2,995 research outputs found
On generalized Kneser hypergraph colorings
In Ziegler (2002), the second author presented a lower bound for the
chromatic numbers of hypergraphs \KG{r}{\pmb s}{\calS}, "generalized
-uniform Kneser hypergraphs with intersection multiplicities ." It
generalized previous lower bounds by Kriz (1992/2000) for the case without intersection multiplicities, and by Sarkaria (1990) for
\calS=\tbinom{[n]}k. Here we discuss subtleties and difficulties that arise
for intersection multiplicities :
1. In the presence of intersection multiplicities, there are two different
versions of a "Kneser hypergraph," depending on whether one admits hypergraph
edges that are multisets rather than sets. We show that the chromatic numbers
are substantially different for the two concepts of hypergraphs. The lower
bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset
version.
2. The reductions to the case of prime in the proofs Sarkaria and by
Ziegler work only if the intersection multiplicities are strictly smaller than
the largest prime factor of . Currently we have no valid proof for the lower
bound result in the other cases.
We also show that all uniform hypergraphs without multiset edges can be
represented as generalized Kneser hypergraphs.Comment: 9 pages; added examples in Section 2; added reference ([11]),
corrected minor typos; to appear in J. Combinatorial Theory, Series
Effect of a polymer additive on heat transport in turbulent Rayleigh-B\'enard convection
Measurements of heat transport, as expressed by the Nusselt number , are
reported for turbulent Rayleigh-B\'enard convection of water containing up to
120 ppm by weight of poly-[ethylene oxide] with a molecular weight of
g/mole. Over the Rayleigh number range 5\times 10^9 \alt Ra
\alt 7 \times 10^{10} is smaller than it is for pure water by up to 10%.Comment: 3 pages, 2 figure
All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions
For Gorenstein quotient spaces , a direct generalization of the
classical McKay correspondence in dimensions would primarily demand
the existence of projective, crepant desingularizations. Since this turned out
to be not always possible, Reid asked about special classes of such quotient
spaces which would satisfy the above property. We prove that the underlying
spaces of all Gorenstein abelian quotient singularities, which are embeddable
as complete intersections of hypersurfaces in an affine space, have
torus-equivariant projective crepant resolutions in all dimensions. We use
techniques from toric and discrete geometry.Comment: revised version of MPI-preprint 97/4, 35 pages, 13 figures,
latex2e-file (preprint.tex), macro packages and eps-file
Competition between Spiral-Defect Chaos and Rolls in Rayleigh-Benard Convection
We present experimental results for pattern formation in Rayleigh-Benard
convection of a fluid with a Prandtl number, Pr~ 4. We find that the
spiral-defect-chaos (SDC) attractor which exists for Pr~1 has become unstable.
Gradually increasing the temperature difference from below to well above its
critical value no longer leads to SDC. A sudden jump of temperature difference
from below to above onset causes convection to grow from thermal fluctuations
and does yield SDC. However, the SDC is a transient; it coarsens and forms a
single cell-filling spiral which then drifts toward the cell wall and
disappears.Comment: 9 pages(RevTeX), 5 jpg figures, To appear as Rapid Communication in
PR
Vertex-Facet Incidences of Unbounded Polyhedra
How much of the combinatorial structure of a pointed polyhedron is contained
in its vertex-facet incidences? Not too much, in general, as we demonstrate by
examples. However, one can tell from the incidence data whether the polyhedron
is bounded. In the case of a polyhedron that is simple and "simplicial," i.e.,
a d-dimensional polyhedron that has d facets through each vertex and d vertices
on each facet, we derive from the structure of the vertex-facet incidence
matrix that the polyhedron is necessarily bounded. In particular, this yields a
characterization of those polyhedra that have circulants as vertex-facet
incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure
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