Nous considérons le problème d’un électron sur des pavages quasipériodiques en une et deux dimensions. Nous introduisons tout d’abord les pavages quasipériodiques d’un point de vue géométrique, et défendons en particulier l’idée que ces pavages sont les pavages apériodiques les plus proches de la périodicité. Nous concentrant plus particulièrement sur l’un des pavages quasipériodiques les plus simples, la chaîne de Fibonacci, nous montrons à l’aide d’un groupe de renormalisation que la multifractalité des états électroniques découle directement de l’invariance d’échelle de la chaîne. Élargissant ensuite notre champ d’étude à un ensemble de chaînes quasipériodiques, nous nous intéressons au théorème de label des gaps, qui décrit comment la géométrie d’une chaîne donnée contraint les valeurs que peut prendre la densité d’états intégrée dans les gaps du spectre électronique. Plus précisément, nous nous intéressons à la façon dont l’énoncé de ce théorème est modifié lorsque l’on considère une séquence d’approximants périodiques approchant une chaîne quasipériodique. Enfin, nous montrons comment des champs de hauteurs géométriques peuvent être utilisés pour construire des états électroniques exacts sur des pavages en une et deux dimensions. Ces états sont robustes aux perturbations du hamiltonien, sous réserve que ces dernières respectent les symétries du pavage sous-jacent. Nous relions les dimensions fractales de ces états à la distribution de probabilités des hauteurs, que nous calculons de façon exacte. Dans le cas des chaînes quasipériodiques, nous montrons que la conductivité suit une loi d’échelle de la taille de l’échantillon, dont l’exposant est relié à cette même distribution de probabilités. / We consider the problem of a single electron on one and two-dimensional quasiperiodic tilings. We first introduce quasiperiodic tilings from a geometrical point of view, and point out that among aperiodic tilings, they are the closest to being periodic. Focusing on one of the simplest one-dimensional quasiperiodic tilings, the Fibonacci chain, we show, with the help of a renormalization group analysis, that the multifractality of the electronic states is a direct consequence of the scale invariance of the chain. Considering now a broader class of quasiperiodic chains, we study the gap labeling theorem, which relates the geometry of a given chain to the set of values the integrated density of states can take in the gaps of the electronic spectrum. More precisely, we study how this theorem is modified when considering a sequence of approximant chains approaching a quasiperiodic one. Finally, we show how geometrical height fields can be used to construct exact eigenstates on one and two-dimensional quasiperiodic tilings. These states are robust to perturbations of the Hamiltonian, provided that they respect the symmetries of the underlying tiling. These states are critical, and we relate their fractal dimensions to the probability distribution of the height field, which we compute exactly. In the case of quasiperiodic chains, we show that the conductivity follows a scaling law, with an exponent given by the same probability distribution.

Cette thèse est consacrée à la simulation du mélange de scalaires passifs à grand nombre de Schmidt (faible diffusion), au moyen d’un modèle de sous-maille structurel pour la Simulation aux Grandes Echelles (LES pour Large Eddy Simulation) reposant sur le caractère multifractal des champs de gradient en turbulence. L’analyse multifractale des champs de dissipation scalaire permet, à l’aide d’une description statistique des singularités, de prendre en compte l’intermittence inhérente à ces champs. Des simulations numériques directes du mélange à différents nombres de Schmidt supérieurs à l’unité sont mises en oeuvre. Une analyse multifractale au moyen de différentes méthodes est menée afin d’obtenir les spectres de singularités de la dissipation scalaire. Une implantation du modèle de sous-maille multifractal pour la vitesse, proposé par Burton et al., est d’abord réalisée dans le code volumes finis YALES2.Une modification du modèle équivalent pour les scalaires, reposant sur une cascade multiplicative pour reconstruire la dissipation scalaire de sous-maille, est proposée afin de prendre en compte le micro-mélange à grand nombre de Schmidt. Ce modèle de sous-maille est alors évalué au moyen de tests a priori. / This thesis is focused on the simulation of turbulent mixing of passive scalars at high Schmidt numbers (low diffusivity). The modeling work is based on a structural subgrid-scale model for Large Eddy Simulation relying on the multifractal nature of gradient fields in turbulence.The multifractal formalism provides a mean to handle the characteristic intermittency of scalar dissipation fields through a statistical description of their singularities. Direct Numerical Simulations of mixing at several Schmidt numbers above unity are run with a dedicated code. Different methods are used to perform a multifractal analysis of scalar dissipation. The multifractal subgrid-scale model of Burton et al. for velocity is implemented in the Finite Volume code YALES2. A modification of the equivalent multifractal model for scalars is proposed to take into account micro-mixing at high Schmidt numbers. The model shows satisfactory results when tested a priori against direct simulations.

No description available.

The presence of rain over oceans interferes with the measurement of sea surface wind speed and direction from the Sea Winds scatterometer and as a result wind measurements contain biases in rain regions. In past research at the Central Florida Remote Sensing Lab, it has been observed that rain has multi-fractal behavior. In this report we present an algorithm to detect the presence of rain so that rain regions are flagged. The forward and aft views of the horizontal polarization σ0 are used for the extraction of textural information with the help of multi-fractals. A single negated multi-fractal exponent is computed to discriminate between wind and rain. Pixels with exponent value above a threshold are classified as rain pixels and those that do not meet the threshold are further examined with the help of correlation of the multi-fractal exponent within a predefined neighborhood of individual pixels. It was observed that the rain has less correlation within a neighborhood compared to wind. This property is utilized for reactivation of the pixels that fall below a certain threshold of correlation. An advantage of the algorithm is that it requires no training, that is, once a threshold is set, it does not need any further adjustments. Validation results are presented through comparison with the Tropical Rainfall Measurement Mission Microwave Imager (TMI) 2A12 rain retrieval product for one whole day. The results show that the algorithm is efficient in suppressing non-rain (wind) pixels. Also algorithm deficiencies are discussed, for high wind speed regions. Comparisons with other proposed approaches will also be presented.

No description available.

The collisions frequency of dispersed phases (particles, droplets, bubbles) in a turbulent carrier phase is a fundamental quantity that is needed for modeling multiphase flows with applications to chemical processes, minerals flotation, food science, and many other industries. In this dissertation, numerical simulations are performed to determine collisions frequency of bi-dispersed particles (solid particles and bubbles) in homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and Large Eddy simulations (LES) are conducted to determine velocity fluctuations of the carrier phase. The DNS results are used to validate existing theoretical models as well as the LES results. The dissertation also presents a CFD-based flotation model for predicting the pulp recovery rate in froth flotation machines. In the direct numerical simulations work, particles and bubbles suspended in homogeneous isotropic turbulence are tracked and their collisions frequency is determined as a function of particle Stokes number. The effects of the dispersed phases on the carrier phase are neglected. Particles and bubbles of sizes on the order of Kolmogorov length scale are treated as point masses. Equations of motion of dispersed phases are integrated simultaneously with the equations of the carrier phase using the same time stepping scheme. In addition to Stokes drag, the pressure gradient in the carrier phase and added-mass forces are also included. The collision model used here allows overlap of particles and bubbles. Collisions kernel, radial relative velocity, and radial distribution function found by DNS are compared to theoretical models over a range of particle Stokes number. In general, good agreement between DNS and recent theoretical models is obtained for radial relative velocity for both particle-particle and particle-bubble collisions. The DNS results show that around Stokes number of unity particles of the same group undergo expected preferential concentration while particles and bubbles are segregated. The segregation behavior of particles and bubbles leads to a radial distribution function that is less than one. Existing theoretical models do not account for effects of this segregation behavior of particles and bubbles on the radial distribution function. In the large-eddy simulations efforts, the dissertation addresses the importance of the subgrid fluctuations on the collisions frequency and investigates techniques for predicting those fluctuations. The cases studied are of particles-particles and particles-bubbles collisions at Reynolds number Re_λ = 96. A study is conducted first by neglecting the effects of subgrid velocity fluctuations on particles and bubbles motions. It is found that around Stokes number of unity solid particles of the same group undergo the well known preferential concentration as observed in the DNS. Effects of pressure gradient on the particles are negligible due to their small sizes. Bubbles as a low inertia particles are very sensitive to subgrid velocity and acceleration fields where the effects of pressure gradient in the carrier phase are dominant. However, particle-bubble radial distribution functions from LES are not as low as that from DNS. To account for the effects of subgrid field on the dispersion of particles and bubbles, a new multifractal methodology has been developed to construct a subgrid vorticity field from the resolved vorticity field in frame work of LES. A Poisson's solver is used to obtain the subgrid velocity field from the subgrid vorticity field. Accounting for the subgrid velocity fluctuations (but neglecting pressure gradient) produced minor changes in the radial distribution function for particle-particle and particle-bubble collisions. We conclude from this study that for accurate particle tracking in LES the subgrid velocity fluctuations must be dynamically realizable field (temporally and spatially correlated with the large scale motion). Adding random SGS velocity fluctuations is not enough to capture the correct radial distribution functions of dispersed phases especially for bubbles-particles collisions where the pressure gradient term ( or acceleration Du_f′/Dt) is responsible for particle-bubble segregation around particle Stokes number near one. A CFD-based model for minerals flotation machines has been developed in this dissertation. The objective of flotation models is to predict the recovery rate of minerals from a flotation cell. The developed model advances the state-of-the-art of pulp recovery rate prediction by incorporating validated theoretical collisions frequency models and detailed hydrodynamics from two-phase flow simulations. Spatial distributions of dissipation rate and air volume fraction are determined by the two-phase hydrodynamic simulations. Knowing these parameters throughout the machine is essential in understanding the effectiveness of different components of flotation machine (rotor, stator or disperser, jets) on the flotation efficiency. The developed model not only predicts the average pulp recovery rate but also it indicates regions of high/low recovery rates. The CFD-based flotation model presented here can be used to determine the dependence of recovery rate constant at any locality within the pulp based on particle diameter, particle specfic gravity, contact angle, and surface tension. / Ph. D.

In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

On étudie certaines propriétés d'une classe de processus aléatoires réels à temps continu, les marches aléatoires multifractales. Une particularité remarquable de ces processus tient en leur propriété d'autosimilarité : la loi du processus à petite échelle est identique à celle à grande échelle moyennant un facteur aléatoire multiplicatif indépendant du processus. La première partie de la thèse se consacre à la question de la convergence du moment empirique de l'accroissement du processus dans une asymptotique assez générale, où le pas de l'accroissement peut tendre vers zéro en même temps que l'horizon d'observation tend vers l'infini. La deuxième partie propose une famille de tests non-paramétriques qui distinguent entre marches aléatoires multifractales et semi-martingales d'Itô. Après avoir montré la consistance de ces tests, on étudie leur comportement sur des données simulées. On construit dans la troisième partie un processus de marche aléatoire multifractale asymétrique tel que l'accroissement passé soit négativement corrélé avec le carré de l'accroissement futur. Ce type d'effet levier est notamment observé sur les prix d'actions et d'indices financiers. On compare les propriétés empiriques du processus obtenu avec des données réelles. La quatrième partie concerne l'estimation des paramètres du processus. On commence par montrer que sous certaines conditions, deux des trois paramètres ne peuvent être estimés. On étudie ensuite les performances théoriques et empiriques de différents estimateurs du troisième paramètre, le coefficient d'intermittence, dans un cas gaussien / We study some properties of a class of real-valued, continuous-time random processes, namely multifractal random walks. A striking feature of these processes lie in their scaling property : the distribution of the process at small scale is the same as the distribution at large scale, given some random multiplicative factor independent of the process. The first part of the dissertation deals with the convergence of the empirical moment of the increment of the process in a rather general asymptotic setting where the step of the increment may go to zero while the observation horizon may also go to infinity. In the second part, we propose a family of nonparametric tests that separate multifractal random walks from Itô semi-martingales. After showing the consistency of these tests, we study their behavior on simulations.In the third part, we build a skewed multifractal random walk process, such that the past increment is negatively correlated with the future squared increment. Such a "leverage effect" is notably seen on financial stock and index prices. We compare the empirical properties of this process with real data. The fourth part deals with the parametric estimation of the process. We first show that under certain conditions, one can not estimate two of the three parameters, even if the sample path is continuously observed on some interval. We next study the theoretical and empirical performances of some estimators of the third parameter, the intermittency coefficient, in a Gaussian case

The present work aims to provide a contribution to improve the efficiency the design flow of integrated systems, focusing, specifically, on the performance evaluation of its communication structures. The use of Transaction Level Modeling (TLM) is proposed, in order to take advantage of the reduction of design effort and time. Within the performance evaluation approaches, the utilization of traffic generators instead of full system simulations started to be adopted due to its higher time efficiency. Initial works on on-chip traffic generation focused on Poisson processes and classic Markovian models, which are unable to capture Long Range Dependence (LRD). This fact led to the adoption of fractal/self-similar models. Later advancements have shown that the traffic produced in multiprocessed systems can show higher degrees of complexity, what can be attributed to the presence multifractal characteristics. In this work, a methodology to evaluate the on-chip traffic and to the development of a transaction level traffic generator is proposed. The main contributions of this work are a detailed analysis of traffic time series obtained by TLM simulations and the study of the effects of the traffic generator on these simulations, concerning, mainly, the speedup-accuracy trade-off. The proposed analysis follow the multifractal paradigm, allowing system developers to (1) understand the statistical nature of on-chip traffic, (2) to obtain accurate representations of this traffic and (3) to build traffic generators that mimic processing elements realistically. Another contribution of this work is a comparison of the performance, considering the accuracy of the obtained synthetic traffic time series, between monofractal and multifractal models. All of the mentioned contributions were grouped throughout the detailed methodology presented on the present document, for which experiments were carried out. / O presente trabalho visa oferecer uma contribuição para o aumentar a eficiência do fluxo de projeto de sistemas integrados, focando, especificamente, na avaliação do desempenho de suas estruturas de comunicação. É proposta a utilização de simulações com modelos no nível de transações (TLM), com o objetivo de se obter vantagens da redução de esforço e tempo de projeto oferecidos por esta abordagem. Dentro das propostas de análise de desempenho, a utilização de geradores de tráfego ao invés simulações de sistema completo tem sido adotada devido a sua maior eficiência no tempo. Trabalhos iniciais na geração de tráfego intrachip focaram-se em processos de Poisson e em modelos de Markov clássicos, os quais não capturam Dependência de Longa Duração (LRD). Este fato levou a adoção de modelos fractais/auto-similares. Avanços posteriores mostraram que o tráfego produzido pelos elementos de sistemas multiprocessados podem apresentar maior grau de complexidade, que pode ser atribuída à presença de características multifractais. Neste trabalho, é proposta uma metodologia para a avaliação de tráfego intrachip para o desenvolvimento de um gerador de tráfego TLM. As principais contribuições deste trabalho são uma análise detalhada das séries temporais de tráfego obtidas nas simulações TLM e o estudo dos efeitos que o gerador de tráfego exerce sobre estas simulações, se concentrando, principalmente, na relação entre precisão e aceleração da simulação. As análises propostas se baseiam no paradigma multifractal, o qual permite (1) um maior entendimento da natureza estatística do tráfego pelos desenvolvedores de sistemas, (2) a obtenção de uma representação precisa deste tráfego e (3) a construção de geradores de tráfego que substituam elementos processantes de maneira realista. Outra contribuição deste trabalho é a comparação do desempenho, no que concerne a precisão das séries de tráfego sintéticas obtidas, de modelos monofractais e multifractais. Todas as contribuições mencionadas foram agrupadas na metodologia detalhada, apresentada no presente documento, sobre a qual experimentos foram realizados.