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On scaled hyperbolic numbers induced by scaled hyperbolic rings

Daniel Alpay, Ilwoo Cho

Abstract

In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of $\mathbb{R}$, the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers. If a scale $t$ is not positive in $\mathbb{R}$, then our $t$-scaled hyperbolic numbers have similar numerical structures with those of the complex numbers, however, if a scale is positive in $\mathbb{R}$, then their numerical properties are similar to those of the classical hyperbolic numbers. We here understand scaled-hyperbolic numbers as elements of the scaled-hypercomplex rings $\{\mathbb{H}_t\}_{t\in \mathbb{R}}$, introduced in [1]. This scaled-hyperbolic analysis is done by algebra, analysis, operator theory, operator-algebra theory and free probability on scaled-hypercomplex numbers

On scaled hyperbolic numbers induced by scaled hyperbolic rings

Abstract

In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of , the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers. If a scale is not positive in , then our -scaled hyperbolic numbers have similar numerical structures with those of the complex numbers, however, if a scale is positive in , then their numerical properties are similar to those of the classical hyperbolic numbers. We here understand scaled-hyperbolic numbers as elements of the scaled-hypercomplex rings , introduced in [1]. This scaled-hyperbolic analysis is done by algebra, analysis, operator theory, operator-algebra theory and free probability on scaled-hypercomplex numbers
Paper Structure (16 sections, 55 theorems, 294 equations)

This paper contains 16 sections, 55 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview
  4. The Hypercomplex Numbers Scaled by $\mathbb{R}$
  5. Scaled Hypercomplex Rings
  6. Algebra on $\mathbb{H}_{t}$
  7. Spectral Properties of Scaled Hypercomplex Numbers
  8. The $t$-Scaled-Spectral Relation on $\mathbb{H}_{t}$
  9. Scaled-Hypercomplex Conjugation
  10. Scaled Hypercomplex Rings $\mathbb{H}_{t}$ as Topological Vector Spaces
  11. Certain Operators on $\mathbf{X}_{t}=\left(\mathbb{H}_{t},\left\langle ,\right\rangle _{t}\right)$
  12. Scaled Hyperbolic Numbers
  13. Motivation: The Hyperbolic Numbers
  14. Scaled Hyperbolic Numbers
  15. Unit Elements of $\mathbb{D}_{t}$
  16. ...and 1 more sections

Key Result

Proposition 2.1

The algebraic structure $\left(\mathbb{C}^{2},+,\cdot_{t}\right)$ forms a unital ring with its unity, or the ($\cdot_{t}$)-identity, $\left(1,0\right)$, where ($+$) is the usual vector addition on $\mathbb{C}^{2}$, and ($\cdot_{t}$) is the vector multiplication (2.1.1).

Theorems & Definitions (127)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 117 more