2018
Том 70
№ 8

# Volume 53, № 1, 2001

Article (Russian)

### On Some Properties of Orthogonal Polynomials over an Area in Domains of the Complex Plane. II

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 3-13

We investigate polynomials that are orthonormal with weight over the area of a domain with quasiconformal boundary. We obtain new exact estimates for the growth rate of these polynomials.

Article (Russian)

### On the Asymptotics of the Sojourn Probability of a Poisson Process between Two Nonlinear Boundaries That Move Away from One Another

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 14-22

We obtain the complete asymptotic decomposition of the sojourn probability of a homogeneous Poisson process inside a domain with curvilinear boundaries. The coefficients of this decomposition are determined by the solutions of parabolic problems with one and two boundaries.

Article (Ukrainian)

### Best Orthogonal Trigonometric Approximations of Classes of Functions of Many Variables $L^{ψ}_{β, p}$

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 23-29

We obtain order estimates for the best orthogonal trigonometric approximations of classes of functions of many variables L β, p ψ in the space L q, 1 < p < q < ∞, q > 2.

Article (Ukrainian)

### Differentiability of Fractional Integrals Whose Kernels Contain Fractional Brownian Motions

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 30-40

We prove the stochastic Fubini theorem for Wiener integrals with respect to fractional Brownian motions. By using this theorem, we establish conditions for the mean-square and pathwise differentiability of fractional integrals whose kernels contain fractional Brownian motions.

Article (Ukrainian)

### Limit Theorems for Random Elements in Ideals of Order-Bounded Elements of Functional Banach Lattices

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 41-49

For a sequence of independent random elements belonging to an ideal of order-bounded elements of a Banach lattice, we investigate the asymptotic relative stability of extremal values, the law of large numbers for the pth powers, and the central limit theorem.

Article (Russian)

### Methods for the Elimination of Unknowns from Systems of Linear Inequalities and Their Applications

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 50-56

We study methods for the elimination of an unknown or a group of unknowns from systems of linear inequalities. We justify these methods by using the Helly theorem. The methods considered are applied to the calculation of streams in networks with a generalized conservation law.

Article (Ukrainian)

### First-Order Equations of Motion in the Supersymmetric Yang–Mills Theory with a Scalar Multiplet

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 57-63

We propose a system of first-order equations of motion all solutions of which are solutions of a system of second-order equations of motion for the supersymmetric Yang–Mills theory with a scalar multiplet. We find N = 1 transformations under which the systems of first- and second-order equations of motion are invariant.

Article (Russian)

### On the Existence of Local Smooth Solutions of Systems of Nonlinear Functional Equations with Deviations Dependent on Unknown Functions

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 64-77

We obtain conditions for the existence of a local differentiable solution of a system of nonlinear functional equations with nonlinear deviations of an argument.

Article (Ukrainian)

### On the Stability of Invariant Sets of Discontinuous Dynamical Systems

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 78-84

We establish sufficient conditions for the stability, asymptotic stability, and instability of invariant sets of discontinuous dynamical systems.

Article (Russian)

### Ultrafilters and Decompositions of Abelian Groups

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 85-93

We prove that every PS-ultrafilter on a group without second-order elements is a Ramsey ultrafilter. For an arbitrary PS-ultrafilter ϕ on a countable group G, we construct a mapping f: G → ω such that f(ϕ) is a P-point in the space ω*. We determine a new class of subselective ultrafilters, which is considerably wider than the class of PS-ultrafilters.

Article (Russian)

### On the Periods of Periodic Motions in Autonomous Systems

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 94-112

We obtain certain estimates for the periods of periodic motions in Lipschitz dynamical systems.

Brief Communications (Russian)

### On τ-Closed Formations of n-Ary Group

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 113-116

We prove that if G is a nonsingle-element n-ary finite group that belongs to a τ-closed formation $\mathfrak{F}$ , then $G/{\text{soc(}}G{\text{)}} \in \Phi _\tau (\mathfrak{F})$ , where $\Phi _\tau (\mathfrak{F})$ is the intersection of all maximal τ-closed subformations of the τ-closed formation of n-ary groups $\mathfrak{F}$ .

Article (Ukrainian)

### Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 117-122

We show that a special choice of the Cameron–Martin direction in the characterization of the Wiener measure via the formula of integration by parts leads to a set of natural representations for derivatives of nonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singularities in the Malliavin calculus.

Brief Communications (Ukrainian)

### A Mixed Problem for One Pseudoparabolic System in an Unbounded Domain

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 123-129

We prove the existence and uniqueness of a solution of a mixed problem for a system of pseudoparabolic equations in an unbounded (with respect to space variables) domain.

Brief Communications (Ukrainian)

### On Proximity of Correlation Functions of Homogeneous and Isotropic Random Fields Whose Spectral Functions Coincide on a Certain Set

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 130-134

We give examples of application of the mean-value theorem to finding various estimates of the proximity of correlation functions in the case where their spectral functions coincide on a certain set.

Article (English)

### Expansions for the Fundamental Hermite Interpolation Polynomials in Terms of Chebyshev Polynomials

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 135-143

We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial.