Doctor of Philosophy / Department of Mathematics / David R. Auckly / We study geometric variational problems for a class of models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. The Hopfions exhibit distinct localized knot-like structure and received a lot of attention lately in both mathematical and physical literature. High non-linearity of the energy functional presents both analytical and algebraic difficulties for studying it. In particular we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homotopy for smooth and continuous maps. As the spaces in question are neither linear nor even convex we take advantage of the algebraic structure on homogeneous spaces to represent maps by gauge potentials that form a linear space and reformulate the problem in terms of these potentials. However this representation of maps introduces some gauge ambiguity into the picture and we work out 'gauge calculus' for the principal bundles involved to apply the gauge-fixing techniques that eliminate the ambiguity. These bundles arise as pullbacks of the structure bundles H[arrow pointing right with hook on tail]G[arrow pointing right]G/H of homogeneous spaces and we study their topology and geometry that are of independent interest. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers only in each 2-homotopy class.

Quantum Chromo Dynamics (QCD) is a relativistic field theory of a non-abelian gauge field coupled to several flavors of fermions. Two dimensional (one space and one time) QCD serves as an interesting toy model that shares several features with the four dimensional physically relevant theory. The main aim of the research is to study two dimensional QCD using the lattice regularization. Two dimensional QCD without any fermion content is solved analytically using lattice regularization. Explicit expressions for the expectation values of Wilson loops and the correlation of two Polyakov loops oriented in two different directions are obtained. Physics of the QCD vacuum is explained using these results. The Hamiltonian formalism of lattice QCD with fermion content serves as an approach to study quark excitations out of the vacuum. The formalism is first developed and techniques to numerically evaluate the spectrum of physical particles, namely, meson and baryons are described. The Hybrid Monte Carlo technique was used to numerically extract the lowest meson and baryon masses as a function of the quark masses. It is shown that neither the lowest meson mass nor the lowest baryon mass goes to zero as the quark mass is taken to zero. This numerically establishes the presence of a mass gap in two dimensional QCD.

Thesis ( M.Sc.(Agricultural Economics)) --University of Limpopo, 2019. / Potato is perceived as an excellent crop in the fight against hunger and poverty. The recent high potato price in South Africa has pushed the vegetable out of reach of the poorest of the poor. The study attempts to analyse potato price volatility in South Africa and furthermore assess how various factors were responsible for the recent potato price volatility. Quarterly data for potato price, number of hectares planted, rainfall and temperature levels from 2006q1 to 2017q4 was collected from various sources and were used for analysis. The total observation of 48. The volatility in the series was determined by performing ARCH/GARCH model. GARCH model indicates an evidence of GARCH effect in the series, meaning that GARCH model influences potato price volatility in South Africa. The Johansen cointegration used both trace and eigenvalue to test the existence of a long run relationship between potato price and various variables. The cointegration results were positive indicating that there exists long run relationship amongst variables. The study further used Johansen cointegration as well as standard error to determine the number of cointegrating variables in the long run. The results indicated that the number of hectares planted and rainfall level have significant relationship with potato price. Wald tests was used to check whether the past values of number of hectares planted and rainfall level influenced the current value of potato price. The Walt test results concluded that there is no evidence of short run causality running from number of hectares planted and rainfall level to potato price. In the study, ECM model was used to forecast the potato price fluctuation in South Africa. The study recommends that farmers need to engage in contract market so as to minimize the risk of potato price volatility. The Department of Agriculture should forecast agricultural commodities price volatility and make information accessible to the farmers so that they are able to adopt strategies that will assist them to overcome crisis.

The purpose of this dissertation is to consider whether centralised bargaining, through bargaining councils, is a suitable mechanism for determining minimum wages in South Africa. In addressing this issue, the minimum wage fixing mechanisms currently available in South Africa, the impact they have on the labour markets and whether there is a need for reformation of our labour laws relating to the setting of minimum wages will be considered. The dissertation focuses on the various philosophical perspectives on labour law, the international development of collective labour law, international wage-fixing mechanisms and the development of South African labour law from the Industrial Conciliations Act 11 of 1924 to the current Labour Relations Act 66 of 1995. The current levels of collective bargaining available in South African, focusing on the establishment and functioning of bargaining councils, the extension of and exemption from collective agreements, as well as the use of collective bargaining to set minimum wages are discussed. The advantages and disadvantages of our current minimum wage fixing mechanisms are also discussed. For the purpose of comparison, reference is also made to wage fixing though sectoral determinations, although the focus of the dissertation is on collective labour law. In the international comparison, the development and functioning of the Australian and French wage-setting regulations are discussed, as well as policies that could be considered for application in South Africa. Collective bargaining, and in particular centralised collective bargaining, plays a significant role in South African labour law. Since South Africa does not have a national minimum wage, centralised bargaining remains the main form of fixing minimum wages, apart from sectoral determinations. In the conclusion and recommendations, possible solutions to the shortcomings in our centralised ii bargaining system, as well as alternative means of setting minimum wages are considered. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mercantile Law / unrestricted

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A field experiment was conducted in the Ridge and Valley region of Virginia near Blacksburg during the 1989 through 1991 corn-rye growing seasons. The treatments in this experiment consisted of varying amounts of ¹⁵N-depleted fertilizer N applied to corn (Zea mays L.) at planting followed by a winter rye (Secale cereale L.) cover crop treatment. The research was divided into four studies. The first study was conducted to evaluate an analytical procedure that could be used for the diffusion of low masses of ¹⁵N-labeled NH₄ in 2M KCI and subsequent analysis for N recovery and ¹⁵N concentrations in soil by direct combustion mass spectrometry. Diffusion was found to be a suitable technique for preparing low-mass N samples for automated ¹⁵N analysis by Automated Nitrogen-Carbon Analysis/Mass Spectrometry (ANCA/MS). Recoveries of low masses of added N were quantitative, and accurate ¹⁵N concentrations were obtained when the results were corrected for isotope dilution due to background or contaminant N. The second study was conducted to determine if ¹⁵N-depleted fertilizer N could be satisfactorily used as a tracer of residual fertilizer N in plant tissue and various soil N fractions through a corn-winter rye crop rotation. Fertilizer-derived N in the soil NO₃-N fraction following corn harvest was clearly detectable and distinguishable from natural abundance to a 90-cm depth. Detection of fertilizer N in the total N pool below the 30-cm depth was not reliable, particularly at the lower N rates. Clay-fixation of fertilizer N measured at corn harvest was not detected by ¹⁵N analysis. Inconclusive results indicate that further research is needed to determine the feasibility of using depleted material for measuring clay-fixation of fertilizer-derived NH₄⁺-N. Nitrogen uptake by a winter rye cover crop reduced soil NO₃-N levels below that required for accurate isotope-ratio analysis. Following winter fallow (approx. 1 yr after fertilizer application) residual ¹⁵N-depleted fertilizer N was still detectable in plant tissue and the soil NO₃-N fraction. The objectives of the third study were to measure plant uptake and soil distribution of fertilizer N applied to corn at varying N rates and to determine the relationships between economic optimum N rate, fertilizer-use efficiency, and potential leaching loss of residual fertilizer N to groundwater. Plant recovery of fertilizer N in 1989 ranged from 33 to 47% even though no grain yield and fertilizer N uptake response resulted from N fertilization. Greatest accumulation of residual fertilizer N was found in the surface 30-cm both years following corn harvest. The economic optimum N rate for 1990 corn planted into a rye mulch (218 kg N ha⁻¹) corresponded closely with the rate (224 kg N ha⁻¹) resulting in the highest fertilizer-use efficiency. Low levels of residual fertilizer-derived NO₃ in the 60-90-cm depth following the 1990 corn harvest provides evidence to support the use of the economic optimum N rate concept from both economic and environmental viewpoints. The fourth study was designed to measure the effectiveness of a winter rye cover crop for recovering residual fertilizer N from the previous application of varying N rates to corn. Recovery of fertilizer N by winter rye increased with increasing N rate applied to the previous corn crop and ranged from 3.5 to 35.9 kg N ha⁻¹ in 1990 and 2.3 to 25.7 kg N ha⁻¹ in 1991. Residual fertilizer N recovery in 1991 was higher in rye plots where the previous corn crop had been planted no-till into rye stubble as compared to corn planted no-till into rye mulch. Little or no fertilizer-derived mineral N was measured in the soil to a final depth of 90-cm following a winter rye cover crop. Amounts of fertilizer-derived mineral N increased with depth and previous fertilizer N rate applied to corn following winter fallow. These results provide evidence to support the use of a winter rye cover crop on a silt loam soil to recover residual fertilizer-derived mineral N that might otherwise be lost to groundwater. / Ph. D.

