We discuss the foreseeable sensitivity to Z'-boson effects of -pair production cross sections at the Large Hadron Collider (LHC). The results of the model dependent analysis are expressed in terms of constraints on such Z'-boson parameters as Z–Z'-mixing angle and its mass . It was shown that the process under investigation in the framework of the ATLAS experiment has the sensitivity to the mixing angle , which is similar or even better than that obtained during resonant experiments at LEP1 and SLC accelerators in processes.

Keywords: Z'-boson, Z –Z'-mixing, gauge cancellation.

Andreev Vasili V. – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

Pankov Alexander – Department of Physics, P.O. Sukhoi Gomel State Technical University, Gomel, Belarus

The Lagrangian in the covariant form and the equation of motion of the pion in the electromagnetic field, taking into account its electric and magnetic polarizabilities were defined in the formalism of Duffin – Kemmer on the basis of the principle of gauge invariance. The amplitude of Compton scattering was defined on the basis of the solution of the equation of the interaction of the pion with the electromagnetic field produced by the method of Green's function, taking into account the impact and polarizabilities of the pion.

Keywords: Compton scattering amplitude, field-theoretical approach, Lagrangian, electric polarizability, magnetic polari-zability, formalism of Duffin – Kemmer.

E.V. Vakulina – I.G. Petrovsky Bryansk State University, Novozibkov, Russia

N.V. Maksimenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

Creating low-dimensional systems and nanostructured composites in the form of functional, special glasses and oxide films based on the solid matrix and uniformly distributed in them highly organized semiconductors with sizes of 5–100 nm, is one of the most important areas of research in the field of synthesis and characterization of inorganic materials and dispersed systems. Nanostructures based on Ge–Si, despite the difference in lattice parameters of 4%, attracted the attention of engineers because of great strides in creating new advanced devices using quantum effects. There are light-emitting and photo detecting silicon-germanium devices in silicon technology to compete successfully with traditional optoelectronic materials. An effective way of obtaining inorganic materials with controlled optical-mechanical and physico-chemical characteristics is sol-gel process of formation of nanoscale particles and clusters dispersed in the silicate matrices of different types: monoliths, films, and quasi-spherical structures.

Keywords: sol-gel technology, GeO₂ sol-gel films, SiO₂ sol-gel films, optical properties, structure.

V.E. Gaishun – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

D.L. Kovalenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.V. Semchenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

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

The surface morphology and microstructure of copper Ì1 treated by the compression plasma flow have been investigated by the methods of optical microscopy and X-ray structural analysis. The correlation between the structure, microhardness and the amount of energy absorbed by the sample surface (5–18 J/cm²) has been found. The thickness of the modified layer is about 36 (at the absorbed energy density – 18 J/cm²).

Keywords: copper, compression plasma flow, surface morphology, microstructure.

A.P. Laskovnev – Physical-Technical Institute of the National Academy of Sciences of Belarus, Minsk, Belarus

N.N. Cherenda – Belarusian State University, Minsk, Belarus

A.V. Basalai – Physical-Technical Institute of the National Academy of Sciences of Belarus, Minsk, Belarus

V.V. Uglov – Belarusian State University, Minsk, Belarus

V.M. Anischik – Belarusian State University, Minsk, Belarus

V.M. Astashynski – A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Minsk, Belarus

A.M. Kuzmitski – A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Minsk, Belarus

In the framework of thermodynamic method the modifications of the Peng – Robinson equations of state are considered in the paper. Critical parameters of the modified equations are calculated and their reduced forms are found as well. On this basis the Joule – Tomson process is analyzed and the explicit forms of effect sign inversion curves are built.

Keywords: critical parameters, the Peng – Robinson equation of state, modified equation, reduced variables, the Joule – Tomson process, inversion curve, positive effect.

O.V. Novikova – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

G.Yu. Tyumenkov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

Computer simulation was carried out in the microwave range based on the finite element method for the frequency dependence of the ellipticity coefficient of reflected wave for single-turn and double-turn helices and two-dimensional arrays consisting of the same helices. The possibility of the transformation of the incident linearly polarized electromagnetic waves in a circularly polarized wave reflected from the two-dimensional array on the basis of a single-turn and double-turn helices was shown. Reflected circularly polarized wave is observed at frequencies close to the resonance and doesn’t depend on the orientation of the plane polarization of incident wave relative to the axis of the helices.

Keywords: simulation, transformation of polarization, single-turn and double-turn helices, ellipticity coefficient.

A.L. Samofalov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

I.A. Faniayeu – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

I.V. Semchenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

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

This paper presents finite element analysis based on the remeshing technique to predict laser-induced crack propagation in quartz glass. The simulation process is accomplished by the analysis of the coupled temperature and stress fields together with the criteria of fracture mechanics of the crack growth.

