[pt] Uma abordagem experimental foi utilizada para o estudo da eficiência de varredura de fluidos viscoplásticos na célula de Hele-Shaw , que representa um meio poroso com porosidade constante. A análise baseia-se em um fluido empurrando outro onde, pela razão de viscosidades acima do punto de referência , manifesta-se a instabilidade de Saffman-Taylor, que desencadeia em uma interface instável com divisões sucessivas conhecidas como viscous fingers. A formação da instabilidade é considerada uma condição indesejável, e no caso da indústria petrolífera, torna-se de interesse na invasão da formação pelo fluido de perfuração, pasta de cimento, e no deslocamento de petróleo pesado em reservatórios entre outros. Apesar de que o fenômeno é vastamente estudado utilizando a célula de Hele-Shaw, a maioria dos trabalhos concentram-se no deslocamento de líquidos newtonianos por ar, deixando uma ampla gama de possibilidades de estudo, particularmente no que diz respeito aos fluidos não newtonianos. Utilizando equipamentos que permitem assegurar uma vazão constante de injeção, testou-se três concentrações diferentes de solução aquosa de carbopol em duas configurações diferentes onde o líquido viscoplástico é o deslocador e vice-versa. Fazendo uso do processamento digital de imagens, avaliou-se a forma da interface em função do desvio padrão, além da velocidade e da eficiência de varredura no deslocamento. Determinaram-se parâmetros adimensionais em função da geometria da bancada e dos fatores dinâmicos e reológicos dos fluidos onde se obtiveram duas zonas vem marcadas de varredura. Observou-se que a razão de viscosidades e o número de capilaridade modificada crítica são os indicadores que governam o deslocamento. / [en] An experimental approach has been used to study the sweep efficiency of viscoplastic fluid in a Hele-Shaw cell, which represents a porous medium with constant porosity. The analysis is based on a lower viscosity fluid pushing a more viscous through a geometry, which manifests the instability of Saffman-Taylor, that result in an unstable interface with successive divisions known as viscous fingers. The occurrence of instability is considered an undesirable condition, and in the petroleum oil industry, it becomes of interest in the reservoir oil invasion and heavy oil displacement in reservoirs. Despite being widely studied, most studies focus on the displacement of Newtonian liquids by air, leaving the field of study of non-Newtonian fluids without being evaluated where none of viscosities of the fluids involved can be considered negligible. Using equipment that ensures a constant flow rate of injection three different aqueous concentrations of carbopol were tested in two different configurations: carbopol as the displacing fluid and vice-versa. Using digital image processing, the shape interface was evaluated according to their standard deviation, average velocity and efficiency sweep during the displacement. Determined dimensionless parameters was evaluated in function of the experimental geometry, dynamic and rheological factors of the fluids in order to get benchmark that allow separating interfaces zones that tend to 100 per cent or plugs and the zone where the viscous fingers reveal themselves. It was found that the viscosity ratio and modified critical capillary govern the displacement.

Modern numerical ordinary differential equation initial-value problem (ODE-IVP) solvers compute a piecewise polynomial approximate solution to the mathematical problem. Evaluating the mathematical problem at this approximate solution defines the defect. Corless and Corliss proposed rigorous defect control of numerical ODE-IVP. This thesis automates rigorous defect control for explicit, first-order, nonlinear ODE-IVP. Defect control is residual-based backward error analysis for ODE, a special case of Wilkinson's backward error analysis. This thesis describes a complete software implementation of the Corless and Corliss algorithm and extensive numerical studies. Basic time-stepping software is adapted to defect control and implemented. Advances in software developed for validated computing applications and advances in programming languages supporting operator overloading enable the computation of a tight rigorous enclosure of the defect evaluated at the approximate solution with Taylor models. Rigorously bounding a norm of the defect, the Corless and Corliss algorithm controls to mathematical certainty the norm of the defect to be less than a user specified tolerance over the integration interval. The validated computing software used in this thesis happens to compute a rigorous supremum norm. The defect of an approximate solution to the mathematical problem is associated with a new problem, the perturbed reference problem. This approximate solution is often the product of a numerical procedure. Nonetheless, it solves exactly the new problem including all errors. Defect control accepts the approximate solution whenever the sup-norm of the defect is less than a user specified tolerance. A user must be satisfied that the new problem is an acceptable model. / Thesis / Master of Science (MSc) / Many processes in our daily lives evolve in time, even the weather. Scientists want to predict the future makeup of the process. To do so they build models to model physical reality. Scientists design algorithms to solve these models, and the algorithm implemented in this project was designed over 25 years ago. Recent advances in mathematics and software enabled this algorithm to be implemented. Scientific software implements mathematical algorithms, and sometimes there is more than one software solution to apply to the model. The software tools developed in this project enable scientists to objectively compare solution techniques. There are two forces at play; models and software solutions. This project build software to automate the construction of the exact solution of a nearby model. That's cool.

