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Probabilistic well-posedness for the cubic wave equation

Nicolas Burq, Nikolay Tzvetkov (2014)

Journal of the European Mathematical Society

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...

⊗-product of Markov matrices.

J. P. Lampreia, A. Rica da Silva, J. Sousa Ramos (1988)

Stochastica

In this paper we introduce a ⊗-operation over Markov transition matrices, in the context of subshift of finite type, reproducing symbolic properties of the iterates of the critical point on a one-parameter family of unimodal maps. To the *-product between kneading sequences we associate a ⊗-product between the corresponding Markov matrices.

Progressions arithmétiques dans les nombres premiers

Bernard Host (2004/2005)

Séminaire Bourbaki

Récemment, B. Green et T. Tao ont montré que : l’ensemble des nombres premiers contient des progressions arithmétiques de toutes longueurs répondant ainsi à une question ancienne à la formulation particulièrement simple. La démonstration n’utilise aucune des méthodes “transcendantes” ni aucun des grands théorèmes de la théorie analytique des nombres. Elle est écrite dans un esprit proche de celui de la théorie ergodique, en particulier de celui de la preuve par Furstenberg du théorème de Szemerédi,...

Projectively Anosov flows with differentiable (un)stable foliations

Takeo Noda (2000)

Annales de l'institut Fourier

We consider projectively Anosov flows with differentiable stable and unstable foliations. We characterize the flows on T 2 which can be extended on a neighbourhood of T 2 into a projectively Anosov flow so that T 2 is a compact leaf of the stable foliation. Furthermore, to realize this extension on an arbitrary closed 3-manifold, the topology of this manifold plays an essential role. Thus, we give the classification of projectively Anosov flows on T 3 . In this case, the only flows on T 2 which extend to T 3 ...

Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds

Gloria Marí Beffa (2008)

Annales de l’institut Fourier

In this paper we describe moving frames and differential invariants for curves in two different | 1 | -graded parabolic manifolds G / H , G = O ( p + 1 , q + 1 ) and G = O ( 2 m , 2 m ) , and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in G / H inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in G / H can be reduced...

Prolongational centers and their depths

Boyang Ding, Changming Ding (2016)

Fundamenta Mathematicae

In 1926 Birkhoff defined the center depth, one of the fundamental invariants that characterize the topological structure of a dynamical system. In this paper, we introduce the concepts of prolongational centers and their depths, which lead to a complete family of topological invariants. Some basic properties of the prolongational centers and their depths are established. Also, we construct a dynamical system in which the depth of a prolongational center is a prescribed countable ordinal.

Proof of the Treves theorem on the KdV hierarchy

Leonid Dickey (2005)

Annales de l’institut Fourier

A new, shorter, proof of the Treves theorem on an algebraic criterion for the first integrals of the KdV hierarchy is given, along with an addition to the theorem.

Propagation des singularités pour les opérateurs différentiels de type principal localement résolubles à coefficients analytiques en dimension 2

Paul Godin (1979)

Annales de l'institut Fourier

Sur une variété analytique paracompacte de dimension 2, on considère un opérateur différentiel P à symbole principal p m analytique vérifiant la condition ( 𝒫 ) de Nirenberg et Treves. En ajoutant une nouvelle variable et en utilisant des estimations a priori de type Carleman, on montre qu’il y a propagation des singularités pour P , dans p m - 1 ( 0 ) , le long des feuilles intégrales du système différentiel engendré par les champs hamiltoniens de Re p m et Im p m .

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

François Castella, Guillaume Dujardin (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study the linear Schrödinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are...

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