作者puberty (飄泊)
看板ck47th320
標題Re: [問題] 數學問題
時間Mon Mar 15 16:31:08 2004
※ 引述《changkh (月光華華)》之銘言：
: 最近我朋友碰上了一題數學題不知道怎麼解，不知道有沒有人學過呢？
: 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對ln s的偏微分)
: 謝謝啦。
(a) N
Q = 1 + sigma * sum (N - i + 1) s^i
i=1
N j
= 1 + sigma * sum sum s^i
j=1 i=1
N s^(j+1) - s
= 1 + sigma * sum -------------
j=1 s - 1
sigma N
= 1 + ------- sum [ s^(j+1) - s ]
s - 1 j=1
sigma s^(N+1) - s
= 1 + ------- [ s * ------------- - Ns ]
s - 1 s - 1
sigma * s
= 1 + ------------- [ s^(N+1) - s - N ( s - 1 ) ]
( s - 1 )^2
sigma * s
= 1 + ------------- [ s^(N+1) - (N+1)*s + N ]
( s - 1 )^2
(b) 不知道定義....:P
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