93 research outputs found
Embedding Four-directional Paths on Convex Point Sets
A directed path whose edges are assigned labels "up", "down", "right", or
"left" is called \emph{four-directional}, and \emph{three-directional} if at
most three out of the four labels are used. A \emph{direction-consistent
embedding} of an \mbox{-vertex} four-directional path on a set of
points in the plane is a straight-line drawing of where each vertex of
is mapped to a distinct point of and every edge points to the direction
specified by its label. We study planar direction-consistent embeddings of
three- and four-directional paths and provide a complete picture of the problem
for convex point sets.Comment: 11 pages, full conference version including all proof
Hamiltonicity of 3-arc graphs
An arc of a graph is an oriented edge and a 3-arc is a 4-tuple of
vertices such that both and are paths of length two. The
3-arc graph of a graph is defined to have vertices the arcs of such
that two arcs are adjacent if and only if is a 3-arc of
. In this paper we prove that any connected 3-arc graph is Hamiltonian, and
all iterative 3-arc graphs of any connected graph of minimum degree at least
three are Hamiltonian. As a consequence we obtain that if a vertex-transitive
graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of
degree at least three, then it is Hamiltonian. This confirms the well known
conjecture, that all vertex-transitive graphs with finitely many exceptions are
Hamiltonian, for a large family of vertex-transitive graphs. We also prove that
if a graph with at least four vertices is Hamilton-connected, then so are its
iterative 3-arc graphs.Comment: in press Graphs and Combinatorics, 201
Shift invariant preduals of ℓ<sub>1</sub>(ℤ)
The Banach space ℓ<sub>1</sub>(ℤ) admits many non-isomorphic preduals, for
example, C(K) for any compact countable space K, along with many more
exotic Banach spaces. In this paper, we impose an extra condition: the predual
must make the bilateral shift on ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is
equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ)
separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach
algebra. We call such preduals <i>shift-invariant</i>. It is known that the
only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K)
(for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide
an explicit construction of an uncountable family of distinct preduals which do
make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we
show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. We then
build some theory to study such preduals, showing that they arise from certain
semigroup compactifications of ℤ. This allows us to produce a large number
of other examples, including non-isometric preduals, and preduals which are not
Banach space isomorphic to c<sub>0</sub>
A comparison of variational and Markov chain Monte Carlo methods for inference in partially observed stochastic dynamic systems
In recent work we have developed a novel variational inference method for partially observed systems governed by stochastic differential equations. In this paper we provide a comparison of the Variational Gaussian Process Smoother with an exact solution computed using a Hybrid Monte Carlo approach to path sampling, applied to a stochastic double well potential model. It is demonstrated that the variational smoother provides us a very accurate estimate of mean path while conditional variance is slightly underestimated. We conclude with some remarks as to the advantages and disadvantages of the variational smoother. Ā© 2008 Springer Science + Business Media LLC
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