Visnyk of Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

Contents and Abstracts

Volume 582,   Issue 52,   2003


Gordevskyy V.D., Buznitska E.M. Non-stationary States of a Gas for the Bryan-Pidduck Model , P. 3-9

The Bryan-Pidduck model is used to describe the evolution of a gas. In this model each of molecules has not only linear, but angular velocity, too, and rotates about its axis. Non-stationary distributions of a special form are proposed which correspond to the vortical states of a gas, i.e. the flows with transitional and rotational degrees of freedom. The bimodal distributions with the vortical modes are constructed for approximate description of the interaction of two flows. The behaviour of the integral remainder between the sides of the Boltzmann equation is studied under the different suppositions on hydrodynamic parameters. 2000 Mathematics Subject Classification 76P05. Top of the page.

Kengne E., Tayou Simo J. On well-posedness of multi-point integral boundary value problems in a stripe , P. 10-14.

In a domain, which is the Cartesian product of a segment by the entire real line (stripe), we investigate on well-posedness of a multi-point boundary value problem with an integral in the boundary condition for partial differential equations. Conditions of the well-posedness of the problem under consideration are established in the class of bounded smooth functions. 2000 Mathematics Subject Classification: 35B30, 35E20, 35A05. Top of the page.

Zagorodnyuk S.M. On a characteristic property of polynomials satysfying a difference equation of 4 th order , P. 15-25.

In this paper we study systems of polynomials $\{ p_n (\ld) \}_{n=0}^\infty$ satysfying the relation: $J_5 p(\ld) = \ld^2 p(\ld)$, where $J_5$ are five-diagonal, hermitian, semi-infinite matrices and $p(\ld) = ( p_0 (\ld), p_1 (\ld), p_2 (\ld), ... )^T$, and also a corresponding moments problem. The corrected versions of the criteria of solvability of the moments problem and of the analog of Favard's theorem for such systems are proved. It is shown that having of orthonormality relations of a special form is a characteristic property of such systems of polynomials. The connection with orthogonal matrix polynomials on the real line is shown. 2000 Mathematics Subject Classification: 42C05, 39A05, 44A60. Top of the page.

Lyubarsky M.G. Calculation of the Lyapunov exponent of Linear Ordinary Equation with Rapidly Oscillating Coefficients , P. 26-48.

We study perturbed exponentially decomposable system assuming the integral norm $ \{B\} \equiv\sup_{s,t} \Bigl |\int^{t+s}_t B(t)dt\Bigr |$ ($0\leq t\leq\infty$, $0\leq s\leq 1$) of the perturbation $B(t)$ to be small. If $B(t)$ is a rapidly oscillating matrix this condition can be fulfilled without making the values of $B(t)$ small. We give an iteration method for evaluation of the Lyapunov and Bohl exponents of the system. This may have many applications in physics. As an example we evaluate the increment of plasma beam instability. One more application is an averaging theorem for system of linear differential equations with small parameter. 2000 Mathematics Subject Classification: 34D30, 58F10, 58F15. Top of the page.

Rozumenko O.V. Nagy -- Foias functional model of bounded operator , P. 49-61.

The constructing of functional model for any bounded operator $T$ (contracting or not) in Hilbert space $H$ is done. This model is considered as an analogue to the B. Sz.-Nagy and C. Foias functional model. 2000 Mathematics Subject Classification: 47A45. Top of the page.

Dyukarev Yu.M. The canonical N-extremal and main solutions for the generalised interpolation problem for Stieltjes functions , P. 62-70.

In this paper canonical N-extremal and main solutions for Stieltjes operator-functions are introduced for investigation. It is proved that the set of canonical solutions of the first (the second) kind coincides the set of N-extremal solutions of the first (the second) kind. The main solutions for the generalised interpolation problem were studied, and explicit formula for them were obtained. 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

Marzeniuk V.P., Ladyka R.B., Kovalchuk O.Ya. Approach for Problem of Chemotherapy Scheme Search , P.71-80.

In this work the model of chemiotherapy for treatment of oncologic diseases is presented. We suggest the methodology of optimal treatment regimes verifying of which can be reduced to the problem of solvability of the system of algebraic inequalities. The optimal process of the system is constructed in the explicit form by use of special functions. 2000 Mathematics Subject Classification: 93C10, 92C50. Top of the page.

Patsegon N.F., Popova L.N., Sviridenko S.A. A qualitative analysis of the structurization processes in a magnetic fluid , P.81-89.

The structurization processes in a magnetic fluid, i.e. generation and destruction of the belonging to the fluid aggregates of the magnetic particles, are considered. With the assumption that the diffusion processes in the fluid may be neglected, an one-parametric dynamic system of second order, describing evolution of a magnetic state of the fluid, being in a equilibrium in a constant homogeneous magnetic field, is investigated. The equilibriums and bifurcating values of a parameter - a magnetic field strength, were found. The phase portraits of the system at different values of this parameter are drawn. 2000 Mathematics Subject Classification: 76W05. Top of the page.

