20,579 research outputs found

    New mathematical structures in renormalizable quantum field theories

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    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

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    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

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    Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<10<p<1 be constant and let GGn,pG\sim G_{n,p}. Let odd(G)odd(G) be the number of odd degree vertices in GG. Then a.a.s. the following hold: (i) GG can be decomposed into Δ(G)/2\lfloor\Delta(G)/2\rfloor cycles and a matching of size odd(G)/2odd(G)/2. (ii) GG can be decomposed into max{odd(G)/2,Δ(G)/2}\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\} paths. (iii) GG can be decomposed into Δ(G)/2\lceil\Delta(G)/2\rceil 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

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    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

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    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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