Please find the referenced videos attached / The physical ends of eukaryotic chromosomes consist of repetitive DNA sequences, which are associated with specialized proteins forming a nucleoprotein structure essential for the integrity of the linear chromosomes, and are known as telomeres. Telomerase is an enzyme responsible for the maintenance of the telomeric repeats at the end of the chromosomes. Telomerase is a ribonucleoprotein, which contains a catalytic subunit that possesses reverse transcriptase activity, and a RNA subunit that acts as a template, since it possess the telomeric repeat sequences necessary to amplify telomere ends. Telomeres are transcribed in most eukaryotes into a non-coding RNA know as TERRA (Telomeric repeats-containing RNA). It has been proposed that TERRA may act as a regulator of telomere homeostasis, and as an inhibitor of telomerase, however, its specific function is still unknown. In Saccharomyces cerevisiae, TERRA is rapidly degraded by the 5’-3’ Rat1 exonuclease, which has hampered its study by classic biochemical experiments in yeast. In this thesis, we report the use of cytological approaches to study TERRA in budding yeast. Two different approaches were used for this purpose: the fluorescent in-situ hybridization (FISH) and the labeling of TERRA by the MS2-GFP system, which allow the visualization of TERRA transcripts form a single telomere in living cells. With these two approaches, we observed that TERRA is expressed from a single telomere and accumulates as a single perinuclear foci, in a small percentage of cells population. We also demonstrate that TERRA expression occurs due to telomere shortening. We demonstrate that TERRA interacts in vivo with the telomerase RNA (TLC1) in yeast. Telomere elongation depends on the action of several telomerase molecules that are visible as clusters, which associate with telomeres in late S phase in yeast, and mammalian cells. In adidition, we show that TERRA stimulates the nucleation of telomerase clusters. By performing time course experiments of TERRA and TLC1 RNA in live cells, we observed that TERRA acts as a scaffold for generating telomerase clusters, which are then recruited in late S phase to the telomere from which TERRA molecules originated. The recruitment of TERRA to its telomere of origin is dependent on factors that control telomerase recruitment at telomeres like: Mre11, Tel1 and the yKu complex. We propose that a short telomere expresses TERRA to assemble and organize telomerase molecules, which later on allows their recruitment at the short telomere, where elongation is needed. Finally we showed an up-regulation of TERRA, and telomerase RNA TLC1, accompanied by a predominant cytoplasmic localization as cell growth progresses from exponential growth to diauxic shift, and stationary phase. In these conditions, TERRA foci co-localize with TLC1 RNA foci, suggesting that the function of TERRA as a scaffold molecule to generate telomerase cluster is necessary for this yeast cell growth phases. / Les télomères à l’extrémité des chromosomes constituent une structure d’ADN et de protéines essentielle à l’intégrité de ces chromosomes. La télomérase est l’enzyme responsable du maintien des répétitions télomériques à l’extrémité des chromosomes. Cette enzyme est constituée d’une sous-unité catalytique, qui possède une activité de transcriptase réverse, et d’une sous-unité d’ARN, qui fourni la matrice nécessaire à la synthèse des répétitions télomériques. Les ARN contenant des répétions télomériques (ou Telomeric repeats-containing RNA; TERRA) constitue une nouvelle classe d’ARN non-codants transcrits à partir des télomères et conservée chez la plupart des eucaryotes. TERRA a été proposé d’agir comme un régulateur de l‘homéostasie des télomères et comme inhibiteur de la télomérase, mais sa fonction spécifique reste inconnue. De plus, chez la levure Saccharomyces cerevisiae, TERRA est rapidement dégradé par l’exonucléase 5’-3’ Rat1, ce qui complique l’étude de cet ARN par les méthodes biochimiques classiques. Dans cette thèse, nous rapportons l‘utilisation d’une approche cytologique pour étudier TERRA dans les cellules de levures. Deux approches sont utilisées : l’hybridation in situ en fluorescence (FISH) et l’étiquetage de TERRA à l’aide du système MS2-GFP, qui nous permet de visualiser l’expression de TERRA transcrit d’un seul télomère dans des cellules vivantes. Avec ces deux approches, nous observons que TERRA exprimé à partir d’un seul télomère s’accumule dans un faible nombre de cellules, sous la forme d’un focus périnucléaire. De plus, nous montrons que TERRA est exprimé lorsque son télomère