# Доклады Национальной академии наук Беларуси, 2003, Том 47, No.5

/ Издания академии / Научные журналы

Доклады Национальной академии наук Беларуси, 2003, Том 47, No.5

## СОДЕРЖАНИЕ

**МАТЕМАТИКА**

*Uss A. T.*
**On boundary-value problems for four-dimensional analogues of the Cauchy -- Riemann system with real coefficients.** pp. 5--8

*Summary:* A homogeneous system of four differential equations of order 1 with real constant coefficients in **R**^{4} is called Cauchy--Riemann's analog if each component of the solutions obeys the Laplace equation in W. Suppose W Ì**R**^{4} is a region with a
smooth boundary ¶W. It is proved that for any Cauchy-Riemann's analog in W, no such ¶W values exist, at which the boundary-value problem obeys the Lopatinskii condition.

*Suprunenko I. D.*

**The minimal polynomials of unipotent elements in irreducible representations of the special linear group over a field of characteristic two.** pp. 9--13

*Summary:* The minimal polynomials of the images of umpotent elements in rational irreducible representations of the special linear group over a field of characteristic 2 are found It appears that if the image of a representation does not coincide with the standard realization of the group, then the degrees of these polynomials are almost always equal to the orders of the relevant elements All exceptions are indicated explicitly The solution of a similar problem for the classical algebraic groups in odd characteristic has been announced by the author in this journal (2001. Vol. 45, N 3. pp. 41--44).

*Gromak E. V., Kilbas A. A.*

**Abelian theorems for integral transformations with Bessel function of the second kind and Struve function in the kernels.** pp. 14--18

*Summary:* Integral transformations containing Bessel function of the second kind and Struve function in the kernels are studied. It is proved that the functions represented by these integral transformations have power asymptotic behavior at infinity and zero provided that their density has power asymptotic at zero and infinity, respectively.

*Rasolko G. A.*

**Application of Chebyshev polynomials in direct method of solution of the vector singular equation with Cauchy kernel.** pp. 19--24

*Summary:* The following vector singular integral equation with cauchy-type kernel
A(x)j(x) + 1/pi ò^{1}_{-1}M(x,t) j(t)/(t-x) dt = f(x), -1 < x < 1,
is considered. In this equation A(x) = (a_{kj}(x))^{2}_{k,j}, M(x,t) = (m_{kj}(x,t))^{2}_{k,j} are complex-valued matrices given on the segment [-1, 1] with the elements being continuous by Helder, f(x) = (f_{1} (x),f_{2}(x))^{T} -- is vector from Helder class given on the segment [-1, 1], j(x) = (j_{1} (x),j_{2}(x))^{T} -- is an unknown vector.
The uniqueness conditions of the solution existence are found in this work. The computational scheme, that substantially uses the method of the decomposition of vector characteristic operator with the nonzero special indices by Chebyshev polynomials U_{n}(x), is constructed.

*Emelichev V. A., Kuzmin K. G.*

**Stability of effective solution of the vector combinatorial problem in the metric ***l*_{1}. pp. 25--28

*Summary:* The analysis of sensitivity of the vector combinatorial problem to perturbations of the parameters of the linear partial criteria in the metric *l*_{1} was earned out.

*Alekhno E. A.*

**Strictly positive functionals on Banach lattices with the countable supe property with unit.** pp. 29--32

*Summary:* The main result is the following: a Banach lattice *E* with a weak order unit, such that closure of the ideal generated by *E* in the Banach lattice *E*** is space with the countable sup property, admits a strictly positive linear functional and if only *E* is a space with the countable sup property. In particular, spaces of continuous functions on compacts with the countable sup property are considered.

*Levakov A. A.*

**Existence theorems for solutions of stochastic differential equations with discontinuous right side.** pp. 33--38

*Summary:* Various definitions of weak and strong solutions for stochastic differential equations with discontinuous right side are formulated. Existence theorems for such solutions are obtained. Examples are considered.

*Karatash H.*

**Normal diffential operators with constant -- operator coefficents.** pp. 39--41

*Summary:* In Hilbert space of vector-functions on the finite interval, the connection between constant-operator coefficents of linear formally normal differential operators for any order is investigated.

