Presented on October 2, 2017 at 11:00 a.m. in the Klaus Advanced Computing Building, room 1116E.
As for Swendsen--Wang, comparison estimates due to Ullrich @cite_2 @cite_39 imply that its inverse gap is at most that of Glauber dynamics on any graph and at any temperature (see Theorem ); thus for @math on @math it also has @math for all @math and for all @math thanks to duality, and similarly at @math it has @math .
@cite_39 @cite_2
1980404829 1982190141
A new bound (Theorem ) for the duration of the chip-firing game with @math chips on a @math -vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Bj "o rner, Lov ' a sz and Shor. In some cases the improvement is dramatic. For instance: for strongly regular graphs the classic and the new bounds reduce to @math and @math , respectively. For dense regular graphs - @math - the classic and the new bounds reduce to @math and @math , respectively. This is a snapshot of a work in progress, so further results in this vein are in the works. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .