Nous étudions expérimentalement deux mémoires quantiques pour la lumière utilisant la transparence électromagnétiquement induite (EIT) dans des nuages froids de césium.Nous expliquons la pertinence des mémoires quantiques pour le développement de réseaux quantiques à longue distance, et décrivons la théorie de l’EIT en soulignant les paramètres essentiels pour l’implémentation de mémoires quantiques.Notre premier cas d’étude est un piège magnéto-optique en espace libre. Notre principal résultat est la démonstration du caractère multimode de ce système pour le stockage quantique de la lumière. Pour cela, nous utilisons des faisceaux de Laguerre-Gauss (LG), porteurs de moment angulaire orbital (OAM). Dans une première étape, nous avons montré que l’état de moment orbital d’impulsions lumineuses en régime de photons uniques est préservé lors du stockage dans la mémoire. Ensuite, nous avons implémenté un bit quantique comme une superposition de modes LG ayant des hélicités opposées. Nous avons développé un système original pour mesurer ces bits quantiques qui nous a permis de caractériser l’action de la mémoire. Nous avons ainsi pu montrer que le stockage quantique de ces bits quantiques.Le second système, également un nuage d’atomes froids, a la particularité que les atomes sont piégés optiquement autour d’un nano-guide d’onde. Ce design innovant permet une plus grande interaction entre lumière et matière, et facilite l’interfaçage des photons dans et hors de la mémoire. Nous décrivons la construction de ce dispositif et les premiers pas vers son utilisation en tant que mémoire quantique. / We present an experimental study of two optical quantum memory systems based on electromagnetically induced transparency (EIT) in cold cesium atoms.We explain the relevance of quantum memories for the development of large-scale quantum networks, we give a comprehensive theory of the EIT phenomenon and underline the role of relevant parameters regarding the implementation of quantum memories.The first system under study is prepared in a free-space magneto-optical trap. The main result of this thesis is the demonstration of the spatial multimode capability of this system at the quantum level. For this, we used Laguerre-Gaussian (LG) light beams, i.e. beams possessing a non-zero value of orbital angular momentum (OAM). In a first step, we showed that the orbital angular momentum of stored light pulses is preserved by the memory, deep in the single photon regime. In a second step, we encoded information in the orbital angular momentum state of a weak light pulse and defined a qubit using two LG beams of opposite helicities. We developed an original setup for the measurement of this OAM qubit and used it to characterize the action of the memory during the storage of such a light pulse. Our results show that the memory performs the quantum storage of such a qubit.The second system under study, also a cloud of cold atoms, has the specificity that the atoms are trapped optically in the vicinity of a nano-waveguide. This innovative design ensures a higher light-matter interaction and facilitates the interfacing of photons into and out of the memory. We describe the building of this setup and the first steps towards quantum memory implementations.

This final thesis deals with the approximation of time delay in time invariant systems. First, the generalized Laguerre functions and their characteristics are presented. After this, the approximation of the Dirac delta function with the help of these functions is shown. Also, the choice of the free parameters is discussed and the results are evaluated with the help of energy. In the final part of the thesis, the approximations of systems with generalized and simple Laguerre functions are compared.

Statistical aspects of random mosaics have not been heretofore given enough attention. This thesis deals with the derivation of estimators and statistical tests in a three-dimensional Poisson-Voronoi mosaic model. The first chapter compiles elementary results in the fields of point processes, random closed sets and particle processes. These are used in a second chapter to deduce geometric properties of random mosaics. The third chapter introduces the statistical research itself, estimators and model tests. Horvitz- Thompson estimator is introduced in order to correct statistics calculated on a reduced sample. Own results are tried in a computer simulation and compared to existing research in the last chapter. Mainly, the quality of estimators and the power of proposed tests is observed. 1

Cette thèse regroupe plusieurs travaux de combinatoire énumérative sur les permutations et permutations d'ensemble. Elle comporte 4 parties.Dans la première partie, nous répondons aux conjectures de Steingrimsson sur les partitions ordonnées d'ensemble. Plus précisément, nous montrons que les statistiques de Steingrimsson sur les partitions ordonnées d'ensemble ont la distribution euler-mahonienne. Dans la deuxième partie, nous introduisons et étudions une nouvelle classe de statistiques sur les mots : les statistiques "maj-inv". Ces dernières sont des interpolations graphiques des célèbres statistiques "indice majeur" et "nombre d'inversions". Dans la troisième partie, nous montrons que la distribution conjointe des statistiques"nombre de croisements" et "nombre d'imbrications" sur les partitions d'ensemble est symétrique. Nous étendrons aussi ce dernier résultat dans le cadre beaucoup plus large des 01-remplissages de "polyominoes lunaires".La quatrième et dernière partie est consacrée à l'étude combinatoire des q-polynômes de Laguerre d'Al-Salam-Chihara. Nous donnerons une interprétation combinatoire de la suite de moments et des coefficients de linéarisations de ces polynômes. / This thesis consists of four chapters, each on a different topic in enumerative combinatorics, all related in some way to the enumeration of permutations or set partitions. In the first chapter, we prove and generalize Steingrimsson's conjectures on Euler-Mahonian statistics on ordered set partitions. In the second chapter, we introduce and study a new class of statistics on words: the "maj-inv" statistics. These are graphical interpolation of the well-known "major index" and "inversion number".In the third chapter, we show that the joint distribution of the numbers of crossings and nestings on set partitions is symmetric. We also put this result in the larger context of enumeration of increasing and decreasing chains in 01-fillings of moon polyominoes.In the last chapter, we decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients.

Dans ce travail nous avons résolu deux problèmes principaux de la mise en forme topologique de faisceau paraxial pour les composants plans : la modalité et le polychromatisme.Nous les résolvons en introduisant de nouveaux concepts d’éléments optiques à interaction spin orbite,à savoir la “q-plate modale” et la “q-plate Bragg-Berry”. D’un côté, la q-plate modale convertit un faisceau gaussien incident en un faisceau de Laguerre-Gauss pour un indice radial et un indice d’azimut donnés, ce qui par conséquent dépasse les capacités des q-plates conventionnelles qui ne modifient que le degré de liberté azimutal, c.à.d. le moment orbital angulaire de la lumière. À des fins expérimentales, deux approches ont été développées : une basée sur des lames de verres nanostructurées artificiellement, l’autre sur des défauts topologiques de cristaux liquides auto-organisés naturellement. D’un autre côté, la q-plate Bragg-Berry consiste en une fine couche inhomogène de cristaux liquides chiraux (cholestériques) devant un miroir, ce qui fournit une mise en forme de faisceau spin-orbite pleinement efficace sur une large bande spectrale du faisceau incident, contrairement au q-plates conventionnelles qui ne sont fabriqués que pour une longueur d’onde donnée. Par ailleurs, nous obtenons une mise en forme de faisceau spin-orbite ultra-large bande en induisant une modulation de la structure supramoléculaire torsadée des cristaux liquides cholestériques selon la direction de propagation de la lumière. Nous montrons également que la présence du miroir derrière permet un puissant contrôle spatio-temporel des propriétés vectorielles de la polarisation du champ lumineux générées par la q-plate Bragg-Berry. / It is well-known that paraxial coherent electromagnetic fields can be completelycharacterized in terms of their radial and azimuthal spatial degrees of freedom in the transverse planethat add to the polarization degree of freedom and wavelength. In this work we address two mainissues of paraxial beam shaping that are the modality and the polychromaticity in the context of flatopticsthat we address by introducing novel concepts of spin-orbit optical elements. Namely, the‘modal q-plate’ and the ‘Bragg-Berry q-plate’. On the one hand, modal q-plate converts an incidentfundamental Gaussian beam into a Laguerre-Gaussian beam of given radial and azimuthal indices,hence going beyond the capabilities of conventional q-plates that only control the azimuthal degreeof freedom, i.e. the orbital angular momentum content of light. Towards experimental realization ofmodal q-plates, two approaches are developed: one based on artificially nanostructured glasses andanother based on naturally self-organized liquid crystal topological defects. On the other hand,Bragg-Berry q-plate consist of mirror-backed inhomogeneous thin film of chiral liquid crystal(cholesteric) that provides fully efficient spin-orbit beam shaping over broad spectral range of theincident beam, in contrast to the conventional q-plates that are designed for single wavelength.Furthermore, ultra-broadband spin-orbit beam shaping is achieved by inducing an extra modulationof the supramolecular twisted structure of the cholesteric liquid crystal along the propagationdirection. We also show that the presence of a back-mirror allows a powerful spatio-temporal controlof the polarization vectorial properties of the light fields generated by Bragg-Berry q-plate.

This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Ce travail est dédié à l’étude des inégalités de type Landau-Kolmogorov en normes L2. Les mesures utilisées sont celles d’Hermite, de Laguerre-Sonin et de Jacobi. Ces inégalités sont obtenues en utilisant une méthode variationnelle. Elles font intervenir la norme d’un polynômes p et celles de ces dérivées. Dans un premier temps, on s'intéresse aux inégalités en une variable réelle qui font intervenir un nombre quelconque de normes. Les constantes correspondantes sont prises dans le domaine où une certaine forme bilinéaire est définie positive. Ensuite, on généralise ces résultats aux polynômes à plusieurs variables réelles en utilisant le produit tensoriel dans L2 et en faisant intervenir au plus les dérivées partielles secondes. Pour les mesures d'Hermite et de Laguerre-Sonin, ces inégalités sont étendues à toutes les fonctions d'un espace de Sobolev. Pour la mesure de Jacobi on donne des inégalités uniquement pour les polynômes d'un degré fixé par rapport à chaque variable. / This thesis is devoted to Landau-Kolmogorov type inequalities in L2 norm. The measures which are used, are the Hermite, the Laguerre-Sonin and the Jacobi ones. These inequalities are obtained by using a variational method and the involved the square norms of a polynomial p and some of its derivatives. Initially, we focused on inequalities in one real variable that involve any number of norms. The corresponding constants are taken in the domain where a certain biblinear form is positive definite. Then we generalize these results to polynomials in several real variables using the tensor product in L2 and involving at most the second partial derivatives. For the Hermite and Laguerrre-Sonin cases, these inequalities are extended to all functions of a Sobolev space. For the Jacobi case inequalities are given only for polynomials of degree fixed with respect to each variable.

Markov-type inequalities provide upper bounds on the norm of the (higher order) derivative of an algebraic polynomial in terms of the norm of the polynomial itself. The present thesis considers the cases in which the norms are of the Laguerre, Gegenbauer, or Hermite type, with respective weights chosen differently on both sides of the inequality. An answer is given to the question on the best constant so that such an inequality is valid for every polynomial of degree at most n. The demanded best constant turns out to be the operator norm of the differential operator. The latter conicides with the tractable spectral norm of its matrix representation in an appropriate set of orthonormal bases. The methods to determine these norms vary tremendously, depending on the difference of the parameters accompanying the weights. Up to a very small gap in the parameter range, asymptotics for the best constant in each of the aforementioned cases are given. / Markovungleichungen liefern obere Schranken an die Norm einer (höheren) Ableitung eines algebraischen Polynoms in Bezug auf die Norm des Polynoms selbst. Diese vorliegende Arbeit betrachtet den Fall, dass die Normen vom Laguerre-, Gegenbauer- oder Hermitetyp sind, wobei die entsprechenden Gewichte auf beiden Seiten unterschiedlich gewählt werden. Es wird die kleinste Konstante bestimmt, sodass diese Ungleichung für jedes Polynom vom Grad höchstens n erfüllt ist. Die gesuchte kleinste Konstante kann als die Operatornorm des Differentialoperators dargestellt werden. Diese fällt aber mit der Spektralnorm der Matrixdarstellung in einem Paar geeignet gewählter Orthonormalbasen zusammen und kann daher gut behandelt werden. Zur Abschätzung dieser Normen kommen verschiedene Methoden zum Einsatz, die durch die Differenz der in den Gewichten auftretenden Parameter bestimmt werden. Bis auch eine kleine Lücke im Parameterbereich wird das asymptotische Verhalten der kleinsten Konstanten in jedem der betrachteten Fälle ermittelt.

We present an implicit approach for constructing smooth isolated or interconnected 3-D inscribed volumes which can be employed for volumetric modeling of various kinds of spongy or porous structures, such as volcanic rocks, pumice stones, Cancellus bones *, liquid or dry foam, radiolarians, cheese, and other similar materials. The inscribed volumes can be represented in their normal or positive forms to model natural pebbles or pearls, or in their inverted or negative forms to be used in porous structures, but regardless of their types, their smoothness and sizes are controlled by the user without losing the consistency of the shapes. We introduce two techniques for blending and creating interconnections between these inscribed volumes to achieve a great flexibility to adapt our approach to different types of porous structures, whether they are regular or irregular. We begin with a set of convex polytopes such as 3-D Voronoi diagram cells and compute inscribed volumes bounded by the cells. The cells can be irregular in shape, scale, and topology, and this irregularity transfers to the inscribed volumes, producing natural-looking spongy structures. Describing the inscribed volumes with implicit functions gives us a freedom to exploit volumetric surface combinations and deformations operations effortlessly / Graduate

Submitted by Luciclea Silva (luci@ufpa.br) on 2018-06-28T13:22:37Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_controlempcmultivariavel.pdf: 17090263 bytes, checksum: 9c35a8e3be627f9a64e83d1d833824c8 (MD5) / Approved for entry into archive by Luciclea Silva (luci@ufpa.br) on 2018-06-28T13:23:07Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_controlempcmultivariavel.pdf: 17090263 bytes, checksum: 9c35a8e3be627f9a64e83d1d833824c8 (MD5) / Made available in DSpace on 2018-06-28T13:23:07Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_controlempcmultivariavel.pdf: 17090263 bytes, checksum: 9c35a8e3be627f9a64e83d1d833824c8 (MD5) Previous issue date: 2018-03-01 / FAPESPA - Fundação Amazônia de Amparo a Estudos e Pesquisas / Neste trabalho apresenta-se o projeto de um controlador preditivo multivariável baseado em modelo, com restrições, usando as Funções de Laguerre, tendo o intuito de demonstrar os benefícios e facilidades da aplicação deste tipo de controlador em sistemas MIMO (MultipleInput and Multiple-Output) com restrições. O controlador proposto apresenta a vantagem de diminuir a carga computacional utilizada para resolver o problema de otimização envolvido no projeto, isto porque utiliza uma rede de filtros de funções ortonormais de Laguerre para obter a trajetória futura do sinal de controle dentro de um horizonte de predição, além de melhorar o compromisso entre a viabilidade do sinal de controle e o desempenho de malha fechada do sistema para os casos com restrições, no qual as funções de Laguerre são utilizadas em conjunto com a Programação Quadrática de Hildreth para encontrar a solução ótima do sinal de controle com restrições. Este controlador apresenta grandes vantagens se comparado com o controle preditivo baseado em modelo em sua abordagem clássica, em que os operadores de avanço de tempo são utilizados para predizer a trajetória futura do sinal de controle, o que leva à soluções, em alguns casos, pouco satisfatórias, e a uma alta carga computacional para casos onde o sinal de controle requer um longo horizonte de predição e uma alto desempenho em malha fechada. Este trabalho também relata testes experimentais com um manipulador robótico configurado como um sistema MIMO com três entradas e três saídas e testes simulados com a coluna de destilação binária de Wood e Berry que é um sistema MIMO com duas entradas e duas saídas, contendo atrasos de transporte. Os testes têm como objetivo comparar os resultados do controlador apresentado com o controlador que usa a abordagem tradicional e com isso demonstrar as vantagens do método usando as funções de Laguerre e sua eficiência para sistemas MIMO. / This work presents a constrained multivariable model predictive controller using Laguerre Functions. This controller uses a set of orthonormal Laguerre networks for representation of the control trajectory within a control horizon. In order to demonstrate the advantages of applying this type of controller in MIMO (Multiple-Input and Multiple-Output) systems, the Laguerre Functions Functions are used to decrease the computational load used to calculate the optimal control. In addition, It improves the compromise between control signal viability and closed-loop performance of the system. The Laguerre Functions are also used in conjunction with Hildreth’s Quadratic Programming to find the optimal solution for the case where the control signal is constrained. The proposed controller presents advantages when compared to the classical model predictive control approach, where forward shift operators are used to predict the future trajectory of the control signal, leading to unsatisfactory solutions and a high computational load for cases where the control signal demands a long prediction horizon and a high closed-loop performance.It is also reported the practical testes with a robotic manipulator configured as a MIMO system with three inputs and three outputs and tests simulated with the Wood and Berry binary distillation column which is a MIMO system with two inputs and two outputs, also containing transport time delays. The tests aim to compare the controller results presented with the traditional predictive control approach and thereby demonstrate the advantages of the method using the Laguerre functions and their efficiency for MIMO systems.