raccourcit. Par immunoprécipitation, nous montrons que TERRA interagit in vivo avec l’ARN de la télomérase de levure, TLC1. L’élongation des télomères dépend de l‘action de multiples molécules de télomérase, qui sont visibles sous la forme de clusters de télomérases, qui s‘associent en phase S avec les télomères chez la levure et les cellules de mammifère. Nous démontrons que TERRA stimule la nucléation de ces clusters de télomérase. Par imagerie en temps réel de TERRA et de l’ARN TLC1, nous observons que TERRA agit comme molécule d’échafaudage pour générer des clusters de télomérases, qui sont par la suite recrutés, en phase S, au télomère duquel TERRA a été exprimé. Le recrutement d’un focus de TERRA à son télomère d’origine dépend des facteurs contrôlant le recrutement de la télomérase aux télomères : Mre11, Tel1 et le complexe yKu. Nous proposons qu’un télomère court exprime TERRA pour assembler et organiser les molécules de télomérase, afin que celles-ci soit puissent être recrutées au télomère court pour permettre son élongation. Enfin, nous observons une surexpression de l’ARN de la télomérase TLC1 et de TERRA, ainsi qu’une accumulation cytoplasmique de ceux-ci sous la forme de foci, lorsque la cellule passe de la phase de croissance exponentiel à la phase diauxique, puis à la phase stationnaire. Dans ces conditions, les foci d’ARN TLC1 colocalisent avec les foci de TERRA, suggérant que la fonction de TERRA comme molécule d’échafaudage pour générer des foci de télomérase est aussi nécessaire durant ces phases du cycle de croissance des levures.

Extracting useful knowledge from data sets is a key concept in modern information systems. Consequently, the need of efficient techniques to extract the desired knowledge has been growing over time. Machine learning is a research field dedicated to the development of techniques capable of enabling a machine to \"learn\" from data. Many techniques have been proposed so far, but there are still issues to be unveiled specially in interdisciplinary research. In this thesis, we explore the advantages of network data representation to develop machine learning techniques based on dynamical processes on networks. The network representation unifies the structure, dynamics and functions of the system it represents, and thus is capable of capturing the spatial, topological and functional relations of the data sets under analysis. We develop network-based techniques for the three machine learning paradigms: supervised, semi-supervised and unsupervised. The random walk dynamical process is used to characterize the access of unlabeled data to data classes, configuring a new heuristic we call ease of access in the supervised paradigm. We also propose a classification technique which combines the high-level view of the data, via network topological characterization, and the low-level relations, via similarity measures, in a general framework. Still in the supervised setting, the modularity and Katz centrality network measures are applied to classify multiple observation sets, and an evolving network construction method is applied to the dimensionality reduction problem. The semi-supervised paradigm is covered by extending the ease of access heuristic to the cases in which just a few labeled data samples and many unlabeled samples are available. A semi-supervised technique based on interacting forces is also proposed, for which we provide parameter heuristics and stability analysis via a Lyapunov function. Finally, an unsupervised network-based technique uses the concepts of pinning control and consensus time from dynamical processes to derive a similarity measure used to cluster data. The data is represented by a connected and sparse network in which nodes are dynamical elements. Simulations on benchmark data sets and comparisons to well-known machine learning techniques are provided for all proposed techniques. Advantages of network data representation and dynamical processes for machine learning are highlighted in all cases / A extração de conhecimento útil a partir de conjuntos de dados é um conceito chave em sistemas de informação modernos. Por conseguinte, a necessidade de técnicas eficientes para extrair o conhecimento desejado vem crescendo ao longo do tempo. Aprendizado de máquina é uma área de pesquisa dedicada ao desenvolvimento de técnicas capazes de permitir que uma máquina \"aprenda\" a partir de conjuntos de dados. Muitas técnicas já foram propostas, mas ainda há questões a serem reveladas especialmente em pesquisas interdisciplinares. Nesta tese, exploramos as vantagens da representação de dados em rede para desenvolver técnicas de