In this essay we give an elementary introduction to topology so that we can prove Tychonoff’s theorem, and also its equivalence with the axiom of choice. / Denna uppsats tillhandahåller en grundläggande introduktion till topologi för att sedan bevisa Tychonoff’s theorem, samt dess ekvivalens med urvalsaxiomet.

by Tong Shiu Sing Dominic. / Parallel title in Chinese characters. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 236-241). / Acknowledgements --- p.iv / Abstract --- p.v / Chapter 1 --- Open Systems and Quasinormal Modes --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Non-Hermitian Systems --- p.1 / Chapter 1.1.2 --- Optical Cavities as Open Systems --- p.3 / Chapter 1.1.3 --- Outline of this Thesis --- p.6 / Chapter 1.2 --- Simple Models of Open Systems --- p.10 / Chapter 1.3 --- Contributions of the Author --- p.14 / Chapter 2 --- Completeness and Orthogonality --- p.16 / Chapter 2.1 --- Introduction --- p.16 / Chapter 2.2 --- Green's Function of the Open System --- p.19 / Chapter 2.3 --- High Frequency Behaviour of the Green's Function --- p.24 / Chapter 2.4 --- Completeness of Quasinormal Modes --- p.29 / Chapter 2. 5 --- Method of Projection --- p.31 / Chapter 2.5.1 --- Problems with the Usual Method of Projection --- p.31 / Chapter 2.5.2 --- Modified Method of Projection --- p.33 / Chapter 2.6 --- Uniqueness of Representation --- p.38 / Chapter 2.7 --- Definition of Inner Product and Quasi-Stationary States --- p.39 / Chapter 2.7.1 --- Orthogonal Relation of Quasinormal Modes --- p.39 / Chapter 2.7.2 --- Definition of Hilbert Space and State Vectors --- p.41 / Chapter 2.8 --- Hermitian Limits --- p.43 / Chapter 2.9 --- Numerical Examples --- p.45 / Chapter 3 --- Time-Independent Perturbation --- p.58 / Chapter 3.1 --- Introduction --- p.58 / Chapter 3.2 --- Formalism --- p.60 / Chapter 3.2.1 --- Expansion of the Perturbed Quasi-Stationary States --- p.60 / Chapter 3.2.2 --- Formal Solution --- p.62 / Chapter 3.2.3 --- Perturbative Series --- p.66 / Chapter 3.3 --- Diagrammatic Perturbation --- p.70 / Chapter 3.3.1 --- Series Representation of the Green's Function --- p.70 / Chapter 3.3.2 --- Eigenfrequencies --- p.73 / Chapter 3.3.3 --- Eigenfunctions --- p.75 / Chapter 3.4 --- Numerical Examples --- p.77 / Chapter 4 --- Method of Diagonization --- p.81 / Chapter 4.1 --- Introduction --- p.81 / Chapter 4.2 --- Formalism --- p.82 / Chapter 4.2.1 --- Matrix Equation with Non-unique Solution --- p.82 / Chapter 4.2.2 --- Matrix Equation with a Unique Solution --- p.88 / Chapter 4.3 --- Numerical Examples --- p.91 / Chapter 5 --- Evolution of the Open System --- p.97 / Chapter 5.1 --- Introduction --- p.97 / Chapter 5.2 --- Evolution with Arbitrary Initial Conditions --- p.99 / Chapter 5.3 --- Evolution with the Outgoing Plane Wave Condition --- p.106 / Chapter 5.3.1 --- Evolution Inside the Cavity --- p.106 / Chapter 5.3.2 --- Evolution Outside the Cavity --- p.110 / Chapter 5.4 --- Physical Implications --- p.112 / Chapter 6 --- Time-Dependent Perturbation --- p.114 / Chapter 6.1 --- Introduction --- p.114 / Chapter 6.2 --- Inhomogeneous Wave Equation --- p.117 / Chapter 6.3 --- Perturbative Scheme --- p.120 / Chapter 6.4 --- Energy Changes due to the Perturbation --- p.128 / Chapter 6.5 --- Numerical Examples --- p.131 / Chapter 7 --- Adiabatic Approximation --- p.150 / Chapter 7.1 --- Introduction --- p.150 / Chapter 7.2 --- The Effect of a Varying Refractive Index --- p.153 / Chapter 7.3 --- Adiabatic Expansion --- p.156 / Chapter 7.4 --- Numerical Examples --- p.167 / Chapter 8 --- Generalization of the Formalism --- p.176 / Chapter 8. 1 --- Introduction --- p.176 / Chapter 8.2 --- Generalization of the Orthogonal Relation --- p.180 / Chapter 8.3 --- Evolution with the Outgong Wave Condition --- p.183 / Chapter 8.4 --- Uniform Convergence of the Series Representation --- p.193 / Chapter 8.5 --- Uniqueness of Representation --- p.200 / Chapter 8.6 --- Generalization of Standard Calculations --- p.202 / Chapter 8.6.1 --- Time-Independent Perturbation --- p.203 / Chapter 8.6.2 --- Method of Diagonization --- p.206 / Chapter 8.6.3 --- Remarks on Dynamical Calculations --- p.208 / Appendix A --- p.209 / Appendix B --- p.213 / Appendix C --- p.225 / Appendix D --- p.231 / Appendix E --- p.234 / References --- p.236

South Africa’s sluggish economic growth and limited structural transformation are at the core of the country’s high levels of unemployment and poverty. Why some countries are able to grow and others not has been the subject of much academic debate, with recent literature throwing fresh light on the phenomenon by exploring the relationship between growth and a country’s productive structure. It is argued that countries’ heterogeneous productive structures stem from differences in entrenched capabilities and know-how in producing and marketing a very specific set of products. Investigating and comparing what countries actually produce and how this supports economic development yield valuable insights into why countries differ and what their transformation priorities should be. In this study a new analytical approach is adopted to determine the potential growth paths of South Africa’s broader agricultural sector, the so-called agro-complex. Comprising 1 456 primary and processed products as well as production inputs within the food, feed and fibre segments, this broad sector makes an important contribution to economic development through its direct and indirect linkages with other economic sectors as well as its relatively high labour absorption rate. The wide-ranging economic challenges that South Africa faces have clearly gripped the country’s agro-complex. A growing dependency on imports of higher value, processed food has led to deteriorating terms of trade, while limited innovation and diversification in agricultural exports (coupled with a high proportion of re-exports) is seriously constraining the positive impact that the agro-complex could have on the country’s growth rate. In this study, the productive structure of the agro-complex is analysed through the application of the ‘product space’ framework. This methodology investigates the relatedness between products as reflected in the conditional probability of a product being exported in tandem with another product, considering all global trade flows. The proximity value, the measurement of relatedness, between all 1 456 products is used to arrive at the structure and visible evidence of an agricultural product space network. This structure allows potential diversification pathways to be analysed. Research has shown that a country’s diversification process involves moves along pathways to nearby - and thus related – ‘new’ products, which simplifies the redeployment of existing productive capabilities and knowledge. These pathways are identified according to three strategic values, namely structural transformation, market potential and employment creation. The potential for structural transformation is determined by using a product-level measure of complexity, with higher level complexity (i.e. upgrading) going to the core of structural transformation. The potential market demand is determined by examining the prospects for import substitution; producing re-exported products locally; producing new, high-potential export products; and diversifying into new markets with existing export products (with the latter two categories evident in the results of a Decision Support Model (DSM)). Finally, the potential for employment creation is analysed in the light of labour and human capital intensities at product level. The study found that South Africa’s position in the agricultural product space is relatively dispersed. This position is characterised by a proportionally high level of core competencies within the primary agricultural cluster and a limited amount of vertical linkages. Furthermore the analyses showed that, especially the products for which South Africa has developed a significant high level of specialisation such as plums and wines are located in the sparser, less promising, parts of the agricultural product space. Notwithstanding, a total of 60 realistic diversification opportunities are within “reach” from the countries core competencies in the agro-complex. The analysis on structural transformation established that the level of product complexity within the agro-complex is more important for economic development than product diversity per se. It was revealed that South Africa only ranked 69th globally with regards to the complexity of its agro-complex. It was determined that upgrading within the forestry and agro-processing of food clusters hold the most potential for improving this position. The investigation into the potential for diversification driven by market potential revealed fore mostly that the potential for substituting imports as well as re-exports with local production is limited. The market prospects for exporting “new” products and expanding the export destinations of existing products within the agro-complex is much more favourable. It was estimated that most employment creation could be derived from diversification within the primary agricultural cluster although its number of opportunities are limited. Diversification within both the primary agricultural and agro-processing of food clusters was found the have to most favourable prospects for developing the level of human capital within the agro-complex. Based on all three strategic values, the most promising diversification opportunities are located within the forestry cluster. The range of realistic opportunities within the five clusters of the agro-complex form the core for developing product-level diversification strategies. This will ultimately strengthen the position of the agro-complex and boost South Africa’s growth path for the next decade. Hence, the main contribution of this study includes the determination of the structure of South Africa’s agro-complex from a product perspective and its implications for potential growth and development. / PhD (International Trade), North-West University, Potchefstroom Campus, 2014

South Africa’s sluggish economic growth and limited structural transformation are at the core of the country’s high levels of unemployment and poverty. Why some countries are able to grow and others not has been the subject of much academic debate, with recent literature throwing fresh light on the phenomenon by exploring the relationship between growth and a country’s productive structure. It is argued that countries’ heterogeneous productive structures stem from differences in entrenched capabilities and know-how in producing and marketing a very specific set of products. Investigating and comparing what countries actually produce and how this supports economic development yield valuable insights into why countries differ and what their transformation priorities should be. In this study a new analytical approach is adopted to determine the potential growth paths of South Africa’s broader agricultural sector, the so-called agro-complex. Comprising 1 456 primary and processed products as well as production inputs within the food, feed and fibre segments, this broad sector makes an important contribution to economic development through its direct and indirect linkages with other economic sectors as well as its relatively high labour absorption rate. The wide-ranging economic challenges that South Africa faces have clearly gripped the country’s agro-complex. A growing dependency on imports of higher value, processed food has led to deteriorating terms of trade, while limited innovation and diversification in agricultural exports (coupled with a high proportion of re-exports) is seriously constraining the positive impact that the agro-complex could have on the country’s growth rate. In this study, the productive structure of the agro-complex is analysed through the application of the ‘product space’ framework. This methodology investigates the relatedness between products as reflected in the conditional probability of a product being exported in tandem with another product, considering all global trade flows. The proximity value, the measurement of relatedness, between all 1 456 products is used to arrive at the structure and visible evidence of an agricultural product space network. This structure allows potential diversification pathways to be analysed. Research has shown that a country’s diversification process involves moves along pathways to nearby - and thus related – ‘new’ products, which simplifies the redeployment of existing productive capabilities and knowledge. These pathways are identified according to three strategic values, namely structural transformation, market potential and employment creation. The potential for structural transformation is determined by using a product-level measure of complexity, with higher level complexity (i.e. upgrading) going to the core of structural transformation. The potential market demand is determined by examining the prospects for import substitution; producing re-exported products locally; producing new, high-potential export products; and diversifying into new markets with existing export products (with the latter two categories evident in the results of a Decision Support Model (DSM)). Finally, the potential for employment creation is analysed in the light of labour and human capital intensities at product level. The study found that South Africa’s position in the agricultural product space is relatively dispersed. This position is characterised by a proportionally high level of core competencies within the primary agricultural cluster and a limited amount of vertical linkages. Furthermore the analyses showed that, especially the products for which South Africa has developed a significant high level of specialisation such as plums and wines are located in the sparser, less promising, parts of the agricultural product space. Notwithstanding, a total of 60 realistic diversification opportunities are within “reach” from the countries core competencies in the agro-complex. The analysis on structural transformation established that the level of product complexity within the agro-complex is more important for economic development than product diversity per se. It was revealed that South Africa only ranked 69th globally with regards to the complexity of its agro-complex. It was determined that upgrading within the forestry and agro-processing of food clusters hold the most potential for improving this position. The investigation into the potential for diversification driven by market potential revealed fore mostly that the potential for substituting imports as well as re-exports with local production is limited. The market prospects for exporting “new” products and expanding the export destinations of existing products within the agro-complex is much more favourable. It was estimated that most employment creation could be derived from diversification within the primary agricultural cluster although its number of opportunities are limited. Diversification within both the primary agricultural and agro-processing of food clusters was found the have to most favourable prospects for developing the level of human capital within the agro-complex. Based on all three strategic values, the most promising diversification opportunities are located within the forestry cluster. The range of realistic opportunities within the five clusters of the agro-complex form the core for developing product-level diversification strategies. This will ultimately strengthen the position of the agro-complex and boost South Africa’s growth path for the next decade. Hence, the main contribution of this study includes the determination of the structure of South Africa’s agro-complex from a product perspective and its implications for potential growth and development. / PhD (International Trade), North-West University, Potchefstroom Campus, 2014

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We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.

