South Africa's North West Province is dependent on gold and platinum exports for its economic growth and employment. Whether this specialisation is optimal from an economic growth, employment creation and welfare point of view, or whether greater export diversification should be encouraged, can most appropriately be evaluated using a general equilibrium framework. Moreover, the answer of whether such specialisation or rather diversification is appropriate may differ depending on the spatial level of analysis that is taken. For instance, export diversification or specialisation on a national level might differ from export diversification or specialisation on sub-national (e.g. provincial) level. A general equilibrium framework is most appropriate for investigating the potential differences between these levels as it takes into consideration a region's economic structure, and the economy-wide linkages between demand, supply, production structure and income distribution. In this light this thesis proposes a regional applied general equilibrium (RAGE) model for the North West Province to address these issues. The question asked is, how can a RAGE model be formulated and implemented for the North West Province of South Africa in order to study the differential impacts of greater export specialisation versus greater export diversification? In answering this question this thesis provides policy makers at regional/provincial level with access to a potentially useful modelling tool, to analyse regional economic issues; it contributes to better understanding the spatial concentration and spatial dynamics of economic activity in the North West Province; and contributes toward the practical policy debate in South Africa. This thesis reviews the literature on export diversification and specialisation, by investigating the extent of export diversification and specialisation in the North West Province over the period 1995-2006 and its relationship to GDP per capita, and using a RAGE model to investigate the economy-wide impacts of greater export diversification versus greater export speciaRsation. These results are contrasted by investigating the economy-wide impacts of greater export diversification versus greater export specialisation on the national level, using a country-wide applied general equilibrium (AGE) model for South Africa. It is found that greater export diversification results in a more substantial increase in exports (of between 0.95 and 1.73 per cent) than in the case of greater export specialisation. In fact it was found that if the North West Province were to specialise in mining exports, such exports would need to grow or increase from the base year by approximately 78 per cent (with no increase in the export demand for other sectors) to result in the same level of growth of total export volumes as is found under export diversification. Avenues for further research are identified. These are a need for more research on elasticities and parameters at regional level, an expansion of the transport services sector in the model, incorporating Tourism Satellite Accounts, BEE and SMME aspects into the model, etc. In conclusion, the thesis implemented the first provincial regional applied general equilibrium model for South Africa, and illustrated that it can make an important contribution to the policy debate on a provincial level in South Africa. The construction of Social Accounting Matrices (SAMs) for all of South Africa's provinces, and the maintenance of these provincial databases could therefore make further contributions to the quantitative assessment of policy options facing government on both the national and provincial level. / Thesis (Ph.D. (Economics))--North-West University, Potchefstroom Campus, 2008.

Dans cette thèse nous développons de nouveaux algorithmes de calcul numérique pour les matrices polynomiales. Nous abordons le problème du calcul de la structure propre (rang, espace nul, structures finie et infinie) d'une matrice polynomiale et nous appliquons les résultats obtenus au calcul de la factorisation J-spectrale des matrices polynomiales. Nous présentons également quelques applications de ces algorithmes en théorie de la commande. Tous les nouveaux algorithmes décrits ici sont basés sur le calcul d'espaces nuls constants de matrices bloc Toeplitz associées à la matrice polynomiale analysée. Pour calculer ces espaces nuls nous utilisons des méthodes standard de l'algèbre linéaire numérique comme la décomposition en valeurs singulières ou la factorisation QR. Nous étudions aussi l'application de méthodes rapides comme la méthode généralisée de Schur pour les matrices structurées. Nous analysons les algorithmes présentés au niveau complexité algorithmique et stabilité numérique, et effectuons des comparaisons avec d'autres algorithmes existants dans la littérature.

Cette thèse étudie des matrices d'alimentations. Dans un premier temps, l'intérêt se porte sur la matrice de Butler symétrique de taille N x N. Des formules permettant d'accéder à sa matrice [S] de manière analytique sont trouvées et démontrées. Dans un deuxième temps, la source focale d'une antenne multifaisceaux basée sur une géométrie originale constituée de coupleurs directionnels tridimensionnels est étudiée. Cette structure possède un fort niveau de ré- utilisation de ses éléments rayonnants et donc limite son encombrement global. La conception est détaillée. Les validations expérimentales effectuées sont montrées. Enfin, un système antennaire hybride mécanique / électronique pour de la poursuite est étudié. Sa conception est divisée en plusieurs parties. Les dépendances entre ces différents éléments sont mises en évidence, ainsi que la complexité de construction et les choix et compromis à effectuer.

