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# Doklady of the National Academy of Sciences of Belarus, 1998, Vol.42, No.5

/ Publications / Scientific Journals

Doklady of the National Academy of Sciences of Belarus, 1998, Vol.42, No.5

## CONTENTS

### MATHEMATICS

Zakrevskii A. D.
Theorems on parallel decomposition of Boolean functions. pp. 5--7

Summary: Necessary and sufficient conditions are formulated for existence of the composition j(m(u), l(v)), which realizes the partial Boolean function f(x) for given u Ì x, v Ì x, and the Boolean functions j, m and l are defined, when this composition exists

Sotskova N. Yu., Tanaev V. S.
On realization of an optimal schedule with uncertain durations of operations. pp. 8--12

Summary: Scheduling problem is modelled in terms of a mixed graph. In contrast to well studied deterministic setting, it is assumed that only structural input data (i. e., precedence and capacity constraints) are fixed while for the operation processing times only their lower and upper bounds are known before scheduling. The main problem is to select and realize an optimal schedule under uncertain numerical input data

Lazakovich N. V., Lesnevskii V. E., Stashulenok S. P.
The stochastic q-approximation of the Poisson process. pp. 13--17

Summary: Problems connected with classification of the methods of approximation of the Poisson process in the convolution algebra are investigated. The Ito formula for q-differentials of the Poisson process is obtained

Lebedev A. V.
The Kveselava -- Vekua type theorems and calculation of the index of G-singular operators with piecewise-continuous coefficients. pp. 18--20

Summary: Let S be the Cauchy singular integral operator dealing in the space L2(G) where G is a simple closed oriented Lyapunov curve. As it is known. S2 = I and, therefore, the operators P± = (I ± S)/2 are projections. Let T be the transformation of an argument operator (Tf)(x) = |a'|1/2f(a(x)) associated with a diffeomorphism a : G ® G. If a preserves the orientation of G and satisfies the Holder condition on G then the known Kveselava -- Vekua theorem implies that the operator P+T+P- is the Fredholm operator and has a zero index. This theorem plays the most important role in calculation of the index of G-singular integral operators, that is, the operators generated by the operators S, T and the operators of multiplication by some functions. The condition that a' is a Holder function plays the fundamental role in the proof of the Kveselava -- Vekua theorem since it essentially uses the equality TS -- ST = K, where K is a compact operator (which is no longer true without this condition). The present article is devoted to the Kveselava -- Vekua type theorems in the situation, where 1) a' is not a Holder function but only a piecewise Holder one. 2) a not necessarily preserves the orientation of G (in the situation, where a changes the orientation the Kveselava -- Vekua type theorem turns out to have the form which is qualitatively different from the classical one)

Tretiakova N. N.
Resulting functions and representations of the Euclidean motion group. pp. 21--25

Summary: A concept of resulting functions or R-functions of the Euclidean motion group ISO(n) is introduced. It is proved that the R-functions are similar to Bessel functions of an integer or a half-integer order. The studied functions form kernels of the Fourier integrals of the fields of arbitrary quantities defined in En

Shilin A. P.
Convolution-type integral equations reducible to mixed boundary-value problems. pp. 26--29

Summary: A solution of two new integral equations of a convolution type on a real axis is presented. The Fourier transformation was used to reduce the equations to mixed boundary-value problems for analytical functions

Moroz V. B.
A class of semilinear boundary-value problems. pp. 30--33

Summary: In this work we consider a semilinear Dirichlet boundary-value problem with nonlinearity superlinear at zero without any assumptions of symmetry. Using methods of Morse's theory for isolated critical points, we prove the existence of multiple solutions with various assumptions of asymptotic behaviour of the nonlinearity

Gorenflo R., Kilbas A. A., Rogozin S. V.
On properties of the generalized Mittag-Leffler type functions. pp. 34--39

Summary: Properties of the special function Ea, m, l(z) generalizing Mittag-Leffler functions are studied. It is proved that it is an integer function of the order r = 1/a and of the type s = 1/m. Recurrent relations are derived for the generalized Mittag-Leffler type function. It is shown that for a = n Î N, this function can be expressed in the form of a generalized hypergeometric series. Differentiation formulas for En, m, l(z) are obtained

