Let R, S be rings with identity and M be a unitary (R, S)-bimodule. We characterize homomorphisms and derivations of the generalized matrix ring T =(or sm), and provide a Triangular representation of the differential polynomial ring T[b; d].

Necessary and sufficient conditions are found on modules over a formal Triangular matrix ring to have Krull and/or Noetherian dimension. Also, critical and atomic modules are characterized.

Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.

The Travelling Salesman Problem is one of the fundamental operational research problems where the objective is to generate the cheapest route for a salesman starting from a given city, visiting all the other cities only once and finally returning to the starting city. In this paper, we study the Travelling Salesman Problem in uncertain environment. Particularly, the single valued Triangular neutrosophic environment is considered viewing that it is more realistic and general in real-world industrial problems. Each element in the distance matrix of the Travelling Salesman Problem is presented as a single valued Triangular neutrosophic number. To solve this problem, we enhance our novel column-row heuristic Dhouib-matrix-TSP1 by the means of the center of gravity ranking function and the standard deviation metric. In fact, the center of gravity ranking function is applied for defuzzification in order to convert the single valued Triangular neutrosophic number to crisp number.A stepwise application of several numerical Travelling Salesman Problems on the single valued Triangular neutrosophic environment shows that the optimal or a near optimal solution can be easily reached thanks to the Dhouib-matrix-TSP1 heuristic enriched with the center of gravity ranking function and the standard deviation metric.