Discreteness of interior transmission eigenvalues revisited

Hoài-Minh Nguyên, Quoc Hung Nguyen
Springer Verlag, 2017
Article

Résumé

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary.

Official source

https://infoscience.epfl.ch/record/227923?ln=fr

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés

Chargement

Publications associées

Chargement

Fast continuous Fourier and Haar transforms of rectilinear polygons from very-large-scale integration layouts

Amina Chebira, Paul Hurley, Robin Scheibler

We propose two new fast algorithms for the computation of the continuous Fourier series and the continuous Haar transform of rectilinear polygons such as those of mask layouts in optical lithography. These algorithms outperform their discrete counterparts traditionally used. Not only are continuous transforms closer to the underlying continuous physical reality, but they also avoid the inherent inaccuracies introduced by the sampling or rasterization of the polygons in the discrete case. Moreover, massive amounts of data and the intense processing methods used in lithography require efficient algorithms at every step of the process. We derive the complexity of each algorithm and compare it to that of the corresponding discrete transform. For the practical very-large-scale integration (VLSI) layouts, we find significant reduction in the complexity because the number of polygon vertices is substantially smaller than the corresponding discrete image. This analysis is completed by an implementation and a benchmark of the continuous algorithms and their discrete counterparts. We run extensive experiments and show that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25 times faster, while the fast continuous Fourier series is 1.5 to 3 times faster than their discrete counterparts.

2013

Chargement

, ,

Particular intra-aneurysmal blood flow conditions, created naturally by the growth of an aneurysm or induced artificially by implantation of a flow diverter stent (FDS), can potentiate intra-aneurysmal thrombosis. The aim of this study was to identify hemodynamic indicators, relevant to this process, which could be used as a prediction of the success of a preventive endovascular treatment. A cross sectional study on 21 patients was carried out to investigate the possible association between intra-aneurysmal spontaneous thrombus volume and the dome to neck aspect ratio (AR) of the aneurysm. The mechanistic link between these two parameters was further investigated through a Fourier analysis of the intra-aneurysmal shear rate (SR) obtained by computational fluid dynamics (CFD). This analysis was first applied to 10 additional patients (4 with and 6 without spontaneous thrombosis) and later to 3 patients whose intracranial aneurysms only thrombosed after FDS implantation. The cross sectional study revealed an association between intra-aneurysmal spontaneous thrombus volume and the AR of the aneurysm (R(2)=0.67, p

2015

Chargement

Fourier uniqueness and interpolation in Euclidean space

Martin Peter Stoller

We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only Schwartz function which, together with its Fourier transform, vanishes on these spheres, is the zero function. We show that this remains true if we replace the spheres by surfaces or discrete sets of points which are sufficiently small perturbations of these spheres. In a complementary, opposite direction, we construct infinite dimensional spaces of Fourier eigenfunctions vanishing on on certain discrete subsets of those spheres. The proofs combine harmonic analysis, the theory of modular forms and algebraic number theory.

EPFL2022

Chargement