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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 Butcher group ： ウィキペディア英語版 Butcher group In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was , prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. pointed out that the Butcher group is the group of characters of the Hopf algebra of rooted trees that had arisen independently in their own work on renormalization in quantum field theory and Connes' work with Moscovici on local index theorems. This Hopf algebra, often called the ''Connes-Kreimer algebra'', is essentially equivalent to the Butcher group, since its dual can be identified with the universal enveloping algebra of the Lie algebra of the Butcher group. As they commented: ==Differentials and rooted trees== A rooted tree is a graph with a distinguished node, called the ''root'', in which every other node is connected to the root by a unique path. If the root of a tree t is removed and the nodes connected to the original node by a single bond are taken as new roots, the tree t breaks up into rooted trees t1, t2, ... Reversing this process a new tree t = (t2, ... ) can be constructed by joining the roots of the trees to a new common root. The number of nodes in a tree is denoted by |t|. A ''heap-ordering'' of a rooted tree t is an allocation of the numbers 1 through |t| to the nodes so that the numbers increase on any path going away from the root. Two heap orderings are ''equivalent'', if there is an automorphism of rooted trees mapping one of them on the other. The number of equivalence classes of heap-orderings on a particular tree is denoted by α(t) and can be computed using the Butcher's formula: :$\backslash displaystyle\; \backslash alpha(t)=\; ,$ where ''S''t denotes the symmetry group of t and the tree factorial is defined recursively by :$()!\; =\; |()|\; \backslash cdot\; t\_1!\; \backslash cdots\; t\_n!$ with the tree factorial of an isolated root defined to be 1 :$\backslash bullet\; !\; =1.$ The ordinary differential equation for the flow of a vector field on an open subset ''U'' of RN can be written :$\backslash displaystyle\; =\; f(x(s)),\backslash ,\backslash ,\; x(0)=x\_0,$ where ''x''(''s'') takes values in ''U'', ''f'' is a smooth function from ''U'' to RN and ''x''0 is the starting point of the flow at time ''s'' = 0. gave a method to compute the higher order derivatives ''x''(''m'')(''s'') in terms of rooted trees. His formula can be conveniently expressed using the ''elementary differentials'' introduced by Butcher. These are defined inductively by :$\backslash delta\_\backslash bullet^i=\; f^i,\; \backslash ,\backslash ,\backslash ,\; \backslash delta^i\_\; =\; \backslash sum\_^N\; (\backslash delta^\_\; \backslash cdots\; \backslash delta^\_)\backslash partial\_\; \backslash cdots\; \backslash partial\_\; f^i.$ With this notation :$=\; \backslash sum\_\; \backslash alpha(t)\; \backslash delta\_t,$ giving the power series expansion :$\backslash displaystyle\; x(s)\; =\; x\_0\; +\; \backslash sum\_\; \backslash alpha(t)\; \backslash delta\_t(0).$ As an example when ''N'' = 1, so that ''x'' and ''f'' are real-valued functions of a single real variable, the formula yields :$x^\; =\; f^f^3\; +\; 3\; f^f^\; f^2\; +\; f^f^\; f^2\; +(f^\backslash prime)^3\; f,$ where the four terms correspond to the four rooted trees from left to right in Figure 3 above. In a single variable this formula is the same as Faà di Bruno's formula of 1855; however in several variables it has to be written more carefully in the form :$x^\; =\; f^(f,f,f)\; +\; 3f^(f,f^\backslash prime(f))\; +\; f^\backslash prime(f^(f,f))\; +f^\backslash prime(f^\backslash prime(f^\backslash prime(f))),$ where the tree structure is crucial. 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「Butcher group」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.