Scale free SL(2,R) analysis and the Picard's existence and uniqueness theorem

TL;DR

This paper reveals a nonlinear SL(2,R) structure in differential analysis through higher derivative solutions, extending Picard's theorem and showing the real numbers form a measure Cantor set.

Contribution

It introduces a novel SL(2,R) framework in analysis, extending Picard's theorem and demonstrating the Cantor set structure of real numbers.

Findings

Higher derivative solutions exhibit nonlinear SL(2,R) symmetry.

Real numbers form a positive Lebesgue measure Cantor set.

Extended Picard's theorem in the context of this new structure.

Abstract

The existence of higher derivative discontinuous solutions to a first order ordinary differential equation is shown to reveal a nonlinear SL(2,R) structure of analysis in the sense that a real variable $t$ can now accomplish changes not only by linear translations $t \to t + h$ but also by inversions $t \to 1/t$. We show that the real number set has the structure of a positive Lebesgue measure Cantor set. We also present an extension of the Picard's theorem in this new light.