The Boneh-Boyen signature scheme is a short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman (SDH) assumption. The primary objective of this thesis is to examine the relationship between the Boneh-Boyen signature scheme and SDH. The secondary objective is to survey surrounding topics such as the generic group model, related signature schemes, intractability assumptions, and the relationship to identity-based encryption (IBE) schemes. Along these lines, we analyze the plausibility of the SDH assumption using the generic bilinear group model. We present the security proofs for the Boneh-Boyen signature scheme, with the addition of a small improvement in one of the probability bounds. Our main contribution is to give the reduction in the reverse direction; that is, to show that if the SDH problem can be solved then the Boneh-Boyen signature scheme can be forged. This contribution represents the first known proof of equivalence between the SDH problem and Boneh-Boyen signatures. We also discuss the algorithm of Cheon for solving the SDH problem. We analyze the implications of Cheon's algorithm for the security of the Boneh-Boyen signature scheme, accompanied by a brief discussion on how to counter the attack.

The Boneh-Boyen signature scheme is a short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman (SDH) assumption. The primary objective of this thesis is to examine the relationship between the Boneh-Boyen signature scheme and SDH. The secondary objective is to survey surrounding topics such as the generic group model, related signature schemes, intractability assumptions, and the relationship to identity-based encryption (IBE) schemes. Along these lines, we analyze the plausibility of the SDH assumption using the generic bilinear group model. We present the security proofs for the Boneh-Boyen signature scheme, with the addition of a small improvement in one of the probability bounds. Our main contribution is to give the reduction in the reverse direction; that is, to show that if the SDH problem can be solved then the Boneh-Boyen signature scheme can be forged. This contribution represents the first known proof of equivalence between the SDH problem and Boneh-Boyen signatures. We also discuss the algorithm of Cheon for solving the SDH problem. We analyze the implications of Cheon's algorithm for the security of the Boneh-Boyen signature scheme, accompanied by a brief discussion on how to counter the attack.

Ever since the discovery of exotic phenomena like high temperature (Tc) superconductivity in the cuprates and colossal magnetoresistance in the manganites, strongly correlated electron systems have become the center of attention in the field of condensed matter physics research. This renewed interest has been further kindled by the rapid development of sophisticated experimental techniques and tremendous computational power. Computation plays a pivotal role in the theoretical investigation of these systems, because one cannot explain their complicated phase diagrams by simple, exactly solvable models. Among the plethora of experimental techniques, various kinds of high energy electron spectroscopies are fast gaining importance due to the multitude of physical properties and phenomena which they can access. However the physical processes involved and the interpretation of the spectra obtained from these spectroscopies are extremely complex and require extensive theoretical modelling. This thesis is concerned with the theoretical modelling of a certain class of high energy electron spectroscopies, viz. the core-level electron spectroscopies, for strongly correlated systems of various kinds. The spectroscopies covered are Auger electron spectroscopy (AES), core-level photoemission spectroscopy (core-level PES) and X-ray absorption spec- troscopy (XAS), which provide non-magnetic information, and also X-ray magnetic circular and linear dichroism (XMCD and XMLD), which provide magnetic information. .

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work. / In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben.

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Die vorliegende Arbeit beschäftigt sich mit den Transporteigenschaften defektbehafteter Kohlenstoffnanoröhrchen (CNTs). Als Beispiel werden dabei einfache und doppelte Fehlstellen betrachtet. Der Fokus liegt auf der Berechnung des Transmissionsspektrums und der Leitfähigkeit mit einem schnellen, linear skalierenden rekursiven Greenfunktions-Formalismus, mit dem große Systeme quantenmechanisch behandelt werden können. Als Grundlage wird ein dichtefunktionalbasiertes Tight-Binding-Modell verwendet. Der Einfluss der Defektdichte und des CNT-Durchmessers wird im Rahmen einer statistischen Analyse untersucht. Es wird gezeigt, dass im Grenzfall kleiner Transmission die Leitfähigkeit exponentiell mit der Defektanzahl skaliert. Das System befindet sich im Regime starker Lokalisierung, wobei die Lokalisierungslänge von der Defektdichte und dem CNT-Durchmesser abhängt.

La dialectique pluralisme religieux/incertitude religieuse, sur laquelle porte cette thèse, se révèle être un thème majeur dans la pensée de Peter L. Berger, en sociologie des religions et en théologie protestante. Une analyse systématique et détaillée des concepts-clés qui la constituent débouche sur la question des rapports entre sociologie et théologie à laquelle Berger lui-même s’est confronté. Abordée sous l’angle de l’idée du principe protestant, cette question s’est résolue, dès la fin des années 1960, en un certain « mariage » entre son approche de la sociologie de la connaissance et son approche théologique libérale. Les concepts de foi et théologie « inductives », de « voie médiane entre le fondamentalisme et le relativisme », semblent jaillir de cette dialectique et de ce « mariage ». Si néanmoins cette dialectique se retrace dans la pensée de Berger dès ses premières œuvres, la défense d’une via media théologique appliquée à toutes les religions se révèle être la conséquence de l’abandon (dès 1967), de sa posture théologique néo-orthodoxe. Dans cette posture, la dialectique bergérienne s’appliquait à toutes les religions mais laissait la foi chrétienne intouchée et pensée en termes de certitude. Or, une analyse critique de sa pensée permet de situer au moins à trois niveaux un certain nombre de problèmes : le niveau de sa conception de la religion manifestant une ambiguïté; le niveau des rapports entre sociologie et théologie révélant un biais libéral et une absence de contenu religieux concret pour le principe protestant; enfin le niveau de sa critique des quêtes contemporaines de certitudes religieuses, critique dont le fondement sur sa dialectique peut être questionné par des exemples de conception différente de la religion et de la certitude religieuse. Pour ces trois niveaux, l’exemple de la conception de la certitude religieuse par les protestants évangéliques permet au moins une ébauche d’un tel questionnement. Cette conception, surtout dans son idée de l’« assurance du salut», se fonde, dans son approche surnaturelle de la certitude religieuse, sur une adhésion et une confiance fortes quant aux contenus traditionnels de la foi chrétienne. Si les arguments avancés dans cette perspective demeurent discutables, ils semblent assez pertinents puisque la vitalité contemporaine de la religion à l’ère du pluralisme religieux (voir notamment le protestantisme évangélique aux États-Unis) constitue une indication que la validité empirique de la dialectique bergérienne, et la critique qu’elle fonde, sont largement problématiques si l’on tient compte de l’auto-compréhension des groupes religieux eux-mêmes. / The dialectic religious pluralism/religious uncertainty, with which deals this dissertation, reveals itself as a major theme in Peter L. Berger’s thought, in sociology of religion and in protestant theology. A systematic and detailed analysis of the key concepts which constitute that dialectic leads to the question of the relationship between sociology and theology, which has been confronted by Berger himself. It is at the time Berger studied that question from the point of view of the idea of the protestant principle, during the late sixties, that he solved it by a kind of ‘wedding’ between his approach in the sociology of knowledge and his liberal theological approach. Concepts as ‘inductive faith and theology’, ‘middle position between fundamentalism and relativism’, seem to emerge both from that dialectic and from that ‘wedding’. Nevertheless, while that dialectic can be retraced in Berger’s thought since his earlier works, the defence of a theological via media applied to all religions, appears to be the consequence of his rejection (since 1967), of his earlier theological stance deployed from the neo-orthodox approach. In that stance, the bergerian dialectic was applied to all religions but not to the Christian faith, thought in terms of certainty. But, a critical analysis of Berger’s thought allows one to identify some problems at least at three levels: the level of his concept of religion which evidentiates an ambiguity ; the level of the relationship between sociology and theology which reveals a liberal bias and a lack of specific religious content for the protestant principle; and finally, the level of his critique of the contemporary religious certainty impulses; a critique whose base on his dialectic can be questioned by some examples of different conceptions of religion and of religious certainty. About those three levels, the example of the conception of religious certainty by the Evangelical Protestants allows at least a draft of that questioning. It deploys a supernatural conception of the religious certainty, especially by the notion of « assurance of salvation », based on strong adherence and confidence in the traditional contents of the Christian faith. While the arguments of that conception can be subjected to questions, they seem enough relevant since the contemporary vitality of religion at the pluralistic era (cf. Evangelical Protestantism in U.S.A for example) constitutes an indication that the empirical validity of the bergerian dialectic, and of the critique based on it, is largely problematic if one takes into account the self-understanding of the religious groups themselves.

