"Dualit\u00E9 (math\u00E9matiques)"@fr . "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 \u00E4hnlich bezeichnet, um die Abh\u00E4ngigkeit von zum Ausdruck zu bringen. Wendet man dieselbe (oder eine \u00E4hnliche) Konstruktion auf an, erh\u00E4lt man daraus ein mit bezeichnetes Objekt. H\u00E4ufig stehen und in einer engen Beziehung, sind z. B. gleich oder isomorph, weshalb Informationen \u00FCber enthalten muss. Man nennt dann das zu duale und das biduale Objekt. In der zugeh\u00F6rigen mathematischen Dualit\u00E4tstheorie untersucht man dann, wie Eigenschaften von zu Eigenschaften von \u00FCbersetzt werden k\u00F6nnen und umgekehrt."@de . . . . . . . . . . "\u0414\u0443\u0430\u0301\u043B\u044C\u043D\u0456\u0441\u0442\u044C (\u0434\u0432\u043E\u0457\u0441\u0442\u0456\u0441\u0442\u044C) \u2014 \u043F\u0440\u0438\u043D\u0446\u0438\u043F, \u0449\u043E \u0441\u0444\u043E\u0440\u043C\u0443\u043B\u044C\u043E\u0432\u0430\u043D\u0438\u0439 \u0443 \u0434\u0435\u044F\u043A\u0438\u0445 \u0440\u043E\u0437\u0434\u0456\u043B\u0430\u0445 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438 \u0456 \u043F\u043E\u043B\u044F\u0433\u0430\u0454 \u0432 \u0442\u043E\u043C\u0443, \u0449\u043E \u043A\u043E\u0436\u043D\u043E\u043C\u0443 \u043F\u0440\u0430\u0432\u0438\u043B\u044C\u043D\u043E\u043C\u0443 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044E \u0446\u044C\u043E\u0433\u043E \u0440\u043E\u0437\u0434\u0456\u043B\u0443 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u0456\u043D\u0448\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F, \u044F\u043A\u0435 \u043C\u043E\u0436\u043D\u0430 \u043E\u0442\u0440\u0438\u043C\u0430\u0442\u0438 \u0437 \u043F\u0435\u0440\u0448\u043E\u0433\u043E \u0437\u0430\u043C\u0456\u043D\u043E\u044E \u043F\u043E\u043D\u044F\u0442\u044C, \u044F\u043A\u0456 \u0432\u0445\u043E\u0434\u044F\u0442\u044C \u0434\u043E \u043D\u044C\u043E\u0433\u043E, \u0456\u043D\u0448\u0438\u043C\u0438, \u0442\u0430\u043A \u0437\u0432\u0430\u043D\u0438\u043C\u0438 \u0434\u0443\u0430\u043B\u044C\u043D\u0438\u043C\u0438 \u0434\u043E \u043D\u0438\u0445 \u043F\u043E\u043D\u044F\u0442\u0442\u044F\u043C\u0438."@uk . . . . . . . . . "In mathematics, a duality 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 mathematical contexts, duality has numerous meanings. It has been described as \"a very pervasive and important concept in (modern) mathematics\" and \"an important general theme that has manifestations in almost every area of mathematics\". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincar\u00E9 duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V \u2192 W its dual f\u2217: W\u2217 \u2192 V\u2217."@en . . . . . . . . . . . . . . . . . . . . . . "En math\u00E9matiques, le mot dualit\u00E9 a de nombreuses utilisations. Une dualit\u00E9 est d\u00E9finie \u00E0 l'int\u00E9rieur d'une famille F d'objets math\u00E9matiques, c'est-\u00E0-dire qu'\u00E0 tout objet X de F on associe un autre objet Y de F. On dit que Y est le dual de X et que X est le primal[r\u00E9f. n\u00E9cessaire] de Y. Si X\u2009=\u2009Y (par = on peut sous-entendre des relations d'isomorphies complexes), on dit que X est autodual. Dans de nombreux cas de dualit\u00E9, le dual du dual est le primal. Ainsi, par exemple, le concept de compl\u00E9mentaire d'un ensemble pourrait \u00EAtre vu comme le premier des concepts de dualit\u00E9."