In this paper, a brief overview of the foundational concepts of fractal calculus is presented. First-order linear fractal differential equations are solved, and the Neumann method is employed to address fractal linear integral equations. Moreover, the Neumann method is extended to fractal operator equations, offering a systematic procedure for deriving approximate solutions. The analytical findings are supported by numerical simulations, and the corresponding solutions are plotted to illustrate the behavior of functions defined on fractal sets.
In this article, we introduce a new function, termed the β -Crescent function, which gives rise to a new geometric domain. Based on this function, we define and investigate a Ma–Minda subclass of analytic bi-univalent functions associated with the β -Crescent-shaped domain. Within this framework, we find the best possible estimates for the initial coefficients, as well as for the Fekete–Szegö functional. Furthermore, we establish the best possible upper bound for the second Hankel determinant |H_2(2)| for functions belonging to this class. The study opens several directions for future research, particularly concerning geometric properties and functional inequalities related to this newly generated domain.
We develop a theory of prime ideals and m-systems for 3-Lie algebras based on the ternary product of ideals. Complements of prime ideals are characterized as m-systems, both in terms of ideals and elementwise through the multiplication algebra. We prove a separation theorem, a Levitzki-type description of semiprime ideals, an elementwise description of the prime radical, and a correspondence between maximal m-systems and minimal prime ideals. We then study the current 3-Lie algebra obtained by tensoring a finite-dimensional central simple 3-Lie algebra with a commutative algebra R. Every ideal of this algebra is induced from a unique ideal of R; the ternary product of ideals corresponds to the ordinary product of ideals of R, and the prime spectrum is homeomorphic to the prime spectrum of R. Central simplicity cannot be weakened to simplicity, and for simple algebras with a nontrivial centroid we identify the correct coefficient ring. Using content ideals, we show that m-systems of the current algebra correspond to multiplicative families of finitely generated ideals of R. The main result is an idempotent-core theorem: every m-system that is a finite union of unit orbits contains an orbit whose content is generated by a nonzero idempotent, and these idempotents are pairwise non-orthogonal. Hence, over a ring with connected spectrum, such m-systems consist exactly of unimodular orbits, and the finiteness assumption is sharp. The results extend to twisted forms of current algebras. The four-dimensional Filippov algebra and its twisted forms serve as the main examples.
Let U_q^+(𝔤) be the multiparameter quantum Borel algebra, where 𝔤 is a simple Lie algebra. The purpose of this paper is to prove that, in the case that 𝔤 is a simple Lie algebra of type F_4 , the algebra U_q^+(𝔤) can be constructed as a quadratic-linear algebra, but it is not a quadratic-linear Koszul algebra.
