# Volume 65, № 10, 2013

### Approximation of Periodic Functions of Many Variables in Metric Spaces by Piecewise-Constant Functions

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1303–1314

We prove the direct and inverse Jackson- and Bernstein-type theorems for averaged approximations of periodic functions of many variables by piecewise-constant functions with uniform partition of the period torus in metric spaces with integral metric given by a function ψ of the type of modulus of continuity.

### Two-Dimensional Generalized Moment Representations and Padé Approximations for Some Humbert Series

Chernets’ka L. O., Golub A. P.

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1315–1331

By extending Dzyadyk’s method of generalized moment representations to the case of two-dimensional number sequences, we construct and study Padé approximants for some confluent Humbert hypergeometric series.

### Genera of the Torsion-Free Polyhedra

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1332–1341

We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of stable homotopy classes in these genera.

### Approximation by Finite Potentials

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1342–1349

We consider an infinite system of point particles whose interaction is described by a stable two-body interaction potential *ϕ* of infinite range. A sequence of finite interaction potentials *ϕ* _{ R } pointwise convergent to *ϕ* as *R* → ∞ is introduced. It is shown that the corresponding sequence of correlation functions *ρ* _{ R } converges to *ρ* in the norm of the Ruelle space *E* _{ ξ}.

### Conditions for the Existence of Local Solutions of Set-Valued Differential Equations with Generalized Derivative

Plotnikov A. V., Skripnik N. V.

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1350–1362

We consider a generalized set-valued differential equation with generalized derivative and prove the theorems on existence and uniqueness of its solution for the cases of interval-valued and set-valued mappings.

### Method of Lines for Quasilinear Functional Differential Equations

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1363–1387

We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given.

### Solvability of the First Boundary-Value Problem for the Heat-Conduction Equation with Nonlinear Sources and Strong Power Singularities

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1388–1407

By using the Schauder principle and the principle of contracting mappings, we study the character of point power singularities for the solution of the generalized first boundary-value problem for the heat-conduction equation with nonlinear boundary conditions. We establish sufficient conditions for the solvability of the analyzed problem.

### Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Approximation of Smooth Functions by Weighted Means of *N*-Point Padé Approximants

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1410–1419

Let *f* be a function we wish to approximate on the interval [*x* _{1} *,x* _{ N }] knowing *p* _{1} *>* 1*,p* _{2} *, . . . ,p* _{ N } coefficients of expansion of *f* at the points *x* _{1} *,x* _{2} *, . . . ,x* _{ N } *.* We start by computing two neighboring *N* -point Padé approximants (NPAs) of *f,* namely *f* _{1} = [*m/n*] and *f* _{2} = [*m −* 1*/n*] of *f.* The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of *f* at *x* _{1} *.* We assume that *f* is sufficiently smooth, (e.g. convex-like function), and (this is essential) that *f* _{1} and *f* _{2} bound *f* in each interval]*x* _{ i } *,x* _{ i+1}[ on the opposite sides (we call the existence of such two-sided approximants the two-sided estimates property of *f* ). Whether this is the case for a given function *f* is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest. In this case, further steps become relatively simple. We select a known function *s* having the two-sided estimates property with values *s*(*x* _{ i }) as close as possible to the values *f*(*x* _{ i })*.* We than compute the approximants *s* _{1} = [*m/n*] and *s* _{2} = [*m −* 1*/n*] using the values at points *x* _{ i } and determine for all *x* the weight function *α* from the equation *s* = *αs* _{1} + (1 *− α*)*s* _{2} *.* Applying this weight to calculate the weighted mean *αf* _{1} + (1 *− α*)*f* _{2} we obtain significantly improved approximation of *f.*

### Estimates for Growth of Derivatives of Analytic Functions Along the Radius

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1420–1426

We study the radial boundary behavior of functions analytic in a unit disk of the complex plane.

### Local Maxima of the Potential Energy on Spheres

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1427–1429

Let *S* ^{ d } be a unit sphere in ℝ^{ d+1}, and let *α* be a positive real number. For pairwise different points *x* _{1},*x* _{2}, . . . ,*x* _{ N } ∈ *S* ^{ d }, we consider a functional *E* _{ α }(*x* _{1},*x* _{2}, . . . ,*x* _{ N }) = Σ_{ i≠j } ||*x* _{ i } − *x* _{ j }||^{−α }. The following theorem is proved: for *α* ≥ *d* − 2, the functional *E* _{ α }(*x* _{1},*x* _{2}, . . . ,*x* _{ N }) does not have local maxima.

### Locally *ϕ*-Symmetric Generalized Sasakian-Space Forms

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1430–1438

The aim of the present paper is to find necessary and sufficient conditions for locally *ϕ*-symmetric generalized Sasakian-space forms to have constant scalar curvature, *η* -parallel Ricci tensor, and cyclic parallel Ricci tensor. Illustrative examples are given.

### International scientific conference “Bogolyubov's Readings, DIF-2013. Differential Equations, Theory of Functions, and Their Approximations” devoted to the 75th birthday of academician A. M. Samoilenko

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1439-1440