187,257 research outputs found
Equivariant semidefinite lifts of regular polygons
Given a polytope P in , we say that P has a positive
semidefinite lift (psd lift) of size d if one can express P as the linear
projection of an affine slice of the positive semidefinite cone
. If a polytope P has symmetry, we can consider equivariant psd
lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest
families of polytopes with interesting symmetries are regular polygons in the
plane, which have played an important role in the study of linear programming
lifts (or extended formulations). In this paper we study equivariant psd lifts
of regular polygons. We first show that the standard Lasserre/sum-of-squares
hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus
yields an equivariant psd lift of size linear in N. In contrast we show that
one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1,
which is exponentially smaller than the psd lift of the sum-of-squares
hierarchy. Our construction relies on finding a sparse sum-of-squares
certificate for the facet-defining inequalities of the regular 2^n-gon, i.e.,
one that only uses a small (logarithmic) number of monomials. Since any
equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the
first example of a polytope with an exponential gap between sizes of
equivariant LP lifts and equivariant psd lifts. Finally we prove that our
construction is essentially optimal by showing that any equivariant psd lift of
the regular N-gon must have size at least logarithmic in N.Comment: 29 page
On the Gromov width of polygon spaces
For generic the space
of --gons in with edges of lengths is a
smooth, symplectic manifold. We investigate its Gromov width and prove that the
expression is the Gromov width of all (smooth) --gon spaces and of
--gon spaces, under some condition on . The same
formula constitutes a lower bound for all (smooth) spaces of --gons.
Moreover, we prove that the Gromov width of is given by the
above expression when is symplectomorphic to
, for any .Comment: 39 pages, 14 figures, to appear on Transformation Group
Systematic analysis of transverse momentum distribution and non-extensive thermodynamics theory
A systematic analysis of transverse momentum distribution of hadrons produced
in ultra-relativistic and collisions is presented. We investigate
the effective temperature and the entropic parameter from the non-extensive
thermodynamic theory of strong interaction. We conclude that the existence of a
limiting effective temperature and of a limiting entropic parameter is in
accordance with experimental data.Comment: 7 pages, XII Hadron Physics Conference, Bento Gon\c{c}alves - Brazi
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
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