On scaled hyperbolic numbers induced by scaled hyperbolic rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916078887632896 |
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| author | Alpay, Daniel Cho, Ilwoo |
| author_facet | Alpay, Daniel Cho, Ilwoo |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00957 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On scaled hyperbolic numbers induced by scaled hyperbolic rings Alpay, Daniel Cho, Ilwoo Rings and Algebras Functional Analysis 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 |
| title | On scaled hyperbolic numbers induced by scaled hyperbolic rings |
| topic | Rings and Algebras Functional Analysis |
| url | https://arxiv.org/abs/2401.00957 |