# Volume 66, № 12, 2014

### Exponential Dichotomy and Bounded Solutions of Differential Equations in the Fréchet Space

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1587-1597

We establish necessary and sufficient conditions for the existence of bounded solutions of linear differential equations in the Fréchet space. The solutions are constructed with the use of a strong generalized inverse operator.

### On the Best Polynomial Approximations of Entire Transcendental Functions of Many Complex Variables in Some Banach Spaces

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1598–1614

For the entire transcendental functions $f$ of many complex variables $m (m ≥ 2)$ of finite generalized order of growth $ρ_m (f; α, β)$, we obtain the limiting relations between the indicated characteristic of growth and the sequences of best polynomial approximations of $f$ in the Hardy Banach spaces $H q (U^m )$ and in the Banach spaces $Bm(p, q, ⋋)$ studied by Gvaradze. The presented results are extensions of the corresponding assertions made by Varga, Batyrev, Shah, Reddy, Ibragimov, and Shikhaliev to the multidimensional case.

### On Rings with Weakly Prime Centers

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1615–1622

We introduce a class of rings obtained as a generalization of rings with prime centers. A ring $R$ is called *weakly prime center* (or simply $WPC$) if $ab \in Z(R)$ ($R$) implies that $aRb$ is an ideal of $R$ where $Z(R)$ stands for the center of $R$. The structure and properties of these rings are studied and the relationships between prime center rings, strongly regular rings, and *WPC* rings are discussed, parallel with the relationship between the $WPC$ and commutativity.

### Corrected $T(q)$-Likelihood Estimator in a Generalized Linear Structural Regression Model with Measurement Errors

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1623–1639

We study a generalized linear structural regression model with measurement errors. The dispersion parameter is assumed to be known. The corrected *T* (*q*) -likelihood estimator for the regression coefficients is constructed. In the case where *q* depends on the sample size and approaches 1 as the sample size infinitely increases, we establish sufficient conditions or the strong consistency and asymptotic normality of the estimator.

### Asymptotic Multiphase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg–de-Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1640–1657

We describe the set of initial conditions under which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic multiphase solitonlike solution. The notion of manifold of initial values for which the above-mentioned solution exists is proposed for the analyzed Cauchy problem. The statements on the estimation of the difference between the exact and constructed asymptotic solutions are proved for the Cauchy problem.

### Order Estimates for the Best Approximations and Approximations by Fourier Sums in the Classes of Convolutions of Periodic Functions of Low Smoothness in the Uniform Metric

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1658–1675

We obtain the exact-order estimates for the best uniform approximations and uniform approximations by Fourier sums in the classes of convolutions of periodic functions from the unit balls of the spaces $L_p, 1 ≤ p < ∞$, with generating kernel $Ψ_{β}$ for which the absolute values of its Fourier coefficients $ψ(k)$ are such that $∑_{k = 1}^{∞} ψ_p ′(k)k^{p ′ − 2} < ∞,\; \frac 1p + \frac 1{p′} = 1$, and the product $ψ(n)n^{1/p}$ cannot tend to zero faster than power functions.

### Critical Points Approaches to Elliptic Problems Driven by a p(x)-Laplacian

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1676-1693

We establish the existence of at least three solutions for elliptic problems driven by a *p*(*x*)-Laplacian. The existence of at least one nontrivial solution is also proved. The approaches are based on the variational methods and critical-point theory.

### Cockcroft–Swan Theorem for Projective Crossed Chain Complexes

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1694-1704

The Cockcroft–Swan theorem is proved for $n$-dimensional projective crossed chain complexes $(Pi,G, ∂_i)$, where $G = A * F$ is a free product of a fixed group $A$ by a free finitely generated group $F$.

### Value-Sharing and Uniqueness of Entire Functions

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1705–1717

We study the uniqueness of entire functions sharing a nonzero value and obtain some results improving the results obtained by Fang, J. F. Chen, X.Y. Zhang and W. C. Lin, et al.

### $C2$ Property of Column Finite Matrix Rings

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1718–1722

A ring $R$ is called a right $C2$ ring if any right ideal of $R$ isomorphic to a direct summand of $R_R$ is also a direct summand. The ring $R$ is called a right $C3$ ring if any sum of two independent summands of $R$ is also a direct summand. It is well known that a right $C2$ ring must be a right $C3$ ring but the converse assertion is not true. The ring $R$ is called $J$ -regular if $R/J(R)$ is von Neumann regular, where $J(R)$ is the Jacobson radical of $R$. Let $ℕ$ be the set of natural numbers and let $Λ$ be any infinite set. The following assertions are proved to be equivalent for a ring $R$:

(1) $ℂFMFM_{ℕ} (R)$ is a right $C2$ ring;

(2) $ℂFMFM_{Λ}(R)$ is a right $C2$ ring;

(3) $ℂFMFM_{ℕ}(R)$ is a right $C3$ ring;

(4) $ℂFMFM_{Λ}(R)$ is a right $C3$ ring;

(5) $ℂFMFM_{ℕ}(R)$ is a $J$ -regular ring and $M_n(R)$ is a right $C2$ (or right $C3$) ring for all integers $n ≥ 1$.

### Index of volume 66 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1723-1728