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Andronov I.V.,Saint Petersburg State Polytechnic University | Bouche D.P.,CEA DAM Ile-de-France | Durufle M.,CNRS Luminy Institute of Mathematics (IML)
IEEE Transactions on Antennas and Propagation | Year: 2012

An asymptotic formula for the problem of diffraction by a strongly elongated body of revolution is constructed. Its uniform nature with respect to the parameter that characterizes the rate of elongation is demonstrated. The results are in good agreement with numerical simulations. © 2012 IEEE.

Feher L.,WIGNER RCP | Feher L.,University of Szeged | Klimcik C.,CNRS Luminy Institute of Mathematics (IML)
Nuclear Physics B | Year: 2012

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III b system from a quasi-Hamiltonian reduction of the internally fused double SU(n)×SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the III b phase space CP(n-1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the III b system in terms of flat SU(n) connections on the one-holed torus. © 2012 Elsevier B.V.

Troubetzkoy S.,CNRS Luminy Institute of Mathematics (IML)
Journal of Statistical Physics | Year: 2010

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas: the typical Lorentz gas, in the sense of Baire, is recurrent. These Lorentz gases need not be of finite horizon! © 2010 Springer Science+Business Media, LLC.

Girard J.-Y.,CNRS Luminy Institute of Mathematics (IML)
Theoretical Computer Science | Year: 2011

Geometry of Interaction is a transcendental syntax developed in the framework of operator algebras. This fifth installment of the program takes place inside a von Neumann algebra, the hyperfinite factor. It provides a built-in interpretation of cut-elimination as well as an explanation for light, i.e., complexity sensitive, logics. © 2010 Elsevier B.V. All rights reserved.

Squellari R.,CNRS Luminy Institute of Mathematics (IML) | Squellari R.,CNRS Theoretical and High Energy Physics
Nuclear Physics B | Year: 2014

We study the quantum properties at one-loop of the Yang-Baxter σ-models introduced by Klimčík [1,2]. The proof of the one-loop renormalizability is given, the one-loop renormalization flow is investigated and the quantum equivalence is studied. © 2014 The Authors.

Rolland R.,CNRS Luminy Institute of Mathematics (IML)
Cryptography and Communications | Year: 2010

The second weight of the Generalized Reed-Muller code of length q n and order d over the finite field with q elements is now known for d ≤ q and d ≥ (n - 1)(q - 1). In this paper, we determine the second weight for the other values of d which are not multiples of q - 1 plus 1. For the special case d = a(q - 1) + 1 we give an estimate. © 2009 Springer Science + Business Media, LLC.

Kohel D.,CNRS Luminy Institute of Mathematics (IML)
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

We present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time 2M + 5S + 2mc and for addition of points in time 7M + 2S, where M is the cost of multiplication, S the cost of squaring, and mc the cost of multiplication by a constant. By a study of the Kummer curves script K = E/{[±1]}, we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with 4M + 4S + 2mc + mt per bit where mt is multiplication by a constant that depends on P. © Springer-Verlag 2012.

Girard J.-Y.,CNRS Luminy Institute of Mathematics (IML)
Leibniz International Proceedings in Informatics, LIPIcs | Year: 2013

Whether we deal with foundations or computation, logic relates questions and answers, typically formulas and proofs: a very entangled relation due to the abuse of presuppositions. In order to analyse syntax, we should step out from language, which is quite impossible. However, it is enough to step out from meaning: this is why our first lighting of logic is that of answers: it is possible to deal with them as meaningless artifacts assuming two basic states, implicit and explicit. The process of explicitation (a.k.a. normalisation, execution), which aims at making explicit what is only implicit, is fundamentally hazardous. The second light is that of questions whose choice involves a formatting ensuring the convergence of explicitation, i.e., the existence of "normal forms". This formatting can be seen as the emergence of meaning. It is indeed a necessary nuisance; either too laxist or too coercitive, there is no just format. Logic should avoid the pitfall of Prussian, axiomatic, formats by trying to understand which deontic dialogue is hidden behind logical restrictions. The third lighting, certainty deals with the adequation between answers and questions: how do we know that an answer actually matches a question? Apodictic certainty - beyond a reasonable doubt - is out of reach: we can only hope for epidictic, i.e., limited, reasonable, certainty. Under the second light (questions), we see that the format is made of two opposite parts, namely rights and duties, and that logical deduction relies on a strict balance between these two opposite terms, expressed by the identity group "A is A and conversely". The issue of certainty thus becomes the interrogation: "Can we afford the rights of our duties?", © Jean-Yves Girard.

Didier G.,CNRS Luminy Institute of Mathematics (IML)
Bulletin of Mathematical Biology | Year: 2011

We give a formal study of the relationships between the transition cost parameters and the generalized maximum parsimonious reconstructions of unknown (ancestral) binary character states {0, 1} over a phylogenetic tree. As a main result, we show there are two thresholds λ n 1 and λ n 0 generally confounded, associated to each node n of the phylogenetic tree and such that there exists a maximum parsimonious reconstruction associating state 1 to n (resp. state 0 to n) if the ratio "10-cost"/"01-cost" is smaller than λ n 1. We propose a dynamic programming algorithm computing these thresholds in a quadratic time with the size of tree. We briefly illustrate some possible applications of this work over a biological dataset. In particular, the thresholds provide a natural way to quantify the degree of support for states reconstructed as well as to determine what kind of evolutionary assumptions in terms of costs are necessary to a given reconstruction. © Society for Mathematical Biology 2010.

Feher L.,KFKI Research Institute | Feher L.,University of Szeged | Klimcik C.,CNRS Luminy Institute of Mathematics (IML)
Communications in Mathematical Physics | Year: 2011

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of 'free' Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars' proof of the crucial symplectomorphism property of the duality map. © 2010 Springer-Verlag.

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