Udodova O.I. Holomorphic almost periodic functions in various metrics , P. 90-106.

In this work the classical Bohr's theorem for holomorphic almost periodic functions in a strip is extended to holomorphic almost periodic functions in a tube domain of finite-dimensional vector space. Except for the uniform metric we consider Stepanoff, Weil, Besikovitch metrics. 2000 Mathematics Subject Classification: 42A75, 30B50. Top of the page.

Kantor B.Ya., Kizilova N.N. Stress-strain state investigation in bidimensional growing biological material at growth restriction , P. 107-120.

The results of stress-strain state investigation in a plate of growing biological material at full or partial growth restriction at the perimeter are given. The material is regarded as viscoelastic, the restriction is considered as a flexible non-stretchable string. At the case of full restriction the solution was obtained as a series expansion. The numerical calculations were carried out on basis of finite elements method. The stress distributions are obtained and the deformations of the plate are investigated at different growth stages. It was shown that the stress distributions and stress tensor principals are essentially changed near the region of restriction. The experimental method of investigation of mechanical stress field influence on the biological materials growth is proposed. 2000 Mathematics Subject Classification: 76A10. Top of the page.

Rezunenko V.A. Multiple regularization of the diffraction operator for the axis-simmetrical TM wave problems on the sphere with aperture , P.121-134.

The static and principal dynamical parts of the operator of diffraction problem for axially symmetric waves on the sphere with circular aperture are isolated and inverted. The inversion of Abel integral operator and an auxiliary Cauchy problem are used. The system of linear algebraic equations of second kind with compact in $l^2$ operator is obtained. The system is effectively solved numerically and analytically. 2000 Mathematics Subject Classification: 65N12, 35A25, 78A45. Top of the page.

Grishin A.F., Poedintseva I.V. Two integrals , P. 135-142

We give values of two definite integrals. These integrals have been raised in investigations of authors. We did not find them in table books of integrals that we know. The problem book of M.A.Evgrafov and coauthors in the theory of analytic functions has more types of integrals then others. One can view the note as an addition to this book. 2000 Mathematics Subject Classification 30E20. Top of the page.

Poletaev G.S. Some results on the conjugate equations in rings with factorization pairs , P.143-149.

Abstract conjugate equations of a general type with unknowns from a ring possessing factorization pair of subrings are considered. The existence and uniquenece theorem with the formulas of a solution is established under the assumption of the regularity of a factorization for some elements built by coefficients. 2000 Mathematics Subject Classification: 45N05. Top of the page.

Dimitrova S.D., Dimitrov D.B. About one generalization of the Bohl-Bohr theorem for abstract functions, given on group , P.150-161.

The difference equation $\Phi(hx)-\Phi(x)=F_h(x)$ is considered. A solution $\Phi(x)$ and right members $F_h(x)$ are functions, which on the Cartesian product of groups have been given. Them values are considered in the Frechet space not containing a subspaces isomorphic to the space $c_0$. Have been proved that if there is a bounded solution $\Phi(x)$ on a relatively dense set, then it is almost periodic (almost periodic by Levitan, almost automorphic) function as soon as a right member is a.p.f. (L-a.p.f., a..f.). 2000 Mathematics Subject Classification: 43A60. Top of the page.

Rekalo A.M. Inertial manifolds for reaction-diffusion equation on thin two-layer domains , P. 162-178.

The paper is concerned with the asymptotic behavior of solutions to semilinear parabolic equation on 2D thin two-layer domains. The problems of such kind may arise as models for reaction-diffusion processes in thin sandwich films separated with permeable membranes. It is shown that there exists an invariant finite dimensional $C^1$-manifold which exponentially attracts as $t\to\infty$ every solution to the problem. In other words, the initial infinite-dimensional dynamical system asymptotically can be described in terms of a finite system of ordinary differential equations. 2000 Mathematics Subject Classification: 35K57. Top of the page.

Lagno, V.I., Magda O.V. Group classification of one class of quasilinear wave equations , P. 179-191.

The problem of group classification of the quasilinear equations of the type $ u _ {tt} = u _ {xx} +g (t, x, u) u_x+f (t, x, u), \ g_u \not =0$ is decided. The complete list of the representatives of nonequivalent classes of the equations having nontrivial symmetric properties is obtained. 2000 Mathematics Subject Classification: 42A70. Top of the page.

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Visnyk Kharkivs'koho natsional'noho universytetu imeni V. N. Karazina, Seriya Matematyka, prykladna matematyka i mekhanika

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