### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Matthew Russell**, Rutgers University, russell2 {at} math [dot] rutgers [dot] edu)
** Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: "Magic" numbers in Smale's 7th problem

Speaker: **Michael Kiessling**, Rutgers University

Date: Thursday, October 17, 2013 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Smale's 7th problem concerns N-point configurations on the
2-sphere which minimize the logarithmic pair-energy V(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(r) may
be replaced by the standardized Riesz pair-energy. In a recent paper
with Brauchart and Nerattini we inquired into the concavity of the map
from the integers >1 into the minimal average standardized Riesz
pair-energies of the N-point configurations on the sphere. It is known
that this map is strictly increasing and in some Riesz parameter range
bounded above, hence ``overall concave.'' It is (easily) proved that
for the Riesz parameter 2 it is even locally strictly concave. By
analyzing computer-experimental data of putatively minimal average
Riesz pair-energies for the Riesz parameters -1,0,1,2,3 and N up to
200 we found that the map in question is locally strictly concave for
parameter -1, while not always locally strictly concave for the other
parameter values. It is found that the empirical map from the Riesz
parameter into the set of convex defect-N is set-theoretically
increasing; moreover, the percentage of odd numbers in the range is
found to increase with the Riesz parameter. They form a curious
sequence of numbers for the logarithmic kernel, reminiscent of the
``magic numbers'' in nuclear physics; it is conjectured that the
``magic numbers'' in Smale's 7th problem are associated with optimally
symmetric optimal-energy configurations. The talk emphasizes the role
of computer experiments, in particular also of Maple, in our
investigation.

See: http://www.math.rutgers.edu/~russell2/expmath/