# Volume 65, № 8, 2013

### Evaluation Fibrations and Path-Components of the Mapping Space $M\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ for $8 ≤ k ≤ 13$

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1023-1034

Let $M\left( {{{\mathbb{S}}^{m}},{{\mathbb{S}}^n}} \right)$ be the space of maps from the $m$-sphere ${\mathbb{S}}^{m}$ into the $n$-sphere ${\mathbb{S}}^{n}$ with $m,n ≥ 1$. We estimate the number of homotopy types of path-components $M_{\alpha}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)$ and fiber homotopy types of the evaluation fibrations ${\omega_{\alpha }}:{M_{\alpha }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)\to {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ and $\alpha \in {\pi_{n+k }}\left( {{{\mathbb{S}}^n}} \right)$ extending the results of [Mat. Stud. - 2009. - 31, № 2. -P. 189-194]. Further, the number of strong homotopy types of ${\omega_{\alpha }}:{M_{\alpha }}\left( {{{\mathbb{S}}^{n+k }},{{\mathbb{S}}^n}} \right)\to {{\mathbb{S}}^n}$ for $8 ≤ k ≤ 13$ is determined and some improvements of the results from [Mat. Stud. - 2009. - 31, № 2. - P. 189-194] are obtained.

### Two-Dimensional Generalized Moment Representations and Rational Approximations of Functions of Two Variables

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1035–1058

The Dzyadyk method of generalized moment representations is extended to the case of two-dimensional sequences and used to construct Padé approximants for functions of two variables.

### Scattering Theory for 0-Perturbed $\mathcal{P}\mathcal{T}$ -Symmetric Operators

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1059–1079

The aim of the present work is to develop the scattering theory for 0-perturbed $\mathcal{P}\mathcal{T}$ -symmetric operators by using the Lax–Phillips method. The presence of a stable $\mathcal{C}$ -symmetry leading to the property of selfadjointness (with proper choice of the inner product) for these PTPT -symmetric operators is described in terms of the corresponding $\mathcal{C}$ -matrix (scattering matrix).

### A Pursuit Problem in an Infinite System of Second-Order Differential Equations

Allahabi F., Ibragimov G. I., Kuchkarov A.

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1080–1091

We study a pursuit differential game problem for an infinite system of second-order differential equations. The control functions of players, i.e., a pursuer and an evader are subject to integral constraints. The pursuit is completed if *z*(τ) = \( \dot{z} \) (τ) = 0 at some τ > 0, where *z*(*t*) is the state of the system. The pursuer tries to complete the pursuit and the evader tries to avoid this. A sufficient condition is obtained for completing the pursuit in the differential game when the control recourse of the pursuer is greater than the control recourse of the evader. To construct the strategy of the pursuer, we assume that the instantaneous control used by the evader is known to the pursuer.

### Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1092–1103

The paper deals with the global existence of weak solutions for a weakly dissipative μ-Hunter–Saxton equation. The problem is analyzed by using smooth data approximating the initial data and Helly’s theorem.

### q-Apostol–Euler Polynomials and q-Alternating Sums

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1104–1117

Basic properties are established and generating functions are obtained for the q -Apostol–Euler polynomials. We define q -alternating sums and obtain q -extensions of some formulas from Integral Transform. Spectr. Funct., 20, 377–391 (2009). We also deduce an explicit relationship between the q -Apostol–Euler polynomials and the q -Hurwitz–Lerch zeta-function.

### On Lie Ideals and Generalized Jordan Left Derivations of Prime Rings

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1118–1125

Let *R* be a prime ring with characteristic different from 2 and *U* be a Lie ideal of *R*. In the paper, we initiate the study of generalized Jordan left derivations on Lie ideals of *R* and prove that every generalized Jordan left derivation on *U* is a generalized left derivation on *U*. Further, it is shown that generalized Jordan left biderivation associated with the left biderivation on *U* is either *U* ⊆ *Z*(*R*) or a right bicentralizer on *U*.

### Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1126–1140

By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form $M = F × B$; where the base $B$ is a onedimensional Riemannian manifold and the fiber $F$ has the form $F = F_1 × F_2$ where $F_i ; i = 1, 2$, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then they are isometric to $S_3$ with constant curvature $k > 0$ in the class of special warped-like product metrics admitting the (weak) $G_2$ holonomy determined by the fundamental 3-form.

### Major Pylypovych Timan (on his 90th birthday)

Babenko V. F., Motornyi V. P., Peleshenko B. I., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Trigub R. M., Vakarchuk S. B.

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1141-1144

### On the *β*-Dual of Banach-Space-Valued Difference Sequence Spaces

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1145–1151

The main object of the paper is to introduce Banach-space-valued difference sequence spaces *ℓ* _{∞}(*X*, Δ), c(*X*, Δ); and c_{0}(*X*, Δ) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient conditions for (*A* _{ k }) ∈ *E* ^{ β }(X, Δ); where *E* ∈ {*ℓ* _{∞}, *c*, *c* _{0}} and (*A* _{ k }) is a sequence of linear operators from a Banach space X into another Banach space Y: Necessary conditions for (*A* _{ k }) ∈ *E* ^{ β }(*X*, Δ) are also investigated.

### Nonergodic Quadratic Operators for a Two-Sex Population

Ganikhodzhaev N. N., Mukhitdinov R. T., Zhamilov U. U.

Ukr. Mat. Zh. - 2013. - 65, № 8. - pp. 1152–1160

We describe the structure of quadratic operators of a two-sex population that differs from the model studied by Lyubich and give an example of nonergodic quadratic operator for a two-sex population.