aprendizado de máquina baseadas em processos dinâmicos em redes. A representação em rede unifica a estrutura, a dinâmica e as funções do sistema representado e, portanto, é capaz de capturar as relações espaciais, topológicas e funcionais dos conjuntos de dados sob análise. Desenvolvemos técnicas baseadas em rede para os três paradigmas de aprendizado de máquina: supervisionado, semissupervisionado e não supervisionado. O processo dinâmico de passeio aleatório é utilizado para caracterizar o acesso de dados não rotulados às classes de dados configurando uma nova heurística no paradigma supervisionado, a qual chamamos de facilidade de acesso. Também propomos uma técnica de classificação de dados que combina a visão de alto nível dos dados, por meio da caracterização topológica de rede, com relações de baixo nível, por meio de medidas de similaridade, em uma estrutura geral. Ainda no aprendizado supervisionado, as medidas de rede modularidade e centralidade Katz são aplicadas para classificar conjuntos de múltiplas observações, e um método de construção evolutiva de rede é aplicado ao problema de redução de dimensionalidade. O paradigma semissupervisionado é abordado por meio da extensão da heurística de facilidade de acesso para os casos em que apenas algumas amostras de dados rotuladas e muitas amostras não rotuladas estão disponíveis. É também proposta uma técnica semissupervisionada baseada em forças de interação, para a qual fornecemos heurísticas para selecionar parâmetros e uma análise de estabilidade mediante uma função de Lyapunov. Finalmente, uma técnica não supervisionada baseada em rede utiliza os conceitos de controle pontual e tempo de consenso de processos dinâmicos para derivar uma medida de similaridade usada para agrupar dados. Os dados são representados por uma rede conectada e esparsa na qual os vértices são elementos dinâmicos. Simulações com dados de referência e comparações com técnicas de aprendizado de máquina conhecidas são fornecidos para todas as técnicas propostas. As vantagens da representação de dados em rede e de processos dinâmicos para o aprendizado de máquina são evidenciadas em todos os casos

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.

Die Besiedlung von Gewässern durch Fische, ist neben abiotischen Lebensraumbedingungen auch von der Erreichbarkeit d.h. von der art-spezifischen Ausbreitungsfähigkeit sowie von Wanderhindernissen abhängig. Der erste Teil dieser Arbeit bietet die erste umfangreiche quantitative Analyse von Ausbreitungsmustern und -distanzen von Flussfischen. Aus der Fachliteratur wurden 160 empirische Datensätze aus 71 wissenschaftlichen Studien zur Ausbreitung von 62 Fischarten in Flüssen extrahiert und an leptokurse Wahrscheinlichkeits-Dichte-Funktionen (Dispersal kernel) angepasst. Es konnte bei Fischpopulationen zwischen einer stationären (ca. 2/3) und einer mobilen Komponente (ca. 1/3) unterschieden werden deren Ausbreitungsdistanzen von vier Faktoren abhängig sind: Fischlänge, Form der Schwanzflosse, Fließgewässergröße, betrachtete Zeitspanne. Der zweite Teil dieser Arbeit widmet sich dem neu entwickelten Fischausbreitungsmodell FIDIMO einem GIS-Softwareprogramm zur Modellierung und Simulation der räumlichen und zeitlichen Ausbreitungsmuster von Fischen in Flüssen unter Berücksichtigung von Wanderhindernissen. FIDIMO verknüpft konzeptionelle Überlegungen zu Ausbreitungsmodellen in verzweigten Fließgewässernetzwerken mit empirisch bestimmten leptokursen Fischausbreitungskurven unter ausschließlicher Verwendung von Free and Open Source Software. Im dritten Teil der Arbeit wurde FIDIMO zur Modellierung der Ausbreitung von 17 Fischarten angewendet um die Einflüsse von (i) Habitatqualität, (ii) Ausbreitungsfähigkeit und (iii) Fließgewässer-Fragmentierung auf die Besiedlungsmuster durch Fische zu bestimmen. Die Ergebnisse zeigen, dass die artspezifische Habitatqualität und Ausbreitungsfähigkeit die Besiedlung maßgeblich bestimmen. Dagegen wurde kein signifikanter Einfluss von Barrieren auf das Vorkommen einer Art gefunden. Über längere Zeiträume sinkt der Einfluss von Fischausbreitung auf das lokale Vorkommen einer Fischart während die Habitatqualität relativ wichtiger wird. / The colonisation of rivers by fishes is directly linked to abiotic habitat conditions but often impaired by dispersal abilities of fishes and movement constraints such as barriers. The first part of this thesis provides the first comprehensive quantitative analysis of freshwater fish movement while considering fish populations consisting of differently mobile specimens. 