The Golden mean, scale free extension of Real number system, fuzzy sets and $1/f$ spectrum in Physics and Biology

Abstract

We show that the generic $1/f$ spectrum problem acquires a natural explanation in a class of scale free solutions to the ordinary differential equations. We prove the existence and uniqueness of this class of solutions and show how this leads to a nonstandard, fuzzy extension of the ordinary framework of calculus, and hence, that of the classical dynamics and quantum mechanics. The exceptional role of the golden mean irrational number is also explained.