204,439 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
Describability via ubiquity and eutaxy in Diophantine approximation
We present a comprehensive framework for the study of the size and large
intersection properties of sets of limsup type that arise naturally in
Diophantine approximation and multifractal analysis. This setting encompasses
the classical ubiquity techniques, as well as the mass and the large
intersection transference principles, thereby leading to a thorough description
of the properties in terms of Hausdorff measures and large intersection classes
associated with general gauge functions. The sets issued from eutaxic sequences
of points and optimal regular systems may naturally be described within this
framework. The discussed applications include the classical homogeneous and
inhomogeneous approximation, the approximation by algebraic numbers, the
approximation by fractional parts, the study of uniform and Poisson random
coverings, and the multifractal analysis of L{\'e}vy processes.Comment: 94 pages. Notes based on lectures given during the 2012 Program on
Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the
2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring
School in Analysis held at Universite Blaise Pasca
Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density
We investigate Lefschetz thimble structure of the complexified
path-integration in the one-dimensional lattice massive Thirring model with
finite chemical potential. The lattice model is formulated with staggered
fermions and a compact auxiliary vector boson (a link field), and the whole set
of the critical points (the complex saddle points) are sorted out, where each
critical point turns out to be in a one-to-one correspondence with a singular
point of the effective action (or a zero point of the fermion determinant). For
a subset of critical point solutions in the uniform-field subspace, we examine
the upward and downward cycles and the Stokes phenomenon with varying the
chemical potential, and we identify the intersection numbers to determine the
thimbles contributing to the path-integration of the partition function. We
show that the original integration path becomes equivalent to a single
Lefschetz thimble at small and large chemical potentials, while in the
crossover region multi thimbles must contribute to the path integration.
Finally, reducing the model to a uniform field space, we study the relative
importance of multiple thimble contributions and their behavior toward
continuum and low-temperature limits quantitatively, and see how the rapid
crossover behavior is recovered by adding the multi thimble contributions at
low temperatures. Those findings will be useful for performing Monte-Carlo
simulations on the Lefschetz thimbles.Comment: 32 pages, 14 figures (typo etc. corrected
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