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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 Borsuk–Ulam theorem ： ウィキペディア英語版 Borsuk–Ulam theorem In mathematics, the Borsuk–Ulam theorem (BUT), states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Formally: if $f:\; S^n\; \backslash to\; R^n$ is continuous then there exists an $x\backslash in\; S^n$ such that: $f(-x)=f(x)$. The case $n=1$ can be illustrated by saying that there always exist a pair of opposite points on the earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously. The case $n=2$ is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures. BUT has several equivalent statements in terms of odd functions. Recall that $S^n$ is the ''n''-sphere and $B^n$ is the ''n''-ball: * If $g:\; S^n\; \backslash to\; R^n$ is a continuous odd function, then there exists an $x\backslash in\; S^n$ such that: $g(x)=0$. * If $g:\; B^n\; \backslash to\; R^n$ is a continuous function which is odd on $S^$ (the boundary of $B^n$), then there exists an $x\backslash in\; B^n$ such that: $g(x)=0$. ==History== According to , the first historical mention of the statement of BUT appears in . The first proof was given by , where the formulation of the problem was attributed to Stanislaw Ulam. Since then, many alternative proofs have been found by various authors, as collected by . 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「Borsuk–Ulam theorem」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.