Volume 31, Issue 2

According to Posner’s second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems’ hypothesis were not unnecessary.

The purpose of this work is to prove the non-triviality of anisotropic Roumieu Gelfand–Shilov spaces S { N } { M } ⁢ ( ℝ n ) {S^{\{M\}}_{\{N\}}(\mathbb{R}^{n})} , and to establish the inclusion between them.

We develop a new generalized form of the fractional kinetic equation involving the generalized Mathieu series. By using the Sumudu transform, a solution of these generalized fractional kinetic equation is obtained in terms of the Mittag-Leffler function. The numerical results and graphical interpretation are also presented.

We present new results for the generalized Drazin inverse of 2 × 2 {2\times 2} anti-triangular matrices with commuting entries over a Banach algebra. As an application, the g -Drazin invertibility of block-operator matrices is obtained under new wider conditions.

Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 𝐏 ⁢ ( i ⁢ ∂ x ) = ( P 1 ⁢ ( i ⁢ ∂ x ) ) m 1 ⁢ ⋯ ⁢ ( P l ⁢ ( i ⁢ ∂ x ) ) m l \mathbf{P}(i\partial_{x})=(P_{1}(i\partial_{x}))^{m_{1}}\cdots(P_{l}(i\partial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 𝐏 ⁢ ( i ⁢ ∂ x ) ⁢ u = f {\mathbf{P}(i\partial_{x})u=f} in ℝ n {\mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.

Three-dimensional boundary-value problem describing a generalized non-Newtonian fluid with nonlinear Tresca friction type in a thin domain Ω ε {\Omega^{\varepsilon}} are considered. We study the asymptotic behavior when one dimension of the fluid domain tends to zero. We prove some weak convergence of the velocity and the pressure of the fluid. Then the limit problem in two-dimensional domain and the specific Reynolds equation are obtained.

We first present a notion of a double lacunary sequence on product time scales. Using this notion, we define the notions of the double lacunary statistical convergence and double lacunary strongly p -Cesàro summability of 2-multiple functions on product time scales and we study some fundamental properties of both notions. We also present a theorem that connects the above-mentioned two concepts. Furthermore, we define a refinement of a double lacunary sequence on product time scales and provide some fundamental properties as well as inclusion theorems for a refined and a non-refined double lacunary sequence on product time scales.

In this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated.

It is known that if B and B ′ {B^{\prime}} are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then B and B ′ {B^{\prime}} differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of B and B ′ {B^{\prime}} in view of their positive equivalence.

Let G be a group. Let R be a G -graded commutative ring and let M be a graded R -module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r ∈ h ⁢ ( R ) {r\in h(R)} and m ∈ h ⁢ ( M ) {m\in h(M)} with r ⁢ m ∈ Q {rm\in Q} , then either r ∈ Gr ( ( Q : R M ) ) {r\in\operatorname{Gr}((Q:_{R}M))} or m ∈ Gr M ⁡ ( Q ) {m\in\operatorname{Gr}_{M}(Q)} . The graded quasi-primary spectrum qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} is defined to be the set of all graded quasi-primary submodules of M . In this paper, we introduce and study a topology on qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which ( qp . Spec g ( M ) , q . τ g ) {(\mathop{\rm qp.Spec}\nolimits_{g}(M),q.\tau^{g})} is a Noetherian, spectral space.

In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G -permutation degree 𝖲𝖯 G n {\operatorname{\sf SP}^{n}_{G}} . We prove: (a) if a space X is an r -space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (b) if X is a cosmic space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (c) if a space X is a C ⁢ ( κ ) {C(\kappa)} -cosmic, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (d) if a space X is an α-space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} .

This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X , thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity cap H ⁡ ( ⋅ ) {\operatorname{cap}_{H}(\,\cdot\,)} (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with cap H ⁡ ( ⋅ ) {\operatorname{cap}_{H}(\,\cdot\,)} is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L 1 ⁢ ( X , γ ) {L^{1}(X,\gamma)} .

In the present paper, a new technique is presented to study the problem of invertibility of unbounded block 3 × 3 {3\times 3} operator matrices defined with diagonal domain. Sufficient criteria are established to guarantee our interest and to prove some interaction between such a model of an operator matrix and its diagonal operator entries. The effectiveness of the proposed new technique is shown by a physical example of an integro differential equation named the neutron transport equation with partly elastic collision operators. In particular, the obtained results answer the question in [H. Zguitti, A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math. 10 2013, 3, 1497–1507] and the conjecture in [A. Bahloul and I. Walha, Generalized Drazin invertibility of operator matrices, Numer. Funct. Anal. Optim. 43 2022, 16, 1836–1847].

The numerical solution of the nonlinear system of equations resulting from a real engineering problem is discussed. We use the approximate solution of a system of two nonlinear integrodifferential equations to build the nonlinear system of equations. This system can be solved by Newton’s method if the solution is differentiable, or using some derivative-free methods, such as Steffensen’s method. Here we show that Steffensen’s method does not always converge and secant method requires more iterations than Traub’s method and Newton’s method. We recommend Traub’s method in case the solution is not differentiable.

We define and study the Stockwell transform S g \mathscr{S}_{g} associated to the Dunkl–Weinstein operator Δ k , β \Delta_{k,\beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function f Δ f_{\Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} . Moreover, we define the localization operators L g ⁢ ( σ ) \mathcal{L}_{g}(\sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function F η , k ∗ := ( η ⁢ I + S g ∗ ⁢ S g ) − 1 ⁢ S g ∗ ⁢ ( k ) F^{\ast}_{\eta,\smash{k}}:=(\eta I+\mathscr{S}^{\ast}_{g}\mathscr{S}_{g})^{-1}\mathscr{S}^{\ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} on the Sobolev space H k , β s ⁢ ( R + d + 1 ) \mathscr{H}^{s}_{k,\beta}(\mathbb{R}_{+}^{d+1}) .