Petal B. Mokryn | Searcharxiv

arXiv2024

A Topological View on Integration and Exterior Calculus

A construction of integration, function calculus, and exterior calculus is made, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann integral, geometric product integral, and Lebesgue integral are shown as special cases. Notions similar to chain complexes are developed to allow this form of integration to define notions of exterior derivative for differential forms, and of derivatives of functions as well. Resulting conclusions on integration, orientation, dimension, and differentiation are discussed. Applications include calculus on fractals, stochastic analysis, discrete analysis, and other novel forms of calculus.

math.DG↗

arXiv2024

A Numerical System For Nested Spaces -- Defining An Intuitive, Universal Coordinate System For Self-Similarity

Analysis on fractals is a growing field, with hints of potential for widespread applicability across all of STEM. One of the most heavily researched type of fractals are the nested fractals, fractal shapes defined by virtue of being made of smaller copies of themselves. In this paper I introduce a simple, universal numerical system for describing all self-similar nested spaces. For a nested space made of $N>1$ smaller copies of itself, the location of all points can be specified using the double shift space $\lbrace 0,1,...,N-1 \rbrace^{\mathbb{Z}}$, endowed with the appropriate nested topology. Connectivity rules are established to fully define the topology of the nested space, and immediate consequences are investigated using the numerical system defined. I introduce the numerical system and demonstrate how it unifies previous existing works under a single, simple framework, to argue for its adoption as standard in discussions of nested fractals.

math-ph↗