The unified formalism which allows to deduce expressions for paraxial two-dimensional Gaussian-like light Kummer-Gaussian beams and Helmholtz-Gaussian beams is offered. This formalism also allows to discover correlations between them. Simple conditions of their physical realizability are formulated. New types of Kummer-Gaussian beams are discovered. Such beams are presented by the Gaussian product on the function Kummer of the complex argument and nonnegative integer index n.

Keywords: paraxial beams, Hermite-Gaussian beams, Kummer-Gaussian beams, Helmholtz-Gaussian beams, Gaussian-like beams.

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

The electrodynamical matter equations for continuous chiral medium in strong gravitational field are constructed. The influence of gravitation on propagation of electromagnetic waves in the medium with natural gyrotropy is considered. The received solutions for non-uniform electromagnetic waves allow to model the propagation of electromagnetic radiation in the matter of neutron stars.

Keywords: gravitational interaction, electromagnetic waves.

A.N. Serdyukov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.N. Yegorov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

All groups considered are finite. Let be a class of groups, be a local formation. We denote by the lattice of local formations concluded between and . If has finite length , then is called the -defect of . A local formation is called reducible if , where is the set of all nontrivial local subformation of . In this paper we obtain the exact description of irreducible soluble local formations with p-decomposable defect 3.

Keywords: finite group, class of groups, local formation, lattice, length of lattice, local chain, p-decomposable group, irreducible local formation, soluble local formation.

V.V. Aniskov – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The Mironenko V.I. reflecting function is applied to investigate the solution periodicity of differential systems.

Keywords: reflecting function, differential system, periodic solution, in period-transformation.

Å.V. Varenikova – Department of Physics and Mathematics I.G. Petrovski Briansk State University, Briansk, Russia

We investigate a finite group such that its -Hall subgroup is permutable with some subgroups of the group. For instance, we establish -solvability of such groups in two cases: a -Hall subgroup is 2-nilpotent; a -Hall subgroup is solvable and . Besides, we establish solvability of a finite group in the case when a 2-nilpotent -Hall subgroup of the group has odd index.

Keywords: finite group, seminormal subgroup, -Hall subgroup, -solvable group.

V.N. Kniahina – Gomel Engineering Institute of the Ministry for Emergency Situations of the Republic of Belarus, Gomel

The product of all normal subgroups of whose all non-Frattini -chief factors are cyclic is called the -hypercentre of . The following theorem is proved. Theorem. Let be soluble normal subgroups of . Suppose that every maximal subgroup of every Sylow subgroup of conditionally covers or avoids each maximal pair (M,G), where . If is either or , then .

Keywords: -hypercentre, supersoluble group, maximal pair, (conditionally) cover-avoidance property of subgroups, -subgroup.

V.A. Kovalyova – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.N. Skiba – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The theory of reflecting function is applied to investigate Riccati equation.

Keywords: differential equation, periodic solution, in period-transformation.

V.I. Mironenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

In this paper we study the inverse theorems of the theory of rational approximations. We prove the analogues of these theorems in the Bergman space of analytical functions in the circle.

Keywords: rational functions, the best rational approximation, Bergman space, rational approximation.

V.R. Misiuk – Department of Mathematics, Y. Kupala Grodno State University, Grodno

The questions of simulation models integration into the customers information environment is described. Simulation models are used as an example of the use and development of different input/output containers. The format of the output results conversion is also described.

Keywords: simulation model, containers, information environment, data exchange formats.

V.D. Liauchuk – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

P.L. Chechat – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

Transactional-process approach to the formalization of distributed information processing network in a multiprocessor computing system is described. The problem of distributed processing is represented as a network schedule. Dynamics of the interaction of network components with the use of the resources of its nodes is studied with the help of two types of simulation models: a model of network equipment and model of the workload, which has the type of a network schedule.

Keywords: simulation model, workload, distributed processing, multiprocessor computer system.

S.F. Maslovich – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

New approach to the research of probabilistic technological processes with discrete character of the technological cycle in which temporal intervals of separate operations implementation are random quantities is examined. The violation of the production process rhythm can become the cause of serious emergency situation. The method of formalization of probabilistic technological processes and imitation of the processes of technological cycle management on the basis of the aggregate system of imitation design automation is expounded.

Keywords: probabilistic processes, systems controls, probabilistic network schedule, simulation.

V.S. Smorodin – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.N. Goncharov – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.V. Klimenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.V. Zhihar – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The derived and nilpotent lengths, -length, -length, chief rank and -rank and etc. are invariants of finite solvable group. The review of the results connected with invariants of finite solvable groups is given in this article. These results adjoin to the researches of authors. Section 1 contains the list of used designations and definitions. The data about invariants of solvable groups with given structure of Sylow subgroups are collected in Section 2. The estimations of the invariants depending on indexes of the maximal subgroups are listed in Section 3. The information about nilpotent -length of -solvable groups are contained in Section 4. Open questions are formulated.

Keywords: solvable group, derived length, nilpotent length, -length, -length, nilpotent -length, chief rank, -rank.

V.S. Monakhov – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A.A. Trofimuk – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus