20,579 research outputs found
New mathematical structures in renormalizable quantum field theories
Computations in renormalizable perturbative quantum field theories reveal
mathematical structures which go way beyond the formal structure which is
usually taken as underlying quantum field theory. We review these new
structures and the role they can play in future developments.Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos
correcte
Cauchy problem for integrable discrete equations on quad-graphs
Initial value problems for the integrable discrete equations on quad-graphs
are investigated. A geometric criterion of the well-posedness of such a problem
is found. The effects of the interaction of the solutions with the localized
defects in the regular square lattice are discussed for the discrete potential
KdV and linear wave equations. The examples of kinks and solitons on various
quad-graphs, including quasiperiodic tilings, are presented.Comment: Corrected version with the assumption of nonsingularity of solutions
explicitly state
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
Local Difference Measures between Complex Networks for Dynamical System Model Evaluation
Acknowledgments We thank Reik V. Donner for inspiring suggestions that initialized the work presented herein. Jan H. Feldhoff is credited for providing us with the STARS simulation data and for his contributions to fruitful discussions. Comments by the anonymous reviewers are gratefully acknowledged as they led to substantial improvements of the manuscript.Peer reviewedPublisher PD
An efficient and principled method for detecting communities in networks
A fundamental problem in the analysis of network data is the detection of
network communities, groups of densely interconnected nodes, which may be
overlapping or disjoint. Here we describe a method for finding overlapping
communities based on a principled statistical approach using generative network
models. We show how the method can be implemented using a fast, closed-form
expectation-maximization algorithm that allows us to analyze networks of
millions of nodes in reasonable running times. We test the method both on
real-world networks and on synthetic benchmarks and find that it gives results
competitive with previous methods. We also show that the same approach can be
used to extract nonoverlapping community divisions via a relaxation method, and
demonstrate that the algorithm is competitively fast and accurate for the
nonoverlapping problem.Comment: 14 pages, 5 figures, 1 tabl
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