45,163 research outputs found
Mediatic graphs
Any medium can be represented as an isometric subgraph of the hypercube, with
each token of the medium represented by a particular equivalence class of arcs
of the subgraph. Such a representation, although useful, is not especially
revealing of the structure of a particular medium. We propose an axiomatic
definition of the concept of a `mediatic graph'. We prove that the graph of any
medium is a mediatic graph. We also show that, for any non-necessarily finite
set S, there exists a bijection from the collection M of all the media on a
given set S (of states) onto the collection G of all the mediatic graphs on S.Comment: Four axioms replaced by two; two references added; Fig.6 correcte
Partition Function Expansion on Region-Graphs and Message-Passing Equations
Disordered and frustrated graphical systems are ubiquitous in physics,
biology, and information science. For models on complete graphs or random
graphs, deep understanding has been achieved through the mean-field replica and
cavity methods. But finite-dimensional `real' systems persist to be very
challenging because of the abundance of short loops and strong local
correlations. A statistical mechanics theory is constructed in this paper for
finite-dimensional models based on the mathematical framework of partition
function expansion and the concept of region-graphs. Rigorous expressions for
the free energy and grand free energy are derived. Message-passing equations on
the region-graph, such as belief-propagation and survey-propagation, are also
derived rigorously.Comment: 10 pages including two figures. New theoretical and numerical results
added. Will be published by JSTAT as a lette
Representing convex geometries by almost-circles
Finite convex geometries are combinatorial structures. It follows from a
recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set
of planar convex polygons such that with respect to geometric
convex hulls is a locally convex geometry and every finite convex geometry can
be represented by restricting the structure of to a finite subset in a
natural way. An \emph{almost-circle of accuracy} is a
differentiable convex simple closed curve in the plane having an inscribed
circle of radius and a circumscribed circle of radius such that
the ratio is at least . % Motivated by Richter and
Rogers' result, we construct a set such that (1) contains
all points of the plane as degenerate singleton circles and all of its
non-singleton members are differentiable convex simple closed planar curves;
(2) with respect to the geometric convex hull operator is a locally
convex geometry; (3) as opposed to , is closed with respect
to non-degenerate affine transformations; and (4) for every (small) positive
and for every finite convex geometry, there are continuum
many pairwise affine-disjoint finite subsets of such that each
consists of almost-circles of accuracy and the convex geometry
in question is represented by restricting the convex hull operator to . The
affine-disjointness of and means that, in addition to , even is disjoint from for every
non-degenerate affine transformation .Comment: 18 pages, 6 figure
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