两张相同的地图

野菜花 · 发表于 2005-4-20 01:45:16

将一张揉成一团，随意地扔在另一张地图上，问揉皱的地图上是否有一个地点正好在另一张地图的同一地点的上面？请说明原因。

fzy · 发表于 2005-4-20 02:03:25

Which fixed point theorem are we allowed to use?

野菜花 · 发表于 2005-4-20 02:13:22

Any is allowed, please tell the statement of the theorem.

Your question has already killed this problem.

fzy · 发表于 2005-4-20 18:00:21

Brouwer Fixed Point Theorem

Any continuous function has a fixed point, where is the unit n-ball. www.ddhw.com

Is there a elemetary way to prove it? Or we have to use some results from topology? I did not want to kill the problem, just dont know what to do.

野菜花 · 发表于 2005-4-20 21:39:33

约一年半前 WXC 有人贴了这题：

将一块桌布揉成一团，随意地扔在桌上，桌布上是否一定有一点在桌面上的投影位子没有变。(我把它改头换面了一下 )

没有人用初等方法做出来。我就是用了 Brouwer 不动点定理。

现在我想是否可以用二维的区间套定理 (当然也不算初等方法 )

这题的另一个版本：

将一张小地图钉在一张按比例放大的大地图上，小地图上必有一个地址与大地图重合。

野菜花 · 发表于 2005-4-20 22:57:38

要用二维的区间套定理，揉皱的地图可能还是需要比例小一点的，否则无限交可能是一个区域，还是不能证明至少有一个不动点。

fzy · 发表于 2005-4-21 01:14:59

When there is no compression for the function, compactness alone cannot guarantee a fixed point. We also need simple connectivity of the region. So any solution would have to explicitly or implicitly use some topology. BTW, do we know the maps are simply connected, no hole in the middle?

野菜花 · 发表于 2005-4-21 01:38:22

If there is a hole in the middle of the map, the one on the top could be put within the hole, the conclusion, of course, fails!

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