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Doklady of the National Academy of Sciences of Belarus, 2009, Vol.53, No.6

Homepage / Publications / Scientific Journals

Doklady of the National Academy of Sciences of Belarus, 2009, Vol.53, No.6

(Doklady Natsionalnoi Akademii Nauk Belarusi)

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

Volume 53, Number 6; November-December, 2009

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Sobolevsky S. L.
Painleve-classification of ordinary differential equations of arbitrary order with quadratic right-hand side. pp. 5--10
E-mail: stansobolevsky(at)

Summary: All ordinary differential equations in the complex domain of arbitrary order with quadratic in the dependent variable and its derivatives right-hand side possessing the strong Painleve property have been found. The classification contains 7 classes of equations, only one of which allows the order six or higher.

Rovba E. A., Mikulich E. G.
Constants in the approximation of |x| using the rational interpolation processes. pp. 11--15
E-mail: rovba(at)

Summary: Rational interpolation processes for |x| on [-1;1] and the behavior of their uniform approximations are investigated. Previously in 1989, in the articles by Rovba E.A. the similar approximation formulas were already investigated. In the present article we find exact constants in the above-mentioned approximation formulas. In order to find exact constants in the asymptotical behavior of uniform approximations Laplace's method was used.

Astrovskii A. I.
Transformation of linear time-varying observation systems with the scalar output to the Frobenius canonical forms. pp. 16--21

Summary: The necessary and sufficient conditions for linear time-varying observation systems with the scalar output to be transformed to the Frobenius canonical form under the action of the linear time-varying group of class C1 are obtained.

Kushel O. Yu.
Spectrum and approximations of one class of irreducible matrices. pp. 22--26
E-mail: kushel(at)

Summary: A new class of matrices with a certain sign-symmetric structure is introduced in this article. Such matrices are named J-sign-symmetric. The spectrum of a nonnegative irreducible matrix is studied under the assumptions that its second compound matrix is J-sign-symmetric and also irreducible. The conditions, when such matrices have complex eigenvalues and when all their eigenvalues are positive, are given. The present article also considers the question, when the approximation of a nonnegative matrix with a J-sign-symmetric compound matrix by strictly positive matrices with strictly J-sign-symmetric compound matrices is possible.

Shlyk V. A.
Combinatorial operations for generating vertices of integer partition polytopes. pp. 27--32
E-mail: v.shlyk(at)

Summary: The vertices of the integer partition polytope of a given integer form a generating subset for a set of all partitions and thus constitute a kind of basis for the latter set relative to taking convex combinations. We show that this "basis" can be considerably reduced. We introduce two combinatorial operations on partitions that recursively generate all vertices starting from the set of support vertices. The essence of the both operations is that some parts of a partition are combined into a new bigger part. So, we show that, from the combinatorial point of view, the vertices of the integer partition polytope are of different importance. The calculation testifies to the number of support vertices being considerably less than the number of vertices, which in its turn, is much less than the total number of partitions.

Zabreiko P. P., Korots Yu. V.
Analysis of the implicit successive approximations. pp. 33--38
E-mail: yulya.korots(at)

Summary: The theorem of convergence of implicit successive approximations to the fixed point for the non-linear operator is given. In the article the ball of non-existence of the fixed point for the non-linear operator, which satisfies the Lipshits condition on the family of embedded in each other balls, is found. The fact that the theorem mentioned above implies Lemarie-Rieusset theorem is stated.

Matysik O. V., Savchuk V. F.
Iteration procedure of an implicit type for solution of the operator equations in the Hilbert space. pp. 39--44
E-mail: matysik(at)

Summary: An implicit iteration procedure for solution of the operator equations in the Hilbert space is proposed. Convergence of the method is proved in the case of an a priori choice of a number of iterations, error estimates are obtained. The cases of nonuniqueness and convergence of the method in power norm are investigated. For the method under consideration, the opportunity of application of the rule neighboring approximations and of the rule residual stop is studied.



Zhestkov S. V.
Constructing soliton-like solutions of Zakharov's systems with the logarithmic laws of nonlinearity. pp. 45--49
E-mail: zhestkov_s(at)

Summary: The method of constructing soliton-like solutions of Zakharov's systems is developed.

Kudryashov V. V., Kurochkin Yu. A., Ovsiyuk E. M., Red'kov V. M.
Motion of a particle in the magnetic field in the Lobachevsky space. pp. 50--53
E-mail: kudryash(at); y.kurochkin(at); mozlena(at); redkov(at)

Summary: We study the motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field that is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motions are found and the equations of motion are solved exactly in the cylindrical coordinates. The motion on the surface of the cylinder of constant radius is considered in detail.

Onischuk A. G., Pegasin D. V., Tolkachev E. A.
Stability groups in the quadripoles theory. pp. 54--59
E-mail: tea(at)

Summary: It is shown that the classes of the quadripole power converter equivalence are determined by small groups of the R3.1 vectors. The projective operator realizing the optimal power conversion is constructed. In the framework of the biquaternion analogue of the Lorentz group vector parameterization the algebraic formalism is derived allowing the calculation of physical parameters of sequences of different converters in explicit form.

