翻訳と辞書 Words near each other ・ SPIN World Cricket Monthly Magazine ・ Spin(7)-manifold ・ Spin-destruction collision ・ Spin-exchange interaction ・ Spin-flip ・ Spin-off ・ Spin-Off (game show) ・ Spin-off (media) ・ Spin-orbital (disambiguation) ・ Spin-polarized electron energy loss spectroscopy ・ Spin-stabilisation ・ Spin-stabilized magnetic levitation ・ Spin-stabilized satellite ・ Spin-transfer torque ・ Spin-up ・ Spin-weighted spherical harmonics ・ Spin-½ ・ Spin.FX ・ SPIN1 ・ Spina ・ Spina bifida ・ Spina di Gualdo ・ Spina helicis ・ Spina Nuova ・ Spina vestibuli ・ Spina, Minnesota ・ Spinacanthidae ・ Spinacanthus ・ Spinacaropus ・ Spinacetin 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 Spin-weighted spherical harmonics ： ウィキペディア英語版 Spin-weighted spherical harmonics In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree ℓ, just like ordinary spherical harmonics, but have an additional spin weight ''s'' that reflects the additional ''U''(1) symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics $Y\_$, and are typically denoted by $\_0Y\_\; =\; Y\_\backslash \; .$ Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group . They were subsequently and independently rediscovered by and applied to describe gravitational radiation, and again by as so-called "monopole harmonics" in the study of Dirac monopoles. ==Spin-weighted functions== Regard the sphere ''S''2 as embedded into the three-dimensional Euclidean space R3. At a point x on the sphere, a positively oriented orthonormal basis of tangent vectors at x is a pair a, b of vectors such that :$$ \begin \mathbf\cdot\mathbf &= \mathbf\cdot\mathbf = 0\\ \mathbf\cdot\mathbf &= \mathbf\cdot\mathbf=1\\ \mathbf\cdot\mathbf &= 0\\ \mathbf\cdot (\mathbf\times\mathbf) &> 0, \end where the first pair of equations states that a and b are tangent at x, the second pair states that a and b are unit vectors, the penultimate equation that a and b are orthogonal, and the final equation that (x, a, b) is a right-handed basis of R3. A spin-weight ''s'' function ''f'' is a function accepting as input a point x of ''S''2 and a positively oriented orthonormal basis of tangent vectors at x, such that :$f(\backslash mathbf\; x,\backslash cos\backslash theta\backslash mathbf-\backslash sin\backslash theta\backslash mathbf,\; \backslash sin\backslash theta\backslash mathbf\; +\; \backslash cos\backslash theta\backslash mathbf)\; =\; e^f(\backslash mathbf\; x,\backslash mathbf,\backslash mathbf)$ for every rotation angle θ. Following , denote the collection of all spin-weight ''s'' functions by B(''s''). Concretely, these are understood as functions ''f'' on C2\ satisfying the following homogeneity law under complex scaling :$f(\backslash lambda\; z,\backslash overline\backslash bar)\; =\; \backslash left(\backslash frac\backslash right)^s\; f(z,\backslash bar).$ This makes sense provided ''s'' is a half-integer. Abstractly, B(''s'') is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle $\backslash overline$ of the Serre twist on the complex projective line CP1. A section of the latter bundle is a function ''g'' on C2\ satisfying :$g(\backslash lambda\; z,\backslash overline\backslash bar)\; =\; \backslash overline^\; g(z,\backslash bar).$ Given such a ''g'', we may produce a spin-weight ''s'' function by multiplying by a suitable power of the hermitian form :$P(z,\backslash bar)\; =\; z\backslash cdot\backslash bar.$ Specifically, ''f'' = ''P''−''s''''g'' is a spin-weight ''s'' function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism. 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「Spin-weighted spherical harmonics」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.