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Apodictic discourse and the Cauchy-Bunyakovsky-Schwarz inequality ; Discours apodictique et inégalité de Cauchy-Bunyakovsky-Schwarz
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In: ISSN: 0970-0307 ; Ganita Bharati (Indian Mathematics): Journal of the Indian Society for History of Mathematics ; https://hal.archives-ouvertes.fr/hal-03643571 ; Ganita Bharati (Indian Mathematics): Journal of the Indian Society for History of Mathematics, Prints Publications Pvt., In press (2022)
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Abstract:
International audience ; 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-Hölder 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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Keyword:
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]; [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]; [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO]; [SHS.HISPHILSO]Humanities and Social Sciences/History; algebraic identities; Analysis; Apodicticité; Cauchy-Schwarz; derivations; Discourse Analysis; history of mathematics; invariant theory; Philosophy and Sociology of Sciences; Saint-Petersburg
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URL: https://hal.archives-ouvertes.fr/hal-03643571
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Semiotic analysis of the scientific measurement of a quantity ; Analyse sémiotique de la mesure scientifique d'une grandeur
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In: La Sémiotique et ses horizons ; https://hal.archives-ouvertes.fr/hal-03556362 ; La Sémiotique et ses horizons, 2022 (2022)
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Plurilinguisme et enseignement des mathématiques ; Plurilinguisme et enseignement des mathématiques: Mise en perspectives des Journées Plurimaths 2020
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In: https://hal.archives-ouvertes.fr/hal-03553842 ; Éditions Lambert-Lucas, 2022 (2022)
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Understanding Teachers’ Professional Development Through their Interactions with Resources: a Multilingual Project
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In: Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education ; https://hal.archives-ouvertes.fr/hal-03090727 ; Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, PME-NA, Jun 2021, Mazatlan, Mexico. pp.143-154, ⟨10.51272/pmena.42.2020-8⟩ ; https:/doi.org/10.51272/pmena.42.20 (2021)
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A workshop on the epistemology and didactics of mathematical structuralism
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In: Advances in the Anthropological Theory of the Didactic and their Consequences in Curricula and in Teacher Education ; https://hal.archives-ouvertes.fr/hal-03275633 ; Yves Chevallard; Berta Barquero Farràs; Marianna Bosch; Ignasi Florensa; Josep Gascón; Pedro Nicolás; Noemí Ruiz-Munzón. Advances in the Anthropological Theory of the Didactic and their Consequences in Curricula and in Teacher Education, Springer Nature Switzerland AG, 2021 (2021)
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Apodictic discourse in ancient and modern Mathematics ; Le discours apodictique dans les mathématiques anciennes et modernes
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In: International Workshop “Srinivasa Ramanujan: The Man Beyond Infinity” ; https://hal.archives-ouvertes.fr/hal-03088818 ; International Workshop “Srinivasa Ramanujan: The Man Beyond Infinity”, Dec 2020, Dharamshala, India (2020)
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Mathematical works of Vladimir A. Uspensky: a commentary (Enlgish and Russian texts) ; Математические работы Владимира Андреевича Успенского: комментарии (английский и русский текст)
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In: Vladimir A. Uspensky. Non-mathematical works collection (Russian version) ; https://hal-lirmm.ccsd.cnrs.fr/lirmm-03059688 ; Vladimir A. Uspensky. Non-mathematical works collection (Russian version), 2020, 978-5-94282-675-8 (2020)
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The documentational approach to didactics
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In: The Documentational Approach to Didactics Multilingual project ; https://hal.archives-ouvertes.fr/hal-02494035 ; The Documentational Approach to Didactics Multilingual project, 2020, DAD-MULTILINGUAL/, ⟨10.1007/978-3-319-77487-9_100011-1⟩ (2020)
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The documentational approach to didactics ; L'approche documentaire du didactique
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02512596 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; Документальний підхід до дидактики
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02905655 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; 文献纪录教学论
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02891008 ; DAD-Multilingual, 2020, ⟨10.1007/978-3-319-77487-9_100011-1⟩ (2020)
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The documentational approach to didactics ; L'approccio documentale alla didattica
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02896989 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; גישת לדידקטיקה התיעוד
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02925177 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; Dokumenteringstilnærmingen til didaktikk
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02908383 ; DAD-Multilingual, 2020 (2020)
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El enfoque documental de lo didáctico
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02557744 ; DAD-Multilingual, 2020 (2020)
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Θεωρία διδακτικής τεκµηρίωσης ; The documentational approach to didactics
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02941052 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; A dokumentum-fejlesztés didaktikai elmélete
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02907944 ; DAD-Multilingual, 2020 (2020)
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Der dokumentarische Ansatz zur Didaktik ; The documentational approach to didactics
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02962875 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; 教授文書活動研究法
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02870657 ; DAD-Multilingual, 2020 (2020)
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The documentational approach to didactics ; Didaktiğe Dokümantal Yaklaşım
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In: DAD-Multilingual ; https://hal.archives-ouvertes.fr/hal-02725882 ; DAD-Multilingual, 2020, ⟨10.1007/978-3-319-77487-9_100011-⟩ (2020)
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