# Volume 52, № 6, 2000

### Rank criteria for the controllability of a boundary-value problem for a linear system of integro-differential equations with pulse influence

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 723–730

We determine necessary and sufficient conditions for the solvability of boundary-value problems for a linear system of integro-differential equations with pulse influence. We prove theorems on the existence and integral representation of solutions of linear second order integral-sum Volterra equations and linear systems of integro-differential equations with pulse influence at fixed times.

### On weak compactness of bounded sets in Banach and locally convex spaces

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 731-739

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero.

### Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 740–747

We show that a subset of the product of*n* metrizable spaces is the set of discontinuity points of some separately continuous function if and only if this subset can be represented in the form of the union of a sequence of*F* _{σ}-sets each, of which is locally projectively a set of the first category.

### Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. II

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 748–757

We develop new methods for the solution of boundary-value problems in the meridian plane of an axisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit form.

### Irresolvable left topological groups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 758–765

We prove that an irresolvable left topological group is of the first category. The pseudocharacter of an irresolvable left topological group*G* is countable, provided that*G* is Abelian or its cardinality is nonmeasurable. Some other cardinal invariants of an irresolvable left topological group are also determined.

### Trigonometric series with uniformly distributed coefficients

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 766–772

We prove the Hölder condition for the sums of series obtained by the substitution of uniformly distributed sequences for random variables in the Fourier expansion of a wiener process.

### Approximation method in problems of potential theory

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 773–782

We investigate the properties of the Dzyadyk approximation method in the case of a binomial function. We study conditions under which an estimate for the relative error of approximation in the uniform metric is exact. The results obtained are applied to certain problems in potential theory.

### Multiply $\mathfrak{L}$ formations of finite groupsformations of finite groups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 783-787

We study $\mathfrak{L}$ formations of finite groups.

### Lebesgue inequalities for poisson integrals

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 798–808

We obtain estimates for the deviations of the Fourier partial sums on the sets of the Poisson integrals of functions from the space*L* _{ p },*p*≥1, that are expressed in terms of the values of the best approximations of such functions by trigonometric polynomials in the metric of*L* _{ p }. We show that the estimates obtained are unimprovable on some important functional subsets.

### On finite solvable groups decomposable into the product of two nilpotent subgroups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 809–819

We establish a series of results concerning various properties of a finite solvable group*G*=*AB* with nilpotent subgroups*A* and*B*.

### Nonlinear d'alembert equation in the pseudo-euclidean space $R_{2,n}$ and its solutions

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 820-827

We investigate the nonlinear D'Alembert equation in the pseudo-Euclidean space $R_{2,n}$ and construct new exact solutions containing arbitrary functions.

### On a bounded almost periodic solution of a semilinear parabolic equation

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 828–830

We obtain sufficient conditions for the existence and uniqueness of a bounded almost periodic solution of a semilinear parabolic equation.

### Investigation of one class of diophantine equations

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 831–836

We consider the problem of existence of solutions of the equation \(\frac{X}{Y} + \frac{Y}{Z} + \frac{Z}{X} = m\) in natural numbers for different*m*∈*N*. We prove that this equation possesses solutions in natural numbers for*m*=*a* ^{2}+5,*a*∈*Z*, and does not have solutions if*m*=4*p* ^{2},*p*∈*N*, and*p* is not divisible by 3. We also prove that, for*n*≥12, the equation $$\frac{{b_1 }}{{b_2 }} + \frac{{b_2 }}{{b_3 }} + \cdots + \frac{{b_{n - 1} }}{{b_n }} + \frac{{b_n }}{{b_1 }} = m$$ possesses solutions in natural numbers if and only if*m*≥*n*,*m*∈*N*.

### On the integration of one class of*N*-dimensional trigonometric series

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 837–840

We prove new sufficient conditions for the integrability of*N*-dimensional trigonometric series, which follow from unimprovable Sidon-type inequalities.

### Completely monotone functions on lie semigroups

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 841–845

We obtain an integral representation of a completely monotone function on a Lie semigroup and prove the equivalence of “difference” and “differential” definitions of this function.

### $Q$-conditional symmetry of a nonlinear two-dimensional heat-conduction equation

Andreeva N. V., Serov N. I., Tulupova L. O.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 846-849

We investigate the $Q$-conditional symmetry of a nonlinear two-dimensional heat-conduction equation. By using ansatzes, we obtain reduced equations.

### Best*m*-term trigonometric approximations of classes of (Ψ, β)-differentiable functions of one variable

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 850–855

We obtain an estimate exact in order for the best trigonometric approximation of classes \(L_{\beta ,p}^\psi \) of functions of one variable in the space*L* _{ q } in the case where 1<*p*≤2≤*q*<∞.

### Approximation of classes $C_\infty ^{\bar \psi }$ by zygmund sumsby zygmund sums

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 856-860

We consider the approximation of functions of the classes of $\bar \psi$ by Zygmund sums. In papticular, we present asymptotic equalities for the quantities $\varepsilon _n (C_\infty ^{\bar \psi } ;Z_n )_C$ under various conditions imposed on functions $ψ_1(·)$ and $ψ_2(·)$.

### A criterion for the solvability of A linear boundary-value problem for A system of the second order

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 861–864

We obtain necessary and sufficient conditions for the solvability of a two-point boundary-value problem for systems of linear differential equations of the second order in the critical case where the corresponding homogeneous boundary-value problem has nontrivial solutions. We construct the general solution of the considered boundary-value problem.