翻訳と辞書 Words near each other ・ Signature crime ・ Signature defect ・ Signature dish ・ Signature Drinks ・ Signature file ・ Signature forgery ・ Signature Half-Step a Retrospective 2000-2014 ・ Signature in the Cell ・ Signature island ・ Signature line of credit ・ Signature mark ・ Signature matrix ・ Signature move ・ Signature of a knot ・ Signature of Divine (Yahweh) ・ Signature operator ・ Signature Place ・ Signature program ・ Signature recognition ・ Signature Record Type Definition ・ Signature Records ・ Signature School ・ Signature Select ・ Signature song ・ Signature Sound ・ Signature Sounds Recordings ・ Signature tag ・ Signature Team ・ Signature Theatre ・ Signature Theatre (Arlington, Virginia) 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 Signature operator ： ウィキペディア英語版 Signature operator In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator. ==Definition in the even-dimensional case== Let $M$ be a compact Riemannian manifold of even dimension $2l$. Let :$d\; :\; \backslash Omega^p(M)\backslash rightarrow\; \backslash Omega^(M)$ be the exterior derivative on $i$-th order differential forms on $M$. The Riemannian metric on $M$ allows us to define the Hodge star operator $\backslash star$ and with it the inner product :$\backslash langle\backslash omega,\backslash eta\backslash rangle=\backslash int\_M\backslash omega\backslash wedge\backslash star\backslash eta$ on forms. Denote by :$d^$ *: \Omega^(M)\rightarrow \Omega^p(M) the adjoint operator of the exterior differential $d$. This operator can be expressed purely in terms of the Hodge star operator as follows: :$d^$ *= (-1)^ \star d \star= - \star d \star Now consider $d\; +\; d^$ * acting on the space of all forms $\backslash Omega(M)=\backslash bigoplus\_^\backslash Omega^(M)$. One way to consider this as a graded operator is the following: Let $\backslash tau$ be an involution on the space of ''all'' forms defined by: :$\backslash tau(\backslash omega)=i^\backslash star\; \backslash omega\backslash quad,\backslash quad\backslash omega\; \backslash in\; \backslash Omega^p(M)$ It is verified that $d\; +\; d^$ * anti-commutes with $\backslash tau$ and, consequently, switches the $(\backslash pm\; 1)$-eigenspaces $\backslash Omega\_(M)$ of $\backslash tau$ Consequently, :$d\; +\; d^$ * = \begin 0 & D \\ D^ * & 0 \end Definition: The operator $d\; +\; d^$ * with the above grading respectively the above operator $D:\; \backslash Omega\_+(M)\; \backslash rightarrow\; \backslash Omega\_-(M)$ is called the signature operator of $M$. 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「Signature operator」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.