Ziel dieser Arbeit ist die explizite Berechnung gewisser geometrischer Zykel in Modulräumen von Kurven. In den letzten Jahren wurden Divisoren auf $\Mbar_{g,n}$ ausgiebig untersucht. Durch die Berechnung von Klassen in Kodimension 1 konnten wichtige Ergebnisse in der birationalen Geometrie der Räume $\Mbar_{g,n}$ erzielt werden. In Kapitel 1 geben wir einen Überblick über dieses Thema. Im Gegensatz dazu sind Klassen in Kodimension 2 im Großen und Ganzen unerforscht. In Kapitel 2 betrachten wir den Ort, der im Modulraum der Kurven vom Geschlecht 2k durch die Kurven mit einem Büschel vom Grad k definiert wird. Da die Brill-Noether-Zahl hier -2 ist, hat ein solcher Ort die Kodimension 2. Mittels der Methode der Testflächen berechnen wir die Klasse seines Abschlusses im Modulraum der stabilen Kurven. Das Ziel von Kapitel 3 ist es, die Klasse des Abschlusses des effektiven Divisors in $\Mbar_{6,1}$ zu berechnen, der durch punktierte Kurven [C, p] gegeben ist, für die ein ebenes Modell vom Grad 6 existiert, bei dem p auf einen Doppelpunkt abgebildet wird. Wie Jensen gezeigt hat, erzeugt dieser Divisor einen extremalen Strahl im pseudoeffektiven Kegel von $\Mbar_{6,1}$. Ein allgemeines Ergebnis über gewisse Familien von Linearsystemen mit angepasster Brill-Noether-Zahl 0 oder -1 wird eingeführt, um die Berechnung zu vervollständigen. / The aim of this thesis is the explicit computation of certain geometric cycles in moduli spaces of curves. In recent years, divisors of $\Mbar_{g,n}$ have been extensively studied. Computing classes in codimension one has yielded important results on the birational geometry of the spaces $\Mbar_{g,n}$. We give an overview of the subject in Chapter 1. On the contrary, classes in codimension two are basically unexplored. In Chapter 2 we consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves. The aim of Chapter 3 is to compute the class of the closure of the effective divisor in $\M_{6,1}$ given by pointed curves [C,p] with a sextic plane model mapping p to a double point. Such a divisor generates an extremal ray in the pseudoeffective cone of $\Mbar_{6,1}$ as shown by Jensen. A general result on some families of linear series with adjusted Brill-Noether number 0 or -1 is introduced to complete the computation.

Text in Afrikaans / "Free people expect much from their police. In such societies the police stand at the point of balance on the one hand securing human rights on the other exercising their lawful powers given to them by Governments in the name of the people, to protect people and their institutions" (J Alderson Human Rights and Police Rights. Publication of the Council of Europe.) This is no small expectation. It means that the police are expected to maintain and secure the principles of democracy and human rights, the principles upon which our Constitution is based. At the same time it is the duty of a police service to maintain law and order which sometimes require the exercise of power and the use of force which on the face of it may appear to violate human dignity and certain rights and freedoms which a police force is expected to maintain in a democratic society. The manner of exercising that power has an impact on the credibility and effectiveness of the police. Human rights law internationally accepts that a police service needs to be given the power to, at times restrict certain individual freedoms in the interests of the security of the community at large. These restrictions may take place only in a constitutional way. If it takes place in an unconstitutional way the courts must have the right to exclude evidence which is unconstitutionally obtained. At this stage the courts have to formulate such a qualified exclusionary rule, but the ultimate goal must be to include such an exclusionary rule in a future Constitution. / Department of Criminal & Procedural Law / LL.M.

Cette thèse est consacrée à l'étude des revêtements galoisiens et à la recherche du revêtement universel et du groupe fondamental pour les algèbres de dimension finie, connexes et basiques sur un corps algébriquement clos. Pour ce faire, nous partons d'une construction déjà existante: le groupe fondamental associé à toute présentation d'une telle algèbre A par son carquois ordinaire Q et des relations admissibles. Nous commençons par comparer les différentes présentations de A. Les automorphismes de l'algèbre kQ des chemins de Q permettent de relier deux présentations de A et parmi ceux-là, nous distinguons les dilatations et les transvections: elles engendrent le groupe des automorphismes de kQ, en outre, les groupes fondamentaux de deux présentations de A reliées par une dilatation ou une transvection sont liés entre eux par un passage au quotient. Ceci permet d'exhiber un groupe fondamental pour A lorsque le corps de base est de caractéristique nulle et lorsque Q n'a pas de double raccourci. Ces raisonnements se transposent à l'étude des revêtements galoisiens de A puisqu'à chaque présentation de A est associé un revêtement galoisien de A et de groupe le groupe fondamental de la présentation. Ainsi, sous les hypothèses précédentes fournissant le groupe fondamental de A, un revêtement universel de A existe. Ce dernier résultat est également démontré pour un corps de caractéristique quelconque, lorsque A est monomiale et lorsque Q n'a ni flèches multiples ni cycle orienté tout en admettant d'éventuels double raccourcis.

