185,891 research outputs found
On products in the coarse shape categories
The paper is devoted to the study of coarse shape of Cartesian products of
topological spaces. If the Cartesian product of two spaces and admits
an HPol-expansion, which is the Cartesian product of HPol-expansions of these
spaces, then is a product in the coarse shape category. As a
consequence, the Cartesian product of two compact Hausdorff spaces is a product
in the coarse shape category. Finally, we show that the shape groups and the
coarse shape groups commute with products under some conditions.Comment: 11 page
Extensions of a result of Elekes and R\'onyai
Many problems in combinatorial geometry can be formulated in terms of curves
or surfaces containing many points of a cartesian product. In 2000, Elekes and
R\'onyai proved that if the graph of a polynomial contains points of an
cartesian product in , then the polynomial
has the form or . They used this to
prove a conjecture of Purdy which states that given two lines in
and points on each line, if the number of distinct distances between pairs
of points, one on each line, is at most , then the lines are parallel or
orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian
product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on
an cartesian product and an asymmetric cartesian
product. We give a proof of a variation of Purdy's conjecture with fewer points
on one of the lines. We finish with a lower bound for our main result in one
dimension higher with asymmetric cartesian product, showing that it is
near-optimal.Comment: 23 page
Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
It is well-known that all finite connected graphs have a unique prime factor
decomposition (PFD) with respect to the strong graph product which can be
computed in polynomial time. Essential for the PFD computation is the
construction of the so-called Cartesian skeleton of the graphs under
investigation.
In this contribution, we show that every connected thin hypergraph H has a
unique prime factorization with respect to the normal and strong (hypergraph)
product. Both products coincide with the usual strong graph product whenever H
is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as
a natural generalization of the Cartesian skeleton of graphs and prove that it
is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian
skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can
be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree
and bounded rank
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