The possibility of the visualization of the defects of a separate component of the thin composite diffractive optical element has been demonstrated. This element represents a thin substrate composition and a pair of combined elementary gratings. Possible variants of the defects visualization as the substrate and of the separate elementary gratings of the diffraction optical element have been considered. In-terference methods well developed in holographic interferometry of phase objects have been used. Experimental results have been presented.

Keywords: diffractive optical element, diffraction grating, light wave, visualization of the defect, interference pattern.

N.T. Avlasevich – Y. Kupala Grodno State University

A.M. Lyalikov – Y. Kupala Grodno State University

The expressions for vectors of the field, density of energy and density of energy flows are received for vector vortex beams of Laplace – Gauss. Analytically spatial structure of longitudinal and track densities of energy flows is investigated. The executed graphic modeling of crossflows of energy supplements analytical conclusions.

Keywords: vector beams, Laplace – Gauss beams, vortex beams, power properties.

S.S. Girgel – Department of Physics, F. Skorina Gomel State University

On the base of the matrix 10-dimensional formalism, behavior of a vector particle with a quadrupole moment is studied in presence of external uniform magnetic field. Three series of the energy levels related to bound states of the particle are found. To assign them physical sense for all values of the main quantum number n = 0,1,2,... one must impose special restrictions on quadrupole moment – they are formulated explicitly. Also, the problem of a neutral vector particle with quadrupole moment in presence of magnetic field is studied. At this, solutions of three types are constructed in terms of Bessel functions, in each case the influence of the quadrupole moment consists in presence of a scale factor in the argument of the functions, which depends on the quadrupole moment parameter and mag-nitude of the uniform magnetic field.

Keywords: vector particle, quadrupole moment, magnetic field, Duffin – Kemmer formalism, separation of the variables, exact solutions.

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

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

Y.A. Voynova – Kochishchansky secondary school

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

The liquid-phase method for obtaining ZnO nanostructures is presented. To form nanorods of zinc oxide, substrates with the seed coat were used. The influence of synthesis conditions and modifying substances on the morphology of ZnO nanorods was studied.

Keywords: zinc oxide, sol-gel method, seed layer, hydrothermal synthesis, nanorods.

A.V. Semchenko – 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

V.V. Sidsky – 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

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

V.A. Solodukha – JSC “INTEGRAL”, Minsk

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

N.S. Kovalchuk – JSC “INTEGRAL”, Minsk

Features of doping influence by nitrogen on optical properties of coatings TiO₂, ZrO₂, SiO₂ are defined. Differences of spectral depend-ences of transmittances and reflection, a degree of polarization of the coatings received under various conditions are established.

Keywords: electron beam evaporation, coatings of oxides, morphology, reflectivity, index of refraction, degree of polarization, nitrogen doping.

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

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

V.E. Agabekov – Institute of Chemistry of New Materials of the NAS of Belarus

D.G. Piliptsou – 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

For nonempty radical formation and a finite group G the following statement was proved: if there exist maximal subgroups of G containing but not containing that is and the factor group is solvable, then In particular, if and is solvable, then The corresponding consequences were obtained for products of nonempty radical formations, in particular for n is any natural number.

Keywords: radical formations of finite groups, products of radical formations, -radicals, intersections of maximal subgroups.

L.M. Belokon – Mogilev State University of Food Technologies

Finite groups having three subgroups with pairwise relatively prime indices are studied. In particular, the structure of a group is established with such subgroups that are supersoluble, have a nilpotent commutator subgroup, and so on.

Keywords: finite group, supersoluble group, hereditary saturated formation, -subnormal subgroup.

A.F. Vasil'ev – F. Skorina Gomel State University

T.I. Vasil'eva – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

K.L. Parfiankou – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

The structure of Hall subgroups of a finite group is considered. All Schmidt subgroups have rank 4.

Keywords: finite group, group of Schmidt, rank of the group, Hall subgroup.

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

A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Ô(G) is nilpotent. The main properties of B-groups are established and the group factorized by a primary and a B-group is studied. In particular, it is proved that if G=HK is the product of a B-subgroup H with a primary subgroup K, and if the order of the non-normal Sylow subgroup of H is not equal to 3 or 7, then G is solvable.

Keywords: finite group, B-group, primary group, product of subgroups.

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

It is proved that the derived subgroup of a finite group is nilpotent if and only if for all primary commutators a and b of co-prime orders.

Keywords: finite group, commutator, derived subgroup, nilpotent subgroup.

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

Let be some partition of the set of all primes and let G be a finite group. G is said to be -full if G has a Hall -subgroup for all i. A subgroup A of G is said to be -permutable in G if G is -full and A permutes with all Hall -subgroups H of G (that is, AH=HA) for all i. In this paper, we give a survey of some recent results on -permutable subgroups of finite groups.

Keywords: finite group, a Robinson -complex of a group, -permutable subgroup, -soluble group, -supersoluble group, -CS-group.

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

A.N. Skiba – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

A subgroup H of a group G is called nÔ-subgroup in G if there exists a normal subgroup K of G such that G=HK and It has been proved that any formation admits characterization by certain nÔ-subgroups of prime orders.

Keywords: finite group, normal subgroup, Sylow subgroup, complement, -coradical, nilpotent group, Abelian group.

D.A. Khadanovich – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

Some new estimates related to the gradient and derivative of an analytic or harmonic function in Bergman type spaces of analytic and harmonic functions in selfconnected domains are presented. Some new results related with L^p-norm estimates of the n-th order derivative or with the gradient of an analytic or harmonic function in Bergman type spaces of analytic and harmonic functions or such type multifunctional spaces in the case of simply connected domain in the complex plane are considered. New inequalities of a similar type are also presented, in which not only simple simply-connected domains, but their Cartesian product, are involved. Proofs of inequalities of a more complex type are derived either directly from simpler inequalities of the same type, or are completely based on some interesting estimates obtained in the course of their proofs. Such inequalities attracted the attention of various authors in recent years. The theorems given in the article can have various interesting applications in the theory of function of both one and several complex variables.

Keywords: gradient, derivative of analytic function, simply connected domains, product of domains, multi functional spaces, Whitney decomposition.

R.F. Shamoyan – Bryansk State University

S.P. Maksakov – Bryansk State University

The original architecture of the software package «Admission committee of F. Skorina Gomel State University» is considered, which is a set of desktop and web applications using a common database hosted on a group of SQL-servers.

Keywords: software package, SQL-server, desktop application, web application, framework, admission committee.

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

L.I. Korotkevich – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

E.U. Lukashov-Fursikov – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University

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

Problems of development, tool filling, the organization of information streams, usages of the integrated program complex of the composer of digital geological and geoecological models are considered. Examples illustrate advantages of an offered process engineering of assemblage of components of the complex by synthesis of program units of computer algebra systems and geographic information systems, possibilities of interactive graphics visualization, comparison of results.

Keywords: digital geological model, scattered set of points, approximating digital field, Computer Algebra System Mathematica, Geographic Information System Surfer, interactive graphics visualization.

V.B. Taranchuk – Belarusian State University, Minsk

An algorithm for faces recognition is described. The algorithm has two stages. The aim of the first stage is to detect faces on source images; the aim of the second stage is to create attributes describing detected faces. Those attributes are used to compare faces and decide whether there is the same or different person on different images.

Keywords: face recognition, Viola-Jones method, principal component analysis.

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

N.B. Osipenko – Faculty of Mathematics and Programming Technologies, F. Skorina Gomel State University