[ LA DUALITE COMME NOTION "FUGS" EN SCIENCES MATHEMATIQUES ]
Volume 20, Issue 1, January 2016, Pages 210–232
David MAPENDANO BAHAGAZE1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d'Idjwi (ISP/IDJWI), Idjwi, Sud-kivu, RD Congo
Original language: French
Copyright © 2016 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The mathematical notion of "dual" and "duality" or "proposal dual" in its federating release, unifying, generalizing and simplifying (fugs, in acronym) studied in Linear Algebra (specifically in the areas of theory vector), is a concept that is a common denominator in several branches of mathematics. It appears, in fact, under diverse forms and with apparent differences on the semantic level, both in projective geometry, set theory, topology in differential geometry and crystallography, to name but a few illustrative examples. It is legitimate to ask about the relationship between these different aspects of duality and our purpose in this article is precisely to show that the duality which we call algebraic - that is to say one that is studied in universities in the theory of vector spaces - is one to which identify themselves in one way or another, all other forms of duality studied mathematics. While it must be acknowledged here that, in some cases, the relationship between some of these forms of dualities with algebraic duality is not always easy to establish at first.
Author Keywords: Duality, dual dual, vector application, crystallography.
Volume 20, Issue 1, January 2016, Pages 210–232
David MAPENDANO BAHAGAZE1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d'Idjwi (ISP/IDJWI), Idjwi, Sud-kivu, RD Congo
Original language: French
Copyright © 2016 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The mathematical notion of "dual" and "duality" or "proposal dual" in its federating release, unifying, generalizing and simplifying (fugs, in acronym) studied in Linear Algebra (specifically in the areas of theory vector), is a concept that is a common denominator in several branches of mathematics. It appears, in fact, under diverse forms and with apparent differences on the semantic level, both in projective geometry, set theory, topology in differential geometry and crystallography, to name but a few illustrative examples. It is legitimate to ask about the relationship between these different aspects of duality and our purpose in this article is precisely to show that the duality which we call algebraic - that is to say one that is studied in universities in the theory of vector spaces - is one to which identify themselves in one way or another, all other forms of duality studied mathematics. While it must be acknowledged here that, in some cases, the relationship between some of these forms of dualities with algebraic duality is not always easy to establish at first.
Author Keywords: Duality, dual dual, vector application, crystallography.
Abstract: (french)
La notion mathématique de « dual », de « dualité » ou de « proposition duale », dans sa version fédératrice, unificatrice, généralisatrice et simplificatrice (FUGS, en sigle) étudiée en algèbre linéaire (spécifiquement dans la théorie des espaces vectoriels), est une notion qui constitue un dénominateur commun à plusieurs branches des mathématiques. Elle apparaît, en effet, sous des formes diversifiées et avec des différences apparentes sur le plan sémantique, aussi bien en géométrie projective, en théorie des ensembles, en topologie, en géométrie différentielle et en cristallographie, pour ne citer que ces quelques exemples illustratifs . Il est légitime de se poser la question sur le rapport existant entre ces différents aspects de la dualité et notre propos, dans le présent article, est de montrer justement que la dualité que nous qualifions d'algébrique – c'est-à-dire celle qui est étudiée dans nos universités dans la théorie des espaces vectoriels – est celle à laquelle s'identifient, d'une manière ou d'une autre, toutes les autres formes de dualité étudiées en mathématiques. Même s'il faut reconnaître ici que, dans certains cas, le rapport entre certaines de ces formes de dualités avec la dualité algébrique n'est pas toujours facile à établir de prime abord.
Author Keywords: Dualité, bi duale, vecteur, application, cristallographie.
How to Cite this Article
David MAPENDANO BAHAGAZE, “THE DUALITY CONCEPT AS "FUGS" IN MATHEMATICAL SCIENCES,” International Journal of Innovation and Scientific Research, vol. 20, no. 1, pp. 210–232, January 2016.