Title: Hierarchical Problems with Evolutionary Equilibrium Constraints Author: Lukáš Adam Supervisor: Prof. Jiří Outrata Abstract: In the presented thesis, we are interested in hierarchical models with evolutionary equilibrium constraints. Such models arise naturally when a time-dependent problem is to be controlled or if parameters in such a model are to be identified. We intend to discretize the problem and solve it on the basis of the so-called implicit programming approach. This technique requires knowledge of a generalized derivative of the solution mapping which assigns the state variable to the control variable/parameter. The computation of this generalized derivative amounts equivalently to the computation of (limiting) normal cone to the graph of the solution mapping. In the first part we summarize known techniques for computation of the normal cone to the set which can be represented as a finite union of convex polyhedra. Then we propose a new approach based on the so-called normally admissible stratification and simplify the obtained formulas for the case of time-dependent problems. The theoretical results are then applied first to deriving a criterion for the sensitivity analysis of the solution mapping and then to the solution of two practically motivated problems. The first one concerns optimal...

On s'intéresse à une compactification, due à Ekedahl, Lando, Shapiro et Vainshtein, du champ des courbes lisses munies de fonctions méromorphes d'ordres fixés. Celle-ci est obtenue comme une adhérence du champ de départ dans un champ propre. On commence par en donner deux constructions alternatives et on étudie les déformations de ses points. On la relie par la suite à la compactification à la Harris-Mumford par les revêtements admissibles et on donne une interprétation modulaire des points du bord. / We study a compactification, due to Ekedahl, Lando, Shapiro and Vainshtein, of the stack of smooth curves endowed with meromorphic functions having fixed orders. The original compactification is obtained as the closure of the initial stack in a proper substack. We start by giving two alternative constructions of the E.L.S.V compactification and by studying the deformation theory of its points. We finally link it to the Harris-Mumford compactification by admissible covers and give a modular interpretation of boundary points.

Text in Afrikaans / "Free people expect much from their police. In such societies the police stand at the point of balance on the one hand securing human rights on the other exercising their lawful powers given to them by Governments in the name of the people, to protect people and their institutions" (J Alderson Human Rights and Police Rights. Publication of the Council of Europe.) This is no small expectation. It means that the police are expected to maintain and secure the principles of democracy and human rights, the principles upon which our Constitution is based. At the same time it is the duty of a police service to maintain law and order which sometimes require the exercise of power and the use of force which on the face of it may appear to violate human dignity and certain rights and freedoms which a police force is expected to maintain in a democratic society. The manner of exercising that power has an impact on the credibility and effectiveness of the police. Human rights law internationally accepts that a police service needs to be given the power to, at times restrict certain individual freedoms in the interests of the security of the community at large. These restrictions may take place only in a constitutional way. If it takes place in an unconstitutional way the courts must have the right to exclude evidence which is unconstitutionally obtained. At this stage the courts have to formulate such a qualified exclusionary rule, but the ultimate goal must be to include such an exclusionary rule in a future Constitution. / Department of Criminal and Procedural Law / LL.M.

Given a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L²</i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H²</i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M^Ã)</i> which together form a direct-sum decomposition of <i>L²</i>. In the case where the pair <i>(M,M^Ã)</i> are finite-dimensional perturbations of the Hardy space <i>H²</i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M^Ã)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs <i>(M,M^Ã)</i> of the <i>L²</i> spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans. / Ph. D.

Text in Afrikaans / Artikel 35(5) van die Grondwet 108 van 1996 handel oor die uitsluiting van ongrondwetlike getuienis en bepaal dat sodanige getuienis uitgesluit moet word indien toelating daarvan sal lei tot 'n onbillike verhoor of tot nadeel sal strek vir die regspleging. Uit die bewoording van die artikel blyk dit dat die howe geen diskresie het ten opsigte van die toelaatbaarheidsvraag nie en 'n streng uitsluitingbenadering moet volg. Die doel van hierdie verha• ndeling is om ondersoek in te stel na die mate van diskresie .en die wyse ·waarop diskresie toepas word in hierdie besluitnemingsproses. Twee benaderings is deur die howe gevolg, naamlik 'n benadering waar 'n wye diskresie uitgeoefen word en 'n benadering waar 'n beperkte diskresie uitgeoefen word, dus 'n gekwalifiseerde uitsluitingsbenadering. Die skrywer doen aan die hand dat beide gronde vir uitsluiting van belang is en dat die howe verkeie faktore moet oorweeg ten einde 'n beslissing te vel oor die insluiting of uitsluiting van ongrondwetlike getuienis. 'n Balans moet dus gehandhaaf word tussen die belang van die beskuldigde op 'n billike verhoor en die belang van die gemeenskap daarin dat regspleging nie benadeel moet word nie en dat reg en geregtigheid moet geskied / Section 35(3) of the Constitution Act 108 of 1996 deals with the exclusion of unconstitutionally obtained evidence and stipulates that such evidence must be excluded if the admission would render the trial unfair or be detrimental to the administration of justice. From the wording of the section it seems that the courts have no jurisdiction in regard to the admissibility question and that a strict exclusionary approach must be followed. The purpose of this dissertation is to investigate the amount of discretion that the Courts have, and the manner in which this discretion is applied in the process of decision making. Two approaches were followed by the courts namely a wide discretionary approach and an approach where a strict discretion was applied. It is submitted that botR grounds for exclusion are of importance and that the courts must consider a variety of factors in deciding the question on the inclusion or exclusion of unconstitutionally obtained evidence. A balance must be struck between the interest of the accused in a fair trial and the interest of the community that the administration of justice must not be prejudiced and that justice must prevail. / Criminal & Procedural Law / LL.M. (Law)

