7,680 research outputs found
Determining full conditional independence by low-order conditioning
A concentration graph associated with a random vector is an undirected graph
where each vertex corresponds to one random variable in the vector. The absence
of an edge between any pair of vertices (or variables) is equivalent to full
conditional independence between these two variables given all the other
variables. In the multivariate Gaussian case, the absence of an edge
corresponds to a zero coefficient in the precision matrix, which is the inverse
of the covariance matrix. It is well known that this concentration graph
represents some of the conditional independencies in the distribution of the
associated random vector. These conditional independencies correspond to the
"separations" or absence of edges in that graph. In this paper we assume that
there are no other independencies present in the probability distribution than
those represented by the graph. This property is called the perfect
Markovianity of the probability distribution with respect to the associated
concentration graph. We prove in this paper that this particular concentration
graph, the one associated with a perfect Markov distribution, can be determined
by only conditioning on a limited number of variables. We demonstrate that this
number is equal to the maximum size of the minimal separators in the
concentration graph.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ193 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the profinite topology of right-angled Artin groups
We give necessary and sufficient conditions on the graph of a right-angled
Artin group that determine whether the group is subgroup separable or not.
Moreover, we investigate the profinite topology of the direct product of two
free groups. We show that the profinite topology of the above group is strongly
connected with the profinite topology of the free group of rank two.Comment: The previous version had an incomplete proof that right-angled Artin
groups are conjugacy separable. A much more general result is proved by
Minasyan in arXiv:0905.128
Partial duals of plane graphs, separability and the graphs of knots
There is a well-known way to describe a link diagram as a (signed) plane
graph, called its Tait graph. This concept was recently extended, providing a
way to associate a set of embedded graphs (or ribbon graphs) to a link diagram.
While every plane graph arises as a Tait graph of a unique link diagram, not
every embedded graph represents a link diagram. Furthermore, although a Tait
graph describes a unique link diagram, the same embedded graph can represent
many different link diagrams. One is then led to ask which embedded graphs
represent link diagrams, and how link diagrams presented by the same embedded
graphs are related to one another. Here we answer these questions by
characterizing the class of embedded graphs that represent link diagrams, and
then using this characterization to find a move that relates all of the link
diagrams that are presented by the same set of embedded graphs.Comment: v2: major change
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