[EN] The present PhD thesis is focused on numerical analysis and computing of finite difference schemes for several relevant option pricing models that generalize the Black-Scholes model. A careful analysis of desirable properties for the numerical solutions of option pricing models as the positivity, stability and consistency, is provided. In order to handle the free boundary that arises in American option pricing problems, various transformation techniques based on front-fixing method are applied and studied. Special attention is paid to multi-asset option pricing, such as exchange or spread option. Appropriate transformation allows eliminating of the cross derivative term. Transformation techniques of partial differential equations to remove convection and reaction terms are studied in order to simplify the models and avoid possible troubles of stability. This thesis consists of six chapters. The first chapter is an introduction containing definitions of option and related terms and derivation of the Black-Scholes equation as well as general aspects of theory of finite difference schemes, including preliminaries on numerical analysis. Chapter 2 is devoted to solve linear Black-Scholes model for American put and call options. A Landau transformation and a new front-fixing transformation are applied to the free boundary value problem. It leads to non-linear partial differential equation (PDE) in a fixed domain. Stable and consistent explicit numerical schemes are proposed preserving positivity and monotonicity of the solution in accordance with the behaviour of the exact solution. Efficiency of the front-fixing method demonstrated in Chapter 2 has motivated us to apply the method to some more complicated nonlinear models. A new change of variables resulting in a time dependent boundary instead of fixed one, is applied to nonlinear Black-Scholes model for American options, such as Barles and Soner and Risk Adjusted Pricing models. Chapter 4 provides a new alternative approach for solving American option pricing problem based on rationality of investor. There exists an intensity function that can be reduced in the simplest case to penalty approach. Chapter 5 deals with multi-asset option pricing. Appropriate transformation allows eliminating of the cross derivative term avoiding computational drawbacks and possible troubles of stability. Concluding remarks are given in Chapter 6. All the considered models and numerical methods are accompanied by several examples and simulations. The convergence rate is computed confirming the theoretical study of consistency. Stability conditions are tested by numerical examples. Results are compared with known relevant methods in the literature showing efficiency of the proposed methods. / [ES] La presente tesis doctoral se centra en la construcción de esquemas en diferencias finitas y el análisis numérico de relevantes modelos de valoración de opciones que generalizan el modelo de Black-Scholes. Se proporciona un análisis cuidadoso de las propiedades de las soluciones numéricas tales como la positividad, la estabilidad y la consistencia. Con el fin de manejar la frontera libre que surge en los problemas de valoración de opciones Americanas, se aplican y se estudian diversas técnicas de transformación basadas en el método de fijación de las fronteras (front-fixing). Se presta especial atención a la valoración de opciones de múltiples activos, como son las opciones ''exchange'' y ''spread''. Esta tesis se compone de seis capítulos. El primer capítulo es una introducción que contiene las definiciones de opción y términos relacionados y la derivación de la ecuación de Black-Scholes, así como aspectos generales de la teoría de los esquemas en diferencias finitas, incluyendo preliminares de análisis numérico. El capítulo 2 está dedicado a resolver el modelo lineal de Black-Scholes para opciones Americanas put y call. Para fijar las fronteras del problema de frontera libre se aplican transformaciones como la de Landau y un nuevo cambio de variable propuesto. La eficiencia del método front-fixing mostrada en el capítulo 2 ha motivado el estudio de su aplicación a algunos modelos no lineales más complicados. En particular, se propone un cambio de variables que lleva a una nueva frontera dependiente del tiempo en lugar de una fija. Este cambio se aplica a modelos no lineales de Black-Scholes para opciones Americanas, como son el de Barles y Soner y el modelo RAPM (Risk Adjusted Pricing Methodology). El capítulo 4 ofrece una nueva técnica para la resolución de problemas de valoración de opciones Americanas basada en la racionalidad de los inversores. Aparece una función de la intensidad que se puede reducir en el caso más simple a la técnica de penalización (penalty method). Este enfoque tiene en cuenta el posible comportamiento irracional de los inversores. En la sección 4.2 se aplica esta técnica al modelo de cambio de regímenes lo que lleva a un nuevo modelo que tiene en cuenta el posible ejercicio irracional, así como varios estados