Article Frontmatter was published on June 1, 2021 in the journal Georgian Mathematical Journal (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 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

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.

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

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

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.

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

Article Frontmatter was published on April 1, 2021 in the journal Georgian Mathematical Journal (volume 28, issue 2).

In this study, given two crossed complexes 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of associative R -algebras and a crossed complex morphism f:𝒞→𝒟{f\colon\mathcal{C}\to\mathcal{D}}, we construct a homotopy as a pair (H,f){(H,f\/)}, where H=(Hn){H=(H_{n})} is a sequence of R -linear maps Hn:Cn→Dn+1{H_{n}\colon C_{n}\to D_{n+1}}. Then we show that for a fixed pair 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of

For certain classes of functions of Λ-bounded variation on [-π,π]{[-\pi,\pi]}, Waterman and D’Antonio constructed a sequence of summability kernels such that the corresponding integral means converge for any function from the class, but not for any function from a broader Λ′⁢BV{\Lambda^{\prime}\mathrm{BV}} class. In our paper, we extend this result to the case of functions of two variables.

Let ℛ{\mathcal{R}} be a prime ring with center Z⁢(ℛ){Z(\mathcal{R})} and *{*} an involution of ℛ{\mathcal{R}}. Suppose that ℛ{\mathcal{R}} admits generalized derivations F , G and H associated with a nonzero derivation f , g and h of ℛ{\mathcal{R}}, respectively. In the present paper, we investigate the commutativity of a prime ring ℛ{\mathcal{R}} satisfying any of the following identities: (i) [F⁢(x)

We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle

In this paper, by means of the generalized Olsen type inequality, related to the Riesz potential and its commutators, the author establishes the interior estimates on the generalized Morrey space for Schrödinger type elliptic equations with potentials satisfying the reverse Hölder condition.

Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

We extend some numerical radius inequalities for adjointable operators on Hilbert C*{C^{*}}-modules. A new refinement of a numerical radius inequality for some Hilbert space operators is given. More precisely, we prove that if T∈ℬ⁢(ℋ){T\in\mathcal{B}(\mathcal{H})} is an invertible operator, then ∥T∥2≤∥T∥2+1∥T-1∥22≤ω⁢(T).\frac{\|T\|}{2}\leq\frac{\sqrt{\|T\|^{2}+\frac{1}{\|T^{-1}\|^{2}}}}{2}\leq% \omega(T)

In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials

In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic

It is known that if f⁢(z)=zp+∑n=p+1∞an⁢zn{f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain D⊂ℂ{D\subset\mathbb{C}} and, for some real α, we have ℜ⁢𝔢⁡{exp⁡(i⁢α)⁢f(p)⁢(z)}>0{\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0}, z∈D{z\in D}, then f⁢(z){f(z)} is at most p -valent in D . This Ozaki condition is a generalization of the well-known Noshiro–Warschawski

The procedure of positioning of pipes, for example for the automobile exhaust construction, by the use of 3D CAD systems is difficult and time consuming because of obstacles in an engine compartment. There are also further strong technical requirements and restrictions. An automatic generation procedure based on the level set method and the numerical solution of the Eikonal equation is proposed. The

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S . In the classical Black–Sholes model for a stock price, Theorem

In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales.

Article Frontmatter was published on February 1, 2021 in the journal Georgian Mathematical Journal (volume 28, issue 1).

A progressive iterative approximation technique for rational least squares fitting curves is developed. The format is interesting in CAGD (Computer Aided Geometric Design) and improves the recent algorithms. An improved chord method for the root finding based on rational operators is also presented.

Effective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter

We prove that a completely regular locale L is realcompact if and only if the “remainder” β⁢L∖L{\beta L\smallsetminus L} is the join of the zero-sublocales of β⁢L{\beta L} that miss L . This extends a result of Mrówka which characterizes realcompact spaces in terms of their remainders in Stone–Čech compactifications. We prove that β⁢L∖L{\beta L\smallsetminus L} is Lindelöf if and only if L is of countable

The purpose of the present paper is to obtain the degree of approximation in terms of a Lipschitz type maximal function for the Kantorovich type modification of Jakimovski–Leviatan operators based on multiple Appell polynomials. Also, we study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation. A Voronvskaja

By using a purely probabilistic argumentation, two theorems are proved. They simplify the existing methods of analysis for the M/G/1{M/G/1} queuing system by means of the supplementary variables method.

In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives

The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra

In this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers and several integer sequences related to them have been studied. In the present paper, new sets of Bell–Sheffer polynomials are introduced. Connections with Bell numbers are shown.

The main goal of the present paper is to study some results concerning generalized derivations of prime rings with involution. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

The nonlinear fractional differential equation with nonlocal fractional integro-differential boundary conditions in Banach spaces is studied, an existence result is obtained by using the method associated with the technique of measures of noncompactness and an appropriate fixed point theorem. An example is given to illustrate the theory.

We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R , the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) M2⁢(R){M_{2}(R)} is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that R/Sl{R/S_{l}}

In this paper, we study a conformally flat 3-space 𝔽3 which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1F2, where F⁢(x)=e-x12-x22 for x=(x1,x2,x3)∈ℝ3. In particular, we construct all helicoidal surfaces in 𝔽3 by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal

In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

In this paper, we define consistent criteria for hypotheses testing, such that the probability of any kind of error is zero for a given criterion. The necessary and sufficient conditions for the existence of such criteria are given.

This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type -div⁡(a⁢(x,u,∇⁡u))+H⁢(x,u,∇⁡u)=f.

The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

Abstract In this note, using some conditions on the weight function K, we investigate the inner functions as multipliers in weighted Dirichlet spaces, and we also discuss zero sets.

Abstract Let 𝒜 = 𝒜 p {\mathcal{A}=\mathcal{A}_{p}} be the mod ⁢ p {\mathrm{mod}\,p} Steenrod algebra, where p is a fixed prime and let 𝒜 ′ {\mathcal{A}^{\prime}} denote the Bockstein-free part of 𝒜 {\mathcal{A}} at odd primes. Being a connected graded Hopf algebra, 𝒜 {\mathcal{A}} has the canonical conjugation χ. Using this map, we introduce a relationship between the X- and Z-bases of 𝒜 ′ {

Abstract We establish off-diagonal extrapolation on mixed variable Lebesgue spaces. As its applications, we obtain the boundedness for strong fractional maximal operators. The vector-valued analogies are also considered. Additionally, the Littlewood–Paley characterization for mixed variable Lebesgue spaces is also established with the help of weighted norm inequalities and extrapolation.

Abstract The purpose of this work is to investigate the existence of solutions for various Neumann boundary value problems associated to the Laplacian-type operators. The main results are obtained using the extension of Mawhin’s continuation theorem.

Abstract For the nonlinear controlled functional differential equations with several constant delays, the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected.

Abstract Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

Abstract In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

Abstract The “oldest quartic” functional equation f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 4 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] - 6 ⁢ f ⁢ ( x ) + 24 ⁢ f ⁢ ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly

Abstract In this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential I α {I_{\alpha}} in the local Morrey–Lorentz spaces M p , q ; λ loc ⁢ ( ℝ n ) {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})} . This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional

Abstract We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider

Abstract In this paper, for 0 < γ < 1 {0<\gamma<1} and b ∈ I γ ⁢ ( BMO ) {b\in I_{\gamma}(\mathrm{BMO})} , the authors give the L 2 ⁢ ( ℝ n ) {L^{2}({\mathbb{R}}^{n})} boundedness of μ γ ; b {\mu_{\gamma;b}} , the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.