# Volume 53, № 5, 2001

### Hereditary Critical Ω-Compositional Formations

Koptyukh D. G., Vedernikov V. A.

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 579-588

We present a solution of the Shemetkov problem (on the investigation of critical formations) for piecewise-compositional hereditary formations.

### Simultaneous Determination of Two Coefficients of a Parabolic Equation in the Case of Nonlocal and Integral Conditions

Ivanov M. I., Pabyrivs’ka N. V.

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 589-596

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with two unknown time-dependent coefficients in the case of nonlocal boundary conditions and integral overdetermination conditions.

### Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 597-609

We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for *p*, *q* ∈ (0, ∞], *q* > *p*, and *s* ∈ [*p*, *q*], we prove that functions \(x \in L_\infty ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ_{ r } is the perfect Euler spline of order *r* and *c* _{ s + 1}(*x*) is the constant of the best approximation of the function *x* in the space *L* _{ s + 1}. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed *n*, namely, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where *k* ∈ ** N**,

*p*∈ (0, 1], and

*q*∈ [1, ∞]. We also consider other applications of the inequality indicated.

### Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure

Kubichka A. A., Parasyuk I. O.

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 610-624

We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.

### On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 625-630

We investigate the relationship between the norm *N* _{G}(∞) of infinite subgroups of an infinite group *G* and the structure of this group. We prove that *N* _{G}(∞) is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if *N* _{G}(∞) is a non-Dedekind group. In both cases, we describe the structure of the group *G* under the condition that the subgroup *N* _{G}(∞) has finite index in *G*.

### On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. I

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 631-646

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

### Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 647-661

We obtain order estimates for linear widths of the Besov classes \(B_{p,\theta}^r\) of periodic functions of many variables in the space *L* _{q} for certain values of the parameters *p* and *q*.

### Generalization of the Cramer Formula

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 662-679

We apply the method of parametrized continued fractions to the solution of systems of linear algebraic equations on the basis of their Liouville–Neumann formal power series. We construct an analog of the Cramer formula, which is also applicable to the cases of singular, ill-posed, and rectangular matrices.

### Existence of Small Periodic Solutions of Nonlinear Systems of Ordinary Differential Equations

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 680-687

We investigate the case where conditions for the existence of a nonzero periodic solution of a system of ordinary differential equations are determined by the properties of elements of the matrix of linear approximation and the properties of nonlinear terms.

### Investigation of One Linear Differential Equation by Using Generalized Functions with Values in a Banach Space

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 688-693

We present a generalization of some facts of the theory of generalized functions of slow growth to the case of operator-valued test functions. We propose a construction of regular generalized functions with values in a Banach space. The results obtained are used for the description of slowly increasing solutions of linear homogeneous differential equations with shifted arguments and bounded operator coefficients in a Banach space.

### Symmetric Subsets and Colorings of Connected Compact Groups

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 694-697

We find upper and lower bounds for the Haar measure of a monochromatic symmetric subset, which can be found in every measurable *r*-coloring of a connected compact group.

### On the Hyperspace of Convex Compact Subsets of the Tikhonov Cube

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 698-701

We prove that the hyperspace of compact convex subsets of the Tikhonov cube \(I^{\omega_1}\) is homeomorphic to \(I^{\omega_1}\) .

### On the Time of Completion of Pursuit in One Nonlinear Differential Game

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 702-706

For a nonlinear antagonistic two-person differential game on a manifold, we propose a method for the solution of the pursuit problem and determine the time of guaranteed capture.

### Asymptotic Properties of Functions Holomorphic in the Unit Disk

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 707-714

We study the behavior of the sum of a power series near the boundary of the disk of convergence.

### Semirotational Tree Factorizations of Complete Graphs

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 715-721

We select the class of so-called semisymmetric trees and prove that every tree *T* from this class admits a *T*-factorization of a special form in the case where *T* is of order *n* = 2*k* ≤ 16. We formulate the conjecture that every semisymmetric tree *T* admits a *T*-factorization. We establish the existence of a *T*-factorization for semisymmetric trees of certain classes.