@fr . . . . . . . . . . . . . . . . . . . . . . . . . . "\uC30D\uB300\uC131(\u96D9\u5C0D\u6027; duality)\uC740 \uC218\uD559\uACFC \uBB3C\uB9AC\uD559\uC5D0\uC11C \uC790\uC8FC \uB4F1\uC7A5\uD558\uB294 \uD45C\uD604\uC774\uB2E4. \uBCF4\uD1B5 \uC5B4\uB5A4 \uC218\uD559\uC801 \uAD6C\uC870\uC758 \uC30D\uB300(\u96D9\u5C0D; dual)\uB780 \uADF8 \uAD6C\uC870\uB97C \u2018\uB4A4\uC9D1\uC5B4\uC11C\u2019 \uAD6C\uC131\uD55C \uAC83\uC744 \uB9D0\uD558\uB294\uB370, \uC5C4\uBC00\uD55C \uC815\uC758\uB294 \uC138\uBD80 \uBD84\uC57C\uC640 \uB300\uC0C1\uC5D0 \uB530\uB77C \uAC01\uAC01 \uB2E4\uB974\uB2E4. \uC30D\uB300\uC758 \uC30D\uB300\uB294 \uC790\uAE30 \uC790\uC2E0\uC774\uBBC0\uB85C \uC5B4\uB5A4 \uB300\uC0C1\uACFC \uADF8 \uC30D\uB300\uB294 \uC11C\uB85C \uC77C\uC885\uC758 \uD55C \u2018\uCF24\uB808\u2019\uB97C \uC774\uB8EC\uB2E4\uACE0 \uD560 \uC218 \uC788\uC73C\uBA70, \uC774\uB97C \uC30D\uB300\uAD00\uACC4(\u96D9\u5C0D\u95DC\u4FC2)\uB77C\uACE0 \uD55C\uB2E4."@ko . . . "A. I."@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "En math\u00E9matiques, le mot dualit\u00E9 a de nombreuses utilisations. Une dualit\u00E9 est d\u00E9finie \u00E0 l'int\u00E9rieur d'une famille F d'objets math\u00E9matiques, c'est-\u00E0-dire qu'\u00E0 tout objet X de F on associe un autre objet Y de F. On dit que Y est le dual de X et que X est le primal[r\u00E9f. n\u00E9cessaire] de Y. Si X\u2009=\u2009Y (par = on peut sous-entendre des relations d'isomorphies complexes), on dit que X est autodual. Dans de nombreux cas de dualit\u00E9, le dual du dual est le primal. Ainsi, par exemple, le concept de compl\u00E9mentaire d'un ensemble pourrait \u00EAtre vu comme le premier des concepts de dualit\u00E9."@fr . . . . . . "D/d034120"@en . . . . . . . . . . . . . . . . . . . . . . "\u5728\u6570\u5B66\u9886\u57DF\u4E2D\uFF0C\u5BF9\u5076\u4E00\u822C\u6765\u8BF4\u662F\u4EE5\u4E00\u5BF9\u4E00\u7684\u65B9\u5F0F\uFF0C\u5E38\u5E38\uFF08\u4F46\u5E76\u4E0D\u603B\u662F\uFF09\u901A\u8FC7\u67D0\u4E2A\u5BF9\u5408\u7B97\u5B50\uFF0C\u628A\u4E00\u79CD\u6982\u5FF5\u3001\u516C\u7406\u6216\u6570\u5B66\u7ED3\u6784\u8F6C\u5316\u4E3A\u53E6\u4E00\u79CD\u6982\u5FF5\u3001\u516C\u7406\u6216\u6570\u5B66\u7ED3\u6784\uFF1A\u5982\u679CA\u7684\u5BF9\u5076\u662FB\uFF0C\u90A3\u4E48B\u7684\u5BF9\u5076\u662FA\u3002\u7531\u4E8E\u5BF9\u5408\u6709\u65F6\u5019\u4F1A\u5B58\u5728\u4E0D\u52A8\u70B9\uFF0C\u56E0\u6B64A\u7684\u5BF9\u5076\u6709\u65F6\u5019\u4F1A\u662FA\u81EA\u8EAB\u3002\u6BD4\u5982\u5C04\u5F71\u51E0\u4F55\u4E2D\u7684\u7B1B\u6C99\u683C\u5B9A\u7406\uFF0C\u5373\u662F\u5728\u8FD9\u4E00\u610F\u4E49\u4E0B\u7684\u81EA\u5BF9\u5076\u3002 \u5BF9\u5076\u5728\u6570\u5B66\u80CC\u666F\u5F53\u4E2D\u5177\u6709\u5F88\u591A\u79CD\u610F\u4E49\uFF0C\u800C\u4E14\uFF0C\u5C3D\u7BA1\u5B83\u662F\u201C\u73B0\u4EE3\u6570\u5B66\u4E2D\u6781\u4E3A\u666E\u904D\u4E14\u91CD\u8981\u7684\u6982\u5FF5\uFF08a very pervasive and important concept in (modern) mathematics\uFF09\u201D\u5E76\u4E14\u662F\u201C\u5728\u6570\u5B66\u51E0\u4E4E\u6BCF\u4E00\u4E2A\u5206\u652F\u4E2D\u90FD\u4F1A\u51FA\u73B0\u7684\u91CD\u8981\u7684\u4E00\u822C\u6027\u4E3B\u9898\uFF08an important general theme that has manifestations in almost every area of mathematics\uFF09\u201D\uFF0C\u4F46\u4ECD\u7136\u6CA1\u6709\u4E00\u4E2A\u80FD\u628A\u5BF9\u5076\u7684\u6240\u6709\u6982\u5FF5\u7EDF\u4E00\u8D77\u6765\u7684\u666E\u9002\u5B9A\u4E49\u3002 \u5728\u4E24\u7C7B\u5BF9\u8C61\u4E4B\u95F4\u7684\u5BF9\u5076\u5F88\u591A\u90FD\u548C\uFF08pairing\uFF09\uFF0C\u4E5F\u5C31\u662F\u628A\u4E00\u7C7B\u5BF9\u8C61\u548C\u53E6\u4E00\u7C7B\u5BF9\u8C61\u6620\u5C04\u5230\u67D0\u4E00\u65CF\u6807\u91CF\u4E0A\u7684\u53CC\u7EBF\u6027\u51FD\u6570\u76F8\u5BF9\u5E94\u3002\u4F8B\u5982\uFF0C\u7EBF\u6027\u4EE3\u6570\u7684\u5BF9\u5076\u5BF9\u5E94\u7740\u628A\u7EBF\u6027\u7A7A\u95F4\u4E2D\u7684\u5411\u91CF\u5BF9\u53CC\u7EBF\u6027\u6620\u5C04\u5230\u6807\u91CF\u4E0A\uFF0C\u5E7F\u4E49\u51FD\u6570\u53CA\u5176\u76F8\u5173\u7684\u4E5F\u5BF9\u5E94\u7740\u4E00\u4E2A\u914D\u5BF9\u4E14\u5728\u8BE5\u914D\u5BF9\u4E2D\u53EF\u7528\u8BD5\u9A8C\u51FD\u6570\u6765\u5BF9\u5E7F\u4E49\u51FD\u6570\u8FDB\u884C\u79EF\u5206\uFF0C\u5E9E\u52A0\u83B1\u5BF9\u5076\u4ECE\u7ED9\u5B9A\u6D41\u5F62\u7684\u5B50\u6D41\u5F62\u4E4B\u95F4\u7684\u914D\u5BF9\u7684\u89D2\u5EA6\u770B\u540C\u6837\u4E5F\u5BF9\u5E94\u7740\u3002"@zh . . . . . . . . "\u5BF9\u5076 (\u6570\u5B66)"@zh . "\u0414\u0443\u0430\u0301\u043B\u044C\u043D\u0456\u0441\u0442\u044C (\u0434\u0432\u043E\u0457\u0441\u0442\u0456\u0441\u0442\u044C) \u2014 \u043F\u0440\u0438\u043D\u0446\u0438\u043F, \u0449\u043E \u0441\u0444\u043E\u0440\u043C\u0443\u043B\u044C\u043E\u0432\u0430\u043D\u0438\u0439 \u0443 \u0434\u0435\u044F\u043A\u0438\u0445 \u0440\u043E\u0437\u0434\u0456\u043B\u0430\u0445 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438 \u0456 \u043F\u043E\u043B\u044F\u0433\u0430\u0454 \u0432 \u0442\u043E\u043C\u0443, \u0449\u043E \u043A\u043E\u0436\u043D\u043E\u043C\u0443 \u043F\u0440\u0430\u0432\u0438\u043B\u044C\u043D\u043E\u043C\u0443 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044E \u0446\u044C\u043E\u0433\u043E \u0440\u043E\u0437\u0434\u0456\u043B\u0443 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u0454 \u0456\u043D\u0448\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F, \u044F\u043A\u0435 \u043C\u043E\u0436\u043D\u0430 \u043E\u0442\u0440\u0438\u043C\u0430\u0442\u0438 \u0437 \u043F\u0435\u0440\u0448\u043E\u0433\u043E \u0437\u0430\u043C\u0456\u043D\u043E\u044E \u043F\u043E\u043D\u044F\u0442\u044C, \u044F\u043A\u0456 \u0432\u0445\u043E\u0434\u044F\u0442\u044C \u0434\u043E \u043D\u044C\u043E\u0433\u043E, \u0456\u043D\u0448\u0438\u043C\u0438, \u0442\u0430\u043A \u0437\u0432\u0430\u043D\u0438\u043C\u0438 \u0434\u0443\u0430\u043B\u044C\u043D\u0438\u043C\u0438 \u0434\u043E \u043D\u0438\u0445 \u043F\u043E\u043D\u044F\u0442\u0442\u044F\u043C\u0438."