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Atoms in optical lattices are promising candidates to implement quantum information processing. Their behaviour is well understood on a microscopic level, they exhibit excellent coherence properties, and they can be easily manipulated using external fields. In very deep optical lattices, each atom is restricted to a single lattice site and can be used as a qubit. If the lattice is shallow enough such that the atoms can move, their properties can be used to simulate certain condensed matter phenomena such as superconductivity. In this thesis, we show how technical problems of optical lattices such as restricted decoherence times, or fundamental shortcomings such as the lack of phonons or strong spin interactions, can be overcome by using current or near-future experimental techniques. We introduce a scheme that makes it possible to simulate model Hamiltonians known from high-temperature superconductivity. For this purpose, previous simulation schemes to realise the spin interaction terms are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. This scheme makes a large range of parameters accessible, which is difficult to cover with a condensed matter setup. We also investigate the properties of optical lattices submerged into a Bose-Einstein condensate (BEC). A weak-coupling expansion in the BEC-impurity interaction strength is used to derive a model that describes the lattice atoms in terms of polarons, i.e.~atoms dressed by Bogoliubov phonons. This is analogous to the description of electrons in solids, and we observe similar effects such as a crossover from coherent to incoherent transport for increasing temperatures. Moreover, the condensate mediates an attractive off-site interaction, which leads to macroscopic clusters at experimentally realistic parameters. Since the atoms in the lattice can also be used as a quantum register with the BEC mediating a two-qubit gate, we derive a quantum master equation to examine the coherence properties of the atomic qubits. We show that the system exhibits sub- and superdecoherence and that a fast implementation of the two-qubit gate competes with dephasing. Finally, we show how to realise the encoding of qubits in a decoherence-free subspace (DFS) using optical lattices. We develop methods for implementing robust gate operations on qubits encoded in a DFS exploiting collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of the gate operations and show that a robust implementation of quantum repeaters can be achieved using our setup. We compare the robust repeater scheme to one that makes use of conventional qubits only, and show the conditions under which one outperforms the other.

The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

Le but des modèles traditionnels en classification (comme les partitions et les hiérarchies de parties) est de permettre de discriminer sans ambiguïté et donc de produire des classes non empiétantes (i.e. l’intersection de deux classes est vide ou une classe est incluse dans l'autre). Cependant, cette exigence de non ambiguïté peut conduire à occulter de l’information. Dans le cas des plantes hybrides en biologie par exemple ou encore de textes appartenant à plusieurs genres en analyse textuelle. Les modèles généraux comme les hypergraphes ou les treillis permettent de prendre en compte l’empiétance entre les classes. Plus précisément, les modèles dits "totalement équilibrés" autorisent l'empiétance tout en conservant certaines contraintes utiles en classification.En apprentissage automatique, les arbres de décision, très utilisés pour leur simplicité d'utilisation et de compréhension réalisent à chaque étape un partitionnement d'un ensemble en deux sous-ensembles.Nous montrons dans ce travail différents liens entre la classification traditionnelle et l'apprentissage automatique supervisé et montrons certains apports que chacun des deux mondes peut faire à l'autre.Nous proposons deux méthodes de classification mêlant les deux univers puis étendons la notion de binarité, très utilisée dans le cas des arbres, aux hypergraphes et aux treillis. Nous montrons alors l'équivalence entre les systèmes binarisables et les systèmes totalement équilibrés, faisant de ces derniers de parfaits candidats à la réalisation de modèles de classification en classes empiétantes. Nous proposons également diverses approximations de systèmes par des systèmes totalement équilibrés. / Traditionally, classification models (such as partitions and hierarchies) aim at separating without ambiguities and produce non-overlapping clusters (i.e two clusters are either disjoint or one is included in the other). However, this non ambiguity may lead to mask information such as in the case of hybrid plants in biology or of texts which belong to two (or more) different genres in textual analysis for instance. General models like hypergraphs or lattices allow to take into account overlapping clusters. More precisely, "totally balanced" models allows class infringement and presents some useful constraints for classification.In machine learning, decision trees are a widely used model as they are simple to use and understand. They are also based on the idea of partition of sets.We show in this work different links between traditional classification and supervised machine learning and show what each world can bring to the other.We propose two methods of classification which link the two universes. We then extend the notion of binarity, widely-used for trees, to hypergraphs and lattices. We show the equivalence between binarizable systems and totally balanced systems, which makes of totally balanced structures a great candidate for classification models with class infringement. We also propose some approximation methods of any system (lattice, hypergraph, dissimilarity) by a totally balanced one.

