# Volume 49, № 5, 1997

### On additive inequalities for intermediate derivatives of functions given on a finite interval

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 619–628

We present a general scheme for deducing additive inequalities of Landau-Hadamard type. As a consequence, we prove several new inequalities for the norms of intermediate derivatives of functions given on a finite interval with an exact constant with the norm of a function.

### Convergence rates and finite-dimensional approximation for a class of ill-posed variational inequalities

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 629–637

The purpose of this paper is to investigate an operator version of Tikhonov regularization for a class of ill-posed variational inequalities under arbitrary perturbation operators. Aspects of convergence rate and finite-dimensional approximations are considered. An example in the theory of generalized eigenvectors is given for illustration.

### Small deviations of solutions of stochastic differential equations in tube domains

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 638–650

We present an algorithm for the determination of a complete asymptotic decomposition of the sojourn probability of a one-dimensional diffusion process in a thin domain with curvilinear boundary.

### Topologically free subsets of universal algebras

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 651–661

We prove that every infinite Abelian algebra and every countable field contain infinite topologically free subsets.

### Carleman problem for two pairs of functions in a ring

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 662–671

We consider a particular case of the matrix Carleman problem for two pairs of functions in a ring and find a constructive solution of this problem. In addition, we propose an algorithm for the construction of solutions for two infinite systems of smooth transition and for a system of two singular equations of special type.

### On asymptotic decompositions of *o*-solutions in the theory of quasilinear systems of difference equations

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 672–677

We consider a quasilinear system of difference equations with certain conditions. We prove that there exists a formal partial *o*-solution of this system in the form of functional series of special type. We also prove a theorem on the asymptotic behavior of this solution.

### On infinite groups whose noncyclic norm has a finite index

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 678–684

We study groups in which the intersection of normalizers of all noncyclic subgroups (noncyclic norm) has a finite index. We prove that if the noncyclic norm of an infinite noncyclic group is locally graded and has a finite index in the group, then this group is central-by-finite and its noncyclic norm is a Dedekind group.

### Estimate of an error of the averaging method on a semiaxis for a multifrequency resonance system

Petryshyn R. I., Pokhyla O. M.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 685–690

In this paper, we justify the averaging method for a multifrequency resonance system on a semiaxis under the assumption that the normal fundamental matrix of a variational system of averaged equations for slow variables exponentially tends to zero. We also study the quantitative dependence of the estimates on the magnitude of a small parameter.

### Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 691–698

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the *j* th component of the effective potential for *n* particles equals the partial derivative with respect to the coordinate of the *j*th particle of “magnetic potential energy” of *n* particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for *d*-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy).

### On the Lie algebra structures connected with Hamiltonian dynamical systems

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 699–705

We construct the hierarchies of master symmetries constituting Virasoro-type algebras for the Hamiltonian vector fields preserving a recursion operator. Similarly, repeatedly contracting a Hamiltonian vector field with the corresponding recursion operator, we define an Abelian Lie algebra of the thus obtained hierarchy of vector fields. The approach is shown to be applicable for the Volterra and Toda lattices.

### On the order of local approximation of functions by trigonometric polynomials that are partial sums of averaging operators

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 706–714

We study the order of polynomial approximations of periodic functions on intervals which are internal with respect to the main interval of periodicity and on which these functions are sufficiently smooth. The estimates obtained contain parameters which characterize the smoothness and alternation of signs of nuclear functions and parameters that determine classes of approximated functions.

### Existence, uniqueness, and dependence on a parameter of solutions of differential-functional equations with ordinary and partial derivatives

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 715–719

For a system of quasilinear hyperbolic equations with a system of differential equations with lag, we prove theorems on the existence and uniqueness of a solution of the Cauchy problem and its continuous dependence on the initial conditions.

### Critical cases of stability of one nonautonomous essentially nonlinear equation of the *n*th order

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 720–724

We establish sufficient conditions of the Lyapunov stability of the trivial solution of a nonautonomous ordinary differential equation of the *n*th order in the case where its characteristic equation has a multiple zero root. The stability is determined by nonlinear terms.

### An example of a complete orthonormal system in a Hilbert space of generalized functions

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 725–728

We construct a complete orthonormal system of generalized functions in a Hilbert space *W* ^{−1}. We obtain an estimate of the error of approximation in *W* ^{−1}, which is expressed in terms of the integral modulus of continuity of a function from *L* _{2}.

### Periodic groups are not *-wild

Kalutskii S. A., Samoilenko Yu. S.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 729–730

We consider certain properties of *-wild groups and prove that periodic groups are not *-wild.

### Analytic representation of periodic solutions of linear differential systems

Kenzhebaev K. K., Laptinskii V. N.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 731–735

We study the problem of periodic solutions of linear differential systems with small parameter. We establish new conditions for the existence and uniqueness of periodic solutions of these systems, which can be efficiently verified.

### Wilhelm Illich Fushchich

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 736