# Volume 64, № 3, 2012

### Simple strongly connected quivers and their eigenvectors

Dudchenko I. V., Kirichenko V. V., Plakhotnyk M. V.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 291-306

We study the relationship between the isomorphism of quivers and properties of their spectra. It is proved that two simple strongly connected quivers with at most four vertices are isomorphic to one another if and only if their characteristic polynomials coincide and their left and right normalized positive eigenvectors that correspond to the index can be obtained from one another by the permutation of their coordinates. An example showing that this statement is not true for quivers with five vertices is given.

### Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 307-317

We show that a hyperquadric $M$ in $\mathbb{R}^4_2$ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric $M$. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Morever, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in $\mathbb{R}^4_2$.

### Approximation of Urysohn operator with operator polynomials of Stancu type

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 318-343

We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is a "rectangular isosceles triangle". As a special case, Bernstein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined.

### Balleans and *G* -spaces

Petrenko O. V., Protasov I. V.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 344-350

We show that every ballean (equivalently, coarse structure) on a set $X$ can be determined by some group $G$ of permutations of $X$ and some group ideal $\mathcal{I}$ on $G$. We refine this characterization for some basic classes of balleans: metrizable, cellular, graph, locally finite, and uniformly locally finite. Then we show that a free ultrafilter $\mathcal{U}$ on $\omega$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $\omega$ if and only if $\mathcal{U}$ is a $Q$-point. The paper is concluded with а list of open questions.

### Inverse Jackson theorems in spaces with integral metric

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 351-362

In the spaces $L_{\Psi}(T)$ of periodic functions with metric $\rho(f, 0)_{\Psi} = \int_T \Psi(|f(x)|)dx$, where $\Psi$ is a function of the modulus-of-continuity type, we investigate the inverse Jackson theorems in the case of approximation by trigonometric polynomials. It is proved that the inverse Jackson theorem is true if and only if the lower dilation exponent of the function $\Psi$ is not equal to zero.

### Lipschitzian invariant tori of indefinite monotone system

Lagoda V. A., Parasyuk I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 363-383

We consider a nonlinear system in the direct product of a torus and a Euclidean space. For this system, under the conditions of indefinite coercivity and indefinite monotonicity, we establish the existence of a Lipschitzian invariant section.

### On groups with a strongly imbedded subgroup having an almost layer-finite periodic part

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 384-391

We study Shunkov groups with the following condition: the normalizer of any finite nonunit subgroup has an almost layer-finite periodic part. It is proved that such a group has an almost layer-finite periodic part if it has a strongly imbedded subgroup with almost layer-finite periodic part.

### Extension of holomorphic mappings for few moving hypersurfaces

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 392-403

We prove the big Picard theorem for holomorphic curves from a punctured disc into $P^n(C)$ with $n + 2$ hypersurfaces. We also prove a theorem on the extension of holomorphic mappings in several complex variables into a submanifold of$P^n(C)$ with several moving hypersurfaces.

### Weak $\alpha$-skew Armendariz ideal

Nikmehr M. J., Pazoki M., Tavallaee H. A.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 404-414

We introduce the concept of weak $\alpha$-skew Armendariz ideals and investigate their properties. Moreover, we prove that $I$ is a weak $\alpha$-skew Armendariz ideal if and only if $I[x]$ is a weak $\alpha$-skew Armendariz ideal. As a consequence, we show that $R$ is a weak $\alpha$-skew Armendariz ring if and only if $R[x]$ is a weak $\alpha$-skew Armendariz ring.

### Quasi-unit regularity and $QB$-rings

Li Jianghua, Shangping Wang, Xiaoqin Shen, Xiaoqing Sun

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 415-425

Some relations for quasiunit regular rings and $QB$-rings, as well as for pseudounit regular rings and $QB_{\infty}$-rings, are obtained. In the first part of the paper, we prove that (an exchange ring $R$ is a $QB$-ring) (whenever $x \in R$ is regular, there exists a quasiunit regular element $w \in R$ such that $x = xyx = xyw$ for some $y \in R$) — (whenever $aR + bR = dR$ in $R$, there exists a quasiunit regular element $w \in R$ such that $a + bz = dw$ for some $z \in R$). Similarly, we also give necessary and sufficient conditions for $QB_{\infty}$-rings in the second part of the paper.

### Well-posedness of the Dirichlet and Poincare problems for a multidimensional Gellerstedt equation in a cylindric domain

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 426-432

We prove the unique solvability of the Dirichlet and Poincare problems for a multidimensional Gellerstedt equation in a ´cylindric domain. We also obtain a criterion for the unique solvability of these problems.