Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2009, No.3
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Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2009, No.3
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Proceedings of the National Academy of Sciences of Belarus
(Vestsi Natsiyanalnai Akademii Navuk Belarusi)
SERIES OF PHYSICAL-MATHEMATICAL SCIENCESPublished by Belaruskaya Navuka Publishing House, Minsk, Republic of Belarus |
Number 3, 2009
CONTENTS
MATHEMATICS
Zhuk E. E.
Statistical analysis of random Brownian motion processes returning to the fixed level. pp. 4--9
Summary: To investigate random processes returning to the fixed level the models of Brownian motion type are used. The maximum likelihood estimators of unknown parameters are constructed and their statistical properties are analytically investigated. The procedure of adequacy testing and forecasting is described.
Malugin V. I.
Asymptotic analysis of the risk of nonparametric classification in the case of essentially dependent features. pp. 10--23
Summary: The problem of classification of multivariate observations in the case of essentially dependent features is considered. The decision «plug-in» rules based on the nonparametric kernel density estimators with fixed and adaptive Gaussian kernel are suggested. Analytical investigations of decision rules are conducted for the case when the essential dependence of the features is described by the multivariate linear regression models with univariate distributed exogenous variables and Gaussian disturbances. Alternative decision rules are compared by means of asymptotic expansions of the conditional risk. The preference of the adaptive decision rule in the asymptotic of strengthening dependence of features and growing learning sample size is established.
Rasolko G. A.
Chebyshev's polynomials application for solution of the first-kind singular integral equation with Cauchy's kernel and a special right hand-side. pp. 24--30
Summary:
Samusenko A. V., Frolova S. V.
Multipoint schemes of the longitudinal variant of the method of lines of high order of accuracy for solution of some problems of mathematical physics. pp. 31--39
Summary: This article introduces differential-difference schemes of high order of accuracy for some problems of mathematical physics with one variable in space. The problem of approximation near boundary nodes is solved. For that, Hermite--Birkhoff's interpolation and a special type of interpolation near boundary nodes are used.
Vas'kovskii M. M.
Theorem of existence of b-martingale solutions to parabolic stochastic evolution equations with discontinuous right hand-sides. pp. 40--50
Summary:
Kukso O. S.
Properties of integer polynomials possessing small values on the sets of positive measure. pp. 51--53
Summary: In the article it is shown that the Diophantine approximation of zero by values of integer polynomials with small values of chosen coefficients is possible for the set with the measure less than the half of the given interval.
Shamukova N. V.
Relationship of resonance phenomena and hyperbolic Diophantine approximations. pp. 54--56
Summary: The article is devoted to the proof of the metric theorem on zero approximations by values of integer linear combinations of non-singular functions provided the convergence of the series that depends on the product of absolute values of these integer numbers.
Babul O. V.
Construction of the system of linear equations in the number field sieve algorithm for solution of the discrete logarithm problem. pp. 57--61
Summary: Theoretical justification for construction of the system of linear equations in the number field sieve algorithm for solution of the discrete logarithm problem is presented.
Galmak A. M.
Frattini's lemma for N-ary groups. pp. 62--67
Summary: The n-ary analogies of Frattini's Lemma are proved.
Korzyuk V. I., Konopelko O. A.
Weak solutions of boundary-value problems in tube domains for the fourth-order equation of composite type. pp. 68--75
Summary: One-valued solvability of boundary-value problems in tube domains for the fourth-order equation of composite type is proved in the article.
PHYSICS
Andreev V. V., Shishkina T. V.
Possibility of discrimination of non-minimal gauge models in the annihilation processes of polarized leptons. pp. 76--83
Summary: The matrix element of the annihilation process of polarized leptons was obtained in the lowest order of perturbative theory of electroweak interaction. An annihilation process of additional diagram, which corresponds to possible Z' - exchange, was included. It was reduced to scalar functions using the method of basis spinors. Differential and total cross sections, polarization and other combined types of asymmetries were calculated and analyzed. Several non-minimal gauge models were considered and analyzed. Probable kinematic and energy regions of deviations from the Standard Model were investigated and possible directions of deviations were suggested. Probable methods of models discrimination were considered.
Ovsiyuk E. M., Tokarevskaya N. G., Red'kov V. M.
Analog of plane waves in the spherical Riemann space and complex horispherical coordinates. pp. 84--89
Summary: In the space of a constant positive curvature S3, a complex analog of horispherical coordinates of the Lobachevsky space H3 is introduced. To parameterize a real space S3, two complex coordinates (r, z) must obey additional restriction. The metrical tensor of the space S3 is expressed in terms of (r, z) with additional constraint, or through pairs of conjugate variables (r, r*) or (z,z*); correspondingly, there exist three different representations for the Schrodinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of the Schrodinger equation in the complex horisperical coordinates of S3. Solutions constructed are single-valued, finite, and continuous functions in the spherical space and correspond to discrete energy levels.
Goncharenko A. M.
Propagation of optical solitons in absorbing media. pp. 90--92
Summary: Some propeties of the optical solitons in the absorbing media are considered.
Gusak N. A., Petrov N. S.
Kinetics of space-charge gratings in the photorefractive crystal with time variation of the exciting light field pattern. pp. 93--97
Summary: It is shown that an influence of an inhomogeneons light field changing with some freguency on a photorefractive crystal gives a space-charge grating changing with the same freguency. In spite of analogy to the oscilation process of the mechanical system, however in this case the resonance phenomenon is absent at any correlation of electric parameters of the crystal.
Ryabushko A. P., Zhur T. A.
Pioneer anomaly as the relict acceleration of a test body in the Solar system. pp. 98--104
Summary: The affirmation is substantiated: Pioneer anomaly is the relict acceleration of a test body in the Solar system.
INFORMATICS
Chebakov S. V.
Algorithms of organizing parallel calculations for some combinatorial tasks. pp. 105--113
Summary: The article deals with organizing parallel calculations for finding solutions of optimizational combinatorial tasks on the finite sets of the initial data. The solution of these tasks represents some subset of the initial data, and the inclusion of alternatives into the optimal subset is given by the definite rule which is used for forming the given task solution algorithm. The general scheme for the parallel calculations algorithm has been developed, and the condition of its optimization has been formulated. The realization of the proposed mathematical models has been considered by an example of solving two classical optimization tasks where different approaches to their application have been shown.
Benediktovitch V. I.
Albertson--Boutin conjecture concerning a distinguishing number of a clique. pp. 114--119
Summary: In this article we have found the conditions, under which the Albertson--Boutin conjecture concerning a distinguishing number of a geometric clique is valid.
SHORT NOTES
Likhoded N. A.
Program system for investigation of the parallel structure of algorithms. pp. 120--121
Summary: A brief information about the ParLoc system (the system is developed at the Institute of Mathematics of the National Academy of Sciences of Belarus) is given. The ParLoc system is a tool for parallelization and locality enhancement of loop nests.
SCIENTISTS OF BELARUS
Pavel Andreevich Apanasevich (To the 80th Anniversary of Birthday). pp. 122--123
Click here to read this article in Russian (Win 1251)
Yurii Semenovich Kharin (To the 60th Anniversary of Birthday). pp. 124--125
Click here to read this article in Russian (Win 1251)
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