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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2006, No.3

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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2006, No.3

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

Published by Belaruskaya Navuka Publishing House, Minsk, Republic of Belarus

Number 3, 2006

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CONTENTS


MATHEMATICS

Kharin Yu. S., Maevskiy V. V.
On risk of forecasting under bilinear distortions of autoregressive time series. pp. 5--11

Summary: Sensitivity of forecasting under the first order autoregression model with bilinear distortions is analyzed in this paper. Asymptotic expansions for the risk of forecasting, guaranteed risk, instability coefficient and d-admissible distortion level are given.

Bazyleu D. F., Prokopchuk A. V., Tikhonov S. V., Shishkevich A. A., Yanchevskii V. I.
Computation of the cardinalities of the sets of rational points of the jacobians of certain curves. pp. 12--14

Summary: Explicit formulas for the cardinalities of the sets of rational points of the jacobians of certain curves are obtained.

Galmak A. M.
On covering group of polyadic group. pp. 15--20

Summary: The covering group of polyadic group is considered and studied.

Zadarazhniuk M. V.
On minimal t-closed w-composition non-nilpotent formations. pp. 21--24

Summary: A description of minimal t-closed w-composition non-nilpotent formations is obtained.

Vasilyeva T. I., Prokopenko A. I.
Finite groups with formationaly subnormal subgroups. pp. 25--30

Summary: Formations which contains groups with F-accessible Sylow subgroups are studied.

Alekhno E. A.
On some properties of the weak topology in space I>L¥. pp. 31--37

Tykoun A. S.
On a method of evaluation of the vector field rotation. pp. 38--44

Summary: A formula calculating rotation of the vector field is proved. This formula allows to reduce calculation of n-dimensional rofation to the calculation of (n - 1) dimensional one. As corollary of this formula the Poincare formula for 2D -rotation and the Rybakowsky formula for Conley index are obtained.

Chetyrkina Z. N.
About symmetryc spaces by Egorov of second and third lacunarity. pp. 45--47

Summary: It has been found all symmetrycal spaces Vn of the not fixed sign metrics by Egorov, operators for entire groups of movements and homotheties in them and some characteristic properties of the spaces.

Kaluguina M. A.
About the number of rational points near certain 2-dimensions surfaces. pp. 48--50

Summary: The lower and upper bounds are obtained for the number of rational points (with denominators not bigger a certain given limit) near a hyperbolic paraboloid.

Navakhrost V. G.
Differential equations with generalized coefficients in the algebra of mnemofunctions. pp. 51--56

Summary: Differential equations with generalized coefficients are considered. Existence and uniqueness theorems for the equations under investigation in the sense of differential inclusions are proved. Constructed differential inclusions are investigated in the algebra of mnemofunctions.

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PHYSICS

Bogush A. A., Redkov V. M.
On the four-dimensional vector parametrization of the group GL(4.C) and of some its subgroup. pp. 57--63

Summary: A way to parameterize the linear group GL(4.C) with the help of four independent four-dimensional complex vector parameters G = G(m, l,n, k) is firstly proposed. As a starting point, the linear decomposition of the transformation 4 Ч 4-matrices in terms of basic elements of the Dirac’s matrix algebra is used. When restricting to a simple subgroup G = G(m, 0, 0, k) the parametrization of the spinor covering for complex orthogonal group SO(4.C) is obtained. The latter is closely related with the known Fedorov’s parameterizing of the Lorentz group through two complex vectors. Connection between Einstein-Mayer study (1932) of semi-vectors and Fedorov’s treatment of the Lorentz group SO(3.1) theory are established firstly.

Ryabushko А. P., Nemanova I. Т., Zhur Т. А.
Relativistic equations of motion of test body in the gravitational field of a heterogeneous gas-dust-like globe with an attracting centre. pp. 64--71

Summary: The post-Newtonian equations of motion of test body in the gravitational field inside and outside of a heterogeneous gas-dust-like globe with an attracting centre have been derived.

