翻訳と辞書 Words near each other ・ Invasion of the Body Snatchas! ・ Invasion of the Body Snatchers ・ Invasion of the Body Snatchers (1978 film) ・ Invasion of the Booty Snatchers ・ Invasion of the Bunny Snatchers ・ Invariance of domain ・ Invariance principle (linguistics) ・ Invariance theorem ・ Invariances ・ Invariant ・ Invariant (computer science) ・ Invariant (mathematics) ・ Invariant (physics) ・ Invariant basis number ・ Invariant convex cone ・ Invariant differential operator ・ Invariant estimator ・ Invariant extended Kalman filter ・ Invariant factor ・ Invariant factorization of LPDOs ・ Invariant manifold ・ Invariant mass ・ Invariant measure ・ Invariant of a binary form ・ Invariant polynomial ・ Invariant set postulate ・ Invariant speed ・ Invariant subspace ・ Invariant subspace problem ・ Invariant theory 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 Invariant differential operator ： ウィキペディア英語版 Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on $\backslash mathbb^n$, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator $D$, the term ''differential operator'' indicates that the value $Df$ of the map depends only on $f(x)$ and the derivatives of $f$ in $x$. The word ''invariant'' indicates that the operator contains some symmetry. This means that there is a group $G$ with a group action on the functions (or other objects in question) and this action is preserved by the operator: :$D(g\backslash cdot\; f)=g\backslash cdot\; (Df).$ Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates. ==Invariance on homogeneous spaces== Let ''M'' = ''G''/''H'' be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation $\backslash rho:H\backslash rightarrow\backslash mathrm(\backslash mathbb)$ gives rise to a vector bundle :$V=G\backslash times\_\backslash mathbb\backslash ;\backslash text\backslash ;(gh,v)\backslash sim(g,\backslash rho(h)v)\backslash ;\backslash forall\backslash ;g\backslash in\; G,\backslash ;h\backslash in\; H\backslash ;\backslash text\backslash ;v\backslash in\backslash mathbb.$ Sections $\backslash varphi\backslash in\backslash Gamma(V)$ can be identified with :$\backslash Gamma(V)=\backslash )\backslash varphi(g)\backslash ;\backslash forall\backslash ;g\backslash in\; G,\backslash ;\; h\backslash in\; H\backslash \}.$ In this form the group ''G'' acts on sections via :$(\backslash ell\_g\; \backslash varphi)(g\text{'})=\backslash varphi(g^g\text{'}).$ Now let ''V'' and ''W'' be two vector bundles over ''M''. Then a differential operator :$d:\backslash Gamma(V)\backslash rightarrow\backslash Gamma(W)$ that maps sections of ''V'' to sections of ''W'' is called invariant if :$d(\backslash ell\_g\; \backslash varphi)\; =\; \backslash ell\_g\; (d\backslash varphi).$ for all sections $\backslash varphi$ in $\backslash Gamma(V)$ and elements ''g'' in ''G''. All linear invariant differential operators on homogeneous parabolic geometries, i.e. when ''G'' is semi-simple and ''H'' is a parabolic subgroup, are given dually by homomorphisms of generalized Verma modules. 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「Invariant differential 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.