Publications

2014

• The lexicographic degree of the first two-bridge knots
  
  • Brugallé Erwan
  • Koseleff Pierre-Vincent
  • Pecker Daniel
  , 2014. We study the degree of polynomial representations of knots. We give here the lexicographic degree of all knots with eight or fewer crossings. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, isotopies on trigonal curves and explicit parametrizations obtained by perturbing a triple point.
• The Boltzmann equation over $\mathbf{R}^D$: dispersion versus dissipation
  
  • Golse François
  , 2016, 162. The Boltzmann equation of the kinetic theory of gases involves two competing processes. Dissipation (or entropy production) due to the collisions between gas molecules drives the gas towards local thermodynamic (Maxwellian) equilibrium. If the spatial domain is the Euclidean space $\mathbf{R}^D$, the ballistic transport of gas molecules between collisions results in a dispersion effect which enhances the rarefaction of the gas, and offsets the effect of dissipation. The competition between these two effects leads to a scattering regime for the Boltzmann equation over $\mathbf{R}^D$ with molecular interaction satisfying Grad's angular cutoff assumption. The present paper reports on results in this direction obtained in collaboration with Bardos, Gamba and Levermore [Comm. Math. Phys. 346 (2016), 435–467] and discusses a few open questions related to this work. (10.1007/978-3-319-32144-8_7)
  DOI : 10.1007/978-3-319-32144-8_7
• COEFFICIENTS OF SYLVESTER'S DENUMERANT
  
  • Baldoni V
  • Berline N
  • de Loera Jesús A.
  • Dutra Brandon E.
  • Koeppe Matthias
  • Vergne Michele
  , 2014. For a given sequence α = [α 1 , α 2 , . . . , α N +1 ] of N + 1 positive integers, we consider the combinatorial function E(α)(t) that counts the non-negative integer solutions of the equation α 1 x 1 + α 2 x 2 + · · · + α N x N + α N +1 x N +1 = t, where the right-hand side t is a varying non-negative integer. It is well-known that E(α)(t) is a quasi-polynomial function in the variable t of degree N . In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number k, computes in polyno-mial time the highest k + 1 coefficients of the quasi-polynomial E(α)(t) as step polynomials of t (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for E(α)(t) and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several compu-tational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.
• INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY
  
  • Baldoni V
  • Berline N
  • Koeppe Matthias
  • Vergne M.
  , 2014. We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvi-nok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytope p and a rational subspace L, we integrate a given polyno-mial function h over all lattice slices of the polytope p parallel to the subspace L and sum up the integrals. We first develop an al-gorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomi-als. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory. (10.1112/S0025579312000101)
  DOI : 10.1112/S0025579312000101
• Collisions of vortex filament pairs
  
  • Banica Valeria
  • Faou Erwan
  • Miot Evelyne
  Journal of Nonlinear Science, Springer Verlag, 2014, 24 (6), pp.1263-1284. We consider the problem of collisions of vortex filaments for a model introduced by Klein, Majda and Damodaran, and Zakharov to describe the interaction of almost parallel vortex filaments in three-dimensional fluids. Since the results of Crow examples of collisions are searched as perturbations of antiparallel translating pairs of filaments, with initial perturbations related to the unstable mode of the linearized problem; most results are numerical calculations. In this article we first consider a related model for the evolution of pairs of filaments and we display another type of initial perturbation leading to collision in finite time. Moreover we give numerical evidence that it also leads to collision through the initial model. We finally study the self-similar solutions of the model. (10.1007/s00332-014-9218-5)
  DOI : 10.1007/s00332-014-9218-5
• INTERMEDIATE SUMS ON POLYHEDRA II:BIDEGREE AND POISSON FORMULA
  
  • Baldoni Velleda
  • Berline Nicole
  • de Loera Jesús A.
  • Koeppe Matthias
  • Vergne Michele
  , 2014. Abstract. We continue our study of intermediate sums over polyhedra,interpolating between integrals and discrete sums, whichwere introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to considerthe case of affine cones s+c, where s is an arbitrary real vertex andc is a rational polyhedral cone. For a given rational subspace L,we integrate a given polynomial function h over all lattice slicesof the affine cone s + c parallel to the subspace L and sum up theintegrals. We study these intermediate sums by means of the intermediategenerating functions SL(s+c)(ξ), and expose the bidegreestructure in parameters s and ξ, which was implicitly used in thealgorithms in our papers [Computation of the highest coefficients ofweighted Ehrhart quasi-polynomials of rational polyhedra, Found.Comput. Math. 12 (2012), 435–469] and [Intermediate sums onpolyhedra: Computation and real Ehrhart theory, Mathematika 59(2013), 1–22]. The bidegree structure is key to a new proof for theBaldoni–Berline–Vergne approximation theorem for discrete generatingfunctions [Local Euler–Maclaurin expansion of Barvinokvaluations and Ehrhart coefficients of rational polytopes, Contemp.Math. 452 (2008), 15–33], using the Fourier analysis with respectto the parameter s and a continuity argument. Our study alsoenables a forthcoming paper, in which we study intermediate sumsover multi-parameter families of polytopes.
• On long time dynamics of perturbed KdV equations
  
  • Huang Guan
  , 2014.
• Long-time dynamics of resonant weakly nonlinear CGL equations
  
  • Huang Guan
  , 2014.
• Pseudoholomorphic simple Harnack curves
  
  • Brugallé Erwan
  , 2014.
• On modular k-free sets
  
  • Lambert Victor
  , 2014. Let $n$ and $k$ be integers. A set $A\subset\mathbb{Z}/n\mathbb{Z}$ is $k$-free if for all $x$ in $A$, $kx\notin A$. We determine the maximal cardinality of such a set when $k$ and $n$ are coprime. We also study several particular cases and we propose an efficient algorithm for solving the general case. We finally give the asymptotic behaviour of the minimal size of a $k$-free set in $\left[ 1,n\right]$ which is maximal for inclusion.
• Regularity of some invariant distributions on nice symmetric pairs
  
  • Harinck Pascale
  , 2014.
• Deformations of Extremal Toric Manifolds
  
  • Rollin Yann
  • Tipler Carl
  The Journal of Geometric Analysis, Springer, 2014, 24 (4), pp.1929-1958. (10.1007/s12220-013-9403-z)
  DOI : 10.1007/s12220-013-9403-z
• Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics
  
  • Brenier Yann
  , 2014. There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savaré (for the heat equation), we provide for the initial value problem a concept of generalized ''dissipative'' solutions which always exist globally in time and are unique whenever theyare smooth.
• LOFAR low-band antenna observations of the 3C295 and Bootes fields: source counts and ultra-steep spectrum sources
  
