翻訳と辞書 Words near each other ・ Extinction paradox ・ Extinction probability ・ Extinction ratio ・ Extinction risk from global warming ・ Extinction threshold ・ Extinction vortex ・ Extinctioners ・ Extinctionism ・ Extinguishment ・ Extirpation ・ Extize ・ EXTL1 ・ EXTL2 ・ EXTL3 ・ External (disambiguation) ・ External (mathematics) ・ External Active Thermal Control System ・ External Affairs (film) ・ External anal sphincter ・ External arcuate fibers ・ External association ・ External auditor ・ External bacterial infection (fish) ・ External ballistics ・ External beam radiotherapy ・ External border of the European Union ・ External Bus Interface ・ External Bus Interface 2 ・ External capsule ・ External cardinal Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク External (mathematics) ： ウィキペディア英語版 External (mathematics) The term external is useful for describing certain algebraic structures. The term comes from the concept of an external binary operation which is a binary operation that draws from some ''external set''. To be more specific, a left external binary operation on ''S'' over ''R'' is a function $f\; :\; R\; \backslash times\; S\; \backslash rightarrow\; S$ and a right external binary operation on ''S'' over ''R'' is a function $f\; :\; S\; \backslash times\; R\; \backslash rightarrow\; S$ where ''S'' is the set the operation is defined on, and ''R'' is the external set (the set the operation is defined ''over''). == Generalizations == The ''external'' concept is a generalization rather than a specialization, and as such, it is different from many terms in mathematics. A similar but opposite concept is that of an ''internal binary function'' from ''R'' to ''S'', defined as a function $f\; :\; R\; \backslash times\; R\; \backslash rightarrow\; S$. Internal binary functions are like binary functions, but are a form of specialization, so they only accept a subset of the domains of binary functions. Here we list these terms with the function signatures they imply, along with some examples: * $f\; :\; Q\; \backslash times\; R\; \backslash rightarrow\; S$ (binary function) * * Example: exponentiation ($z^q\; :\; \backslash Bbb\; \backslash times\; \backslash Bbb\; \backslash rightarrow\; \backslash Bbb$ as in $^\; =\; i$), * * Example: set membership ($(\backslash in)\; :\; S\; \backslash times\; \backslash mathbf\; \backslash rightarrow\; \backslash Bbb$ where $\backslash mathbf$ is the category of sets) * * Examples: matrix multiplication, the tensor product, and the Cartesian product * $f\; :\; R\; \backslash times\; R\; \backslash rightarrow\; S$ (internal binary function) * * Example: internal binary relations ($(\backslash le)\; :\; R\; \backslash times\; R\; \backslash rightarrow\; \backslash Bbb$) * * Examples: the dot product, the inner product, and metrics. * $f\; :\; R\; \backslash times\; S\; \backslash rightarrow\; S$ (external binary operation) * * Examples: dynamical system flows, group actions, projection maps, and scalar multiplication. * $f\; :\; S\; \backslash times\; S\; \backslash rightarrow\; S$ (binary operation). * * Examples: addition, multiplication, permutations, and the cross product. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「External (mathematics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.