Volume 32, Issue 5

In this paper, we monitor the behavior of a prime ring with involution of second kind. Such rings are studied through the prism of higher derivations satisfying certain differential identities. Precisely, we prove that for a higher derivation D = ( d i ) i ∈ ℕ ∪ { 0 } {D=(d_{i})_{i\in\mathbb{N}\cup\{{0}\}}} , if we are able to establish the identity, d n ⁢ [ x , x * ] ∈ Z ⁢ ( R ) {d_{n}[x,x^{*}]\in Z(R)} for a single positive integer n , then the structure exhibits certain interesting properties. Some similar looking results are also presented.

Suppose that Ω ∈ L 1 ⁢ ( 𝕊 n - 1 × 𝕊 n - 1 ) {\Omega\in L^{1}(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1})} is a homogeneous function of degree zero in the sense Ω ⁢ ( t ⁢ x , s ⁢ y ) = Ω ⁢ ( x , y )   for any  ⁢ t , s > 0 \Omega(tx,sy)=\Omega(x,y)\quad\mbox{for any }t,s>0 satisfying the cancellation property ∫ 𝕊 n - 1 Ω ⁢ ( u , ⋅ ) ⁢ 𝑑 σ n ⁢ ( u ) = ∫ 𝕊 m - 1 Ω ⁢ ( ⋅ , v ) ⁢ 𝑑 σ m ⁢ ( v ) = 0 . \int_{\mathbb{S}^{n-1}}\Omega(u,\cdot\,)\,d\sigma_{n}(u)=\int_{\mathbb{S}^{m-1% }}\Omega(\,\cdot\,,v)\,d\sigma_{m}(v)=0. Under suitable growth conditions on the mappings ϕ 1 , ϕ 2 : ( 0 , ∞ ) → ℝ {\phi_{1},\phi_{2}:(0,\infty)\rightarrow\mathbb{R}} , we prove that the singular integral operator T Ω ( ϕ 1 ↫ ϕ 2 ) ⁢ ( f ) ⁢ ( x , y ) = p.v. ⁢ ∫ ℝ n × ℝ m f ⁢ ( ( x - ϕ 1 ⁢ ( | v | ) ⁢ u , y - ϕ 2 ⁢ ( | u | ) ⁢ v ) ) ⁢ Ω ⁢ ( u , v ) | u | n ⁢ | v | m ⁢ 𝑑 u ⁢ 𝑑 v T_{\Omega}^{(\phi_{1}\looparrowleft\phi_{2})}(f)(x,y)=\textup{p.v.}\int_{% \mathbb{R}^{n}\times\mathbb{R}^{m}}f((x-\phi_{1}(|v|)u,y-\phi_{2}(|u|)v))\frac% {\Omega(u,v)}{|u|^{n}|v|^{m}}\,du\,dv is bounded on L p ⁢ ( ℝ n × ℝ m ) {L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m})} , 1 < p < ∞ {1<p<\infty} , provided the kernel function Ω satisfies weak size conditions. Furthermore, we prove the boundedness of the related truncated maximal singular integral operators.

For a bounded linear operator T on a complex Hilbert space ℋ {\mathcal{H}} , we define the weighted real and imaginary parts of the operator T as ℜ t ⁢ ( T ) := ( 1 - t ) ⁢ T ∗ + t ⁢ T {\mathfrak{R}_{t}(T):=(1-t)T^{\ast}+tT} and ℑ t ⁢ ( T ) := ( 1 - t ) ⁢ T - t ⁢ T ∗ i {\mathfrak{I}_{t}(T):=\frac{(1-t)T-tT^{\ast}}{i}} for t ∈ [ 0 , 1 ] {t\in[0,1]} , where T ∗ {T^{\ast}} denotes the adjoint operator of T . By setting T t := ℜ t ⁢ ( T ) + i ⁢ ℑ t ⁢ ( T ) {T_{t}:=\mathfrak{R}_{t}(T)+i\mathfrak{I}_{t}(T)} , we define the weighted norm ∥ T ∥ t := ∥ T t ∥ {\|T\|_{t}:=\|T_{t}\|} . Another weighted numerical radius is given by w ~ t ⁢ ( T ) = sup ⁡ { ∥ ℜ t ⁢ ( e i ⁢ φ ⁢ T ) ∥ : φ ∈ ℝ } {\widetilde{w}_{t}(T)=\sup\{\|\mathfrak{R}_{t}(e^{i\varphi}T)\|:\varphi\in% \mathbb{R}\}} for t ∈ [ 0 , 1 ] {t\in[0,1]} . In this paper, we present new power inequalities for the weighted operator norm ∥ ⋅ ∥ t {\|\cdot\|_{t}} and the weighted numerical radius w ~ t ⁢ ( ⋅ ) {\widetilde{w}_{t}(\,\cdot\,)} . Additionally, we discuss the special cases of the classical numerical radius and norm of Hilbert space operators.

