We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces Z s,r = H s (R) ∩ L 2 (|x| 2r dx), s ∈ R, s 1 and s r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.
We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish persistence properties of the solution flow in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties of solutions to this equation. In particular, we shall establish optimal decay rate for the solutions of this model. RésuméNous étudions le problème de Cauchy associé à l'équation de Benjamin-Ono avec dispersion généralisée. Notre objectif est d'établir les propriétés de persistance de la solution dans des espaces de Sobolev avec poids et d'en déduire quelques propriétés de prolongement unique pour ses solutions. En particulier, nous établirons un taux de décroissance optimal pour les solutions de ce modèle.
In the polytope membership problem, a convex polytope K in R d is given, and the objective is to preprocess K into a data structure so that, given a query point q ∈ R d , it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting and assume that d is a constant. Given an approximation parameter ε > 0, the query can be answered either way if the distance from q to K's boundary is at most ε times K's diameter. Previous solutions to the problem were on the form of a space-time tradeoff, where logarithmic query time demands O(1/ε d−1 ) storage, whereas storage O(1/ε (d−1)/2 ) admits roughly O(1/ε (d−1)/8 ) query time. In this paper, we present a data structure that achieves logarithmic query time with storage of only O(1/ε (d−1)/2 ), which matches the worst-case lower bound on the complexity of any ε-approximating polytope. Our data structure is based on a new technique, a hierarchy of ellipsoids defined as approximations to Macbeath regions.As an application, we obtain major improvements to approximate Euclidean nearest neighbor searching. Notably, the storage needed to answer ε-approximate nearest neighbor queries for a set of n points in O(log n ε ) time is reduced to O(n/ε d/2 ). This halves the exponent in the ε-dependency of the existing space bound of roughly O(n/ε d ), which has stood for 15 years (HarPeled, 2001).
In this work we continue our study initiated in [10] on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in [10] do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in [10] under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.1991 Mathematics Subject Classification. Primary: 35B05. Secondary: 35B60.
Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.Comment: In Proceedings of the 32nd International Symposium Computational Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and Computational Geometr
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