Feranchuk I. D., Skoromnik O. D.
Regularization of Mott scattering cross section in the Coulomb field. pp. 60--67
E-mail: OlEGSKOR(at)

Summary: It is shown in the present paper that the singularity of the differential cross section of relativistic electrons in the Coulomb field of the nuclei is appeared due to incorrect use of the asymptotic form of wave function within the small angle range. The correct solution of the Dirac equation has been used in the small angle range (Furry--Zommerfeld--Maye approximation). It leads to the regularization of the cross section in the entire angle range. The polarization of the scattered relativistic particles proves to be different from the polarization of the incident beam. It gives a principal possibility to distinguish the scattered particles from the incident ones and to measure experimentally the diffraction effects in the scattering of the charged particles.



Rudakov D. A., Shirokii V. L.,Knizhnikov V. A., Potkin V. I., Maier N. A., Starikova Z. A.
Electrochemical synthesis of tetramethylammonium salt 7,4'-dimethyl-8,8'-dibrom-bis(1,2-dicarbollyl)iron(III). pp. 68--71
E-mail: potkin(at)

Summary: Tetramethylammonium salt of 7,4'-dimethyl-8,8'-dibrom-bis(1,2-dicarbollyl)iron(III) was obtained with a preparative yield by the electrochemical method. The structure of substances was confirmed by the X-ray analysis.

Dikhtievskaya L. V., Makarevich N. A., Mozheiko F. F.
Influence of surfactants and inorganic electrolytes on interfacial tension in the oil/water system. pp. 72--76
E-mail: secretar(at)

Summary: The interfacial tension in the oil/water system in the presence of ionic surfactants and inorganic electrolytes has been investigated. The peculiarity of the influence of the nature of surfactants, oil phase and the structure of their molecules and electrolytes with ions of different radius, valency, and hydration were determined. The criteria and conditions of formation of model emulsions based on paraffin hydrocarbons and technical emulsions based on vegetable oils were established.

Vasilevskaya A. V., Sergeev G. V., Gilep A. A., Usanov S. A.
PCR-targets for detection of the Mycobaterium tuberculosis complex species. pp. 77--81
E-mail: vasilevskaya(at)

Summary: The Mycobacterium genus includes a number of species. But just a few species are known as the causative agent of human tuberculosis and belong to the Mycobacterium tuberculosis complex. Therefore the main aim of tuberculosis diagnostics is the specific detection of specimens of the Mycobacterium tuberculosis complex. In this study we analyzed possible targets for detection of the Mycobacterium tuberculosis complex using PCR. The target gene loci were IS-elements (IS1081, IS6110), the RD-region, and the sequences of conservative genes MPB70 and Rv0577. The result of this work is the development of the PCR-procedure for detection of pathogens of the Mycobacterium tuberculosis complex.

Khripach V. A., Litvinovskaya R. P., Drach S. V., Averkava M. A., Zhabinskii V. N., Sviridov O. V., Pryadko A. G., Novik T. V., Matveentsev V. D. †
Immunoenzyme assay of (24S)-methylbrassinosteroids. pp. 82--85
E-mail: litvin(at)

Summary: Starting from castasterone phytohormone, the synthesis of haptenes was carried out for preparing castasterone-horseradish peroxidase and castasterone-bovine serum albumin conjugates. On the base of castasterone-bovine serum albumin conjugate polyclonal antibodies were obtained. The immunoenzyme test-system was developed for quantitative determination of 24S-methylbrassinosteroids (castasterone and brassinolide).



Guzenko E. V., Lemesh V. A., Khotyleva L. V.
Rhizogenesis in the in vitro culture in fiber flax (Linum usitatissimum L.) cultivars. pp. 86--89
E-mail: E.Guzenko(at)

Summary: The technology for rhizogenesis induction in the in vitro culture was developed in fiber flax cultivars by creating certain culturing conditions stimulating rooting. Regenerated shoots 2 cm in length were transferred to a rhizogenesis medium (MS with a half set of macro- and microsalts). Flax shoots were not deepened into the nutrient medium and were cultured on the boundary - nutrient medium/air. A basic period of rooting for all the studied genotypes amounted to 12 days and was observed on 7--18 days. The mean value of the rhizogenesis efficiency made up 51.6% for the fiber flax cultivars investigated. Rhizogenesis induction according to the developed technology allows successful production of functional roots in plant-regenerants of fiber flax.

