# Volume 53, № 1, 2001

### 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.

### 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.

### 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.

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

Krvavich Yu. V., Mishura Yu. S.

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.

### 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 *p*th powers, and the central limit theorem.

### 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.

### 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.

### 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.

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

Chernikova O. S., Perestyuk N. A.

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.

### 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.

### 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.

### 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 \(\mathfrak{F}\) .

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

Antoniouk A. Val., Antoniouk A. Vict.

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.

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

Domans'ka G. P., Lavrenyuk S. P.

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.

### 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.

### 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 *n*th-degree Chebyshev polynomial.