Étant donne un mélange de plusieurs signaux sources, par exemple un morceau et plusieurs instruments, ou un entretien radiophonique et plusieurs interlocuteurs, la séparation de source mono-canal consiste a' estimer chacun des signaux sources a' partir d'un enregistrement avec un seul microphone. Puisqu'il y a moins de capteurs que de sources, il y a a priori une infinité de solutions sans rapport avec les sources originales. Il faut alors trouver quelle information supplémentaire permet de rendre le problème bien pose. Au cours des dix dernières années, la factorisation en matrices positives (NMF) est devenue un composant majeurs des systèmes de séparation de sources. En langage profane, la NMF permet de d'écrire un ensemble de signaux audio a ́ partir de combinaisons d' éléments sonores simples (les atomes), formant un dictionnaire. Les systèmes de séparation de sources reposent alors sur la capacité a trouver des atomes qui puissent être assignes de fa con univoque 'a chaque source sonore. En d'autres termes, ils doivent être interprétables. Nous proposons dans cette thèse trois contributions principales aux méthodes d'apprentissage de dictionnaire. La première est un critère de parcimonie par groupes adapte a la NMF lorsque la mesure de distorsion choisie est la divergence d'Itakura-Saito. Dans la plupart des signaux de musique on peut trouver de longs intervalles ou' seulement une source est active (des soli). Le critère de parcimonie par groupe que nous proposons permet de trouver automatiquement de tels segments et d'apprendre un dictionnaire adapte a chaque source. Ces dictionnaires permettent ensuite d'effectuer la tache de séparation dans les intervalles ou' les sources sont mélangées. Ces deux taches d'identification et de séparation sont effectuées simultanément en une seule passe de l'algorithme que nous proposons. Notre deuxième contribution est un algorithme en ligne pour apprendre le dictionnaire a grande échelle, sur des signaux de plusieurs heures, ce qui était impossible auparavant. L'espace mémoire requis par une NMF estimée en ligne est constant alors qu'il croit linéairement avec la taille des signaux fournis dans la version standard, ce qui est impraticable pour des signaux de plus d'une heure. Notre troisième contribution touche a' l'interaction avec l'utilisateur. Pour des signaux courts, l'apprentissage aveugle est particulièrement difficile, et l'apport d'information spécifique au signal traite est indispensable. Notre contribution est similaire à l'inpainting et permet de prendre en compte des annotations temps-fréquence. Elle repose sur l'observation que la quasi-totalite du spectro- gramme peut être divise en régions spécifiquement assignées a' chaque source. Nous d'éecrivons une extension de NMF pour prendre en compte cette information et discutons la possibilité d'inférer cette information automatiquement avec des outils d'apprentissage statistique simples.

South Africa's North West Province is dependent on gold and platinum exports for its economic growth and employment. Whether this specialisation is optimal from an economic growth, employment creation and welfare point of view, or whether greater export diversification should be encouraged, can most appropriately be evaluated using a general equilibrium framework. Moreover, the answer of whether such specialisation or rather diversification is appropriate may differ depending on the spatial level of analysis that is taken. For instance, export diversification or specialisation on a national level might differ from export diversification or specialisation on sub-national (e.g. provincial) level. A general equilibrium framework is most appropriate for investigating the potential differences between these levels as it takes into consideration a region's economic structure, and the economy-wide linkages between demand, supply, production structure and income distribution. In this light this thesis proposes a regional applied general equilibrium (RAGE) model for the North West Province to address these issues. The question asked is, how can a RAGE model be formulated and implemented for the North West Province of South Africa in order to study the differential impacts of greater export specialisation versus greater export diversification? In answering this question this thesis provides policy makers at regional/provincial level with access to a potentially useful modelling tool, to analyse regional economic issues; it contributes to better understanding the spatial concentration and spatial dynamics of economic activity in the North West Province; and contributes toward the practical policy debate in South Africa. This thesis reviews the literature on export diversification and specialisation, by investigating the extent of export diversification and specialisation in the North West Province over the period 1995-2006 and its relationship to GDP per capita, and using a RAGE model to investigate the economy-wide impacts of greater export diversification versus greater export speciaRsation. These results are contrasted by investigating the economy-wide impacts of greater export diversification versus greater export specialisation on the national level, using a country-wide applied general equilibrium (AGE) model for South Africa. It is found that greater export diversification results in a more substantial increase in exports (of between 0.95 and 1.73 per cent) than in the case of greater export specialisation. In fact it was found that if the North West Province were to specialise in mining exports, such exports would need to grow or increase from the base year by approximately 78 per cent (with no increase in the export demand for other sectors) to result in the same level of growth of total export volumes as is found under export diversification. Avenues for further research are identified. These are a need for more research on elasticities and parameters at regional level, an expansion of the transport services sector in the model, incorporating Tourism Satellite Accounts, BEE and SMME aspects into the model, etc. In conclusion, the thesis implemented the first provincial regional applied general equilibrium model for South Africa, and illustrated that it can make an important contribution to the policy debate on a provincial level in South Africa. The construction of Social Accounting Matrices (SAMs) for all of South Africa's provinces, and the maintenance of these provincial databases could therefore make further contributions to the quantitative assessment of policy options facing government on both the national and provincial level. / Thesis (Ph.D. (Economics))--North-West University, Potchefstroom Campus, 2008.

