# Volume 46, № 8, 1994

### Equations of optimal nonlinear filtration and interpolation for partially observed Markov processes

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 971–976

Recurrence relations are obtained for problems of optimal filtration and interpolation of partially observed discrete Markov chains. We present the system of differential equations for problems of optimal nonlinear filtration for Markov processes with continuous time and the system of inverse differential equations for problems of optimal nonlinear interpolation.

### Some properties of biorthogonal polynomials and their application to Padé approximations

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 977–984

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders $[N−1/N],\; N \in ℕ,$ for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ where $f(z)$ is a function with known Padé approximants of the indicated orders, $T_j [f;z]$ are Taylor polynomials of degreej for the function $f(z)$, and $α_{ m, M} = \overline {1,M}$ are constants.

### Finite-difference approximation of first-order partial differential-functional equations

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 985–996

We consider initial boundary-value problems of Dirichlet type for nonlinear equations. We give sufficient conditions for the convergence of a general class of one-step difference methods. We assume that the right-hand side of the equation satisfies an estimate of Perron type with respect to the functional argument.

### On the rate of convergence of stochastic approximation procedures

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 997–1002

The rate of convergence of a linear stochastic approximation procedure in $R^d$ is studied under fairly general assumptions on the coefficients of the equation.

### Limiting distributions of the solutions of the many-dimensional Bürgers equation with random initial data. II

Leonenko N. N., Orsingher E., Rybasov K. V.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1003–1010

We find non-Gaussian limiting distributions of the solutions of the many-dimensional Burgers equation with the initial condition given by a homogeneous isotropic Gaussian random $χ^2$-type field with strong dependence.

### Asymptotic methods in the theory of nonlinear random oscillations

Kolomiyets V. G., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1011–1016

We study applications of asymptotic methods of nonlinear mechanics and the method of Fokker-Planck-Kolmogorov equations to the investigation of random multifrequency oscillations in systems with many degrees of freedom.

### Extension and approximation of functions subharmonic in a half plane. Impossibility of extension of plurisubharmonic functions

Rashkovskii A. Yu., Ronkin L. I.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1017–1030

We construct different extensions of functions subharmonic in a half plane to the whole plane. The results obtained are applied to the approximation of subharmonic functions of finite order in a half plane by the logarithm of the modulus of an entire function. It is shown that the problem of extension of a plurisubharmonic function may have no solution.

### Investigation and solution of boundary-value problems with parameters by numerical-analytic method

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1031–1042

We suggest a modification of the numerical-analytic iteration method. This method is used for studying the problem of existence of solutions and for constructing approximate solutions of nonlinear two-point boundary-value problems for ordinary differential equations with unknown parameters both in the equation and in boundary conditions.

### Quasiinvariant deformations of invariant submanifolds of Hamiltonian dynamical systems

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1043–1054

We study quasiinvariant deformations of invariant submanifolds of nonlinear Hamiltonian dynamical systems and their small perturbations.

### Gelfand pair associated with a hoph algebra and a coideal

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1055–1066

We consider a pair of a compact quantum group and a coideal in its dual Hopf *-algebra and introduce the notions of Gelfand pair and strict Gelfand pair. For a strict Gelfand pair, we construct two hyper-complex systems dual to each other. As an example, we consider the quantum analog of the pair (U*(n), SO(n)*).

### Green-Samoilenko function and existence of integral sets of linear extensions of nonautonomous equations

Asrorov F. A., Perestyuk N. A.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1067–1071

An integral invariant set is constructed for systems of differential equations by using the Green-Samoilenko function. The problem of asymptotic stability of this set is studied.

### On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1072–1079

For $t \uparrow \omega, \quad \omega \leq +\infty$, we obtain sufficient conditions for Lyapunov stability of the zero solution of a specific nonautonomous quasilinear differential system in the case where the matrix of the first-degree approximation has the Jordan form with triangular blocks. Methods to reduce certain classes of general differential systems to differential systems of special type are given.

### Approximation of functions subharmonic in a disk by the logarithm of the modulus of an analytic function

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1080–1083

Yulmukhametov's result concerning the approximation of a function subharmonic in a bounded domain by the logarithm of the modulus of an analytic function is supplemented with an estimate of the exceptional set in the important case of a disk. We show that this approximation is unimprovable in a certain sense.

### On coercive solvability of operator-differential equations in $B$-spaces

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1084–1087

We study the coercive solvability of operator-differential equations in anisotropic *B*-spaces of vector-functions.

### Jacobi polynomials and Lax representation for completely integrable dynamical systems

Enolskii V. Z., Kondratev A. Yu.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1088–1091

We consider a method aimed at the investigation of completely integrable dynamical systems by using the Lax representation of their equations of motion. The Lax representations are found for the integrable case of the Henon-Heiles system and for an anisotropic oscillator.

### Central limit theorem for special classes of functions of ergodic chains

Moskal'tsova N. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1092–1094

A central limit theorem is proved for *E*-finite bounded functions of ergodic Markov chains. Two useful corollaries are presented.

### Approximation of periodic functions by constants in the metric spaces $ϕp(L)$

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1095–1098

By using the best approximations of functions by constants, we establish necessary conditions for the moduli of continuity of periodic functions in metric spaces with integral metric and find the Young constants of these spaces.

### A criterion for Banach manifolds with a separable model to be finite-dimensional

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1099–1103

We give an example of a continuous bijective mapping with a discontinuous inverse which acts in a separable Banach space and differs from the identical mapping only in an open unit ball. A criterion for Banach manifolds with a separable model to be finite-dimensional is established in terms of the continuity of inverse operators.

### On the second Bogolyubov theorem for systems with random perturbations

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1104–1109

For systems of differential equations with random right-hand sides, we establish conditions for the existence of periodic solutions in the neighborhoods of equilibrium points of the averaged system.

### To the memory of Valentin Anatol'evich Zmorovich

Baranovskii F. T., Berezansky Yu. M., Buldygin V. V., Daletskii Yu. L., Dobrovol'skii V. A., Dzyadyk V. K., Lozovik V. G., Mitropolskiy Yu. A., Samoilenko A. M., Skrypnik I. V., Tamrazov P. M., Yaremchuk F. P.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1110–1111