Volume 32, Issue 3

In this paper, we generalize some results from [M. Ashraf and S. Ali, On left multipliers and the commutativity of prime rings, Demonstr. Math. 41 2008, 4, 763–771] and present our contribution to the study of endomorphisms. In particular, we prove that a prime ring must be an integral domain if it admits endomorphisms and multipliers satisfying some algebraic identities on non-zero ideals. As a consequence of our main theorems, many known results can either be generalized or deduced. Furthermore, an example is given to show that the necessity of the primeness assumption imposed on the hypotheses of various theorems cannot be unnecessary.

In this paper, we investigate the fractional powers of the Laplace–Bessel operator Δ ν {\Delta_{\nu}} defined on the positive space ℝ + n {\mathbb{R}^{n}_{+}} : Δ ν = ∑ i = 1 n ∂ 2 ∂ ⁡ x i 2 + ν i x i ⁢ ∂ ∂ ⁡ x i , ν i ≥ 0 . \Delta_{\nu}=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}+\frac{\nu_{% i}}{x_{i}}\frac{\partial}{\partial x_{i}},\quad\nu_{i}\geq 0. Initially, we establish a Pizzetti-type formula for the Laplace–Bessel operator. We then advance our investigation to explore the fractional powers of this operator, developing a new normalized pointwise formula for the fractional operator ( - Δ ν ) s {(-\Delta_{\nu})^{s}} within the Schwartz space, specifically valid for all s > 0 {s>0} , s ≠ 1 , 2 , … s\neq 1,2,\dots  .

In this paper, we show that every solution of hypoelliptic differential equations of constant strength with coefficients in Roumieu spaces is in some Roumieu space.

We present new characterizations of the rings additively generalized certain periodic elements (idempotent, tripotent and nilpotent). We prove that such rings are completely determined by the additive decompositions of their square elements. This improves the results of Chen and Sheibani [ J. Algebra Appl. 16 (2017), 1750178] and Zhou [ J. Algebra Appl. 16 (2017), 1850009].

We define the class of CUSC rings, which are rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called USC rings, introduced by Chen–Wang–Zhou in J. Pure Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called CUC rings, defined by Călugăreanu and Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. We establish that a ring is USC if and only if it is simultaneously CUSC and potent. Some other interesting relationships with CUC rings are obtained as well.

Assume that h : G → ℂ {h:G\rightarrow\mathbb{C}} is analytic on the convex domain G and x ∈ ℒ ⁢ ( ℬ ; E , 𝒜 , μ ) {x\in\mathcal{L}(\mathcal{B};E,\mathcal{A},\mu)} , the set of Bochner-integrable functions on a measurable space ( E , 𝒜 , μ ) {(E,\mathcal{A},\mu)} endowed with a countably-additive scalar measure μ on a σ-algebra 𝒜 {\mathcal{A}} of subsets of E and with values in the Banach algebra ℬ {\mathcal{B}} . If the spectrum σ ⁢ ( x ⁢ ( t ) ) ⊂ G {\sigma(x(t))\subset G} for all t ∈ E {t\in E} and γ ⊂ G {\gamma\subset G} is taken to be close rectifiable curve in G such that σ ⁢ ( x ⁢ ( t ) ) ⊂ ins ⁡ ( γ ) {\sigma(x(t))\subset\operatorname*{ins}(\gamma)} for all t ∈ E {t\in E} , then, in this paper, we show among others that ∥ ∫ E ( h ∘ x ) ⁢ ( u ) ⁢ 𝑑 μ ⁢ ( u ) - h ⁢ ( ∫ E x ⁢ ( u ) ⁢ 𝑑 μ ⁢ ( u ) ) ∥ ≤ 1 2 ⁢ π ⁢ ∫ E ∥ x ¯ E - x ⁢ ( v ) ∥ ⁢ ( ∫ γ | h ⁢ ( ξ ) | ( | ξ | - ∥ x ¯ E ∥ ) ⁢ ( | ξ | - ∥ x ⁢ ( v ) ∥ ) ⁢ | d ⁢ ξ | ) ⁢ 𝑑 μ ⁢ ( v ) , \Bigg{\|}\int_{E}(h\circ x)(u)\,d\mu(u)-h\Bigg{(}\int_{E}x(u)\,d\mu(u)\Bigg{)}% \Bigg{\|}\leq\frac{1}{2\pi}\int_{E}\|\bar{x}_{E}-x(v)\|\Bigg{(}\int_{\gamma}% \frac{|h(\xi)|}{(|\xi|-\|\bar{x}_{E}\|)(|\xi|-\|x(v)\|)}|\,d\xi|\Bigg{)}\,d\mu% (v), where x ¯ E := ∫ E x ⁢ ( u ) ⁢ 𝑑 μ ⁢ ( u ) . \bar{x}_{E}:=\int_{E}x(u)\,d\mu(u). Some examples for exponential function in Banach algebras are also given. Applications for discrete inequalities and Hermite–Hadamard-type inequalities are provided as well.

