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    Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II

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    We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849]).Comment: for Part I see arXiv:0712.084

    Zeroth and Second Laws of Thermodynamics Simultaneously Questioned in the Quantum Microworld

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    Several models of quantum open systems are known at present to violate, according to principles of the standard quantum theory of open systems, the second law of thermodynamics. Here, a new and rather trivial model of another type is suggested describing mechanism that violates, according to the same principles, the zeroth and the second laws of thermodynamics simultaneously. Up to a technically minor modification, the model resembles some models already known, solved by standard means, and properly understood. Universal validity of two basic principles of thermodynamics in strictly quantum situations is thus simultaneously called in question.Comment: Extended reference list and acknowledgement

    Simple heteroclinic cycles in R^4

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    In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in R^n the minimal dimension for which such robust heteroclinic cycles can exist is n=3. In this case the list of admissible symmetry groups is short and well-known. The situation is different and more interesting when n=4. In this paper we list all finite groups Gamma such that an open set of smooth Gamma-equivariant dynamical systems in R^4 possess a very simple heteroclinic cycle (a structurally stable heteroclinic cycle satisfying certain additional constraints). This work extends the results which were obtained by Sottocornola in the case when all equilibria in the heteroclinic cycle belong to the same Gamma-orbit (in this case one speaks of homoclinic cycles).Comment: 43 pages; submitted to "Nonlinearity

    Eclipsing Binaries in Open Clusters

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    Detached eclipsing binaries are very useful objects for calibrating theoretical stellar models and checking their predictions. Detached eclipsing binaries in open clusters are particularly important because of the additional constraints on their age and chemical composition from their membership of the cluster. I compile a list containing absolute parameters of well-studied eclipsing binaries in open clusters, and present new observational data on the B-type systems V1481 Cyg and V2263 Cyg which are members of the young open cluster NGC 7128.Comment: 4 pages, 2 colour figures. Poster presentation for IAUS 240 (Binary Stars as Critical Tools and Tests in Contemporary Astrophysics), Prague, August 2006. The poster itself can be dowloaded in ppt and pdf versions from http://www.astro.keele.ac.uk/~jkt/pubs.htm