A Study on Generalized Real Analysis and Its Applications

Exploring Simplified Real Analysis and its Practical Applications

Abstract

In this paper there are emphasized some of the benefits of a simplified real analysis (called pseudo-analysis) based on some real operations which are taken instead of the normal addition and product of reals. Namely, there are covered by one philosophy and so with centralized approaches several issues (usually nonlinear) from several fields (system theory, optimization, control theory, differential equations, difference equations, etc.). There are provided several essential actual aggregation functions as triangular norms and triangular conorms and a real semiring with pseudo-operations. First it is presented how these operations arise as simple operations in the philosophy of fuzzy logics and fuzzy sets and there is seen a generalization of the utility theory represented by hybrid probabilistic – possibilistic calculation. The real semirings serve as a base for pseudo-additive measures, pseudo-integrals, pseudo-convolutions which shape the pseudo-analysis. There are provided some of the implementations by broad variance theorem, nonlinear Hamilton – Jacobi equation, cumulative prospect theory.