The structural, morphological, and optical properties CdS thin films were studied. X-ray diffraction patterns showed the peaks (111), (220), and (311) of CdS. The values of the band gap are in the range 2,39–2,48 eV, which corresponds to the published data. The use of methods allows the formation of thin films.

Keywords: thin films, CdS, physical properties.

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

V.F. Gremenok – Scientific and Practical Center for Materials of National Academy of Sciences of Belarus, Minsk

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

O.B. Aksyonov – Belarusian State University of Informatics and Radioelectronics, Minsk

V.K. Li – Belarusian State University of Informatics and Radioelectronics, Minsk

The intersections of maximal subgroups close to -abnormal, whose indices are not divisible by some prime numbers are investigated.

Keywords: finite group, abnormal subgroup, -residual.

R.V. Borodich – F. Scorina Gomel State University

R.A. Kucherov – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

The concept of an X-abnormal subgroup of finite groups is introduced and studied. Several attributes of the supersolubility of finite groups in terms of products of X-abnormal subgroups are established. In particular, the supersolubility of the group G=AB=CD is proved, where A and B are subnormal, and C and D are X-abnormal supersoluble subgroups in G.

Keywords: finite group, supersoluble subgroup, abnormal subgroup, X-abnormal subgroup.

A.F. Vasil'ev – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

Let be a set of primes. In this article, the operation on the formations of finite groups is introduced. If is a non-empty formation, then is the class of all groups G such that and every Sylow q-subgroup of G is strongly K--subnormal in G for The properties of are obtained, in particular, for hereditary formations . Hereditary saturated formations for which coincides with have been found.

Keywords: finite group, Sylow subgroup, normalizer of Sylow subgroup, hereditary formation, -subnormal subgroup, strongly K--subnormal subgroup.

T.I. Vasilyeva – Belarusian State University of Transport, Gomel

A.G. Koranchuk – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

The article goes on with a study of skew elements in polyadic groups of special form, that is in polyadic groups with l-ary operation , that is called polyadic operation of special form and is defined on Cartesian power of A_k n-ary group by substitution which order divides l–1 and n-ary operation In particular a theorem has been proved that allows us to determine a skew element for each element of ln-ary group on Cartesian power of which the given l-ary group is constructed.

Keywords: polyadic operation, n-ary group, skew element, inverse sequence.

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

The coincidence of generalized modules of smoothness of order k, determined with the help of the generalized shift operator, with dif-ferent and identical shifts is proved.

Keywords: generalized shift operator, generalized modulus of smoothness.

G.N. Kazimirov – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

The finite simple non-abelian groups all the maximal subgroups of which have odd index are studied.

Keywords: finite group, maximal subgroup, odd index, simple non-abelian group.

S.F. Kamornikov – F. Scorina Gomel State University

V.N. Tyutyanov – Gomel Branch of International University «ÌIÒSλ, Gomel

A finite group G such that in every maximal subgroup of G there is a subnormal in G subgroup is studied. In particular, we proved that the derived subgroup of G is nilpotent.

Keywords: finite group, subnormal subgroup, 2-maximal subgroup.

M.N. Konovalova – Russian Presidential Academy of National Economy and Public Administration, Bryansk

V.S. Monakhov – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

One way of application the reflecting function to the solution of the center-focus problem is given.

Keywords: reflecting function, differential equation, periodic solution, centre-focus problem.

V.I. Mironenko – Faculty of Mathematics and Programming Technologies, F. Scorina Gomel State University

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

New concepts are introduced in the work: an admissible index and an almost perfect system of functions. Using these concepts for an arbitrary system of power series of Laurent type a criterion for the uniqueness of an associated with this system of a polyorthogonal polynomial is formulated and proved. The explicit form of this polynomial is found, as well as the explicit form of polynomials standing in the numerator and denominator of the corresponding of Pade approximations. The propositions proved complement the well-known results the in theory of polyorthogonal polynomials and Pade approximations.

Keywords: Pade approximations, polyorthogonal polynomials, normal index, perfect system, Hankel determinant.

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

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

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

A mathematical model of query synthesis and web search of alternatives in the information support system for decision-making processes is proposed. Within the framework of the model, the problems of calculating the information content of words, sentences and texts, as well as the information content of the verbal association between them are solved. Descriptions of the algorithms for query synthesis, search for web pages and factual information are described.

Keywords: indexing, informativeness, information retrieval, mathematical model, decision making, query synthesis.

S.F. Lipnitsky – United Institute of Informatics Problems of the National Academy of Sciences of Belarus, Minsk