62 research outputs found
Two-Planar Graphs Are Quasiplanar
It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further show that quasiplanarity is witnessed by a simple topological drawing, that is, any two edges cross at most once and adjacent edges do not cross
Notes on large angle crossing graphs
A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges
in G intersect at an angle of at least a. The concept of right angle crossing
(RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown
that any RAC graph with n vertices has at most 4n-10 edges and that there are
infinitely many values of n for which there exists a RAC graph with n vertices
and 4n-10 edges. In this paper, we give upper and lower bounds for the number
of edges in aAC graphs for all 0 < a < Pi/2
The shuffle Hopf algebra and quasiplanar Wick products
The operator valued distributions which arise in quantum field theory on the
noncommutative Minkowski space can be symbolized by a generalization of chord
diagrams, the dotted chord diagrams. In this framework, the combinatorial
aspects of quasiplanar Wick products are understood in terms of the shuffle
Hopf algebra of dotted chord diagrams, leading to an algebraic characterization
of quasiplanar Wick products as a convolution. Moreover, it is shown that the
distributions do not provide a weight system for universal knot invariants.Comment: 16 pages, prepared for the conference proceedings "Non commutative
Geometry and Physics", Laboratoire de physique th\'eorique d'Orsay, April
23-27, 200
Divergences in QFT on the Noncommutative Minkowski Space with Grosse-Wulkenhaar potential
We study quantum field theory on the two-dimensional Noncommutative Minkoswki
space with a Grosse-Wulkenhaar potential. We explicitly construct the retarded
propagator and show that it is not a tempered distribution. This leads to
problems when trying to define planar products of such distributions, as they
appear in the Yang-Feldman series. At and above the self-dual point, these can
no longer be defined, not even at different points. This shows that we do not
deal with an ordinary ultraviolet divergence.Comment: 8 pages, Contribution to the proceedings of the Corfu Summer
Institute on Elementary Particles and Physics 201
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