In this paper, we use the q -Fibonacci band matrix F ^ q = ( F ^ n ⁢ k ⁢ ( q ) ) {\widehat{F}_{q}=(\widehat{F}_{nk}(q))} to introduce the notions of q -Fibonacci summability, q -Fibonacci statistical convergence, strongly F ^ q ⁢ [ p ] {\widehat{F}_{q}[p]} -Cesàro summability and give some inclusion relations about these concepts.

The scope of this paper is to introduce and study new classes of operators, namely, left and right generalized Fredholm operators. We also investigate the spectrum of left and right generalized Fredholm and generalized Weyl operators. At the end of the work, the invariance of the left (right) generalized Fredholm and generalized Weyl spectra is studied.

In the present paper, we examine the generating properties of Sidki’s weak commutativity group. More precisely, if G and G φ {G^{\varphi}} are two isomorphic groups, the weak commutativity group χ ⁢ ( G ) {\chi(G)} is the group generated by G and G φ {G^{\varphi}} subject to the relations [ g , g φ ] = 1 {[g,g^{\varphi}]=1} for all g ∈ G {g\in G} . Here we provide bounds for the number of generators of some subgroups of χ ⁢ ( G ) {\chi(G)} when G is a p -group of odd order and either G is powerful or D ⁢ ( G ) {D(G)} is abelian.

The aim of this paper is to investigate the solvability of both direct and inverse problems related to the heat equation involving the Dunkl operator and Laplace operator in a punctured domain. We establish the existence and uniqueness of solutions for both classes of problems. Furthermore, explicit solution formulas for the problems under consideration are derived.

In this paper, we introduce a ring in which every finitely generated regular ideal is principal, and we call it a regular Bézout ring. We investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions, pullbacks and amalgamated duplication of a ring along an ideal. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.

This paper discusses the boundedness of pseudodifferential operator on weak Hardy space and provides a sufficient condition such that the operator is continuous from the weak Hardy space into the weak Lebesgue space.

In this note we show that in the simplicial setting, the classifying space construction converts short exact sequences of groups not just to homotopy fibrations, but in fact to fibre bundles.

This work is concerned with the stability analysis of nonlinear time-varying systems by using a generalized integral inequality of Gamidov type. The approach is based on the use of the notion of logarithmic norm. We give sufficient conditions for exponential stability of the perturbed system, including Lyapunov-type stability criteria and eigenvalue conditions on “slowly varying systems”. The effectiveness of the proposed approach is illustrated by numerical examples.

This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant a = 1 {a=1} . The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber k ∈ ( 0 , 2 ⁢ 2 ) {k\in(0,2\sqrt{2})} , where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green’s function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green’s representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed two-dimensional inductor-capacitor metamaterial.

In the paper we consider semilinear parabolic equation with a nonlinear term ε ⁢ f ⁢ ( u ) {\varepsilon f(u)} , whose trajectories undergo impulsive perturbations when they meet the impulsive set M = { ∥ u ∥ = R } {M=\{\|u\|=R\}} . We prove that for sufficiently small ε this problem generates an impulsive dynamical system which possesses a compact uniform attractor in the phase space L 2 {L^{2}} .

An autonomous three-dimensional quadratic system of ordinary differential equations with two parameters is considered. This system exhibits hidden oscillations at certain parameter values (it has a limit cycle, but does not have equilibrium points). We are looking for non-autonomous perturbations of this autonomous system that do not change the Mironenko reflecting function. Such so-called admissible perturbations preserve some qualitative properties of the solutions of the original system. For perturbed non-autonomous systems, we derived the Lyapunov stability of equilibrium solutions, proved the existence of the limit cycle and periodic solutions, as well as their stability (instability).

The principal goal of this paper is to present one-sided versions of the extended g -Drazin inverse as weaker kinds of the extended g -Drazin inverse. Precisely, we define left and right extended g -Drazin inverses for Banach algebra elements. A number of characterizations for left and right extended g -Drazin inverses are developed. Since we obtain equivalence between concepts of left (or right) extended g -Drazin invertibility and left (or right) generalized Drazin invertibility, in fact, we get new characterizations for left (or right) generalized Drazin invertibility as consequences. As particular cases of one-sided extended g -Drazin inverses, we investigate one-sided extended Drazin inverses.

In this paper, we introduce matrix A p , q {A_{p,q}} weights on spaces of homogeneous type in the sense of Coifman and Weiss, and prove some basic properties of these weights. As applications, the necessary and sufficient condition for the boundedness of matrix-weighted fractional maximal operators on Lebesgue spaces is established, and the boundedness of matrix-weighted fractional integral operators on Lebesgue spaces is obtained.