Том 69
№ 4

All Issues

Volume 57, № 10, 2005

Article (Russian)

Majorant estimates for the percolation threshold of a Bernoulli field on a square lattice

Tolmacheva Yu. A., Virchenko Yu. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1315–1326

We suggest a method for obtaining a monotonically decreasing sequence of upper bounds of percolation threshold of the Bernoulli random field on $Z^2$. On the basis of this sequence, we obtain a method of constructing approximations with the guaranteed exactness estimate for a percolation probability. We compute the first term $c_2 = 0,74683$ of the considered sequence.

Article (Russian)

Some remarks on a Wiener flow with coalescence

Dorogovtsev A. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1327–1333

We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time.

Article (Russian)

Degenerate Nevanlinna-Pick problem

Dyukarev Yu. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1334–1343

A general solution of the degenerate Nevanlinna-Pick problem is described in terms of fractional-linear transformations. A resolvent matrix of the problem is obtained in the form of a J-expanding matrix of full rank.

Article (Russian)

Qualitative investigation of a singular Cauchy problem for a functional differential equation

Chaichuk O. R., Zernov A. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1344–1358

We consider the singular Cauchy problem $$txprime(t) = f(t,x(t),x(g(t)),xprime(t),xprime(h(t))), x(0) = 0,$$ where $x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t$, and $h(t) ≤ t, t ∈ (0, τ)$, for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set of continuously differentiable solutions $x: (0, ρ] → ℝ$ ($ρ$ is sufficiently small) with required asymptotic properties.

Article (Russian)

On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks

Kadankov V. F., Kadankova T. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1359–1384

For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution of the supremum, infimum, and value of the process.

Article (Ukrainian)

On properties of subdifferential mappings in Fréchet spaces

Kasyanov P. O., Mel'nik V. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1385–1394

We present conditions under which the subdifferential of a proper convex lower-semicontinuous functional in a Fréchet space is a bounded upper-semicontinuous mapping. The theorem on the boundedness of a subdifferential is also new for Banach spaces. We prove a generalized Weierstrass theorem in Fréchet spaces and study a variational inequality with a set-valued mapping.

Article (Ukrainian)

Approximation of classes of analytic functions by Fourier sums in the metric of the space $L_p$

Serdyuk A. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1395–1408

Asymptotic equalities are established for upper bounds of approximants by Fourier partial sums in a metric of spaces $L_p,\quad 1 \leq p \leq \infty$ on classes of the Poisson integrals of periodic functions belonging to the unit ball of the space $L_1$. The results obtained are generalized to the classes of $(\psi, \overline{\beta})$-differentiable functions (in the Stepanets sense) that admit the analytical extension to a fixed strip of the complex plane.

Article (Ukrainian)

Exact order of relative widths of classes $W^r_1$ in the space $L_1$

Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1409–1417

As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions.

Anniversaries (Ukrainian)

Ivan Oleksandrovych Lukovs'kyi (on his 70-th birthday)

Korenovskii A. A., Korolyuk V. S., Koshlyakov V. N., Samoilenko A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1418-1419

Brief Communications (Russian)

On domains with regular sections

Zelinskii Yu. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1420–1423

We prove the generalized convexity of domains satisfying the condition of acyclicity of their sections by a certain continuously parametrized family of two-dimensional planes.

Brief Communications (Russian)

On one problem for comonotone approximation

Nesterenko A. N., Petrova T. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1424–1429

For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function.

Brief Communications (Russian)

On one extremal problem for numerical series

Radzievskaya E. I., Radzievskii G. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1430–1434

Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\| {\nu (\alpha \eta )_\gamma } \right\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{\eta :\left\| \eta \right\|_p = 1} \inf _{\gamma \in \Gamma } \left\| {\nu (\alpha \eta )_\gamma } \right\|_1$ in the case where $1 < p < ∞$.

Brief Communications (Russian)

Finite-dimensionality and growth of algebras specified by polylinearly interrelated generators

Redchuk I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1435–1440

We investigate the finite-dimensionality and growth of algebras specified by a system of polylinearly interrelated generators. The results obtained are formulated in terms of a function $\rho$.