A generalised Gauss circle problem and integrated density of states

Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schroedinger operator.Comment: 17 page

Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8\pi$ for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $2\pi$. This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus 0 in the unit ball with even larger area. The first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they verify a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois--El Soufi--Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.Comment: 30 pages, 5 figures, 1 tabl

Concession, restriction et opposition : l’apport du québécois à la description des connecteurs français

L’examen de quelques connecteurs particuliers du français du Québec dans le domaine des adversatives (concession, restriction, opposition : quand même que, comment que, par exemple, pareil) en comparaison avec les faits du français standard (même si, avoir beau, quand même, quoique) nous amène à constater des ressemblances et des différences importantes. À l’aide de tests syntaxiques et de la compatibilité avec diverses opérations, nous tentons

From Steklov to Neumann via homogenisation

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.Comment: 34 pages, 1 figure. v3: added some references, simplification of proofs and expositio

Sharp isoperimetric upper bounds for planar Steklov eigenvalues

We solve the isoperimetric problem for the first and second nonzero Steklov eigenvalues of planar domains, without any assumption on the number of connected components of the boundary. Our approach uses the known sharp upper bounds for the weighted Neumann eigenvalues, and a homogenisation method allowing to approximate these eigenvalues by the Steklov eigenvalues of appropriately chosen perforated subdomains

Asymptotiques spectrales et géométrie des nombres

Dans cette thèse, nous étudions le spectre du laplacien ainsi que celui d’autres opérateurs qui lui sont associés. Sur une variété riemannienne compacte M fermée, ou possédant un bord et munie de conditions frontières auto-adjointes, le laplacien a un spectre réel, discret 1(M) ≤ 2≤ (M)… ↗∞ ne s’accumulant qu’à l’infini, où les j (M) sont les nombres réels pour lesquels il existe une solution non-triviale à l’équation Δ + = 0. Nous nous sommes particulièrement intéressé au comportement asymptotique de la fonction de compte N(; M) ≔ #{ j (M) ˂ }. Hermann Weyl a démontré en 1911 [80] ce qui s’appelle aujourd’hui la loi de Weyl, N(; M) ω^d/〖(2π)〗_d Vol(M)d/2, où ωd est le volume de la boune unité en dimension d. Nous cherchons à déterminer la taille de R(; M) ≔ N(; M) − ω^d/〖(2π)〗_d Vol(M)d/2. Dans les contextes que nous avons étudiés, nous avons traduit ce problème dans les termes de la géométrie des nombres, i.e. l’étude de l’interaction entre les points de réseaux, par exemple ℤd, et les ensembles convexes. Dans le premier chapitre, nous décrivons précisément les problèmes à l’étude ainsi que les liens qu’ils possèdent avec la géométrie des nombres, et décrivons plus en détails les principales techniques utilisées. Le second chapitre, intitulé On a generalised Gauss circle problem and integrated density of states [54], est le fruit d’une collaboration avec Leonid Parnovski. Nous y étudions le spectre du laplacien sur un produit d’un tore plat et de l’espace euclidien. Dans ce cas le spectre n’est pas discret mais nous étudions une quantité, la densité intégrée des états, qui remplit le rôle de la fonction de compte des valeurs propres et qui suit elle-même une loi de Weyl. Nous obtenons des bornes supérieures et inférieures sur R() dans ce contexte, qui dépendent des dimensions relatives du tore et de l’espace euclidien. Nous obtenons que lorsque la dimension du tore est strictement inférieure à celle de l’espace euclidien, nos bornes inférieures et supérieures sont dumême ordre polynomial.Nous obtenons aussi un développement asymptotique jusqu’à l’ordre constant pour la densité d’états intégrée de l’opérateur de Schrödinger magnétique avec potentiel constant. Le troisième chapitre, intitulé The Steklov spectrum of cuboids [26] provient d’une collaboration avec Alexandre Girouard, Iosif Polterovich et Alessandro Savo. Nous y étudions le problème aux valeurs propres de Steklov sur des cuboïdes en toute dimension. Cet opérateur a été peu étudié sur des domaines dont la frontière n’est pas lisse et nous utilisons le cuboïde comme premier modèle d’un tel cas. Le spectre reste discret et ne s’accumule qu’à l’infini, nous obtenons une loi de Weyl à deux termes ainsi qu’une inégalité isopérimétrique pour la première valeur propre non triviale. Finalement, nous y obtenons aussi que certaines suites de fonctions propres se concentrent asymptotiquement sur des ensembles de mesure nulle, un comportement qu’on appelle la cicatrisation. Dans le dernier chapitre, intitulé Eigenvalue optimisation on flat tori and lattice points in anisotropically expanding domains [53], nous étudions le spectre du laplacien sur des tores plats. Nous obtenons des bornes pour R(;M) dépendant du rayon d’injectivité. Nous utilisons ensuite ces bornes pour démontrer que toute suite de tores plats K maximisant la k-ième valeur propre du laplacien doit dégénérer lorsque la dimension est inférieure à 10. Pour ce faire, nous avons ramené le problème à celui de compter les points de ℤd dans un domaine qui croît de façon anisotrope, généralisant des résultats obtenus par Yuri Kordyukov et Andrei Yakovlev [49].In this thesis, we study the spectrum of the Laplacian and of other related operators. When defined on either a closed compact Riemannian manifold, or a manifold with boundary and self-adjoint boundary conditions, the Laplacian Δ has a real and discrete spectrum 1(M) ≤ 2≤ (M)… ↗∞ accumulating only at ∞ The numbers j (M) are those for which there is a non-trivial solution to the equation Δ + = 0. We are more specifically interested in the asymptotic behaviour of the counting function N(; M) ≔ #{ j (M) ˂ }. Hermann Weyl has shown in 1911 [80] what is now known as Weyl’s law, R(; M) ≔ N(; M) − ω^d/〖(2π)〗_d Vol(M)d/2 as ⟶ ∞ where ωd is the volume of the unit ball in dimension d. We want to determine the size of R(; M) ≔ N(; M) − ω^d/〖(2π)〗_d Vol(M)d/2. In the context at hand, we have translated this problem in terms of the geometry of numbers, the study of the interaction between lattice points, e.g. ℤd and convex sets. In the first chapter, we make a precise description of the problems studied and how they can be linked to the geometry of numbers. Furthermore, we describe in more detail themain techniques that we have used. The second chapter, titled On a generalised Gauss circle problem and integrated density of states [54], has been written in collaboration with Leonid Parnovski. There, we study the spectrum of the Laplacian on a product of a flat torus and Euclidean space. In this case, the spectrum is not discrete. However, we study the integrated density of states, which takes the role of the eigenvalue counting function and also satisfies Weyl’s law. We obtain upper and lower bounds on R() in this context, which depend on the relative dimensions of the flat torus and Euclidean space. When the dimension of the torus is strictly smaller than that of the Euclidean space the upper and lower bound share the same polynomial order. We also obtain an asymptotic expansion up to constant order for the integrated density of states of a magnetic Schrödinger operator with constant potential. The third chapter, titled The Steklov spectrum of cuboids [26] has been written together with Alexandre Girouard, Iosif Polterovich and Alessandro Savo. We study the Steklov spectrum, i.e.the spectrum of the Dirichlet-to-Neumann operator on cuboids of any dimension. Eigenvalue asymptotics for this operator had not been very much studied on domains whose boundaries are not smooth and cuboids provide a first example of such domains. The spectrum is discrete and only accumulates at infinity, and we obtain a two-term Weyl’s law for the Steklov spectrum. We also obtain an isoperimetric inequality for the first non-trivial eigenvalue. Finally, we prove that some sequence of eigenfunctions concentrates along edges, which are subsets of measure zero, a phenomenon named scarring. In the last chapter, titled Eigenvalue optimisation on flat tori and lattice points in anisotropically expanding domains [53], we turn our attention to the spectrum of the Laplacian on flat tori. We obtain bounds on R() depending on the injectivity radius. We then use those bounds to obtain that any sequence of flat tori K maximising the kth eigenvalue of the Laplacian must degenerate when dimension is inferior or equal to 10. To do so, we have stated the problem at hand in terms of counting points of ℤd inside anisotropically expanding domains, generalising results of Yuri Kordyukov and Andrei Yakovlev [49]