The major part of this thesis has been devoted to path and motion generation synthesis of spatial mechanisms. For the first time kinematic synthesis methods have been developed based on the algebra of exponential rotation matrices. Besides modeling spatial pairs such as spheric, cylindric and Hook&#039 / s joints by combinations of revolute and prismatic joints and applying Denavit-Hartenberg&#039 / s convention, general loop closure equations have been presented for path and motion generation synthesis of any spatial mechanism with lower kinematic pairs. In comparison to the existing synthesis methods the main advantage of the methods presented in this thesis is that, general loop closure equations have been presented for any kind of spatial mechanism with lower kinematic pairs. Besides these methods enable the designer to benefit the advantages of the algebra of exponential rotation matrices. In order to verify the applicability of the synthesis methods presented in this thesis, the general loop closure equations of RSHR, RCCR and RSSR-SC mechanisms have been determined and then using these equations six numerical examples have been solved. Some tables have been presented based on the determined loop closure equations which reveal useful information about the number of precision points or positions that can be considered for the kinematic synthesis of the above mentioned mechanisms and the number of free parameters. In numerical examples, the mechanisms have been synthesized based on the general loop closure equations and the synthesis algorithms presented in the thesis. Although in some cases semi-analytical solutions have been obtained, in most of the cases, the loop closure equations were solved by computer programs written by Mathcad. The input angle-output angle diagrams drawn at the end of each numerical example illustrate the motion continuity of the mechanisms and that branching has been avoided. Detailed information has been given about the computer programs and the difficulties which may arise while synthesizing spatial mechanisms. In addition to the above mentioned points, a mobility analysis has been done for the RCCR mechanism and some inequalities have been obtained in terms of the link lengths. The swing angle diagram of the RCCR linkage has been drawn too.

This thesis focuses on the use of extremal techniques in analyzing problems that historically have been associated with other areas of discrete mathematics. We establish new techniques for analyzing combinatorial problems with two different types of nonlinear orders, and then use them to solve important previously-open problems in mathematics. In addition, we use entropy techniques to establish a variety of bounds in the theory of sumsets. In the second chapter, we examine a problem of Furedi and Hajnal regarding forbidden patterns in (0,1)-matrices. We introduce a new technique that gives an asymptotically tight bound on the number of 1-entries that a (0,1)-matrix can contain while avoiding a fixed permutation matrix. We use this result to solve the Stanley-Wilf conjecture, a well-studied open problem in enumerative combinatorics. We then generalize the technique to give results on d-dimensional matrices. In the third chapter, we examine a problem of Pinchasi and Radoicic on cyclically order sets. To do so, we prove an upper bound on the sizes of such sets, given that their orders have the intersection reverse property. We then use this to give an upper bound on the number of edges that a graph can have, assuming that the graph can be drawn so that no cycle of length four has intersecting edges. This improves the previously best known bound and (up to a log-factor) matches the best known lower bound. This result implies improved bounds on a number of well-studied problems in geometric combinatorics, most notably the complexity of pseudo-circle arrangements. In the final chapter, we use entropy techniques to establish new bounds in the theory of sumsets. We show that such sets behave fractionally submultiplicatively, which in turn provides new Plunecke-type inequalities of the form first introduced by Gyarmati, Matolcsi, and Ruzsa.

The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.