Normed modules, integral sequences, and integrals with variable upper limits
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909597351018496 |
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| author | Liu, Miantao Liu, Yu-Zhe Liu, Shengda |
| author_facet | Liu, Miantao Liu, Yu-Zhe Liu, Shengda |
| contents | This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathitΛ$ and the category $\mathscr{A}^p_{\mathitΛ}$ associated with $\mathitΛ$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normed modules, integral sequences, and integrals with variable upper limits Liu, Miantao Liu, Yu-Zhe Liu, Shengda Category Theory Classical Analysis and ODEs 16D10, 16G10, 46H25 This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathitΛ$ and the category $\mathscr{A}^p_{\mathitΛ}$ associated with $\mathitΛ$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras. |
| title | Normed modules, integral sequences, and integrals with variable upper limits |
| topic | Category Theory Classical Analysis and ODEs 16D10, 16G10, 46H25 |
| url | https://arxiv.org/abs/2411.19904 |