1,414 research outputs found
Disimplicial arcs, transitive vertices, and disimplicial eliminations
In this article we deal with the problems of finding the disimplicial arcs of
a digraph and recognizing some interesting graph classes defined by their
existence. A diclique of a digraph is a pair of sets of vertices such
that is an arc for every and . An arc is
disimplicial when is a diclique. We show that the problem
of finding the disimplicial arcs is equivalent, in terms of time and space
complexity, to that of locating the transitive vertices. As a result, an
efficient algorithm to find the bisimplicial edges of bipartite graphs is
obtained. Then, we develop simple algorithms to build disimplicial elimination
schemes, which can be used to generate bisimplicial elimination schemes for
bipartite graphs. Finally, we study two classes related to perfect disimplicial
elimination digraphs, namely weakly diclique irreducible digraphs and diclique
irreducible digraphs. The former class is associated to finite posets, while
the latter corresponds to dedekind complete finite posets.Comment: 17 pags., 3 fig
Distal and non-distal NIP theories
We study one way in which stable phenomena can exist in an NIP theory. We
start by defining a notion of 'pure instability' that we call 'distality' in
which no such phenomenon occurs. O-minimal theories and the p-adics for example
are distal. Next, we try to understand what happens when distality fails. Given
a type p over a sufficiently saturated model, we extract, in some sense, the
stable part of p and define a notion of stable-independence which is implied by
non-forking and has bounded weight. As an application, we show that the
expansion of a model by traces of externally definable sets from some adequate
indiscernible sequence eliminates quantifiers
Localic completion of uniform spaces
We extend the notion of localic completion of generalised metric spaces by
Steven Vickers to the setting of generalised uniform spaces. A generalised
uniform space (gus) is a set X equipped with a family of generalised metrics on
X, where a generalised metric on X is a map from the product of X to the upper
reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its
generalised uniform structure to a point-free generalised uniform structure.
This point-free structure induces a complete generalised uniform structure on
the set of formal points of the localic completion that gives the standard
completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the
category of locally compact uniform spaces into that of overt locally compact
completely regular formal topologies. Moreover, we give an elementary
characterisation of the cover of the localic completion of a locally compact
uniform space that simplifies the existing characterisation for metric spaces.
These results generalise the corresponding results for metric spaces by Erik
Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is
equivalent to the point-free completion of the uniform formal topology
associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension
Axiom. Some of our results also require Countable Choice.Comment: 39 page
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