1,378 research outputs found
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
Towards a characterization of convergent sequences of -line graphs
Let and be graphs such that has at least 3 vertices and is
connected. The -line graph of , denoted by , is that graph whose
vertices are the edges of and where two vertices of are adjacent if
they are adjacent in and lie in a common copy of . For each nonnegative
integer , let denote the -th iteration of the -line graph
of . We say that the sequence converges if there exists a
positive integer such that , and for
we set as the set of all graphs whose sequence
converges when . The sets and
have been characterized. To progress towards the characterization of
in general, this paper defines and studies the following property:
a graph is minimally -convergent if but no proper
subgraph of is in . In addition, prove conditions that imply
divergence, and use these results to develop some of the properties of
minimally -convergent graphs.Comment: 11 pages, 11 figure
Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models
The cluster variation method (CVM) is a hierarchy of approximate variational
techniques for discrete (Ising--like) models in equilibrium statistical
mechanics, improving on the mean--field approximation and the Bethe--Peierls
approximation, which can be regarded as the lowest level of the CVM. In recent
years it has been applied both in statistical physics and to inference and
optimization problems formulated in terms of probabilistic graphical models.
The foundations of the CVM are briefly reviewed, and the relations with
similar techniques are discussed. The main properties of the method are
considered, with emphasis on its exactness for particular models and on its
asymptotic properties.
The problem of the minimization of the variational free energy, which arises
in the CVM, is also addressed, and recent results about both provably
convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
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