Nicolas Raymond

Monique Dauge, Yvon Lafranche, Nicolas Raymond

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to 0 (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.

Tommy Löfstedt, Fouad Hadj-Selem, Vincent Guillemot et al.

Regularised canonical correlation analysis was recently extended to more than two sets of variables by the multiblock method Regularised generalised canonical correlation analysis (RGCCA). Further, Sparse GCCA (SGCCA) was proposed to address the issue of variable selection. However, for technical reasons, the variable selection offered by SGCCA was restricted to a covariance link between the blocks (i.e., with $τ=1$). One of the main contributions of this paper is to go beyond the covariance link and to propose an extension of SGCCA for the full RGCCA model (i.e., with $τ\in[0, 1]$). In addition, we propose an extension of SGCCA that exploits structural relationships between variables within blocks. Specifically, we propose an algorithm that allows structured and sparsity-inducing penalties to be included in the RGCCA optimisation problem. The proposed multiblock method is illustrated on a real three-block high-grade glioma data set, where the aim is to predict the location of the brain tumours, and on a simulated data set, where the aim is to illustrate the method's ability to reconstruct the true underlying weight vectors.

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond

The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for any aperture, the eigenvalues accumulate below the thresh-old of the essential spectrum: For a small distance from the essential spectrum, the number of eigenvalues farther from the threshold than this distance behaves like the logarithm of the distance. On the other hand, in the small aperture regime, we provide a two-term asymptotics of the first eigenvalues thanks to a priori localization estimates for the associated eigenfunctions. We prove that these eigenfunctions are localized in the conical cap at a scale of order the cubic root of the aperture angle and that they get into the other part of the layer at a scale involving the logarithm of the aperture angle.