2019
Том 71
№ 8

All Issues

Article (Russian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (Ukrainian)

Limit theorems for systems of the type M θ/G/1/b with resume level of input stream

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (English)

Multiplicative relations with conjugate algebraic numbers

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (Russian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 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) 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.

Article (Russian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (Ukrainian)

On some properties of convex functions

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 7. - pp. 920–938

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

Article (Ukrainian)

Controllability problems for the string equation

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (Ukrainian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Article (Ukrainian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 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 $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$ $ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$.

Anniversaries (Russian)

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

Ukr. Mat. Zh. - 2007νmber=6. - 59, № 7. - pp. 996-1000

Brief Communications (Ukrainian)

Linearly ordered compact sets and co-Namioka spaces

Ukr. Mat. Zh. - 2007νmber=6. - 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.

Brief Communications (English)