591 research outputs found
Symplectic capacity and short periodic billiard trajectory
We prove that a bounded domain in with smooth boundary has a
periodic billiard trajectory with at most bounce times and of length less
than , where is a positive constant which depends only on
, and is the supremum of radius of balls in . This
result improves the result by C.Viterbo, which asserts that 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
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
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
Technical Note: Comparing and ranking soil drought indices performance over Europe, through remote-sensing of vegetation
In the past years there have been many attempts to produce and improve global soil-moisture datasets and drought indices. However, comparing and validating these various datasets is not straightforward. Here, interannual variations in drought indices are compared to interannual changes in vegetation, as captured by NDVI. By comparing the correlations of the different indices with NDVI we evaluated which drought index describes most realistically the actual changes in vegetation. Strong correlation between NDVI and the drought indices were found in areas that are classified as warm temperate climate with hot or warm dry summers. In these areas we ranked the PDSI, PSDI-SC, SPI3, and NSM indices, based on the interannual correlation with NDVI, and found that NSM outperformed the rest. Using this best performing index, and the ICA (Independent Component Analysis) technique, we analyzed the response of vegetation to temperature and soil-moisture stresses over Europe
Algebraic lattice constellations: bounds on performance
In this work, we give a bound on performance of any full-diversity lattice constellation constructed from algebraic number fields. We show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain. Furthermore, we discuss constructions, minimum product distance, and bounds for full-diversity complex rotated Z[i]/sup n/-lattices for any dimension n, which avoid the need of component interleaving
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