Volume 46, № 8, 1994
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.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 977–984
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.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 997–1002
Limiting distributions of the solutions of the many-dimensional Bürgers equation with random initial data. II
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 ?2-type field with strong dependence.
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
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.
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.
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.
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
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
We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form ast ? ?, ? ? + ?. For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular blocks. We indicate methods that enable one to reduce certain classes of differential systems of the general form to special differential systems.
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.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1084–1087
We study the coercive solvability of operator-differential equations in anisotropicB-spaces of vector-functions.
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.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1092–1094
A central limit theorem is proved forE-finite bounded functions of ergodic Markov chains. Two useful corollaries are presented.
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.
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.
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.
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