Soit G un groupe de Lie résoluble connexe et H un de ses sous-groupes fermés connexes d'algèbres de Lie g et h respectivement. On note g* (resp. h*) le dual linéaire de g (resp. h) ). Le sujet de ma thèse consiste à étudier la restriction d'une série discrète π de G, associée à une orbite coadjointe Ω C g*, à H. Si la restriction de π à H se décompose en somme directe de représentations de H avec multiplicités finies, on dit que π est H-admissible. Notons Pg,n : Ω → h* l'application restriction. Il s'agit de démontrer la conjecture suivante due à Michel Duflo : 1. La représentation π est H-admissible si et seulement si l'application moment Pg,n est propre sur l'image. 2. Si π est H-admissible, et si T est une série discrète de H alors sa multiplicité dans la restriction de π à H doit pouvoir se calculer en « quantifiant » l'espace réduit correspondant (qui est compact dans ce cas). Dans cette thèse, nous apportons une réponse positive à cette conjecture dans deux situations, à savoir :(i) Le groupe G est résoluble exponentiel. (ii) Le groupe G est le produit semi direct d'un tore compact par le groupe de Heisenberg et H est un sous-groupe algébrique connexe. / Let G be a connected solvable Lie group and H a closed connected subgroup with Lie algebra g and h respectively. We denote g* (resp. h*) the dual of g (resp. h). The aim of my thesis is to study the restriction of a discrete series π of G, associated with a coadjoint orbit Ω C g* to H. If the restriction of π to H can be decomposed in to a direct sum of representations of H with finite multiplicities, we say that π is H-admissible. Let Pg,n : Ω → h* denote the restriction map. My objective is to show the following conjecture due to Michel Duflo : 1. The representation π i s H-admissible if and only if the moment application Pg,n is proper on the image. 2. If π is H-admissible, and if T is a discrete series of H then it s multiplicity in the restriction of π to H must be calculated by « quantifying » the corresponding reduced space (that is compact in this case). In this thesis, we provide a positive response to this conjecture in two situations, namely when: (i) G is exponential solvable Lie group. (ii) G is the semi direct product of a compact torus and the Heisenberg group and H is a connected algebraic subgroup.

This thesis concerns the wake control and flow dynamic analysis for a flow around a circular cylinder at different Reynolds numbers using reduced models. The wake control and dynamics in the reduced space were addressed using the sensitivity theory and the adjoint method. In the case of wake control, it was possible to predict the physical parameters of the active and passive controllers on the wake of the main cylinder. On the other hand, in the construction of the reduced space, a new shift mode calculated from a perturbation of the mean flow was proposed using the sensitivity to base flow modifications. The mathematical basis of the reduced space was constructed using a Fourier modal decomposition of the flow enriched by the shift mode. The proposed reduced space made possible the recomposition of the flow and the comparison with the physical parameters calculated in the physical space. Additionally, using the reduced space, it was possible to determine the transition dynamics between the equilibrium point of the Navier Stokes equation and the non-linear saturation state using the Landau coefficients obtained in the reduced model, opening the possibility of solving the flow around a 2D and 3D cylinder with low computational cost. / Esta tese trata sobre o controle de esteira assim como a análise dinâmica do escoamento em torno de um cilindro a diferentes números de Reynolds usando modelos reduzidos. O controle de esteira e a dinâmica no espaço reduzido foram abordados usando a teoria da sensibilidade e o método adjunto. No caso de controle de esteira, foi possível prever os parâmetros físicos dos controladores ativos e passivos no escoamento do cilindro principal. Por outro lado, na construção do espaço reduzido, foi proposto um novo modo de deslocamento (shift mode) calculado a partir de uma perturbação do campo médio usando a sensibilidade às modificações do campo base. A base matemática do espaço reduzido foi construída usando uma decomposição modal de Fourier do escoamento enriquecido pelo modo de deslocamento (shift mode). O espaço reduzido proposto possibilitou a recomposição do escoamento e a comparação com os parâmetros físicos calculados no espaço físico. Além disso, usando o espaço reduzido, foi possível determinar a dinâmica de transição entre o ponto de equilíbrio da equação de Navier Stokes e o estado de saturação não linear usando os coeficientes de Landau obtidos no modelo reduzido, abrindo a possibilidade de resolver o escoamento em torno de um cilindro 2D e 3D com baixo custo computacional

This thesis concerns the wake control and flow dynamic analysis for a flow around a circular cylinder at different Reynolds numbers using reduced models. The wake control and dynamics in the reduced space were addressed using the sensitivity theory and the adjoint method. In the case of wake control, it was possible to predict the physical parameters of the active and passive controllers on the wake of the main cylinder. On the other hand, in the construction of the reduced space, a new shift mode calculated from a perturbation of the mean flow was proposed using the sensitivity to base flow modifications. The mathematical basis of the reduced space was constructed using a Fourier modal decomposition of the flow enriched by the shift mode. The proposed reduced space made possible the recomposition of the flow and the comparison with the physical parameters calculated in the physical space. Additionally, using the reduced space, it was possible to determine the transition dynamics between the equilibrium point of the Navier Stokes equation and the non-linear saturation state using the Landau coefficients obtained in the reduced model, opening the possibility of solving the flow around a 2D and 3D cylinder with low computational cost. / Esta tese trata sobre o controle de esteira assim como a análise dinâmica do escoamento em torno de um cilindro a diferentes números de Reynolds usando modelos reduzidos. O controle de esteira e a dinâmica no espaço reduzido foram abordados usando a teoria da sensibilidade e o método adjunto. No caso de controle de esteira, foi possível prever os parâmetros físicos dos controladores ativos e passivos no escoamento do cilindro principal. Por outro lado, na construção do espaço reduzido, foi proposto um novo modo de deslocamento (shift mode) calculado a partir de uma perturbação do campo médio usando a sensibilidade às modificações do campo base. A base matemática do espaço reduzido foi construída usando uma decomposição modal de Fourier do escoamento enriquecido pelo modo de deslocamento (shift mode). O espaço reduzido proposto possibilitou a recomposição do escoamento e a comparação com os parâmetros físicos calculados no espaço físico. Além disso, usando o espaço reduzido, foi possível determinar a dinâmica de transição entre o ponto de equilíbrio da equação de Navier Stokes e o estado de saturação não linear usando os coeficientes de Landau obtidos no modelo reduzido, abrindo a possibilidade de resolver o escoamento em torno de um cilindro 2D e 3D com baixo custo computacional

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text