  • van Weeren R. J.
  • Williams W. L.
  • Tasse C.
  • Rottgering H. J. A.
  • Rafferty D. A.
  • van Der Tol S.
  • Heald G.
  • White G. J.
  • Shulevski A.
  • Best P.
  • Intema H. T.
  • Bhatnagar S.
  • Reich W.
  • Steinmetz M.
  • van Velzen S.
  • Ensslin T. A.
  • Prandoni I.
  • de Gasperin F.
  • Jamrozy M.
  • Brunetti G.
  • Jarvis M. J.
  • Mckean J. P.
  • Wise M. W.
  • Ferrari C.
  • Harwood J.
  • Oonk J. B. R.
  • Hoeft M.
  • Kunert-Bajraszewska M.
  • Horellou C.
  • Wucknitz O.
  • Bonafede A.
  • Mohan N. R.
  • Scaife A. M. M.
  • Klockner H. -R.
  • van Bemmel I. M.
  • Merloni A.
  • Chyzy K. T.
  • Engels D.
  • Falcke H.
  • Pandey-Pommier M.
  • Alexov A.
  • Anderson J.
  • Avruch I. M.
  • Beck R.
  • Bell M. E.
  • Bentum M. J.
  • Bernardi G.
  • Breitling F.
  • Broderick J.
  • Brouw W. N.
  • Bruggen M.
  • Butcher H. R.
  • Ciardi B.
  • de Geus E.
  • de Vos M.
  • Deller A.
  • Duscha S.
  • Eisloffel J.
  • Fallows R. A.
  • Frieswijk W.
  • Garrett M. A.
  • Griessmeier Jean-Mathias
  • Gunst A. W.
  • Hamaker J. P.
  • Hassall T. E.
  • Horandel J.
  • van Der Horst A.
  • Iacobelli M.
  • Jackson N. J.
  • Juette E.
  • Kondratiev V. I.
  • Kuniyoshi M.
  • Maat P.
  • Mann G.
  • Mckay-Bukowski D.
  • Mevius M.
  • Morganti R.
  • Munk H.
  • Offringa A. R.
  • Orru E.
  • Paas H.
  • Pandey V. N.
  • Pietka G.
  • Pizzo R.
  • Polatidis A. G.
  • Renting A.
  • Rowlinson A.
  • Schwarz D.
  • Serylak M.
  • Sluman J.
  • Smirnov O.
  • Stappers B. W.
  • Stewart A.
  • Swinbank J.
  • Tagger Michel
  • Tang Y.
  • Thoudam S.
  • Toribio C.
  • Vermeulen R.
  • Vocks C.
  • Zarka P.
  The Astrophysical Journal, American Astronomical Society, 2014, 793 (2), pp.22 pages. We present LOFAR Low Band observations of the Bootes and 3C295 fields. Our images made at 34, 46, and 62 MHz reach noise levels of 12, 8, and 5 mJy beam$^{-1}$, making them the deepest images ever obtained in this frequency range. In total, we detect between 300 and 400 sources in each of these images, covering an area of 17 to 52 deg$^{2}$. From the observations we derive Euclidean-normalized differential source counts. The 62 MHz source counts agree with previous GMRT 153 MHz and VLA 74 MHz differential source counts, scaling with a spectral index of $-0.7$. We find that a spectral index scaling of $-0.5$ is required to match up the LOFAR 34 MHz source counts. This result is also in agreement with source counts from the 38 MHz 8C survey, indicating that the average spectral index of radio sources flattens towards lower frequencies. We also find evidence for spectral flattening using the individual flux measurements of sources between 34 and 1400 MHz and by calculating the spectral index averaged over the source population. To select ultra-steep spectrum ($\alpha < -1.1$) radio sources, that could be associated with massive high redshift radio galaxies, we compute spectral indices between 62 MHz, 153 MHz and 1.4 GHz for sources in the Bo\"otes field. We cross-correlate these radio sources with optical and infrared catalogues and fit the spectral energy distribution to obtain photometric redshifts. We find that most of these ultra-steep spectrum sources are located in the $ 0.7 \lesssim z \lesssim 2.5$ range. (10.1088/0004-637X/793/2/82)
  DOI : 10.1088/0004-637X/793/2/82
• Formes automorphes et voisins de Kneser des réseaux de Niemeier
  
  • Chenevier Gaëtan
  • Lannes Jean
  , 2014. In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal and symplectic groups of small rank. Our guide is the question of determining the number of p-neighborhoods, in the sense of M. Kneser, between two isometry classes of such lattices. We prove a formula for this number, in which occur certain Siegel modular forms of genus 1 and 2. It has several applications, such as the proof of a conjecture of G. Nebe and B. Venkov about the linear span of the higher genus theta series of the Niemeier lattices, the computation of the p-neighborhoods graphs of the Niemeier lattices (the case p = 2 being due to Borcherds), or the proof of a congruence conjectured by G. Harder. Classical arguments reduce the problem to the description of the automorphic representations of a suitable integral form of the euclidean orthogonal group of R^24 which are unramified at each finite prime and trivial at the archimedean prime. The recent results of J. Arthur suggest several new approaches to this type of questions. This is the other main theme that we develop in this memoir. We give a number of other applications, for instance to the classification of Siegel modular cuspforms of weight at most 12 for the full Siegel modular group.
• A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system
  
