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This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables. Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property. Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property. In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given. In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain. Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set. Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.

Cette thèse porte sur les phénomènes critiques survenant dans les modèles bidimensionnels sur réseau. Les résultats sont l'objet de deux articles : le premier porte sur la mesure d'exposants critiques décrivant des objets géométriques du réseau et, le second, sur la construction d'idempotents projetant sur des modules indécomposables de l'algèbre de Temperley-Lieb pour la chaîne de spins XXZ. Le premier article présente des expériences numériques Monte Carlo effectuées pour une famille de modèles de boucles en phase diluée. Baptisés "dilute loop models (DLM)", ceux-ci sont inspirés du modèle O(n) introduit par Nienhuis (1990). La famille est étiquetée par les entiers relativement premiers p et p' ainsi que par un paramètre d'anisotropie. Dans la limite thermodynamique, il est pressenti que le modèle DLM(p,p') soit décrit par une théorie logarithmique des champs conformes de charge centrale c(\kappa)=13-6(\kappa+1/\kappa), où \kappa=p/p' est lié à la fugacité du gaz de boucles \beta=-2\cos\pi/\kappa, pour toute valeur du paramètre d'anisotropie. Les mesures portent sur les exposants critiques représentant la loi d'échelle des objets géométriques suivants : l'interface, le périmètre externe et les liens rouges. L'algorithme Metropolis-Hastings employé, pour lequel nous avons introduit de nombreuses améliorations spécifiques aux modèles dilués, est détaillé. Un traitement statistique rigoureux des données permet des extrapolations coïncidant avec les prédictions théoriques à trois ou quatre chiffres significatifs, malgré des courbes d'extrapolation aux pentes abruptes. Le deuxième article porte sur la décomposition de l'espace de Hilbert \otimes^nC^2 sur lequel la chaîne XXZ de n spins 1/2 agit. La version étudiée ici (Pasquier et Saleur (1990)) est décrite par un hamiltonien H_{XXZ}(q) dépendant d'un paramètre q\in C^\times et s'exprimant comme une somme d'éléments de l'algèbre de Temperley-Lieb TL_n(q). Comme pour les modèles dilués, le spectre de la limite continue de H_{XXZ}(q) semble relié aux théories des champs conformes, le paramètre q déterminant la charge centrale. Les idempotents primitifs de End_{TL_n}\otimes^nC^2 sont obtenus, pour tout q, en termes d'éléments de l'algèbre quantique U_qsl_2 (ou d'une extension) par la dualité de Schur-Weyl quantique. Ces idempotents permettent de construire explicitement les TL_n-modules indécomposables de \otimes^nC^2. Ceux-ci sont tous irréductibles, sauf si q est une racine de l'unité. Cette exception est traitée séparément du cas où q est générique. Les problèmes résolus par ces articles nécessitent une grande variété de résultats et d'outils. Pour cette raison, la thèse comporte plusieurs chapitres préparatoires. Sa structure est la suivante. Le premier chapitre introduit certains concepts communs aux deux articles, notamment une description des phénomènes critiques et de la théorie des champs conformes. Le deuxième chapitre aborde brièvement la question des champs logarithmiques, l'évolution de Schramm-Loewner ainsi que l'algorithme de Metropolis-Hastings. Ces sujets sont nécessaires à la lecture de l'article "Geometric Exponents of Dilute Loop Models" au chapitre 3. Le quatrième chapitre présente les outils algébriques utilisés dans le deuxième article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2", constituant le chapitre 5. La thèse conclut par un résumé des résultats importants et la proposition d'avenues de recherche qui en découlent. / This thesis is concerned with the study of critical phenomena for two-dimensional models on the lattice. Its results are contained in two articles: A first one, devoted to measuring geometric exponents, and a second one to the construction of idempotents for the XXZ spin chain projecting on indecomposable modules of the Temperley-Lieb algebra. Monte Carlo experiments, for a family of loop models in their dilute phase, are presented in the first article. Coined "dilute loop models (DLM)", this family is based upon an O(n) model introduced by Nienhuis (1990). It is defined by two coprime integers p,p' and an anisotropy parameter. In the continuum limit, DLM(p,p') is expected to yield a logarithmic conformal field theory of central charge c(\kappa)=13-6(\kappa+1/\kappa), where the ratio \kappa=p/p' is related to the loop gas fugacity \beta=-2\cos\pi/\kappa. Critical exponents pertaining to valuable geometrical objects, namely the hull, external perimeter and red bonds, were measured. The Metropolis-Hastings algorithm, as well as several methods improving its efficiency, are presented. Despite the extrapolation of curves presenting large slopes, values as close as three to four digits from the theoretical predictions were attained through rigorous statistical analysis. The second article describes the decomposition of the XXZ spin chain Hilbert space \otimes^nC^2 using idempotents. The model of interest (Pasquier & Saleur (1990)) is described by a parameter-dependent Hamiltonian H_{XXZ}(q), q\in C^\times, expressible as a sum of elements of the Temperley-Lieb algebra TL_n(q). The spectrum of H_{XXZ}(q) in the continuum limit is also believed to be related to conformal field theories whose central charge is set by q. Using the quantum Schur-Weyl duality, an expression for the primitive idempotents of End_{TL_n}\otimes^nC^2, involving U_qsl_2 elements, is obtained. These idempotents allow for the explicit construction of the indecomposable TL_n-modules of \otimes^nC^2, all of which are irreducible except when q is a root of unity. This case, and the case where q is generic, are treated separately. Since a wide variety of results and tools are required to tackle the problems stated above, this thesis contains many introductory chapters. Its layout is as follows. The first chapter introduces theoretical concepts common to both articles, in particular an overview of critical phenomena and conformal field theory. Before proceeding to the article entitled \emph{Geometric Exponents of Dilute Loop Models} constituting Chapter 3, the second chapter deals briefly with logarithmic conformal fields, Schramm-Loewner evolution and the Metropolis-Hastings algorithm. The fourth chapter defines some algebraic concepts used in the second article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2" of Chapter 5. A summary of the main results, as well as paths to unexplored questions, are suggested in a final chapter.

Why has the resurrection once again become the centre point of a new storm brewing in both popular and academic culture? Because of the combination of a realisation of death, and of human beings’ need to interpret its (death’s) mysteries; a question innate to the human experience. In a fear-filled world where war, terrorism, and economic collapse bring the question of death (and the afterlife) to the fore, people are asking – perhaps more than ever – what happens after we die. This popular fascination with the end, with death, and with what (if anything) lies beyond it, has also influenced the theme and the direction of academic work in the theological field. For this reason an informed analysis of the resurrection debate has become necessary – a process of analysing the different strata of understanding as it relates to current resurrection research. Any consideration given to gender or power, birth or burial, money or food is made in an effort to situate the debates being studied. Could a reason for these still varied conclusions on the subject be that those writing on it are not equipped for the task of analysing and interpreting history and historical method? In order to be able to begin answering this question, one of this study's main objectives is to learn and apply the approach of historians – outside of the community of Biblical scholars – to the question of whether Jesus of Nazareth rose from the dead; thus providing interaction with philosophers of history related to hermeneutical and methodological considerations. The method proposed here is a combination of historiography and an ethics of understanding, with the use of Correspondence theory (in which history is described as knowable, and some hypotheses as truer than others in a correspondence sense). This study wants to address both the different questions and analyses of the debate by asking: What if we see things differently? What if we were to ask a different set of questions? In order for this to be possible, we need to develop an ethics of interpretation – instead of asking the expected questions, this study aims to ask: What interests and frameworks inform the questions we ask and the way in which we interpret our sources? How does scholarship echo (and even participate in) contemporary public discourses about Christian identity? These questions will be attended to through three intersecting practices – critical reflexivity, complemented by the use of the two related practices of textual re-reading and public debate. However, these are not methodical steps in a linear progression, they are mutually interacting practices that draw on each other; raising new possibilities for the way in which we historically reconstruct the Jesus movement, allowing us to enter into the public debate about Jesus and eschatology in a way that takes the ethical possibilities and consequences of our reconstructions of Christian origins and identity seriously. For, though fragmentary and broken human words may be, they nevertheless possess a capacity to function as the medium through which God is able to disclose himself. Copyright / Dissertation (MTh)--University of Pretoria, 2010. / New Testament Studies / unrestricted

Cette thèse s'inscrit dans l'étude mathématique de la percolation, qui regroupe une famille de modèles présentant une transition de phase. Des avancées majeures au cours des quinze dernières années, notamment l'invention du SLE et la preuve de l'invariance conforme de la percolation de Bernoulli critique, nous permettent aujourd'hui d'avoir une image très complète de la percolation de Bernoulli sur le réseau triangulaire. Cependant, de nombreuses questions demeurent ouvertes, et ont motivé notre travail.La première d'entre elle est l'universalité de la percolation plane, qui affirme que les propriétés macroscopiques de la percolation plane critique ne devraient pas dépendre du réseau sous-jacent à sa définition. Nous montrons, dans le cadre de la percolation Divide and Color, un résultat qui va dans le sens de cette universalité et identifions, dans ce contexte, des phénomènes macroscopiques indépendants du réseau microscopique. Une version plus faible d'universalité est donnée par la théorie de Russo-Seymour-Welsh (RSW), et sa validité est connue pour la percolation de Bernoulli (sans dépendance) sur les réseaux plans suffisamment symétriques. Nous étudions de nouveaux arguments de type RSW pour des modèles de percolation avec dépendance. La deuxième question que nous avons abordée est celle de l'absence d'une composante connexe ouverte infinie au point critique, une question importante du point de vue physique, puisqu'elle traduit la continuité de la transition de phase. Dans deux travaux en collaboration avec Hugo Duminil-Copin et Vladas Sidoravicius, nous montrons que la transition de phase est continue pour la percolation de Bernoulli sur le graphe Z^2x{0,...,k}, et pour la percolation FK sur le réseau carré avec paramètre q inférieur ou égal à 4. Enfin, la dernière question qui nous a guidés est la localité du point critique : la donnée des boules de grands rayons d'un graphe suffit-elle à identifier avec une bonne précision la valeur du point critique? Dans un travail en collaboration avec Sébastien Martineau, nous répondons de manière affirmative à cette question dans le cadre des graphes de Cayley de groupes abéliens. / This thesis is part of the mathematical study of percolation theory, which includes a family of models with a phase transition. Major advances in the 2000s, including the invention of SLE and the proof of conformal invariance of critical Bernoulli percolation, provide us with a very complete picture of the Bernoulli percolation process on the triangular lattice. Fortunately, many questions remain open, and motivated our work.The first of these is the universality of planar percolation, which states that the macroscopic properties of critical planar percolation should not depend on the underlying graph. We study this question in the framework of Divide and Color percolation, and prove in this context a result that goes in the direction of universality. A weaker universality statement is given by the theory of Russo-Seymour-Welsh (RSW), which is known to hold for planar Bernoulli percolation (without dependence) on sufficiently symmetric graphs. We study new RSW-type arguments for percolation models with dependence.The second question is the absence of an infinite cluster at the critical point, an important question from a physical point of view, equivalent to the continuity of the phase transition. In two different joint works with Hugo Duminil-Copin and Vladas Sidoravicius, we show that the phase transition is continuous for Bernoulli percolation on the graph Z^2 x {0,...,k} and for FK percolation on the square lattice with parameter q smaller than or equal to 4.Finally, the last question that guided us is the locality of the critical point: is it possible to determine with good accuracy the critical value for Bernoulli percolation on a graph if we know only the balls with large radii? Jointly with Sébastien Martineau, we answer positively to this question in the framework of Cayley graphs of abelian groups.

Diese Dissertation untersucht Aspekte des Zusammenspiels von ergodischem Langzeitver- halten und der Glättungseigenschaft dynamischer Systeme, die von stochastischen Differen- tialgleichungen (SDEs) mit Sprüngen erzeugt sind. Im Speziellen werden SDEs getrieben von Lévy-Prozessen und der Marcusschen kanonischen Gleichung untersucht. Ein vari- ationeller Ansatz für den Malliavin-Kalkül liefert eine partielle Integration, sodass eine Variation im Raum in eine Variation im Wahrscheinlichkeitsmaß überführt werden kann. Damit lässt sich die starke Feller-Eigenschaft und die Existenz glatter Dichten der zuge- hörigen Markov-Halbgruppe aus einer nichtstandard Elliptizitätsbedingung an eine Kom- bination aus Gaußscher und Sprung-Kovarianz ableiten. Resultate für Sprungdiffusionen auf Untermannigfaltigkeiten werden aus dem umgebenden Euklidischen Raum hergeleitet. Diese Resultate werden dann auf zufällige dynamische Systeme angewandt, die von lin- earen stochastischen Differentialgleichungen erzeugt sind. Ruelles Integrierbarkeitsbedin- gung entspricht einer Integrierbarkeitsbedingung an das Lévy-Maß und gewährleistet die Gültigkeit von Oseledets multiplikativem Ergodentheorem. Damit folgt die Existenz eines Lyapunov-Spektrums. Schließlich wird der top Lyapunov-Exponent über eine Formel der Art von Furstenberg–Khasminsikii als ein ergodisches Mittel der infinitesimalen Wachs- tumsrate über die Einheitssphäre dargestellt. / The present thesis investigates certain aspects of the interplay between the ergodic long time behavior and the smoothing property of dynamical systems generated by stochastic differential equations (SDEs) with jumps, in particular SDEs driven by Lévy processes and the Marcus’ canonical equation. A variational approach to the Malliavin calculus generates an integration-by-parts formula that allows to transfer spatial variation to variation in the probability measure. The strong Feller property of the associated Markov semigroup and the existence of smooth transition densities are deduced from a non-standard ellipticity condition on a combination of the Gaussian and a jump covariance. Similar results on submanifolds are inferred from the ambient Euclidean space. These results are then applied to random dynamical systems generated by linear stochas- tic differential equations. Ruelle’s integrability condition translates into an integrability condition for the Lévy measure and ensures the validity of the multiplicative ergodic theo- rem (MET) of Oseledets. Hence the exponential growth rate is governed by the Lyapunov spectrum. Finally the top Lyapunov exponent is represented by a formula of Furstenberg– Khasminskii–type as an ergodic average of the infinitesimal growth rate over the unit sphere.

Dizertační práce se zabývá elektronicky nastavitelnými oscilátory, studiem nelineárních vlastností spojených s použitými aktivními prvky a posouzením možnosti vzniku chaotického signálu v harmonických oscilátorech. Jednotlivé příklady vzniku podivných atraktorů jsou detailně diskutovány. V doktorské práci je dále prezentováno modelování reálných fyzikálních a biologických systémů vykazujících chaotické chování pomocí analogových elektronických obvodů a moderních aktivních prvků (OTA, MO-OTA, CCII ±, DVCC ±, atd.), včetně experimentálního ověření navržených struktur. Další část práce se zabývá možnostmi v oblasti analogově – digitální syntézy nelineárních dynamických systémů, studiem změny matematických modelů a odpovídajícím řešením. Na závěr je uvedena analýza vlivu a dopadu parazitních vlastností aktivních prvků z hlediska kvalitativních změn v globálním dynamickém chování jednotlivých systémů s možností zániku chaosu v důsledku parazitních vlastností použitých aktivních prvků.