Les méthodes de décomposition de domaine en espace ont prouvé leur utilité dans le cadre des architectures parallèles. Pour les problèmes d’évolution en temps, il est nécessaire d’introduire une dimension supplémentaire de parallélisme dans la direction du temps. Ceci peut alors être couplé avec des méthodes de type optimisé Schwarz waveform relaxation. Nous nous intéressons dans cette thèse aux méthodes directes de décomposition en temps. Nous en étudions particulièrement deux. Dans une première partie nous étudions la méthode de produit tensoriel, introduite par R. E. Lynch, J. R. Rice, et D. H. Thomas in 1963. Nous proposons une méthode d’optimisation des pas de temps, basée sur une étude d’erreur en variable de Fourier en temps. Nous menons cette étude sur les schémas d’Euler et de Newmark pour la discrétisation en temps de l’équation de la chaleur. Nous présentons ensuite des tests numériques établissant la validité de cette approche. Dans la seconde partie, nous étudions les méthodes dites de Bloc, introduites par Amodio et Brugnano en 1997. Nous comparons diverses implémentations de la méthode, basées sur différentes approximations de l’exponentielle de matrice. Nous traitons l’équation de la chaleur et l’équation des ondes, et montrons par une étude numérique bidimensionnelle la puissance de la méthode. / Domain decomposition methods in space applied to Partial Differential Equations (PDEs) expanded considerably thanks to their effectiveness (memory costs, calculation costs, better local conditioned problems) and this related to the development of massively parallel machines. Domain decomposition in space-time brings an extra dimension to this optimization. In this work, we study two different direct time-parallel methods for the resolution of Partial Differential Equations. The first part of this work is devoted to the Tensor-product space-time method introduced by R.E. Lynch, J. R. Rice, and D. H. Thomas in 1963. We analyze it in depth for Euler and Crank-Nicolson schemes in time applied to the heat equation. The method needs all time steps to be different, while accuracy is optimal when they are all equal (in the Euler case). Furthermore, when they are close to each other, the condition number of the linear problems involved becomes very big. We thus give for each scheme an algorithm to compute optimal time steps, and present numerical evidences of the quality of the method. The second part of this work deals with the numerical implementation of the Block method of Amodio and Brugnano presented in 1997 to solve the heat equation with Euler and Crank- Nicolson time schemes and the elasticity equation with Euler and Gear time schemes. Our implementation shows how the method is accurate and scalable.

Based upon the export product space, this paper shows the main features of the export product composition of Peru and its regions. With respect to export baskets, these include: a high con- centration in few commodities; a relatively low degree of similarity between products; relatively high levels of productivity at the international level, and; enormous undiscovered or untapped export potential. In addition, this paper analyzes the hypothesis that the natural resource curse affects regional growth, employment and export diversification. Subject to the limitations of the data and the heterogeneous panel methods employed, estimation results indicate that the share of natural capital out of the total tangible wealth and natural capital per capita seems, on average and in a relatively statistically robust way, to spur growth, primary employment and export concentration in a few products. Consequently, the natural resource curse for the Peruvian regions is not about growth but about low regional development, since such resources do not have a significant impact on secondary and tertiary employment or on export product diversification. / Basado en la red de espacio de productos de exportación, este trabajo muestra las principales características de la composición de productos de exportación del Perú y sus regiones. Entre ellas: la de ser altamente concentrada en pocos productos intensivos en el uso de recursos naturales (RN), muy distantes unos de otros, de altos niveles de productividad internacional relativos al valor agregado real per cápita de cada región y de la economía en general, y de un enorme potencial de exportación todavía no explotado o descubierto. De otro lado, se analiza la validez de la hipótesis de la maldición de los efectos de los RN sobre el crecimiento y complementariamente sobre el empleo y la diversificación de los productos de exportación a nivel regional. Sujeto a las limitacio- nes de la información y métodos de paneles heterogéneos empleados, la evidencia relativamente robusta estadísticamente indica que la participación del capital natural del total de riqueza tangible y el capital natural per cápita han incidido en promedio positivamente en el crecimiento regional, el empleo relativo del sector primario, y la concentración de los principales productos de exportación. En consecuencia, la ‘maldición’ radica en la dependencia de las regiones en los RN para crecer retardando así el desarrollo regional por los efectos no significativos sobre elempleo secundario y terciario, y el proceso de diversificación de exportaciones.

This thesis presents work around 3 themes: dark energy, gravitational waves and Bayesian inference. Both dark energy and gravitational wave physics are not yet well constrained. They present interesting challenges for Bayesian inference, which attempts to quantify our knowledge of the universe given our astrophysical data. A dark energy equation of state reconstruction analysis finds that the data favours the vacuum dark energy equation of state $w {=} -1$ model. Deviations from vacuum dark energy are shown to favour the super-negative ‘phantom’ dark energy regime of $w {< } -1$, but at low statistical significance. The constraining power of various datasets is quantified, finding that data constraints peak around redshift $z = 0.2$ due to baryonic acoustic oscillation and supernovae data constraints, whilst cosmic microwave background radiation and Lyman-$\alpha$ forest constraints are less significant. Specific models with a conformal time symmetry in the Friedmann equation and with an additional dark energy component are tested and shown to be competitive to the vacuum dark energy model by Bayesian model selection analysis: that they are not ruled out is believed to be largely due to poor data quality for deciding between existing models. Recent detections of gravitational waves by the LIGO collaboration enable the first gravitational wave tests of general relativity. An existing test in the literature is used and sped up significantly by a novel method developed in this thesis. The test computes posterior odds ratios, and the new method is shown to compute these accurately and efficiently. Compared to computing evidences, the method presented provides an approximate 100 times reduction in the number of likelihood calculations required to compute evidences at a given accuracy. Further testing may identify a significant advance in Bayesian model selection using nested sampling, as the method is completely general and straightforward to implement. We note that efficiency gains are not guaranteed and may be problem specific: further research is needed.