Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded
Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/11/1936 |
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| author | Brian Villegas-Villalpando Jorge E. Macías-Díaz Qin Sheng |
| author_facet | Brian Villegas-Villalpando Jorge E. Macías-Díaz Qin Sheng |
| author_sort | Brian Villegas-Villalpando |
| collection | DOAJ |
| description | Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number. |
| first_indexed | 2024-03-10T01:05:45Z |
| format | Article |
| id | doaj.art-bbbe091b10db438cbc344359656aaad0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:05:45Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bbbe091b10db438cbc344359656aaad02023-11-23T14:27:12ZengMDPI AGMathematics2227-73902022-06-011011193610.3390/math10111936Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are BoundedBrian Villegas-Villalpando0Jorge E. Macías-Díaz1Qin Sheng2Centro de Ciencias Básicas, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, MexicoDepartment of Mathematics and Didactics of Mathematics, Tallinn University, 10120 Tallinn, EstoniaDepartment of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX 76706, USABoundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number.https://www.mdpi.com/2227-7390/10/11/1936bounded solutionsintegrable functionsreal function spacescomplete characterization |
| spellingShingle | Brian Villegas-Villalpando Jorge E. Macías-Díaz Qin Sheng Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded Mathematics bounded solutions integrable functions real function spaces complete characterization |
| title | Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded |
| title_full | Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded |
| title_fullStr | Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded |
| title_full_unstemmed | Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded |
| title_short | Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded |
| title_sort | solution spaces associated to continuous or numerical models for which integrable functions are bounded |
| topic | bounded solutions integrable functions real function spaces complete characterization |
| url | https://www.mdpi.com/2227-7390/10/11/1936 |
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