**ФИЗИКА**

*Kukharchyk P. D., Serdyuk V. M., Titovitsky I. A., Purovsky M. O.*
**Theoretical and experimental study of grain dielectric properties using coaxial microwave resonator.** pp. 42--45

*Summary:* A macroscopic theoretical model of calculation of grain material permittivity depending on its moisture in microwave range is proposed. The method of variation of grain form coefficients is employed. These values are established by the best approximation of theoretical results to experimental data. A device with a volume coaxial cylindrical microwave resonator was used for measurements on wheat grain samples of different moisture. The possibility of employment of such device for grain material moisture measuring in continuous working is shown.

*Sevruk B. B.*

**Numerical modeling of nonlinear conversion of wave beams.** pp. 46--49

*Summary:* A semianalytical method for numerical modeling of the formation of diffracting cylindrical wave beams during three-frequency interaction in quadratic nonlinear media is developed. The process of a sum frequency irradiation generation is considered in the regime of angular phase matching the coupled cylindrical modes. The near-field radial intensity profiles of light beams and their far-field angular spectra are calculated for the interaction in KTP crystal. It is shown that self-trapping spatial wave structures of the soliton-type with angular divergence of the second harmonic component at the diffraction-limited level are formed in the field of Bessel pump beams.

*Pozdnyakov D. V., Borzdov V. M., Komarov F. F.*

**Density of states in semiconductor quantum structures with low-dimensional electron gas and broadened energy levels.** pp. 50--53

*Summary:* An approach to calculation of the density of states in semiconductor quantum structures with low-dimensional electron gas taking into account broadening of the energy levels was developed. The thermal flotations of atoms in the crystal lattice and roughnesses of the confining quantum system surfaces were considered as mechanisms of the broadening. It is shown that the dependence of the density of states on the energy for quantum wire has no singular points.

*Bogush A. A., Kurochkin Yu. A., Otchik V. S.*

**Scattering by Coulomb field in the Lobachevsky space .** pp. 54--57

*Summary:* The quantum-mechanical problem on particle scattering by the Coulomb center in the three-dimensional Lobachevsky space is formulated and solved. The analog of the plane wave in the Lobachevsky space is used for description of the particle in absence of the Coulomb field. An exact solution of the Schroedinger equation in the coordinate system, which is an analog of the parabolic coordinates of flat space, is used for definition of the scattering amplitude. At large distances from the Coulomb center, this solution behaves like a superposition of the incident and scattered waves. An expression for the differential cross section is obtained.

*Gurskii L. I., Ivanov A. A., Feranchuk I. D.*

**On possibility of application of operator method in statistical physics.** pp. 58--61

*Summary:* The operator method and the cumulant expansion are used for the approximate calculation of the partition function and the free energy of the system of quantum anharmonic oscillators It is shown that the zeroth approximation of the method proposed interpolates these values with rather good accuracy in the entire range of both the Hamiltonian parameters and temperature. The method also allows one to calculate the corrections to the zeroth order.

**ХИМИЯ**

*Ratko A. I., Romanenkov V. E., Bolotnikova E. V., Krupenkina Zh. V.*
**Effect of bayerite thermal dehydration on adsorption-structural properties and mechanical strength of porous composite Al/Al**_{2}O_{3}. pp. 62--65

*Summary:* It has been established that cyclic treatment, every cycle of which includes treatment by water at 100 °C and thermal treatment at 550 °C, allows removing diffusion limitations and forming a porous composite having desired adsorption-structural characteristics and mechanical strength.

*Matulis Vadim E., Ivashkevich O. A., Gaponik P. N.*

**Quantum-chemical study ab initio of p-electron delocalization energy in 1-R-5-aminotetrazolium cations.** pp. 66--71

*Summary:* Quantum-chemical study of the molecular and electronic structures and p-electron delocalization energy in protonated on N_{(4)} atom molecules of 1-R-5-aminotetrazoles, where R = H (1), CH_{3} (2), NH_{2} (3), has been carried out using MP2/6-31G* and B3LYP/6-31G* levels of theory. 5-Amino group in the cations lies in the plane of the tetrazole ring and valence angles around the N atom are close to 120°, which is in a good agreement with the results of X-ray analysis of 1-methyl-5-amino- and 1,5-diamino-tetrazolium picrates. p-Electrons in the cations are delocalized between the all C--N bonds. The p-electron delocalization energy in 5-aminotetrazole molecule, estimated from the calculated rotational barrier of the 5-amino group, is approximately 15 KJ/mol lower than this one in cation 1. The reason is that the p-electron delocalization in cations 1--3, in contrast to neutral molecules, also includes the positive charge delocalization, which results in decreasing of electrostatic energy. The rotational barrier in cation 3 is more than 20 KJ/mol higher than the values for cations 1 and 2. The considerable increasing of the rotational barrier indicates the presence of intramolecular hydrogen bonding in cation 3. The geometrical characteristics of the hydrogen bond have also been studied.

**БИОЛОГИЯ**

*Brychkova G. G., Sorokin A. P., Maneshina T. V., Kurman P. V., Krasovskaya L. I., Jones Jonathan J. D., Kartel N. A.*
**Growth of transgenic Arabidopsis thaliana plants for effective remediation of sites contaminated with oil hydrocarbons.** pp. 72--75

*Summary:* Environmental biotechnology is currently faced with a challenge of growth of territories With the help of rhlA and rhlB genes, which contribute to the biosynthesis of rhamnolipids, we have grown transgenic Arabidopsis thaliana plants that efficiently degrade crude oil Unlike the control plants, the plants with rhlB gene and those with rhlA gene help degrade large crude oil volumes with minimal accumulation of oil hydrocarbons.

*Sinyushkina S. S., Sholukh A. M., Baranova L. A., Volotovsky I. D.*

**Properties and localization of protein tyrosine phosphatase in retinal rod outer segments.** pp. 76--79

*Summary:* A protein tyrosine phosphatase (FTP) activity in retinal rod outer segments showed a strong dependence on pH, time of reaction, and concentration of *p*-nitrophenyl phosphate. The FTP activity was found in the ROS discs, cytosolic fraction, and in the plasma membrane fraction. The plasma membrane fraction demonstrated the highest level of FTP activity. The PTP activity was significantly inhibited by orthovanadate in all fractions of ROS. The lihgt controls work of FTP in retinal ROS increasing its activity.

*Korik E. O., Naumova M. V., Naydun S. N., Reshetnikov V. N., Yurin V. M., Semak I. V.*

**Influence of aqueous-alcoholic extracts from the seeds of Silybum, Origanum, Lophanthus, and Aconogonon on system of biotransformation of xenobiotics in the liver of rats with extrahepatic cholestasis.** pp. 80--84

*Summary:* Influence of aqueus-alcoholic extracts from the seeds of Silybum marianum, Aconogonon Weyrichii, Lophanthus virescens, Origanum vulgare on content and activity of enzymes of detoxication of xenobiotics in the liver of bile duct ligated rats was studied. It was shown, that development of extrahepatic cholestasis is accompanied by significant decrease in activity of cytozolic glutathion S-transferases as well as in activity of NADPH-cytochrome c-reductase and content of cytochrome P-450 in microsomal fraction of the liver. After treatment of bile duct ligated rats by extracts, expressed to different extent reduction of activity of glutathion S-transferases and NADPH-cytochrome c-reductase is observed. At the same time, extracts did not cause statistically significant changes in content of cytochrome P-450.
Analysis of these findings testifies to the fact that extract from Lophanthus virescens seeds has the largest modulating activity in respect to studied enzymes of system of biotransformation of xenobiotics. It has been known that its efficiency exceeds one of Silybum marianum that is traditionally used for treatment of liver deseases.

**НАУКИ О ЗЕМЛЕ**

*Shkuratov V. I., Makhnach A. S.*
**Paleogeographic conditions of sedimentation in Vendian Valdaian time in Belarus.** pp. 85--87

*Summary:* In Valdaian time of the Late Proterozoic, almost all the territory of Belarus was covered with water with normal solinity showing reduction conditions and only in the marginal parts, showing oxydative conditions of sedimentation. The Baltic and Sarmatian crystalline uplifted massif of the crystalline basement was a zone of the clastic mamerial supply into the marine basin.

*Rylova T. B., Korol G. G., Savchenko I. E.*

**Age and paleogeographic conditions of accumulation of Pleistocene lacustrine deposits near Bobruisk.** pp. 88--93

*Summary:* The results of palynological investigations of ancient lacustrine deposits removed from the boreholes 132 and 40 near Bobruisk show that they were laid down at the end of the Narev Glaciation and throughout the Byelovezhian Interglacial. The presence in the composition of the studied diatom flora of extinct taxa, such as Cyclotella reczickiae, C. recyckiae var. diversa, C. comta var. pliocaenica, C. comta var. lichvinensis, Stephanodiscus niagarae var. insuetus, etc. supports its byelovezhian age.