Neste trabalho, apresentamos estudos sobre o magnetotransporte em um sistema de bicamadas com uma rede de antipontos triangulares em campos magnéticos baixos sob a aplicação de campos elétricos externos, que são produzidos por voltagens de porta. A bicamada é feita em poços quânticos largos (wide quantum well) de alta densidade eletrônica, formado em heteroestruturas semicondutoras de AlxGa1xAs=GaAs. Oscila- ções magneto-inter-sub-banda (MIS) são observadas em poços quânticos largos de alta densidade eletrônica com duas sub-bandas ocupadas. Estas são originadas pelo espalhamento inter-sub-bandas e tem um máximo para campos magnéticos B que satisfazem a condição de alinhamento entre os leques dos níveis de Landau de cada sub-banda. Oscila- ções de comensurabilidade são observadas na magnetoresistência que é sensível ao arranjo do potencial dos antipontos. A aplicação de campos elétricos faz diminuir o número de oscilações na magnetoresistência para campos magnéticos compreendidos entre 0; 1T e 0; 4T, observando-se uma transição das oscilações MIS aos efeitos de comensurabilidade. Aplicando voltagens de porta podemos variar a amplitude do potencial dos antipontos. / In this work, we present studying about magnetotransport in a bilayer system with triangular antidot lattices in low magnetic elds under the application of external electric eld. The bilayer forms inside a wide quantum well of high electron density in semiconductor heterostructures formed by AlxGa1xAs=GaAs. Magneto-inter-subband (MIS) oscillations are observed in a wide quantum wells of high electron density with two subbands occupied, and they are caused by intersubband scattering and have a maximum for a magnetic eld B that satises the alignment condition between the staircase of Landau level. Commensurability oscillations are observed in magnetoresistance, which is sensitive to the potential of antidot arrangements. The application of electric elds decrease the number of oscillations in the magnetoresistance for magnetic elds between 0; 1T and 0:4T, showing a transition of MIS oscillations to commensurability oscillations. We varied the amplitude of the potential of the antidots applying dierent gate voltages.

A Análise de Conceitos Formais (FCA) é uma teoria matemática que formaliza a noção de conceitos e hierarquias conceituais. De importância central a esta teoria é uma estrutura algébrica denominada reticulado de conceitos. Esta estrutura é definida em função de um conjunto de objetos, outro de atributos e uma relação que indica os atributos apresentados por cada objeto. Uma representação gráfica de um reticulado de conceitos, por meio de uma interface computacional, é capaz de expor regularidades presentes em dados a um usuário, e este pode então realizar tarefas de análise exploratória de dados. Este tipo de aplicação de FCA vem sendo empregado em dezenas de projetos pertencentes a áreas diversas, como medicina, serviços de inteligência, engenharia de software e bioinformática. Mostramos neste trabalho um sistema de análise exploratória de dados baseado em FCA, e sua utilização sobre dados reais. Também é mostrado como reticulados de conceitos podem ser empregados em interfaces de recuperação de informação. Do ponto de vista algorítmico, analisamos métodos computacionais para a determinação do reticulado de conceitos, e também de uma subestrutura simplificada, o conjunto de conceitos. O tamanho de um reticulado de conceitos pode ser exponencial em função dos tamanhos dos conjuntos de objetos e de atributos. Assim, é de vital interesse o estabelecimento de cotas superiores para o número de conceitos de um reticulado. Neste trabalho, apresentamos as cotas já conhecidas presentes na literatura. Também estabelecemos uma nova cota superior, e mostramos famílias de casos em que nossa cota superior é mais justa que as demais. Para algumas famílias particulares, nossa cota é polinomial, enquanto que as demais são exponenciais. / Formal Concept Analysis (FCA) is a mathematical theory that formalizes the notion of concepts and conceptual hierarchies. Of central importance to this theory is an algebraic structure termed concept lattice. Such structure becomes defined after being given one set of objects, one of attributes, and an incidence relation describing the attributes held by each object. A graphical representation of a concept lattice, by means of a computational interface, is capable of unfolding regularities present in data to an user, who is then able to conduct exploratory data analysis tasks. This sort of FCA application is currently deployed in tens of projects belonging to a wide range of areas, such as medicine, intelligence services, software engineering and bioinformatics. We show in this work an FCA-based system of exploratory data analysis, and its use over real data. Moreover, it is shown how concept lattices can be employed in information retrieval interfaces. From the algorithmic viewpoint, we analyse computational methods for the determination of a concept lattice, and also of a simplified substructure, the concept set. The size of a concept lattice can be exponential when compared to the size of the objects and the attributes sets. Therefore, it is of paramount interest the establishment of upper bounds for the number of concepts of a lattice. In this work, we present the upper bounds already known in the literature. We also establish a new upper bound, and show families of cases in which our bound is sharper than the others. For particular families, our bound is polynomial, whereas the other bounds are exponential.

Made available in DSpace on 2017-07-21T19:25:57Z (GMT). No. of bitstreams: 1 Romeu Szmoski.pdf: 12436289 bytes, checksum: 6e754552b30a293cd49e5368fbb3bfd0 (MD5) Previous issue date: 2009-03-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Brownian motion is one important topic of the non-equilibrium statistical mechanics and it is related to many natural phenomena. The first observations and theories on this motion were essential for understand the microscopic behavior of the nature and its influence on macroscopics observables. In this dissertation, we studied the dynamics of a system composed of several interacting Brownian particle from the point of view of coupled maps lattices. We use a map with a direct correlation to the abovementioned motion and we employ four different kinds of couplings in order to represent several ways of interaction among the particles. Using nonlinear dynamics tools, we observe the situations in which the particles velocities synchronize or show a tendency to the synchronized state. We also obtain algebrics expressions for the Lyapunov spectra of lattices with regular couplings whose interactions decays with distance as a power-law and we raise two hypotheses about Lyapunov exponents of a lattice with the coupling probability decreasing with the distance, as follows: the exponents of this lattice converge to the exponents of the lattice whose interactions decay with the distance in agreement to a power-law when the number of particles is very large; and the Lyapunov exponents of this lattice are given by the sum of the probabilities products of the each coupling matrix by eigenvalues of these matrixes. The values obtained for the Lyapunov exponents by means of the expressions deducted are in agreement with those obtained by numerical approximations techniques. Regarding distributions of the velocities of the particles, we observed that occur an aproximation to a Gaussian distribuition when the intensity of the coupling tends to its maximum. / O movimento browniano e um dos assuntos mais intrigantes da mecanica estatıstica de nao-equilıbrio e explica uma serie de fenomenos observados na natureza. As primeiras observaçoes a respeito deste movimento e as teorias propostas para descreve-lo foram fundamentais para entender o comportamento microscópico da natureza e a influência deste sobre observáveis macroscópicos. Nesta dissertação, estudamos a dinâmica de um sistema composto por várias partículas brownianas interagentes a partir de modelos de redes de mapas acoplados. Utilizamos um mapa que possui uma correspondência física direta com o movimento mencionado e empregamos quatro formas distintas de acoplamentos a fim de representar as várias formas de interação entre as partículas. Por meio de ferramentas da dinâmica não ao linear, observamos as situações em que as velocidades das partículas sincronizam ou tendem para o estado sincronizado. Também em obtivemos expressões exatas para determinar os expoentes de Lyapunov das redes com acoplamentos regulares cujas interações decaem com a distância segundo uma lei de potência e levantamos duas hipóteses sobre os expoentes de Lyapunov de uma rede com probabilidade de acoplamento decaindo com a distância, a saber: que os expoentes desta rede convergem para os expoentes da rede cujas interações decaem com a distância segundo uma lei de potência quando o número de partículas é muito grande; e que os expoentes de Lyapunov desta rede são dados pela soma dos produtos da probabilidade de ocorrer cada matriz de acoplamento pelos respectivos autovalores destas matrizes. Os valores obtidos para os expoentes de Lyapunov por meio das expressões deduzidas mostraram-se em acordo com aqueles obtidos por técnicas de aproximações numéricas. Em relação às distribuições das velocidades das partículas, observamos que elas se aproximam de uma gaussiana quando a intensidade do acoplamento tende a seu valor máximo.

Les comportement épistémiques, mobiles et spatiaux sont omniprésent dans les systèmes distribués aujourd’hui. La nature intrinsèque épistémique de ces types de systèmes provient des interactions des éleménts qui en font parties. La plupart des gens sont familiarisés avec des systèmes numériques où les utilisateurs peuvent partager ses croyances, opinions et même des mensonges intentionnels (des canulars). Aussi, les modèles de ces systèmes doivent tenir compte des interactions avec d’autres de même que leur nature distribués. Ces comportements spatiaux et mobiles font part d’applications où les données se déplacent dans des espaces (peut-être imbriqués) qui sont définis par, par exemple, cercles d’amis, des groupes, ou des dossiers partagés. Nous pensons donc qu’une solide compréhension des notion d’espaces, de mobilité spatial ainsi que le flux d’information épistémique est cruciale dans la plupart des modèles de systèmes distribués de nos jours.Les systèmes de contrainte (sc) fournissent les domaines et les opérations de base pour les fondements sémantiques de la famille de modèles déclaratifs formels de la théorie de la concurrence connu sous le nom de programmation concurrent par contraintes (pcc). Les systèmes des contraintes spatiales (scs) représentent des structures algébriques qui étendent sc pour raisonner sur les comportement spatiaux et épistémiques de base tel que croyance et l’extrusion. Les assertions spatiales et épistémiques peuvent être vues comme des modalités spécifiques. D’autres modalités peuvent être utilisées pour les assertions concernant le temps, les connaissances et même pour l’analyse des groupes entre autres concepts utilisés dans la spécification et la vérification des systèmes concurrents.Dans cette thèse nous étudions l’expressivité des systèmes de contraintes spatiales dans la perspective générale du comportement modal et épistémique. Nous montrerons que les systèmes de contraintes spatiales sont assez robustes pour capturer des modalités inverses et pour obtenir de nouveaux résultats pour les logiques modales. Également, nous montrerons que nous pouvons utiliser les scs pour exprimer un comportement épistémique fondamental comme connaissance. Finalement, nous donnerons une caractérisation algébrique de la notion de l’information distribuée au moyen de constructions sur scs. / Epistemic, mobile and spatial behaviour are common place in today’s distributed systems. The intrinsic epistemic nature of these systems arises from the interactions of the elements taking part of them. Most people are familiar with digital systems where users share their beliefs, opinions and even intentional lies (hoaxes). Models of those systems must take into account the interactions with others as well as the distributed quality these systems present. Spatial and mobile behaviour are exhibited by applications and data moving across (possibly nested) spaces defined by, for example, friend circles, groups, and shared folders. We therefore believe that a solid understanding of the notion of space and spatial mobility as well as the flow of epistemic information is relevant in many models of today’s distributed systems.Constraint systems (cs’s) provide the basic domains and opera- tions for the semantic foundations of the family of formal declarative models from concurrency theory known as concurrent constraint programming (ccp). Spatial constraint systems (scs’s) are algebraic structures that extend cs’s for reasoning about basic spatial and epistemic behaviour such as belief and extrusion. Both spatial and epistemic assertions can be viewed as specific modalities. Other modalities can be used for assertions about time, knowledge and even the analysis of groups among other concepts used in the specification and verification of concurrent systems.In this thesis we study the expressiveness of spatial constraint systems in the broader perspective of modal and epistemic behaviour. We shall show that spatial constraint systems are sufficiently robust to capture inverse modalities and to derive new results for modal logics. We shall show that we can use scs’s to express a fundamental epistemic behaviour such as knowledge. Finally we shall give an algebraic characterization of the notion of distributed information by means of constructors over scs’s.