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Motivation and background The enormous amount of capabilities that every human learns throughout his life, is probably among the most remarkable and fascinating aspects of life. Learning has therefore drawn lots of interest from scientists working in very different fields like philosophy, biology, sociology, educational sciences, computer sciences and mathematics. This thesis focuses on the information theoretical and mathematical aspects of learning. We are interested in the learning process of an agent (which can be for example a human, an animal, a robot, an economical institution or a state) that interacts with its environment. Common models for this interaction are Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Learning is then considered to be the maximization of the expectation of a predefined reward function. In order to formulate general principles (like a formal definition of curiosity-driven learning or avoidance of unpleasant situation) in a rigorous way, it might be desirable to have a theoretical framework for the optimization of more complex functionals of the underlying process law. This might include the entropy of certain sensor values or their mutual information. An optimization of the latter quantity (also known as predictive information) has been investigated intensively both theoretically and experimentally using computer simulations by N. Ay, R. Der, K Zahedi and G. Martius. In this thesis, we develop a mathematical theory for learning in the sensorimotor loop beyond expected reward maximization. Approaches and results This thesis covers four different topics related to the theory of learning in the sensorimotor loop. First of all, we need to specify the model of an agent interacting with the environment, either with learning or without learning. This interaction naturally results in complex causal dependencies. Since we are interested in asymptotic properties of learning algorithms, it is necessary to consider infinite time horizons. It turns out that the well-understood theory of causal networks known from the machine learning literature is not powerful enough for our purpose. Therefore we extend important theorems on causal networks to infinite graphs and general state spaces using analytical methods from measure theoretic probability theory and the theory of discrete time stochastic processes. Furthermore, we prove a generalization of the strong Markov property from Markov processes to infinite causal networks. Secondly, we develop a new idea for a projected stochastic constraint optimization algorithm. Generally a discrete gradient ascent algorithm can be used to generate an iterative sequence that converges to the stationary points of a given optimization problem. Whenever the optimization takes place over a compact subset of a vector space, it is possible that the iterative sequence leaves the constraint set. One possibility to cope with this problem is to project all points to the constraint set using Euclidean best-approximation. The latter is sometimes difficult to calculate. A concrete example is an optimization over the unit ball in a matrix space equipped with operator norm. Our idea consists of a back-projection using quasi-projectors different from the Euclidean best-approximation. In the matrix example, there is another canonical way to force the iterative sequence to stay in the constraint set: Whenever a point leaves the unit ball, it is divided by its norm. For a given target function, this procedure might introduce spurious stationary points on the boundary. We show that this problem can be circumvented by using a gradient that is tailored to the quasi-projector used for back-projection. We state a general technical compatibility condition between a quasi-projector and a metric used for gradient ascent, prove convergence of stochastic iterative sequences and provide an appropriate metric for the unit-ball example. Thirdly, a class of learning problems in the sensorimotor loop is defined and motivated. This class of problems is more general than the usual expected reward maximization and is illustrated by numerous examples (like expected reward maximization, maximization of the predictive information, maximization of the entropy and minimization of the variance of a given reward function). We also provide stationarity conditions together with appropriate gradient formulas. Last but not least, we prove convergence of a stochastic optimization algorithm (as considered in the second topic) applied to a general learning problem (as considered in the third topic). It is shown that the learning algorithm converges to the set of stationary points. Among others, the proof covers the convergence of an improved version of an algorithm for the maximization of the predictive information as proposed by N. Ay, R. Der and K. Zahedi. We also investigate an application to a linear Gaussian dynamic, where the policies are encoded by the unit-ball in a space of matrices equipped with operator norm.

In the environmental context banks face direct, indirect and reputational risks from their internal operations and their external business activities. The current specific focus on the protection of the environment makes it essential for banks and their directors to be aware and stay on top of potential risks and liabilities. This is especially so because banks’ directors can be criminally prosecuted for environmental crimes. The application and effect of the Prevention of Organised Crime Act 121 of 1998 (POCA) on persons convicted of an environmental crime or crimes has been identified as a possible new or added risk for banks and their directors. Banks in addition to their normal environmental risk and liabilities also need to contend with the possibility of lender liability. Existing legislation pertinent to lender liability does not expressly or specifically deal with lender liability. Absence of judgements on lender liability further exacerbates the risks and the uncertainty for banks in South Africa. Therefore, banks remain subject to legal uncertainty and associated risks. The issue of lender liability specifically with regard to the implication of “the person in control” requires clarification. Hence, it is recommended that legislation relevant to lender liability (National Environmental Management Act 107 of 1998; National Water Act 36 of 1998 and the National Environmental Management: Waste Act 59 of 2008) be revised to specifically accommodate and protect lenders (lending banks) in certain distinct circumstances. The role of banks is that of an intermediary between borrowers and lenders of money. Therefore, it influences the direction and pace of economic development and by default steers and promotes either sustainable or non-sustainable development. Currently, mainstream banks are in effect financing a brown economy and hence subscribe to a weak form of sustainability. It would seem that mainstream banks are more concerned with managing the impact that environmental risk may have on bank lending than the impact of bank lending on the environment. The evolving nature of sustainability (from weak to strong and from a brown to green economy) demands a fundamental policy change for banks. It is expected that mainstream banks will be put under even greater pressure than before to make the transition from weak to strong sustainability. Hence, banks’ current environmental risk management systems will not be sufficient to cater for new environmental risks and liabilities that the move to stronger sustainability (in the form of the green economy) will present. Banks should adopt the stronger version of sustainability; formulate environmental principles that the bank will adhere to; incorporate these environmental principles into all aspects of its lending cycle, develop an environmental risk management system that should include as a minimum the identification of all the applicable legislation pertaining to the specific financing or lending of capital, risk identification, assessment of the specific risk, implementation of risk control measures, mitigation of the risk, risk monitoring and auditing. / LLM (Environmental Law and Governance), North-West University, Potchefstroom Campus, 2014