Volume 51, № 10, 1999
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1299–1310
We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1311–1316
We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1317–1323
We consider new eigenvalue problems with discontinuous eigenfunctions and construct computational algorithms whose accuracy is not worse than the accuracy of analogous known algorithms for problems with smooth eigenfunctions.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1324–1333
We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups.
Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1334–1341
Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1342–1351
We consider a system of nonlinear parabolic equations with transformed argument and prove the existence of integral manifolds. We investigate the bifurcation of an invariant torus from the state of equilibrium.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1352–1359
We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1360–1367
We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion. We obtain the asymptotics of the amount of Shannon information contained in an observation and in the Bayes criterion.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1368–1378
We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium.
Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1379–1390
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1391–1397
By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1398–1410
We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal.
On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1411–1416
For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system).
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1417–1424
A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved.
On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1425–1430
We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index.
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1431–1432
In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds.
On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1433–1441
We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations.