# Volume 57, № 10, 2005

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

Tolmacheva Yu. A., Virchenko Yu. P.

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.

### Some remarks on a Wiener flow with coalescence

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.

### Degenerate Nevanlinna-Pick problem

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.

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

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.

### 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.

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.

### On properties of subdifferential mappings in Fréchet spaces

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.

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

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.

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

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.

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

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

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

### On domains with regular sections

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.

### On one problem for comonotone approximation

Nesterenko A. N., Petrova T. O.

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.

### On one extremal problem for numerical series

Radzievskaya E. I., Radzievskii G. V.

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 $1 < p < ∞$.

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

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 $\rho$.