2,757 research outputs found
Gender and School Achievement in the Caribbean
Teaching/Communication/Extension/Profession,
On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs
We prove an arithmetic removal result for all compact abelian groups,
generalizing a finitary removal result of Kr\'al', Serra and the third author.
To this end, we consider infinite measurable hypergraphs that are invariant
under certain group actions, and for these hypergraphs we prove a
symmetry-preserving removal lemma, which extends a finitary result of the same
name by the second author. We deduce our arithmetic removal result by applying
this lemma to a specific type of invariant measurable hypergraph. As a direct
application, we obtain the following generalization of Szemer\'edi's theorem:
for any compact abelian group , any measurable set with Haar
probability satisfies
where the constant is valid uniformly for all . This
result is shown to hold more generally for any translation-invariant system of
linear equations given by an integer matrix with coprime
minors.Comment: 36 pages. Minor changes. To appear in Annals of Combinatoric
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Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure
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