37 research outputs found
"Magic" numbers in Smale's 7th problem
Smale's 7-th problem concerns N-point configurations on the 2-dim sphere
which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the
pairs in a configuration; here, r is the chordal distance between the points
forming a pair. More generally, V_0(r) may be replaced by the standardized
Riesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to
0, and the sphere may be replaced by other compact manifolds. This paper
inquires into the concavity of the map from the integers N>1 into the minimal
average standardized Riesz pair-energies v_s(N) of the N-point configurations
on the 2-sphere for various real s. It is known that v_s(N) is strictly
increasing for each real s, and for s<2 also bounded above, hence "overall
concave." It is (easily) proved that v_{-2}(N) is even locally strictly
concave, and that so is v_s(2n) for s<-2. By analyzing computer-experimental
data of putatively minimal average Riesz pair-energies v_s^x(N) for s in
{-1,0,1,2,3} and N in {2,...,200}, it is found that {v}_{-1}^x(N) is locally
strictly concave, while v_s^x(N) is not always locally strictly concave for s
in {0,1,2,3}: concavity defects occur whenever N in C^{x}_+(s) (an s-specific
empirical set of integers). It is found that the empirical map C^{x}_+(s), with
s in {-2,-1,0,1,2,3}, is set-theoretically increasing; moreover, the percentage
of odd numbers in C^{x}_+(s), s in {0,1,2,3}, is found to increase with s. The
integers in C^{x}_+(0) are few and far between, forming a curious sequence of
numbers, reminiscent of the "magic numbers" in nuclear physics. It is
conjectured that the "magic numbers" in Smale's 7-th problem are associated
with optimally symmetric optimal-energy configurations.Comment: 109 pages, of which 30 are numerical data tables. Thoroughly revised
version, to appear in J. Stat. Phys. under the different title: `Optimal N
point configurations on the sphere: "Magic" numbers and Smale's 7th problem