# Volume 56, № 7, 2004

### Some new conditions for the solvability of the cauchy problem for systems of linear functional-differential equations

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 867–884

We establish efficient conditions sufficient for the unique solvability of certain classes of Cauchy problems for systems of linear functional-differential equations. The conditions obtained are optimal in a certain sense.

### On one condition of weak compactness of a family of measure-valued processes

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 885–891

We present a criterion for the weak compactness of continuous measure-valued processes in terms of the weak compactness of families of certain space integrals of these processes.

### Random attractors for ambiguously solvable systems dissipative with respect to probability

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 892–900

We prove a theorem on the existence of a random attractor for a multivalued random dynamical system dissipative with respect to probability. Abstract results are used for the analysis of the qualitative behavior of solutions of a system of ordinary differential equations with continuous right-hand side perturbed by a stationary random process. In terms of the Lyapunov function, for an unperturbed system, we give sufficient conditions for the existence of a random attractor.

### Shape-preserving kolmogorov widths of classes of *s*-monotone integrable functions

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 901–926

Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences $[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, where $B_p$ is the unit ball of the space $L_p$, and let $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. For every $s \geq 3$ and $1 \leq q \leq p \leq \infty$, exact orders of the shape-preserving Kolmogorov widths $$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$ are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n \leq n$ and $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$.

### Invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures

Kharchenko N. V., Koshmanenko V. D.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 927–938

We prove a theorem on the existence and structure of invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures.

### Lie-algebraic structure of (2 + 1)-dimensional Lax-type integrable nonlinear dynamical systems

Hentosh О. Ye., Prikarpatskii A. K.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 939–946

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable two-metrizable systems is studied.

### Best approximation of reproducing kernels of spaces of analytic functions

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 947–959

We obtain exact values for the best approximation of a reproducing kernel of a system of *p*-Faber polynomials by functions of the Hardy space *H* _{q}, *p* ^{-1} + *q* ^{-1} = 1 and a Szegö reproducing kernel of the space *E* ^{2}(Ω) in a simply connected domain Ω with rectifiable boundary.

### Approximation of the $\bar {\Psi}$ -integrals of functions defined on the real axis by Fourier operators

Sokolenko I. V., Stepanets O. I.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 960–965

We find asymptotic formulas for the least upper bounds of the deviations of Fourier operators on classes of functions locally summable on the entire real axis and defined by $\bar {\Psi}$-integrals. On these classes, we also obtain asymptotic equalities for the upper bounds of functionals that characterize the simultaneous approximation of several functions.

### Conditions for the maximal dissipativity of almost bounded perturbations of smooth restrictions of operators adjoint to symmetric ones

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 966–976

We establish conditions for the maximal dissipativity of one class of densely-defined closed linear operators in a Hilbert space. The results obtained are applied to the investigation of some special differential boundary operators.

### Spaces appearing in the construction of infinite-dimensional analysis according to the biorthogonal scheme

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 977–990

We study properties of annihilation operators of infinite order that act in spaces of test functions. The results obtained are used for establishing the coincidence of spaces of test functions.

### Generalized moment representations and multipoint padé approximants

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 991–995

We establish conditions for the maximal dissipativity of one class of densely-defined closed linear operators in a Hilbert space. The results obtained are applied to the investigation of some special differential boundary operators.

### On coxeter functors for some classes of algebras generated by idempotents

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 996–1001

We study properties of annihilation operators of infinite order that act in spaces of test functions. The results obtained are used for establishing the coincidence of spaces of test functions.

### First-passage probabilities for randomly excited mechanical systems by a selective Monte-Carlo simulation method

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 1002–1008

In this paper, Monte-Carlo methods used for the reliability assessment of structures under stochastic excitations are further advanced, e.g., by leading the generated samples towards the low probability range which is practically not assessable by direct Monte-Carlo methods. Based on criteria denoting the realizations that lead most likely to failure, a simulation technique called the “Russian Roulette and Splitting” (RR&S) is presented and discussed briefly. In a numerical example, the RR&S procedure is compared with the direct Monte-Carlo simulation method (MCS), demonstrating comparative accuracy.