3,621 research outputs found
End compactifications in non-locally-finite graphs
There are different definitions of ends in non-locally-finite graphs which
are all equivalent in the locally finite case. We prove the compactness of the
end-topology that is based on the principle of removing finite sets of vertices
and give a proof of the compactness of the end-topology that is constructed by
the principle of removing finite sets of edges. For the latter case there
exists already a proof in \cite{cartwright93martin}, which only works on graphs
with countably infinite vertex sets and in contrast to which we do not use the
Theorem of Tychonoff. We also construct a new topology of ends that arises from
the principle of removing sets of vertices with finite diameter and give
applications that underline the advantages of this new definition.Comment: 17 pages, to appear in Math. Proc. Cambridge Philos. So
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; additionally, it can be proven that sequences of
approximately Einstein manifolds and sequences of approximately Einstein
conifolds both converge to Einstein conifolds. Consequently, generalized
Euclidean functional integrals based on these conifold histories yield
semiclassical amplitudes for sequences of both manifold and conifold histories
that approach a stationary point of the Einstein action. Therefore sums over
conifold histories provide a useful and self-consistent starting point for
further study of topological effects in quantum gravity. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.
Typical dynamics of plane rational maps with equal degrees
Let be a rational map with
algebraic and topological degrees both equal to . Little is known in
general about the ergodic properties of such maps. We show here, however, that
for an open set of automorphisms , the
perturbed map admits exactly two ergodic measures of maximal entropy
, one of saddle and one of repelling type. Neither measure is supported
in an algebraic curve, and is `fully two dimensional' in the sense
that it does not preserve any singular holomorphic foliation. Absence of an
invariant foliation extends to all outside a countable union of algebraic
subsets. Finally, we illustrate all of our results in a more concrete
particular instance connected with a two dimensional version of the well-known
quadratic Chebyshev map.Comment: Many small changes in accord with referee comments and suggestion
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