Goncharenko I. A., Esman A. K., Kuleshov V. K., Pilipovich V. A.
Analog-digital conversion of optical signals by means of ring waveguide microresonators. pp. 72--78

Summary: We propose the fast method of analog-digital conversion of optical signals with the use of waveguide ring resonators of micron size. The results of calculation allowing optimizing parameters of optical analog-digital converter based on the method proposed for specific applications are presented. It is shown that conversion operating speed of such an ADC is defined by the time of steady output signal establishing. For microresonator of 10 µm radius this time is about 10 ps. Optical loss in microresonator results in reduction of its switching time. However, the switching contrast also reduces. To keep switching contrast constant it is desirable increase the step of discretization of the analog signal levels. The analysis of proposed optical method of analog-digital conversion shows that their use makes it possible to achieve frequencies of discretization above 109.

Minkevich A. A., Feranchuk I. D.
X-ray diffraction from superlattices. pp. 79--83

Summary: The analytical solution of recurrent equations for amplitudes of electromagnetic field is found for description of X-ray diffraction from periodical multilayered media. The method proposed uses the Bloch eigenwaves approach for periodical structure, which reduces considerably the computer time required for simulation of diffracted x-ray intensity and, therefore, accelerates the fitting trial-and-error procedure for sample model parameters. A new parameter describing the fluctuation of superlattice period is introduced and its influence on experimental data interpretation is discussed.

Zhestkov S. V.
Influence of dispersion and nonlinearity of higher orders on propagation of pulses in optical waveguides. pp. 84--87

Summary: For extension of mathematical possibilities of simulation of processes propagation of femtosecond optical pulses two new model are suggested. The method of constructing of soliton solutions is worked out. The necessary and sufficient conditions of existence of bright and dark optical solitons are obtained.

Akimov A. I., Letko A. K.
Preparation and investigation of properties of lead zinc niobate-based ceramics. pp. 88--91

Summary: 0.91Pb[(Zn1/3Nb2/3)0.91Ti0.09]O3 ceramics with the perovskite structure can be stabilized by substituting 6 mol.% BaTiO3 or PbZrO3. Specimens prepared by the conventional solid-state reaction of consistent oxides were investigated. Dielectric properties of the PZN – PT – BT ceramics when measured at a frequency of 1000 Hz were dielectric constant of 3400; dielectric loss of 0.38; Curie temperature of 170°C and of the PZN – PT – PZ ceramics 1200; 0.21; 270°C respectively.

Shipilo V. B. †, Dutov A. G., Shipilo N. V., Komar V. A.
Crystallization of diamonds in system Mn -- Ni -- C -- iron, Fe2O3. pp. 92--94

Summary: Behavior of degree and speed of the phase change of graphite in diamond, formations fractional structure of diamonds and strength in systems Mn -- Ni -- C iron, Fe2O3 depending on the synthesis time is established.

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INFORMATICS

Adutskevich E. V.
Algorithms of organization of data exchange during realization affine loop nests on distributed memory parallel computers. pp. 95--102

Summary: The problem of organization of communications for implementation programs on distributed memory parallel computers is considered. Algorithms to define communications (point-to-point, broadcast, translation) and to place operations of data sending and receiving in the code of parallel program are proposed. The algorithms have formalized form and can be automatized.

Khomitskaya T. G.
Open-loop and closed-loop solutions of linear-convex optimal control problem with terminal restriction. pp. 103--109

Summary: An optimal control problem for a linear nonstationary system is under consideration. The behavior of this system is estimated by the values of convex objective function on its terminal state that belong to the bounded convex set. The method of construction of optimal open-loop solution is proposed. It consists from the linearization of the considered problem and special finishing procedure. On the base of this method the algorithm of optimal controller work is realized. It allows to synthesis an optimal feedback control. Results are illustrated on the example of fourth-order dynamical system.

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SCIENTISTS OF BELARUS

Fyodor Ivanovich Fyodorov (On the occasion of the 95th birthday). pp. 110--112
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here to read this article in Russian (Win 1251)

Vladimir Gennadievich Sprindzhuk (On the occasion of the 70th birthday). pp. 113--115
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Faina Mikhailovna Kirillova (On the occasion of the 75th birthday). pp. 116--118
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Ivan Vasilevich Gaishun (On the occasion of the 60th birthday). pp. 119--121
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Sergey Vladimirovich Ablameyko (On the occasion of the 50th birthday). pp. 122--124
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