@uk . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u062A\u062D\u0648\u0651\u0644 \u0627\u0644\u0645\u062B\u0646\u0648\u064A\u0629 \u0627\u0644\u0645\u0641\u0627\u0647\u064A\u0645 \u0623\u0648 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0627\u062A \u0623\u0648 \u0627\u0644\u0647\u064A\u0627\u0643\u0644 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629 \u0625\u0644\u0649 \u0645\u0641\u0627\u0647\u064A\u0645\u064E \u0648\u0645\u0628\u0631\u0647\u0646\u0627\u062A \u0648\u0647\u064A\u0627\u0643\u0644 \u0623\u062E\u0631\u0649\u060C \u0639\u0646 \u0637\u0631\u064A\u0642 \u062F\u0627\u0644\u0629 \u0645\u062A\u0628\u0627\u064A\u0646\u0629\u060C \u0648\u063A\u0627\u0644\u0628\u064B\u0627 \u0639\u0646 \u0637\u0631\u064A\u0642 \u062F\u0627\u0644\u0629 \u0627\u0631\u062A\u062F\u0627\u062F\u064A\u0629: \u0625\u0630\u0627 \u0643\u0627\u0646\u062A A \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 B \u060C \u0641\u0625\u0646\u0651 B \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 A. \u0642\u062F \u062A\u062D\u062A\u0648\u064A \u0645\u062B\u0644 \u0647\u0630\u0647 \u0627\u0644\u0627\u0631\u062A\u062F\u0627\u062F\u0627\u062A \u0639\u0644\u0649 \u0646\u0642\u0627\u0637 \u062B\u0627\u0628\u062A\u0629\u060C \u0628\u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0645\u062B\u0646\u0648\u064A\u0629 A \u0647\u064A \u0646\u0641\u0633\u0647\u0627 A. \u0645\u062B\u0644\u0627\u064B \u0645\u0628\u0631\u0647\u0646\u0629 \u062F\u064A\u0632\u0627\u0631\u063A \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 \u0630\u0627\u062A\u064A\u0627\u064B \u0641\u064A \u0638\u0644 \u0627\u0644\u0627\u0632\u062F\u0648\u0627\u062C\u064A\u0629 \u0627\u0644\u0642\u064A\u0627\u0633\u064A\u0629 \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0633\u0642\u0627\u0637\u064A\u0629. \u0641\u064A \u0627\u0644\u0633\u064A\u0627\u0642\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0644\u0644\u0645\u062B\u0646\u0648\u064A\u0629 \u0645\u0639\u0627\u0646\u064D \u0639\u062F\u064A\u062F\u0629. \u0648\u0642\u062F \u0648\u0635\u0641 \u0628\u0623\u0646\u0647 \u00AB\u0645\u0641\u0647\u0648\u0645 \u0648\u0627\u0633\u0639 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631 \u0648\u0645\u0647\u0645 \u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A (\u0627\u0644\u062D\u062F\u064A\u062B\u0629)\u00BB \u0648 \u00AB\u0645\u0648\u0636\u0648\u0639 \u0639\u0627\u0645 \u0645\u0647\u0645 \u0644\u0647 \u0645\u0638\u0627\u0647\u0631 \u0641\u064A \u0643\u0644 \u0645\u062C\u0627\u0644 \u0645\u0646 \u0645\u062C\u0627\u0644\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u062A\u0642\u0631\u064A\u0628\u064B\u0627\u00BB."@ar . . . . . . . "Dualit\u00E4t (Mathematik)"@de . . . . . . . . . . . . . . . "\u0645\u062B\u0646\u0648\u064A\u0629 (\u0631\u064A\u0627\u0636\u064A\u0627\u062A)"@ar . . . "En matem\u00E1ticas, una dualidad, en t\u00E9rminos generales, traduce conceptos, teoremas o estructuras matem\u00E1ticas en otros conceptos, teoremas o estructuras, mediante una correspondencia uno a uno, a menudo (pero no siempre) por medio de una operaci\u00F3n de involuci\u00F3n: si el dual de A es B, entonces el dual de B es A. Tales involuciones a veces tienen puntos fijos, de modo que el dual de A es A en s\u00ED mismo. Por ejemplo, el teorema de Desargues expresa una condici\u00F3n auto dual en este sentido bajo el concepto de dualidad en geometr\u00EDa proyectiva. En contextos matem\u00E1ticos, el t\u00E9rmino dualidad tiene numerosos significados,\u200B aunque es \u00ABun concepto muy dominante e importante en matem\u00E1ticas (modernas)\u00BB\u200B y \u00ABun tema general de gran inter\u00E9s que tiene manifestaciones en casi todas las \u00E1reas de las matem\u00E1ticas\u00BB.\u200B Muchas dualidades matem\u00E1ticas entre objetos de dos tipos corresponden a emparejamientos, que mediante operadores bilineales relacionan un objeto de un tipo y otro objeto de un segundo tipo a una familia de escalares. Por ejemplo, la \u00ABdualidad en \u00E1lgebra lineal\u00BB se corresponde de esta manera con aplicaciones bilineales de pares de espacios de vectores a escalares, la \u00ABdualidad entre distribuciones y las funciones de prueba asociadas\u00BB corresponde al emparejamiento en el que se integra una distribuci\u00F3n con una funci\u00F3n de prueba, y la dualidad de Poincar\u00E9 corresponde de manera similar al , visto como un emparejamiento entre subvariedades de una colecci\u00F3n de objetos matem\u00E1ticos determinada.\u200B Desde el punto de vista de la teor\u00EDa de categor\u00EDas, la dualidad tambi\u00E9n se puede ver como un funtor, al menos en el \u00E1mbito de los espacios vectoriales. Este funtor asigna a cada espacio su espacio dual, y la construcci\u00F3n de asigna a cada flecha f: V \u2192 W su dual f\u2217: W\u2217 \u2192 V\u2217 En teor\u00EDa de conjuntos y en l\u00F3gica matem\u00E1tica el concepto de dualidad tambi\u00E9n desempe\u00F1a un papel esencial."@es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "\uC30D\uB300\uC131(\u96D9\u5C0D\u6027; duality)\uC740 \uC218\uD559\uACFC \uBB3C\uB9AC\uD559\uC5D0\uC11C \uC790\uC8FC \uB4F1\uC7A5\uD558\uB294 \uD45C\uD604\uC774\uB2E4. \uBCF4\uD1B5 \uC5B4\uB5A4 \uC218\uD559\uC801 \uAD6C\uC870\uC758 \uC30D\uB300(\u96D9\u5C0D; dual)\uB780 \uADF8 \uAD6C\uC870\uB97C \u2018\uB4A4\uC9D1\uC5B4\uC11C\u2019 \uAD6C\uC131\uD55C \uAC83\uC744 \uB9D0\uD558\uB294\uB370, \uC5C4\uBC00\uD55C \uC815\uC758\uB294 \uC138\uBD80 \uBD84\uC57C\uC640 \uB300\uC0C1\uC5D0 \uB530\uB77C \uAC01\uAC01 \uB2E4\uB974\uB2E4. \uC30D\uB300\uC758 \uC30D\uB300\uB294 \uC790\uAE30 \uC790\uC2E0\uC774\uBBC0\uB85C \uC5B4\uB5A4 \uB300\uC0C1\uACFC \uADF8 \uC30D\uB300\uB294 \uC11C\uB85C \uC77C\uC885\uC758 \uD55C \u2018\uCF24\uB808\u2019\uB97C \uC774\uB8EC\uB2E4\uACE0 \uD560 \uC218 \uC788\uC73C\uBA70, \uC774\uB97C \uC30D\uB300\uAD00\uACC4(\u96D9\u5C0D\u95DC\u4FC2)\uB77C\uACE0 \uD55C\uB2E4."@ko . . . . . . . . . . "September 2016"@en . . . . . . . . . . . . . . "\uC30D\uB300\uC131"@ko . . . . . "Dualidad (matem\u00E1tica)"@es . . . "In matematica il tema della dualit\u00E0 \u00E8 importante e pervasivo,ma non vi \u00E8 una definizione universalmente accettata in grado di unificaretutte le sue accezioni. In linea generale si pu\u00F2 dire che una dualit\u00E0 \u00E8 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. In gran parte dei casi una dualit\u00E0 consiste in una involuzione, ma non sempre. Si possono quindi distinguere le dualit\u00E0 involutorie dalle non involutorie. Nel seguito di questo articolo, dato che esaminiamo soprattutto le involutorie, le chiameremo semplicemente dualit\u00E0. Nei casi pi\u00F9 semplicemente definiti una dualit\u00E0 \u00E8 una involuzione entro un insieme di formule (ad esempio entro l'insieme delle uguaglianze per i sottoinsiemi di un insieme ambiente) o entro un insieme di strutture (ad esempio l'insieme dei poliedri convessi). Il trasformato B di una nozione A da parte di una dualit\u00E0 involutoria d, B:=d(A), si dice duale di A; per il carattere involutorio della endofunzione d(d(A)) = A. In taluni contesti una tale nozione A viene detta primale della B. Una nozione che coincide con la propria duale viene detta autoduale: ad esempio sono autoduali l'operazione della complementazione dei sottoinsiemi di un dato insieme e la classe dei tetraedri rispetto alla trasformazione di un poliedro nel suo duale. L'importanza di una dualit\u00E0 entro una teoria riguarda il fatto che facendo riferimento ad essa la teoria stessa pu\u00F2 essere sviluppata pi\u00F9 economicamente (si possono risparmiare dimostrazioni di teoremi duali) e pu\u00F2 essere esposta pi\u00F9 organicamente."@it . "In matematica il tema della dualit\u00E0 \u00E8 importante e pervasivo,ma non vi \u00E8 una definizione universalmente accettata in grado di unificaretutte le sue accezioni. In linea generale si pu\u00F2 dire che una dualit\u00E0 \u00E8 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."@it . . . . . . . . . . . . . "Dualit\u00E0 (matematica)"@it . "\u0414\u0443\u0430\u043B\u044C\u043D\u0456\u0441\u0442\u044C"@uk . . . . . . "50552"^^ . . . . . . . . . . . . . "Duality (mathematics)"@en . . "Kostrikin"@en . . . . "Duality"@en . . . . . "\u0414\u0432\u043E\u0439\u0441\u0442\u0432\u0435\u043D\u043D\u043E\u0441\u0442\u044C"@ru . . . . . . . . . "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 \u00E4hnlich bezeichnet, um die Abh\u00E4ngigkeit von zum Ausdruck zu bringen. Wendet man dieselbe (oder eine \u00E4hnliche) Konstruktion auf an, erh\u00E4lt man daraus ein mit bezeichnetes Objekt. H\u00E4ufig stehen und in einer engen Beziehung, sind z. B. gleich oder isomorph, weshalb Informationen \u00FCber enthalten muss. Man nennt dann das zu duale und das biduale Objekt. In der zugeh\u00F6rigen mathematischen Dualit\u00E4tstheorie untersucht man dann, wie Eigenschaften von zu Eigenschaften von \u00FCbersetzt werden k\u00F6nnen und umgekehrt."@de . . . . . . . . . . "In mathematics, a duality 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."@en . . . . . . . . . . "1092848557"^^ . . . "609737"^^ . . . . . . . . . . "En matem\u00E1ticas, una dualidad, en t\u00E9rminos generales, traduce conceptos, teoremas o estructuras matem\u00E1ticas en otros conceptos, teoremas o estructuras, mediante una correspondencia uno a uno, a menudo (pero no siempre) por medio de una operaci\u00F3n de involuci\u00F3n: si el dual de A es B, entonces el dual de B es A. Tales involuciones a veces tienen puntos fijos, de modo que el dual de A es A en s\u00ED mismo. Por ejemplo, el teorema de Desargues expresa una condici\u00F3n auto dual en este sentido bajo el concepto de dualidad en geometr\u00EDa proyectiva. f: V \u2192 W su dual f\u2217: W\u2217 \u2192 V\u2217"@es . . . . . . . . . . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u062A\u062D\u0648\u0651\u0644 \u0627\u0644\u0645\u062B\u0646\u0648\u064A\u0629 \u0627\u0644\u0645\u0641\u0627\u0647\u064A\u0645 \u0623\u0648 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0627\u062A \u0623\u0648 \u0627\u0644\u0647\u064A\u0627\u0643\u0644 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629 \u0625\u0644\u0649 \u0645\u0641\u0627\u0647\u064A\u0645\u064E \u0648\u0645\u0628\u0631\u0647\u0646\u0627\u062A \u0648\u0647\u064A\u0627\u0643\u0644 \u0623\u062E\u0631\u0649\u060C \u0639\u0646 \u0637\u0631\u064A\u0642 \u062F\u0627\u0644\u0629 \u0645\u062A\u0628\u0627\u064A\u0646\u0629\u060C \u0648\u063A\u0627\u0644\u0628\u064B\u0627 \u0639\u0646 \u0637\u0631\u064A\u0642 \u062F\u0627\u0644\u0629 \u0627\u0631\u062A\u062F\u0627\u062F\u064A\u0629: \u0625\u0630\u0627 \u0643\u0627\u0646\u062A A \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 B \u060C \u0641\u0625\u0646\u0651 B \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 A. \u0642\u062F \u062A\u062D\u062A\u0648\u064A \u0645\u062B\u0644 \u0647\u0630\u0647 \u0627\u0644\u0627\u0631\u062A\u062F\u0627\u062F\u0627\u062A \u0639\u0644\u0649 \u0646\u0642\u0627\u0637 \u062B\u0627\u0628\u062A\u0629\u060C \u0628\u062D\u064A\u062B \u062A\u0643\u0648\u0646 \u0645\u062B\u0646\u0648\u064A\u0629 A \u0647\u064A \u0646\u0641\u0633\u0647\u0627 A. \u0645\u062B\u0644\u0627\u064B \u0645\u0628\u0631\u0647\u0646\u0629 \u062F\u064A\u0632\u0627\u0631\u063A \u0647\u064A \u0645\u062B\u0646\u0648\u064A\u0629 \u0630\u0627\u062A\u064A\u0627\u064B \u0641\u064A \u0638\u0644 \u0627\u0644\u0627\u0632\u062F\u0648\u0627\u062C\u064A\u0629 \u0627\u0644\u0642\u064A\u0627\u0633\u064A\u0629 \u0641\u064A \u0627\u0644\u0647\u0646\u062F\u0633\u0629 \u0627\u0644\u0625\u0633\u0642\u0627\u0637\u064A\u0629. \u0641\u064A \u0627\u0644\u0633\u064A\u0627\u0642\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0629\u060C \u0644\u0644\u0645\u062B\u0646\u0648\u064A\u0629 \u0645\u0639\u0627\u0646\u064D \u0639\u062F\u064A\u062F\u0629. \u0648\u0642\u062F \u0648\u0635\u0641 \u0628\u0623\u0646\u0647 \u00AB\u0645\u0641\u0647\u0648\u0645 \u0648\u0627\u0633\u0639 \u0627\u0644\u0627\u0646\u062A\u0634\u0627\u0631 \u0648\u0645\u0647\u0645 \u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A (\u0627\u0644\u062D\u062F\u064A\u062B\u0629)\u00BB \u0648 \u00AB\u0645\u0648\u0636\u0648\u0639 \u0639\u0627\u0645 \u0645\u0647\u0645 \u0644\u0647 \u0645\u0638\u0627\u0647\u0631 \u0641\u064A \u0643\u0644 \u0645\u062C\u0627\u0644 \u0645\u0646 \u0645\u062C\u0627\u0644\u0627\u062A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u062A\u0642\u0631\u064A\u0628\u064B\u0627\u00BB."@ar . "\u5728\u6570\u5B66\u9886\u57DF\u4E2D\uFF0C\u5BF9\u5076\u4E00\u822C\u6765\u8BF4\u662F\u4EE5\u4E00\u5BF9\u4E00\u7684\u65B9\u5F0F\uFF0C\u5E38\u5E38\uFF08\u4F46\u5E76\u4E0D\u603B\u662F\uFF09\u901A\u8FC7\u67D0\u4E2A\u5BF9\u5408\u7B97\u5B50\uFF0C\u628A\u4E00\u79CD\u6982\u5FF5\u3001\u516C\u7406\u6216\u6570\u5B66\u7ED3\u6784\u8F6C\u5316\u4E3A\u53E6\u4E00\u79CD\u6982\u5FF5\u3001\u516C\u7406\u6216\u6570\u5B66\u7ED3\u6784\uFF1A\u5982\u679CA\u7684\u5BF9\u5076\u662FB\uFF0C\u90A3\u4E48B\u7684\u5BF9\u5076\u662FA\u3002\u7531\u4E8E\u5BF9\u5408\u6709\u65F6\u5019\u4F1A\u5B58\u5728\u4E0D\u52A8\u70B9\uFF0C\u56E0\u6B64A\u7684\u5BF9\u5076\u6709\u65F6\u5019\u4F1A\u662FA\u81EA\u8EAB\u3002\u6BD4\u5982\u5C04\u5F71\u51E0\u4F55\u4E2D\u7684\u7B1B\u6C99\u683C\u5B9A\u7406\uFF0C\u5373\u662F\u5728\u8FD9\u4E00\u610F\u4E49\u4E0B\u7684\u81EA\u5BF9\u5076\u3002 \u5BF9\u5076\u5728\u6570\u5B66\u80CC\u666F\u5F53\u4E2D\u5177\u6709\u5F88\u591A\u79CD\u610F\u4E49\uFF0C\u800C\u4E14\uFF0C\u5C3D\u7BA1\u5B83\u662F\u201C\u73B0\u4EE3\u6570\u5B66\u4E2D\u6781\u4E3A\u666E\u904D\u4E14\u91CD\u8981\u7684\u6982\u5FF5\uFF08a very pervasive and important concept in (modern) mathematics\uFF09\u201D\u5E76\u4E14\u662F\u201C\u5728\u6570\u5B66\u51E0\u4E4E\u6BCF\u4E00\u4E2A\u5206\u652F\u4E2D\u90FD\u4F1A\u51FA\u73B0\u7684\u91CD\u8981\u7684\u4E00\u822C\u6027\u4E3B\u9898\uFF08an important general theme that has manifestations in almost every area of mathematics\uFF09\u201D\uFF0C\u4F46\u4ECD\u7136\u6CA1\u6709\u4E00\u4E2A\u80FD\u628A\u5BF9\u5076\u7684\u6240\u6709\u6982\u5FF5\u7EDF\u4E00\u8D77\u6765\u7684\u666E\u9002\u5B9A\u4E49\u3002 \u5728\u4E24\u7C7B\u5BF9\u8C61\u4E4B\u95F4\u7684\u5BF9\u5076\u5F88\u591A\u90FD\u548C\uFF08pairing\uFF09\uFF0C\u4E5F\u5C31\u662F\u628A\u4E00\u7C7B\u5BF9\u8C61\u548C\u53E6\u4E00\u7C7B\u5BF9\u8C61\u6620\u5C04\u5230\u67D0\u4E00\u65CF\u6807\u91CF\u4E0A\u7684\u53CC\u7EBF\u6027\u51FD\u6570\u76F8\u5BF9\u5E94\u3002\u4F8B\u5982\uFF0C\u7EBF\u6027\u4EE3\u6570\u7684\u5BF9\u5076\u5BF9\u5E94\u7740\u628A\u7EBF\u6027\u7A7A\u95F4\u4E2D\u7684\u5411\u91CF\u5BF9\u53CC\u7EBF\u6027\u6620\u5C04\u5230\u6807\u91CF\u4E0A\uFF0C\u5E7F\u4E49\u51FD\u6570\u53CA\u5176\u76F8\u5173\u7684\u4E5F\u5BF9\u5E94\u7740\u4E00\u4E2A\u914D\u5BF9\u4E14\u5728\u8BE5\u914D\u5BF9\u4E2D\u53EF\u7528\u8BD5\u9A8C\u51FD\u6570\u6765\u5BF9\u5E7F\u4E49\u51FD\u6570\u8FDB\u884C\u79EF\u5206\uFF0C\u5E9E\u52A0\u83B1\u5BF9\u5076\u4ECE\u7ED9\u5B9A\u6D41\u5F62\u7684\u5B50\u6D41\u5F62\u4E4B\u95F4\u7684\u914D\u5BF9\u7684\u89D2\u5EA6\u770B\u540C\u6837\u4E5F\u5BF9\u5E94\u7740\u3002"@zh . . . "Alexei Kostrikin"@en . . . . . . . . . . . . . "What is Z in the formula?"@en . . . . . . . . .