Apodictic discourse and the Cauchy-Bunyakovsky- Schwarz inequality
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| Názov: | Apodictic discourse and the Cauchy-Bunyakovsky- Schwarz inequality |
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| Autori: | Kichenassamy, Satyanad |
| Prispievatelia: | Kichenassamy, Satyanad |
| Zdroj: | GANITA BHARATI. 42:129-147 |
| Publication Status: | Preprint |
| Informácie o vydavateľovi: | Printspublications Private Limited, 2022. |
| Rok vydania: | 2022 |
| Predmety: | Discourse Analysis, Mathematics - History and Overview, History and Overview (math.HO), derivations, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Apodicticité, 16. Peace & justice, Cauchy-Schwarz, [MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA], invariant theory, Functional Analysis (math.FA), Mathematics - Functional Analysis, algebraic identities, Mathematics - Analysis of PDEs, Saint-Petersburg, [SHS.HISPHILSO] Humanities and Social Sciences/History, Philosophy and Sociology of Sciences, [MATH.MATH-HO] Mathematics [math]/History and Overview [math.HO], FOS: Mathematics, history of mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Analysis, Analysis of PDEs (math.AP) |
| Popis: | Bunyakovsky's integral inequality (1859) is one of the familiar tools of modern Analysis. We try and understand what Bunyakovsky did, why he did it, why others did not follow the same path, and explore some of the mathematical (re)interpretations of his inequalities. This is achieved by treating the texts as discourses that provide motivation and proofs by their very discursive structure, in addition to what meets the eye at first reading. Bunyakovsky paper is an outgrowth of the mathematical theory of mean-values in Cauchy's work (1821), but viewed from the point of view of Probability and Statistics. Liouville (1836) gave a result that implies Bunyakovsky's inequality, but did not identify it as significant because his interests lay elsewhere. Grassmann (1862) stated the inequality in abstract form but did not prove it for reasons that can be identified. Finally, by relating the result to quadratic binary forms, Schwarz (1885) opened the way to a geometric interpretation of the inequality that became important in the theory of integral equations. His argument is the source of one of the proofs most commonly taught nowadays. At about the same time, the Rogers-H{ö}lder inequality suggested generalizations of Cauchy's and Bunyakovsky's results in an entirely different direction. Later extensions and reinterpretations show that no single result, even now, subsumes all known generalizations. |
| Druh dokumentu: | Article |
| Popis súboru: | application/pdf |
| ISSN: | 0970-0307 |
| DOI: | 10.32381/gb.2020.42.1-2.5 |
| DOI: | 10.48550/arxiv.2504.19543 |
| Prístupová URL adresa: | http://arxiv.org/abs/2504.19543 https://hal.science/hal-03643571v1 https://hal.science/hal-03643571v1/document https://doi.org/10.32381/gb.2020.42.1-2.5 |
| Rights: | arXiv Non-Exclusive Distribution |
| Prístupové číslo: | edsair.doi.dedup.....bae20efbebc95f3367b7d3e2e8c98989 |
| Databáza: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | Bunyakovsky's integral inequality (1859) is one of the familiar tools of modern Analysis. We try and understand what Bunyakovsky did, why he did it, why others did not follow the same path, and explore some of the mathematical (re)interpretations of his inequalities. This is achieved by treating the texts as discourses that provide motivation and proofs by their very discursive structure, in addition to what meets the eye at first reading. Bunyakovsky paper is an outgrowth of the mathematical theory of mean-values in Cauchy's work (1821), but viewed from the point of view of Probability and Statistics. Liouville (1836) gave a result that implies Bunyakovsky's inequality, but did not identify it as significant because his interests lay elsewhere. Grassmann (1862) stated the inequality in abstract form but did not prove it for reasons that can be identified. Finally, by relating the result to quadratic binary forms, Schwarz (1885) opened the way to a geometric interpretation of the inequality that became important in the theory of integral equations. His argument is the source of one of the proofs most commonly taught nowadays. At about the same time, the Rogers-H{ö}lder inequality suggested generalizations of Cauchy's and Bunyakovsky's results in an entirely different direction. Later extensions and reinterpretations show that no single result, even now, subsumes all known generalizations. |
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| ISSN: | 09700307 |
| DOI: | 10.32381/gb.2020.42.1-2.5 |