  • Miot Évelyne
  , 2014.
• Deux applications arithmétiques des travaux d'Arthur
  
  • Taïbi Olivier
  , 2014. Nous proposons deux applications à l'arithmétique des travaux récents de James Arthur sur la classification endoscopique du spectre discret des groupes symplectiques et orthogonaux. La première consiste à ôter une hypothèse d'irréductibilité dans un résultat de Richard Taylor décrivant l'image des conjugaisons complexes par les représentations galoisiennes p-adiques associées aux représentations automorphes cuspidales algébriques régulières essentiellement autoduales pour le groupe GL_{2n+1} sur un corps totalement réel. Nous l'étendons également au cas de GL_{2n}, sous une hypothèse de parité du caractère multiplicatif. Nous utilisons un résultat de déformation p-adique. Plus précisément, nous montrons l'abondance de points correspondant à des représentations galoisiennes (quasi-)irréductibles sur les variétés de Hecke pour les groupes symplectiques et orthogonaux pairs. La classification d'Arthur est utilisée à la fois pour définir les représentations galoisiennes et pour transférer des représentations automorphes autoduales (pas nécessairement cuspidales) de groupes linéaires aux groupes symplectiques et orthogonaux. La deuxième application concerne le calcul explicite de dimensions d'espaces de formes automorphes ou modulaires. Notre contribution principale est un algorithme calculant les intégrales orbitales aux éléments de torsion des groupes classiques p-adiques non ramifiés, pour l'unité de l'algèbre de Hecke non ramifiée. Cela permet le calcul du côté géométrique de la formule des traces d'Arthur, et donc celui de la caractéristique d'Euler du spectre discret en niveau un. La classification d'Arthur permet l'analyse fine de cette caractéristique d'Euler, jusqu'à en déduire les dimensions des espaces de formes automorphes. De là il n'est pas difficile d'apporter une réponse à un problème plus classique: déterminer les dimensions des espaces de formes modulaires de Siegel à valeurs vectorielles.
• The Hilali conjecture for hyperelliptic spaces
  
  • Fernández de Bobadilla Javier
  • Fresán Javier
  • Muñoz Vicente
  • Murillo Aniceto
  , 2014.
• Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules
  
  • Renard David
  , 2014.
• Derivation of a homogenized two-temperature model from the heat equation
  
  • Desvillettes Laurent
  • Golse François
  • Ricci Valeria
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2014, 48 (2014), pp.1583-1613. This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat: Collège de France Seminar vol. 2 (Paris 1979-1980) Res. Notes in Math. vol. 60, pp. 98-138. Pitman, Boston, London, 1982.] (10.1051/m2an/2014011)
  DOI : 10.1051/m2an/2014011
• An averaging theorem for nonlinear Schrödinger equations with small nonlinearities
  
  • Huang Guan
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2014, 34 (9), pp.3555-3574. (10.3934/dcds.2014.34.3555)
  DOI : 10.3934/dcds.2014.34.3555
• A refinement of Izumi's Theorem
  
  • Boucksom S.
  • Favre Charles
  • Jonsson Mattias
  , 2014, pp.55–81. We improve Izumi's inequality, which states that any divisorial valuation v centered at a closed point 0 on an algebraic variety Y is controlled by the order of vanishing at 0. More precisely, as v ranges through valuations that are monomial with respect to coordinates in a fixed birational model X dominating Y, we show that for any regular function f on Y at 0, the function v--> v(f)/\ord_0(f) is uniformly Lipschitz continuous as a function of the weight defining v. As a consequence, the volume of v is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.
• Minimal time for the bilinear control of Schrödinger equations
  
  • Beauchard Karine
  • Coron Jean-Michel
  • Teismann Holger
  Systems and Control Letters, Elsevier, 2014, 71, pp.1-6. We consider a quantum particle in a potential V (x) (x in R^N) subject to a (spatially homogeneous) time-dependent electric field E(t), which plays the role of the control. Under generic assumptions on V , this system is approximately controllable on the L2(R^N;C)-sphere, in su ffiently large times T, as proved by Boscain, Caponigro, Chambrion and Sigalotti. In the present article, we show that this approximate controllability result is false in small time. As a consequence, the result by Boscain et al. is, in some sense, optimal with respect to the control time T. (10.1016/j.sysconle.2014.06.009)
  DOI : 10.1016/j.sysconle.2014.06.009
• Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation
  
  • Brenier Yann
  Communications in Mathematical Physics, Springer Verlag, 2014, 330 (2), pp.757-770. The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of "dissipative solutions", which shares common features with both P.-L. Lions' dissipative solutions to the Euler equations and the concept of "curves of maximal slopes", a la De Giorgi, recently used to study the scalar heat equation in very general metric spaces. (10.1007/s00220-014-1967-3)
  DOI : 10.1007/s00220-014-1967-3
• Controlling the Galois images in one-dimensional families of ℓ-adic representations
  
  • Cadoret Anna
  • Tamagawa Akio
  Journal of Algebra, Elsevier, 2014, 412, pp.189-206. (10.1016/j.jalgebra.2014.04.024)
  DOI : 10.1016/j.jalgebra.2014.04.024