Cette thèse est consacrée à deux aspects du programme de Langlands local p-adique et de la compatibilité local-global p-adique.Dans la première partie, j'étudie la question de savoir comment extraire, d'un certain sous-espace Hecke-isotypique de formes automorphes modulo p, suffisament d'invariants d'une représentation galoisienne. Soient p un nombre premier, n>2 un entier, et F un corps à multiplication complexe dans lequel p est complètement décomposé. Supposons qu'une représentation galoisienne automorphe continue r-:Gal(Q-/F)→GLn(F-p) est triangulaire supérieure et suffisament générique ( dans un certain sens ) en une place w au-dessus de p. On montre, en admettant un résultat d'élimination de poids de Serre prouvé dans [LLMPQ], que la classe d'isomorphisme de r-|_Gal(Q-p/Fw) est déterminée par l'action de GLn(Fw) sur un espace de formes automorphes modulo p découpé par l'idéal maximal associée à r- dans une algèbre de Hecke. En particulier, on montre que la partie sauvagement ramifiée de r-|_Gal(Q-p/Fw) est déterminée par l'action de sommes de Jacobi ( vus comme éléments de Fp[GLn(Fp)] ) sur cet espace.La deuxième partie de ma thèse vise à établir une relation entre les résultats précédents de [Schr11], [Bre17] and [BD18]. Soient E une extension finie de Qp suffisamment grande et ρp: Gal(Q-p/Qp)→GL3(E) une représentation p-adique semi-stable telle que la représentation de Weil-Deligne WD(ρp) associée a un opérateur de monodromie N de rang 2 et que la filtration de Hodge associée est non-critique. On sait que la filtration de Hodge de ρp dépend de trois invariants dans E. On construit une famille de représentations localement analytiques Σ^min(λ, L1, L2, L3) qui dépend de trois invariants L1, L2, L3 dans E et telle que chaque représentation contient la représentation localement algébrique Algotimes Steinberg déterminée par ρp. Quand ρp provient, pour un groupe unitaire convenable G/Q, d'une représentation automorphe π de G(A_Q) avec un niveau fixé U^p premier avec p, on montre ( sous quelques hypothèses techniques ) qu'il existe une unique représentation localement analytique dans la famille ci-dessus qui est une sous-représentation du sous-espace Hecke-isotypique associé dans la cohomologie complétée de niveau U^p. On rappelle que [Bre17] a construit une famille de représentations localement analytiques qui dépend de quatre invariants (voir (4) dans [Bre17]) avec une propriété similaire. On donne un critère purement de théorie de représentation: si une représentation Π dans la famille de Breuil se plonge dans un certain sous-espace Hecke-isotypique de la cohomologie complétée, alors elle se plonge nécessairement dans une Σ^min(λ, L1, L2, L3) pour certains choix de L1, L2, L3 dans E qui sont déterminés explicitement par Π. De plus, certains sous-quotients naturels de Σ^min(λ, L1, L2, L3) permettent de construite un complexe de représentations localement analytiques qui "réalise" l'objet dérivé abstrait Σ(λ, underline{L}) defini dans [Schr11]. / This thesis is devoted to two aspects of the p-adic local Langlands program and p-adic local-global compatibility.In the first part, I study the problem of how to capture enough invariants of a local Galois representation from a certain Hecke-isotypic subspace of mod p automorphic forms. Let p be a prime number, n>2 an integer, and F a CM field in which p splits completely. Assume that a continuous automorphic Galois representation r-:Gal(Q-/F)→GLn(F-p) is upper-triangular and satisfies certain genericity conditions at a place w above p, and that every subquotient of r-|_Gal(Q-p/Fw) of dimension >2 is Fontaine-Laffaille generic. We show that the isomorphism class of r-|_Gal(Q-p/Fw) is determined by GLn(Fw)-action on a space of mod p algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to r-, assuming a weight elimination result which is now a theorem to appear in [LLMPQ]. In particular, we show that the wildly ramified part of r-|_Gal(Q-p/Fw) is determined by the action of Jacobi sum operators ( seen as elements of Fp[GLn(Fp)] ) on this space.The second part of my thesis aims at clarifying the relation between previous results in [Schr11], [Bre17] and [BD18]. Let E be a sufficiently large finite extension of Qp and ρp be a p-adic semi-stable representation Gal(Q-p/Qp)→GL3(E) such that the Weil-Deligne representation WD(ρp) associated with it has rank two monodromy operator N and the Hodge filtration associated with it is non-critical. We know that the Hodge filtration of ρp depends on three invariants in E. We construct a family of locally analytic representations Σ^min(λ, L1, L2, L3) of GL3(Qp) depending on three invariants L1, L2, L3 in E with each of the representation containing the locally algebraic representation Algotimes Steinberg determined by ρp. When ρp comes from an automorphic representation π of G(A_Q) with a fixed level U^p prime to p for a suitable unitary group G/Q, we show ( under some technical assumption ) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the associated Hecke-isotypic subspace in the completed cohomology with level U^p. We recall that [Bre17] constructed a family of locally analytic representations depending on four invariants ( cf. (4) in [Bre17] ) with a similar property. We give a purely representation theoretic criterion: if a representation Π in Breuil's family embeds into a certain Hecke-isotypic subspace of completed cohomology, then it must equally embed into Σ^min(λ, L1, L2, L3) for certain choices of L1, L2, L3 in E determined explicitly by Π. Moreover, certain natural subquotients of Σ^min(λ, L1, L2, L3) give a true complex of locally analytic representations that realizes the derived object Σ(λ, underline{L}) [Schr11]. Consequently, the family of locally analytic representations Σ^min(λ, L1, L2, L3) give a relation between the higher L-invariants studied in [Bre17] as well as [BD18] and the p-adic dilogarithm function which appears in the construction of Σ^min(λ, L1, L2, L3) in [Schr11].

[pt] Ao longo das últimas décadas, mais intensamente, a microscopia de força atômica (AFM) tem sido utilizada em atividades de pesquisa e desenvolvimento, notadamente para a caracterização de materiais. Não obstante os avanços e sofisticações da técnica, a determinação do módulo de elasticidade de materiais poliméricos continua sendo um desafio dada a complexidade de caracterização de materiais complexos, comumente gerando expressivo espalhamento nos resultados de medição. Esta pesquisa de mestrado apresenta mapeamentos da morfologia e medições do módulo de elasticidade de celulose comercial (fibras de bambu e de eucalipto) e lignocelulósico, referenciados a materiais poliméricos sintéticos já caracterizados (considerados como padrão). A microscopia de força atômica (AFM), no modo peak force tapping, alternativamente fundamentada no modelo Derajaguin-Muller-Toporo, DMT (que é o modelo incorporado no software da Bruker, fabricante do equipamento AFM) e no modelo JohnsonKendall-Robert, JKR (proposto neste trabalho, fazendo-se uso de um algoritmo Matlab), foi utilizada como técnica de medição do módulo de elasticidade dos materiais investigados. A partir dos ensaios realizados, foi possível avaliar o método de medição com base em critérios metrológicos e determinar o modelo de cálculo mais apropriado para a caracterização dos materiais investigados. Para os materiais padrão utilizados como referência, e cujos valores do módulo de elasticidade são informados pelos respectivos fornecedores (E igual 2,7 mais ou menos 0,1 GPa, para o poliestireno e E igual 1,8 GPa, para a poliamida), o modelo JKR mostrou-se mais adequado para fundamentar o cálculo do módulo de elasticidade. Enquanto o modelo JKR reproduziu o valor do fabricante com 6,3 porcento, para o caso do poliestireno (E igual 2,540754 mais ou menos 0,056 GPa) e com 12,5 porcento, para o caso da poliamida (E igual 1,680050 mais ou menos 0,041 GPa), o modelo DMT produziu resultados com desvios superiores a 50 porcento do valor de referência considerado como padrão. Fundamentado nesses mesmos modelos DMT e JKR, o trabalho avaliou o módulo de elasticidade de outros materiais, mais especificamente: (i) filme de celulose extraída do eucalipto; (ii) filme de celulose microfobrilada comercial; (iii) filme de celulose microfibrilada, depositada no laboratório pela técnica de ultrassom e (iv) amostras de bambu (Dendrocalamus Giganteus), extraídas da seção externa e interna do colmo d o bambu. Os resultados são discutidos em conformidade ao rigor metrológico dos componentes estocásticos e determinísticos da incerteza associada às medições realizadas, confirmando que a microscopia de força atômica (AFM), no modo peak force tapping se apresenta como técnica promissora para ser utilizada como ferramenta metrológica para a caracterização de materiais poliméricos na escala nanométrica. / [en] Over the past few decades, more intensely, atomic force microscopy (AFM) has been used in research and development activities, notably for the characterization of materials. Despite the advances and sophistications of the technique, the determination of the modulus of elasticity of polymeric materials remains a challenge given the complexity of characterizing complex materials, commonly generating significant spread in the measurement results. This research presents mappings of the morphology and measurements of the elasticity modulus of commercial cellulose (bamboo and eucalyptus fibers) and lignocellulosic, referenced to already characterized synthetic polymeric materials (considered as standard). Atomic force microscopy (AFM), in the peak force tapping mode, alternatively based on the Derajaguin-Muller-Toporo model, DMT (which is the model incorporated by Bruker in its AFM software) and the Johnson-KendallRobert model, JKR (proposed in this work, using a Matlab algorithm), was used as a technique for measuring the elasticity module of the investigated materials. From the tests performed, it was possible to evaluate the measurement method based on metrological criteria and determine the most appropriate calculation model for the characterization of the investigated materials. For the standard materials used as a reference, whose elasticity modulus values are reported by the respective suppliers (E equal 2.7 more or less 0.1 GPa, for polystyrene and E equal 1.8 GPa, for polyamide), the JKR model proved to be most suitable to support the calculation of the modulus of elasticity. While the JKR model reproduced the manufacturer s value with 6.3 percent, for the case of polystyrene (E equal 2.540754 more or less 0.056 GPa) and with 12.5 percent, for the case of polyamide (E equal 1.680050 more or less 0.041 GPa), the DMT model produced results with deviations greater than 50 percent from the reference value taken as standard reference values. Based on these DMT and JKR models, the work evaluated the elasticity module of other materials, more specifically: (i) cellulose film extracted from eucalyptus; (ii) commercial microfiltered cellulose film; (iii) microfibrillated cellulose film, deposited in the laboratory by and ultrasound technique and (iv) samples of bamboo (Dendrocalamus Giganteus), extracted from the external and internal section of the bamboo stem. The results are discussed in accordance with the metrological rigor of the stochastic and deterministic components of the uncertainty associated with the measurements performed, confirming that the atomic force microscopy (AFM), in the peak force tapping mode, presents itself as a promising technique to be used as a metrological tool for the characterization of polymeric materials at the nanoscale.

Model predictive control is a popular research focus in electric motor control as it allows designers to specify optimization goals and exhibits fast transient response. Availability of faster and more affordable computers makes it possible to implement these algorithms in real-time. Real-time implementation is not without challenges however as these algorithms exhibit high computational complexity. Field-programmable gate arrays are a potential solution to the high computational requirements. However, they can be time-consuming to develop for. In this thesis, we present a methodology that reduces the size and development time of field-programmable gate array based fixed-point model predictive motor controllers using automated numerical analysis, optimization and code generation. The methods can be applied to other domains where model predictive control is used. Here, we demonstrate the benefits of our methodology by using it to build a motor controller at various sampling rates for an interior permanent magnet synchronous motor, tested in simulation at up to 125 kHz. Performance is then evaluated on a physical test bench with sampling rates up to 35 kHz, limited by the inverter. Our results show that the low latency achievable in our design allows for the exclusion of delay compensation common in other implementations and that automated reduction of numerical precision can allow the controller design to be compacted. / Thesis / Master of Applied Science (MASc)

The multiple constant multiplication (MCM) operation is a fundamental operation in digital signal processing (DSP) and digital image processing (DIP). Examples of the MCM are in finite impulse response (FIR) and infinite impulse response (IIR) filters, matrix multiplication, and transforms. The aim of this work is minimizing the complexity of the MCM operation using common subexpression elimination (CSE) technique and redundant number representations. The CSE technique searches and eliminates common digit patterns (subexpressions) among MCM coefficients. More common subexpressions can be found by representing the MCM coefficients using redundant number representations. A CSE algorithm is proposed that works on a type of redundant numbers called the zero-dominant set (ZDS). The ZDS is an extension over the representations of minimum number of non-zero digits called minimum Hamming weight (MHW). Using the ZDS improves CSE algorithms' performance as compared with using the MHW representations. The disadvantage of using the ZDS is it increases the possibility of overlapping patterns (digit collisions). In this case, one or more digits are shared between a number of patterns. Eliminating a pattern results in losing other patterns because of eliminating the common digits. A pattern preservation algorithm (PPA) is developed to resolve the overlapping patterns in the representations. A tree and graph encoders are proposed to generate a larger space of number representations. The algorithms generate redundant representations of a value for a given digit set, radix, and wordlength. The tree encoder is modified to search for common subexpressions simultaneously with generating of the representation tree. A complexity measure is proposed to compare between the subexpressions at each node. The algorithm terminates generating the rest of the representation tree when it finds subexpressions with maximum sharing. This reduces the search space while minimizes the hardware complexity. A combinatoric model of the MCM problem is proposed in this work. The model is obtained by enumerating all the possible solutions of the MCM that resemble a graph called the demand graph. Arc routing on this graph gives the solutions of the MCM problem. A similar arc routing is found in the capacitated arc routing such as the winter salting problem. Ant colony optimization (ACO) meta-heuristics is proposed to traverse the demand graph. The ACO is simulated on a PC using Python programming language. This is to verify the model correctness and the work of the ACO. A parallel simulation of the ACO is carried out on a multi-core super computer using C++ boost graph library.

Le sol présente une variabilité spatiale des propriétés physiques et mécaniques dont les effets sur des structures légères avec semelles filantes et sur les conduites enterrées ne sont pas bien pris en compte dans leur dimensionnement. Cette variabilité naturelle peut être très importante dans le cas de ces ouvrages car elle induit des tassements différentiels, dont les conséquences peuvent être dommageables : fissures dans les murs, les poutres ou encore des fuites dans les réseaux d’assainissement. La variabilité naturelle du sol et l'incertitude liée à la connaissance imparfaite des propriétés du sol et/ou du béton ou de l'acier de la structure sont les principales sources d'incertitude dans le choix des paramètres de calcul pour le dimensionnement de ces structures. Dans cette thèse, une approche analytique avec les méthodes probabilistes (FOSM et SOSM) et le modèle de Winkler, puis numérique avec le couplage de la méthode des éléments finis avec des approches géostatistiques ont été successivement menées pour modéliser le comportement des semelles filantes et des conduites enterrés lorsque les incertitudes sur les propriétés mécaniques du sol et de la structure sont prises en compte dans leur dimensionnement. Il apparait ainsi, l’importance du comportement longitudinal de ces ouvrages et du poids des incertitudes dans leur dimensionnement. / Soil exhibits spatial heterogeneities resulting from the history of its deposition and aggregation processes that occur in different physical and chemical environments. This inherent or natural variability can be very important in the case of the superficial geotechnical works inducing differential settlements, whose consequences on structural response can be harmful: local failures, cracking in beams or walls, leakage in sewers. Natural variability of soil and uncertainty related to imperfect knowledge in soil properties and/or of concrete or steel of the structure, are the major source of uncertainty in the choice of the design parameters. In this thesis the probabilistic methods in geotechnical engineering, the analytical Winkler model and the coupling of the finite element method with geostatistical approaches were successively used to model the behavior of shallow foundations and buried pipe networks when soil and structure uncertainties are considered in their design.