# Volume 47, № 11, 1995

### Elementary and multielementary representations of vectroids

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

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1451–1477

We prove that every finitely represented vectroid is determined, up to an isomorphism, by its completed biordered set. Elementary and multielementary representations of such vectroids (which play a central role for biinvolutive posets) are described.

### Periodic solutions of autonomous systems with pulse influence in critical cases

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1478–1484

We obtain existence conditions and iterative schemes for constructing periodic solution of weakly nonlinear autonomous systems with pulse influence in critical cases.

### On asymptotic normality of estimates for correlation functions of stationary Gaussian processes in the space of continuous functions

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1485–1497

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.

### Uniqueness theorems for algebraic functions that take the number of algebraic elements into account

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1498–1505

We obtain uniqueness theorems for algebraic functions that take into account not only *A*-points but also the number of covering elements located above them.

### Information widths

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1506–1518

We introduce the notions of adaptive information widths of a set in a metric space and consider the problem of comparing them with nonadaptive widths. Exact results are obtained for one class of continuous functions that is not centrally symmetric.

### To the problem of continuity of many-valued mappings

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1519–1525

We study the problem of the upper and lower semicontinuity of the union and intersection for a family of many-valued mappings. We establish new conditions of lower semicontinuity for the intersection of a family of lower semicontinuous mappings.

### Averaging of differential inclusions with many-valued pulses

Plotnikov V. A., Plotnikova L. I.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1526–1532

We justify a method of complete and partial averaging on finite and infinite intervals for differential inclusions with many-valued pulses.

### On the reducibility of countable systems of linear difference equations

Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1533–1541

We solve the problem of reducibility of a countable linear system of standard difference equations with unbounded right-hand sides by the method of construction of iterations with accelerated convergence. For systems of this type with bounded right-hand sides, this problem is reduced to a finite-dimensional case.

### Integration of a singularly perturbed degenerate linear system

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1542–1548

We present a method for finding a solution of a linear homogeneous system with degenerate matrix and a small parameter with the derivative.

### Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1549–1557

For the class*B* _{ p } ^{ρ} , 0 ≤ ρ < 1, 1≤p ≤ ∞, of 2π-periodic functions of the form f(*t*)=u(ρ,*t*), where*u* (ρ,*t*) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K_{ρ}(*t*) of the convolution f= K_{ρ} **g*, ∥g∥_{ρ}≤l, with respect to the metric of L_{1}. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.

### Asymptotic stability of solutions of systems of stochastic differential equations in the critical case

Yasinskaya L. I., Yurchenko I. V.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1558–1565

We consider the problem of mean square stability and instability of trivial solutions of systems of stochastic differential equations with random operators in the critical case.

### On one problem of stochastic control

Sverdan M. L., Yasinsky V. K., Yurchenko I. V.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1566–1573

We prove a comparison theorem for solutions of nonlinear stochastic differential equations with Poisson perturbations and delay. We consider one problem of stochastic control.

### On majorants in the hardy-littlewood theorem for higher derivatives

Gorbaichuk V. I., Piddubnyi O. M.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1574–1576

We establish conditions for majorants under which the classical Hardy-Littlewood theorem for the class of functions analytic in a disk is true in terms of derivatives of arbitrary fixed order.

### Hypercentral groups of finite subnormal rank

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1577–1580

We introduce the notion of subnormal rank of a group and study hypercentral groups of finite subnormal rank. We construct an example of a hypercentral group that has a finite subnormal rank and infinite (special) rank.

### To the problem of multiplicativity of canonical diagonal forms of matrices over the domain of principal ideals

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1581–1584

We study the structure of nonsingular matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms. In particular, we establish necessary and sufficient conditions of multiplicativity of canonical diagonal forms of nonsingular matrices over this domain.