Volume 32, Issue 1

In the framework of Clifford analysis, we study higher order Dirac operators constructed with k -vectors. We find a necessary and sufficient condition to determine whether a function cancels them.

Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research.

In this paper, we give new results on demicompact linear operators, study some properties and some results on Fredholm and upper semi-Fredholm relations involving demicompact operators. Our results are used to provide a fine description of the essential pseudospectra.

This article discusses the rotations (windings) of solutions to the basic system of equations y ′ = F 1 ⁢ ( x , y ) {y^{\prime}=F_{1}(x,y)} and x ′ = F 2 ⁢ ( x , y ) {x^{\prime}=F_{2}(x,y)} . This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles.

In this paper, we are interested to formulate new assumptions on the entries of an unbounded 3 × 3 3\times 3 block operator matrix defined with a maximal domain on the product of Banach spaces guaranteeing its corresponding Frobenius–Schur formula. Our approach allows us to derive some original stability results intervening in the theory of perturbed lower semi-Fredholm operators involving the concept of a measure of non-strict cosingularity perturbation. A new technique is presented to investigate the Weidmann and defect essential spectra of the closure of such model of operator matrix via new criterion of perturbation.

Given any regular T 0 {T_{0}} (equivalently, regular T 1 {T_{1}} ) space X , the question of whether X being Lindelöf implies X being a D -space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff T 0 {T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D -space. Also given is a class of handy examples of a second countable hyperconnected T 0 {T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D -space.

In this article, we introduce Fibo-Pascal sequence spaces P p F {P_{p}^{F}} , 0 < p < ∞ {0<p<\infty} , and P ∞ F {P_{\infty}^{F}} through the utilization of the Fibo-Pascal matrix P F {P^{F}} . We establish that both P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} are BK -spaces, enjoying a linear isomorphism with the classical spaces ℓ p {\ell_{p}} and ℓ ∞ {\ell_{\infty}} , respectively. Further contributing to the depth of our investigation, we proceed to derive the Schauder basis of the space P p F {P_{p}^{F}} , alongside an exhaustive computation of the α-, β-, and γ-duals for both spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . Additionally, we undertake the task of characterizing certain classes of matrix mappings pertaining to the spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . The final section of this study is dedicated to the meticulous characterization of compact operators acting in the spaces P 1 F {P_{1}^{F}} , P p F {P_{p}^{F}} , and P ∞ F {P_{\infty}^{F}} by using Hausdorff measure of non-compactness.

Let 𝒦 + ⁢ ( σ ) = { ℙ ( ϑ , σ ) ⁢ ( d ⁢ ζ ) : ϑ ∈ ( 0 , ϑ + ⁢ ( σ ) ) } {{\mathcal{K}_{+}}(\sigma)=\{\mathbb{P}_{(\vartheta,\sigma)}(d\zeta):\vartheta% \in(0,\vartheta_{+}(\sigma))\}} be the Cauchy–Stieltjes Kernel (CSK) family generated by a probability measure σ which is non degenerate and has support bounded from above. Consider the concept of V a {V_{a}} -transformation of measures introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545] for a ∈ ℝ {a\in\mathbb{R}} . We prove that V a ⁢ ( ℙ ( ϑ , σ ) ) ∈ 𝒦 + ⁢ ( σ ) {V_{a}(\mathbb{P}_{(\vartheta,\sigma)})\in{\mathcal{K}_{+}}(\sigma)} for all a ∈ ℝ ∖ { 0 } {a\in\mathbb{R}\setminus\{0\}} if and only if the measure σ is of the free Gaussian (semicircle) type law up to affinity.

Let H be a subgroup of a group G . We say that H is c -subnormal in G if there exists a subnormal subgroup T of G such that H ⁢ T = G {HT=G} and H ∩ T ⩽ H G {H\cap T\leqslant H_{G}} , where H G {H_{G}} is the maximal normal subgroup of G which is contained in H . In this paper, we investigate the structure of a finite group G under the assumption that all maximal subgroups are c -subnormal subgroups and present some new conditions for supersolvability.

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

We are interested in the solution of the Björling problem for timelike surfaces in R 1 4 \mathbb{R}_{1}^{4} . The main contribution of the paper is to present new and many examples of timelike zero mean curvature surfaces and give their explicit parametric equations. In particular cases, one observes that the parametric equations of these surfaces coincide with the timelike minimal surfaces in Lorentz–Minkowski 3-space.

It is shown that there exists a Mazurkiewicz subset of the Euclidean plane containing the graph of some Sierpiński–Zygmund function.

The paper discusses corrected Simpson-type inequalities on fractal sets. Based on an introduced identity, we establish some error bounds for the considered formula using the generalized s -convexity and s -concavity of the local fractional derivative. Finally, we present some graphical representations justifying the established theoretical framework as well as some applications.

This paper is devoted to studying the regularity properties for the new maximal operator M φ {M_{\varphi}} and the fractional new maximal operator M φ , β {M_{\varphi,\beta}} in the local case. Some new pointwise gradient estimates of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {M_{\varphi,\beta,\Omega}} are given. Moreover, the boundedness of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {M_{\varphi,\beta,\Omega}} on Sobolev space is established. As applications, we also obtain the bounds of the above operators on Sobolev space with zero boundary values.

In this paper, we investigate the existence of singular directions, singular radii and indirect singular points of holomorphic curves in P n ⁢ ( ℂ ) {P^{n}(\mathbb{C})} .

A surface is minimal if the mean curvature ℋ mean {\mathcal{H}_{\rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} . In particular, we completely classify minimal surfaces.

In this paper, mixed moduli of smoothness of functions of two variables are studied. We prove Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in different metrics. Estimates for the mixed moduli of smoothness of the derivative of a function are also obtained in terms of the mixed moduli of smoothness of the function itself.

In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform ℬ α {\mathscr{B}_{\alpha}} in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space ℱ α ⁢ ( ℂ d ) {\mathscr{F}_{\alpha}(\mathbb{C}^{d})} .

The main purpose of this paper is to scrutinize the deportment of generalized derivations of R satisfying some functional * {*} -identities involving the center of the factor ring R / P {R/P} where P is a prime ideal of the ring R . Moreover, we suggest to give generalization of some well known results.

In this note, we point out some errors in [M. Zerra, K. Bouchannafa and L. Oukhtite, On generalized derivations in factor rings, Georgian Math. J. 32 2025, 1, 175–182]. Also, we give some insights on the open problems given in the same paper.