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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2003, No.2

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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2003, No.2

Proceedings of the National Academy of Sciences of Belarus
(Vestsi Natsiyanalnai Akademii Navuk Belarusi)
SERIES OF PHYSICAL-MATHEMATICAL SCIENCES

Published by The Institute of Mathematics of NASB, Minsk, Republic of Belarus

Number 2, 2003

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CONTENTS


MATHEMATICS

Beresnevich V. V., Vasilyev D. V.
On prime number generation. pp. 5--10

Summary: The improvement of some methods for generating prime number is proposed.

Bazylev D. F.
Indices of W-algebras with special ramification. pp. 11--13

Summary: It is proved that any W-algebra over the field R((t))(x) have the index which do not exceed 2.

Bazylev D. F., van Geel J., Yanchevskii V. I.
A special case of the local Pfister conjecture. pp. 14--16

Summary: Finite-dimensional central simple algebras of exponent 2 over the field R((t))(x) which are trivial at all real closures of their centers are considered. For special ramification point of such algebras the local Phister conjecture is proved.

Romanovskii Yu. Ya.
Invariant metric structures on homogeneous spaces, which are generated by proper Poincare group. pp. 17--21

Summary: The global pare (G, G) for proper Poincare group B+­ is constructed. G is universal covering group of B+­ and G is a semigroup of automorphisms of G. Homogeneous spaces which are generated by the global pare (G, G) are isomorphic to the spaces of figures in Minkowski space R14 . The problem of finding of invariant metric structures on such spaces has been solved.

Vorobei L. A.
On the properties of the lattice of transitive regular subgroup functors. pp. 22--24

Summary: The construction of group G is studied in case when width of lattice Regtr (F) with F = form G do not exceed of the number of prime divisor of the group G.

Gal'mak A. M.
n-Ary subgroup of identityes. pp. 25--30

Zherelo A. V.
Approximation formula of one class of continual integrals on quasimeasure in cylinder domain. pp. 31--36

Summary: The approximation formula for the some class of continual integrals on the quasimeasure in the cylinder domain is constructed.

Al'sevich V. V.
On the presentation of the solution of differential equations with the variable delay argument of the neutral type. pp. 37--44

Summary: The form of the presentation of solutions for the linear neutral differential system with the variable delay is given.

Levakov A. A.
Application of the method of Lyapunov functions of constant signs for the investigation of the stability of the semidynamical systems. pp. 45--51

Summary: It is proved that systems which are generated by the functional and generalized differential equation, evolutionary parabolic equations are so-called G-systems. G-systems are some class of the semicontinuous and semidynamical systems. The asymptotic stability criterion for the G-systems is derived. It is proved that from the asymptotic stability of the G-system follows the asymptotic stability of the any system which tends to the given G-system as t ® + µ.

Rasol'ko G. A.
The direct method of the approximate solution of singular integral equation with Cauchy-type kernel using the Chebyshev polynomials. pp. 52--58

Karandzhulov L. I., Stoyanova Ya. P.
Boundary value problem for singular perturbed impulse systems in a critical case. pp. 59--65

Summary: The asymptotic expansion for the linear singular perturbed differential equations with impulse actions effects and linear boundary conditions in a critical case is constructed.

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PHYSICS

Pilipovich V. A., Esman A. K., Goncharenko I. A., Kuleshov V. K.
Parallel optical calculations in binary sign bit system. pp. 66--69

Summary: The algorithms of parallel performing of basic arithmetic operations in optical computers on the base of spectrally sensitive fibre Fabry -- Perrot resonators in binary sign bit system with two-channel binary coding of digits has been considered. The proposed algorithms make it possible to perform the operations of adding, subtraction, multiplication without ripple-through carries and in parallel both on digits, and on operand matrices. Peak performance of data processing could be as high as 1012 operations per second.

Red'kov V. M.
Linear representations of spinor coverings of the full Lorentz group and intrinsic space-time parity of a fermion. pp. 70--76

Summary: The known ambiguity in determining intrinsic fermion parities is considered on the base of investigating possible linear single-valued representations of spinor coverings of the extended Lorentz group, describing in accordance with Cartan approach by matrix operations without complex conjugation. It is shown that every of four different groups, coverings of the full Lorentz group, has only two non-equivalent, 4-component, complex, spinor representations, which can be related to certain two sorts of physical fermions. In is shown that spinor coverings of partly extended Lorentz groups L+-­ and L+­¯ have only one type of non-equivalent representations, therefore in the framework of such partly extended spinor symmetry any determination for an intrinsic fermion parity cannot be found. It is established that only one group from four coverings of the full Lorentz group can be reduced by a similarity transformation to a real-valued form and therefore just this covering permits existence of Majorana fermions.

