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- R Alexandre, L Desvillettes, C Villani, And B Wennberg
- 2000

Abstract. We study Boltzmann’s collision operator for long-range interactions, i.e. without Grad’s angular cut-off assumption. We establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function, in a precise sense. Our estimate is optimal, and gives a unified treatment of both the linear and the… (More)

- R Alexandre, C Villani
- 2002

We study the Boltzmann equation without Grad's angular cut-oo assumption. We introduce a suitable renormalized formulation , which allows the cross-section to be singular in both the angular and the relative velocity variables. This situation occurs as soon as one is interested in long-range interactions. Together with several new estimates, this enables us… (More)

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially… (More)

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem… (More)

In this work, we continue the study of precise functional properties of a linear operator linked with Boltzmann quadratic operator, started in Part I. This is done for singular cross-sections. In particular, we show Calderon-Zygmund type estimates.

- El-Hadji Samba Diop, Radjesvarane Alexandre, Abdel-Ouahab Boudraa
- IEEE Signal Processing Letters
- 2010

The empirical mode decomposition is a powerful tool for signal processing. Because of its original algorithmic, recent works have contributed to its theoretical framework. Following these works, some mathematical contributions on its comprehension and formalism are provided. In this paper, the so called <i>local mean</i> is computed in such a way that it… (More)

- El-Hadji Samba Diop, Radjesvarane Alexandre, Abdel-Ouahab Boudraa
- 2009 16th IEEE International Conference on Image…
- 2009

In this paper, we provide some theoretical contributions on the study of the 2D empirical mode decomposition (EMD). For doing this, we model the 2D sifting process (SP) in a suitable way, which helps us prove its convergence. Indeed, we prove that the 2D SP converges towards the solution of a fourth order partial differential equation (PDE). We show that 2D… (More)

- El-Hadji Samba Diop, Radjesvarane Alexandre, Lionel Moisan
- Computer Vision and Image Understanding
- 2012

- El-Hadji Samba Diop, Radjesvarane Alexandre, Abdel-Ouahab Boudraa
- 2009 IEEE International Conference on Acoustics…
- 2009

For the first time, a proof of the sifting process (SP) and so the empirical mode decomposition (EMD), is given. For doing this, lower and upper envelopes are modeled in a more convenient way that helps us prove the convergence of the SP towards a solution of a partial differential equation (PDE). We also prove that such a PDE has a unique solution, which… (More)