Anderson localization for generic deterministic operators

We consider a class of ensembles of lattice Schrödinger operators with deterministic potentials, including quasi-periodic potentials with Diophantine frequencies, depending upon an infinite number of parameters in an auxiliary measurable space. Using a variant of the Multi-Scale Analysis, we prove Anderson localization for generic ensembles in the strong disorder regime and establish an analog of Minami-type bounds for spectral spacings.     

Positive Lyapunov exponents for a class of ergodic orthogonal polynomials on the unit circle

Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C¹ function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent.     

Rescaled Lévy-Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE(κ) as limits.     

Anderson localization and Lifshits tails for random surface potentials

We consider Schrödinger operators on L²(R^d) with a random potential concentrated near the surface R^d₁×{0}⊂R^d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87-97] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tails relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.     

<Emphasis Type="Italic">e</Emphasis>-Principal numberings

We prove the existence of the computable families of finite sets and general recursive functions with no e-principal numbering. We give a series of examples of e-degrees such that the p-degrees of their computable numberings include no top p-degree.     

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.     

Lower bounds for the Chvátal-Gomory rank in the 0/1 cube

We revisit the method of Chvátal, Cook, and Hartmann to establish lower bounds on the Chvátal-Gomory rank, and develop a simpler method. We provide new families of polytopes in the 0/1 cube with high rank, and we describe a deterministic family achieving a rank of at least (1+1/e)n−1>n. Finally, we show how integrality gaps lead to lower bounds.     

Endpoint Strichartz estimates for the magnetic Schrödinger equation

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n?3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.     

Positive Presentations of Families Relative to <Emphasis Type="Italic">e</Emphasis>-Oracles

We introduce the notion of A-numbering which generalizes the classical notion of numbering. All main attributes of classical numberings are carried over to the objects considered here. The problem is investigated of the existence of positive and decidable computable A-numberings for the natural families of sets e-reducible to a fixed set. We prove that, for every computable A-family containing an inclusion-greatest set, there also exists a positive computable A-numbering. Furthermore, for certain families we construct a decidable (and even single-valued) computable total A-numbering when A is a low set; we also consider a relativization containing all cases of total sets (this in fact corresponds to computability with a usual oracle).     

On the power of lookahead in online lot-sizing

We propose an online algorithm for an economic lot-sizing (ELS) problem with lookahead, which achieves asymptotically optimal worst-case performance for increasing lookahead. Although intuitive, this result is interesting since deterministic algorithms for previously studied online ELS problems have unbounded competitive ratio. We also prove lookahead-dependent lower bounds for deterministic algorithms.     

On localization for the Schrödinger operator with a Poisson random potential

We prove exponential localization for the Schrödinger operator with a Poisson random potential at the bottom of the spectrum in any dimension. We also prove exponential localization in a prescribed interval for all large Poisson densities. In addition, we obtain dynamical localization and finite multiplicity of the eigenvalues. To cite this article: F. Germinet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

About <Emphasis Type="Italic">A</Emphasis>-properties of some elastic three-webs

We consider two families of multidimensional three-webs and prove that these webs are webs E, i.e., in their coordinate loops the elasticity identity holds true. We also show that these webs have A-properties.     

Poisson thickening

Let × be a Poisson point process of intensity λ on the real line. A thickening of it is a (deterministic) measurable function f such that X∪f(X) is a Poisson point process of intensity λ′ where λ′ > λ. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo. We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.     

Rational integrability of trigonometric polynomial potentials on the flat torus

We consider a lattice ? ? ? ⁿ and a trigonometric potential V with frequencies k ∈ ?. We then prove a strong rational integrability condition on V, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions 2 and 3 and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.     

Operator norm localization property of relative hyperbolic group and graph of groups

In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.     

Layer potentials C*-algebras of domains with conical points

To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ?Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.     

Wiener chaos and uniqueness for stochastic transport equation

We prove a uniqueness result for the stochastic transport linear equation (STLE), without any $W^{1,1}$ or BV hypothesis on the coefficient, which is needed for the corresponding deterministic equation. We use Wiener chaos decomposition to pass from the STLE to a deterministic second-order transport equation with uniqueness property.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

Admissible values of one parameter for maximal Sperner families of subsets of the type (<Emphasis Type="Italic">k,k</Emphasis> + 1)

In this paper we generalize one assertion (obtained by us earlier) on admissible values of a certain parameter for partial maximal Sperner families (m. S. f.) of subsets of a finite set of the type (k, k + 1). We also prove that the minimal value of the parameter under consideration for all m. S. f. of the type (k, k + 1), except for two families, is less than \(\left( {_k^{n - 1} } \right)\) ? 1.