21,139 research outputs found
A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
Convergent expansions for Random Cluster Model with q>0 on infinite graphs
In this paper we extend our previous results on the connectivity functions
and pressure of the Random Cluster Model in the highly subcritical phase and in
the highly supercritical phase, originally proved only on the cubic lattice
, to a much wider class of infinite graphs. In particular, concerning the
subcritical regime, we show that the connectivity functions are analytic and
decay exponentially in any bounded degree graph. In the supercritical phase, we
are able to prove the analyticity of finite connectivity functions in a smaller
class of graphs, namely, bounded degree graphs with the so called minimal
cut-set property and satisfying a (very mild) isoperimetric inequality. On the
other hand we show that the large distances decay of finite connectivity in the
supercritical regime can be polynomially slow depending on the topological
structure of the graph. Analogous analyticity results are obtained for the
pressure of the Random Cluster Model on an infinite graph, but with the further
assumptions of amenability and quasi-transitivity of the graph.Comment: In this new version the introduction has been revised, some
references have been added, and many typos have been corrected. 37 pages, to
appear in Communications on Pure and Applied Analysi
The realization problem for tail correlation functions
For a stochastic process with identical one-dimensional
margins and upper endpoint its tail correlation function
(TCF) is defined through . It is a popular bivariate summary measure
that has been frequently used in the literature in order to assess tail
dependence. In this article, we study its realization problem. We show that the
set of all TCFs on coincides with the set of TCFs stemming from a
subclass of max-stable processes and can be completely characterized by a
system of affine inequalities. Basic closure properties of the set of TCFs and
regularity implications of the continuity of are derived. If is
finite, the set of TCFs on forms a convex polytope of matrices. Several general results reveal its
complex geometric structure. Up to a reduced system of
necessary and sufficient conditions for being a TCF is determined. None of
these conditions will become obsolete as grows.Comment: 42 pages, 7 Table
Generalized modularity matrices
Various modularity matrices appeared in the recent literature on network
analysis and algebraic graph theory. Their purpose is to allow writing as
quadratic forms certain combinatorial functions appearing in the framework of
graph clustering problems. In this paper we put in evidence certain common
traits of various modularity matrices and shed light on their spectral
properties that are at the basis of various theoretical results and practical
spectral-type algorithms for community detection
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