Shiv Datt Kumar

On the Involutivity of the Characteristic Variety

In this paper, we give a new simple algebraic proof of the involutivity of the characteristic variety after Gabber, which is one of the most important results of D-module theory.

Finite Majid Algebras Over the Klein Group

We provide a classification of finite-dimensional connected coradically graded pointed Majid algebras, generated in degrees 0 and 1, over the Klein group on the field of complex numbers.

Fuzzy efficient and pareto — Optimal solution for multiobjective linear plus linear fractional programming problem

Many practical optimization problems usually have several conflicting objectives. In those multiobjective optimization, no solution optimizing all the objective functions simultaneously exists in general. Instead, pareto — optimal solutions, which are efficient in terms of all objective functions, are introduced. In general we have many optimal solutions. Therefore we need to decide a final solutions among pareto — optimal solutions taking in to account the balance among objective functions. In this paper we find fuzzy efficient and pareto — optimal solution to the multiobjective linear plus linear fractional programming problem and show that, in the case that, when any goal is fully achieved, then fuzzy efficient solution may or may not be pareto — optimal solution and therefore we propose a procedure to obtain fuzzy efficient solution which is pareto — optimal also and review some results. In the proposed approach each objective function is transformed into linear functions by using Taylors theorem. Then the MOLLFP is changed into equivalent multiobjective linear programming problem (MOLP) and then find fuzzy efficient and pareto — optimal solution in finite number of steps. Efficiency of proposed method is verified by numerical examples. To explore the potential use of the proposed method, three numerical examples are solved. AMS 2000 Subject Classification: 90C29, 90C32

On the classification of finite quasi-quantum groups

We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

Mathematical counterpart of the heisenberg uncertainty principle

In this paper we provide an exposition on integrability theorem and relation between D -modules and symplectic geometry. We prove that the characteristic ideal of a module over the Weyl algebra with Bernstein and order filtrations is closed under the Poisson bracket. This result can be viewed as a mathematical counterpart of the heisenberg uncertainty principle. Several examples have been explained in detail to understand this important deep result and related concepts.

Fuzzy efficient and Pareto-optimal solution for multi-objective linear fractional programming problems

Many practical optimisation problems usually have several conflicting objectives. In these multi-objective optimisation problems, solution optimising all the objective functions simultaneously does not exist, in general. Instead, Pareto-optimal solutions, which are efficient in terms of all objective functions, are introduced. Nevertheless, many optimal solutions exist. A final solution among Pareto-optimal solutions is to be selected based on the balance among objective functions. In this paper, we find fuzzy efficient and Pareto-optimal solution to the multi-objective linear fractional programming problem (MOLFP). It has shown that when any fuzzy goal is fully achieved, the fuzzy efficient solution may or may not be Pareto-optimal. Therefore, a procedure is proposed to obtain fuzzy efficient solution which is also Pareto-optimal. The efficiency of proposed method is verified by numerical examples and a practical application in production planning.