La présente thèse porte sur les fonctions propres du laplacien et d’opérateurs de Schrödinger en dimension quelconque. Plus précisément, pour une variété (M,g) de dimension d et une fonction V : M → R, on considère les solutions de l’équation suivante: (∆_g + V ) f_λ = λ f_λ . On appelle l’opérateur ∆_g + V un opérateur de Schrödinger et V le potentiel. Le cas le plus simple et le plus étudié est le laplacien (on pose V ≡ 0 sur M ). Si M est compacte et sans bord, alors il existe une suite 0 = λ_0 < λ_1 ≤ λ_2 -> +∞ qui forme le spectre de ∆_g et une suite de fonctions propres f_n qui satisfont à ∆_g f_n = λ_n f_n . Cette propriété est aussi respectée pour beaucoup de potentiels et de variétés. Premièrement, nous avons étudié le nombre de domaines nodaux des fonctions propres quand la valeur propre tend vers l’infini. Les domaines nodaux d’une fonction f sur M sont les composantes connexes de l’ensemble M \f^{−1} (0). Ils nous permettent de mesurer le caractère oscillatoire de f en comptant le nombre de fois où f change de signe. L’objectif principal de la thèse était de généraliser le théorème de Pleijel [52] sur le nombre de domaines nodaux des fonctions propres du laplacien à d’autre opérateurs de Schrödinger. Dans l’article [2], nous avons montré que la borne du théorème de Pleijel s’applique aussi à l’oscillateur harmonique quantique dans R^d . De plus, nous avons remarqué que cette borne pouvait être améliorée en fonction de la forme quadratique qui définit le potentiel. Ensuite, dans l’article [3], nous avons généralisé le résultat obtenu dans [2] à une large classe de potentiels radiaux, incluant des potentiels qui tendent vers zéro à l’infini ou ayant une singularité à l’origine. Cela inclut le potentiel de Coulomb, qui modélise un atome d’hydrogène isolé dans l’espace. Pour ces potentiels, nous considérons les valeurs propres strictement inférieures au spectre essentiel. Nous avons aussi étudié les points critiques des fonctions propres du laplacien. Jusqu’à tout récemment, il y avait seulement une borne inférieure sur le nombre de points critiques pour certaines variétés [36], mais il n’y avait pas de borne supérieure connue. En 2019, Buhovsky, Logunov et Sodin ont construit une métrique sur T^2 et une suite de fonctions propres du laplacien qui ont toutes une infinité de points critiques. Dans l’article [4], nous utilisons une nouvelle méthode pour construire des métriques sur T^2 et S^2 et des fonctions propres pour ces métriques qui ont une infinité de points critiques. De plus, nous montrons que ces métriques peuvent être arbitrairement proches de la métrique plate sur T^2 et de la métrique standard sur S^2 . Ces métriques donnent aussi des contre-exemples à la conjecture de Courant-Hermann sur le nombre de domaines nodaux des combinaisons linéaires de fonctions propres du laplacien. / The theme of this thesis is the study of the eigenfunctions of the Laplacian and Schrödinger operators. Let (M,g) be a manifold and V : M → R. We are looking at solutions of the following equation: (∆_g + V ) f_λ = λ f_λ . The operator ∆_g + V is called a Schrödinger operator and V is called the potential. The simplest and most studied example is the Laplacian (we put V ≡ 0 on M ). If M is compact and without boundary, then there exists a sequence 0 = λ_0 < λ_1 ≤ λ_2 -> +∞ that makes the spectrum of ∆_g and a sequence of eigenfunctions f_n such that ∆_g f_n = λ_n f_n . This decomposition also holds for various potentials and manifolds. Firstly, we studied the nodal domains of the eigenfunctions as the eigenvalues tend to infinity. The nodal domains of a function f on M are the connected components of M \f^{−1} (0). They can be used to understand the oscillatory character of eigenfunctions by counting the number of times that f changes sign. The principal goal of this thesis was to generalize Pleijel’s nodal domain theorem [52] to other Schrödinger operators. In the article [2], we showed that the upper bound in Pleijel’s theorem also holds for the quantum harmonic oscillator. Furthermore, this bound can be improved depending on the quadratic form that defines the potential. Afterwards, in the article [3], we generalized the result from [2] to a large class of radial potentials, including ones that tend to zero at infinity. These include the Coulomb potential, which modelizes the hydrogen atom in free space. We also studied the number of critical points of Laplace eigenfunctions. Until recently, there were only known lower bounds for certain manifolds [36], but no upper bound was known. In 2019, Buhovsky, Logunov and Sodin [18] constructed a metric on T^2 and a sequence of Laplace eigenfunctions which all have infinitely many critical points. In our article [4], we used a different method to create metrics on T^2 and S^2 and Laplace eigenfunctions for these metrics that have infinitely many critical points. Furthermore, these metrics can be taken arbitrarily close to the flat metric on T^2 and the round metric on S^2. These constructions also provide strong counterexamples to the Courant-Hermann conjecture on the number of nodal domains of linear combinations of Laplace eigenfunctions.

L'année 1992 marque un tournant historique dans la répression des infractions routières. Désormais, chaque français dispose d'un permis de conduire affecté d'un capital de points s'amenuisant à chaque infraction. Cette loi, autoproclamée pédagogique, dont il est pourtant difficile de contester le caractère répressif est ambitieuse : éradiquer la mortalité routière ; le résultat est partiellement atteint. Mais passer de 15000 à 4000 en 30 ans a un coût : celui d'une répression toujours plus grande / The year 1992 marks an historic turning point in punishing infractions driving (licences) offences. Today each french citizen have a driving licence with a number of points that increases according to the offences. This autoproclamed pedagogical law, that it is hard contest the repressif and ambitions : eradicate roads deaths ; the target is parially achieved. But with more than 15000 death per 4000 in 30 years will a cost : a greater repression

Orientador: Edmea Cassia Baptista / Resumo: O problema de Despacho Econômico com Ponto de Válvula é um importante problema relacionado aos Sistemas Elétricos de Potência, que pode ser formulado como um problema de otimização não linear, não convexo e não diferenciável, o que dificulta sua resolução através de métodos exatos. Pode-se observar na literatura que diversos métodos heurísticos são propostos para a resolução do mesmo, os quais são eficientes e com um baixo custo computacional. Uma das desvantagens desses métodos é o tamanho do espaço de busca para realizer tais testes. Pesquisas realizadas apontam que, na grande maioria das vezes, os pontos ótimos para o problema de Despacho Econômico com Ponto de Válvula se encontram em pontos nos quais a função modular, presente na formulação do problema, possui valor nulo, ou estão na região destes e tais pontos são denominados de Pontos Singulares. Neste trabalho, com o bjetivo de propor um método heurístico com espaço de busca reduzido, é proposto um método de Busca Tabu direcionada a Pontos Singulares, o qual utiliza o método de Busta Tabu para percorrer os pontos nos quais a função modular se anula. O método se mostra eficiente para problemas de DEPV de 3, 13 e 40 geradores, com valores próximos aos valores ótimos obtidos por métodos determinísticos e com baixo custo computacional. / Abstract: The problem of Economic Load Dispatch with Valve Point (EDVP) is an important problem related to Electric Power Systems, that can be formulated as a non-linear, non-convex and non-differentiable optimization problem, that difficults resolution through deterministic methods. We can observe in the literature that many heuristic methods are proposed for the resolution of the same, being efficient with a low computational cost. One of the advantages of this methods is the size of the search space necessary to perform the tests. Researches points out that, in most cases, the optimal points for the Economic Load Dispatch with Valve Point problem are at points where the modular function present in the problem formulation has zero value, or in the region thereof, these points are called Singular Points. In this work is proposed, with the objective to propose a heuristic method with the search space reducted, a Tabu Search Directed to Singular Point Search, which uses he tatbu search method to the points in which the modular function cancels out. The method is efficient for resolution of Economic Load Dispatch with Valve Point problems of 3, 13 and 40 generators unities, with values close to optimal obtained by deterministic methods values and low computational cost. / Mestre

This work presents an overview of several different methods for construct- ing ultrafilters. The first part contains constructions not needing additional assumptions beyond the usual axioms of Set Theory. K. Kunen's method using independent systems for constructing weak P-points is presented. This is followed by a presentation of its application in topology (the proof of the existence of sixteen topological types due to J. van Mill). Finally a new con- struction due to the author is presented together with a proof of his result, the existence of a seventeenth topological type: ω∗ contains a point which is discretely untouchable, is a limit point of a countable set and the countable sets having it as its limit point form a filter. The second part looks at constructions which use additional combina- torial axioms and/or forcing. J. Ketonen's construction of a P-point and A. R. D. Mathias's construction of a Q-point are presented in the first two sections. The next sections concentrate on strong P-points introduced by C. Laflamme. The first of these contains a proof of a new characterization theorem due jointly to the author, A. Blass and M. Hrušák: An ultrafilter is Canjar if and only if it is a strong P-point. A new proof of Canjar's the- orem on the existence of non-dominating filters (Canjar filters) which uses...

Les gens infectés par la rougeole subissent une suppression immunitaire. Ce mémoire porte sur l’influence de cette caractéristique de la rougeole sur un autre pathogène, ici la grippe. Il s’agit donc de modéliser la réponse immunitaire d’une interaction virus-virus, problème d’une grande pertinence durant la pandémie causée par le SRAS-CoV-2 alors que les interactions de celui-ci avec le virus de l’influenza demeurent à déterminer. Nous avons également que la grippe augmente la production d’autres pathogènes. Un modèle de chaque pathogène va être développé et analysé. On cherchera les points fixes, leurs conditions de stabilité et on observera quelques résultats numériques pour constater leurs évolutions dans le temps. Ensuite un modèle suivant l’évolution des deux pathogènes ayant infecté un individu au même moment sera conçu. Dans ce modèle nous inclurons les interactions d’un pathogène l’un sur l’autre pour déterminer théoriquement les effets chez les individus infectés à la fois par la grippe et la rougeole. On pourra par la suite comparer entre les différentes populations lorsqu’il n’y a aucune interaction et avec les différentes interactions entre les deux pathogènes. / People infected with measles experience immune suppression. This work focuses on the influence of this characteristic of measles on another pathogen, here the flu. We also have that the flu will increase the production of other pathogens in a co-infection model. Modeling the immune response to such virus-virus interaction is currently of signficant relevance, given the limited knowledge on SARS-CoV-2/influenza interactions. A model of each pathogen will be developed and analysed. We will look for the fixed points, conditions for their stability and we will observe some numerical results of their evolution over time. Then a model following the evolution of the two pathogens having simultaneously infected an individual will be designed. In this model we will include the interactions of the pathogens on each other to theoretically determine the effects in individuals infected with both influenza and measles. Then we can compare between the different populations when there is no interaction and with the different interactions between the two pathogens.

This paper presents a numerical model for the evaluation of the actual torque in Gerotor units. The model consists of two major modules: the pre-processor module and the HYGESim module. The preprocessor module consists of the geometric and the mechanical module. The geometric pre-processor module considers the CAD geometry of Gerotor with tolerances as input and it provides as output the geometric features needed to evaluate the rotor loading and the flow features. The mechanical preprocessor module evaluates the forces of interaction at the contact points between the rotors. The flow displaced by the unit is evaluated using a lumped parameter model whereas the lubricating gaps are evaluated by solving the Reynolds Equation. The main novel aspects consist of the evaluation of the frictional losses at various interfaces. An Elasto-Hydrodynamic Lubrication (EHL) approach is used to evaluate the frictional losses at the contact points between the rotors. Tests on a prototype Gerotor unit are performed for the model validation, particularly as pertains to the features of the shaft torque. Additionally, the paper comments on the distribution of the different torque loss contributions associated with the operation of the unit taken as reference.