Utilisation de bactériophages pour contrôler les populations de Aeromonas salmonicida résistantes aux antibiotiques

La furonculose, causée par la bactérie Aeromonas salmonicida, représente une des principales causes de mortalité chez les salmonidés d’élevage. L’antibiothérapie constitue l’approche la plus largement répandue pour contrer les effets néfastes de cette maladie. Cependant, le développement de bactéries résistantes aux antibiotiques représente un problème de plus en plus préoccupant. La présente recherche a visé à explorer une nouvelle option pour lutter contre la furonculose, soit la possibilité d’utiliser des bactériophages comme moyen de prévention pour contrôler les populations de A. salmonicida. La sensibilité de 19 souches de A. salmonicida, résistantes à aucun, un, deux ou trois antibiotiques, a été évaluée vis‑à‑vis de 12 bactériophages. Les résultats ont montré que les souches de A. salmonicida résistantes aux antibiotiques utilisés dans l’industrie piscicole canadienne sont aussi sensibles à de nombreux bactériophages, tout comme des souches sensibles aux antibiotiques. Il serait donc possible d’envisager un traitement préventif à base de bactériophages pour lutter contre la furonculose chez les salmonidés d’élevage.Aquaculture represents an increasingly important source of food fish worldwide. The aquaculture industry currently produces between 25 and 30% of all seafood for human consumption. In Canada, salmonids (salmon, rainbow trout, arctic char and brook trout) account for the majority of food fish production. Furonculosis involving the bacterium Aeromonas salmonicida is one of the most important infections observed in salmonid farms. An A. salmonicida infection results either in morbidity and mortality with few clinical signs, or in weakened fish with skin ulcers that make them unmarketable for human consumption. The A. salmonicida bacterium uses a number of mechanisms to counteract the natural barrier of the immune system. Bacterial growth is encouraged by an increase in the ambient temperature and in the concentration of organic matter in the water.During recent years, a relationship between therapeutic failures and the development of bacterial resistance to antibiotics has been reported in salmonid farms. This problem is complicated by the fact that only four antibiotics are authorized for the aquaculture industry in Canada. One consequence of this increasing resistance is a renewed interest in alternative therapies and prevention. Bacteriophages (bacterial viruses) may represent one such alternative. In recent decades, interest in bacteriophages as antibacterial agents has been growing in the Americas and in Asia. Some researchers have tried to exploit the potential of bacteriophages to reduce bacterial populations in infections affecting humans, while others have tried to identify uses in veterinary medicine.The overall objective of this research was to explore a new treatment against furonculosis infection based on the use of bacteriophages to inhibit growth of A. salmonicida cells. In this study, we looked at 19 strains of A. salmonicida, resistant to zero, one, two or three antibiotics, and evaluated their sensitivity to 12 bacteriophages. The results showed that the antibiotic-resistant strains were sensitive to as many bacteriophages as were the bacterial strains sensitive to antibiotics. They also showed that all the A. salmonicida strains were sensitive to several bacteriophages and, conversely, that several bacteriophages were effective against all the A. salmonicida strains. It may thus be possible to consider a preventive treatment using bacteriophages to fight against furonculosis in salmonid farms