# Volume 49, № 11, 1997

### Vladimir Nikolaevich Koshlyakov

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1443

### On V.N. Koshlyakov’s works in mechanics and its applications

Kalinovich V. N., Mitropolskiy Yu. A., Onishchenko S. M., Polishchuk A. N., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1444–1453

We present a survey of the principal results obtained by V. N. Koshlyakov in analytical mechanics, dynamics of solids, and applied theory of gyroscopes.

### On the product of inner radii of symmetric nonoverlapping domains

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1454–1464

Some results concerning extremal problems for nonoverlapping domains with free poles on the unit circle, known for the simply connected case, are generalized to the case of multiply connected domains.

### Finitely represented dyadic sets and their multielementary representations

Belousov K. I., Nazarova L. A., Roiter A. V.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1465–1477

We obtain the direct reduction of representations of a dyadic set *S* such that |Ind *C(S)*| < ∞ to the bipartite case.

### Nonlocal boundary-value problems for systems on linear partial differential equations

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1478–1487

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.

### On extremal problems on classes of functions defined by integral moduli of continuity

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1499–1503

We obtain lower bounds for solutions of some extremal problems on classes of functions *W* ^{r}H _{1} ^{ω} with integral modulus of continuity ω(*t*). Some of these bounds are regarded as exact.

### Bogolyubov theorem for quasidifferential equations with pulses

Kitanov P. M., Plotnikov V. A.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1504–1511

We consider the averaging method for pulse quasidifferential equations in metric spaces.

### Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1512–1518

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.

### On the approximate solution of nonlinear Volterra-Fredholm integral equations on a complex domain by Dzyadyk’s method

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1519–1528

In 1980–1984, V. K. Dzyadyk suggested and modified an iterative approximation method (IA-method) for numerical solution of the Cauchy problem *y′=f(x,y), y(x* _{0})=x_{0}. Particular cases of nonlinear mixed Volterra-Fredholm integral equations of the second kind arise in the mathematical simulation of the space-time development of an epidemic. This paper is concerned with the approximate solution of integral equations of this type by the Dzyadyk method on complex domains. Finally, we test this method numerically by four different examples.

### Unimprovable estimates for solutions of a mixed problem for linear elliptic equations of the second order in a neighborhood of an angular point

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1529–1542

Under the minimal conditions for the smoothness of the coefficients of an equation, unimprovable estimates are obtained for solutions of a mixed problem for linear nondivergent elliptic equations of the second order in a neighborhood of an angular point of the boundary of a domain.

### System of differential equations with a strong turning point

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1543–1547

The uniform asymptotics of a solution of a system of singularly perturbed differential equations with strong turning point is constructed. We study the case where the boundary operator is analytic with respect to a small parameter.

### On complementability of certain generalized hypercenters in Artinian modules

Kurdachenko L. A., Petrenko B. V., Subbotin I. Ya.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1548–1552

We prove that, in an Artinian module, the upper *FC*-hypercenter over an infinite *FC*-hypercentral locally solvable group has a direct complement. Thus, we obtain a generalization of one of Zaitsev’s theorems and one of Duan’s theorems.

### On linearly convex domains with smooth boundaries

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1553–1556

We establish that an arbitrary locally linearly convex domain with a smooth boundary is strongly linearly convex.

### On point interaction in quantum mechanics

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1557–1560

For the Schrödinger operator corresponding to the point interaction, a direct definition is given in terms of a singular perturbation.

### Perturbation of homogeneous parabolic pseudodifferential equations by locally unbounded vector fields

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1561–1566

We construct solutions of homogeneous pseudodifferential equations of parabolic type perturbed by locally unbounded vector fields. We investigate some properties of these solutions.

### On a fiber bundle over a disk with the cantor set as a fiber

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1567–1571

We construct a Pontryagin fiber bundle ξ = (*N, p, S* ^{1}), the total space *N* of which cannot be imbedded into any two-dimensional oriented manifold but can be imbedded into an arbitrary nonoriented two-dimensional manifold.

### On one stochastic model that leads to a stable distribution

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1572–1579

We consider an integral equation describing the contagion phenomenon, in particular, the equation of the state of a hereditarily elastic body, and interpret this equation as a stochastic model in which the Rabotnov exponent of fractional order plays the role of density of probability of random delay time. We invesgigate the approximation of the distribution for sums of values with a given density to the stable distribution law and establish the principal characteristics of the corresponding renewal process.

### On asymptotic approximation of a solution of a boundary-value problem for a nonlinear nonautonomous equation

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1580–1583

On the basis of periodic Ateb functions, in the resonance and nonresonance cases, we construct the asymptotic approximation of one-frequency solutions of a boundary-value problem for a nonlinear nonautonomous equation.

### The second school “Fourier series. Theory and Applications”

Romanyuk A. S., Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1584