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arXiv · 2411.19904

Normed modules, integral sequences, and integrals with variable upper limits

Miantao Liu·Yu-Zhe Liu·Shengda Liu
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Abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathit{\Lambda}$ and the category $\mathscr{A}^p_{\mathit{\Lambda}}$ associated with $\mathit{\Lambda}$. 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.

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Miantao Liu, Yu-Zhe Liu, Shengda Liu. 2024-11-29. Normed modules, integral sequences, and integrals with variable upper limits. https://doi.org/10.1007/s41980-025-00990-4

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