Volume 49, № 10, 1997
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1299–1315
We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1316–1323
We establish necessary and sufficient conditions for a many-dimensional diffusion process to reside in a fixed domain with probability one.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1324–1331
We study non-Abelian locally finite groups and non-Abelian locally solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite.
Integral representation of a solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1332–1336
By using the method of integral equations, we prove the existence and uniqueness of a regular solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1337–1344
We formulate sufficient conditions for the technical stability on given bounded and infinite time intervals and for the asymptotic technical stability of continuously controlled linear dynamical processes with distributed parameters. By using the comparison method and the method of Lagrange multipliers in combination with the Lyapunov direct method, we obtain criteria which define a set of controls providing the technical stability of the output process. We select the optimal control that realizes the least value of the norm corresponding to a given process.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1345–1359
We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1360–1372
We establish necessary conditions for the existence of effects of space localization and stabilization in time that are qualitatively new for evolutionary equations. We suggest constructive methods for the solution of the corresponding one-dimensional problems with free boundaries that appear in ecology and medicine.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1373–1384
Theorems on the existence of vector fields with given sets of indexes of isolated singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated singular point of the gradient field is not greater than one.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1385–1395
We propose a new method for the decomposition of direct products of groups into U subsets. By using this method, we prove the following generalization of the Comfort-van Mill theorem: An arbitrary nondiscrete topological Abelian group with a finite number of second-order elements can be decomposed into a countable number of dense subsets.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1396–1403
We describe certain CDN-groups of order p n with elementary commutant of rank two.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1404–1421
Finite volume grand canonical correlation functions of nonequilibrium systems of d-dimensional Brownian particles, interacting through a regular (long-range) pair potential with integrable first partial derivatives, are expressed in terms of the expectation values of a Gaussian random field. The initial correlation functions coincide with the Gibbs correlation functions corresponding to a more general pair long-range potential. Nonequilibrium Euclidean action is introduced, satisfying a thermodynamic stability property.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1422–1428
Theorems on equilibrium instability of conservative systems with gyroscopic forces are proved. The theorems obtained are nonlinear analogs of the Kelvin theorem. The equilibrium instability of the Chaplygin nonholonomic systems is considered.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1429–1431
We suggest a new method for optimizing solutions of a linear control system, which is based on the solution of the Lyapunov matrix equation.
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1432–1435
We obtained the inequalities for upper bounds of seminorms of classes of 2π-periodic functions, which are determined by a linear differential operator and by the majorant of the modulus of continuity.
On the behavior of solutions of the equation for components of a normal system of differential equations
Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1436–1440
We obtain necessary and sufficient conditions for the existence of a sliding mode and also for the knotting of solutions of the equation for components of a normal system of first-order differential equations.