Nessa tese, o objetivo principal foi verificar a estabilidade de sistemas atômicos condensados, sujeitos a diferentes combinações lineares e não-lineares de redes ópticas bie tridimensionais, considerando algumas situações simétricas e assimétricas. Com esse objetivo, foram realizadas análises variacionais e simulações numéricas exatas da equação não-linear correspondente que descreve sistemas condensados de Bose-Einstein, tipo-Schrödinger, mais conhecida como equação de Gross-Pitaevskii. No caso bidimensional, com redes ópticas cruzadas, linear e não-linear, foi verificada a existência de estabilidade para certas regiões de parâmetros das interações. Observou-se que essa estabilidade desaparece ao se incluir uma terceira dimensão sem a presença de um potencial de confinamento. No caso tridimensional, considerando redes ópticas lineares e não-lineares cruzadas, a estabilidade só ocorre quando consideramos uma interação confinante na terceira dimensão, no caso, uma segunda rede óptica linear. Finalmente, espera-se que nossos resultados venham a ser úteis para estudos experimentais que vêm sendo feitos em laboratórios de átomos ultra-frios. / In this thesis, the main objective was the verification of stability of condensed atomic systems, subject to different combinations of linear and nonlinear bi- and tridimensional optical lattices , considering some symmetric and asymmetric situations. With this objective, were performed variational analyzes and numerical exact simulations of the nonlinear Schrödinger-type equation that describes Bose-Einstein condensate systems, better known as Gross-Pitaevskii equation. In two-dimensional case, with a crossed linear and nonlinear optical lattice, the stability was confirmed for certain parameter regions of the interactions. It was observed that the stability disappears when including a third dimension without the presence of a confinement potential. In the three dimensional case, considering crossed linear and nonlinear optical lattices, stability occurs only when considering an interaction confining the third dimension, in this case a second linear optical lattice. Finally, it is expected that our results will be useful for experimental studies which have been done in the laboratories of ultra-cold atoms. Keywords:

La cryptographie à base de réseaux euclidiens a généré un vif intérêt durant les deux dernièresdécennies grâce à des propriétés intéressantes, incluant une conjecture de résistance àl’ordinateur quantique, de fortes garanties de sécurité provenant d’hypothèses de difficulté sur lepire cas et la construction de schémas de chiffrement pleinement homomorphes. Cela dit, bienqu’elle soit cruciale à bon nombre de schémas à base de réseaux euclidiens, la génération debruit gaussien reste peu étudiée et continue de limiter l’efficacité de cette cryptographie nouvelle.Cette thèse s’attelle dans un premier temps à améliorer l’efficacité des générateurs de bruitgaussien pour les signatures hache-puis-signe à base de réseaux euclidiens. Nous proposons unnouvel algorithme non-centré, avec un compromis temps-mémoire flexible, aussi rapide que savariante centrée pour des tables pré-calculées de tailles acceptables en pratique. Nousemployons également la divergence de Rényi afin de réduire la précision nécessaire à la doubleprécision standard. Notre second propos tient à construire Falcon, un nouveau schéma designature hache-puis-signe, basé sur la méthode théorique de Gentry, Peikert et Vaikuntanathanpour les signatures à base de réseaux euclidiens. Nous instancions cette méthode sur les réseauxNTRU avec un nouvel algorithme de génération de trappes. / Lattice-based cryptography has generated considerable interest in the last two decades due toattractive features, including conjectured security against quantum attacks, strong securityguarantees from worst-case hardness assumptions and constructions of fully homomorphicencryption schemes. On the other hand, even though it is a crucial part of many lattice-basedschemes, Gaussian sampling is still lagging and continues to limit the effectiveness of this newcryptography. The first goal of this thesis is to improve the efficiency of Gaussian sampling forlattice-based hash-and-sign signature schemes. We propose a non-centered algorithm, with aflexible time-memory tradeoff, as fast as its centered variant for practicable size of precomputedtables. We also use the Rényi divergence to bound the precision requirement to the standarddouble precision. Our second objective is to construct Falcon, a new hash-and-sign signaturescheme, based on the theoretical framework of Gentry, Peikert and Vaikuntanathan for latticebasedsignatures. We instantiate that framework over NTRU lattices with a new trapdoor sampler.

Die ultrakalte Atom- und Molekülephysik, zu welcher man zum Beispiel bei der Bose-Einstein-Kondensation von verdünnten Gasen Zugang hat, wurde untersucht. In solchen Systemen dominieren Zwei-Körper-Stöße und ihre detaillierte Untersuchung ist eines der zentralen Themen dieser Arbeit. Diese wurden durchgeführt unter Berücksichtigung von elementaren chemischen Reaktionen, Photoassoziation und magnetischen Feshbach-Resonanzen. Weiterhin wurden Untersuchungen von Atomen in optischen Gittern durchgeführt. Die Viel-Teilchen-Systeme wurden nicht nur mit dem üblichen mean-field Ansatz behandelt, sondern auch darüber hinausgehend, um die voll korrelierte Bewegung zu simulieren. / The ultracold atomic and molecular physics as it is accessible, e.g., in Bose-Einstein condensates of dilute gases was investigated. In such systems two-body collisions are dominant and their detailed study is one of the central topics of this work. They were done considering elementary chemical reactions as photoassociation, and magnetic Feshbach resonances. Additionally, studies of atoms in optical lattice sites were carried out. The many-body systems were not only considered within the usually adopted mean-field approach but also beyond that in order to simulate the fully correlated motion.

Utilizamos o modelo de Bose-Hubbard para estudar as estabilidades dinâmica e termodinâmica dos condensados numa rede óptica periódica circular. O nosso principal objetivo foi investigar a existência de condensados metaestáveis no sistema. Deduzimos e resolvemos a equação de Gross-Pitaevskii e, a partir da análise das soluções, foi possível mostrar que o sistema se condensa em estados com momento modular bem definido. Esses estados formam uma base que diagonaliza o termo que descreve o tunelamento atômico no hamiltoniano de Bose-Hubbard. No contexto da teoria de Bogoliubov deduzimos para cada condensado, o hamiltoniano efetivo cuja diagonalização determina o espectro das excitações coletivas do sistema. Identificamos corretamente o modo de energia zero, conseqüência da violação da conservação do número de átomos, e verificamos que este possui momento modular igual ao do condensado. No estudo da estabilidade vimos que todos os condensados com momento modular nos 2º e 3º quadrantes são termodinamicamente instáveis e as respectivas condições de estabilidade dinâmica dependem dos parâmetros de controle do sistema. Por outro lado os condensados com momento modular nos 1º e 4º quadrantes são todos dinamicamente estáveis enquanto que, nesse caso, é a estabilidade termodinâmica que depende dos parâmetros de controle do sistema. Nessa análise verificamos que o condensado com momento modular zero, que corresponde ao mínimo global da energia, é sempre estável. Determinamos exatamente o intervalo nos parâmetros de controle a partir do qual podemos encontrar condensados metaestáveis no sistema. Examinamos como a competição entre as intensidades dos termos de tunelamento e repulsão local afeta a estabilidade dos condensados. Essa competição define dois regimes distintos: Rabi, onde a coerência entre estados localizados nos sítios é mantida, e Fock, onde não há mais essa coerência e a aplicabilidade da aproximação de Bogoliubov é questionável. / We use the Bose-Hubbard model to study the dynamical and thermodynamical stabilities of condensates in a circular periodic optical lattice. Our main goal was to investigate the existence of metastable condensates in the system. We derive and solve the Gross-Pitaevskii equation, and from the analysis of the solutions it was possible to show that the system condenses in states with well-defined modular momentum. These states constitute a basis that diagonalizes the term of the Bose-Hubbard Hamiltonian which describes the dynamics of atomic tunneling. In the framework of Bogoliubov theory we determine, for each condensate, the effective Hamiltonian whose diagonalization give us the collective excitation spectrum of the system. We show that the mode associated to a zero eigenvalue, which is a consequence of the violation of atoms number conservation, has the same modular momentum of the condensate. The condensates with modular momentum in the 2nd and 3rd quadrants are all thermodynamically unstable whereas the dynamical stability depends on the control parameters. On the other hand, the condensates with modular momentum in the 1st and 4th quadrants are all dynamically stable whereas the thermodynamical stability depends on the control parameters. Our analysis shows that the condensate with modular momentum zero, which corresponds to a global minimum of energy, is always stable independently of the control parameters. We determine, exactly, the range on the control parameters where it is possible to detect metastability in the system. We have studied how the competition between the intensities of the tunneling and local interaction terms affects the stability of the condensates. This competition defines two distinct regimes: Rabi, where the coherence between states localized in the sites is achieved, and Fock, where this coherence is not achieved and the validity of Bogoliubov approximation is questionable.

Made available in DSpace on 2017-07-21T19:26:03Z (GMT). No. of bitstreams: 1 Romeu Miqueias.pdf: 9797233 bytes, checksum: d4b08f71cb22063247e9bb495366dd55 (MD5) Previous issue date: 2013-01-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Synchronization is a dynamical behavior exhibited by a wide range of systems. Neurons, firefly and Josephson junctions are examples of these systems. It is defined as an adjustment of rhythms of oscillating objects due to weak interaction between them, and it is studied using different mathematical models including the coupled map lattices (CMLs). In CML the synchronization corresponds to process in which all state variables become identical at the same instant. Usually we study the CML synchronization by calculating the conditional Lyapunov exponents. However, if the coupling or network topology is time-varying, this exponents are not readily determined. In this work we propose new quantities to study the synchronization in these CMLs. These quantities are defined as weighted averages over all possible topologies and, if the topology is constant, they are equivalent to the usual Lyapunov exponents. We find an analytical expression for the effective quantities when the topology is invariant over translation on the network and demonstrate that an ensemble of short time observations can be used to predict the long-term behavior of the lattice. Also we point that, if network has constant and homogeneous structure, the effective quantities correspond to the projection on the eigenvectors associated with this structure. We show the availability of effective quantities using them to build a lattice with constant topology that exhibits the same synchronization critical properties of the lattice with time-varying topology. Finally, we present a cryptosystem for communication systems based on two replica synchronized networks whose elements are not synchronous. We investigate it as to operation time, robustness and security against intruders. Our results suggest that it is safe and efficient for a wide range of parameters. / A sincronização é um comportamento dinâmico exibido por uma ampla variedade de sistemas naturais tais como neurônios, vaga-lumes e junções Josephson. Ela é definida como um ajuste de ritmos de objetos oscilantes devido a uma fraca interação entre eles, e é estudada usando diferentes modelos matemáticos tais como as redes de mapas acoplados (RMAs). Em uma RMA, o processo de sincronização representa uma evolução conjunta entre todas variáveis de estados. Este processo é geralmente investigado com base nos expoentes de Lyapunov condicionais. No entanto, para redes com topologia variável tais expoentes não são facilmente determinados. Neste trabalho propomos novas quantidades para estudar a sincronização nestas redes. Estas quantidades são definidas como médias ponderadas sobre todas as topologias possíveis e, no caso em que a topologia é constante, equivalem aos expoentes de Lyapunov usuais. Encontramos uma expressão analítica para as quantidades efetivas para o caso em que a topologia é invariante sobre translação na rede e demonstramos que um conjunto de observações sobre um intervalo curto de tempo pode ser usado para predizer o comportamento da rede a longo prazo. Também verificamos que, se a rede possui uma estrutura constante e homogênea, as quantidades efetivas podem ser obtidas considerando a projeção sobre os autovetores associados a esta estrutura. Mostramos a eficácia das quantidades efetivas usando-as para construir uma rede com topologia constante que exibe as mesmas propriedades de sincronização da rede com topologia variável. Por último apresentamos um criptossistema para sistemas de comunicação que é baseado em duas réplicas de redes sincronizadas cujos elementos são não-síncronos. Investigamos este sistema quanto ao tempo de operação, a robustez e a segurança contra intrusos. Nossos resultados sugerem que ele é seguro e eficiente para uma amplo intervalo de parâmetros.

Cette thèse a pour objet l'étude des espaces classifiants pour les actions propres d'un groupe discret. La dimension géométrique propre est la plus petite dimension possible pour un tel espace (qui existe toujours). Nous montrons tout d'abord que pour un réseau dans le groupe d'isométries d'un espace symétrique de type non-compact sans facteur euclidien, la dimension géométrique propre est égale à la dimension cohomologique virtuelle. La preuve utilise le fait que si le rang réel de l'espace est supérieur ou égal à 2 et le réseau est irréductible, alors il est arithmétique. Dans ce cas, nous pouvons calculer explicitement la dimension cohomologique virtuelle à l'aide du rang rationnel. Dans un deuxième temps, nous cherchons à construire concrètement des espaces classifiants pour les actions propres de dimension minimale. Nous essayons d'adapter la construction du "rétract bien équilibré" de Soulé et Ash (pour le cas SL(n,Z)) aux groupes arithmétiques Sp(2n,Z) et Aut(SL(n,Z)). Nous montrons qu'en fait cette construction ne s'étend pas. / In this thesis we study classifying spaces for proper actions of a discrete group. The proper geometric dimension is the smallest dimension of such a space (which always exists). Firstly we prove that for a lattice in the group of isometries of a symmetric space of the non-compact type without euclidean factors, the proper geometric dimension equals the virtual cohomological dimension. The proof relies on the fact that if the space has real rank at least 2 and if the lattice is irreducible, then it is arithmetic. In this case, the virtual cohomological dimension can be explicitly computed with the rational rank. Secondly we want to construct concretely classifying spaces for proper actions of minimal dimension. We try to adapt the construction of the "well-rounded retract" of Soulé and Ash (in the case SL(n,Z)) for the arithmetic groups Sp(2n,Z) and Aut(SL(n,Z)). We show that in fact this construction does not extend.

Made available in DSpace on 2014-12-17T15:41:41Z (GMT). No. of bitstreams: 1 GuymmannCS.pdf: 1754712 bytes, checksum: 731ccc6c9bb4f783f3a118094959297b (MD5) Previous issue date: 2008-02-29 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / Latices based on acrylic acid and ethyl methacrylate, crosslinked with 1,6‐propoxylate‐hexanodiol diacrylate were synthesized via emulsion polymerization with different monomeric compositions. The resultant latices were thickened with different NaOH/(acrylic acid) molar ratios and were characterized by titrimetry, zeta potential measurements, turbidimetry, and capillary viscometry. Intrinsic viscosity was determined for an uncrosslinked copolymer, using toluene as solvent. All the latices were coagulated with NaCl and washed with water at 60?C analyzed by FTIR spectrophotometry, in order to characterize functional groups from the copolymer and crosslinking agent. / L?tices ? base de ?cido acr?lico e metacrilato de etila, reticulados com 1,6‐diacrilato de propoxilato hexanodiol, foram sintetizados via polimeriza??o em emuls?o, com diferentes composi??es monom?ricas. Os l?tices sintetizados foram espessados com diferentes raz?es molares NaOH/(?cido acr?lico) e caracterizados por titrimetria, medidas de potencial zeta, turbidimetria e viscometria capilar. Todos os l?tices foram coagulados com NaCl, lavados com ?gua destilada e em seguida secos a 60 0C, para a an?lise de FTIR. A viscosidade intr?nseca do l?tex n?o reticulado foi determinada utilizando tolueno como solvente. O potencial zeta foi usado para determinar a carga superficial das part?culas e o FTIR para caracterizar os grupos funcionais dos comon?meros.

Esse trabalho visa a investigar a estrutura bi-Hamiltoniana de uma classe de sistemas dinâmicos. Depois de introduzir as ferramentas necessárias, a saber, as noções de variedade de Poisson, de grupo de PoissonLieedenetworknodiscoenoanêl,introduziremosossistemasdinâmicos relevantes nessa dissertação, chamados de reticulados de Coxeter-Toda. Esses sistemas dinâmicos, cujo espaço de fase pode ser identicado com umoportunoquocientedeumacéluladupladeCoxeter-Bruhatdogrupo linear geral, são obtidos por redução do sistema de Toda em GLn. Na parte nal do presente trabalho apresentaremos alguns resultados relacionado à um sistema dinâmico discreto chamado de aplicação do pentagrama, o qual pode ser obtido através uma oportuna discretização do sistema dinâmico de Boussinesq. / This work aims to study the bi-Hamiltonian structure of a class of dynamical systems. After introducing the relevant tools, namely the notions of Poisson manifold, Poisson-Lie group and of network dened in a disc and in an annulus, we will introduce the dynamical systems of interest for this dissertation, i.e., the Coxeter-Toda lattices. These dynamical systems, whose phase-space can be identied with a suitable quotient of a Coxeter double Bruhat cell of the general linear group, are obtained by reduction starting from the Toda ow on GLn. In the nal part of the present work will be presented some results concerning a discrete integrable system close to the so called Pentagram map, which is a discretization of the Boussinesq dynamical system..

du fait de l'explosion actuelle des télécommunications, les opérateurs sont victimes d'une crise de croissance les obligeant à installer toujours plus de relais, à découper les cellules (zone de couverture d'un relais) en micro-cellules dans les grandes villes, afin de faire face à la demande toujours grandissante de communications. Les concepteurs des nouveaux réseaux de transmission sont donc constamment à la recherche d'une utilisation plus efficace des ressources disponibles

Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic Laplacian evolution operator and a coin toss instruction. In quantum theory, an alternative is to use the relativistic Dirac operator. That necessarily introduces an internal degree of freedom (chirality), which may be identified with the coin. The resultant walk spreads quadratically faster than the classical one, and can be applied to a variety of graph theoretical problems. We study in detail the problem of spatial search, i.e. finding a marked site on a hypercubic lattice in d-dimensions. For d=1, the scaling behaviour of classical and quantum spatial search is the same due to the restriction on movement. On the other hand, the restriction on movement hardly matters for d ≥ 3, and scaling behaviour close to Grover’s optimal algorithm(which has no restriction on movement) can be achieved. d=2 is the borderline critical dimension, where infrared divergence in propagation leads to logarithmic slow down that can be minimised using clever chirality flips. In support of these analytic expectations, we present numerical simulation results for d=2 to d=9, using a lattice implementation of the Dirac operator inspired by staggered fermions. We optimise the parameters of the algorithm, and the simulation results demonstrate that the number of binary oracle calls required for d= 2 and d ≥ 3 spatial search problems are O(√NlogN) and O(√N) respectively. Moreover, with increasing d, the results approach the optimal behaviour of Grover’s algorithm(corresponding to mean field theory or d → ∞ limit). In particular, the d = 3 scaling behaviour is only about 25% higher than the optimal value.