Durant ces dernières années, les technologies d’acquisition numériques n’ont cessé de se perfectionner, permettant d’obtenir des données d’une qualité toujours plus fine. Néanmoins, le signal acquis reste corrompu par des défauts qui ne peuvent être corrigés matériellement et nécessitent l’utilisation de méthodes de restauration adaptées. J'usqu’au milieu des années 2000, ces approches s’appuyaient uniquement sur un traitement local du signal détérioré. Avec l’amélioration des performances de calcul, le support du filtre a pu être étendu à l’ensemble des données acquises en exploitant leur caractère autosimilaire. Ces approches non locales ont principalement été utilisées pour restaurer des données régulières et structurées telles que des images. Mais dans le cas extrême de données irrégulières et non structurées comme les nuages de points 3D, leur adaptation est peu étudiée à l’heure actuelle. Avec l’augmentation de la quantité de données échangées sur les réseaux de communication, de nouvelles méthodes non locales ont récemment été proposées. Elles utilisent un modèle a priori extrait à partir de grands ensembles d’échantillons pour améliorer la qualité de la restauration. Néanmoins, ce type de méthode reste actuellement trop coûteux en temps et en mémoire. Dans cette thèse, nous proposons, tout d’abord, d’étendre les méthodes non locales aux nuages de points 3D, en définissant une surface de points capable d’exploiter leur caractère autosimilaire. Nous introduisons ensuite une nouvelle structure de données, le CovTree, flexible et générique, capable d’apprendre les distributions d’un grand ensemble d’échantillons avec une capacité de mémoire limitée. Finalement, nous généralisons les méthodes de restauration collaboratives appliquées aux données 2D et 3D, en utilisant notre CovTree pour apprendre un modèle statistique a priori à partir d’un grand ensemble de données. / In recent years, digital technologies allowing to acquire real world objects or scenes have been significantly improved in order to obtain high quality datasets. However, the acquired signal is corrupted by defects which can not be rectified materially and require the use of adapted restoration methods. Until the middle 2000s, these approaches were only based on a local process applyed on the damaged signal. With the improvement of computing performance, the neighborhood used by the filter has been extended to the entire acquired dataset by exploiting their self-similar nature. These non-local approaches have mainly been used to restore regular and structured data such as images. But in the extreme case of irregular and unstructured data as 3D point sets, their adaptation is few investigated at this time. With the increase amount of exchanged data over the communication networks, new non-local methods have recently been proposed. These can improve the quality of the restoration by using an a priori model extracted from large data sets. However, this kind of method is time and memory consuming. In this thesis, we first propose to extend the non-local methods for 3D point sets by defining a surface of points which exploits their self-similar of the point cloud. We then introduce a new flexible and generic data structure, called the CovTree, allowing to learn the distribution of a large set of samples with a limited memory capacity. Finally, we generalize collaborative restoration methods applied to 2D and 3D data by using our CovTree to learn a statistical a priori model from a large dataset.

Au cours des dernières années, nous avons observé le développement de dispositifs connectéset nomades tels que les téléphones mobiles, tablettes ou même les ordinateurs portablespermettant aux gens d’utiliser dans leur quotidien des services géolocalisés qui sont personnalisésd’après leur position. Néanmoins, les services géolocalisés présentent des risques enterme de vie privée qui ne sont pas forcément perçus par les utilisateurs. Dans cette thèse,nous nous intéressons à comprendre les risques en terme de vie privée liés à la disséminationet collection de données de localisation. Dans ce but, les attaques par inférence que nousavons développé sont l’extraction des points d’intérêts, la prédiction de la prochaine localisationainsi que la désanonymisation de traces de mobilité, grâce à un modèle de mobilité quenous avons appelé les chaînes de Markov de mobilité. Ensuite, nous avons établi un classementdes attaques d’inférence dans le contexte de la géolocalisation se basant sur les objectifsde l’adversaire. De plus, nous avons évalué l’impact de certaines mesures d’assainissement àprémunir l’efficacité de certaines attaques par inférence. En fin nous avons élaboré une plateformeappelé GEoPrivacy Enhanced TOolkit (GEPETO) qui permet de tester les attaques parinférences développées. / In recent years, we have observed the development of connected and nomad devices suchas smartphones, tablets or even laptops allowing individuals to use location-based services(LBSs), which personalize the service they offer according to the positions of users, on a dailybasis. Nonetheless, LBSs raise serious privacy issues, which are often not perceived by the endusers. In this thesis, we are interested in the understanding of the privacy risks related to thedissemination and collection of location data. To address this issue, we developed inferenceattacks such as the extraction of points of interest (POI) and their semantics, the predictionof the next location as well as the de-anonymization of mobility traces, based on a mobilitymodel that we have coined as mobility Markov chain. Afterwards, we proposed a classificationof inference attacks in the context of location data based on the objectives of the adversary.In addition, we evaluated the effectiveness of some sanitization measures in limiting the efficiencyof inference attacks. Finally, we have developed a generic platform called GEPETO (forGEoPrivacy Enhancing Toolkit) that can be used to test the developed inference attacks

No description available.

