Journal of Mathematics

This paper is devoted to present a numerical scheme based on operational matrices to compute approximate solutions to fractional-order boundary value problems. For the mentioned operational matrices, we utilize fractional-order Bernoulli polynomials. Since fractional-order problems are usually difficult to treat for their corresponding analytical or exact solutions, therefore, we need sophisticated methods to find their best numerical solutions. The presented numerical scheme has the ability to reduce the proposed problem to the corresponding algebraic equations. The obtained algebraic equations are then solved by using the computational software MATLAB for the corresponding numerical results. The used method has the ability to save much more time and also is reliable to secure the proper amount of memory. Several examples are solved by using the considered method. Also, the solutions are compared with their exact solution graphically. In addition, the absolute errors for different scale values are presented graphically. In addition, we compare our results with the results of the shifted Legendre polynomials spectral method.

Let be a ring, be the group of all units of , and . In this paper, we investigate for a ring , where is the set of all vertices of the zero-divisor graph of adjacent to . We also investigate the question on zero-divisor graphs posed in the literature such that when the equality holds in a commutative regular ring with identity. Here, is a nonzero idempotent of which is not the identity element of .

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

In this paper, we focus on solving the vector scheduling problem with submodular penalties on parallel machines. We are given jobs and parallel machines, where each job is associated with a -dimensional vector. Each job can either be rejected, incurring a rejection penalty, or accepted and processed on one of the parallel machines. The objective is to minimize the sum of the maximum load overall dimensions of the total vector for all accepted jobs, along with the total penalty for rejected jobs. The penalty is determined by a submodular function. Our main work is to design a -approximation algorithm to solve this problem. Here, denotes the maximum ratio of the maximum load to the minimum load on the -dimensional vectors among all jobs.

In this paper, we establish a correspondence between fuzzy hypergroupoids and certain types of hypergroupoids that possess map-pointed properties. Specifically, we introduce a new type of fuzzy hyperoperations, known as map-pointed fuzzy hyperoperations, and show that this correspondence yields an adjunction.

Innovation ecosystem management is now a new paradigm in innovation management research. Innovation ecosystems are dynamic innovation alliances based on noncontractual governance. In this paper, a traditional cooperative innovation game model and a cooperative innovation game model with noncontractual mechanism are developed and compared. The results show that the cooperation formed only by contractual mechanisms’ model is unstable and the system is likely to collapse. The noncontractual mechanism can be used as a supplement to the contractual mechanism to make the game develop towards a stable “cooperation-cooperation” state. The research conclusion of this paper will make the core companies in the innovation ecosystem realize the important role of noncontractual mechanism and help them make decisions.