Erovenko V. A.
Spectral and Fredholm properties of perturbed differential Euler operators. pp. 40--44

Summary: The paper describes the essential spectra of Euler differential operators and some of their singular perturbations in spaces Lp(a, ¥), 0 < a < ¥, 1 £ p £ ¥

Lepin V. V.
Linear locations of hypergraphs. pp. 45--50

Summary: The problem of an addition to the matrix with a connectedness property is considered. Methods of dynamic programming arc developed for solution of this problem and its extensions

### PHYSICS

Zhestkov S. V., Kuvshinov V. I.
A particular extension of Hirota's method to the multi-dimensional case. pp. 51--54

Summary: It is found that Hirota's method can be used for equations of Zakharov--Kuznetsov's type. Single and double exponential solutions are constructed. Conditions arc obtained at which the double exponential solution describes interaction of two single exponential solutions

Abramov I. I., Danilyuk A. L.
The effect of external actions on coherent electron transport in the metal-insulator-semiconductor structure. pp. 55--59

Summary: Results are presented of simulation of the effect of temperature, light and ionizing radiation on coherent electron transport in the metal-- insulator-- semiconductor structure with thin dielectric

### CHEMISTRY

Krutko N. P., Dubkova V. I., Apanasenok V. I., Alekseenko V. I.
The effect of the magnetic field on the hardening process of the epoxy binder. pp. 60--66

Summary: The study is carried out to investigate the effect of a constant magnetic field with a strength from 227 to 9000 Oe on the process of transformation of epoxy oligomer in a three-dimensional structure with a chemical method of analysis and the continuous electric capacity procedure of controlling the dielectric characteristics of the epoxy binder with an amine hardener. Substantial differences arc found in the hardening kinetic curves of the binder during formation of epoxy polymer under the action of a magnetic field and out of it, which suggests the changed hardening process under the magnetic field. Data on dielectric relaxation of the hardened epoxide heated in the region of positive temperatures and results of thermomechanical analysis indicating the formation of structural elements with different molecular mobility support this suggestion. The influence of the magnetic field is exhibited by an increase in rigidity, hardness and Young's modulus of the hardened binder. It is also found that physical, chemical and dielectric characteristics of the hardened binder depend on the intensity and direction of the magnetic vector

Prokopchuk N. R., Tolkach O. Ya., Paplevko I. G.
On the temperature dependence of the activation energy of mechanical destruction of plastics, fibres and rubbers. pp. 67--71

Summary: The temperature dependence of the activation energy of mechanical destruction of plastics. fibres and rubbers is found. It should be taken into consideration in developing methods of forecasting durability of these materials on the basis of the parameter U0 as a criterion

### BIOLOGY

Laman N. A., Putyrskii I. N., Rosolenko S. I.
A method for estimation of autonomous development of shoots in the cereal shrub. pp. 72--76

Summary: It is proved and experimentally verified that autotomous development of shoots in the shrub can be quantitatively estimated with the number of leaves in them in the order of appearance of the shoots. When the number of leaves in the main stem is N and its apex plays the coordinating role. this ratio can be expressed by the numerical series N : (N-3) : (N-4) : (N-5) : (N-6), while in the case of autonomous development of lateral shoots, at least six leaves are formed in each of them

Abramchik L. M., Serdyuchenko E. V., Chaika M. T. †
A study of regulation aspects of forming the light-harvesting complex of photosystem 2. pp. 77--81

Summary: It is found that there is no coordination in the synthesis of the pigment and protein components of the light-harvesting complex of photosystem 2 (LHC-2) in early stages of biogenesis of the pigment apparatus in postetiolated barley seedlings under inhibition of protein synthesis in cytoplasm

Chernitskii E. A., Senkovich O. A.
The mechanism of rapid hemolysis by an anionic detergent. pp. 82--84

Summary: The effect of a decrease in the detergent concentration at different moments of hemolysis of human erythrocytes by sodium dodecyl sulphate on the shape of kinetic curves of hemolysis is studied. It is shown that rapid hemolysis by sodium dodecyl sulphate does not follow the colloid-osmotic mechanism but is caused by opening large pores sufficient for release of hemoglobin molecules and that kinetics of hemolysis is mainly determined by the time dependence of the probability of opening these pores

