翻訳と辞書 Words near each other ・ Clifford (character) ・ Clifford (film) ・ Clifford (name) ・ Clifford (ward) ・ Clifford A. Henricksen ・ Clifford A. Jones ・ Clifford A. Pickover ・ Clifford A. Wolff ・ Clifford Aboagye ・ Clifford Alexander, Jr. ・ Clifford algebra ・ Clifford Allbutt ・ Clifford Allen ・ Clifford Allen, 1st Baron Allen of Hurtwood ・ Clifford Allison ・ Clifford analysis ・ Clifford Anderson ・ Clifford Ando ・ Clifford Andre ・ Clifford Andrews ・ Clifford Ann Creed ・ Clifford Annex ・ Clifford Antone ・ Clifford B. Harmon ・ Clifford B. Hicks ・ Clifford B. Wilson ・ Clifford Ball ・ Clifford Banks ・ Clifford Barclay ・ Clifford Barker 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 Clifford analysis ： ウィキペディア英語版 Clifford analysis Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, $d+$ *d * on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on $C\_^(\backslash mathbf^)$ and their conformal equivalents on the sphere, the Laplacian in euclidean ''n''-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on Spin''c'' manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations. == Euclidean space == In Euclidean space the Dirac operator has the form :$D=\backslash sum\_^e\_\backslash frac\; =\; -\backslash Delta\_$ where Δ''n'' is the Laplacian in ''n''-euclidean space. The fundamental solution to the euclidean Dirac operator is :$G(x-y):=\backslash frac$ where ω''n'' is the surface area of the unit sphere S''n''−1. Note that :$D\backslash frac\}=G(x-y)$ where :$\backslash frac\}$ is the fundamental solution to Laplace's equation for . The most basic example of a Dirac operator is the Cauchy–Riemann operator :$\backslash frac+i\backslash frac$ in the complex plane. Indeed many basic properties of one variable complex analysis follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem, a Cauchy integral formula, Morera's Theorem, Taylor series, Laurent series and Liouville Theorem. In this case the Cauchy kernel is ''G''(''x''−''y''). The proof of the Cauchy integral formula is the same as in one complex variable and makes use of the fact that each non-zero vector ''x'' in euclidean space has a multiplicative inverse in the Clifford algebra, namely :$-\backslash frac^.$ Up to a sign this inverse is the Kelvin inverse of ''x''. Solutions to the euclidean Dirac equation ''Df'' = 0 are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold. In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When , the Dirac operator is sometimes referred to as the Cauchy–Riemann–Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis. Clifford analysis has analogues of Cauchy transforms, Bergman kernels, Szegő kernels, Plemelj operators, Hardy spaces, a Kerzman–Stein formula and a Π, or Beurling–Ahlfors, transform. These have all found applications in solving boundary value problems, including moving boundary value problems, singular integrals and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain Sobolev spaces, the full water wave problem in 3D. This method works in all dimensions greater than 2. Much of Clifford analysis works if we replace the complex Clifford algebra by a real Clifford algebra, ''C''ℓ''n''. This is not the case though when we need to deal with the interaction between the Dirac operator and the Fourier transform. 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「Clifford analysis」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.