LD2668 .T4 1966 C6 / Master of Science

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, higher order numerical methods for weak approximation of solutions of stochastic differential equations (SDEs) are presented. They are motivated by option pricing problems in finance where the price of a given option can be written as the expectation of a functional of a diffusion process. Numerical methods of order at most one have been the most used so far and higher order methods have been difficult to perform because of the unknown density of iterated integrals of the d-dimensional Brownian motion present in the stochastic Taylor expansion. In 2001, Kusuoka constructed a higher order approximation scheme based on Malliavin calculus. The iterated stochastic integrals are replaced by a family of finitely-valued random variables whose moments up to a certain fixed order are equivalent to moments of iterated Stratonovich integrals of Brownian motion. This method has been shown to outperform the traditional Euler-Maruyama method. In 2004, this method was refined by Lyons and Victoir into Cubature on Wiener space. Lyons and Victoir extended the classical cubature method for approximating integrals in finite dimension to approximating integrals in infinite dimensional Wiener space. Since then, many authors have intensively applied these ideas and the topic is today an active domain of research. Our work is essentially based on the recently developed higher order schemes based on ideas of the Kusuoka approximation and Lyons-Victoir “Cubature on Wiener space” and mostly applied to option pricing. These are the Ninomiya-Victoir (N-V) and Ninomiya- Ninomiya (N-N) approximation schemes. It should be stressed here that many other applications of these schemes have been developed among which is the Alfonsi scheme for the CIR process and the decomposition method presented by Kohatsu and Tanaka for jump driven SDEs. After sketching the main ideas of numerical approximation methods in Chapter 1 , we start Chapter 2 by setting up some essential terminologies and definitions. A discussion on the stochastic Taylor expansion based on iterated Stratonovich integrals is presented, we close this chapter by illustrating this expansion with the Euler-Maruyama approximation scheme. Chapter 3 contains the main ideas of Kusuoka approximation scheme, we concentrate on the implementation of the algorithm. This scheme is applied to the pricing of an Asian call option and numerical results are presented. We start Chapter 4 by taking a look at the classical cubature formulas after which we propose in a simple way the general ideas of “Cubature on Wiener space” also known as the Lyons-Victoir approximation scheme. This is an extension of the classical cubature method. The aim of this scheme is to construct cubature formulas for approximating integrals defined on Wiener space and consequently, to develop higher order numerical schemes. It is based on the stochastic Stratonovich expansion and can be viewed as an extension of the Kusuoka scheme. Applying the ideas of the Kusuoka and Lyons-Victoir approximation schemes, Ninomiya- Victoir and Ninomiya-Ninomiya developed new numerical schemes of order 2, where they transformed the problem of solving SDE into a problem of solving ordinary differential equations (ODEs). In Chapter 5 , we begin by a general presentation of the N-V algorithm. We then apply this algorithm to the pricing of an Asian call option and we also consider the optimal portfolio strategies problem introduced by Fukaya. The implementation and numerical simulation of the algorithm for these problems are performed. We find that the N-V algorithm performs significantly faster than the traditional Euler-Maruyama method. Finally, the