Tropnikova G. K., Azev O. A., Kulchitskii V. A.
The role of medial areas of ventral bulbar nuclei in implementation of vestibular effect on respiration and heart activity. pp. 85--88

Summary: In experiments on anesthetized (nembutal + urethan) rats it was found that an increase in the respiratory amplitude induced by stimulation of vestibular nuclei was abolished by electrolytic destruction of nucleus raphe magnus and nucleus raphe pallidus, while changes in the heart rate were caused by destruction of nucleus raphe magnus and nucleus raphe obscurus. Novocain block of the rostroventrolateral bulbar nucleus did not affect the vestibular stimulation-induced increase in the depth of breathing and bradycardia but eliminated the increase in the rate of respiratory muscle contractions. Selective involvement of ventromedial and ventrolateral bulbar nuclei in regulation of respiration and heart activity upon stimulation of vestibular nuclei is suggested

### EARTH SCIENCES

Velichkevich F. Yu.
On new findings of Belovezhian interglacial floras in Belarus. pp. 89--93

Summary: The Belovezhian age of interglacial deposits from three boreholes in the territory of Mogilev region of Belarus was established on the basis of paleocarpologic investigations. These floras arc very similar to the coeval ones within Belarus and western regions of Russia due to the presence of a large group (15 taxa) of extinct species

Astapenko V. N., Girin R. E.
Crustal anomalies of electrical conductivity and the block structure of the Earth's crust in the western part of the East European Cration. pp. 94--98

Summary: The territory of the western part of the East--European Craton is composed of different types of crustal blocks which arc characterized by regular alteration of their deep structure. The comparison of the block model with the zones of location of middle and lower crustal anomalies of electrical conductivity has revealed their coincidence with the blocks of quite definite types

Konishchev V. S.
Dynamics of the development of breaks of the Pripyat trough in Mesozoic and Cenozoic eras. pp. 99--103

Summary: It is shown that amplitudes of movement along faults of the Pripyat trough in Mesozoic and Cenozoic times range from some tens to several hundreds of metres which equals some per cent of all fault amplitudes. The velocity of movement along the faults in Triassic was 1--6 m/mln years, in Jurassic. 0.06--0.3 m/mln years, in Lower Cretaceous, 0.02--0.73 m/mln years, in Upper Cretaceous. 0.01--0.82 m/mln years, in Cenozoic. 0.05--0.60 m/mln years. It was from hundred to thousand times less than that in the principal phase of the rift stage during Lebedyan-Streshin time. This means that in Mesozoic and Cenozoic times the fault and fold-fault structures of subsalt and intrasalt deposits did not change substantially

Bulkin Yu. S.
Genesis of aplite-formed granites of the Belarusian anteclise. pp. 104--106

Summary: The data presented suggest metasomatic genesis of aplitic granites in the Belarusian anteclise

Khursevich G. K., Fedenya S. A.
Microstratigraphy of the Alexandrian interglacial deposites based on the data of diatom analysis. pp. 107--112

Summary: Biostratigraphic subdivision of the Alexandrian interglacial deposits, stripped in the profiles of Matveev Rov-2 and Gvoznitsa, into local diatom assemblage zones (MR1--MR3 and GV1--GV3, respectively), correlated with pollen zones, is given in the paper

### TECHNICAL SCIENCES

Vysotskii M. S., Evgrafov A. N., Ilin E. V.
Improvement of the flow over the subpan area of a vehicle. pp. 113--116

Summary: Characteristics of the flow over the underpan area of a vehicle are studied. It is shown that aerodynamic drag can be decreased by blowing-off the boundary layer from the floorpan, a negative pitch of the body and optimization of the clearance

Kudina E. F., Lisovskii V. V., Tyurina S. I., Vegero N. V.
Modification of asilicate polymer binder by the action of ultrahigh frequency radiation. pp. 117--121

Summary: The influence of microwave radiation on the structurization processes of an organomineral system based on water-soluble silicates of alkaline metals and epoxy resin is investigated. The relations of structural changes, treatment regime and mechanical properties of the modified compositions are obtained