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Concept

Théorie spectrale

Résumé

En mathématiques, et plus particulièrement en analyse, une théorie spectrale est une théorie étendant à des opérateurs définis sur des espaces fonctionnels généraux la théorie élémentaire des valeurs propres et des vecteurs propres de matrices. Bien que ces idées viennent au départ du développement de l'algèbre linéaire, elles sont également liées à l'étude des fonctions analytiques, parce que les propriétés spectrales d'un opérateur sont liées à celles de fonctions analytiques sur les valeurs de son spectre. Contexte mathématique Le nom de théorie spectrale fut introduit par David Hilbert dans sa formulation initiale de la théorie des espaces de Hilbert, énoncée en termes de formes quadratiques à une infinité de variables. Le théorème spectral initial était donc conçu comme une généralisation du théorème définissant les axes principaux d'un ellipsoïde, dans le cas d'un espace de dimension infinie. L'applicabilité, en mécanique quantique, de la théorie spectrale à l'explica

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We prove that the Kloosterman sum changes sign infinitely often as runs over squarefree moduli with at most 10 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler and Matomaki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.

Springer Verlag2015

Chargement

A fundamental problem in signal processing is to design computationally efficient algorithms to filter signals. In many applications, the signals to filter lie on a sphere. Meaningful examples of data of this kind are weather data on the Earth, or images of the sky. It is then important to design filtering algorithms that are computationally efficient and capable of exploiting the rotational symmetry of the problem. In these applications, given a continuous signal

f: \mathbb S^2 \rightarrow \mathbb R

on a 2-sphere

\mathbb S^2 \subset \mathbb R^3

, we can only know the vector of its sampled values

\mathbf f \in \mathbb R^N:\ (\mathbf f)_i = f(\mathbf x_i)

in a finite set of points

\mathcal P \subset \mathbb S^2,\quad \mathcal P = \{\mathbf x_i\}_{i=0}^{n-1}

where our sensors are located. Perraudin et al. in \cite{DeepSphere} construct a sparse graph

G

on the vertex set

\mathcal P

and then use a polynomial of the corresponding graph Laplacian matrix

\mathbf L \in \mathbb R^{n\times n}

to perform a computationally efficient -

\mathcal O (n)

- filtering of the sampled signal

\mathbf f

. In order to study how well this algorithm respects the symmetry of the problem - i.e it is equivariant to the rotation group SO(3) - it is important to guarantee that the spectrum of

\mathbf L

and spectrum of the Laplace-Beltrami operator

\Delta_\mathbb S^2

are somewhat ``close''. We study the spectral properties of such graph Laplacian matrix in the special case of \cite{DeepSphere} where the sampling

\mathcal P

is the so called HEALPix sampling (acronym for \textbf Hierarchical \textbf Equal \textbf Area iso\textbf Latitude \textbf {Pix}elization) and we show a way to build a graph

G'

such that the corresponding graph Laplacian matrix

\mathbf L'

shows better spectral properties than the one presented in \cite{DeepSphere}. We investigate other different methods of building the matrix

\mathbf L

better suited to non uniform sampling measures. In particular, we studied the Finite Element Method approximation of the Laplace-Beltrami operator on the sphere, and how FEM filtering relates to graph filtering, showing the importance of non symmetric discrete Laplacians when it comes to non uniform sampling measures. We finish by showing how the graph Laplacian

\mathbf L'

proposed in this work improved the performances of DeepSphere in a well known classification task using different sampling schemes of the sphere, and by comparing the different Discrete Laplacians introduced in this work.

2019

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The completeness of the generalized eigenfunctions and an upper bound for the counting function of the transmission eigenvalue problem for Maxwell equations

Jean Louis-Alexandre Fornerod, Hoài-Minh Nguyên

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreten ess of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.

2021

Chargement

f: \mathbb S^2 \rightarrow \mathbb R

on a 2-sphere

\mathbb S^2 \subset \mathbb R^3

, we can only know the vector of its sampled values

\mathbf f \in \mathbb R^N:\ (\mathbf f)_i = f(\mathbf x_i)

in a finite set of points

\mathcal P \subset \mathbb S^2,\quad \mathcal P = \{\mathbf x_i\}_{i=0}^{n-1}

where our sensors are located. Perraudin et al. in \cite{DeepSphere} construct a sparse graph

G

on the vertex set

\mathcal P

and then use a polynomial of the corresponding graph Laplacian matrix

\mathbf L \in \mathbb R^{n\times n}

to perform a computationally efficient -

\mathcal O (n)

- filtering of the sampled signal

\mathbf f

. In order to study how well this algorithm respects the symmetry of the problem - i.e it is equivariant to the rotation group SO(3) - it is important to guarantee that the spectrum of

\mathbf L

and spectrum of the Laplace-Beltrami operator

\Delta_\mathbb S^2

are somewhat ``close''. We study the spectral properties of such graph Laplacian matrix in the special case of \cite{DeepSphere} where the sampling

\mathcal P

is the so called HEALPix sampling (acronym for \textbf Hierarchical \textbf Equal \textbf Area iso\textbf Latitude \textbf {Pix}elization) and we show a way to build a graph

G'

such that the corresponding graph Laplacian matrix

\mathbf L'

shows better spectral properties than the one presented in \cite{DeepSphere}. We investigate other different methods of building the matrix

\mathbf L

\mathbf L'

2019