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

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Contents and Abstracts

Number 444,     May 1999

Sklyar G.M., Velkovsky I.L. A Method of Solutions of the Trigonometric Min-Moment Problem with Periodic Gaps , P. 3-9.

The Markov trigonometric moment problem on a minimal possible interval (min-moment problem) is considered in the case when the moment sequence has periodically repeating gaps. The complete analytical description of the solvability set of the Markov trigonometric min-moment problem with gaps as well as a method of searching of its canonical solutions are given. 1991 Mathematics Subject Classification 42A70. Top of the page.

Korobov V.I., Pavlichkov S.S. The controllability of triangular systems with uniformly bounded perturbations , P. 10-14.

The complete controllability property is proved for triangular non-stationary systems of ordinary differential equations with uniformly bounded perturbations and with controls in class $C([t_0,T];{\bf R}^1)$ via the Brouwer fixed-point theorem. It is assumed that the perturbations satisfy the local Lipschitz condition with respect to $x$ and $u$ and the triangular part satisfies the global one. One example is considered. Top of the page.

Skorik V. A. Analytic inverse of a class of ill-posed matrices arising from the controllability function method , P. 15-23.

According to the controllability function method the construction the controllability function method the synthesis control for linear systems connects closely with some matrices of an arbitrary order. To obtain these matrices it is necessary to invert a given class of special matrices. This class is the class of ill- posed matrices. In the work in this connection we proposed analytic representation of the inverse matrices. Top of the page.

Korobov V. I., Florinsky V. V. On determination of the optimal time and switching points in a time- optimal problem , P. 24-43.

For a linear dynamical time-optimal problem we investigate questions of smooth dependence of the optimal time and switching moments on a matrix spectrum. We also prove continuous dependence of a solution of a nonlinear optimal control problem on initial data and an n-dimensional parameter. In this case the matrix spectrum can be this parameter. For a linear time-optimal problem we propose a numerical method. By the way of variation of a matrix spectrum using a given solution of a time-optimal problem with a given matrix spectrum this method makes it possible to find a solution of a time-optimal control problem with other matrix spectrum. Top of the page.

Bessonov G.O. Lutsenko A. V. On conditions for nonstability of discrete systems , P. 44-49.

We investigate questions of nonstability of the zero solution of the nonlinear diskette system $x_{m+1}=f(m, x_m)$. For continuous systems if the classical Chetaiv's conditions are satisfied in some 1-connected domain that is contained in a neighborhood of the origin and on its bound then this system is nonstable. We proved that existence of at least one point belonging to the bound of the domain such that its image with respect to the map $f$ belongs to the interior of this domain also implies nonstability of this system. Top of the page.

Sokhin A.S. Identification of System with Distributed Parameters Describing Unstationary Transfer Equation with Isotropical Potential , P. 50-60.

The article considers Boltzmann's linear equation in the space of many dimensions with unstationary isotropical collision integral. For the first time we set up the scattering inverse problem that allows to find the density of the scattering matter in space, solving the integral equation of Fredhelm of the second type. The kernel is determined by comparing startingand final streams of neutrons. Top of the page.

Suzikov G.V., Vorobyov I.V., Rozumenko O.V. About one method of the solving the 2-nd order boundary problem , P. 61-70.

The paper studies one new method of the solving the boundary problems for the second order ordinary differential equations, which is that the solution of the "difficult" boundary problem is represented in the form of the limit of the solutions of the "simple" problems -- the Cauchy problems for the same equation. In fact, the addition of one cycle in the program of the solving the Cauchy problem for the differential equation with the simple recalculation after every passing, transforms it in the program of the solving the boundary problem. The method is very simple and effective, has an extensive domain of employment. Top of the page.

Marchenko I. I., Nikolenko I. G. On the quantities of the deviations of meromorphic minimal surfaces , P. 71-88.

The inequalities for the sum of the positive deviations of meromorphic and entire minimal surfaces are obtained. A corresponding example of minimal surfaces that made these inequalities more precise is constructed. Bibliography : 18 ref. Top of the page.

Marchenko I. I. On deficiencies of plurisubharmonic functions , P. 89-93.

It is investigated the class of plurisubharmonic functions $u(w,z)$ of two complex variables that have a logarithmic growth in the variable $w$. We prove that the set of $w$ for which the function $u$ has a positive Valiron's deficiency is one of logarithmic capacity zero. Further we construct a function $u(z,w)$ that has slowly growth in $w$ in the sense Petrenko and such that it has positive Valiron's deficiency for any complex $w$. Top of the page.

Vlasenko L.A., Piven A.L. On Basicness of Solutions of Degenerate Linear Differential Equations , P. 94-100.

In a Hilbert space we consider the differential equation $\sum\limits_{j=0}^{n}A_ju^{(j)}(t)=0$ with linear closed operators. The operator $A_n$ of higher derivative can have a non-trivial kernel. We lay down the conditions under which solutions are expanded to a series in elementary solutions. The results are applied to partial differential equations. Top of the page.

Dyukarev Yu. M. De Brange spaces and interpolation problems in Stieltjes class , P. 101-110.

A quastion on de Brange spaces which are connected with interpolation problems in Stieltjes class is investigated. It is shown that to such problems correspond two de Brange spaces. In addition the interpolation problem in equivalent to a problem of agreed integral images of scalar product in the corresponding de Brange spaces. Top of the page.

Kabalyants P.S., Rutkas A.G. On a Quasi-Linear Differential Equation with Kronecker Type Singularity , P. 111-118.

In a Banach space we consider a quasi-linear differential equation with a singular operator sheaf. The original sheaf and the adjoint sheaf form a singular pair of subspaces. Sufficient existence and uniquness conditions for the Cauchy problem were obtained. Restrictions are formulated in terms of regular and singular projections of nonlinear vector-functions. An example of a singular mixed problem for a partial differential equation is given. Top of the page.

Yatsenko T.U. On Abstract Cauchy Problem, which Connected with One Hydrodynamic Task , P.119-125.

In Hilbert space we consider the differential equation $C \dot x(t) +Fx(t)=f(t)$, where $C$ is the positive defined operator, $F$ is the maxinum ccretive operator. This equation describes the small motions of the homogeneous viscous noncompressible liquid in the vessel, which bounded by the elactic memrane. We build the equation solutions scale. We prove that the Cauchy problem is correct in the energetic norm. Top of the page.

Legejda V. I. Application of method of dual integral equations to the problem of a magnetohydrodynamics , P. 126-136.

In the given work the method of dual integral equations is applied to a solution flat boundary the problem for equations $\delta u-k^2u=0$ with mixed boundary conditions originating in a magnetohydrodynamics at calculation electromagnetic and temperature fields in channels MGD - devices with segmented electrodes. Thus are received new dual integral equations and are transformed them to system of integral equations of Fredholm 2 kind. Top of the page.

Gubareva V. F. Stability of Permanent Rotations of System of Rigid Bodies which is Connected with Hinges , P. 137-141.

Stability of rotation of system of connected rigid bodies is investigated. One of the bodies in each system has a fixed point. With the special change of variables, the system of the motion equations is become the system of the steady-state equations. Its characteristic equations give a possibility to establish the necessary conditions of stability and to compar them with the sufficient conditions. Top of the page.

Tropina A. A. Numerical Investigation of the Heat transfer in the Poorly Liquid with Linear Law of Injection , P. 142-147.

Electroconvection heat transfer in liquid dielectrics placed between plane and cylindrical electrodes has been investigated. It has been assumed that ions are generated due to dissociation of an impurity and an injection of charge carriers from metal electrodes. The temperature boundary layer is computed and a heat flux as a function of applied voltage and electrode diameter is determined. It has been shown that the suppression of heat transfer in the small DC voltages is caused by the formation of heteroions in the non-equilibrium layers near electrodes. For liquid with dissociation--injection conductivity variation of velocity with electrode diameter (which is experimentally observed) is described from the position of non-autonomous level of injection, when the charge density on the electrode is the linear function of an electric field. Top of the page.

Kizilova N. M., Popova N. A. Criterions of optimum performance of branching transport system of alive nature , P. 148-156.

The analysis of optimal criterions which can underlie structures of the branching transport systems of plant and animal tissues is conducted. For different pairs of criterions the solution of appropriate extremum problem is carried out and the ratios for cross-sectional areas of the branching pipeline optimal in a sense of selected criterions are obtained. The comparison of theoretical ratios with similar, obtained by a way of measurements on different types of leaves' vein boundless is conducted. 5 fig. Ref. 8 items. Top of the page.

Cheremskaja N. V. The Sequences in Hilbert Spaces of Infinite Rank Nonstationarity , P. 157-161.

In this article nonstationary sequences in Hilbert Space H of infinite rank nonstationarity are studed. For this sequences the special characteristic was introduced and called correlation difference. Necessary and sufficient conditions were found (in term of correlation difference for sequences type $\xi (n)=A^n\xi_0, A\in [H, H], \xi_0\in H$ in order to operator A was a dissipative operator with discrete spectrum or Voltterra operator. Top of the page.

Geleznakova E. Yu. Linear transformations of sequences in Hilbert spaces , P. 162-166.

Linear transformations of sequences (stationary and nonstationary) in Hilbert are considered. It is obtained necessary and sufficient conditions for a correlation function to be the correlation function of a linear transformation of the corresponding sequence. Top of the page.

Bosov, A. A.; Mukhina, N. A. An algorithm for solving of the canonical problem of the method of the least modulus in the regression analysis , P. 167-170.

In this article we proposed an algorithm of the method of the least modulus for solving a parametric identification problem when the information of an object is a priori given help of results of passive test. 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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