For n a positive integer, let E(n) be the minimal potential energy for any configuration of n points on the unit sphere subject to inverse-square-law repulsion. Must E(n) be algebraic? (That's gotta be true, but I don't see a rigorous argument.). Are there algorithms for computing E(n) (in the sense of finding an algebraic equation that it satisfies, and indicating which real root of the equation is intended)? How fast are these algorithms, and relatedly, what bounds can be given for the degree of E(n)? Presumably this question would be part of a subject called "real algebraic programming" or something like that, but I don't know of any such subject (though I'm aware of Tarski's decision procedure for determining when a system of algebraic equations and inequalities admits a real solution). Jim Propp On Mon, Oct 8, 2012 at 5:06 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Wow -- what an interesting paper! From the 2010 Hyderabad ICM.
Also very easy to read (I'm through about half of it so far).
At < http://arxiv.org/abs/1003.3053 >.
--Dan
Veit wrote:
<< Henry Cohn made an interesting parallel with equiangular line arrangements in complex space:
"These two problems are finely balanced between order and disorder. Any Hadamard matrix or equiangular line configuration must have considerable structure, but in practice they frequently seem to have just enough structure to be tantalizing, without enough to guarantee a clear construction. This contrasts with many of the most symmetrical mathematical objects, which are characterized by their symmetry groups: once you know the full group and the stabilizer of a point, it is often not hard to deduce the structure of the complete object. That seems not to be possible in either of these two problems, and it stands as a challenge to find techniques that can circumvent this difficulty."
from "Order and disorder in energy minimization"
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun