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R. Montalto - One of the best experts on this subject based on the ideXlab platform.

  • Quasi-periodic standing wave solutions of gravity-capillary water waves
    'American Mathematical Society (AMS)', 2020
    Co-Authors: M. Berti, R. Montalto
    Abstract:

    We prove the existence and the linear stability of small amplitude time quasiperiodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel Set of asymptotically full Lebesgue measure

  • Time quasi-periodic gravity water waves in finite depth
    Inventiones mathematicae, 2018
    Co-Authors: Pietro Baldi, M. Berti, Emanuele Haus, R. Montalto
    Abstract:

    We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x —of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel Set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x -derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

Olivier Finkel - One of the best experts on this subject based on the ideXlab platform.

  • Classical and Effective Descriptive Complexities of omega-Powers
    Annals of Pure and Applied Logic, 2009
    Co-Authors: Olivier Finkel, Dominique Lecomte
    Abstract:

    We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel Set as the range of a closed subSet of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some Sets of dictionaries considered by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232].

  • An omega-power of a context-free language which is Borel above Delta^0_omega
    2008
    Co-Authors: Jacques Duparc, Olivier Finkel
    Abstract:

    We use erasers-like basic operations on words to construct a Set that is both Borel and above Delta^0_omega, built as a Set V^\omega where V is a language of finite words accepted by a pushdown automaton. In particular, this gives a first example of an omega-power of a context free language which is a Borel Set of infinite rank.

  • An omega-Power of a Finitary Language Which is a Borel Set of Infinite Rank
    arXiv: Logic in Computer Science, 2008
    Co-Authors: Olivier Finkel
    Abstract:

    Omega-powers of finitary languages are omega languages in the form V^omega, where V is a finitary language over a finite alphabet X. Since the Set of infinite words over X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers naturally arises and has been raised by Niwinski, by Simonnet, and by Staiger. It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel Sets, and that there exists a context free language L such that L^omega is analytic but not Borel. But the question was still open whether there exists a finitary language V such that V^omega is a Borel Set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose omega-power is Borel of infinite rank.

  • There Exist some Omega-Powers of Any Borel Rank
    2007
    Co-Authors: Dominique Lecomte, Olivier Finkel
    Abstract:

    Omega-powers of finitary languages are languages of infinite words (omega-languages) in the form V^omega, where V is a finitary language over a finite alphabet X. They appear very naturally in the characterizaton of regular or context-free omega-languages. Since the Set of infinite words over a finite alphabet X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It has been recently proved that for each integer n > 0 , there exist some omega-powers of context free languages which are Pi^0_n-complete Borel Sets, that there exists a context free language L such that L^omega is analytic but not Borel, and that there exists a finitary language V such that V^omega is a Borel Set of infinite rank. But it was still unknown which could be the possible infinite Borel ranks of omega-powers. We fill this gap here, proving the following very surprising result which shows that omega-powers exhibit a great topological complexity: for each non-null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers.

  • An ω-Power of a Finitary Language Which is a Borel Set of Infinite Rank
    Fundamenta Informaticae, 2004
    Co-Authors: Olivier Finkel
    Abstract:

    ω-powers of finitary languages are ω-languages in the form V ω, where V is a finitary language over a finite alphabet Sigma. Since the Set Σ ω of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [13], by Simonnet [15], and by Staiger [18]. It has been proved in [14] that for each integer n≥1, there exist some ω-powers of context free languages which are Π n 0-complete Borel Sets, and in [5] that there exists a context free language L such that L ω is analytic but not Borel. But the question was still open whether there exists a finitary language V such that V ω is a Borel Set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose ω-power is Borel of infinite rank.

Pietro Baldi - One of the best experts on this subject based on the ideXlab platform.

  • Time quasi-periodic gravity water waves in finite depth
    Inventiones mathematicae, 2018
    Co-Authors: Pietro Baldi, M. Berti, Emanuele Haus, R. Montalto
    Abstract:

    We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x —of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel Set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x -derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

M. Berti - One of the best experts on this subject based on the ideXlab platform.

  • Quasi-periodic standing wave solutions of gravity-capillary water waves
    'American Mathematical Society (AMS)', 2020
    Co-Authors: M. Berti, R. Montalto
    Abstract:

    We prove the existence and the linear stability of small amplitude time quasiperiodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel Set of asymptotically full Lebesgue measure

  • Time quasi-periodic gravity water waves in finite depth
    Inventiones mathematicae, 2018
    Co-Authors: Pietro Baldi, M. Berti, Emanuele Haus, R. Montalto
    Abstract:

    We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x —of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel Set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x -derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

Yimin Xiao - One of the best experts on this subject based on the ideXlab platform.

  • Packing dimensions of the images of Gaussian random fields
    Statistics & Probability Letters, 2015
    Co-Authors: Yali Du, Dongsheng Wu, Junjie Miao, Yimin Xiao
    Abstract:

    Let X={X(t):t∈RN} be a Gaussian random field with values in Rd and let E⊆RN be a Borel Set. We determine the packing dimension of the image Set X(E) in terms of the packing dimension profiles in the canonical metric ρ of X, which are extensions of the packing dimension profiles of Falconer and Howroyd (1997) and the box-counting dimension profiles of Howroyd (2001).

  • Packing dimension results for anisotropic Gaussian random fields
    Communications on Stochastic Analysis, 2011
    Co-Authors: Anne Estrade, Dongsheng Wu, Yimin Xiao
    Abstract:

    Let $X=\{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by $$X(t) = \big(X_1(t), \ldots, X_d(t)\big), \qquad \forall \ t \in \R^N, $$ where $X_1, \ldots, X_d$ are independent copies of a centered real-valued Gaussian random field $X_0$. We consider the case when $X_0$ is anisotropic and study the packing dimension of the range $X(E)$, where $E\subSeteq \R^N$ is a Borel Set. For this purpose we extend the original notion of packing dimension profile due to Falconer and Howroyd (1997) to the anisotropic metric space $(\R^N, \rho)$, where $\rho(s, t) = \sum_{j=1}^N |s_j - t_j|^{H_j}$ and $(H_1, \ldots, H_N) \in (0, 1)^N$ is a given vector. The extended notion of packing dimension profile is of independent interest.

  • Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields
    Bulletin of The London Mathematical Society, 2009
    Co-Authors: Hermine Biermé, Céline Lacaux, Yimin Xiao
    Abstract:

    Let X = {X(t),t∈ R N } be a Gaussian random field with values in R d defined by X(t )= (X1(t),...,Xd(t)), where X1,...,Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions on X0, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X �1 (F ), where F ⊆ R d is a nonrandom Borel Set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed �

  • Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of Anisotropic Gaussian Random Fields
    Bulletin of the London Mathematical Society, 2009
    Co-Authors: Hermine Biermé, Céline Lacaux, Yimin Xiao
    Abstract:

    Let X be a (N,d)-Gaussian random field. Under certain general conditions, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image under X of a non-random Borel Set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion, the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and operator-scaling Gaussian random fields with stationary increments.