Keywords: crack, laser ñleaving, quartz glass.

E.B. Shershnev – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

Yu.V. Nikitjuk – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

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

Right-side resolvents are constructed for discrete weighted shift operator and estimates of their norms are given.

Keywords: discrete weighted shift operator, right-side resolvent, stable vectorial subset.

A.B. Antonevich – Belarusian State University, Minsk, Belarus

E.V. Pantsialeyeva – Belarusian State University, Minsk, Belarus

The paper deals with the method proposed for the inverse of some class of operators. The result is applied to some equations with closed operators. The examples are considered.

Keywords: Markov function, Krein class, functional calculus, closed operator, operator equation.

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

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

A subgroup H of a group G is called modular in G if H is a modular element (in sense of Kurosh) of the lattice L(G) of all subgroups of G. The subgroup of H generated by all modular subgroups of G contained in H is called the modular core of H and denoted by H_mG. In the paper a new criterion of the p-nilpotency of a group was obtained on the basis of the concept of the m-supplemented subgroup which is the extension of concepts of modular and supplemented subgroups respectively.

Keywords: finite group, p-nilpotent group, modular subgroup, modular core, m-supplemented subgroup, maximal subgroup, cyclic subgroup, Sylow p-subgroup.

V.A. Vasil’ev – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

A convolution theorem for the Markov – Stieltjes transformation is stated and its application to a class of singular integral equations is given.

Keywords: integral transform, Markov – Stieltjes function, convolution, integral equation.

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

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

The exponential queueing networks class with Jackson traffic of communications and with possibility of instant bypassing of nodes by communications with probabilities depending on their states is considered. The communications taking over by thenodes generate customers’ batches of random sizes and will be joined to these nodes. Customers’ batch of random size is served simultaneously, the size being determinated at the moment of completing the service due to the assemble-transfer batch service discipline. When customers batch service is finished it leaves the network and sends communication to other nodes or leaves the network with according to some routing matrix. Sufficient conditions are established for the product form stationary network distribution in terms of insulating nodes in fictitious random environment. As an example we considered the case when the sizes of customers’ batches taking over by nodes have a geometric distribution and the network state stationary distribution has the product form of factors which have shift quasigeometric distribution.

Keywords: queueing networks, batches movements, communications, bypassing of nodes, quasireversibility, product-form of shift quasigeometric distributions.

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

Zhao Yue – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The article deals with the cross-system of the two differential equations of the second degree and fourth-degree derivative of the variables. The necessary and sufficient conditions for the Painleve, property for solutions of the system are found.

Keywords: system of the differential equations, property of Painleve, simplified differential equation.

I.P. Martynov – Y. Kupala Grodno State University, Grodno, Belarus

O.N. Parmanchuk – Grodno State Agrarian University, Grodno, Belarus

V.M. Pecevich – Y. Kupala Grodno State University, Grodno, Belarus

Let be a class of finite groups which are the direct products of their Hall -subgroups corresponding to a given partition of a nonempty subset of the set of all primes. This class is a local formation. In this paper the properties of -hypercenter and -residual of a finite group are studied. It was shown that for a finite -group G the intersection of all normalizers of all maximal -subgroups for all i is the -hypercenter of G. As corollaries were obtained well-known properties of hypercenter and nilpotent residual of finite groups.

Keywords: finite group, formation of finite groups, local formation, -hypercenter, -residual.

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

This work is devoted to the study of finite groups with given properties of critical subgroups.

Keywords: group, formation, coradical, generalized subnormal subgroup, Shmidt group.

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

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

The known devices for tremor recording are considered and their drawbacks are analyzed. A system and methods for video recording of man’s extremities tremor are developed. The laboratory researches of medical therapy have been carried out. The tremor at Parkinson’s disease and at essential tremor has been spectrally evaluated.

Keywords: tremor, oscillations, Parkinson’s disease, essential tremor, amplitude-frequency analysis, neurology, video recording, video camera, video analysis.

S.K. Dick – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

A.S. Tereh – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

A.V. Smirnov – Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

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

Different kinds of malfunction or damage of the parts may occur in the operation of electrical equipment. It is therefore very important to detect defects in the early stages. The paper presents the method of identification of electrical equipment on the basis of the analysis of the products current. The results of experimental tests are given.

Keywords: identification, spectral noise, high harmonic, neural network.

D.I. Kuznetsov – Krivoy Rog National University, Krivoy Rog, Ukraine

A.I. Kupin – Krivoy Rog National University, Krivoy Rog, Ukraine