# Volume 59, № 7, 2007

### On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain

Bazalii B. V., Degtyarev S. P.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 867–883

We prove the existence and uniqueness of a classical solution of a singular elliptic boundary-value problem in an angular domain. We construct the corresponding Green function and obtain coercive estimates for the solution in the weighted Hölder classes.

### Limit theorems for systems of the type *M *^{θ}/*G*/1/*b* with resume level of input stream

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 884-889

A finite capacity queueing system of the type *M *^{θ}/*G*/1/*b* is considered in which the input flow is regulated by some threshold level.
Asymptotic properties of the first busy period and the number of customers served for this period are studied.

### Multiplicative relations with conjugate algebraic numbers

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 890–900

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field *K* in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over *K* must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over *K* is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over *K*. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square.

### Direct and inverse theorems on approximation of functions defined on a sphere in the space *S *^{(p,q)}(σ^{ m})

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 901-911

We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space *S *^{(p,q)}(σ^{ m}), * m* > 3, in terms of the best approximations and modules of continuity.
We consider constructive characteristics of functional classes defined by majorants of modules of continuity of their elements.

### Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 912–919

On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of the level zero.

### On some properties of convex functions

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 920–938

We obtain some new results for convex-downward functions vanishing at infinity.

### Controllability problems for the string equation

Fardigola L. V., Khalina K. S.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 939–952

For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability.

### Approximation of (ψ, β)-differentiable functions by Weierstrass integrals

Kalchuk I. V., Kharkevych Yu. I.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 953–978

Asymptotic equalities are obtained for upper bounds of approximations of functions from the classes $C^{\psi}_{\beta \infty}$ and $L^{\psi}_{\beta 1}$ by the Weierstrass integrals.

### On a complete description of the class of functions without zeros analytic in a disk and having given orders

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 979–995

For arbitrary $0 ≤ σ ≤ ρ ≤ σ + 1$, we describe the class $A_{σ}^{ρ}$ of functions $g(z)$ analytic in the unit disk $D = \{z : ∣z∣ < 1\}$ and such that $g(z) ≠ 0,\; ρ_T[g] = σ$, and $ρ_M[g] = ρ$, where $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$ $ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$.

### Aleksandr Mikhailovich Lyapunov (the 150th anniversary of his birth)

Martynyuk A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 996-1000

### Linearly ordered compact sets and co-Namioka spaces

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 1001–1004

It is proved that for any Baire space $X$, linearly ordered compact $Y$, and separately continuous mapping $f:\, X \times Y \rightarrow \mathbb{R}$, there exists a $G_{\delta}$-set $A \subseteq X$ dense in $X$ and such that $f$ is jointly continuous at every point of the set $A \times Y$, i.e., any linearly ordered compact is a co-Namioka space.

*I*-radicals and right perfect rings

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 1005–1008

We determine the rings for which every hereditary torsion theory is an *S*-torsion theory in the sense of Komarnitskiy.
We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings.