作者changkh (月光华华)
看板ck47th320
标题Re: [问题] 数学问题
时间Mon Mar 15 21:55:39 2004
※ 引述《changkh (月光华华)》之铭言：
: 最近我朋友碰上了一题数学题不知道怎麽解，不知道有没有人学过呢？
: In the zipper model for the helix-coil transition, the partition
: function Q is give by:
: N
: Q= 1 + sigma * sum (N - i + 1) s^i
: i=1
: sum is summation.
: Where N is the number of residues in the chain.
: a) using the fact that
: N
: sum s^i = (s^(N+1) - S) / s - 1
: i=1
: show that the partition function in zipper model can be
: evaluated to give:
: Q= 1 + (sigma * s) * (s^(N+1) - (N+1)s + N) / (s - 1)^2
: b) show that the average fraction of helical residue, theta,
: can be obtained from
: theta = 1 / N * (ln Q对ln s的偏微分)
: 谢谢啦。
其实後来我也算出了a)
这题重点是
N
sum i*s^i怎麽算
i=1
我的算法是设
N N s^(N+1) - s
I=sum i*s^i 又令 J=sum s^i = -------------
i=1 i=1 s - 1
I = s + 2s^2 + 3s^3 + ... + (N-1)s^(N-1) + Ns^N
J= s + s^2 + s^3 + ... + S^(N-1) + s^N
则I-J= 0 + s^2 + 2s^3 + ... + (N-2)s^(N-1) + (N-1)s^N = I*s-Ns^(N+1)
所以I-J= Is - Ns^(N+1)
化简可得I
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