/sci/ - Minimizing...

Name: 4tran 2009-05-30 5:59

A friend asked me this math question (presumably popped out of QFT or something)

Consider a set of N elements: {a_i}
It is known that sum(a_i) = 0
and that sum(a_i²) = 1
What is the minimum of sum(a_i⁴)?

The AM-RMS/power mean inequality tells us that sqrt(sum(a_i⁴)/N) >= sum(a_i²)/N ->
sum(a_i⁴) >= 1/N, with equality iff the a_is are the same.

If N is even, then the a_is just alternate as +-1/sqrt(N)
If N is odd, then that trick doesn't work, since then the sum won't = 0. Therein lies the problem. It seems likely that the answer has (N+1)/2 of the a_is equaling one number, and the remainder equaling something else. However, I haven't been able to prove that this is the optimal solution yet.
I tried brute forcing this for N=3, but after an hr of algebra... the answer is always 1/2, regardless of how I pick the a_is.

Name: Anonymous 2009-05-31 3:24

>>1
For

s = \frac{n-1}{2}

sa + (s+1)b = 0

sa^2 + (s+1)b^2 = 1

, I get

a^2 = \frac{s+1}{2s^2 + s}

b^2 = \frac{s}{2s^2 + 3s + 1}

, the results of which seems to match >>3 (Well, I only tested it for n=5...).
I haven't proved it's optimal, but I think it's the best real solution at least.
Does your function try complex numbers, >>3?

Minimizing...

1 Name: 4tran 2009-05-30 5:59

4 Name: Anonymous 2009-05-31 3:24

Name: 4tran 2009-05-30 5:59

Name: Anonymous 2009-05-31 3:24