In this paper, we concern the existence result of the following general eigenvalue problem: { 𝒜 ⁢ ( u ) = λ ⁢ ℬ ⁢ ( u ) in  ⁢ Ω , D α ⁢ ( u ) = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 ⁢ n {2n} and 2 ⁢ ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 𝒜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized N -function, including the Δ 2 {\Delta_{2}} -condition for both the generalized N -function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts.

Let m ≥ 1 {m\geq 1} , n ≥ 3 {n\geq 3} be fixed integers, ℛ {\mathscr{R}} be a unital ring with a nontrivial idempotent, and let ζ m : ℛ → ℛ {\zeta_{m}:\mathscr{R}\to\mathscr{R}} be a multiplicative Jordan n -higher derivation ( n ≥ 3 ) {(n\geq 3)} . In this paper, we prove that if c ⁢ h ⁢ a ⁢ r ⁢ ( ℛ ) ≠ 2 ⁢ 𝑎𝑛𝑑 ≠ n - 1 {char(\mathscr{R})\neq 2~{}\textit{and}~{}\neq n-1} , then ζ m {\zeta_{m}} is an additive Jordan n -higher derivation, more precisely, ζ m ⁢ ( a ) = D m ⁢ ( a ) + π m ⁢ ( a ) {\zeta_{m}(a)=D_{m}(a)+\pi_{m}(a)} for all a ∈ ℛ {a\in\mathscr{R}} , where D m : ℛ → ℛ {D_{m}:\mathscr{R}\to\mathscr{R}} is an additive higher derivation and π m : ℛ → ℛ {\pi_{m}:\mathscr{R}\to\mathscr{R}} is an additive singular Jordan higher derivation.

The main aim of this paper is to study an obvious linear representation of a multiplicative Lie algebra. Also, we find some criteria to determine all possible multiplicative Lie algebra structures on a general linear group and we show that the general linear group on a finite field is a Lie simple group.

In this paper, we study the optimal control problem for integro-differential equations on the semi-axis, which are non-linear with respect to the phase variables and linear with respect to the control. We obtained the sufficient conditions for existence of optimal control in terms of the right-hand side and the quality criterion. Also, we studied the relation between the solutions of the problems on infinite and finite intervals when the length of the interval goes to infinity.

Let S 1 , S 2 , S 3 {S_{1},S_{2},S_{3}} be oriented non-flat minimal surfaces immersed in ℝ n + 1 {{\mathbb{R}}^{n+1}} ( n ≥ 2 ) {(n\geq 2)} with the Gauss maps G 1 , G 2 , G 3 {G_{1},G_{2},G_{3}} into ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} , respectively. Assume that there are conformal diffeomorphisms Φ 2 , Φ 3 {\Phi_{2},\Phi_{3}} of S 1 {S_{1}} onto S 2 , S 3 {S_{2},S_{3}} respectively and let f 1 = G 1 {f^{1}=G_{1}} , f 2 = G 2 ∘ Φ 2 {f^{2}=G_{2}\circ\Phi_{2}} , f 3 = G 3 ∘ Φ 3 {f^{3}=G_{3}\circ\Phi_{3}} . In this paper, we will show that f 1 , f 2 , f 3 {f^{1},f^{2},f^{3}} are algebraic dependence, i.e., f 1 ∧ f 2 ∧ f 3 ≡ 0 {f^{1}\wedge f^{2}\wedge f^{3}\equiv 0} , if they have the same inverse images for a few hyperplanes of ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} in general position with some certain conditions.