Pavlyuchuk N. V., Voronkova E. V., Buloichik A. A., Makhanko V. L., Rusetsky N. V.
Screening of L-virus resistant genotypes of potato (Solatium tuberosum) using PCR-markers. pp. 90--93
E-mail: N.Pavlyichuk(at)

Summary: Screening of 48 collection potato accessions for PLRV resistance was performed by the PCR-marker N127 identifying the locus PLRV.1 related to this trait. Based on the PCR-analysis results, 25 genotypes carrying QTL PLRV.1 were selected. The forms selected for the presence of this locus are recommended to be used in breeding programs as potential sources of resistance to the L-virus.



Tsybovsky I. S., Veremeichyk V. M., Kritskaya S. V., Evmenenko S. A., Lobatsevich S. M., Pavlu-chenko A. V., Kartel N. A., Zhivotovsky L. A.
Autosomal DNA-marker reference database: possibilities of large modern Belarusian population genotypic sets analysis. pp. 94--99
E-mail: tsybovsky(at)

Summary: For the first time, the genotypic data from national forensic database of Belarus, including 9626 genotypes from 109 administrative regions and 13 prisons, were investigated in comparison with the data on ethnic Belarusians, Russians, Ukrainians, etc. The calculated polymorphism parameters showed the lack of statistically significant differences between Eastern Slavs. Comparison of allele frequency distributions in the population of Belarus and those in American Caucasians recommended by Promega and Applied Biosystems found significant differences. This finding demonstrated an inadequacy of using frequencies by these corporations at expert institutions of Belarus. Our results substantiated a necessity for constructing national reference DNA database for needs of forensic DNA-analysis and possibility of using, for this aim, large cumulative sets of the modern Belarusian population genotypes on the assumption of careful genetic data analyses.



Loginov V. F., Karataev G. I.
Tectonophysical nature of squalls in Belarus. pp. 100--104
E-mail: nature(at)

Summary: The comparative analysis of anomalies of squalls and geophysical fields in Belarus show that maxima of squalls are caused by the presence of simultaneously acting factors: vortex structure of the local area of the earth crust, deep fault junction, deep electroconductive zones and considerable gravitational energy.



Taek Won Kim, Gaponenko N. V., Stepanova E. A., Rat'ko A. I.
Thermally induced changes in Ba0.6Sr0.3Ca0.1TIO3 xerogel. pp. 105--109
E-mail: nik(at)

Summary: We investigated by means of FT-IR and XRD measurements thermally induced peculiarities of synthesis the Ba0.6Sr0.3Ca0.1TiO3 xerogel from three sols on the basis of titanium tetraethoxide (Ti(OC2H5)4), titanium tetrabutoxide (Ti(OC3H9)4), and titanium tetraisoproxide Ti(OC3H7)4. Ti(OC3H7)4 was considered as an adequate material for fabrication of ceramic powder because its xerogel revealed better homogeneity than Ti(OC2H5)4 and smaller particle size than Ti(OC3H9)4. Thermal processing at 200°C results in the evaporation of the remaining solvents from xerogel and further oxidation decomposition of the alkoxide radicals results in the appearance of carboxylate. Increasing the annealing temperature to 600°C results in decomposing carboxylate, and 750°C annealing results in complete formation of the perovskite phase.

Romanenkov V. E., Petyushik E. E., Afanasjeva N. A.
Calculation of the diffusion coefficient in the Al/Al(OH)3/H2O system under hydration hardening of pigment aluminum powder. pp. 110--113
E-mail: rom52(at)

Summary: In the present work the diffusion coefficient of the aqueous solution of Al(OH)4- ions from the surface of dissolving aluminum through the porous layer to the growing nanocrystallites is calculated using the physical-chemical kinetic model. It is shown that the use of the physical-chemical model allows one to greatly raise the calculation accuracy of the diffusion coefficient.

Minchenya V. T., Stepanenko D. A.
Investigation of linear vibrations of a two-step waveguide for ultrasonic thrombolysis. pp. 114--119
E-mail: vlad_minch(at); stepd(at)

Summary: The article presents the mathematical model allowing the investigation of linear longitudinal and flexural vibrations of stepped flexible waveguides for ultrasonic thrombolysis. Resonant curves of longitudinal and flexural vibrations of a two-step waveguide are traced for the given vibration frequency. Step lengths values providing the simultaneous resonance of longitudinal and flexural vibrations for the given frequency are determined. The validity of the proposed model is proved by the results of the finite elements modeling using the ANSYS software.



Sheiko I. P., Khodosovsky D. N.
Estimation of boars in viability and productive traits of the derived posterity at industrial pig complexes. pp. 120--123
E-mail: belniig(at)

Summary: An estimation of productivity and viability of meat breed and synthetic lines boars' posterity was carried out in the conditions of a large pig breeding complex. It was determined that the posterity of boar No. 40 of the Landrace breed of the Polish selection had the best characteristics. The average litter weight before slaughtering was 901 kg that was by 19.8% higher than that of the posterity of all the checked boars. Boars of 990 synthetic line and hybrid boar 990 line ´ Pietren had the average litter weight by 2.1--6.7% lower than that, on the average, for all boars.

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