The delivery of IP traffic on the Internet depends on the complex interactions between a set of autonomous systems that exchange routing information about IP prefix destinations utilizing the Border Gateway Protocol (BGP). Autonomous systems are often connected to a route server located at an Internet eXchange Point, which facilitates the administration of BGP peering arrangements for all parties connected to it. One of the most popular open-source implementations of BGP is the FRRouting software suite, making it an important part of the Internet infrastructure. This thesis investigates the performance of FRRouting, configured as a route server, in terms of its capabilities of announcing routing information on the network to a set of peers emulating autonomous systems. The routing information consists of a set of distinct IP prefixes that FRRouting receives from its peers. With various benchmarks of different configurations, we relate the number of received prefixes to the number of prefixes that FRRouting has announced on the network to its peers in a given time span. The output of the thesis is a wide overview of how the performance of FRRouting is impacted by different configurations such as filtering of specific prefixes that are not announced to the peering networks. The obtained results show that there exists a divergence between the number of prefixes that have been received and announced by FRRouting. Specifically, the discrepancy shows that FRRouting, in our benchmarks, is incapable of announcing prefixes at the same rate as it receives these prefixes from its peers. In general, the number of announced prefixes is dependent on how the prefix filter is configured. However, one can question what real-world limitations this may result in. Suggestions for future work include developing a more realistic benchmarking environment that does not rely on emulated peers as well as improving how the routing information is recorded. There also exists a wide variety of other metrics and configurations of FRRouting that may reveal further limitations. / Leveransen av IP-trafik på Internet beror av komplexa interaktioner mellan en uppsättning autonoma system som utbyter dirigeringsinformation med hjälp av Border Gateway Protocol (BGP). Autonoma system är ofta anslutna till en dirigeringsserver belägen vid en Internetknutpunkt vilket underlättar administrationen av BGP-förbindelser mellan de parter som är anslutna till dirigeringsservern. En av de mest populära implementationerna av BGP med öppen källkod är FRRouting vilket gör denna mjukvara till en betydelsefull komponent för Internets infrastruktur. Detta arbete undersöker prestandan av FRRouting konfigurerad som en dirigeringsserver vad gäller dess kapacitet att behandla och via nätverket vidarebefordra dirigeringsinformation till en uppsättning autonoma system. Dirigeringsinformationen består av en samling IP-prefix som FRRouting erhåller från de autonoma systemen. Genom att variera konfigurationen av FRRouting undersöker vi hur antalet mottagna IP-prefix relaterar till den mängd IP-prefix som FRRouting har vidarebefordrat till de autonoma systemen under en given tidsperiod. Arbetet resulterar i en bred genomgång av hur prestandan för FRRouting påverkas av olika konfigurationer såsom filtrering av specifika prefix. De erhållna resultaten visar att antalet vidarebefordrade IP-prefix skiljer sig markant från antalet prefix som dirigeringsservern erhållit från de autonoma systemen. Denna avvikelse visar att FRRouting inte är kapabel att vidarebefordra IP-prefix i samma takt som dessa mottages från de autonoma systemen. I allmänhet beror antalet vidarebefordrade IP-prefix av hur prefixfiltreringen konfigurerats. Det kan dock ifrågasättas vilka verkliga begränsningar detta kan resultera i. Förslag på framtida arbeten inkluderar att utveckla en förbättrad testmiljö som inte förlitar sig på emulerade autonoma system samt att förbättra tekniken för insamling av vidarebefordrade IP-prefix. Det existerar även ett stort antal andra mätvärden och konfigurationer av FRRouting som möjligtvis kan resultera i att ytterligare begränsningar uppdagas.