# Volume 53, № 4, 2001

### New Integral Representations for a Hypergeometric Function

Volchkov V. V., Volchkov V. V.

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 435-440

We obtain new integral representations for a hypergeometric function.

### Independent Linear Statistics on Finite Abelian Groups

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 441-448

We give a complete description of the class of all finite Abelian groups *X* for which the independence of linear statistics *L* _{1} = α_{1}(ξ_{1}) + α_{2}(ξ_{2}) + α_{3}(ξ_{3}) and *L* _{2} = β_{1}(ξ_{1}) + β_{2}(ξ_{2}) + β_{3}(ξ_{3}) (here, ξ_{ j }, *j* = 1, 2, 3, are independent random variables with values in *X* and distributions μ_{ j }; α_{ j } and β_{ j } are automorphisms of *X*) implies that either one, or two, or three of the distributions μ_{ j } are idempotents.

### $\Gamma$-Transformation of Parabolic Kählerian Spaces Related by an Almost Geodesic Mapping π2 $π_2 (e = 0)$

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 449-454

For parabolic Kählerian spaces, we obtain a new form of the main equations and construct a Γ-transformation that enables one to convert a certain pair of related parabolic Kählerian spaces into an infinite sequence of different related parabolic Kählerian spaces.

### On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 455-465

We prove the existence of continuously differentiable solutions with required asymptotic properties as *t* → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ where α: (0, τ) → (0, +∞), *g*: (0, τ) → (0, +∞), and *h*: (0, τ) → (0, +∞) are continuous functions, 0 < *g*(*t*) ≤ *t*, 0 < *h*(*t*) ≤ *t*, *t* ∈ (0, τ), \(\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \\ \end{gathered}\) , and the function ϕ is continuous in a certain domain.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. II

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 466-488

We continue the investigation of the problem of energy minimum for condensers began in the first part of the present work. Condensers are treated in a certain generalized sense. The main attention is given to the case of classes of measures noncompact in the vague topology. In the case of a positive-definite kernel, we develop an approach to this minimum problem based on the use of both strong and vague topologies in the corresponding semimetric spaces of signed Radon measures. We obtain necessary and (or) sufficient conditions for the existence of minimal measures. We describe potentials for properly determined extremal measures.

### On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 489-500

We obtain the exact asymptotics (as *n* → ∞) of the best *L* _{1}-approximations of classes \(W_1^r\) of periodic functions by splines *s* ∈ *S* _{2n, r − 1} and *s* ∈ *S* _{2n, r + k − 1} (*S* _{2n, r } is the set of 2π-periodic polynomial splines of order *r* and defect 1 with nodes at the points *k*π/*n*, *k* ∈ Z) under certain restrictions on their derivatives.

### Dzyadyk's Technique for Ordinary Differential Equations Using Hermitian Interpolating Polynomials

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 501-512

For the case of Hermitian interpolation, we consider the approximation-iterative method introduced by Dzyadyk. We construct a practical algorithm.

### On the Existence of a Unique Green Function for the Linear Extension of a Dynamical System on a Torus

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 513-521

We prove two theorems on the existence of a unique Green function for a linear extension of a dynamical system on a torus. We also give two examples of the construction of this function in explicit form.

### Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 522-530

Let *M* _{f}(*r*) and μ_{ f }(*r*) be, respectively, the maximum of the modulus and the maximum term of an entire function *f* and let *l*(*r*) be a continuously differentiable function convex with respect to ln *r*. We establish that, in order that ln *M* _{f}(*r*) ∼ ln μ_{ f }(*r*), *r* → +∞, for every entire function *f* such that μ_{ f }(*r*) ∼ *l*(*r*), *r* → +∞, it is necessary and sufficient that ln (*rl*′(*r*)) = *o*(*l*(*r*)), *r* → +∞.

### Properties of a Finite Group Representable as the Product of Two Nilpotent Groups

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 531-541

We establish a series of new properties of a finite group *G* = *AB* with nilpotent subgroups *A* and *B*.

### On the Binomial Asymptotics of an Entire Dirichlet Series

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 542-549

Let *M*(σ) be the maximum modulus and let μ(σ) be the maximum term of an entire Dirichlet series with nonnegative exponents λ_{ n } increasing to ∞. We establish a condition for λ_{ n } under which the relations $$\ln {\mu }\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + o\left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$ and $$\ln M\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + \left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$ are equivalent under certain conditions on the functions Φ_{1} and Φ_{2}.

### Finitely Represented $K$-Marked Quivers

Belousov K. I., Nazarova L. A., Roiter A. V.

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 550-555

We present necessary and sufficient conditions for the finite representability of *K*-marked quivers.

### A Condition for the Existence of a Unique Green–Samoilenko Function for the Problem of Invariant Torus

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 556-559

Under the assumption that a linear homogeneous system defined on the direct product of a torus and a Euclidean space is exponentially dichotomous on the semiaxes, we obtain a condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus. We find an expression for this function in terms of projectors that determine the dichotomy on the semiaxes.

### Mixed Stieltjes–Hilbert and Fourier Sine and Cosine Convolutions

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 560-567

We introduce generalized convolutions of Stieltjes, Hilbert, and Fourier sine and cosine transforms and consider their applications to integral equations.

### On the Solution of a Locally Finite System of Linear Inequalities with Graph Structure

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 568-571

We propose a method for the solution of a locally finite system of linear inequalities that arises in the course of solution of problems of control over resource in networks with generalized Kirchhoff law. We present a criterion for a system of inequalities to have the graph structure.

### Conditions for Instability of an Invariant Toroidal Manifold of a Discrete Dynamical System in a Banach Space

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 572-576

We obtain conditions for the instability of an invariant toroidal manifold of a discrete dynamic system.