# Volume 66, № 9, 2014

### On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions

Afrouz G. A., Chung N. T., Naghizadeh Z.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1155–1165

The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using the minimum principle and the Mountain-pass theorem.

### Many-Dimensional Generalized Moment Representations and Padé -Type Approximants for Functions of Many Variables

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1166–1174

We propose an approach to the construction of multidimensional Pad´e-type approximants for analytic functions based on the extension of Dzyadyk’s method of generalized moment representations.

### Associated Branched Continued Fractions with Two Independent Variables

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1175–1184

An algorithm for the expansion of a given formal double power series in the associated branched continued fraction with two independent variables is constructed and the conditions for the existence of this expansion are established.

### Structure of the Systems of Orthogonal Projections Connected with Countable Coxeter Trees

Kirichenko A. A., Samoilenko Yu. S., Tymoshkevych L. M.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1185–1192

The paper is devoted to the investigation of representations of Temperley–Lieb-type algebras generated by orthogonal projections connected with countable Coxeter trees. The theorem on the structure of these systems of orthogonal projections is proved. Some examples are presented.

### Spectral Analysis of Some Graphs with Infinite Rays

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1193–1204

We perform a detailed spectral analysis of countable graphs formed by joining semibounded infinite chains to vertices of a finite graph. The spectrum of a self-adjoint operator generated by the adjacency matrix of the graph is characterized, the spectral measure is constructed, the eigenvectors are presented in the explicit form, and the spectral expansion in eigenvectors is obtained.

### Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus

Feshchenko B. G., Maksimenko S. I.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1205–1212

Let *f* : *T* ^{2} → ℝ be a Morse function on a 2-torus, let *S*(*f*) and \( \mathcal{O} \) (*f*) be, respectively, its stabilizer and orbit with respect to the right action of the group \( \mathcal{D} \) (*T* ^{2}) of diffeomorphisms of *T* ^{2}, let \( \mathcal{D} \) _{id}(*T* ^{2}), be the identity path component of the group \( \mathcal{D} \) (*T* ^{2}), and let *S*′(*f*) = *S*(*f*) ∩ \( \mathcal{D} \) _{id}(*T* ^{2}). We present sufficient conditions under which $$ {\uppi}_1\mathcal{O}(f)={\uppi}_1{\mathcal{D}}_{\mathrm{id}}\left({T}^2\right)\times {\uppi}_0S^{\prime }(f)\equiv {\mathrm{\mathbb{Z}}}^2\times {\uppi}_0S^{\prime }(f). $$ The obtained result is true for a larger class of functions whose critical points are equivalent to homogeneous polynomials without multiple factors.

### Generalized Semicommutative and Skew Armendariz Ideals

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1213–1222

We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts.

### On the Integration of a Nonlinear System of Differential Equations

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1223–1234

We study a system of nonlinear differential equations used as basic for the construction of triangular models for commutative systems of linear nonself-adjoint bounded operators.

### A Class of *p*-Valent Meromorphic Functions Defined by the Liu–Srivastava Operator

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1235–1243

We introduce a subclass of *p*-valent meromorphic functions involving the Lui–Srivastava operator and investigate various properties of this subclass. We also indicate the relationships between various results presented in the paper and the results obtained in earlier works.

### Estimations of the Best Approximations for the Classes of Infinitely Differentiable Functions in Uniform and Integral Metrics

Serdyuk A. S., Stepanyuk T. A.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1244–1256

We establish uniform (with respect to the parameter *p*, 1 ≤ *p* ≤ ∞) upper estimations of the best approximations by trigonometric polynomials for the classes *C* _{ β,p } ^{ ψ } of periodic functions generated by sequences *ψ*(*k*) vanishing faster than any power function. The obtained estimations are exact in order and contain constants expressed in the explicit form and depending solely on the function *ψ*. Similar estimations are obtained for the best approximations of the classes *L* _{ β,1} ^{ ψ } in metrics of the spaces *L* _{ s }, 1 ≤ *s* ≤ ∞.

### A Note on Solymosi’s Sum-Product Estimate for Ordered Fields

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1257–1261

It is proved that Solymosi’s sum-product estimate max{|*A* + *A*|, |*A* · *A*|} ≫ |*A*|^{4/3}/(log |*A*|)^{1/3} holds for any finite set *A* in an ordered field *F*.

### A Note on Semialgebraically Proper Maps

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1262–1268

We prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application of this assertion, we compare the semialgebraically proper actions with proper actions in a sense of Palais.

### On Elliptic Boundary-Value Problems with Small Parameter and Additional Functions on the Boundary of a Domain

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1269–1275

We consider elliptic boundary-value problems in which the elliptic operator is a polynomial function of a small parameter and the boundary conditions contain additional unknown functions. It is shown that the condition of ellipticity with small parameter is not only sufficient but also necessary for the *a priori* estimation of the solutions to this problem in the corresponding special functional spaces depending on the parameter.

### Holomorphic Transformation to a Miniversal Deformation not Always Exists Under *Congruence

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1276–1279

In 1971, Arnold constructed miniversal deformations of square complex matrices under the similarity transformation. Similar miniversal deformations were constructed for matrices under congruence and under *congruence. For matrices under similarity and under congruence, the holomorphic transformations to their miniversal deformations always exist. We prove that this is not true for matrices under *congruence.

### Superfractal Approximation of Functions

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1280–1285

The methods of superfractal approximation of sets introduced in 2005–2011 by M. Barnsley, et al. are modified for the approximation of functions. Nonlinear operators are introduced in the space of bounded functions. The limit behavior of this operator sequence is investigated in a function space (in a sense of pointwise and uniform convergence). We consider a nonhyperbolic case in which not all plane maps specifying the operator in the function space are contractive and propose sufficient conditions for the convergence of approximations and estimates of the errors for this kind of approximation (similar to the collage theorem for fractal approximation).

### Malmquist Theorem for the Solutions of Differential Equations in the Vicinity of a Branching Point

Mokhon'ko A. Z., Mokhon'ko O. A.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1286–1290

An analog of the Malmquist theorem on the growth of solutions of the differential equation $f' = P(z, f)/Q(z, f)$, where $P(z, f)$ and $Q(z, f)$ are polynomials in all variables, is proved for the case where the coefficients and solutions of this equation have a branching point in infinity (e.g., a logarithmic singularity).

### Yetter–Drinfel’d Hopf Algebras on Basic Cycle

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1291–1296

A class of Yetter–Drinfel’d Hopf algebras on basic cycle is constructed.