Tunyak U. M.
On the gravitational theory with tensor tetrahedron-like potential. pp. 77--79

Summary: The gravitational field in Minkowski space is described in terms of tensor tetrahedron-like potential. The expression for the metric energy-momentum tensor of gravitational field and non-gravitational matter is obtained.

Sinitsyn G. V., Khodasevich M. A.
Entropy analysis of information-rate density of passive optical communication systems. pp. 80--85

Summary: Information-rate densities in optical communication systems are investigated on the base of the number-state model. Results of numerical simulation show that the considered characteristics increases monotonically with increasing signal power, reducing passband width and environment temperature. For wide-band optical communication channels quantum-statistical peculiarities of information carriers should be taken into account at small enough signal powers when fluctuations can not be neglected.

Brinkevich D. I., Prosolovich V. S., Yankovskii Yu. N.
Interaction of oxygen impurity with rareearth elements in silicon. pp. 86--89

Summary: The processes of interaction of oxygen with rare-earth elements (Er, Yb, Gd, Ho, Lu, Dy) in Si have been studied by the methods of scanning electron microscopy, photoluminecence as well as of Hall effect measurements and infra-red absorption. Two types of rare-earth elements inclusions were detected for all lantanoids: aggregations of cell structures with dimensions > (1 - 5) µ having well defined boundary and blurred inclusions of less diameters. Oxygen concentration increases with the approaching to the inclusion boundary meanwhile the correlation versus the change of lantanoid concentration.

Petrashko V.V., Shishonok N. A., Gatal'skii G.V., Leusenko A. A., Kurochkin L. A., Vyshatko N. P.
Properties Bi1.6Pb0.4Sr2-xKxCa2Cu3Oy superconductor which obtained in the different synthesis conditions. pp. 90--93

Summary: The influence of different synthesis conditions on phase composition and superconducting properties of Bi1.6Pb0.4Sr2-xKxCa2Cu3Oу superconductor are investigated. The maximum value of critical current is reached with using of high pressure treatment of precursor on the appointed stage of synthesis process. Potassium input in content of started mixture improves of intergrain boundaries and critical current increase.

Akimov A. I., Savchuk G. K., Rubzov V. A., Letko A. K.
Rietveld method refinement of the crystal structure of the ceramic titanate prepared by the method of the cold compacting under a high pressure. pp. 94--98

Summary: The influence of the high pressure of cold compacting on the change of crystal structure parameters of lead titanate ferrielectric ceramics is investigated. The decrease of the value of the displacements of titanium atoms as well as of the volume and tetragonality degree of the unit cell, and the increase of the distortion of the oxygen TiO6-octahedra with increasing high pressure of cold compacting were established. It is shown that the increase of the high pressure of cold compacting causes the monotonic decrease of the temperature of the ferrielectric phase transition.

Shipilo V. B., Dutov A. G., Shipilo N. V., Komar V. A., Azarko I. I.
Crystallization of diamonds in a melt of metal at use of annealed graphite in the carbonic gas atmosphere. pp. 99--103

Summary: It is established that the bakeout of graphite in atmosphere of carbonic gas (470 - 1070 l) diminishes a degree and speed of transformation of graphite in diamond in a system Mn -- Ni -- C at P = 5.5 GPa, T = 1620 K and t = 180 s. It is shown that a bakeout of graphite up to ~ 470 K reduces the contents in diamonds of manganese and nickel, augments their durability and density of paramagnetic imperfections Np1 and Npp. The obtained experimental data are considered from a point of view of the colloidal theory of diamonds synthesis and contact with change at a bakeout in a crystallizing melt of the contents CO2 or products of its interaction with components of a melt.

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INFORMATICS

Tiunchik A. A.
Fine-grained systolic implementation of discrete logarithm problem. pp. 104--108

Summary: Pure systolic array (suitable for VLSI or FPGA implementation) such that the whole procedure of the Pollard's r-method can be carried out entirely by the single systolic unit is constructed.

Demidenko V. M., Metelskii N. N.
Quadratic assignment problem and packing graphs. pp. 109--112

Summary: The approach to construction of new heuristic algorithms for the quadratic assignment problem is developed. The approach is based on the results about packing graphs and shortest trees under constrains on their configuration.

Matskevich A. N.
Mathematical models of the determined non-stationary perturbations in the class of regular processes. pp. 113--119

Summary: The techniques of synthesis and mathematical models of the determined non-stationary perturbations in the class of regular processes are developed.

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ABSTRACTS OF DEPOSITED PAPERS

Lukshin E. V.
On non-integer vertices of a class of three-index axial transportation polytopes. p. 120

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SCIENTISTS OF BELARUS

Boris Ivanovich Stepanov (On the occasion of the 90th birthday). pp. 121--123
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