In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

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 and 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 and Procedural Law / LL.M. (Law)

Dans cette thèse, nous définissons un nouvel invariant des algèbres de Lie quadratiques et des superalgèbres de Lie quadratiques et donnons une étude et classification complète des algèbres de Lie quadratiques singulières et des superalgèbres de Lie quadratiques singulières, i.e. celles pour lesquelles l’invariant n’est pas nul. La classification est en relation avec les orbites adjointes des algèbres de Lie o(m) et sp(2n). Aussi, nous donnons une caractérisation isomorphe des algèbres de Lie quadratiques 2-nilpotentes et des superalgèbres de Lie quadratiques quasi-singulières pour le but d’exhaustivité. Nous étudions les algèbres de Jordan pseudoeuclidiennes qui sont obtenues des extensions doubles d’un espace vectoriel quadratique par une algèbre d’une dimension et les algèbres de Jordan pseudo-euclidienne 2-nilpotentes, de la même manière que cela a été fait pour les algèbres de Lie quadratiques singulières et des algèbres de Lie quadratiques 2-nilpotentes. Enfin, nous nous concentrons sur le cas d’une algèbre de Novikov symétrique et l’étudions à dimension 7. / In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

With the technological advancement in computer science, Geographic Information Science (GIScience), and transportation, more and more complex path finding queries including category based queries are proposed and studied across diverse disciplines. A category based query, such as Optimal Sequenced Routing (OSR) queries and Trip Planning Queries (TPQ), asks for a minimum-cost path that traverses a set of categories with or without a predefined order in a graph. Due to the extensive computing time required to process these complex queries in a large scale environment, efficient algorithms are highly desirable whenever processing time is a consideration. In Artificial Intelligence (AI), a best first search is an informed heuristic path finding algorithm that uses domain knowledge as heuristics to expedite the search process. Traditional best first searches are single-variate in terms of the number of variables to describe a state, and thus not appropriate to process these queries in a graph. In this dissertation, 1) two new types of category based queries, Category Sequence Traversal Query (CSTQ) and Optimal Sequence Traversal Query (OSTQ), are proposed; 2) the existing single-variate best first searches are extended to multivariate best first searches in terms of the state specified, and a class of new concepts--state graph, sub state graph, sub state graph space, local heuristic, local admissibility, local consistency, global heuristic, global admissibility, and global consistency--is introduced into best first searches; 3) two bivariate best first search algorithms, C* and O*, are developed to process CSTQ and OSTQ in a graph, respectively; 4) for each of C* and O*, theorems on optimality and optimal efficiency in a sub state graph space are developed and identified; 5) a family of algorithms including C*-P, C-Dijkstra, O*-MST, O*-SCDMST, O*- Dijkstra, and O*-Greedy is identified, and case studies are performed on path finding in transportation networks, and/or fully connected graphs, either directed or undirected; and 6) O*- SCDMST is adopted to efficiently retrieve optimal solutions for OSTQ using network distance metric in a large transportation network. / Ph. D.