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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Busemann function ： ウィキペディア英語版 Busemann function Busemann functions were introduced by Busemann to study the large-scale geometry of metric spaces in his seminal ''The Geometry of Geodesics''.〔Busemann, Herbert. The geometry of geodesics. Vol. 6. DoverPublications. com, 1985.〕 More recently, Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation.〔Hoffman, Christopher. "Coexistence for Richardson type competing spatial growth models." The Annals of Applied Probability 15.1B (2005): 739-747.〕〔Damron, Michael, and Jack Hanson. "Busemann functions and infinite geodesics in two-dimensional first-passage percolation." arXiv preprint arXiv:1209.3036 (2012).〕 == Definition == Let $(X,d)$ be a metric space. A ray is a path $\backslash gamma\; :\; [0,\backslash infty)\; \backslash to\; X$ which minimizes distance everywhere along its length. i.e., for all $t,t\text{'}\; \backslash in\; [0,\backslash infty)$, :$d\backslash big(\backslash gamma(t),\; \backslash gamma(t\text{'})\; \backslash big)\; =\; \backslash big|\; t\; -\; t\text{'}\; \backslash big|$. Equivalently, a ray is an isometry from the "canonical ray" (the set $[0,\backslash infty)$ equipped with the Euclidean metric) into the metric space ''X''. Given a ray ''γ'', the Busemann function $B\_\backslash gamma\; :\; X\; \backslash to\; \backslash mathbb\; R$ is defined by :$B\_\backslash gamma(x)=\backslash lim\_\backslash big(\; d\backslash big(\; \backslash gamma(t),\; x\; \backslash big)\; -\; t\; \backslash big)$ That is, when ''t'' is very large, the distance $d\backslash big(\; \backslash gamma(t),\; x\; \backslash big)$ is approximately equal to $B\_\backslash gamma(x)\; +\; t$. Given a ray ''γ'', its Busemann function is always well-defined. Loosely speaking, a Busemann function can be thought of as a "distance to infinity" along the ray ''γ''. 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「Busemann function」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.