Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.Math. Sci., 2004, No.3
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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.Math. Sci., 2004, No.3

Proceedings of the National Academy of Sciences of Belarus (Vestsi Natsiyanalnai Akademii Navuk Belarusi)
SERIES OF PHYSICALMATHEMATICAL SCIENCESPublished by The Institute of Mathematics of NASB, Minsk, Republic of Belarus 
CONTENTS
MATHEMATICS
Kilbas A. A., Marzan S. A.
Approximate method for solution of a class of differential equations of fractional order. pp. 59
Summary: The Cauchytype problem for a nonlinear differential equation of fractional order with the Riemann  Liouville fractional derivative in a weighted space of Holderian functions is investigated. The method based on an approximation of the solution by summator polynomial operators, is applied to obtain an approximate solution of this problem. The estimate for an error of the approximate solution is established and an example is given.
Rybachenko I. V.
Approximation of functions with integrable derivatives by interpolational and rational operators. pp. 1016
Summary: The estimation of deviation of special rational interpolating operator in uniform metrics for functions, which have fractional derivative from L_{p}, was found. This estimation has the order which better than the best polynomial approximation.
Gromak E. V.
Generalized Y_{h}transform in L_{v,r}spaces. pp. 1721
Summary: Generalized integral transform involving Bessel function of the second kind in the kernel is studied in the weighted spaces of rsummable functions. Mapping properties such as the boundedness, the representations and the range of the considered transform are given, and the inversion formulae are established.
Monastyrnyi P. I., Shvakel A. I.
Datedriven orthogonal sweep method for net equations with strongly varying coefficients. pp. 2228
Summary: The construction and justification of the datedriven orthogonal method allowing to extend class of the solving net boundary problems with strongly varying coefficients and to provide the stability of the method under more weak than traditional restrictions are given.
Sabuts A. V.
Criterions of existence of even integral positive definite quadratic forms. II. pp. 2934
Summary: It is established, that nary even integral positive definite quadratic forms with invariant factors 1 ... , 1, q_{1}, q_{1}q_{2} exist if only if the values n, q_{1} and q_{2} satisfy one from follow conditions. First, n, q_{1} is odd, q_{2} is even, and q_{1} or 2q_{2} is not square. Second, n = ± l(mod8), q_{1} = (2k  1)^{2}, q_{2} = 2s^{2}, k = 1, 2,. .., s = 1,2, .. . Third, n = 0(mod4), q_{1} is odd, q_{2} = ±l(mod4), and q_{1} or q_{2} is not square. Fourth, n = 0(mod 8), q_{1} = (2k  1)^{2}, q_{2} = (2s  1)^{2}, k = 1, 2, ... , s = 1, 2, . . . Fifth, n = 0(mod4), q_{1} is even, q_{2} = ±l(mod4). Sixth, n = 2(mod4), q_{1} is arbitrary, q_{2} =  1(mod4). Seventh, n, q_{1}, q_{2} is even.
Kharin Yu. S., Maevskiy V. V.
Localmedian method of regression forecasting under outliers. pp. 3542
Summary: The localmedian method of forecasting under the regression model with outliers is analyzed. The breakdown point is evaluated, the asymptotic expansion for the probability distribution function and the risk of the localmedian forecast are given. Comparison of the robustness of the least square method and the localmedian method is made.
Yudin M. D.
On indispensable conditions of convergence of distributions of the sums of dependent random vectors with unrestricted dispersion. pp. 4346.
Summary: There are indispensable conditions of convergence of distributions of sums of dependent random vectors in case of unrestricted dispersion. The canonical representation of limiting distributions under the formula extending the formula Levy  Khinchin is indicated.
Vilanski Y. V., Lepin V. V.
Information leakage in mappings with different length images. pp. 4753
Summary: The cryptographic strength of mappings f : {0, l}^{n} ® U^{m}_{i=r}{0, 1}^{i}, is investigated. The exact expressions and estimations for the expected value of the information leakage of randomly selected mappings are presented.
Melnikov O. I., Vopnyarskiy A. V.
Realization of hypergraphs by trees of minimal diameter. pp. 5457
Summary: The algorithm which allow to construct the realization of hypergraphs by trees of minimal diameter with complexity O(min(VH,EH)EH å_{ei ÎHE} e_{i} proposed.
PHYSICS
Kovalev A. A, Zhdanovich S. N.
Features of the reliefphase holograms development on photothermoplastic materials by laser method. pp. 5862
Summary: It is shown that the developing of the reliefphase holograms on photothermoplastic materials with pulsed solidstate Nd:YAGlaser it is necessary to use intermediate temperature level situated between initial and recording temperature. The dependence of the hologram diffraction efficiency on laser radiation energy density is established.
Kats P. B., Feranchuk I. D.
Regularization of the total scattering crosssection at the Coulomb potential. pp. 6369
Summary: Angular distribution of the particles scattered by the Coulomb potential is considered on the basis of nonasymptotic representation of the exact solution of the Schrodinger equation. It is shown that in this case the differential scattering crosssection has no singularity at zero angle. The total and transport crosssections are calculated and proved to be finite values even for nonscreening Coulomb potential. The results are important for transport processes in plasma and impurity semiconductors.
Minkevich A. A., Feranchuk I. D.
