In a recent paper, I began with four classical results due to Jacobi, Dirichlet and Lorenz which give the number of representations of a number by various forms involving squares in terms of divisor functions. I deduced another sixteen similar representation theorems involving triangular numbers and/or squares, including celebrated results of Legendre and Ramanujan (and omitted a seventeenth, g...

In this paper, we consider a two persons zero sum game with imprecise values in payoff matrix. All the imprecise values are assumed to be octagonal fuzzy numbers. An approach for solving problems by using ranking of the fuzzy numbers has been considered to solve the fuzzy game problem. By using ranking to the payoffs we convert the fuzzy valued game problem to crisp valued game problem, which c...

The vague sum (product) of a finite number of real numbers and the vague integral multiples (powers) of real numbers based on a perfect vague addition (multiplication) operation are introduced, and some of their fundamental properties are pointed out. Furthermore, some examples are designed to show how they can be practically implemented. Keyword: Fuzzy arithmetic, Vague arithmetic, Vague arith...