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In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

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  • Duality (mathematics)
  • Dualität (Mathematik)
  • Dualité (mathématiques)
  • Dualità (matematica)
  • 对偶 (数学)
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  • 在数学领域中,对偶一般来说是以一对一的方式,常常(但并不总是)通过某个对合算子,把一种概念、公理或数学结构转化为另一种概念、公理或数学结构:如果A的对偶是B,那么B的对偶是A。由于对合有时候会存在不动点,因此A的对偶有时候会是A自身。比如射影几何中的笛沙格定理,即是在这一意义下的自对偶。 对偶在数学背景当中具有很多种意义,而且,尽管它是“现代数学中极为普遍且重要的概念(a very pervasive and important concept in (modern) mathematics)”并且是“在数学几乎每一个分支中都会出现的重要的一般性主题(an important general theme that has manifestations in almost every area of mathematics)”,但仍然没有一个能把对偶的所有概念统一起来的普适定义。 在两类对象之间的对偶很多都和配对(pairing),也就是把一类对象和另一类对象映射到某一族标量上的双线性函数相对应。例如,线性代数的对偶对应着把线性空间中的向量对双线性映射到标量上,广义函数及其相关的试验函数也对应着一个配对且在该配对中可用试验函数来对广义函数进行积分,庞加莱对偶从给定流形的子流形之间的配对的角度看同样也对应着交数。
  • In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
  • In vielen Bereichen der Mathematik kommt es oft vor, dass man zu jedem Objekt der jeweils betrachteten Klasse ein weiteres Objekt konstruieren und zur Untersuchung von heranziehen kann. Dieses Objekt wird dann mit oder ähnlich bezeichnet, um die Abhängigkeit von zum Ausdruck zu bringen. Dazu gibt es ferner dieselbe, eine ähnliche oder eine umgekehrte Konstruktion, die auf angewendet werden kann, man erhält daraus ein mit bezeichnetes Objekt. Häufig stehen und in einer engen Beziehung, weshalb Informationen über enthalten muss. Man nennt dann das zu duale und zu Eigenschaften von
  • En mathématiques, le mot dualité a de nombreuses utilisations. * En algèbre : dual d'un ensemble ordonné. * En algèbre linéaire : espace dual et espace bidual, voir aussi base duale. * En analyse convexe : cône dual, paire duale, ensemble polaire. * En analyse fonctionnelle : dual topologique. * En analyse harmonique : dualité de Pontryagin, dualité de Tannaka-Krein (en) * En géométrie : dual d'un polyèdre et dual d'un polygone. * En géométrie algébrique : dualité de Serre (en) * En géométrie projective : dualité. * En optimisation : problème d'optimisation dual. * En théorie des catégories : catégorie duale. * En théorie des graphes : graphe dual. * En théorie des groupes : représentation duale. * En topologie algébrique : dualité de Spanier-Whitehead (en). * En
  • In matematica il tema della dualità è importante e pervasivo,ma non vi è una definizione universalmente accettata in grado di unificaretutte le sue accezioni. In linea generale si può dire che una dualità è una endofunzione che agisce suuna teoria matematica, da intendersi come un sistema logicamente coerente di definizioni,teoremi e strutture, in modo da trasformare tali componenti in altre definizioni, teoremi e strutture.
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