Modification of Debye  Waller factor at large scattering angles. pp. 7075
Summary: The modified expression for Debye  Waller factor is derived for scattering of xrays from rough surfaces and interfaces. For the large values of scattering vector, the nonGaussian behavior of this factor is predicted. This effect is caused by natural limitation of roughness fluctuation amplitude and is found can be essential for interpretation of xray reflectivity and highorder xray diffraction data. Some experimental xray reflectivity measurements are interpreted using the expression derived.
Sevruk B. B.
Tuning performance of the sumfrequency irradiation generation in nonlinear KTP crystal. pp. 7680
Summary: The collinear threefrequency interaction of light waves in the biaxial KTP crystal is considered. Using analytical expressions for the angular dependence of eigen mode refractive indices and the Sellmeier dispersion formulas, the wavelengths satisfied phasematching conditions, are found. The tuning curves are constructed for the generation of sumfrequency irradiation in the X_{1}X_{2} and X_{2}X_{3} planes at different directions of the wave normal vector. The angular dependencies of pump and second harmonic wavelengths, as well as effective nonlinear coefficients are calculated.
Chorny A. D., Babenko V. A., Hierro J.
Gradient magnitude of turbulent passive scalar field fluctuations and its onepoint PDF. pp. 8186
Summary: The closed equation of scalar gradient magnitude PDF for turbulent mixing problem is obtained. The closure is built on base of assumption about Gaussian conditional joint PDF of components of velocity gradient and scalar gradient tensors. Numerical integration of the model equation is carried out and the obtained results are compared with direct numerical simulation of turbulent scalar field.
Karolik A. S., Luhvich A. A.
Finding the contribution of dislocations to the thermoelectric power of copper and silver. pp. 8790
Summary: On measurements the thermoelectric power induced by tensile plastic deformation the change of thermoelectric power per unit of dislocation density DS_{d}/N_{d} is fined for copper (99.998%) and silver (99.99%) at temperature close to the room one. The given measurements are the first attempt of defining the value DS_{d}/N_{d} in these metals and are in the rather good agreement with the earlier performed calculations.
Makovetskii G. I., Galias A. I., Demidenko O. F.
Structure, density and magnetic susceptibility of Mn_{1x}Fe_{x}Se (x £ 0.45) and Mn_{1x}Co_{x}Se (x £ 0.1) solid solutions. pp. 9193.
Summary: By solid state reaction method the Mn_{1x}Fe_{x}Se (x £ 0.45) and Mn_{1x}Co_{x}Se (x £ 0.1) solid solutions were synthesized. All samples have the cubic lattice like MnSe. The lattice parameter a decreases when the concentration of iron and cobalt increases. The density of all samples is calculated from Xray measurements and measured experimentally. The magnetic susceptibility for all solid solutions is determined in temperature interval 77  1000 K. The Neel temperatures of all investigated samples are about 180 K.
INFORMATICS
Benediktovich V. I.
Noncrossing kfactors in geometric graphs. I. pp. 9499.
Summary: It is proved that deciding the existence of a noncrossing 1factor in orthogonal geometric graph G with r(G) ³ 1, the existence of a noncrossing 3factor in 4regular geometric graph G with r(G) ³ 1 and the existence of a noncrossing 2factor in geometric graph G with r(G) ³ 1, where r(G) is the index of intersection of geometric graph G, are NPcomplete.
Adutskevich E. V., Likhoded N. A.
Necessary and sufficient conditions for preservation of dependencies when paralleling of algorithms. pp. 100105
Summary: Necessary and sufficient conditions for preservation of dependences when paralleling of algorithms are obtained. In many cases the approach suggested allows us to get less constraints which do not depend on external variables then the known ones.
Lis A. V., Sidorenko L. A., Goshko D A.
The infinite time ruin probability in the model of collective risk with Levy processes. pp. 106111
Summary: The infinite time ruin probability is considered. The nonclassical feature where the income premium process is stochastic process with independent increments as well as the aggregate claim. Under these conditions the theorem about infinite time ruin probability is proved, the equation for an adjustment coefficient is obtained. Special cases of stochastic processes with independent increments are considered. The expressions for an adjustment coefficients are obtained.
SHORT NOTES
Zhestkov S. V., Zabreiko P. P.
One way of constructing of invariant Banach space and global theorems of solvability Cauchy problem. pp. 112114
Summary: The way of constructing of invariant Banach space for matrix systems of ordinary differential equations is suggested. The global theorems of solvability Cauchy problem on the base of principles Banach — Cattioppoli and Kantorovich are obtained.
Sinkevich D. V.
On the calculation of the estimate of the initial state vector of the regular Markov chain. pp. 115117
Summary: Some statistical results about estimation of the initial state vector on the basis of the aggregated information are given.
Krushevski A. E., Kondratyuk V. F.
On the problem of equilibrium of the elastic rectangular with mixed boundary conditions. pp. 118120
Summary: Original solution of one problem of elasticity theory with Galiorkin method is presented.
ABSTRACTS OF DEPOSITED PAPERS
Shmuratko A. S.
Degree of convergence in the central limit theorem for the weakly dependent random variables. p. 121
SCIENTISTS OF BELARUS
Vitaliy Antonovich Tolkachev (On the occasion of the 70th birthday). pp. 122123
Click here to read this article in Russian (Win 1251)
Pavel Andreevich Apanasevich (On the occasion of the 75th birthday). pp. 124125
Click here to read this article in Russian (Win 1251)
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