del mercado. El enfoque del parámetro de racionalidad junto con una transformación logarítmica permiten construir un esquema numérico eficiente sin aplicar el método front-fixing o la conocida formulación de LCP (Linear Complementarity Problem). El capítulo 5 se dedica a la valoración de opciones de activos múltiples. Una transformación apropiada permite la eliminación del término de derivadas cruzadas evitando inconvenientes computacionales y posibles problemas de estabilidad. Las conclusiones se muestran en el capítulo 6. Se pone en relieve varios aspectos de la presente tesis. Todos los modelos considerados y los métodos numéricos van acompañados de varios ejemplos y simulaciones. Se estudia la convergencia numérica que confirma el estudio teórico de la consistencia. Las condiciones de estabilidad son corroboradas con ejemplos numéricos. Los resultados se comparan con métodos relevantes de la bibliografía mostrando la eficiencia de los métodos propuestos. / [CA] La present tesi doctoral se centra en la construcció d'esquemes en diferències finites i l'anàlisi numèrica de rellevants models de valoració d'opcions que generalitzen el model de Black-Scholes. Es proporciona una anàlisi cuidadosa de les propietats de les solucions numèri-ques com ara la positivitat, l'estabilitat i la consistència. A fi de manejar la frontera lliure que sorgix en els problemes de valoració d'opcions Americanes, s'apliquen i s'estudien diverses tècniques de transformació basades en el mètode de fixació de les fronteres (front-fixing). Es presta especial atenció a la valoració d'opcions de múltiples actius, com són les opcions ''exchange'' i ''spread''. Esta tesi es compon de sis capítols. El primer capítol és una introducció que conté les definicions d'opció i termes relacionats i la derivació de l'equació de Black-Scholes, així com aspectes generals de la teoria dels esquemes en diferències finites, incloent aspectes preliminars d'anàlisi numèrica. El 2n capítol està dedicat a resoldre el model lineal de Black-Scholes per a opcions Americanes ''put'' i ''call''. Per a fixar les fronteres del problema de frontera lliure s'apliquen transformacions com la de Landau i s'ha proposat un nou canvi de variable proposat. Açò porta a una equació diferencial en derivades parcials no lineal en un domini fix. L'eficiència del mètode front-fixing mostrada en el 2n capítol ha motivat l'estudi de la seua aplicació a alguns models no lineals més complicats. En particular, es proposa un canvi de variables que porta a una nova frontera dependent del temps en compte d'una fixa. Este canvi s'aplica a models no lineals de Black-Scholes per a opcions Americanes, com són el de Barles i Soner i el model RAPM (Risk Adjusted Pricing Methodology). El 4t capítol oferix una nova tècnica per a la resolució de problemes de valoració d'opcions Americanes basada en la racionalitat dels inversors. Apareix una funció de la intensitat que es pot reduir en el cas més simple a la tècnica de penalització (penal method) . Este enfocament té en compte el possible comportament irracional dels inversors. En la secció 4.2 s'aplica esta tècnica al model de canvi de règims el que porta a un nou model que té en compte el possible exercici irracional, així com diversos estats del mercat. L'enfocament del paràmetre de racionalitat junt amb una transformació logarítmica permeten construir un esquema numèric eficient sense aplicar el mètode front-fixing o la coneguda formulació de LCP (Linear Complementarity Problem). El 5é capítol es dedica a la valoració d'opcions d'actius múltiples. Una transformació apropiada permet l'eliminació del terme de derivades mixtes evitant inconvenients computacionals i possibles problemes d' estabilitat. Les conclusions es mostren al 6é capítol. Es posa en relleu diversos aspectes de la present tesi. Tots els models considerats i els mètodes numèrics van acompanyats de diversos exemples i simulacions. S'estu-dia la convergència numèrica que confirma l'estudi teòric de la consistència. Les condicions d'estabilitat són corroborades amb exemples numèrics. Els resultats es comparen amb mètodes rellevants de la bibliografia mostrant l'eficiència dels mètodes proposats. / Egorova, V. (2016). Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/68501 / Premios Extraordinarios de tesis doctorales

Nitrogenase fixation occurred in a number of habitats in and nearby several antarctic lakes. The observed acetylene reduction occurred in bluegreen algal mats in littoral areas that received maximal sunlight. The benthic bluegreen algal communities in reduced light under 5-6 m of permanent ice showed no detectable nitrogenase activity. The observed nitrogen fixation potential correlated with the presence of heterocystous bluegreen algae considered to be the major nitrogen fixing organisms in these habitats. The relatively low acetylene reduction rates suggest that a small but significant contribution of ammonia to these environments deficient in nitrogen may occur through nitrogen fixation. / Master of Science