*Veretennikov N. V., Laptsevich A. G., Gipnik V. I.*

**Petrographic diagnostics and facies nature of magmatic rocks near Glusha (Belarus).** pp. 94--97

*Summary:* The borehole N 46B6/2 was drilled by the Belarusian geological prospection expedition 25 km WSW of Bobruisk. At a depth of 496 m, the borehole opened porphyritic magmatic rocks that have not been previously found within the studied region and the whole territory of Belarus and drilled them down to a depth of 496 m. The rocks are overlain by tillites of the Vilcha series of the Vendian. Petrographic and chemical evidences suggest that these rocks are represented mainly by alkali feldspathic trachydacites that probably form a small intrusive body: stock or dyke. It breaks there a larger rounded isometric intrusion of granites ranging from the normal and transitional to subalkalic series, which just form the Bobruisk ring structure in the magnetic and gravitational fields.

*Kadatskaya O. V., Sanets E. V., Khomich V. S., Tischikov G. M.*

**Space-time dynamics of sulfates concentrations in the Berezina river.** pp. 98--101

*Summary:* Changes in the long-time average annual and seasonal concentrations of sulphates in the Berezina river waters under the influence of different levels of anthropogenic loading for the period from 1950 to 1999 are shown.

*Komarovsky M. E., Komarovskaya O. A.*

**On structure and nature of the Glubokoye glacial valley.** pp. 102--106

*Summary:* The Glubokoye glacial valley is a polygenetic landform, which was formed due to pressure subglacial meltwater erosion and glacial squeezing-out during the Poozerie glaciation degradation. The valley developed along the axial part of the glacial tongue and had grew rhythmically in the course of the glacier edge regression by attaching new 4--5 tunnel valleys with es-kers and kames in mouths that marked fixing stages of subglacial pool water drawing in the Polotsk lowland.

**ТЕХНИЧЕСКИЕ НАУКИ**

*Ustinovich D. F., Pribylsky V. I.*
**Mathematical simulation of force interaction between single-section brush and flat surface.** pp. 107--111

*Summary:* A mathematical model of force interaction between a single-section brush made from polymer-abrasive composites and flat surface is presented. The model allows analyzing the changes in cutting forces that occur under the action of a single operating element in the process of contact. It also makes it possible to optimize the technological parameters of the processing.

*Stavrov V. P., Karpovich O. I., Sviridyonok A. I.*

**Percolation model of consolidation of reinforced thermoplastic layers.** pp. 112--114

*Summary:* A new model of consolidation of reinforced thermoplastic layers grounded on the supposition, that the anchoring strength of autohesion is determined by percolation of a polymer melt from one layer to another is offered. The adequacy to the model is confirmed by results of tests on unidirectionally reinforced tapes alloyed at different modes on the basis of a polyamide -- 6 secondary, polyethylene LD and polypropylene.

*Zhemzhurov M. L., Serebryany G. Z.*

**Convective diffusion model of radionuclide migration in porous media.** pp. 115--118

*Summary:* A mathematical model of radionuclide transport in porous medium is considered. The solution is presented in generalized form suitable for practical application. The dependence is shown, allowing determining, on set distance, time of attainment of the maximal concentration of radionuclides and coordinates of zone of influence of radioactive source of pollution. It is received Pade approximation of the scaled error function which can be used for the solution of numerous problems.

*Chichko A. N., Sobolev V. F., Yatskevich Y. V., Matochkin V. A., Chichko O. I.*

**Simulation of evolution of thermal and mechanical stresses of steel ingot produced by means of continuous casting at radial drawing.** pp. 119--123

*Summary:* A mathematical model for simulation of stresses in a crystallizing curvilinear ingot-casting pulled out with constant speed is offered. On the basis of the computer calculations of three-dimensional ingot-casting, it is shown that in the zones of secondary cooling, the thermal stresses dominate over mechanical, and in the zone of rectification, the reverse tendency is observed.

/ Издания академии / Научные журналы / К началу страницы

Разработана и поддерживается Николаем Н. Костюковичем. Последнее обновление: *15 ноября 2006 г.*

Создана при участии Игнатия И. Корсака

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