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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.05540 |
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| _version_ | 1866910821598101504 |
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| author | Gao, Xing Guo, Li Rosenkranz, Markus Zhang, Huhu Zhang, Shilong |
| author_facet | Gao, Xing Guo, Li Rosenkranz, Markus Zhang, Huhu Zhang, Shilong |
| contents | In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are themselves an abstraction of classical calculus (incorporating its Fundamental Theorem). The results comprise (set) species as well as linear species. Localization of (set) species leads to the more general structure of modified integro-differential rings, previously employed in the algebraic treatment of Volterra integral equations. Furthermore, the ring homomorphism from species to power series via taking generating series is shown to be a (modified) integro-differential ring homomorphism. As an application, a topology and further algebraic operations are imported to virtual species from the general theory of integro-differential rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integro-differential rings on species and derived structures Gao, Xing Guo, Li Rosenkranz, Markus Zhang, Huhu Zhang, Shilong Combinatorics Classical Analysis and ODEs Rings and Algebras 18M30, 17B38, 45J05, 47G20, 12H05, 34M15 In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are themselves an abstraction of classical calculus (incorporating its Fundamental Theorem). The results comprise (set) species as well as linear species. Localization of (set) species leads to the more general structure of modified integro-differential rings, previously employed in the algebraic treatment of Volterra integral equations. Furthermore, the ring homomorphism from species to power series via taking generating series is shown to be a (modified) integro-differential ring homomorphism. As an application, a topology and further algebraic operations are imported to virtual species from the general theory of integro-differential rings. |
| title | Integro-differential rings on species and derived structures |
| topic | Combinatorics Classical Analysis and ODEs Rings and Algebras 18M30, 17B38, 45J05, 47G20, 12H05, 34M15 |
| url | https://arxiv.org/abs/2501.05540 |