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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 Symmetrization methods ： ウィキペディア英語版 Symmetrization methods In mathematics the symmetrization methods are algorithms of transforming a set $A\backslash subset\; \backslash mathbb^n$ to a ball $B\backslash subset\; \backslash mathbb^n$ with equal volume $\backslash operatorname(B)=\backslash operatorname(A)$ and centered at the origin. ''B'' is called the symmetrized version of ''A'', usually denoted $A^$. These algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has the minimal perimeter (for details see Isoperimetric inequality). The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method (described below). From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality for details). Another problem is that the Newtonian capacity of a set A is minimized by $A^$ and this was proved by Polya and G. Szego (1951) using circular symmetrization (described below). ==Symmetrization== If $\backslash Omega\backslash subset\; \backslash mathbb^n$ is measurable, then it is denoted by $\backslash Omega^$ the symmetrized version of $\backslash Omega$ i.e. a ball $\backslash Omega^:=B\_r(0)\backslash subset\backslash mathbb^n$ such that $\backslash operatorname(\backslash Omega^)=\backslash operatorname(\backslash Omega)$. We denote by $f^$ the symmetric decreasing rearrangement of nonnegative measurable function f and define it as $f^(x):=\backslash int\_0^\backslash infty\; 1\_\}(t)\; \backslash ,\; dt$, where $\backslash ^$ is the symmetrized version of preimage set $\backslash$. The methods described below have been proved to transform $\backslash Omega$ to $\backslash Omega^$ i.e. given a sequence of symmetrization transformations $\backslash$ there is $\backslash lim\backslash limits\_d\_(\backslash Omega^,\; T\_k(K)\; )=0$, where $d\_$ is the Hausdorff distance (for discussion and proofs see ) 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「Symmetrization methods」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.