My last posting turns out to have been complete cobblers --- just one of those "senior moments", I guess. Apologies to everybody, and congratulations to Fred. Now, as I was saying, what about a proof ... WFL On 8/13/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 8/12/09, Fred W. Helenius <fredh@ix.netcom.com> wrote:
...
After checking the rest, it seems your data is produced by the formula
h^2(q) = 2 - q^2/2 - p^2/2,
where p is given by the formula above.
[It wasn't as easy as I make it look above; I've omitted a lot of pointless thrashing about.]
I already had all the clues, but thrashed about even more woefully.
Since h^2(q) = 0 when q = (+/-) sqrt2, it has a factor 2-q^2; since h^2(q) = h^2(p), by symmetry it also has a factor 2-p^2. And since the plot looks suspiciously like a quartic, it must be h^2(q) = c(2-q^2)(2-p^2), where by inspection c = 1.
And p = 2(1-q)/(2-q), so this may be recast in various ways, such as Fred's above.
Now, what about a civilised proof, I wonder ... WFL