空间中的球面

fzy · 发表于 2005-10-11 21:31:05

难度：++到+++

空间中N个球面最多能把空间分成多少部分？

野菜花 · 发表于 2005-10-11 22:15:20

f(N)=1/3*N*(N^2-3N+8)

fzy · 发表于 2005-10-11 23:00:18

No detail,no credit

野菜花 · 发表于 2005-10-11 23:10:25

answer too.

Let F(N)=a*N^3+b*N^2+c*N+d

Use F(1)=2, F(2)=4, F(3)=8, F(4)=16

and undetermine coefficient method, solve the system for a,b,c,d

fzy · 发表于 2005-10-11 23:20:31

You cannot assume it is a polynomial

野菜花 · 发表于 2005-10-11 23:44:30

solution.

乱弹 · 发表于 2005-10-12 03:11:49

Please tell me the thm.. Thanks

野菜花 · 发表于 2005-10-12 04:16:58

I don't remember what the theorem exactly says, I only got this conclusion from it (but don't take it seriously, maybe my impression is not accurate) :

For a k-dim space, let F(N) be the max number of parts that N (k-1)-dim spheres divide the k-dim space into, then F(N) is a polynomial of N and F(N)=O(N^k).

In fact, it is true when k=1,2,3

乱弹 · 发表于 2005-10-12 04:35:29

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A046127

Formula:   a(n)=n(n^2-3n+8)/3 (n>0).
           n hyperspheres divide R^k into at most C(n-1,k) + Sum_{i=0..k} C(n,i)
              regions.

husonghu · 发表于 2005-11-10 03:16:35

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空间中的球面

回复：空间中的球面

No detail,no credit

I learn from you guys, you sometimes only give

You cannot assume it is a polynomial [:((][:X]

I know it from a theorem. I like to see your nicer

Please tell me the thm.. Thanks [@};-][@};-][@};-]

回复：Please tell me the thm.. Thanks [@};-][@};-][@}

查了一下，没查到权威的，不过你说的是对的。[>:D<]

此题标准答案已由fzy给出。有兴趣者请见坛上另贴。

动态微博

空间中的球面

回复：空间中的球面

No detail,no credit

I learn from you guys, you sometimes only give

You cannot assume it is a polynomial [:((][:X]

I know it from a theorem. I like to see your nicer

Please tell me the thm.. Thanks [@};-][@};-][@};-]

回复：Please tell me the thm.. Thanks [@};-][@};-][@}

查了一下， 没查到权威的， 不过你说的是对的。[>:D<]

此题标准答案已由fzy给出。有兴趣者请见坛上另贴。

查了一下，没查到权威的，不过你说的是对的。[>:D<]