385,953 research outputs found
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
Algorithms for detecting dependencies and rigid subsystems for CAD
Geometric constraint systems underly popular Computer Aided Design soft-
ware. Automated approaches for detecting dependencies in a design are critical
for developing robust solvers and providing informative user feedback, and we
provide algorithms for two types of dependencies. First, we give a pebble game
algorithm for detecting generic dependencies. Then, we focus on identifying the
"special positions" of a design in which generically independent constraints
become dependent. We present combinatorial algorithms for identifying subgraphs
associated to factors of a particular polynomial, whose vanishing indicates a
special position and resulting dependency. Further factoring in the Grassmann-
Cayley algebra may allow a geometric interpretation giving conditions (e.g.,
"these two lines being parallel cause a dependency") determining the special
position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract
based on v1
On the Glue Content in Heavy Quarkonia
Starting with two coupled Bethe-Salpeter equations for the quark-antiquark,
and for the quark-glue-antiquark component of the quarkonium, we solve the
bound state equations perturbatively. The resulting admixture of glue can be
partially understood in a semiclassical way, one has, however, to take care of
the different use of time ordered versus retarded Green functions. Subtle
questions concerning the precise definition of the equal time wave function
arise, because the wave function for the Coulomb gluon is discontinuous with
respect to the relative time of the gluon. A striking feature is that a one
loop non abelian graph contributes to the same order as tree graphs, because
the couplings of transverse gluons in the tree graphs are suppressed in the non
relativistic bound state, while the higher order loop graph can couple to
quarks via non suppressed Coulomb gluons. We also calculate the amplitude for
quark and antiquark at zero distance in the quark-glue-antiquark component of
the P-state. This quantity is of importance for annihilation decays of
P-states. It shows a remarkable compensation between the tree graph and the non
abelian loop graph contribution. An extension of our results to include non
perturbative effects is possible.Comment: 15 pages, 8 figure
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