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 PHYSICALMATHEMATICAL SCIENCESPublished by Belaruskaya Navuka Publishing House, Minsk, Republic of Belarus 
CONTENTS
MATHEMATICS
Kharin Yu. S., Maevskiy V. V.
On risk of forecasting under bilinear distortions of autoregressive time series. pp. 511
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 dadmissible 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. 1214
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. 1520
Summary: The covering group of polyadic group is considered and studied.
Zadarazhniuk M. V.
On minimal tclosed wcomposition nonnilpotent formations. pp. 2124
Summary: A description of minimal tclosed wcomposition nonnilpotent formations is obtained.
Vasilyeva T. I., Prokopenko A. I.
Finite groups with formationaly subnormal subgroups. pp. 2530
Summary: Formations which contains groups with Faccessible Sylow subgroups are studied.
Alekhno E. A.
On some properties of the weak topology in space I>L_{¥}. pp. 3137
Tykoun A. S.
On a method of evaluation of the vector field rotation. pp. 3844
Summary: A formula calculating rotation of the vector field is proved. This formula allows to reduce calculation of ndimensional 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. 4547
Summary: It has been found all symmetrycal spaces V_{n} 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 2dimensions surfaces. pp. 4850
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. 5156
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.
PHYSICS
Bogush A. A., Redkov V. M.
On the fourdimensional vector parametrization of the group GL(4.C) and of some its subgroup. pp. 5763
Summary: A way to parameterize the linear group GL(4.C) with the help of four independent fourdimensional complex vector parameters G = G(m, l,n, k) is firstly proposed. As a starting point, the linear decomposition of the transformation 4 Ч 4matrices 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 EinsteinMayer study (1932) of semivectors 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 gasdustlike globe with an attracting centre. pp. 6471
Summary: The postNewtonian equations of motion of test body in the gravitational field inside and outside of a heterogeneous gasdustlike globe with an attracting centre have been derived.
Goncharenko I. A., Esman A. K., Kuleshov V. K., Pilipovich V. A.
Analogdigital conversion of optical signals by means of ring waveguide microresonators. pp. 7278
Summary: We propose the fast method of analogdigital conversion of optical signals with the use of waveguide ring resonators of micron size. The results of calculation allowing optimizing parameters of optical analogdigital 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 analogdigital conversion shows that their use makes it possible to achieve frequencies of discretization above 10^{9}.
Minkevich A. A., Feranchuk I. D.
Xray diffraction from superlattices. pp. 7983
Summary: The analytical solution of recurrent equations for amplitudes of electromagnetic field is found for description of Xray 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 xray intensity and, therefore, accelerates the fitting trialanderror 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. 8487
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 niobatebased ceramics. pp. 8891
Summary: 0.91Pb[(Zn_{1/3}Nb_{2/3})_{0.91}Ti_{0.09}]O_{3} ceramics with the perovskite structure can be stabilized by substituting 6 mol.% BaTiO_{3} or PbZrO_{3}. Specimens prepared by the conventional solidstate 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, Fe_{2}O_{3}. pp. 9294
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, Fe_{2}O_{3} depending on the synthesis time is established.
INFORMATICS
Adutskevich E. V.
Algorithms of organization of data exchange during realization affine loop nests on distributed memory parallel computers. pp. 95102
Summary: The problem of organization of communications for implementation programs on distributed memory parallel computers is considered. Algorithms to define communications (pointtopoint, 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.
Openloop and closedloop solutions of linearconvex optimal control problem with terminal restriction. pp. 103109
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 openloop 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 fourthorder dynamical system.
SCIENTISTS OF BELARUS
Fyodor Ivanovich Fyodorov (On the occasion of the 95^{th} birthday). pp. 110112
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Vladimir Gennadievich Sprindzhuk (On the occasion of the 70^{th} birthday). pp. 113115
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Faina Mikhailovna Kirillova (On the occasion of the 75^{th} birthday). pp. 116118
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Ivan Vasilevich Gaishun (On the occasion of the 60^{th} birthday). pp. 119121
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Sergey Vladimirovich Ablameyko (On the occasion of the 50^{th} birthday). pp. 122124
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