The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.

In an effort to make the aerospace industry more sustainable, GKN Aerospace is heavily investing both time and money into additive manufacturing technologies. A challenging aspect with additive manufacturing is finding a set of process parameters that produces high quality parts and components to a standard that the aerospace industry demands. To aid the process development, turning to simulations is a great alternative and with the prospect of adding to the tools available, the work of this thesis has been focused on developing a mathematical model of the cross sectional deposition geometry. Through a literature study, an initial approach to developing such a model, as well as gaps in knowledge was established. Validation data was gathered by laser scanning additively manufactured builds. A model based on a fourth degree polynomial was developed. The fourth degree polynomial model was validated with the laser scan data using the mean squared error value and coefficient of determination as a quantifiable method of determining the goodness of fit. It was found to be an improvement over the common parabolic model found in the literature based on a second degree polynomial. The improved model manages to capture a wide variety of contact angles and the overlap region better follows the smooth transition between beads.

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.

The main focus of the present thesis is on the homogenization of some selected elliptic and parabolic problems. More precisely, we homogenize: non-periodic linear elliptic problems in two dimensions exhibiting a homothetic scaling property; two types of evolution-multiscale linear parabolic problems, one having two spatial and two temporal microscopic scales where the latter ones are given in terms of a two-parameter family, and one having two spatial and three temporal microscopic scales that are fixed power functions; and, finally, evolution-multiscale monotone parabolic problems with one spatial and an arbitrary number of temporal microscopic scales that are not restricted to be given in terms of power functions. In order to achieve homogenization results for these problems we study and enrich the theory of two-scale convergence and its kins. In particular the concept of very weak two-scale convergence and generalizations is developed, and we study an application of this convergence mode where it is employed to detect scales of heterogeneity. / Huvudsakligt fokus i avhandlingen ligger på homogeniseringen av vissa elliptiska och paraboliska problem. Mer precist så homogeniserar vi: ickeperiodiska linjära elliptiska problem i två dimensioner med homotetisk skalning; två typer av evolutionsmultiskaliga linjära paraboliska problem, en med två mikroskopiska skalor i både rum och tid där de senare ges i form av en tvåparameterfamilj, och en med två mikroskopiska skalor i rum och tre i tid som ges i form av fixa potensfunktioner; samt, slutligen, evolutionsmultiskaliga monotona paraboliska problem med en mikroskopisk skala i rum och ett godtyckligt antal i tid som inte är begränsade till att vara givna i form av potensfunktioner. För att kunna uppnå homogeniseringsresultat för dessa problem så studerar och utvecklar vi teorin för tvåskalekonvergens och besläktade begrepp. Speciellt så utvecklar vi begreppet mycket svag tvåskalekonvergens med generaliseringar, och vi studerar en tillämpningav denna konvergenstyp där den används för att detektera förekomsten av heterogenitetsskalor.

Lubricated friction issues are central to all hydraulic actuation systems undergoing motion and any in-depth understanding of the nature of lubricated friction will advance future component design. The classic friction models of hydraulic actuation systems under steady state conditions and their dependency on velocity and temperature have been studied extensively over the past years. A model which is commonly employed to represent the characteristics of friction is that of Stribeck in which the dependency of the friction force is based on velocity alone. However, experimentally, it has been found that lubricated friction is dependent on acceleration. Thus, the Stribeck model can be considered as a subset of a dynamic friction model in which acceleration is zero. Thus, it can be concluded that the Stribeck model is best applied to cases when the change rate of the velocities is very small. This thesis considers the dependency of lubricated friction on acceleration when pressure and temperature changes are relatively constant. As such, the basic hypothesis for this study was proposed as follows: “Lubricated friction in hydraulic actuation systems is not only a function of velocity, but is also a function of both velocity and acceleration”. In this thesis several terms are defined which facilitate the description under which friction models are developed. For example, the term non-steady state friction is used to account for the effect of acceleration on lubricated friction force while in motion. Further, the lubricated friction models are divided into two groups: steady state friction models and non-steady state friction models. Nonlinear friction modeling and measuring methods are reviewed in this dissertation. This review also includes nonlinear lubricated friction modeling in hydraulic actuation systems. A conclusion from this review was that limited research has been done in documenting and explicitly demonstrating the role of acceleration on lubricated friction. The research first introduced a methodology to experimentally measure friction as a function of acceleration and to demonstrate this dependency in the form of a three dimensional graph. A novel technique to experimentally obtain data for the lubricated friction model was introduced. This allowed the lubricated friction forces to be measured as a function of velocity in a continuous manner, but with acceleration being held constant as a family parameter. Two different valve controlled hydraulic actuation systems (VCHAS) were studied under a wide variety of accelerations at constant temperature and pressure. To enable repeatable data collection for the different friction conditions and to accommodate for the effect of hysteresis, a periodic parabolic displacement waveform was chosen which enabled the acceleration to be a family parameter. The second phase of the research introduced a method of representing the data (lubricated friction model) in a lookup table form. The relationship of lubricated friction (in this work, pressure differential, ΔP across the actuator) as a function of velocity and acceleration was presented in a unique semi-empirical 2D lookup table (2D LUT). Limitations of this experimental approach were identified, but the dependency on acceleration was clearly established. The last phase of the study implemented this 2D LUT model into a practical software model of an actuator and demonstrated its accuracy when compared to its experimental counterpart. The semi-empirical model (2D LUT) was experimentally verified by implementing the semi-empirical and Stribeck models into a real time simulation of an actuator and by comparing the experimental outputs against simulated outputs for a common sinusoidal input. A sinusoidal actuator displacement input was chosen to test the simulations as it was not used in the collection of the original data. The output of the simulation was compared to the experimental results and it was evident that for the range in which data could be collected in developing the model, the proposed 2D LUT model predicted an output that was superior to a model which used a standard Stribeck model. It was concluded that the semi-empirical model could be integrated into a simulation environment and predict outputs in a superior fashion when compared to the Stribeck friction model. Thus it was concluded that the stated hypothesis is consistent with the experimental evidence shown by all hydraulic actuators considered. Further, it was also observed that the traditional Stribeck form (steady state dynamic friction) does change with increasing acceleration to the point that the standard breakaway friction almost disappears. It is evident that the 2D LUT is a viable tool for modeling the non-steady state friction of hydraulic actuation systems. The semi-empirical 2D LUT model so developed is a more global representation of hydraulic actuator lubricated friction. In this research, only linear hydraulic actuators were considered; however, the novel nonlinear semi-empirical 2D LUT lubricated friction model can be applied to any actuator (linear and rotary) and provides a new way in which the dynamic friction can be viewed and modeled.

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

Cette thèse s'intéresse à la cohomologie de fibrés en droite sur le fibré cotangent de variétés projectives. Plus précisément, pour $G$ un groupe algébrique simple, connexe et simplement connexe, $P$ un sous-groupe maximal de $G$ et $\omega$ un générateur dominant du groupe de caractères de $P$, on cherche à comprendre les groupes de cohomologie $H^i(T^*(G/P),\mathcal{L})$ où $\mathcal{L}$ est le faisceau des sections d'un fibré en droite sur $T^*(G/P)$. Sous certaines conditions, nous allons montrer qu'il existe un isomorphisme, à graduation près, entre $H^i(T^*(G/P),\mathcal{L})$ et $H^i(T^*(G/P),\mathcal{L}^{\vee})$ Après avoir travaillé dans un contexte théorique, nous nous intéresserons à certains sous-groupes paraboliques en lien avec les orbites nilpotentes. Dans ce cas, l'algèbre de Lie du radical unipotent de $P$, que nous noterons $\nLie$, a une structure d'espace vectoriel préhomogène. Nous pourrons alors déterminer quels cas vérifient les hypothèses nécessaires à la preuve de l'isomorphisme en montrant l'existence d'un $P$-covariant $f$ dans $\comp[\nLie]$ et en étudiant ses propriétés. Nous nous intéresserons ensuite aux singularités de la variété affine $V(f)$. Nous serons en mesure de montrer que sa normalisation est à singularités rationnelles. / In this thesis, we study the cohomology of line bundles on cotangent bundle of projective varieties. To be more precise, let $G$ be an semisimple algebraic group which is simply connected, $P$ a maximal subgroup and $\omega$ a dominant weight that generates the character group of $P$. Our goal is to understand the cohomology groups $H^i(T^*(G/P),\mathcal{L})$ where $\mathcal{L}$ is the sheaf of sections of a line bundle on $T^*(G/P)$. Under some conditions, we will show that there exists an isomorphism, up to grading, between $H^i(T^*(G/P),\mathcal{L})$ and $H^i(T^*(G/P),\mathcal{L}^{\vee})$. After we worked in a theoretical setting, we will focus on maximal parabolic subgroups related to nilpotent varieties. In this case, the Lie algebra of the unipotent radical of $P$ has a structure of prehomogeneous vector spaces. We will be able to determine which cases verify the hypothesis of the isomorphism by showing the existence of a $P$-covariant $f$ in $\comp[\nLie]$ and by studying its properties. We will be interested by the singularities of the affine variety $V(f)$. We will show that the normalisation of $V(f)$ has rational singularities.

La radiographie X, synchrotron et avec une source de laboratoire, a été appliquée pour étudier des phénomènes en relation avec la solidification directionnelle d’alliages Al-Cu sous champs externes et dépendants du temps. La formation des microstructures de solidification a été analysée durant des campagnes des vols paraboliques dans le cadre du projet XRMON. Nous avons montré que les variations du niveau de gravité modifiait la couche solutale, et donc la surfusion du liquide devant l’interface solide/liquide. En particulier, l’accélération du niveau de gravité induit une transition Colonnaire-Equiaxe pour un alliage affiné. Pour un alliage non-affiné, la vitesse de croissance des dendrites varie suivant les variations du niveau de gravité durant chaque parabole. Le second volet de ma thèse concerne l'effet d'un champ magnétique permanent sur la solidification équiaxe dans un gradient de température. Nous avons établi que le couplage entre le champ magnétique et le gradient de température génère une force Thermo-Electro-Magnétique qui modifie la trajectoire des grains équiaxes au cours de leur sédimentation. L’action de cette force est bien décrite par le modèle analytique de Wang et al., moyennant des corrections associées au confinement et à la morphologie des grains. Certains effets du champ magnétique sur le liquide ont été également révélés. / X-ray radiography with synchrotron and laboratory source was applied to study phenomena related to the directional solidification of Al-Cu alloys under external fields and time-dependent. The solidification microstructures formation has been analyzed during parabolic flight campaigns in the framework of XRMON project. We have shown that the gravity level variations modified the solutal layer and then the liquid undercooling ahead of the solid/liquid interface. In particular, the acceleration of gravity level induces the Columnar-to-Equiaxed transition in a refined alloy. For a non-refined alloy, the dendrite growth velocity changes with respect to the gravity level variations during each parabola. The second part of this thesis relates to the effect of a permanent magnetic field on the equiaxed solidification in a temperature gradient. The coupling between the magnetic field and the temperature gradient generates a Thermo-Electro-Magnetic force that modifies the equiaxed grains trajectory during their sedimentation. The action of this force is well described by an analytical model proposed by Wang et al. with additional corrections associated to the wall confinement and the grain morphology. Effects of the magnetic field on the liquid have been also revealed.

Dans cette thèse, nous classifions représentations typiques pour certaines composants Bernstein. Suite aux travaux de Henniart dans le cas de GL_2(F) et Paskunas pour les composants cuspidales, nous classifions représentations typiques pour les composants de niveau zéro pour GL_n(F) pour n> 2, composants de série principale, composants avec Levi sous-groupe de la forme (n, 1) pour n>1 et certains composants avec sous-groupe de Levi de la forme (2,2). Chacun des composants ci-dessus est traité dans un chapitre distinct. La classification utilise la théorie des types développés par Bushnell-Kutzko d'une manière significative. Nous allons donner la classification en termes de types de Bushnell-Kutzko. / In this thesis we classify typical representations for certain non-cuspidal Bernstein components. Following the work of Henniart in the case of GL_2(F) and Paskunas for the cuspidal components, we classify typical representations for of level-zero components for GL_n(F) for n>2, principal series components, components with Levi subgroup of the form (n, 1) for n>1 and certain components with Levi subgroup of the form (2,2). Each of the above component is treated in a separate chapter. The classification uses the theory of types developed by Bushnell-Kutzko in a significant way. We will give the classification in terms of Bushnell-Kutzko types for a given inertial class.

Pour un grand nombre d'applications, les sources laser impulsionnelles femtoseconde (fs) doivent fournir des puissances toujours plus importantes. En régime impulsionnel, on recherche d'une part une forte puissance crête par impulsion, et d'autre part une forte puissance moyenne, c'est à dire un taux de répétition élevé. Parmi les technologies existantes, les amplificateurs à fibre optique dopée ytterbium présentent de nombreux avantages pour l'obtention de fortes puissances moyennes, cependant le fort confinement des faisceaux dans la fibre sur de grandes longueurs d'interaction induit inévitablement des effets non-linéaires, et limite ainsi la puissance crête accessible. Nous avons étudié lors de cette thèse la combinaison cohérente d'impulsions fs appliquée aux systèmes fibrés.Ayant déjà fait ses preuves dans les régimes d'amplification continu et nanoseconde, la combinaison cohérente de faisceaux (dite combinaison spatiale) permet de diviser une seule et unique source en N voies indépendantes, disposées en parallèle et incluant chacune un amplificateur. Les faisceaux amplifiés sont ensuite recombinés en espace libre en un seul et unique faisceau, qui contient toute la puissance des N amplificateurs sans accumuler les effets non-linéaires. Cette architecture permet théoriquement de monter d'un facteur N le niveau de puissance crête issu des systèmes d'amplification fibrés. Au cours de cette thèse, nous avons démontré la compatibilité et l'efficacité de cette méthode en régime d'amplification fs avec deux amplificateurs, selon différents procédés. Les expériences démontrent d'excellentes efficacités de combinaison ainsi qu'une très bonne préservation des caractéristiques temporelles et spatiales initiales de la source. Les procédés de combinaison cohérente nécessitent cependant un accord de phase entre différents amplificateurs stable dans le temps, assuré en premier lieu par une boucle de rétroaction. Nous avons poursuivi notre étude en concevant une architecture totalement passive, permettant une implémentation plus simple d'un système de combinaison à deux faisceaux sans asservissement électronique. Enfin, une méthode passive de combinaison cohérente dans le domaine temporel est étudiée et caractérisée dans le domaine fs, et implémentée simultanément avec la méthode passive de combinaison spatiale proposée précédemment. Ces expériences démontrent la validité et la variété des concepts proposés, ainsi que leurs nombreuses perspectives pour les systèmes d'amplification fs fibrés. / Applications addressed by femtosecond (fs) laser sources are requiring increasing pulse energies and increasing average powers. Ytterbium-doped fiber amplifiers are excellent candidates to generate high average powers at high repetition rates, but present strong disadvantages in terms of peak power. Indeed, the tight confinement of the beam over long interaction length induces nonlinear effects at high peak-powers that affect the overall performances of fiber systems. This work describes coherent combining methods that can be used to scale the performances of femtosecond laser sources.Coherent beam combining has been widely used in CW regime and more recently in the nanosecond range. It consists in splitting a single seed into N beam replicas, amplified each by independent amplifiers in parallel. Their respective outputs are combined in free space into one single beam that carries the power of the N amplifiers without cumulating nonlinearities. This architecture allows scaling both peak and average powers of the amplification systems. We have studied and demonstrated the efficiency of active coherent beam combining in the fs regime with two fiber amplifiers, which are peak-power limited. The experiments show the preservation of the temporal/spectral/spatial properties of the combined pulses, with high combination efficiencies.Coherent beam combining methods require phase-matching between all the beams to combine. This is usually achieved by an active feedback loop on each amplifier along with a phase detection scheme. We demonstrate that a Sagnac interferometer can be used to ensure perfect and stable phase-matching over time, which considerably simplifies the setup. Finally, another passive combining method known as divided-pulse amplification, acting in the temporal domain, is studied and demonstrated in the fs regime. It is coupled with the passive spatial combining method described above to scale the number of pulse divisions. All these experiments show the compatibility of coherent combining concepts in the fs regime and provide new opportunities for fiber amplifier systems.