Regular Functions on the Scaled Hypercomplex Numbers

TL;DR

This paper investigates the properties of differentiable functions on scaled hypercomplex number systems by analyzing differential operators and their kernels, providing insights into the structure of regular functions across different scales.

Contribution

It introduces a framework for studying regularity of functions on scaled hypercomplex numbers using differential operators and their kernels, extending classical analysis to a scaled hypercomplex setting.

Findings

Characterization of kernels of differential operators on scaled hypercomplex numbers

Extension of regularity concepts to scaled hypercomplex systems

Insights into the structure of differentiable functions on these systems

Abstract

In this paper, we study the regularity of $\mathbb{R}$-differentiable functions on open connected subsets of the scaled hypercomplex numbers $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$ by studying the kernels of suitable differential operators $\left\{ \nabla_{t}\right\} _{t\in\mathbb{R}}$, up to scales in the real field $\mathbb{R}$.