Transforming the South African magistracy : how far have we come?

Word processed copy. Includes bibliographical references (leaves 105-110)

Modeling Support for the Attoyac Bayou Assessment Using SELECT

The Spatially Explicit Load Enrichment Calculation Tool (SELECT) methodology developed by Biological and Agricultural Engineering Department and Spatial Sciences Laboratory at Texas A&M University was used to independently characterize potential E. coli sources and estimate daily potential E. coli loads for the Attoyac Bayou watershed. SELECT is an analytical approach for developing an inventory of potential bacterial sources, particularly nonpoint source contributors, and distributing their potential bacterial loads based on land use and geographical location. A thorough understanding of the watershed and potential contributors that exist is necessary to estimate and assess bacterial load inputs. Land use classification data and data from state agencies, municipal sources, and local stakeholders on the number and distribution of pollution sources are used as inputs in a Geographical Information Systems (GIS) software format. The watershed is divided into multiple smaller subwatersheds based on elevation changes along tributaries and the main segment of the water body. Pollutant sources in the landscape can then be identified and targeted where they are most likely to have significant effects on water quality, rather than looking at contributions on a whole-watershed basis

Sur la stabilité des E-feuilletages

Dans ce travail, nous résolvons un cas particulier du problème standard de la théorie des feuilletages qui est de trouver des conditions pour la stabilité d’une feuille compacte: une feuille compacte est stable si elle possède un système fondamental de voisinages saturés par rapport à la relation d’équivalence définie par les feuilles du feuilletage. Comme de nombreuses variétés différentiables ne possèdent pas de feuilletage, au sens classique, non trivial, nous nous intéressons à des feuilletages avec singularités et plus particulièrement aux E-feuilletages : un E-feuilletage F a la propriété d’être localement décrit par une application holomorphe appelée F-carte qui sépare les feuilles, c’est-à-dire une application qui factorise localement la projection canonique sur l’espace des feuilles. Ces feuilletages ont été étudiés pour la première fois dans la thèse de Egger. Pour une feuille compacte d’un E-feuilletage F, il n’est pas possible de définir un groupe d’holonomie car il n’y a pas de compatibilité biholomorphe entre les Fcartes. La compatibilité entre deux F-cartes fj :Uj −→ Vj, j = 1, 2, est donnée par une famille d’applications (uj :Z −→ Vj)j∈{1,2} appelée mont. Le comportement de F autour d’une feuille compacte est décrit par une famille finie de monts appelée massif que nous obtenons en recouvrant la feuille par un nombre fini de F-cartes. Nous reprenons et continuons dans cette thèse le travail de Egger en proposant une étude catégorique plus poussée des massifs. Ceci nous permet de définir différemment et plus simplement les germes d’holonomie, l’analogue au groupe d’holonomie que Egger a introduit. Les germes d’holonomie, dans le sens o`u nous les avons définis, nous donnent aussi un critère de stabilité. L’étude des germes d’holonomie définie par des E-feuilletages compacts de codimension 1 nous permet alors de démontrer que tous ces feuilletages sont stables: un feuilletage compact est stable si toutes ses feuilles sont stables. Il suit du théorème de Mattei-Moussu et du critère de simplicité de Reiffen que tous les feuilletages holomorphes compacts de codimension 1 définis sur une variété sont des E-feuilletages, nous avons donc trouvé une nouvelle démonstration de la stabilité de ces feuilletages.In this work, we solve a particular case of a standard problem of the theory of foliations which is to find conditions for the stability of a compact leaf. A compact leaf is stable if it possesses a fundamental system of neighborhoods saturated with respect to the equivalence relation defined by the leaves of the foliation. Since a lot of manifolds do not possess any foliation in the classical sense, that is not trivial, we study foliations with singularities and in particular the E-foliations. An E-foliation has the property to be locally described by a holomorphic mapping named F-chart which separates the leaves, i.e. a mapping that factorizes locally the canonical projection on the leafs space. This foliations have been studied first by Egger in his thesis. For a compact leaf of an E-foliation F it is not possible to define a holonomy group since there is no biholomorphic compatibility between the F-charts. The compatibility between two F-charts fj :Uj −→ Vj, j = 1, 2, is given by a family of applications (uj :Z −→ Vj)j∈{1,2} named a mountain. In order to know the behavior around a compact leaf we cover it with a finite number of F-charts and we obtain a finite family of mountains named massif. We develop further and complement the work of Egger proposing a more advanced categorical study of the massifs. This allows us to define the holonomy germs, i.e. the analogue to the holonomy group Egger introduced, in a different and simpler way. The holonomy germs, in the sense we defined them, also give a stability criterion. The study of the holonomy germs defined by 1-codimensional compact E-foliations allows us to prove the stability of these foliations. A compact foliation is stable if all its leaves are stable. The theorem of Mattei-Moussu and the simplicity criterion of Reiffen imply that all the 1-codimensional compact foliations defined on a manifold are E-foliations. Thus we have found a new proof of their stability

Multiscale reconstruction of time series

A new method is proposed which allows a reconstruction of time series based on higher order multiscale statistics given by a hierarchical process. This method is able to model the time series not only on a specific scale but for a range of scales. It is possible to generate complete new time series, or to model the next steps for a given sequence of data. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The results of this approach are shown for a real world dataset, namely for turbulence. The unconditional and conditional probability densities of the original and reconstructed time series are compared and the ability to reproduce both is demonstrated. Therefore in the case of Markov properties the method proposed here is able to generate artificial time series with correct n-point statistics.Comment: 4 pages, 3 figure

Improved estimation of Fokker-Planck equations through optimisation

An improved method for the description of hierarchical complex systems by means of a Fokker-Planck equation is presented. In particular the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for constraint problems (L-BFGS-B) is used to minimize the distance between the numerical solutions of the Fokker-Planck equation and the empirical probability density functions and thus to estimate properly the drift and diffusion term of the Fokker-Planck equation. The optimisation routine is applied to a time series of velocity measurements obtained from a turbulent helium gas jet in order to demonstrate the benefits and to quantify the improvements of this new optimisation routine