# Volume 65, № 6, 2013

### Semigroups of Strong Endomorphisms of Infinite Graphs and Hypergraphs

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 743–754

We define a class of infinite undirected graphs and a class of infinite $n$-regular hypergraphs and prove that any semigroup of all strong endomorphisms of the graphs and hypergraphs from these classes is isomorphic to the wreath product of a transformation monoid and a small category. We establish the criterional conditions for the regularity of the semigroup of strong endomorphisms of infinite $n$-regular hypergraphs.

### On the Average Value of a Generalized Pillai Function over $\mathbb{Z} [i]$ in the Arithmetic Progression

Dadayan Z. Yu., Varbanets P. D.

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 755–764

We construct an asymptotics relation for the average value of the generalized Pillai function in the arithmetic progression.

### Nonlocal Inverse Problem for a Parabolic Equation with Degeneration

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 765–779

We establish conditions for the existence and uniqueness of a classical solution of the inverse problem of determination of the time-dependent coefficient of the higher-order derivative in a parabolic equation with degeneration in the coefficient of the time derivative. We impose boundary conditions of the second kind and a nonlocal overdetermination condition. The case of weak degeneration is investigated.

### On One Class of Factorizable Fundamental Inverse Monoids

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 780–786

Let *G* be an arbitrary group of bijections on a finite set and let *I*(*G*) denote the set of all partial injective transformations each of which is included in a bijection from *G*. The set *I*(*G*) is a fundamental factorizable inverse semigroup. We study various properties of the semigroup *I*(*G*). In particular, we describe the automorphisms of *I*(*G*) and obtain necessary and sufficient conditions for each stable order on *I*(*G*) to be fundamental or antifundamental.

### Fuzzy Functional Differential Equations under Dissipative-Type Conditions

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 787–795

Fuzzy functional differential equations with continuous right-hand sides are studied. The existence and uniqueness of a solution are proved under dissipative-type conditions. The continuous dependence of the solution on the initial conditions is shown. The existence of the solution on an infinite interval and its stability are also analyzed.

### Finiteness Properties of Minimax and $\mathfrak{a}$-Minimax Generalized Local Cohomology Modules

Kianezhad A., Taherizadeh A. J.

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 796–801

Let $R$ be a commutative Noetherian ring with nonzero identity, let $\mathfrak{a}$ be an ideal of $R$, and let $M$ and $N$ be two (finitely generated) $R$-modules. We prove that $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax $\mathfrak{a}$-cofinite $R$-module for all $i < t, t ∈ {{\mathbb{N}}_0}$, if and only if $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax ${R_{\mathfrak{p}}}$ -module for all $i < t$. We also show that, under certain conditions, $\mathrm{Ho}{{\mathrm{m}}_R}\left( {\frac{R}{\mathfrak{a}},H_{\mathfrak{a}}^t\left( {M,N} \right)} \right)$ is minimax $(t ∈ {{\mathbb{N}}_0})$. Finally, we study necessary conditions for $H_{\mathfrak{a}}^i\left( {M,N} \right)$ to be $\mathfrak{a}$-minimax.

### On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 802–813

Motivated by the results of Huang, we deduce a pair of discrete multidimensional Hilbert-type inequalities involving a homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weight functions. In this way, we obtain generalizations of some results known from the literature.

### Fixed-Point Results on Complete *G*-Metric Spaces for Mappings Satisfying an Implicit relation of New Type

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 814–821

We prove general fixed-point theorems (generalizing some recent results) in a complete *G*-metric space.

### Strong Convergence of Two-Dimensional Walsh–Fourier Series

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 822–834

We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space *H* _{ p } to the space *L* _{ p } for 0 < *p* < 1.

### On the Asymptotic Behavior of a Subcritical Branching Process with Immigration

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 835–843

We study a subcritical branching process with inhomogeneous immigration in the case where the mean value and variance of immigration are regularly varying at infinity. We show that a properly normalized subcritical process with immigration weakly approaches a deterministic process and prove the limit theorem for the fluctuation of this process.

### On the Lebesgue Inequality on Classes of $\bar{\psi}$ -Differentiable Functions

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 844–849

We consider the deviations of Fourier sums in the spaces ${C^{\bar{\psi}}}$. The estimates of these deviations are expressed via the best approximations of the $\bar{\psi}$ -derivatives of functions in the Stepanets sense. The sequences $\bar{\psi} = (ψ_1, ψ_2)$ are quasiconvex.

### α-Sasakian 3-Metric as a Ricci Soliton

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 850–856

We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally, we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton.

### On Zeros of Periodic Zeta Functions

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 857–862

We consider zeta functions *ζ*(*s*; \( \mathfrak{a} \) ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions *F*(*ζ*(*s*; \( \mathfrak{a} \) )) have infinitely many zeros in the critical strip. For example, this is true for sin(*ζ*(*s*; \( \mathfrak{a} \) )).

### Derivations on Pseudoquotients

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 863–869

A space of pseudoquotients denoted by *B*(*X, S*) is defined as equivalence classes of pairs (*x*, *f*); where *x* is an element of a nonempty set *X*, *f* is an element of *S*; a commutative semigroup of injective maps from *X* to *X*;
and (*x*, *f*) ~ (*y*, *g*) for *gx* = *fy*: If *X* is a ring and elements of S are ring homomorphisms, then *B*(*X*, *S*) is a ring. We show that, under natural conditions, a derivation on *X* has a unique extension to a derivation on *B*(*X, S*): We also consider (*α*, *β*) -Jordan derivations, inner derivations, and generalized derivations.

### Semiderivations with Power Values on Lie Ideals in Prime Rings

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 870–873

Let *R* be a prime ring, let *L* a noncentral Lie ideal, and let *f* be a nonzero semiderivation associated with an automorphism *σ* such that f(*u*)^{ n } = 0 for all *u* ∈ *L*; where *n* is a fixed positive integer. If either Char *R* > *n* + 1 or Char *R* = 0; then *R* satisfies *s* _{4}; the standard identity in four variables.

### Goursat-Type Problem for a Higher-Order Equation

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 874–880

For a higher-order equation with leading mixed derivative, we consider the Goursat-type problem without consistency conditions. The notion of fundamental solution is introduced. By using this notion, we obtain a representation of the solution of the analyzed problem.