Let 𝔄 be a unital prime * -algebra (over the complex field ℂ ) with a nontrivial projection. We denote the skew Lie product of elements x, y∈𝔄 by [x, y]_♢ =xy-yx^*. In this article, we show that if a mapping ξ : 𝔄→𝔄 (not necessarily linear) satisfies ξ ([[x, y]_♢ , z]_♢ )=[[ξ (x), y]_♢ , z]_♢ =[[x, ξ (y)]_♢ , z]_♢ =[[x, y]_♢ , ξ (z)]_♢ for all x, y, z∈𝔄 with x^*y^*z^*=0, then ξ is an additive * -centralizer. Moreover, ξ is of the form ξ (a)=λ a for all a∈𝔄, where λ∈𝒵(𝔄) , the center of 𝔄. Using this characterization of ξ , we also determine the structures of generalized skew Lie triple derivation-like mappings on * -algebras. Finally, we apply our main results to standard operator algebras and factor von Neumann algebras.
In this paper we study biharmonic Schrödinger equation Δ ^2 u - Δ u + V (x) u - 1/2 u Δ (u^2) = f (x, u) , u ∈ H^2 (ℝ^N) with indefinite and bounded potential V. Under certain conditions including the primitive F (x, t) ≤ A t^2 for some A > 0 on the nonlinearity, multiple solutions are obtained. Then, we also obtain a sequence of solutions when f(x,· ) is odd and 4-superlinear.
In this paper we investigate 12 -derivations, local 12 -derivations, and 2-local 12 -derivations of Lie algebras related to model nilpotent Lie algebras. We first obtain an explicit description of the general form of 12 -derivations on model algebras. We also demonstrate that model algebras admit local and 2-local 12 -derivations which are not 12 -derivations. Further, we study solvable Lie algebras whose nilradical is a model Lie algebra, derive the general form of 12 -derivations for these solvable extensions, and prove a rigidity result: for solvable Lie algebras with model nilradical and maximal dimension of the complementary space, all local and 2-local 12 -derivations coincide with 12 -derivations.
Nonlinear science combined with computer simulations plays a vital role in modern technology. Due to the limitations over linear theories and the chaotic nature of the problems in this technological era, the analysis of nonlinear problems has become extremely essential. In this article, a class of two parameter singularly perturbed nonlinear differential equations with Robin boundary conditions is considered. A fitted mesh method consists of a classical finite difference scheme together with a Shishkin mesh is constructed to solve the class of problems. A computational algorithm involving the continuation technique jointly with the suggested fitted mesh method is established to solve the class of problems numerically. The fitted mesh method is proved to be essentially first order convergent uniformly with respect to the parameters. Numerical experiments support the theoretical results.
The primary purpose of this research is to develop a numerical method to generate approximate solutions to third-order differential equations (ThODEs) with Robin-type Boundary conditions (RTBCs). A novel approach for constructing basis functions is developed in terms of SCPFK by making sure that three homogeneous RTBCs were met a priori. New operational matrices (OMs) for developed polynomials’ derivatives are established. Utilizing these OMs and the collocation method, approximate solutions for ThODEs are generated. In addition to establishing the theoretical convergence and deriving rigorous error bounds, a comprehensive perturbation and numerical stability analysis is conducted, proving that continuous solution perturbations decay super-exponentially under spectral discretization. Finally, seven numerical examples – including physical and engineering applications such as sandwich beam deflection in solid mechanics and the nonlinear Falkner–Skan boundary layer flow in fluid mechanics – are presented to illustrate the validity, usefulness, robustness, and applicability of the established technique. The exact solutions and outcomes of other methods are compared to the numerical results obtained. The tables and figures indicate that the suggested approach yields an excellent agreement between approximate and exact solutions.
