Dans cette thèse, on étudie principalement l’équation de Boltzmann homogène 3D pour les potentiels durs et les potentiels modérément mous et l’équivalence entre une EDS à sauts et l’EDP correspondante. En particulier, on calcule le spectre multifractal de certains processus stochastiques, on étudie le caractère bien-posé et la propagation du chaos pour l’équation de Boltzmann. Dans le premier chapitre, on étudie les propriétés trajectorielle pathologiques du processus stochastique (Vt)t_0 représentant l’évolution de la vitesse d’une particule typique dans un gaz modélisé par l’équation de Boltzmann pour les potentiels durs ou modérément mous. Nous montrons que ce processus est multifractal et qu’il a un spectre déterministe. Pour les potentiels durs, nous donnons aussi le spectre multifractal du processus $X_t =\int_0^t V_s ds$, représentant l’évolution de la position de la particule typique. Dans le deuxième chapitre, nous étudions l’unicité de la solution faible à l’équation de Boltzmann dans la classe de toutes les solutions mesures, pour les potentiels modérément mous. Ceci nous permet aussi d’obtenir un taux quantitatif de propagation du chaos pour le système de particules de Nanbu. / This thesis mainly studies the 3D homogeneous Boltzmann equation for hard potentials and moderately soft potentials and the equivalence between some jumping SDE and the corresponding PDE. In particular, we compute the multifractal spectrum of some stochastic processes, study the well-posedness and the propagation of chaos for the Boltzmann equation. The purpose of the first chapter is to study the pathwise properties of the stochastic process $(V_t)_{t\geq0}, representing the time-evolution of the velocity of a typical particle in a gas modeled by the Boltzmann equation for hard or moderately potentials. We show that this process is multifractal and has a deterministic spectrum. For hard potentials, we also give the multifractal spectrum of the process $X_t =\int_0^t V_s ds$, representing the time-evolution of the position of the typical particle. The second chapter is devoted to study the uniqueness of the weak solution to the Boltzmann equation in the class of all measure solutions, in the case of moderately soft potentials. This allows us to obtain a quantitive rate of propagation of chaos for Nanbu particle system for this singular interaction. Finally in the third chapter, we extend Figalli’s work [19] to study the relation between some jumping SDE and the corresponding Fokker-Planck equation. We prove that for any weak solution $(ft)_{t\in[0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time-marginals are given by the family $(f_t)_{t\in[0,T]$

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

Originality is the most basic requirement for the copyrighting of a work and may be viewed as the mirror image of copyright infringement. In terms of section 2(3) of the Copyright Act, a work remains eligible for copyright even if the making thereof involved an infringement of copyright in some other work. However, a certain aspect or a feature of a work (relating to the “substantial part” requirement for copyright infringement; “substantial part” being understood from a hypertext rather than a linear point of view) cannot be infringing and original. In this thesis, the South African legal understanding of the originality concept is determined from case law. Specific attention is given to the meaning of “skill” and “labour”; the protection afforded to mere labour in South Africa and certain selected other jurisdictions; how the subjective nature of the originality test is affected by the “meritorious distinctiveness” requirement; and the degree of own skill and/or labour required for a work to be original and consequently protected. The present literary concept of originality is derived from literary discussions that appeared in newspapers and other publications in the wake of six “plagiarism scandals” which each caused a furore in Afrikaans literary circles. Even though the terminology used by littérateurs differs from that used in legal circles, originality essentially means the same for littérateurs and lawyers. Skill and/or labour as required by law is reflected in the literary “crucial distance” concept. The fact that a sufficient degree of skill and/or labour is required is reflected in the fact that the literary standard of a work is determined on the basis of how “tightly woven” a work is. Although a general protection of original ideas would negatively influence the free flow of information, measures for the protection of ideas are developing, particularly in the United States of America, where ideas (especially in the film industry) are a very valuable commodity. As Swarth proposes, applying the “novelty” and “concreteness” criteria in inverse ratio to each other could help to create an environment where idea purveyors and prospective buyers felt more free to negotiate and enter into agreements over original ideas. Postmodernism, a contemporary interpretative strategy that reaches into nearly every aspect of modern society, is discussed with specific reference to its interaction with originality. The influences of two phenomena of postmodernism on the originality concept, namely hypertext and Chaos theory, are investigated. Recommendations are made regarding measures aimed at the retention of talented authors and the original content of works in the wake of plagiarism scandals, while still holding the wrongdoer responsible for his actions. Certain suggestions are also made regarding the accessibility of courts and the supplementation of the few available precedents regarding originality in creative literary works. / Thesis (LLD)--University of Pretoria, 2007. / Private Law / unrestricted

The land–atmosphere interactions and the coupling between climate and land surface hydrological processes are gaining interest in the recent past. The increased knowledge in hydro climatology and the global hydrological cycle, with terrestrial and atmospheric feedbacks, led to the utilization of the climate variables and atmospheric tele-connections in modeling the hydrological processes like rainfall and runoff. Numerous statistical and dynamical models employing different combinations of predictor variables and mathematical equations have been developed on this aspect. The relevance of predictor variables is usually measured through the observed linear correlation between the predictor and the predictand. However, many predictor climatic variables are found to have been switching the relationships over time, which demands a replacement of these variables. The unsatisfactory performance of both the statistical and dynamical models demands a more authentic method for assessing the dependency between the climatic variables and hydrologic processes by taking into account the nonlinear causal relationships and the instability due to these nonlinear interactions. The most obvious cause for limited predictability in even a perfect model with high resolution observations is the nonlinearity of the hydrological systems [Bloschl and Zehe, 2005]. This is mainly due to the chaotic nature of the weather and its sensitiveness to initial conditions [Lorenz, 1963], which restricts the predictability of day-to-day weather to only a few days or weeks. The present thesis deals with developing association rules to extract the causal relationships between the climatic variables and rainfall and to unearth the frequent predictor patterns that precede the extreme episodes of rainfall using a time series data mining algorithm. The inherent nonlinearity and uncertainty due to the chaotic nature of hydrologic processes (rainfall and runoff) is modeled through a nonlinear prediction method. Methodologies are developed to increase the predictability and reduce the predictive uncertainty of chaotic hydrologic series. A data mining algorithm making use of the concepts of minimal occurrences with constraints and time lags is used to discover association rules between extreme rainfall events and climatic indices. The algorithm considers only the extreme events as the target episodes (consequents) by separating these from the normal episodes, which are quite frequent and finds the time-lagged relationships with the climatic indices, which are treated as the antecedents. Association rules are generated for all the five homogenous regions of India (as defined by Indian Institute of Tropical Meteorology) and also for All India by making use of the data from 1960-1982. The analysis of the rules shows that strong relationships exist between the extreme rainfall events and the climatic indices chosen, i.e., Darwin Sea Level Pressure (DSLP), North Atlantic Oscillation (NAO), Nino 3.4 and Sea Surface Temperature (SST) values. Validation of the rules using data for the period 1983-2005, clearly shows that most of the rules are repeating and for some rules, even if they are not exactly the same, the combinations of the indices mentioned in these rules are the same during validation period with slight variations in the representative classes taken by the indices. The significance of treating rainfall as a chaotic system instead of a stochastic system for a better understanding of the underlying dynamics has been taken up by various studies recently. However, an important limitation of all these approaches is the dependence on a single method for identifying the chaotic nature and the parameters involved. In the present study, an attempt is made to identify chaos using various techniques and the behaviour of daily rainfall series in different regions. Daily rainfall data of three regions with contrasting characteristics (mainly in the spatial area covered), Malaprabha river basin, Mahanadi river basin and All India for the period 1955 to 2000 are used for the study. Auto-correlation and mutual information methods are used to determine the delay time for the phase space reconstruction. Optimum embedding dimension is determined using correlation dimension, false nearest neighbour algorithm and also nonlinear prediction methods. The low embedding dimensions obtained from these methods indicate the existence of low dimensional chaos in the three rainfall series considered. Correlation dimension method is repeated on the phase randomized and first derivative of the data series to check the existence of any pseudo low-dimensional chaos [Osborne and Provenzale, 1989]. Positive Lyapunov exponents obtained prove the exponential divergence of the trajectories and hence the unpredictability. Surrogate data test is also done to further confirm the nonlinear structure of the rainfall series. A limit in predictability in chaotic system arises mainly due to its sensitivity to the infinitesimal changes in its initial conditions and also due to the ineffectiveness of the model to reveal the underlying dynamics of the system. In the present study, an attempt is made to quantify these uncertainties involved and thereby improve the predictability by adopting a nonlinear ensemble prediction. A range of plausible parameters is used for generating an ensemble of predictions of rainfall for each year separately for the period 1996 to 2000 using the data till the preceding year. For analyzing the sensitiveness to initial conditions, predictions are made from two different months in a year viz., from the beginning of January and June. The reasonably good predictions obtained indicate the efficiency of the nonlinear prediction method for predicting the rainfall series. Also, the rank probability skill score and the rank histograms show that the ensembles generated are reliable with a good spread and skill. A comparison of results of the three regions indicates that although they are chaotic in nature, the spatial averaging over a large area can increase the dimension and improve the predictability, thus destroying the chaotic nature. The predictability of the chaotic daily rainfall series is improved by utilizing information from various climatic indices and adopting a multivariate nonlinear ensemble prediction. Daily rainfall data of Malaprabha river basin, India for the period 1955 to 2000 is used for the study. A multivariate phase space is generated, considering a climate data set of 16 variables. The redundancy, if any, of this atmospheric data set is further removed by employing principal component analysis (PCA) method and thereby reducing it to 8 principal components (PCs). This multivariate series (rainfall along with 8 PCs) are found to exhibit a low dimensional chaotic nature with dimension 10. Nonlinear prediction is done using univariate series (rainfall alone) and multivariate series for different combinations of embedding dimensions and delay times. The uncertainty in initial conditions is thus addressed by reconstructing the phase space using different combinations of parameters. The ensembles generated from multivariate predictions are found to be better than those from univariate predictions. The uncertainty in predictions is reduced or in other words, the predictability is improved by adopting multivariate nonlinear ensemble prediction. The restriction on predictability of a chaotic series can thus be reduced by quantifying the uncertainty in the initial conditions and also by including other possible variables, which may influence the system. Even though, the sensitivity to initial conditions limit the predictability in chaotic systems, a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent. All the traditional chaotic prediction methods are based on local models since these methods model the sudden divergence of the trajectories with different local functions. Conceptually, global models are ineffective in modeling the highly unstable structure of the chaotic attractor [Sivakumar et al., 2002a]. This study focuses on combining a local learning wavelet analysis (decomposition) model with a global feedforward neural network model and its implementation on phase space prediction of chaotic streamflow series. The daily streamflow series at Basantpur station in Mahanadi basin, India is found to exhibit a chaotic nature with dimension varying from 5-7. Quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimension and delay time. Compared with traditional local approximation approach, the total predictive uncertainty in the streamflow is reduced when modeled with wavelet networks for different lead times. Localization property of wavelets, utilizing different dilation and translation parameters, helps in capturing most of the statistical properties of the observed data. The need for bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor is clearly demonstrated.

A evolução de sistemas distribuídos resultou em aumento significativo de complexidade para manutenção e gerenciamento, tornando pouco eficientes técnicas convencionais baseadas em intervenções manuais. Isso motivou pesquisas que deram origem ao paradigma de computação autônoma (Autonomic Computing), que provê aspectos de auto-configuração, auto-recuperação, auto-otimização e auto-proteção a fim de tornar sistemas auto-gerenciáveis. Nesse contexto, esta tese teve como objetivo prover autonomia a ambientes distribuídos, sem a necessidade de mudar o paradigma de programação e as aplicações de usuários. Para isso, propôs-se uma abordagem que emprega técnicas para compreensão e predição de dinâmicas comportamentais de processos, utilizando abordagens de sistemas dinâmicos, inteligência artificial e teoria do caos. Os estudos realizados no decorrer desta pesquisa demonstraram que, ao predizer padrões comportamentais, pode-se otimizar diversos aspectos de computação distribuída, suportando tomadas de decisão autônomas pelos ambientes. Para validar a abordagem proposta, foi desenvolvida uma política de escalonamento distribuído, denominada PredRoute, a qual utiliza o conhecimento sobre o comportamento de processos para otimizar, transparentemente, a alocação de recursos. Experimentos realizados demonstraram que essa política aumenta o desempenho em até 4 ordens de grandeza e apresenta baixo custo computacional, o que permite a sua adoção para escalonamento online de processos / The evolution of distributed systems resulted in a significant growth in management and support complexities, which uncovered the inefficiencies incurred by the usage of conventional management techniques, based in manual interventions. This, therefore, has motivated researches towards the concept of Autonomic Computing, which provides aspects of self-configuration, self-healing, self-optimization and self-protection, aiming at developing computer systems capable of self-management. In this context, this thesis was conceived with the goal of providing autonomy to distributed systems, without changing the programming paradigm or user applications. In order to reach this goal, we proposed an approach which employs techniques capable of modelling and predicting the dynamics of application behavior, using concepts introduced in dynamical systems, artificial intelligence, and chaos theory. The obtained results demonstrated that it is possible to optimize several aspects of distributed computing, providing support for autonomic computing capabilities to distributed environments. In order to validate the proposed approach, a distributed scheduling policy was developed, named PredRoute, which uses the knowledge about the process behavior to transparently optimize the resource allocation. Experimental results demonstrated that this policy can improve the system performance by up to a power of 4, and also requires a considerably low computational cost, which suggests its adoption for online process scheduling in distributed environments

Neste trabalho, desenvolvemos um mapeamento simplético que nos permite estudar o comportamento dinâmico de ressonâncias asteroidais no âmbito do problema dos três corpos restrito, elíptico, espacial. Para obter este mapeamento, combinamos um esquema simplético similar ao desenvolvido por Hadjidemetriou (1986) junto com o desenvolvimento assimétrico da função perturbadora (Ferraz-Mello, 1987), que leva em conta as inclinações do perturbado e do perturbador como sendo referidas a um plano invariante (Roig et al., 1997). Este mapeamento é aplicado aos casos das ressonâncias asteroidais 2/1 e 3/2. Estudam-se um grande número de condições iniciais no espaço de fase, de forma a conseguir tirar conclusões de tipo estatístico sobre os processos envolvidos na geração de mecanismos difusivos que podem agir nessas ressonâncias. / In this work, we developed a symplectic mapping which allow us to study the dynamical behaviour of asteroidal resonances in the frame of the non-planar elliptic restricted three-body problem. To obtain such a mapping we combine a symplectic scheme similar to that of Hadjidemetriou (1986) together with an asymmetric expansion of the disturbing funtion (Ferraz-Mello, 1987) which takes into account the inclinations of both the perturber and the disturbed bodies (Roig et al., 1997). This mapping is applied to the 2/1 and 3/2 mean motion resonances in the asteroidal belt. We explore a wide range of initial conditions in the phase space in order to get a large number of results which allow us to make some statistical conclusions about the generation of diffusion mechanisms acting in these resonances.

Given the state of the planet at present —specifically, the linked global ecological and economic crises that conjure dark imaginings and nihilistic actualities of increasing resource depletion, poisonings, and wide-scale sufferings and extinctions—I ask What might we hope now? What points of intervention offer possibility for transformation? At best, the response can only be partial. The approach this thesis takes initiates from specific pre-discursive assumptions. The first understands current conditions as having been produced, and continuing to be so, through practices that enact and sustain neoliberal relations. Secondly, these practices are expressive of a subjectivity tied to a Cartesian worldview, which, therefore, needs to be interrupted at its foundational roots. Thirdly, the scaffolding that supports this subjectivity draws on Newtonian science and neo-Darwinian narratives deemed to be natural law and, therefore, ontological, immutable reality. Contrary to modernist thinking, I premise that these two strains, subjectivity and science, are neither autonomous nor ontological, but that they are materially and contingently integral. Finally, this thesis presumes that different and life-affirming trajectories are, in fact, desired. An integral framing of science and subjectivity provides a productive method of feminist science studies analysis and theorization. Observing the capitalist Western social imaginary through this lens reveals its philosophical and scientific infrastructures to be outdated and crumbling. Observing how emerging scientific narratives in quantum physics and systems-biology intersect with marginalized theories in process-philosophy and subjectivity reveals a life-affirming imaginary of difference, one that arrests nihilism and sets ethical trajectories in motion. Certain, though not all, percepts of feminist new materialism engage twentieth and twenty-first century sciences successfully to show that ethicality matters. Though many questions remain, this points auspiciously towards the possibility for a transformed politics of justice.

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The master's thesis deals mainly with continuous nonlinear dynamical systems that exhibit chaotic behavior. The main goal is to create algorithms for chaos detection and their subsequent testing on known models. Most of the thesis is devoted to the estimation of the Lyapunov exponents, further it deals with the estimation of the fractal dimension of an attractor and summarizes the 0--1 test. The thesis includes three algorithms created in MATLAB -- an algorithm for estimating the largest Lyapunov exponent and two algorithms for estimating the entire Lyapunov spectra. These algorithms are then tested on five continuous dynamical systems. Especially the error of estimation, speed of these algorithms and properties of Lyapunov exponents in different areas of system behavior are investigated.