Increasing use is being made of various types of scientific evidence in court. The general requirement for the admissibility of such evidence is relevance. Although expert evidence is considered to be opinion evidence, it is admissible if it can assist the court to decide a fact in issue; provided that it is also reliable. In South Africa, the initial wide judicial discretion to either admit or exclude unconstitutionally obtained evidence, has developed into a more narrowly defined discretion under the final Constitution. Examples of scientific evidence, namely, DNA evidence, fingerprints, psychiatric evidence, bite-mark evidence and polygraph evidence are considered and problems inherent in the presentation of such evidence in courts in various jurisdictions are highlighted. An investigation of the presentation and evaluation of evidence in both the accusatorial and inquisitorial systems seems to indicate that the adversarial procedure has a marked influence on the evaluation of evidence / Criminal & Procedural Law / LL.M. (Law)

Ce travail de recherche vise à contribuer au développement de nouveaux outils d'estimation d'erreur globale et locale en ingénierie mécanique. Les estimateurs d'erreur globale étudiés reposent sur le concept d'erreur en relation de comportement à travers des techniques spécifiques de construction de champs admissibles, assurant l'aspect conservatif ou garanti de l'estimation. Une nouvelle méthode de construction de champs admissibles est mise en place et comparée à deux autres méthodes concurrentes, en matière de précision, coût de calcul et facilité d'implémentation dans les codes éléments finis. Une amélioration de cette nouvelle méthode hybride fondée sur une minimisation locale de l'énergie complémentaire est également proposée. Celle-ci conduit à l'introduction et à l'élaboration de critères géométriques et énergétiques judicieux, permettant un choix approprié des régions à sélectionner pour améliorer localement la qualité des champs admissibles. Dans le cadre des estimateurs d'erreur locale basés sur l'utilisation conjointe des outils d'extraction et des estimateurs d'erreur globale, deux nouvelles techniques d'encadrement de l'erreur en quantité d'intérêt sont proposées. Celles-ci sont basées sur le principe de Saint-Venant à travers l'emploi de propriétés spécifiques d'homothétie, afin d'améliorer la précision des bornes d'erreur locale obtenues à partir de la technique d'encadrement classique fondée sur l'inégalité de Cauchy-Schwarz. Les diverses études comparatives sont menées dans le cadre des problèmes d'élasticité linéaire en quasi-statique. Le comportement des différents estimateurs d'erreur est illustré et discuté sur des exemples numériques tirés d'applications industrielles. Les travaux réalisés constituent des éléments de réponse à la problématique de la vérification dans un contexte industriel. / This research work aims at contributing to the development of innovative global and goal-oriented error estimation tools applied to Computational Mechanics. The global error estimators considered rely on the concept of constitutive relation error through specific techniques for constructing admissible fields ensuring the recovery of strict and high-quality error estimates. A new hybrid method for constructing admissible stress fields is set up and compared to two other techniques with respect to three different criteria, namely the quality of associated error estimators, the computational cost and the simplicity of practical implementation into finite element codes. An enhanced version of this new technique based on local minimization of the complementary energy is also proposed. Judicious geometric and energetic criteria are introduced to select the relevant zones for optimizing the quality of the admissible fields locally. In the context of goal-oriented error estimation based on the use of both extraction techniques and global error estimators, two new improved bounding techniques are proposed. They lean on Saint-Venant's principle through specific homotheticity properties in order to obtain guaranteed and relevant bounds of better quality than with the classical bounding technique based on the Cauchy-Schwarz inequality. The various comparative studies are conducted on linear elasticity problems under quasi-static loading conditions. The behaviour of the different error estimators is illustrated and discussed through several numerical experiments carried out on industrial cases. The associated results may open up opportunities and help broaden the field of model verification for both academic research and industrial applications.