Abelian Groups of Fractional Operators
Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Computer Sciences & Mathematics Forum |
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| Online Access: | https://www.mdpi.com/2813-0324/4/1/4 |
| _version_ | 1827750369863663616 |
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| author | Anthony Torres-Hernandez Fernando Brambila-Paz Rafael Ramirez-Melendez |
| author_facet | Anthony Torres-Hernandez Fernando Brambila-Paz Rafael Ramirez-Melendez |
| author_sort | Anthony Torres-Hernandez |
| collection | DOAJ |
| description | Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators. |
| first_indexed | 2024-03-11T06:42:37Z |
| format | Article |
| id | doaj.art-3ed250f52d8e45448e99b55b9f930a1b |
| institution | Directory Open Access Journal |
| issn | 2813-0324 |
| language | English |
| last_indexed | 2024-03-11T06:42:37Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Computer Sciences & Mathematics Forum |
| spelling | doaj.art-3ed250f52d8e45448e99b55b9f930a1b2023-11-17T10:30:22ZengMDPI AGComputer Sciences & Mathematics Forum2813-03242022-12-0141410.3390/cmsf2022004004Abelian Groups of Fractional OperatorsAnthony Torres-Hernandez0Fernando Brambila-Paz1Rafael Ramirez-Melendez2Department of Physics, Faculty of Science, Universidad Nacional Autónoma de México, Mexico City 04510, MexicoDepartment of Mathematics, Faculty of Science, Universidad Nacional Autónoma de México, Mexico City 04510, MexicoMusic and Machine Learning Lab, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, SpainTaking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators.https://www.mdpi.com/2813-0324/4/1/4fractional operatorsset theorygroup theoryfractional calculus of sets |
| spellingShingle | Anthony Torres-Hernandez Fernando Brambila-Paz Rafael Ramirez-Melendez Abelian Groups of Fractional Operators Computer Sciences & Mathematics Forum fractional operators set theory group theory fractional calculus of sets |
| title | Abelian Groups of Fractional Operators |
| title_full | Abelian Groups of Fractional Operators |
| title_fullStr | Abelian Groups of Fractional Operators |
| title_full_unstemmed | Abelian Groups of Fractional Operators |
| title_short | Abelian Groups of Fractional Operators |
| title_sort | abelian groups of fractional operators |
| topic | fractional operators set theory group theory fractional calculus of sets |
| url | https://www.mdpi.com/2813-0324/4/1/4 |
| work_keys_str_mv | AT anthonytorreshernandez abeliangroupsoffractionaloperators AT fernandobrambilapaz abeliangroupsoffractionaloperators AT rafaelramirezmelendez abeliangroupsoffractionaloperators |