160 empirical datasets from 71 studies on the movement of 62 riverine fish species were analysed based on refitted leptokurtic probability-density functions (dispersal kernels). A share of one third and two thirds emerged as a general pattern of the mobile and stationary component of a fish population, respectively. Moreover, four variables were identified primarily determining dispersal distances: fish length, aspect ratio of the caudal fin, river size and time. In the second part of the thesis, the novel fish dispersal model FIDIMO is introduced. FIDIMO provides a GIS-tool for predicting and simulating spatio-temporal patterns of fish dispersal in dendritic river networks considering movement barriers. The fish dispersal model FIDIMO links conceptual considerations on dispersal modelling with empirically observed leptokurtic fish movement patterns and the strengths of geographically explicit modelling in Free and Open Source GIS. In the third part of the thesis, FIDIMO was applied for modelling dispersal of 17 fish species to disentangle the effects of (i) habitat suitability, (ii) dispersal constraints and (iii) network fragmentation on the distribution of river fishes. The results show significant positive effects of both, local-scale habitat quality and species-specific dispersal ability on the distribution of river fishes, whereas no significant effect of barriers influencing the presence of a species could be found. Over longer time periods the importance of dispersal decreased in favour of habitat suitability becoming relatively more relevant in determining species'' presence.

Zusammenfassung Diese Arbeit ist ein Versuch, verschiedene klassische und neuere Methodender glatten bzw. nichtglatten Optimierung zu verallgemeinern und in ihrem Zusammenhang darzustellen. Als Hauptinstrument erweist sich dabei die sogenannte verallgemeinerte Kojima-Funktion. Neben reichlichen Beispielen setzen wir einen besonderen Akzent auf die Betrachtung von Variationsungleichungen, Komplementaritaetsaufgaben und der Standartaufgabeder mathematischen Programmierung. Unter natuerlichen Voraussetzungen an diese Probleme kann man u.a. Barriere-, Straf- und SQP-Typ-Methoden, die auf Newton-Verfahrenbasieren, aber auch Modelle, die sogenannte NCP-Funktionen benutzen, mittelsspezieller Stoerungen der Kojima-Funktion exakt modellieren. Daneben werdendurch explizite und natuerliche Wahl der Stoerungsparameter auch neue Methoden dieser Arten vorgeschlagen. Die Vorteile solcher Modellierungsind ueberzeugend vor allem wegen der direkt moeglichen (auf Stabilitaetseigenschaften der Kojima-Gleichung beruhendenden)Loesungsabschaetzungen und weil die entsprechenden Nullstellen ziemlich einfach als Loesungen bekannter Ersatzprobleme interpretiert werden koennen. Ein weiterer Aspekt der Arbeit besteht in der genaueren Untersuchungder "nichtglatten Faelle". Hier wird die Theorie von verschiedenen verallgemeinerten Ableitungen und dadurch entstehenden verallgemeinerten Newton-Verfahren, die im Buch "Nonsmooth Equations in Optimization" von B. Kummer und D. Klatte vorgeschlagen und untersucht wurde, intensiv benutzt. Entscheidend ist dabei, dass die benutzten verallgemeinerten Ableitungen auch praktisch angewandt werden koennen, da man sie exakt ausrechnen kann. / This work attempts to generalize various classical and new methods of smooth or nonsmooth optimization and to show them in their interrelation. The main tool for doing this is the so-called generalized Kojima-function. In addition to numerous examples we specialy emphasize the consideration of variational inequalities, complementarity problems and the standard problem of mathematical programming. Under natural assumptions on these problems we can model e.g. barrier-, penalty-, and SQP-Type-methods basing on Newton methods, and also methods using the so-called NCP-function exactly by means of special perturbations of the Kojima-function. Furthermore, by the explicit and natural choice of the perturbation parameters new methods of these kinds are introduced. The benefit of such a modelling is obvious, first of all due to the direct solution estimation (basing on stability properties of the Kojima-equation) and because the corresponding zeros can easily be interpreted as solutions of known subproblems. A further aspect considered in this paper is the detailed investigation of "nonsmooth cases". The theory of various generalized derivatives and resulting generalized Newton methods, which is introduced and investigated in the book "Nonsmooth Equations in Optimization" of B. Kummer and D. Klatte, is intensely used here. The crucial point is the applicability of the used generalized derivatives in practice, since they can be calculated exactly.

The increase in electric vehicles and photovoltaic power production may introduce problems to the low-voltage distribution grid. With a higher number of electric vehicles, their accumulated charging power might breach the lowest allowed voltage level of the grid. Photovoltaic-modules can on the other hand exceed the highest allowed voltage level, by producing high accumulated power when the solar irradiance is high. Normally, electric distribution companies in Sweden reinforce the existing grid with more resilient infrastructure, such as stronger and larger cables or transformer stations. This is however a costly and time-consuming solution, which could be solved by using alternative means such as already existing resources. This Master's Thesis investigates how smart charging of batteries can support the low-voltage electrical grid with the increase in electric vehicles and photovoltaic power production. To do this, an optimization tool has been developed in Matlab. An existing model of a low-voltage grid is combined with the developed tool, where controllable batteries and photovoltaic-modules can be placed at specific households in the grid. The controllable batteries belong to either electric vehicles or stationary battery systems, and are intended to support the grid by the means of either reducing peak load powers, voltage variations, or a trade-off between them. Furthermore, this thesis investigates the maximum electric vehicle capability for a specific low-voltage electrical grid in Sweden. From the results, it can be concluded that smart charging of batteries can reduce the peak loads as well as voltage variations. The reduction of voltage variations for the entire low-voltage grid is greatest during the summer, when photovoltaic production generally is at its highest. The results also show that a stationary battery system can reduce the voltage variations to a greater extent, compared to an electric vehicle. Also, the introduction of multiple controllable batteries allows further support of the low-voltage grid. Regarding the maximum electric vehicle capability, the results show that the placement of the vehicles and the charging power strongly affect the maximum number of electric vehicles the low-voltage grid can manage. / Ökningen av elbilar och elproduktion från solceller kan ge problem i lågspänningsnätet. Med ett ökat antal elbilar kan den sammanlagrade effekten vid laddning underskrida den minsta tillåtna spänningsnivån i nätet. Solpaneler kan däremot leda till att den högsta tillåtna spänningsnivån överskrids, genom att producera en hög sammanlagrad effekt när solstrålningen är som högst. Vanligtvis förstärker elnätsbolag i Sverige det befintliga nätet med motståndskraftigare infrastruktur, såsom kraftigare och större kablar eller transformatorstationer. Detta är dock en kostsam och tidskrävande lösning, som skulle kunna lösas med alternativa medel, till exempel redan existerande resurser. Detta examensarbete undersöker hur smart laddning av batterier kan ge stöd till lågspänningsnätet, med en ökning av elbilar samt solcellsproduktion. För att undersöka detta har ett optimeringsverktyg utvecklats i Matlab. En befintlig modell av ett lågspänningsnät har kombinerats med det utvecklade optimeringsverktyget, där styrbara batterier samt solcellsproduktion kan placeras vid specifika hushåll i elnätet. De styrbara batterierna är antingen elbilar eller stationära batterisystem, och är ämnade till att stödja lågspänningsnätet genom att antingen reducera effekttoppar, spänningsvariationer eller en kompromiss av båda. Vidare undersöker detta examensarbete det maximala antalet elbilar som ett specifikt lågspänningsnät i Sverige kan hantera. Resultaten visar att smart laddning av batterier kan reducera effekttoppar samt spänningsvariationer. Reduceringen av spänningsvariationerna för hela lågspänningsnätet visar sig vara högst under sommaren, vilket är då solcellsproduktionen generellt är som högst. Resultaten visar även att stationära batterisystem kan reducera spänningsvariationer ytterligare, jämfört med en elbil. Att introducera flera styrbara batterier tillåter ett ännu större stöd till lågspänningsnätet. Angående det maximala antalet av elbilar som ett lågspänningsnät kan hantera visade resultaten att placeringen av elbilarna samt laddningseffekten har en stor påverkan.

