This thesis concerns measuring, understanding and modelling Internet traffic. We first study the origins of the statistical properties of Internet traffic, in particular its scaling behaviour, and propose a constructive model of packet traffic with physically motivated parameters. We base our analysis on a large amount of empirical data measured on different networks, and use a so called semi-experimental approach to isolate certain features of traffic we seek to model. These results lead to the choice of a particular Poisson cluster process, known as Bartlett-Lewis point process, for a new packet traffic model. This model has a small number of parameters with simple networking meaning, and is mathematically tractable. It allows us to gain valuable insight on the underlying mechanisms creating the observed statistics. / In practice, Internet traffic measurements are limited by the very large amount of data generated by high bandwidth links. This leads us to also investigate traffic sampling strategies and their respective inversion methods. We argue that the packet sampling mechanism currently implemented in Internet routers is not practical when one wants to infer the statistics of the full traffic from partial measurements. We advocate the use of flow sampling for many purposes. We show that such sampling strategy is much easier to invert and can give reasonable estimates of higher order traffic statistics such as distribution of number of packets per flow and spectral density of the packet arrival process. This inversion technique can also be used to fit the Bartlett-Lewis point process model from sampled traffic. / We complete our understanding of Internet traffic by focusing on the small scale behaviour of packet traffic. To do so, we use data from a fully instrumented Tier-1 router and measure the delays experienced by all the packets crossing it. We present a simple router model capable of simply reproducing the measured packet delays, and propose a scheme to export router performance information based on busy periods statistics. We conclude this thesis by showing how the Bartlett-Lewis point process can model the splitting and merging of packet streams in a router.

Les causes de l’activité sismique à l’intérieur des plaques tectoniques sont encore mal comprises, que ce soit l’origine des contraintes responsables des séismes ou leur relation avec la variation dans le temps et dans l’espace de l’activité sismique.<p>Les contraintes à l’intérieur des plaques résultent de l’action de forces de longueurs d’onde différentes, qui se superposent et s’additionnent. En utilisant une nouvelle méthode (Camelbeeck et al. 2013), déterminant les contraintes générées localement (échelle de 10 à 100 km), nous estimons l’importance relative de cette composante locale du champ de contrainte. En comparant ces contraintes locales avec les contraintes déduites des mécanismes au foyer des tremblements de terre en Europe occidentale, nous suggérons que celles-ci semblent jouer un rôle non-négligeable dans l’occurrence de l’activité sismique. C’est le cas dans des régions où les contraintes locales étaient déjà reconnues, mais également dans des régions précédemment identifiées comme dominées par les contraintes à plus grandes longueurs d’onde.<p>Le champ de contrainte généré localement est constant à l’échelle de temps des catalogues sismiques, ce qui ne permet pas d’expliquer l’occurrence dans le temps des séismes. Il est cependant modifié par les variations des contraintes locales générées par l’activité séismique elle-même, ce qui explique les séquences de répliques des séismes de Roermond (13/04/1992, Mw=5.4) et d’Alsdorf (22/07/2002, Mw= 4.7) dans le graben de la Roer. Nous y suggérons également l’importance de ces variations à une échelle de temps plus longue (millier d’années) à partir des données de paléoséismologie.<p>Pour mieux comprendre les relations spatio-temporelles des séismes, nous avons également analysé dans quelle mesure l’occurrence de l’activité sismique dans le graben de la Roer est un processus de Poisson, ou si l’activité future est localisée à proximité des séismes du passé, ou située dans des régions dénuées de séismes à ce jour. L’emploi des méthodes linéaires classiques et de méthodes non-linéaires dans cette région mais aussi dans les îles britanniques et dans le sud de la Norvège montrent que les séismes du passé ne peuvent expliquer les taux d’activité sismique actuellement mesurés. Une partie de l’activité sismique correspond ainsi à une activité de background, indépendante de l’occurrence des séismes du passé. La méthode des multifractales permet aussi de caractériser régionalement l’importance, la variété et continuité des processus responsables de l’activité sismique sans pour autant en identifier les causes. <p>Notre travail nous a permis d’identifier l’importance relative de certaines causes de l’activité sismique, par exemple l’importance des variations locales des contraintes générées par l’activité séismique elle-même, mais n’a pas permis par exemple d’identifier l’origine de l’activité de background, clairement mise en évidence par l’analyse multifractale./<p><p> / Doctorat en sciences, Spécialisation géologie / info:eu-repo/semantics/nonPublished

Le but de cette thèse est de définir puis d'étudier différentes notions d'irrégularité uniforme ou ponctuelle permettant de traduire le fait qu'une fonction peut avoir des 'grands accroissements' à toutes les échelles. Pour cela on 'inverse' les notions de régularité Höldérienne usuelles. L'objectif principal du travail est ensuite de relier ces différentes notions à la théorie des ondelettes. Les critères ondelettes établis vont ainsi permettre de définir des fonctions ou des champs aléatoires dont le comportement est différent suivant la gamme d'échelles considérée. Par ailleurs, si on se place du point de vue ponctuel, une question naturelle est celle de la définition d'une analyse multifractale -dite faible- liée à la notion d'irrégularité ponctuelle. Les ondelettes vont alors permettre de définir des séries d'ondelettes multifractales pour l'irrégularité ponctuelle. Enfin, nous étudions des exemples de champs aléatoires où des propriétés de régularité directionelle apparaissent. Nous nous sommes ainsi centré sur l'étude d'un modèle de champ aléatoire gaussien particulier vérifiant une relation d'autosimilarité matricielle. Nous avons ensuite généralisé ce modèle et introduit des champs gaussiens autosimilaires par rapport à un groupe / The main purpose of this thesis is the definition and the study of different concepts of uniform or pointwise irregularity which enable one to account for the fact that a function may have 'large increments' at any scales. To this end, we 'invert' the usual notions of Hölderian regularity. The main goal is then to relate these different concepts to wavelet theory. The wavelet criteria supplied enable to define functions or random fields the behavior of which differ with respect the family of scales chosen. Moreover, if we consider the pointwise point of view, a natural question is that of the definition of a weak multifractal analysis related to pointwise irregularity. Finally, we study examples of random fields with some properties of directional regularity. Thus we focus on the study of a special model of operator scaling Gaussian field. We then extend this model and introduced group self-similar Gaussian fields

Depuis une dizaine d'années, la transformée en ondelettes a été reconnue comme un outil privilégié d'analyse des objets fractals, en permettant de définir un formalisme multifractal généralisé des mesures aux fonctions. Dans une première partie, nous utilisons la méthode MMTO (Maxima du Module de la Transformée en Ondelettes) 2D, outil d'analyse multifractale en traitement d'images pour étudier des mammographies. On démontre les potentialités de la méthode pour le problème de la segmentation de texture rugueuse et la caractérisation géométrique d'amas de microcalcifications, signes précoces d'apparition du cancer du sein. Dans une deuxième partie méthodologique, nous généralisons la méthode MMTO pour l'analyse multifractale de données 3D scalaires et vectorielles, en détaillant la mise en oeuvre numérique et un introduisant la transformée en ondelettes tensorielle. On démontre en particulier que l'utilisation d'une technique de filtres récursifs permet un gain de 25 a 60 \% en temps de calcul suivant l'ondelette analysatrice choisie par rapport à un filtrage par FFT. La méthode MMTO 3D est appliquée sur des simulations numériques directes (SND) des équations de Navier-Stokes en régime turbulent. On montre que les champs 3D de dissipation et d'enstrophie pour des nombres de Reynolds modérés sont bien modélisés par des processus multiplicatifs de cascades non-conservatifs comme en témoigne la mesure de l'exposant d'extinction $\kappa$ qui diffère significativement de zéro. On observe en outre que celui-ci diminue lorsqu'on augmente le nombre de Reynolds. Enfin, on présente les premiers résultats d'une analyse multifractale pleinement vectorielle des champs de vitesse et de vorticité des mêmes simulations numériques en montrant que la valeur du paramètre d'intermittence $C_2$, mesuré par la méthode MMTO 3D tensorielle, est significativement plus grande que celle obtenue en étudiant les incréments de vitesse longitudinaux 1D.

Dans cette thèse, nous nous intéressons, à la <br />caractérisation, sur des images météorologiques infrarouges, des systèmes convectifs <br />susceptibles d'engendrer de fortes pluies. <br />L'étude des propriétés statistiques des phénomènes <br />observés révèle une évolution chaotique, mise en évidence par l'invariance d'échelle de <br />certaines grandeurs significatives. Pour les étudier, nous introduisons des méthodes <br />multiéchelles de traitement d'image dérivées de concepts <br />thermodynamiques et qui constituent un prolongement des méthodes <br />d'analyse de la turbulence. Nous utilisons tout d'abord un modèle <br />multifractal afin de détecter les singularités du signal et d'extraire, <br />dans une décomposition hiérarchique de l'image, des structures <br />pertinentes pour la compréhension des mécanismes atmosphériques. <br />Nous proposons ensuite une extension de ce modèle <br />permettant d'exhiber les zones de diffusion de la <br />luminance dans l'image et d'identifier les zones de convection associées aux <br />précipitations.

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

The introduced mineral deposit model (MDM) is the product of a trans-disciplinary study, based on Complexity and General Systems Theory. Both investigate the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence. The focus of the research has been on giant, hydrothermal mineral deposits. They constitute <0.001% of the total number of deposits yet contain 70-85% of the world's metal resources. Giants are the definitive exploration targets. They are more profitable to exploit and less susceptible to fluctuations of the market. Consensus has it that the same processes that generate small deposits also form giants but those processes are simply longer, vaster, and larger. Heat is the dominant factor in the genesis of giant mineral deposits. A paleothermal map shows where the vast heat required to generate a giant has been concentrated in a large space, and even allows us to deduce the duration of the process. To generate a paleothermal map acceptable to the scientific community requires reproducibility. Experimentation with various approaches to pattern recognition of geochemical data showed that the AUTOCLUST algorithm not only gave reproducibility but also gave the most consistent, most meaningful results. It automatically extracts boundaries based on Voronoi and Delaunay tessellations. The user does not specify parameters; however, the modeller does have tools to explore the data. This approach is near ideal in that it removes much of the human-generated bias. This algorithm reveals the radial, spatial distribution, of gold deposits in the Lachlan Fold Belt of southeastern Australia at two distinct scales – repeating patterns every ~80 km and ~230 km. Both scales of patterning are reflected in the geology. The ~80 km patterns are nested within the ~230 km patterns revealing a self-similar, geometrical relationship. It is proposed that these patterns originate from Rayleigh-Bénard convection in the mantle. At the Rayleigh Number appropriate for the mantle, the stable planform is the spoke pattern, where hot mantle material is moving upward near the centre of the pattern and outward along the radial arms. Discontinuities in the mantle, Rayleigh-Bénard convection in the mantle, and the spatial distribution of giant mineral deposits, are correlative. The discontinuities in the Earth are acting as platforms from which Rayleigh-Bénard convection can originate. Shallow discontinuities give rise to plumelets, which manifest at the crust as repeating patterns ranging, from ~100 to ~1,000 km in diameter. Deeper discontinuities give rise to plumes, which become apparent at the crust as repeating patterns ranging from >1,000 to ~4,000 km in diameter. The deepest discontinuities give rise to the superplumes, which become detectable at the crust as repeating patterns ranging from >4,000 to >10,000 km in diameter. Rayleigh-Bénard convection concentrates the reservoir of heat in the mantle into specific locations in the crust; thereby providing the vast heat requirements for the processes that generate giant, hydrothermal mineral deposits. The radial spatial distribution patterns observed for gold deposits are also present for base metal deposits. At the supergiant Broken Hill deposit in far western New South Wales, Australia, the higher temperature Broken Hill-type deposits occur in a radial pattern while the lower temperature deposits occur in concentric patterns. The supergiant Broken Hill deposit occurs at the very centre of the pattern. If the supergiant Broken Hill Deposit was buried beneath alluvium, water or younger rocks, it would now be possible to predict its location with accuracy measured in tens of square kilometres. This predictive accuracy is desired by every exploration manager of every exploration company. The giant deposits at Broken Hill, Olympic Dam, and Mount Isa all occur on the edge of an annulus. There are at least two ways of creating an annulus on the Earth's surface. One is through Rayleigh-Bénard convection and the other is through meteor impact. It is likely that only 'large' meteors (those >10 km in diameter) would have any permanent impact on the mantle. Lesser meteors would leave only a superficial scar that would be eroded away. The permanent scars in the mantle act as ‘accidental templates’ consisting of concentric and possibly radial fractures that impose those structures on any rocks that were subsequently laid down or emplaced over the mantle. In southeastern Australia, the proposed Deniliquin Impact structure has been an 'accidental template' providing a 'line-of-least-resistance' for the ascent of the ~2,000 km diameter, offshore, Cape Howe Plume. The western and northwestern radial arms of this plume have created the very geometry of the Lachlan Fold Belt, as well as giving rise to the spatial distribution of the granitic rocks in that belt and ultimately to the gold deposits. The interplay between the templating of the mantle by meteor impacts and the ascent of plumelets, plumes or superplumes from various discontinuities in the mantle is quite possibly the reason that mineral deposits occur where they do.

The introduced mineral deposit model (MDM) is the product of a trans-disciplinary study, based on Complexity and General Systems Theory. Both investigate the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence. The focus of the research has been on giant, hydrothermal mineral deposits. They constitute <0.001% of the total number of deposits yet contain 70-85% of the world's metal resources. Giants are the definitive exploration targets. They are more profitable to exploit and less susceptible to fluctuations of the market. Consensus has it that the same processes that generate small deposits also form giants but those processes are simply longer, vaster, and larger. Heat is the dominant factor in the genesis of giant mineral deposits. A paleothermal map shows where the vast heat required to generate a giant has been concentrated in a large space, and even allows us to deduce the duration of the process. To generate a paleothermal map acceptable to the scientific community requires reproducibility. Experimentation with various approaches to pattern recognition of geochemical data showed that the AUTOCLUST algorithm not only gave reproducibility but also gave the most consistent, most meaningful results. It automatically extracts boundaries based on Voronoi and Delaunay tessellations. The user does not specify parameters; however, the modeller does have tools to explore the data. This approach is near ideal in that it removes much of the human-generated bias. This algorithm reveals the radial, spatial distribution, of gold deposits in the Lachlan Fold Belt of southeastern Australia at two distinct scales – repeating patterns every ~80 km and ~230 km. Both scales of patterning are reflected in the geology. The ~80 km patterns are nested within the ~230 km patterns revealing a self-similar, geometrical relationship. It is proposed that these patterns originate from Rayleigh-Bénard convection in the mantle. At the Rayleigh Number appropriate for the mantle, the stable planform is the spoke pattern, where hot mantle material is moving upward near the centre of the pattern and outward along the radial arms. Discontinuities in the mantle, Rayleigh-Bénard convection in the mantle, and the spatial distribution of giant mineral deposits, are correlative. The discontinuities in the Earth are acting as platforms from which Rayleigh-Bénard convection can originate. Shallow discontinuities give rise to plumelets, which manifest at the crust as repeating patterns ranging, from ~100 to ~1,000 km in diameter. Deeper discontinuities give rise to plumes, which become apparent at the crust as repeating patterns ranging from >1,000 to ~4,000 km in diameter. The deepest discontinuities give rise to the superplumes, which become detectable at the crust as repeating patterns ranging from >4,000 to >10,000 km in diameter. Rayleigh-Bénard convection concentrates the reservoir of heat in the mantle into specific locations in the crust; thereby providing the vast heat requirements for the processes that generate giant, hydrothermal mineral deposits. The radial spatial distribution patterns observed for gold deposits are also present for base metal deposits. At the supergiant Broken Hill deposit in far western New South Wales, Australia, the higher temperature Broken Hill-type deposits occur in a radial pattern while the lower temperature deposits occur in concentric patterns. The supergiant Broken Hill deposit occurs at the very centre of the pattern. If the supergiant Broken Hill Deposit was buried beneath alluvium, water or younger rocks, it would now be possible to predict its location with accuracy measured in tens of square kilometres. This predictive accuracy is desired by every exploration manager of every exploration company. The giant deposits at Broken Hill, Olympic Dam, and Mount Isa all occur on the edge of an annulus. There are at least two ways of creating an annulus on the Earth's surface. One is through Rayleigh-Bénard convection and the other is through meteor impact. It is likely that only 'large' meteors (those >10 km in diameter) would have any permanent impact on the mantle. Lesser meteors would leave only a superficial scar that would be eroded away. The permanent scars in the mantle act as ‘accidental templates’ consisting of concentric and possibly radial fractures that impose those structures on any rocks that were subsequently laid down or emplaced over the mantle. In southeastern Australia, the proposed Deniliquin Impact structure has been an 'accidental template' providing a 'line-of-least-resistance' for the ascent of the ~2,000 km diameter, offshore, Cape Howe Plume. The western and northwestern radial arms of this plume have created the very geometry of the Lachlan Fold Belt, as well as giving rise to the spatial distribution of the granitic rocks in that belt and ultimately to the gold deposits. The interplay between the templating of the mantle by meteor impacts and the ascent of plumelets, plumes or superplumes from various discontinuities in the mantle is quite possibly the reason that mineral deposits occur where they do.

The introduced mineral deposit model (MDM) is the product of a trans-disciplinary study, based on Complexity and General Systems Theory. Both investigate the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence. The focus of the research has been on giant, hydrothermal mineral deposits. They constitute <0.001% of the total number of deposits yet contain 70-85% of the world's metal resources. Giants are the definitive exploration targets. They are more profitable to exploit and less susceptible to fluctuations of the market. Consensus has it that the same processes that generate small deposits also form giants but those processes are simply longer, vaster, and larger. Heat is the dominant factor in the genesis of giant mineral deposits. A paleothermal map shows where the vast heat required to generate a giant has been concentrated in a large space, and even allows us to deduce the duration of the process. To generate a paleothermal map acceptable to the scientific community requires reproducibility. Experimentation with various approaches to pattern recognition of geochemical data showed that the AUTOCLUST algorithm not only gave reproducibility but also gave the most consistent, most meaningful results. It automatically extracts boundaries based on Voronoi and Delaunay tessellations. The user does not specify parameters; however, the modeller does have tools to explore the data. This approach is near ideal in that it removes much of the human-generated bias. This algorithm reveals the radial, spatial distribution, of gold deposits in the Lachlan Fold Belt of southeastern Australia at two distinct scales – repeating patterns every ~80 km and ~230 km. Both scales of patterning are reflected in the geology. The ~80 km patterns are nested within the ~230 km patterns revealing a self-similar, geometrical relationship. It is proposed that these patterns originate from Rayleigh-Bénard convection in the mantle. At the Rayleigh Number appropriate for the mantle, the stable planform is the spoke pattern, where hot mantle material is moving upward near the centre of the pattern and outward along the radial arms. Discontinuities in the mantle, Rayleigh-Bénard convection in the mantle, and the spatial distribution of giant mineral deposits, are correlative. The discontinuities in the Earth are acting as platforms from which Rayleigh-Bénard convection can originate. Shallow discontinuities give rise to plumelets, which manifest at the crust as repeating patterns ranging, from ~100 to ~1,000 km in diameter. Deeper discontinuities give rise to plumes, which become apparent at the crust as repeating patterns ranging from >1,000 to ~4,000 km in diameter. The deepest discontinuities give rise to the superplumes, which become detectable at the crust as repeating patterns ranging from >4,000 to >10,000 km in diameter. Rayleigh-Bénard convection concentrates the reservoir of heat in the mantle into specific locations in the crust; thereby providing the vast heat requirements for the processes that generate giant, hydrothermal mineral deposits. The radial spatial distribution patterns observed for gold deposits are also present for base metal deposits. At the supergiant Broken Hill deposit in far western New South Wales, Australia, the higher temperature Broken Hill-type deposits occur in a radial pattern while the lower temperature deposits occur in concentric patterns. The supergiant Broken Hill deposit occurs at the very centre of the pattern. If the supergiant Broken Hill Deposit was buried beneath alluvium, water or younger rocks, it would now be possible to predict its location with accuracy measured in tens of square kilometres. This predictive accuracy is desired by every exploration manager of every exploration company. The giant deposits at Broken Hill, Olympic Dam, and Mount Isa all occur on the edge of an annulus. There are at least two ways of creating an annulus on the Earth's surface. One is through Rayleigh-Bénard convection and the other is through meteor impact. It is likely that only 'large' meteors (those >10 km in diameter) would have any permanent impact on the mantle. Lesser meteors would leave only a superficial scar that would be eroded away. The permanent scars in the mantle act as ‘accidental templates’ consisting of concentric and possibly radial fractures that impose those structures on any rocks that were subsequently laid down or emplaced over the mantle. In southeastern Australia, the proposed Deniliquin Impact structure has been an 'accidental template' providing a 'line-of-least-resistance' for the ascent of the ~2,000 km diameter, offshore, Cape Howe Plume. The western and northwestern radial arms of this plume have created the very geometry of the Lachlan Fold Belt, as well as giving rise to the spatial distribution of the granitic rocks in that belt and ultimately to the gold deposits. The interplay between the templating of the mantle by meteor impacts and the ascent of plumelets, plumes or superplumes from various discontinuities in the mantle is quite possibly the reason that mineral deposits occur where they do.

The dissertation covers phase transitions in the realm of the Anderson model of localization on topologically disordered Voronoi-Delaunay lattices. The disorder is given by random connections which implies correlations due to the restrictive lattice construction. Strictly speaking, the system features "strong anticorrelation", which is responsible for quenched long-range fluctuations of the coordination number. This attribute leads to violations of universal behavior in various system, e.g. Ising and Potts model, and to modifications of the Harris and the Imry-Ma criteria. In general, these exceptions serve to further understanding of critical phenomena. Hence, the question arises whether such deviations also occur in the realm of the Anderson model of localization in combination with random Voronoi-Delaunay lattice. For this purpose, four cases, which are distinguished by the spatial dimension of the systems and by the presence or absence of a magnetic field, are investigated by means of two different methods, i.e the multifractal analysis and the recursive Green function approach. The behavior is classified by the existence and type of occurring phase transitions and by the critical exponent v of the localization length. The results for the four cases can be summarized as follows. In two-dimensional systems, no phase transitions occur without a magnetic field, and all states are localized as a result of topological disorder. The behavior changes under the influence of the magnetic field. There are so-called quantum Hall transitions, which are phase changes between two localized regions. For low magnetic field strengths, the resulting exponent v ≈ 2.6 coincides with established values in literature. For higher strengths, an increased value, v ≈ 2.9, was determined. The deviations are probably caused by so-called Landau level coupling, where electrons scatter between different Landau levels. In contrast, the principle behavior in three-dimensional systems is equal in both cases. Two localization-delocalization transitions occur in each system. For these transitions the exponents v ≈ 1.58 and v ≈ 1.45 were determined for systems in absence and in presence of a magnetic field, respectively. This behavior and the obtained values agree with known results, and thus no deviation from the universal behavior can be observed. / Diese Dissertation behandelt Phasenübergange im Rahmen des Anderson-Modells der Lokalisierung in topologisch ungeordneten Voronoi-Delaunay-Gittern. Die spezielle Art der Unordnung spiegelt sich u.a. in zufälligen Verknüpfungen wider, welche aufgrund der restriktiven Gitterkonstruktion miteinander korrelieren. Genauer gesagt zeigt das System eine "starke Antikorrelation", die dafür sorgt, dass langreichweitige Fluktuationen der Verknüpfungszahl unterdrückt werden. Diese Eigenschaft hat in anderen Systemen, z.B. im Ising- und Potts-Modell, zur Abweichung vom universellen Verhalten von Phasenübergängen geführt und bewirkt eine Modifikation von allgemeinen Aussagen, wie dem Harris- and Imry-Ma-Kriterium. Die Untersuchung solcher Ausnahmen dient zur Weiterentwicklung des Verständnisses von kritischen Phänomenen. Somit stellt sich die Frage, ob solche Abweichungen auch im Anderson-Modell der Lokalisierung unter Verwendung eines solchen Gitters auftreten. Dafür werden insgesamt vier Fälle, welche durch die Dimension des Gitters und durch die An- bzw. Abwesenheit eines magnetischen Feldes unterschieden werden, mit Hilfe zweier unterschiedlicher Methoden, d.h. der Multifraktalanalyse und der rekursiven Greensfunktionsmethode, untersucht. Das Verhalten wird anhand der Existenz und Art der Phasenübergänge und anhand des kritischen Exponenten v der Lokalisierungslänge unterschieden. Für die vier Fälle lassen sich die Ergebnisse wie folgt zusammenfassen. In zweidimensionalen Systemen treten ohne Magnetfeld keine Phasenübergänge auf und alle Zustände sind infolge der topologischen Unordnung lokalisiert. Unter Einfluss des Magnetfeldes ändert sich das Verhalten. Es kommt zur Ausformung von Landau-Bändern mit sogenannten Quanten-Hall-Übergängen, bei denen ein Phasenwechsel zwischen zwei lokalisierten Bereichen auftritt. Für geringe Magnetfeldstärken stimmen die erzielten Ergebnisse mit den bekannten Exponenten v ≈ 2.6 überein. Allerdings wurde für stärkere magnetische Felder ein höherer Wert, v ≈ 2.9, ermittelt. Die Abweichungen gehen vermutlich auf die zugleich gestiegene Unordnungsstärke zurück, welche dafür sorgt, dass Elektronen zwischen verschiedenen Landau-Bändern streuen können und so nicht das kritische Verhalten eines reinen Quanten-Hall-Überganges repräsentieren. Im Gegensatz dazu ist das Verhalten in dreidimensionalen Systemen für beide Fälle ähnlich. Es treten in jedem System zwei Phasenübergänge zwischen lokalisierten und delokalisierten Bereichen auf. Für diese Übergänge wurde der Exponent v ≈ 1.58 ohne und v ≈ 1.45 unter Einfluss eines magnetischen Feldes ermittelt. Dieses Verhalten und die jeweils ermittelten Werte stimmen mit bekannten Ergebnissen überein. Eine Abweichung vom universellen Verhalten wird somit nicht beobachtet.