翻訳と辞書 Words near each other ・ Quasi-solid ・ Quasi-split group ・ Quasi-star ・ Quasi-syllogism ・ Quasi-synchronous transmission ・ Quasi-tort ・ Quasi-triangular quasi-Hopf algebra ・ Quasi-triangulation ・ Quasi-War ・ Quasi-Zenith Satellite System ・ Quasianosteosaurus ・ Quasiatom ・ Quasicaecilia ・ Quasicircle ・ Quasiconcha ・ Quasiconformal mapping ・ Quasicontraction semigroup ・ Quasiconus ・ Quasiconvex function ・ Quasicooronga ・ Quasicravenoceras ・ Quasicrystal ・ Quasicrystal Research ・ Quasicyclotosaurus ・ Quasideterminant ・ Quasidihedral group ・ Quasielastic neutron scattering ・ Quasielastic scattering ・ Quasieulia ・ Quasieulia endela 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 Quasiconformal mapping ： ウィキペディア英語版 Quasiconformal mapping In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D'' → ''D''′ be an orientation-preserving homeomorphism between open sets in the plane. If ''f'' is continuously differentiable, then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''. ==Definition== Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have continuous partial derivatives, then ''f'' is quasiconformal provided it satisfies the Beltrami equation = \mu(z)\frac,|}} for some complex valued Lebesgue measurable μ satisfying sup |μ| < 1 . This equation admits a geometrical interpretation. Equip ''D'' with the metric tensor :$ds^2\; =\; \backslash Omega(z)^2\backslash left|\; \backslash ,\; dz\; +\; \backslash mu(z)\; \backslash ,\; d\backslash bar\backslash right|^2,$ where Ω(''z'') > 0. Then ''f'' satisfies () precisely when it is a conformal transformation from ''D'' equipped with this metric to the domain ''D''′ equipped with the standard Euclidean metric. The function ''f'' is then called μ-conformal. More generally, the continuous differentiability of ''f'' can be replaced by the weaker condition that ''f'' be in the Sobolev space ''W''1,2(''D'') of functions whose first-order distributional derivatives are in L2(''D''). In this case, ''f'' is required to be a weak solution of (). When μ is zero almost everywhere, any homeomorphism in ''W''1,2(''D'') that is a weak solution of () is conformal. Without appeal to an auxiliary metric, consider the effect of the pullback under ''f'' of the usual Euclidean metric. The resulting metric is then given by :$\backslash left|\backslash frac\backslash right|^2\backslash left|\backslash ,dz+\backslash mu(z)\backslash ,d\backslash bar\backslash right|^2$ which, relative to the background Euclidean metric $dz\; d\backslash bar$, has eigenvalues :$(1+|\backslash mu|)^2\backslash textstyle\backslash right|^2\},\backslash qquad\; (1-|\backslash mu|)^2\backslash textstyle\backslash right|^2\}.$ The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along ''f'' the unit circle in the tangent plane. Accordingly, the ''dilatation'' of ''f'' at a point ''z'' is defined by :$K(z)\; =\; \backslash frac.$ The (essential) supremum of ''K''(''z'') is given by :$K\; =\; \backslash sup\_\; |K(z)|\; =\; \backslash frac$ and is called the dilatation of ''f''. A definition based on the notion of extremal length is as follows. If there is a finite ''K'' such that for every collection Γ of curves in ''D'' the extremal length of Γ is at most ''K'' times the extremal length of . Then ''f'' is ''K''-quasiconformal. If ''f'' is ''K''-quasiconformal for some finite ''K'', then ''f'' is quasiconformal. 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「Quasiconformal mapping」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.