Let 𝒜 {\mathcal{A}} be a unital ∗ {\ast} -algebra over the complex field ℂ {\mathbb{C}} . For any μ 1 , μ 2 , μ 3 , … , μ n ∈ 𝒜 {\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}\in\mathcal{A}} , a product μ 1 ∘ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 \mu_{1}\circ\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1} is called Jordan product and μ 1 ∙ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 ∗ {\mu_{1}\bullet\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1}^{\ast}} is called skew Jordan product. Define P 3 ⁢ ( μ 1 , μ 2 , μ 3 ) = μ 1 ∘ μ 2 ∙ μ 3 P_{3}(\mu_{1},\mu_{2},\mu_{3})=\mu_{1}\circ\mu_{2}\bullet\mu_{3} (called mixed Jordan triple product) and P n ⁢ ( μ 1 , μ 2 , … , μ n ) = μ 1 ∘ μ 2 ∘ ⋯ ∙ μ n P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n})=\mu_{1}\circ\mu_{2}\circ\cdots\bullet\mu% _{n} (called mixed Jordan n -product) for all integer n ≥ 3 {n\geq 3} . In this article, it is shown that a map (called nonlinear mixed Jordan n -derivation) φ : 𝒜 → 𝒜 {\varphi:\mathcal{A}\rightarrow\mathcal{A}} satisfies φ ⁢ ( P n ⁢ ( μ 1 , μ 2 , … , μ n ) ) = ∑ i = 1 n P n ⁢ ( μ 1 , … , ν i - 1 , φ ⁢ ( ν i ) , ν i + 1 , … , ν n ) {\varphi(P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n}))=\sum_{i=1}^{n}P_{n}(\mu_{1},% \ldots,\nu_{i-1},\varphi(\nu_{i}),\nu_{i+1},\ldots,\nu_{n})} for all ν 1 , ν 2 , … , ν n ∈ 𝒜 {\nu_{1},\nu_{2},\ldots,\nu_{n}\in\mathcal{A}} if and only if φ is an additive ∗ {\ast} -derivation. As applications, our main result is applied to several special classes of unital ∗ {\ast} -algebras such as prime ∗ {\ast} -algebras, factor von Neumann algebras and von Neumann algebras with no central summands of type I 1 {I_{1}} .

This study explores the L p {L^{p}} -boundedness and compactness characteristics of SG pseudo-differential operators within the Sobolev space framework. We analyze different attributes of minimal-maximal pseudo-differential operators using the SG symbol σ ⁢ ( x , λ ) {\sigma(x,\lambda)} within the class S m 1 , m 2 {S^{m_{1},m_{2}}} , where m 1 {m_{1}} and m 2 {m_{2}} are real numbers. The investigation delves into various aspects of these operators, and we derive a weak solution for the SG pseudo-differential equation by employing the Dunkl transform as part of the aforementioned theoretical framework.

This work presents a general framework that innovates and explores different mathematical aspects associated with special functions by utilizing the mathematical physics-based idea of monomiality. This study presents a unique family of multivariable Hermite polynomials that are closely related to Frobenius–Genocchi polynomials of Apostol type. The study’s deductions address the differential equation, generating expression, operational formalism, and other characteristics that define these polynomials. The affirmation of the controlling monomiality principle further confirms their mathematical foundations. In addition, the work proves recurrence relations, fractional operators, summation formulae, series representations, operational and symmetric identities, and so on, all of which contribute to our knowledge of these complex polynomials.

The problem of the existence of finite products in the category of algebras with genetic realization is considered.

Our fundamental aim in this paper is to study certain * {*} -identities with pair of derivations acting on a prime ideal P and investigate commutativity of a quotient ring R / P {R/P} where R is any ring. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, examples are given to demonstrate that the restrictions imposed on the hypothesis of our results are not superfluous.