Cette thèse porte sur l'étude probabiliste de modèles écologiques appartenant à la récente théorie des "dynamiques adaptatives". Après avoir précisé et généralisé le cadre et l'heuristique biologique de ces modèles, nous obtenons une justification microscopique d'un modèle d'évolution par sauts à partir d'un système de particules en interaction à valeurs mesure, décrivant la dynamique de la population à l'échelle individuelle. Il s'agit d'un résultat de séparation d'échelles de temps lié à deux asymptotiques : mutations rares et grande population. Ensuite, nous retrouvons une équation différentielle ordinaire connue sous le nom d'"équation canonique des dynamiques adaptatives" en appliquant une asymptotique de petits sauts au processus précédent. Cette asymptotique nous conduit à introduire un modèle d'évolution par diffusion comme approximation diffusion du processus de saut, dont les coefficients présentent une mauvaise régularité : dérive discontinue et diffusion dégénérée aux mêmes points. Nous examinons d'abord l'existence faible, l'unicité en loi et la propriété de Markov forte pour ces processus, questions liées au problème d'atteinte de certains points isolés de l'espace. Enfin, nous démontrons un principe de grandes déviations pour ces diffusions qui permet d'étudier le temps et le lieu de sortie d'un domaine attracteur --- question biologique fondamentale.

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Dans cette thèse, nous étudions l'existence, non existence et multiplicité des ondes stationnairesavec les normes prescrites pour deux types d'équations aux dérivées partiellesnon linéaires elliptiques découlant de différents modèles physiques. La stabilité orbitale desondes stationnaires est également étudiée dans certains cas. Les principales méthodes denos preuves sont des arguments variationnels. Les solutions sont obtenues comme pointscritiques de fonctionnelle associée sur une contrainte.La thèse se compose de sept chapitres. Le Chapitre 1 est l'introduction de la thèse. Dansles Chapitres 2 à 4, nous étudions une classe d'équations de Schrödinger-Poisson-Slaternon linéaires. Nous établissons dans le Chapitre 2 des résultats optimaux non existencede solutions d'énergie minimale ayant une norme L2 prescrite. Dans le Chapitre 3, nousmontrons un résultat d'existence de solutions L2 normalisées, dans une cas où la fonctionnelleassociée n'est pas bornée inférieurement sur la contrainte. Nos solutions sonttrouvées comme des points de selle de la fonctionnelle, mais ils correspondent à des solutionsd'énergée minimale. Nous montrons également que les ondes stationnaires associéessont orbitalement instables. Ici, puisque nos points critiques présumés ne sont pas desminimiseurs globaux, il n'est pas possible d'utiliser de façon systématique les méthodesde compacité par concentration développées par P. L. Lions. Ensuite, dans le Chapitre4, nous montrons que sous les hypothèses du Chapitre 3, il existe une infinité de solutionsayant une norme L2 prescrite. Dans les deux chapitres suivants, nous étudions uneclasse d'équations de Schrödinger quasi-linéaires. Des résultats optimaux non existence desolutions d'énergie minimale sont donnés dans le Chapitre 5. Dans le Chapitre 6, nousprouvons l'existence de deux solutions positives ayant une norme donnée. L'une d'elles,relativement à la contrainte L2, est de type point selle. L'autre est un minimum, soit localou global. Le fait que la fonctionnelle naturelle associée à cette équation n'est pas biendéfinie nécessite l'utilisation d'une méthode de perturbation pour obtenir ces deux pointscritiques. Enfin, au Chapitre 7, nous mentionnons quelques questions que cette thèse asoulevées.

In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy. First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity. Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.

Les résultats présentés dans ce mémoire sont dédiés à l'étude théorique et numérique de modèles en chimiotaxie-fluide motivés par un large éventail de phénomènes biologiques comme la chimiotaxie de populations cellulaires dans un fluide. Les deux premiers chapitres de cette thèse portent sur la chimiotaxie dans un fluide au repos. Au début, on généralise un schéma de volumes finis au cas de modèles isotropes de Keller-Segel avec des coefficients diffusifs scalaires généraux sur des maillages admissibles. Ensuite, on propose et on étudie un schéma monotone combinant les méthodes de volumes finis et d'éléments finis non conformes et permettant une discrétisation efficace et robuste de modèles de Keller-Segel avec des tenseurs diffusifs anisotropes hétérogènes sans imposer des conditions restrictives sur le maillage du domaine en espace. Les deux derniers chapitres sont dédiés à l'étude théorique (existence globale, unicité) et l'étude numérique (extension de la méthode combinée) du système chimiotactisme-fluide complet constitué d'équations chimiotaxiques anisotropes couplées aux équations de Navier-Stokes modélisant un fluide incompressible. Ce couplage s'effectue à travers les termes décrivant d'un part le transport des cellules vivantes et du chimio-attractant par le fluide et d'autre part la force gravitationnelle exercée par ces organismes vivants sur le fluide. Les travaux de cette thèse ont donné lieu à l'écriture d'un code de calcul très développé en Fortran 95 afin de valider nos résultats par des simulations numériques.

Emerging energy storage applications are driving the demand for Li-ion battery positive electrode materials with higher energy densities and lower costs. The recent production of complete pseudo-ternary phase diagrams of the Li-Mn-Ni-O system generated using combinatorial methods has provided a greater understanding of the impact of initial composition, synthesis temperature, and cooling rate on the phases that form in the final materials. This thesis focuses on the synthesis and characterization of gram-scale positive electrode materials in the Li-Mn-Ni-O system. Structural analysis of these samples has resulted in the production of partial pseudo-ternary phase diagrams focusing on the positive electrode materials region of the Li-Mn-Ni-O system at 800°C and 900°C in air for both quenched and slow cooled compositions. These bulk-scale diagrams support the observations of the combinatorial diagrams, and show similar layered and cubic structures contained within several single- and multi-phase regions. The phases that form at each composition are shown to be dependent on both the reaction temperature and cooling rate used during synthesis. The electrochemical characterization of two composition series near Li2MnO3, one quenched and one slow cooled, is presented. The quenched compositions exhibited reversible cycling at 4.4 V, voltage plateaus and small increases in capacity above 4.6 V, and large first cycle irreversible capacity losses at 4.8 V. In the slow cooled series, all but one composition exhibited initial capacities below 100 mAh/g which began to continually increase with cycling, with several compositions exhibiting capacity increases of 300% over 150 cycles at 4.9 V. In both series, analysis of the voltage and differential capacity plots indicated that significant structure rearrangements are taking place in these materials during extended cycling, the possible origins of which are discussed. Finally, high precision coulometry studies of one Li-deficient and two Li-rich single-phase layered compositions are discussed. These materials exhibit minimal oxidation of simple carbonate-based electrolyte when cycled to high potential, with the Li-deficient composition producing less electrolyte oxidation at 4.6 V vs. Li/Li+ than commercial Li[Ni1/3Mn1/3Co1/3]O2 at 4.2 V. The inherent inertness of this composition may make it suitable for use as a thin protective layer in a core-shell particle.

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.

On considère le problème d'advection-diffusion stationnaire v(∇u, ∇v)+( β*∇u, v) = (f, v) et non stationnaire d/dt (u(t), v) + v(∇u, ∇v)+( β*∇u, v) = (g(t), v), ainsi que le problème d'advection (β*∇u, v) = (f, v) sur un domaine polygonal borné du plan. Le terme de diffusion est approché par des éléments de Crouzeix Raviart et le terme de convection par une méthode upwind sur des volumes barycentriques finis avec un maillage triangulaire. Pour le problème stationnaire d'advection-diffusion, la L²-stabilité (c'est-à-dire indépendante du coefficient de diffusion v) est démontrée pour la solution du problème approché obtenue par cette méthode d'éléments finis et de volumes finis. Pour cela une condition sur la géométrie doit être satisfaite. Des exemples de maillages sont donnés. Toujours avec cette condition géométrique sur le maillage, une inégalité de stabilité (où la discrétisation en temps n'est pas couplée à une condition sur la finesse du maillage) est obtenue pour le cas non-stationnaire. La discrétisation en temps y est faite par un schéma d'Euler implicite. Une majoration de l'erreur, proportionnelle au pas en temps et à la finesse du maillage, est ensuite proposée et exprimée explicitement en fonction des données du problème. Pour le problème d'advection, une approche utilisant la théorie des graphes est utilisée pour obtenir l'existence et l'unicité de la solution, ainsi que le résultat de stabilité. Comme pour la stabilité du problème d'advection-diffusion, une condition géométrique - qui est équivalente pour les points intérieurs du maillage à celle du problème d'advection-diffusion - est nécessaire.

A Vontade de Poder no pensamento de Nietzsche é a expressão da vida, sem nenhuma outra conotação, seja metafísica, religiosa, moral ou psicológica. Para seu entendimento como filosofema foram analisados outros conceitos correlatos, tais como vida e existência, sendo a vida a expressão puramente biológica e a existência a construção psicológica que atribui um sentido à vida; ser humano, como o ente que se construiu, criando e desenvolvendo características específicas como a razão, a inteligência, a memória, o querer e a concrescência; o conhecimento, como recurso para construção da existência. Para o entendimento deste ser humano, no pensamento de Nietzsche, é necessário enquadra-lo dentro daquilo que ele chama de espírito histórico, civilização, mundo e forças vitais. Somente dentro deste conjunto de conceitos a Vontade de Poder se sobressai não só como a expressão da vida, mas também como seu enaltecimento. A vida ou se fortalece continuamente ou perece. A maioria dos intérpretes, no entanto, se dedicou a analisar a expressão Vontade de Poder – Wille zur Macht – sob o aspecto metafísico. Aqui são analisados seis entre os principais – Arthur C. Danto, Maudemarie Clark, John Richardson, Peter Poellner, Martin Heidegger e Karl Jaspers – sendo que para alguns a expressão é um princípio metafísico e para outros não. A expressão Vontade de Poder – Wille zur Macht – nas obras de Nietzsche, tanto nas que foram publicadas como nas que não foram, não tem essa conotação metafísica. Nietzsche, pelo menos, não estava preocupado com a questão metafísica da Vontade de Poder. Sua preocupação era com a vida, e a vida neste mundo, que havia, segundo ele, perdido o sentido porque se fundamentou sobre valores supostamente absolutos e eternos, mas que ao perderem sua validade possibilitaram o surgimento do niilismo. Para Nietzsche, portanto, a Vontade de Poder é enaltecimento da vida – e nada mais! / The Will to Power on Nietzsche’s thought is the expression of life, without another connection, such metaphysical, religious, moral or psychological. For his understanding with phylosofema was been analyzed another correlated concepts, such like life and existence; the life purely biological and the existence the psychological construction which ascribes a sense to life; human being, like the being which constructs itself, creating and developing specific characteristics like the reason, the intelligence, the memory, the will and the conscience; the knowledge, as resource for existence’s construction. For the understanding of this human being, on Nietzsche’s thought, is necessary to frame it within of that which he call historical spirit, civilization, world and vital forces. Only within this concept’s complex the Will to Power excels so far the life’s expression as his enhancement. The life either continually grows or perishes. The interpreter’s majority, however, applied oneself to analyze the expression Will to Power – Wille zur Macht – under the metaphysical aspect. Here are analyzed six among the principals, Arthur C. Danto, Maudemarie Clark, John Richardson, Peter Poellner, Martin Heidegger e Karl Jaspers; this expression is a metaphysical principle for some and not for others. The expression Will to Power – Wille zur Macht – in Nietzsche’s works so in published as in unpublished, hasn’t that metaphysical connection. Nietzsche, at least, hasn’t worried with the metaphysical question of will to power. His preoccupation has with the life, and the life in this world, which, according to Nietzsche, has lost the sense because substantiate itself upon values so-called absolute and eternal, but that losing his validity enabled the nihilism’s emergence. Hence for Nietzsche the will to power is life’s enhancement – and nothing besides!

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional (that derivative is called the 􀀀derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold behavior. The results obtained here may be signi cant for the ght and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still a ecting many people in West-Africa and other parts of the world. The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [129, 142], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. Moreover, the explicit form of its solution is, in general, impossible to obtain. We explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with the conventional time derivative with a new parameter. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and prove that the system describing the polymer chain degradation contains complex and simple harmonic poles whose e ects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, di usion or relaxation equations. The question is to say whether or not this is always true! Whence, three generalized evolution equations with an additional fractional parameter are solved analytically with conventional techniques. These are processes related to stationary state system, relaxation and di usion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However, unlike other models, the generalized di usion and relaxation models are proven not to be governed by Mittag-Le er functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called ; can be used to control the stationarity of such generalized models. / Mathematical Sciences / Ph. D. (Applied Mathematics)