# Volume 49, № 10, 1997

### Stefan problem for the laplace equation with regard for the curvature of the free boundary

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.

### Viability of solutions of many-dimensional stochastic differential equations

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.

### Groups of finite non-Abelian sectional rank

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.

### On conditions of technical stability of controlled processes with distributed parameters

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.

### Configurations in finite projective planes

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*.

### Problems with free boundaries for nonlinear parabolic equations

Berezovs'kyi M. A., Berezovsky A. A., Mitropolskiy Yu. A.

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.

### Vector fields with a given set of singular points

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.

### Decompositions of direct products of groups

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.

### On the construction of CDN[]-groups with elementary commutant of rank two

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1396–1403

We describe certain CDN-groups of order *p* ^{ n } with elementary commutant of rank two.

### Sine-gordon transformations in nonequilibrium systems of Brownian particles

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.

### On instability of conservative systems with gyroscopic forces

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.

### Optimization of solutions of a linear control system

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.

### On inequalities for seminorms of certain classes of differentiable periodic functions

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.