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Systeme mit Zeitverzögerung sind dadurch charakterisiert, dass deren zukünftige Entwicklung durch den Zustand zum aktuellen Zeitpunkt nicht eindeutig festgelegt ist. Die Historie des Zustands muss in einem Zeitraum bekannt sein, dessen Länge Totzeit genannt wird und die Gedächtnislänge festlegt. In dieser Arbeit werden fundamentale Effekte untersucht, die sich ergeben, wenn die Totzeit zeitlich variiert wird. Im ersten Teil werden zwei Klassen periodischer Totzeitvariationen eingeführt. Da diese von den dynamischen Eigenschaften einer eindimensionalen iterierten Abbildung abgeleitet werden, die über die Totzeit definiert wird, werden die Klassen entsprechend der zugehörigen Dynamik konservativ oder dissipativ genannt. Systeme mit konservativer Totzeit können in Systeme mit konstanter Totzeit transformiert werden und besitzen gleiche charakteristische Eigenschaften. Dagegen weisen Systeme mit dissipativer Totzeit fundamentale Unterschiede z.B. in der Tangentialraumdynamik auf. Im zweiten Teil werden diese Ergebnisse auf Systeme angewendet, deren Totzeit im Vergleich zur internen Relaxationszeit des Systems groß ist. Es zeigt sich, dass ein durch dissipative Totzeitvariationen induzierter Mechanismus, genannt resonanter Dopplereffekt, unter anderem zu neuen Arten chaotischer Dynamik führt. Diese sind im Vergleich zur bekannten chaotischen Dynamik in Systemen mit konstanter Totzeit sehr niedrig-dimensional. Als Spezialfall wird das so genannte laminare Chaos betrachtet, dessen Zeitreihen durch nahezu konstante Phasen periodischer Dauer gekennzeichnet sind, deren Amplitude chaotisch variiert. Im dritten Teil dieser Arbeit wird auf der Basis experimenteller Daten und durch die Analyse einer nichtlinearen retardierten Langevin-Gleichung gezeigt, dass laminares Chaos robust gegenüber Störungen wie zum Beispiel Rauschen ist und experimentell realisiert werden kann. Es werden Methoden zur Zeitreihenanalyse entwickelt, um laminares Chaos in experimentellen Daten ohne Kenntnis des erzeugenden Systems zu detektieren. Mit diesen Methoden ist selbst dann eine Detektion möglich, wenn das Rauschen so stark ist, dass laminares Chaos mit bloßem Auge nur schwer erkennbar ist.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix / In systems with time-delay, the evolution of a system is not uniquely determined by the state at the current time. The history of the state must be known for a time period of finite duration, where the duration is called delay and determines the memory length of the system. In this work, fundamental effects arising from a temporal variation of the time-delay are investigated. In the first part, two classes of periodically time-varying delays are introduced. They are related to a specific dynamics of a one-dimensional iterated map that is defined by the time-varying delay. Referring to the related map dynamics the classes are called conservative or dissipative. Systems with conservative delay can be transformed into systems with constant delay, and thus have the same characteristic properties as constant delay systems. In contrast, there are fundamental differences, for instance, in the tangent space dynamics, between systems with dissipative delay and systems with constant delay. In the second part, these results are applied to systems with a delay that is considered large compared to the internal relaxation time of the system. It is shown that a mechanism induced by dissipative delays leads to new kinds of regular and chaotic dynamics. The dynamics caused by the so-called resonant Doppler effect is fundamentally different from the behavior known from systems with constant delay. For instance, the chaotic attractors in systems with dissipative delay are very low-dimensional compared to typical ones arising in systems with constant delay. An example of this new kind of low-dimensional dynamics is given by the so-called Laminar Chaos. It is characterized by nearly constant laminar phases of periodic duration, where the amplitude varies chaotically. In the third part of this work, it is shown that Laminar Chaos is a robust phenomenon, which survives perturbations such as noise and can be observed experimentally. Therefore experimental data is provided and a nonlinear delayed Langevin equation is analyzed. Using the robust features that characterize Laminar Chaos, methods for time series analysis are developed, which enable us to detect Laminar Chaos without the knowledge of the specific system that has generated the time series. By these methods Laminar Chaos can be detected even for comparably large noise strengths, where the characteristic properties are nearly invisible to the eye.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix

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