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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 Unfolding (functions) ： ウィキペディア英語版 Unfolding (functions) In mathematics, an unfolding of a function is a certain family of functions. Let $M$ be a smooth manifold and consider a smooth mapping $f\; :\; M\; \backslash to\; \backslash mathbb.$ Let us assume that for given $x\_0\; \backslash in\; M$ and $y\_0\; \backslash in\; \backslash mathbb$ we have $f(x\_0)\; =\; y\_0$. Let $N$ be a smooth $k$-dimensional manifold, and consider the family of mappings (parameterised by $N$) given by $F\; :\; M\; \backslash times\; N\; \backslash to\; \backslash mathbb\; .$ We say that $F$ is a $k$-parameter unfolding of $f$ if $F(x,0)\; =\; f(x)$ for all $x.$ In other words the functions $f\; :\; M\; \backslash to\; \backslash mathbb$ and $F\; :\; M\; \backslash times\; \backslash \; \backslash to\; \backslash mathbb$ are the same: the function $f$ is contained in, or is unfolded by, the family $F.$ Let $f\; :\; \backslash mathbb^2\; \backslash to\; \backslash mathbb$ be given by $f(x,y)\; =\; x^2\; +\; y^5.$ An example of an unfolding of $f$ would be $F\; :\; \backslash mathbb^2\; \backslash times\; \backslash mathbb^3\; \backslash to\; \backslash mathbb$ given by :$F((x,y),(a,b,c))\; =\; x^2\; +\; y^5\; +\; ay\; +\; by^2\; +\; cy^3.$ As is the case with unfoldings, $x$ and $y$ are called variables and $a,$ $b,$ and $c$ are called parameters – since they parameterise the unfolding. In practice we require that the unfoldings have certain nice properties. In $\backslash mathbb$ notice that $f$ is a smooth mapping from $M$ to $\backslash mathbb$ and so belongs to the function space $C^(M,\backslash mathbb).$ As we vary the parameters of the unfolding we get different elements of the function space. Thus, the unfolding induces a function $\backslash Phi\; :\; N\; \backslash to\; C^(M,\backslash mathbb).$ The space $\backslash operatorname(M)\; \backslash times\; \backslash operatorname(\backslash mathbb),$ where $\backslash operatorname(M)$ denotes the group of diffeomorphisms of $M$ etc., acts on $C^(M,\backslash mathbb).$ The action is given by $(\backslash phi,\backslash psi)\; \backslash cdot\; f\; =\; \backslash psi\; \backslash circ\; f\; \backslash circ\; \backslash phi^.$ If $g$ lies in the orbit of $f$ under this action then there is a diffeomorphic change of coordinates in $M$ and $\backslash mathbb$ which takes $g$ to $f$ (and vice versa). One nice property that we may like to impose is that :$\backslash operatorname(\backslash Phi)\; \backslash pitchfork\; \backslash operatorname(f)$ where "$\backslash pitchfork$" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliate $C^\backslash infty\; (M,\backslash mathbb)$ – how the resulting functions will vary. There is an idea of a versal unfolding. Every versal unfolding has the property that $\backslash operatorname(\backslash Phi)\; \backslash pitchfork\; \backslash operatorname(f)$, but the converse is false. Let $x\_1,\backslash ldots,x\_n$ be local coordinates on $M$, and let $\backslash mathcal(x\_1,\backslash ldots,x\_n)$ denote the ring of smooth functions. We define the Jacobian ideal of $f,$ denoted by $J\_f$ as follows: :$J\_f\; :=\; \backslash left\backslash langle\; \backslash frac,\; \backslash ldots,\; \backslash frac\; \backslash right\backslash rangle.$ Then a basis for a versal unfolding of $f$ is given by quotient :$\backslash frac$ This quotient is known as the local algebra of $f.$ The dimension of the local algebra is called the Milnor number of $f$. The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal! Consider the function $f(x,y)\; =\; x^2\; +\; y^5.$ A calculation shows that :$\backslash frac\; =\; \backslash \; \backslash \; .$ This means that $\backslash$ give a basis for a versal unfolding, and that :$F((x,y),(a,b,c))\; =\; x^2\; +\; y^5\; +\; ay\; +\; by^2\; +\; cy^3$ is a versal unfolding. A versal unfolding with the minimum possible number of unfolding parameters is called a miniversal unfolding. Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc. An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities. == References == * V. I. Arnold, S. M. Gussein-Zade & A. N. Varchenko, Singularities of differentiable maps, Volume 1, Birkhäuser, (1985). * J. W. Bruce & P. J. Giblin, Curves & singularities, second edition, Cambridge University press, (1992). 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「Unfolding (functions)」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.