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Berard P.,Institute Fourier | Helffer B.,University Paris - Sud
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2014

Given a bounded open set in Rn (or in a Riemannian manifold), and a partition of ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk( ω) the infimum of maxj λ(ωj) over all k-partitions. A minimal k-partition is a partition that realizes the infimum. Although the analysis of minimal k-partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ(ωj) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition. © 2013 The Author(s) Published by the Royal Society. All rights reserved.

Faure F.,Institute Fourier | Sjostrand J.,University of Burgundy
Communications in Mathematical Physics | Year: 2011

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts. © 2011 Springer-Verlag.

Barkhofen S.,University of Marburg | Barkhofen S.,University of Paderborn | Faure F.,Institute Fourier | Weich T.,University of Marburg
Nonlinearity | Year: 2014

In many non-integrable open systems in physics and mathematics, resonances have been found to be surprisingly ordered along curved lines in the complex plane. In this article we provide a unifying approach to these resonance chains by generalizing dynamical zeta functions. By means of a detailed numerical study we show that these generalized zeta functions explain the mechanism that creates the chains of quantum resonance and classical Ruelle resonances for three-disk systems as well as geometric resonances on Schottky surfaces. We also present a direct system-intrinsic definition of the continuous lines on which the resonances are strung together as a projection of an analytic variety. Additionally, this approach shows that the existence of resonance chains is directly related to a clustering of the classical length spectrum on multiples of a base length. Finally, this link is used to construct new examples where several different structures of resonance chains coexist. © 2014 IOP Publishing Ltd & London Mathematical Society.

Romero A.,University of La Rioja | Sergeraert F.,Institute Fourier
Discrete and Computational Geometry | Year: 2014

This paper is devoted to the Cradle Theorem. It is a recursive description of a discrete vector field on the direct product of simplices Δp × Δq endowed with the standard triangulation. The vector field provides an explicit deformation that is used to establish an algorithm for computing the Bousfield–Kan spectral sequence, more precisely to compute the homotopy groups (Formula presented) for G a 1-reduced simplicial abelian group. © 2014, Springer Science+Business Media New York.

Krcal M.,Charles University | Matousek J.,Charles University | Matousek J.,ETH Zurich | Sergeraert F.,Institute Fourier
Foundations of Computational Mathematics | Year: 2013

In an earlier paper of Čadek, Vokřínek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces. We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg-MacLane space K(ℤ, 1) represented as a simplicial group, can be equipped with polynomial-time homology (this is a polynomial-time version of effective homology considered in previous works of the third author and co-workers). To this end, we construct a suitable discrete vector field, in the sense of Forman's discrete Morse theory, on K(ℤ, 1). The construction is purely combinatorial and it can be understood as a certain procedure for reducing finite sequences of integers, without any reference to topology. The Eilenberg-MacLane spaces are the basic building blocks in a Postnikov system, which is a "layered" representation of a topological space suitable for homotopy-theoretic computations. Employing the result of this paper together with other results on polynomial-time homology, in another paper we obtain, for every fixed k, a polynomial-time algorithm for computing the kth homotopy group π k(X) of a given simply connected space X, as well as the first k stages of a Postnikov system for X, and also a polynomial-time version of the algorithm of Čadek et al. mentioned above. © 2013 SFoCM.

Foucaud F.,University of Bordeaux 1 | Guerrini E.,Institute Fourier | Kovse M.,University of Bordeaux 1 | Naserasr R.,University of Bordeaux 1 | And 2 more authors.
European Journal of Combinatorics | Year: 2011

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A. Lobstein that if a graph on n vertices with at least one edge admits an identifying code, then a minimal identifying code has size at most n-1. They introduced classes of graphs whose smallest identifying code is of size n-1. Few conjectures were formulated to classify the class of all graphs whose minimum identifying code is of size n-1.In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided. © 2011 Elsevier Ltd.

Corwin I.,Courant Institute of Mathematical Sciences | Ferrari P.L.,University of Bonn | Peche S.,Institute Fourier
Journal of Statistical Physics | Year: 2010

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ- and right density ρ+. We study the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of ρ±. We characterize the large time scaling limit of the multipoint fluctuations as a function of the densities ρ± and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices. © 2010 Springer Science+Business Media, LLC.

Mollard M.,Institute Fourier
Journal of Combinatorial Optimization | Year: 2014

Let γ(P m □P n ) be the domination number of the Cartesian product of directed paths P m and P n for m,n≥2. Liu et al. in (J. Comb. Optim. 22(4):651-662, 2011) determined the value of γ(P m □P n ) for arbitrary n and m≤6. In this work we give the exact value of γ(P m □P n ) for any m,n and exhibit dominating sets of minimum cardinality. © 2012 Springer Science+Business Media, LLC.

Jean-Francois A.,Institute Fourier
Nonlinearity | Year: 2012

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back) operator F. Harmonic analysis yields a natural decomposition F = ⊕F j, where j indexes irreducible representation spaces. Using semi-classical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family {F j} when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centred on 0 in the complex plane. © 2012 IOP Publishing Ltd ∧ London Mathematical Society.

We consider a specific family of skew product of partially expanding map on the torus. We study the spectrum of the Ruelle transfer operator and show that in the limit of high frequencies in the neutral direction (this is a semiclassical limit), the spectrum develops a spectral gap, for a generic map. This result has already been obtained by Tsujii (2008 Ergodic Theory Dyn. Syst. 28 291-317). The novelty here is that we use semiclassical analysis which provides a different and quite natural description. We show that the transfer operator is a semiclassical operator with a well-defined 'classical dynamics' on the cotangent space. This classical dynamics has a 'trapped set' which is responsible for the Ruelle resonances spectrum. In particular, we show that the spectral gap is closely related to a specific dynamical property of this trapped set. © 2011 IOP Publishing Ltd & London Mathematical Society.

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