Volume 28, Issue 3

In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.

In the first part of the paper, we study the existence of derivations on a class of 3-prime near-rings. In the second part, we get some commutativity results for prime rings with generalized derivations.

We prove that block spaces defined on ℝn{\mathbb{R}^{n}} with an arbitrary Radon measure, which are known to be the preduals of Morrey spaces, are closed under the first and the second complex interpolation method. The proof of our main theorem uses the duality theorem in the complex interpolation method, the complex interpolation of certain closed subspaces of Morrey spaces, a characterization of the preduals of block spaces, and some formulas related to the Calderón product.

The purpose of this paper is to study the problems of meromorphic functions sharing unique range sets with one or two elements, which can be seen as a new answer to a question due to Gross. The examples also illustrate the accuracy of all of the conditions.

We prove that the set of all k -commutative n -place functions defined on a fixed set forms a Menger algebra, and characterize such algebras by their densely embedded ideals. We also describe all automorphisms of such algebras.

Using the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences: {div⁡(a⁢(x,∇⁡u))=0in ⁢Ω,a⁢(x,∇⁡u)⋅ν=λ⁢m⁢(x)⁢|u|p⁢(x)-2⁢uon ⁢∂⁡Ω,\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}(N≥2){(N\geq 2)} is a bounded domain of smooth boundary ∂⁡Ω{\partial\Omega} and ν is the outward unit normal vector on ∂⁡Ω{\partial\Omega}. The functions m∈L∞⁢(∂⁡Ω){m\in L^{\infty}(\partial\Omega)}, p:Ω¯↦ℝ{p\colon\overline{\Omega}\mapsto\mathbb{R}} and a:Ω¯×ℝN↦ℝN{a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space Lp⁢(⋅){L^{p(\,\cdot\,)}} with min⁡p⁢(⋅)=1{\min p(\,\cdot\,)=1}, which are invariant with respect to Cauchy singular integral operators.

In this paper, we consider a natural extension in the context of d -orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d -orthogonal polynomials, asymptotic expansions in terms of d -Hermite ones. From these expansions, several limits between d -orthogonal polynomials are obtained.

In this paper we show that the dual Riesz transform associated with the generalized Schrödinger operator ℒ{\mathcal{L}} is bounded from BMO{\mathrm{BMO}} into BMOℒ{\mathrm{BMO}_{\mathcal{L}}} and give the Fefferman–Stein-type decomposition of BMOℒ{\mathrm{BMO}_{\mathcal{L}}} functions in terms of Riesz transforms.

We prove McCoy’s property for the zero-divisors of polynomials in semirings, investigate the zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule M⁢[G]{M[G]} has very few zero-divisors if and only if the S -semimodule M does so. The concept of Auslander semimodules is introduced as well. Then we introduce Ohm–Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. Finally, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras.

In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.

Recently, some authors have established a number of inequalities involving the minimum eigenvalue for the Hadamard product of M -matrices. In this paper, we improve these results and give some new lower bounds on the minimum eigenvalue for the Hadamard product of an M -matrix A and its inverse A-1{A^{-1}}. Finally, it is shown by the numerical examples that our bounds are also better than some previous results.

The aim of this work is to offer a new solution to the problem of the classification of endomorphisms with an annihilating polynomial on infinite-dimensional vector spaces. For these endomorphisms we provide a family of invariants that allows us to classify them when the group of automorphisms acts by conjugation. Moreover, the description of a new method to construct Jordan bases is given.

We investigate the asymptotic normality of distributions of the sequence ∑k∈ℤun,k⁢Xk{\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}}, n∈ℕ{n\in\mathbb{N}}, where (Xk,k∈ℤ){(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights (un,k){(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake’s theorem in terms of a variational measure.

In this paper, we investigate the strong summability of two-dimensional Walsh–Fourier series obtained in [F. Weisz, Strong convergence theorems for two-parameter Walsh–Fourier and trigonometric-Fourier series, Studia Math. 117 1996, 2, 173–194] (see Theorem W) and prove the sharpness of this result.

The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.