The 3-XORSAT thresholdLe seuil de 3-XORSAT

We prove the existence of a threshold phenomenon regarding the random 3-XORSAT problem (or more generally k-XORSAT). We provide the value of the threshold as the solution of two transcendental equations. These results confirm rigorously those obtained by physicists using the one-step replica symmetry breaking method and thus give for the first time the proof of the validity of this method for a problem of the class of satisfiability problems. To cite this article: O. Dubois, J. Mandler, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 963–966.     

A p-adic proof of Hodge symmetry for threefolds

We give a p-adic proof of Hodge symmetry for smooth and projective varieties of dimension three over the field of complex numbers. To cite this article: K. Joshi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Periodic solutions for systems of forced coupled pendulum-like equations

The existence of periodic solutions for systems of forced pendulum-like equations was studied in the papers by J. A. Marlin (Internat. J. Nonlinear Mech.3 (1968), 439–447) and J. Mawhin (Internat. J. Nonlinear Mech.5 (1970), 335–339). In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N-coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M. Levi, F. C. Hoppensteadt, and W. L. Miranker (Quart. Appl. Math.36 (1978), 167–198).     

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of \(SL(2,\mathbb {R})\). We address a general problem to find explicit formulæ  for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9) and based on the algebraic Fourier transform for generalized Verma modules.The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulæ  of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.     

Facteurs de transfert pour les intégrales de Kloosterman

We compute the transfer factors which relate absolute Klosterman integrals and Kloosterman integrals relative to a quadratic extension. To cite this article: H. Jacquet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

Generalized scattering phases for asymptotically hyperbolic manifolds

We prove asymptotic expansions of generalized scattering phases asssociated to pairs of Laplacians, for a class of noncompact manifolds with infinite volume and negative curvature near infinity. We use one of these expansions to define relative determinants which appear naturally in this context. To cite this article: J.-M. Bouclet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths

We study the asymptotic behavior of the solution of a diffusion problem posed in the union of a cylinder of small diameter and fixed length with another cylinder with much smaller diameter and length. The Dirichlet condition is assumed to hold at both extremities of this domain. Depending on the relative size of the parameters, we show that the boundary condition of the one-dimensional limit problem is a Dirichlet, Fourier or Neumann condition. We also prove a corrector result for every case. To cite this article: J. Casado-D??az et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Limiting laws associated with Brownian motion perturbated by normalized exponential weights

We perturb Brownian motion on the time interval [0,t] by an exponential weight; we show that for a large class of these weights the corresponding probability laws converge weakly as t→∞. To cite this article: B. Roynette et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Modules de Hodge mixtes à croisements normaux

We study algebraic mixed Hodge modules on the (relative) affine space $\text{A}{}^{\mathrm{n}}\text{×S}$ which are of normal crossing type. Our main result (Theorem 4.3) gives an equivalence between the category of Hodge modules of normal crossing type, and the category of hypercubes of admissible variations on S. If the base S is a point, then 4.3 is the Hodge theoretic analogue of the main result of Galligo et al. (Ann. Inst. Fourier 35 (1985) 1–48). To cite this article: J. Wildeshaus, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Les formes de torsion holomorphes du complexe de de RhamThe holomorphic torsion forms of the de Rham complex

In this Note, we announce the vanishing of the holomorphic torsion forms of the relative de Rham complex of an equivariant fibration. To cite this article: J.-M. Bismut, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 243–247.     

Explicit Plancherel formula for the p-adic group GL(n)

We provide an explicit Plancherel formula for the p-adic group GL(n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL(n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Symmetry analysis of differential equations with Mathematica

We will discuss three different methods for finding symmetry solutions based on the Fréchet derivative common to each procedure. The methods discussed are Lie's standard procedure of symmetry analysis, the nonclassical method, and the derivation of potential symmetries. A ferromagnet in a strong external field represented by a nonlinear telegraph equation serves as an example describing the application of all three methods. The symmetry methods discussed are realized in a Mathematica package called MathLie performing all of the required calculations.     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Utilisation des modules de Drinfeld en cryptologie

We present in this Note a new and efficient public-key cryptosystem based on Drinfeld modules. The details will appear elsewhere. To cite this article: R. Gillard et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,I

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A null series with small anti-analytic part

We show that it is possible for an L² function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space H². A consequence in the uniqueness theory is obtained. To cite this article: G. Kozma, A. Olevskiǐ, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Lemme du mouvement rigide infinitésimal en coordonnées lipschitziennes et application aux coques de régularité minimale

We establish a version of the infinitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz. To cite this article: S. Anicic et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Interpolation et espace de Hardy sur l'arbre homogène dyadique : le cas stationnaire

We define a function theory and a Hardy space on the dyadic tree, which extend in a natural way the notion of analytic functions, Cauchy's formula and the case of the Hardy space of the open unit disk. We define Blaschke factors in this setting and prove an homogeneous interpolation theorem in this setting. To cite this article: D. Alpay, D. Volok, C. R. Acad. Sci. Paris, Ser. I 336 (2003).