# Volume 68, № 10, 2016

### Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(ψ,β)$-differentiable functions in $L_2$. III

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1299-1319

In the classes $L^{\psi}_{\beta ,2}$ of $2\pi$ -periodic $(\psi , \beta)$-differentiable functions for which $f^{\psi}_{\beta} \in L_2$, we determine the exact constants in Jackson-type inequalities for the characteristic of smoothness $\Lambda_{\gamma} (f, t) = \biggl\{\frac1t \int^t_0 \| \Delta^{\gamma}_ h(f)\|^2dh \biggr\}^{1/2}$, $t > 0$, deternined by averaging the norm of the generalized difference relation $\Delta_{ \gamma}h(f)$. For the classes of $(\psi,\beta)$ -differentiable functions defined by using the characteristic of smoothness $\Lambda_{\gamma}$ and the majorant $\Phi$, satisfying numerous conditions, we find the exact values of some $n$-widths in $L_2$.

### On the existence of a surface in the pseudo-Euclidean space with given Grassman image

Grechneva M. A., Stegantseva P. G.

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1320=1329

We study the properties of submanifolds of the Grassmanian manifold of the four-dimensional pseudo-Euclidean space and also the Grassman image of a surface in this space. The theorem on the existence of a surface in the pseudo-Euclidean space with the given Grassman image is formulated and proved.

### Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex mappings

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1330-1347

We obtain some inequalities of the Hermite – Hadamard type for $K$-bounded norm convex mappings between two normed spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained.

### Differentiability of Borel measures along vector fields on Banach manifolds with uniform structure

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1348-1364

We analyze the differentiability of Borel measures on Banach manifolds with uniform structure and prove a criterion of weak differentiability.

### Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I

Abdullayev F. G., Özkartepe N. P.

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1365-1379

We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue space, where the contour and the weight functions have some singularities.

### Limit theorem for coiuntable systems of stochastic differential equations

Pilipenko A. Yu., Tantsiura M. V.

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1380-1402

We consider infinite systems of stochastic differential equations used to describe the motion of interacting particles in random media. It is assumed that mass of each particle tends to zero and the density of particles infinitely increases in a proper way. It is proved that the sequence of the corresponding measure-valued processes converges in distribution. We also prove existence and uniqueness of a strong solution for the limit equation.

### Entropy numbers and the widths of the classes $B_{p,θ}^r$ of periodic functions of many variables

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1403-1417

We establish the exact order estimates for the entropy numbers of the Nikol’skii – Besov classes of periodic functions of two variables in the space $L_\infty$ and use the obtained results in estimating the lower bounds for Kolmogorov, linear, and trigonometric widths. We also study the behavior of similar approximating characteristics of the classes $B_{p,θ}^r$ of periodic functions of many variables in the spaces $L_1$ and $B_{1,1}$.

### Half integer values of order-two harmonic numbers sums

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1418-1429

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closedform representations and integral expressions are developed for the infinite series.

### Approximation of general α-cubic functional equations in 2-Banach spaces

Eskandani G. Z., Rassias J. M.

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1430-1436

We introduce a new \alpha -cubic functional equation and investigate the generalized Hyers – Ulam stability of this functional equation in 2-Banach spaces.

### On the relationships between the central series in some locally finite groups

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1437-1441

It is proved that the class of locally finite divisible-by-bounded groups is both a Schur class and a Baer class.