This dissertation is concerned with the potential multistability of protein concentrations in the cell that can arise in biochemical networks. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. Models of multisite protein phosphorylation have shown that this mechanism is able to exhibit unlimited multistability. Nevertheless, these models have not considered enzyme docking, the binding of the enzymes to one or more substrate docking sites, which are separate from the motif that is chemically modified. Enzyme docking is, however, increasingly being recognised as a method to achieve specificity in protein phosphorylation and dephosphorylation cycles. Most models in the literature for these systems are deterministic i.e. based on Ordinary Differential Equations, despite the fact that these are accurate only in the limit of large molecule numbers. For small molecule numbers, a discrete probabilistic, stochastic, approach is more suitable. However, when compared to the tools available in the deterministic framework, the tools available for stochastic analysis offer inadequate visualisation and intuition. We firstly try to bridge that gap, by developing three tools: a) a discrete `nullclines' construct applicable to stochastic systems - an analogue to the ODE nullcines, b) a stochastic tool based on a Weakly Chained Diagonally Dominant M-matrix formulation of the Chemical Master Equation and c) an algorithm that is able to construct non-reversible Markov chains with desired stationary probability distributions. We subsequently prove that, for multisite protein phosphorylation and similar models, in the deterministic domain, enzyme docking and the consequent substrate enzyme-sequestration must inevitably limit the extent of multistability, ultimately to one steady state. In contrast, bimodality can be obtained in the stochastic domain even in situations where bistability is not possible for large molecule numbers. We finally extend our results to cases where we have an autophosphorylating kinase, as for example is the case with $Ca^{2+}$/calmodulin-dependent protein kinase II (CaMKII), a key enzyme in synaptic plasticity.

Cette thèse s’inscrit dans le domaine des mathématiques appelé optimisation de formes. Plus précisément, nous étudions ici un problème inverse de détection à l’aide du calcul de forme et de l’analyse asymptotique. L’objectif est de localiser un objet immergé dans un fluide visqueux, incompressible et stationnaire. Les questions principales qui ont motivé ce travail sont les suivantes :– peut-on détecter un objet immergé dans un fluide à partir d’une mesure effectuée à la surface ?– peut-on reconstruire numériquement cet objet, i.e. approcher sa position et sa forme, à partir de cette mesure ?– peut-on connaître le nombre d’objets présents dans le fluide en utilisant cette mesure ?Les résultats obtenus sont décrits dans les cinq chapitres de cette thèse :– le premier met en place un cadre mathématique pour démontrer l’existence des dérivées de forme d’ordre un et deux pour les problèmes de détection d’inclusions ;– le deuxième analyse le problème de détection à l’aide de l’optimisation géométrique de forme : un résultat d’identifiabilité est montré, le gradient de forme de plusieurs types de fonctionnelles de forme est caractérisé et l’instabilité de ce problème inverse est enfin démontrée ;– le chapitre 3 utilise nos résultats théoriques pour reconstruire numériquement des objets immergés dans un fluide à l’aide d’un algorithme de gradient de forme ;– le chapitre 4 analyse la localisation de petites inclusions dans un fluide à l’aide de l’optimisation topologique de forme : le gradient topologique d’une fonctionnelle de forme de Kohn-Vogelius est caractérisé ;– le dernier chapitre utilise cette dernière expression théorique pour déterminer numériquement le nombre et la localisation de petits obstacles immergés dans un fluide à l’aide d’un algorithme de gradient topologique. / This dissertation takes place in the mathematic field called shape optimization. More precisely, we focus on a detecting inverse problem using shape calculus and asymptotic analysis. The aim is to localize an object immersed in a viscous, incompressible and stationary fluid. This work was motivated by the following main questions:– can we localize an obstacle immersed in a fluid from a boundary measurement?– can we reconstruct numerically this object, i.e. be close to its localization and its shape, from this measure?– can we know how many objects are included in the fluid using this measure?The results are described in the five chapters of the thesis:– the first one gives a mathematical framework in order to prove the existence of the shape derivatives oforder one and two in the frame of the detection of inclusions;– the second one analyzes the detection problem using geometric shape optimization: an identifiabilityresult is proved, the shape gradient of several shape functionals is characterized and the instability of thisinverse problem is proved;– the chapter 3 uses our theoretical results in order to reconstruct numerically some objets immersed in a fluid using a shape gradient algorithm;– the fourth chapter analyzes the detection of small inclusions in a fluid using the topological shape optimization : the topological gradient of a Kohn-Vogelius shape functional is characterized;– the last chapter uses this theoretical expression in order to determine numerically the number and the location of some small obstacles immersed in a fluid using a topological gradient algorithm.

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.

Dans cette thèse, nous étudions l’équation parabolique non linéaire ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) et avec une donnée initiale u(0) = φ. On établit l’existence et l’unicité locale dans Lq(Rᴺ) et dans Cₒ(Rᴺ). En particulier, la valeur q = N ⍺/(2 − γ) joue un rôle critique. Pour ⍺ > (2 − γ)/N, on montre l’existence de solutions auto-similaires globales avec données initiales φ(x) = ω(x) |x|−(2−γ)/⍺, où ω ∈ L∞(Rᴺ) homogène de degré 0 et ||ω||∞ est suffisamment petite. Nous montrons ainsi que si φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ pour |x| grande, alors la solution est globale et asymptotique dans L∞(Rᴺ) à une solution auto-similaire de l’équation non linéaire. Tandis que si φ(x)∼ω(x) |x| (x)|x|−σ pour des |x| grandes avec (2 − γ)/⍺ < σ < N, alors la solution est globale, mais elle est asymptotique dans L∞(Rᴺ) à eᵗ∆(ω(x) |x|−σ). L’équation avec un potentiel plus général, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), est également étudiée. En particulier, pour des données initiales φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| grande, nous montrons que le comportement à grand temps est linéaire si V est à support compact au voisinage de l’origine, alors qu’il est non linéaire si V est à support compact au voisinage de l’infini. Nous étudions également le système non linéaire ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Sous des conditions sur les paramètres p, q, γ et ρ nous montrons l’existence et l’unicité de solutions globales avec données initiales petites par rapport à certaines normes. En particulier, on montre l’existence de solutions auto-similaires avec donnée initiale Φ = (φ₁, φ₂), où φ₁, φ₂ sont des données initiales homogènes. Nous montrons également que certaines solutions globales sont asymptotiquement auto-similaires. Comme deuxième objectif, nous considérons l’équation de la chaleur non linéaire ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, avec t ≥ 0 et x ∈ Ω, la boule unité de Rᴺ, N ≥ 3, avec des conditions aux limites de Dirichlet. Soit h une solution stationnaire à symétrie radiale avec changement de signe de (E). On montre que la solution de (E) avec donnée initiale λh explose en temps fini si |λ − 1| > 0 est suffisamment petit et si 1 < q < p < Ps = N+2/N−2 et p suffisamment proche de Ps. Ceci prouve que l’ensemble des données initiales pour lesquelles la solution est globale n’est pas étoilé au voisinage de 0. / In this thesis, we study the nonlinear parabolic equation ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) and with initial value u(0) = φ. We establish local well-posedness in Lq(Rᴺ) and in Cₒ(Rᴺ). In particular, the value q = N ⍺/(2 − γ) plays a critical role.For ⍺ > (2 − γ)/N, we show the existence of global self-similar solutions with initial values φ(x) = ω(x) |x|−(2−γ)/⍺, where ω ∈ L∞(Rᴺ) is homogeneous of degree 0 and ||ω||∞ is sufficiently small. We then prove that if φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ for |x| large, then the solution is global and is asymptotic in the L∞-norm to a self-similar solution of the nonlinear equation. While if φ(x)∼ω(x) |x| (x)|x|−σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L∞-norm toe t(ω(x) |x|−σ). The equation with more general potential, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), is also studied. In particular, for initial data φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. we study also the nonlinear parabolic system ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Under conditions on the parameters p, q, γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In particular, we show the existence of self-similar solutions with initial value Φ = (φ₁, φ₂), where φ₁, φ₂ are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions. As a second objective we consider the nonlinear heat equation ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, where t ≥ 0 and x ∈ Ω, the unit ball of Rᴺ, N ≥ 3, with Dirichlet boundary conditions. Let h be a radially symmetric, sign-changing stationary solution of (E). We prove that the solution of (E) with initial value λ h blows up in finite time if |λ − 1| > 0 is sufficiently small and if 1 < q < p < Ps = N+2/N−2 and p sufficiently close to Ps. This proves that the set of initial data for which the solution is global is not star-shaped around 0.