On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows

We show that if a sequence of Hamiltonian flows has a $C^0$ limit, and if the generating Hamiltonians of the sequence have a limit, then this limit is uniquely determned by the limiting $C^0$ flow. This answers a question by Y.G. Oh.Comment: 11 page

Symplectic capacity and short periodic billiard trajectory

We prove that a bounded domain $\Omega$ in $\R^n$ with smooth boundary has a periodic billiard trajectory with at most $n+1$ bounce times and of length less than $C_n r(\Omega)$, where $C_n$ is a positive constant which depends only on $n$, and $r(\Omega)$ is the supremum of radius of balls in $\Omega$. This result improves the result by C.Viterbo, which asserts that $\Omega$ has a periodic billiard trajectory of length less than C'_n \vol(\Omega)^{1/n}. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.Comment: 32 pages, final version with minor modifications. Published online in Mathematische Zeitschrif

Performance of Linear Field Reconstruction Techniques with Noise and Uncertain Sensor Locations

We consider a wireless sensor network, sampling a bandlimited field, described by a limited number of harmonics. Sensor nodes are irregularly deployed over the area of interest or subject to random motion; in addition sensors measurements are affected by noise. Our goal is to obtain a high quality reconstruction of the field, with the mean square error (MSE) of the estimate as performance metric. In particular, we analytically derive the performance of several reconstruction/estimation techniques based on linear filtering. For each technique, we obtain the MSE, as well as its asymptotic expression in the case where the field number of harmonics and the number of sensors grow to infinity, while their ratio is kept constant. Through numerical simulations, we show the validity of the asymptotic analysis, even for a small number of sensors. We provide some novel guidelines for the design of sensor networks when many parameters, such as field bandwidth, number of sensors, reconstruction quality, sensor motion characteristics, and noise level of the measures, have to be traded off

Deformations of symplectic cohomology and exact Lagrangians in ALE spaces

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin diagrams. The proof relies on symplectic cohomology.Comment: 35 pages, 3 figures, minor changes and corrected typo

Pseudographs and Lax-Oleinik semi-group: a geometric and dynamical interpretation

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow of H, we prove that for t > 0 small enough, \phi-t (E (u)) is an exact Lagrangian Lipschitz graph. This provides a geometric interpretation/explanation of a regularization tool that was introduced by P.~Bernard to prove the existence of C 1,1 subsolutions

Reconstruction of Multidimensional Signals from Irregular Noisy Samples

We focus on a multidimensional field with uncorrelated spectrum, and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.Comment: To appear on IEEE Transactions on Signal Processing, 200

Functors and Computations in Floer homology with Applications Part II

Version revised in 2003The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I (GAFA, Geom. funct. anal. Vol. 9 (1999) 985-1033), others will be given in forthcoming papers. The results in this paper had been announced (with indications of proof) in a talk at the ICM 94 in Zürich. Up to typos, this is the revised version from 2003

Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the spectral sequence considerably expanded. Other minor change