708 research outputs found
Polygons as Sections of Higher-Dimensional Polytopes
We show that every heptagon is a section of a 3-polytope with 6 vertices. This
implies that every n-gon with n≥7 can be obtained as a section of a
(2+⌊n7⌋)-dimensional polytope with at most ⌈6n7⌉ vertices; and provides a
geometric proof of the fact that every nonnegative n×m matrix of rank 3 has
nonnegative rank not larger than ⌈6min(n,m)7⌉. This result has been
independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122,
2014)
Existence of symmetric central configurations
Central configurations have been of great interest over many years, with the
earliest examples due to Euler and Lagrange. There are numerous results in the
literature demonstrating the existence of central configurations with specific
symmetry properties, using slightly different techniques in each. The aim here
is to describe a uniform approach by adapting to the symmetric case the
well-known variational argument showing the existence of central
configurations. The principal conclusion is that there is a central
configuration for every possible symmetry type, and for any symmetric choice of
masses. Finally the same argument is applied to the class of balanced
configurations introduced by Albouy and Chenciner.Comment: 14 pages, to appear in Cel Mech and Dyn Ast
Quantum automorphism groups of homogeneous graphs
Associated to a finite graph is its quantum automorphism group . The
main problem is to compute the Poincar\'e series of , meaning the series
whose coefficients are multiplicities of 1 into tensor
powers of the fundamental representation. In this paper we find a duality
between certain quantum groups and planar algebras, which leads to a planar
algebra formulation of the problem. Together with some other results, this
gives for all homogeneous graphs having 8 vertices or less.Comment: 30 page
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