# Volume 45, № 10, 1993

### On sequences that do not increase the number of real roots of polynomials

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1323–1331

A complete description is given for the sequences $\{λ_k}_{k = 0}^{ ∞}$ such that, for an arbitrary real polynomial $f(t) = \sum\nolimits_{k = 0}^n {a_k t^k }$, an arbitrary $A \in (0, +∞)$, and a fixed $C \in (0,+∞)$, the number of roots of the polynomial $(Tf)(t) = \sum\nolimits_{k = 0}^n {a_k \lambda _k t^k }$ on $[0,C]$ does not exceed the number of roots off $(t)$ on $[0, A]$.

### Localization of the Bochner-Riesz means in the Nikol'skii classes

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1331–1344

We investigate the conditions for the localization of the Bochner-Riesz means in the Nikol'skii classes $u_p^α$ for $p \in [1, 2]$.

### The asymptotics of constrained control in optimal singular parabolic problems

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1345–1355

A complete asymptotic solution is constructed and justified for the optimal singular parabolic problems with constrained control and a completely degenerate differential part of an operator.

### Massiveness of the sets of extremal functions in some problems in approximation theory

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1356–1361

It is proved that the sets of extremal functions are massive in some problems in approximation theory.

### On the existence of a cyclic vector for some families of operators

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1362–1370

Under certain restrictions, it is proved that a family of self-adjoint commuting operators $A = (A_{\varphi})_{\varphi \in \Phi}$ where $\Phi$ is a nuclear space, possesses a cyclic vector iff there exists a Hubert space $H \subset \Phi'$ of full operator-valued measure $E$, where $\Phi'$ is the space dual to $\Phi$, $E$ is the joint resolution of the identity of the family $A$.

### Limit theorems for diffusion-type processes in $R^m$

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1371–1378

For a sequence of stochastic equations of diffusion type, conditions that are close to necessary and sufficient ones are established for the weak convergence of measures $\mu _{(\xi ^{(n)} ,w)},\; n = 1, ... ,$ associated with the solutions, to the limiting measure $\mu _{(\xi ^{(0)} ,w)}$. The conditions under which the weak convergence of the solutions of stochastic equations implies their strong convergence are established as well.

### On the Noether property of singular integral equations with Cauchy kernels on a rectifiable curve

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1379–1389

The paper deals with the theory of a complete singular integral equation with a Cauchy kernel. The classes of curves and given functions are extended and generalizations of the classical Noether theorems are proved. As a consequence of these theorems, the Noether property is established for the operators associated with this equation, which act into incomplete normed spaces.

### Periodic solutions of pulse evolutionary systems with unbounded nonlinearities

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1390–1397

The conditions are established, under which periodic solutions of pulse evolutionary systems with unbounded (in *x*) operators on the right-hand sides exist and are stable.

### Simulation of spatial-temporal chaos: The simplest mathematical patterns and computer graphics

Romanenko Ye. Yu., Vereikina M. B.

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1398–1410

The article presents three scenarios of the evolution of spatial-temporal chaos and specifies the corresponding types of chaotic solutions to a certain nonlinear boundary-value problem for PDE. Analytic assertions are illustrated by numerical analysis and computer graphics.

### The best trigonometric and bilinear approximations for functions of many variables from the classes $B^r_{p, \theta}$. II

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1411–1423

The order estimates are obtained for the best trigonometric and bilinear approximations of the classes $B^r_{p, \theta}$ of functions of many variables with respect to the metric $L_q$ when $p$ and $q$ satisfy certain relations.

### On decomposability of countable systems of differential equations

Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1424–1432

Conditions under which there exists a change of variables that decomposes a countable system of differential equations are established for the entire real axis and a semiaxis. Similar problems are investigated for a countable system with pulse influence.

### Conditional symmetry and new representations of the Galilean algebra for nonlinear parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1433–1443

An efficient method for finding operators of conditional symmetry, which form a basis of the Galilean algebra, is suggested for a class of Galilei noninvariant parabolic equations. Additional conditions, under which the symmetry can be extended, are described. For the nonlinear equation under consideration, the antireduction is realized and some exact solutions are found by using the conditional Galilei invariance of its differential consequences.

### On the Pompeiu problem and its generalizations

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1444–1448

Functions are investigated whose integrals over a given collection of sets are zero. Pompeiu sets are described in terms of the approximation of their indicators by linear combinations of the indicators of balls with special radii.

### On the solvability of a complete second-order differential equation in Banach space

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1449–1454

For the complete second-order differential equation with unbounded operator coefficients $u'' + A(t)u' + B(i)u = f, \quad u(0) = u_o, \quad u'(0)=u_1$ the Cauchy problem is studied. By using the "coinmutant method", we construct the coercive solution of this problem is in the Holder space in the case where the operator $В$ has the same "strength" as the operator $А^2$.

### Fourier coefficients of functions from the classes $B$ and $C$. Parseval equality for the class $C$ or for the Fourier-Stieltjes series

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1455–1460

We study restrictions that should be imposed on the numbers sequences $\{α_n\}$ and $\{Β_n\}$ in order to guarantee that the series $\sum\nolimits_{n = 1}^\infty {a_n } \cos nx$ and $\sum\nolimits_{n = 1}^\infty {b_n } \sin nx$ do not belong to the classes $B$ or $C$ for any {a n } and {b n } such that $a_n ≥ α_n, b_n ≥ Β_n,\; n = 1, 2$.

### Existence and asymptotic properties of solutions of a real system of differential equations unresolved with respect to the derivative

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1461–1464

A system of differential equations unresolved with respect to a derivative is studied. Sufficient conditions for the existence of solutions with singular initial data are found. The dimensionality of the set of these solutions is determined and their asymptotic properties are studied.