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Hyperseries in the non-Archimedean ring of Colombeau generalized numbers
This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring [Image: see text] of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series [Formula: see text] of gene...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer Vienna
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8776721/ https://www.ncbi.nlm.nih.gov/pubmed/35125521 http://dx.doi.org/10.1007/s00605-021-01647-0 |
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author | Tiwari, Diksha Giordano, Paolo |
author_facet | Tiwari, Diksha Giordano, Paolo |
author_sort | Tiwari, Diksha |
collection | PubMed |
description | This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring [Image: see text] of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series [Formula: see text] of generalized numbers [Image: see text] is convergent if and only if [Formula: see text] in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results. |
format | Online Article Text |
id | pubmed-8776721 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer Vienna |
record_format | MEDLINE/PubMed |
spelling | pubmed-87767212022-02-02 Hyperseries in the non-Archimedean ring of Colombeau generalized numbers Tiwari, Diksha Giordano, Paolo Mon Hefte Math Article This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring [Image: see text] of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series [Formula: see text] of generalized numbers [Image: see text] is convergent if and only if [Formula: see text] in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results. Springer Vienna 2021-11-28 2022 /pmc/articles/PMC8776721/ /pubmed/35125521 http://dx.doi.org/10.1007/s00605-021-01647-0 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Tiwari, Diksha Giordano, Paolo Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title | Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title_full | Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title_fullStr | Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title_full_unstemmed | Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title_short | Hyperseries in the non-Archimedean ring of Colombeau generalized numbers |
title_sort | hyperseries in the non-archimedean ring of colombeau generalized numbers |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8776721/ https://www.ncbi.nlm.nih.gov/pubmed/35125521 http://dx.doi.org/10.1007/s00605-021-01647-0 |
work_keys_str_mv | AT tiwaridiksha hyperseriesinthenonarchimedeanringofcolombeaugeneralizednumbers AT giordanopaolo hyperseriesinthenonarchimedeanringofcolombeaugeneralizednumbers |