N-N approximation method is introduced. The idea behind this scheme is to construct an ODE-valued random variable whose average approximates the solution of a given SDE. The Runge-Kutta method for ODEs is then applied to the ODE drawn from the random variable and a linear operator is constructed. We derive the general expression for the constructed operator and apply the algorithm to the pricing of an Asian call option under the Heston volatility model. / AFRIKAANSE OPSOMMING: In hierdie proefskrif, word ’n hoërorde numeriese metode vir die swak benadering van oplossings tot stogastiese differensiaalvergelykings (SDV) aangebied. Die motivering vir hierdie werk word gegee deur ’n probleem in finansies, naamlik om opsiepryse vas te stel, waar die prys van ’n gegewe opsie beskryf kan word as die verwagte waarde van ’n funksionaal van ’n diffusie proses. Numeriese metodes van orde, op die meeste een, is tot dus ver in algemene gebruik. Dit is moelik om hoërorde metodes toe te pas as gevolg van die onbekende digtheid van herhaalde integrale van d-dimensionele Brown-beweging teenwoordig in die stogastiese Taylor ontwikkeling. In 2001 het Kusuoka ’n hoërorde benaderings skema gekonstrueer wat gebaseer is op Malliavin calculus. Die herhaalde stogastiese integrale word vervang deur ’n familie van stogastiese veranderlikes met eindige waardes, wat se momente tot ’n sekere vaste orde bestaan. Dit is al gedemonstreer dat hierdie metode die tradisionele Euler-Maruyama metode oortref. In 2004 is hierdie metode verfyn deur Lyons en Victoir na volumeberekening op Wiener ruimtes. Lyons en Victoir het uitgebrei op die klassieke volumeberekening metode om integrale te benader in eindige dimensie na die benadering van integrale in oneindige dimensionele Wiener ruimte. Sedertdien het menige outeurs dié idees intensief toegepas en is die onderwerp vandag ’n aktiewe navorsings gebied. Ons werk is hoofsaaklik gebaseer op die onlangse ontwikkelling van hoërorde skemas, wat op hul beurt gebaseer is op die idees van Kusuoka benadering en Lyons-Victoir "Volumeberekening op Wiener ruimte". Die werk word veral toegepas op die prysvastelling van opsies, naamlik Ninomiya-Victoir en Ninomiya-Ninomiya benaderings skemas. Dit moet hier beklemtoon word dat baie ander toepassings van hierdie skemas al ontwikkel is, onder meer die Alfonsi skema vir die CIR proses en die ontbinding metode wat voorgestel is deur Kohatsu en Tanaka vir sprong aangedrewe SDVs. Na ’n skets van die hoof idees agter metodes van numeriese benadering in Hoofstuk 1 , begin Hoofstuk 2 met die neersetting van noodsaaklike terminologie en definisies. ’n Diskussie oor die stogastiese Taylor ontwikkeling, gebaseer op herhaalde Stratonovich integrale word uiteengeset, waarna die hoofstuk afsluit met ’n illustrasie van dié ontwikkeling met die Euler-Maruyama benaderings skema. Hoofstuk 3 bevat die hoofgedagtes agter die Kusuoka benaderings skema, waar daar ook op die implementering van die algoritme gekonsentreer word. Hierdie skema is van toepassing op die prysvastelling van ’n Asiatiese call-opsie, numeriese resultate word ook aangebied. Ons begin Hoofstuk 4 deur te kyk na klassieke volumeberekenings formules waarna ons op ’n eenvoudige wyse die algemene idees van "Volumeberekening op Wiener ruimtes", ook bekend as die Lyons-Victoir benaderings skema, as ’n uitbreiding van die klassieke volumeberekening metode gebruik. Die doel van hierdie skema is om volumeberekening formules op te stel vir benaderings integrale wat gedefinieer is op Wiener ruimtes en gevolglik, hoërorde numeriese skemas te ontwikkel. Dit is gebaseer op die stogastiese Stratonovich ontwikkeling en kan beskou word as ’n ontwikkeling van die Kusuoka skema. Deur Kusuoka en Lyon-Victoir se idees oor benaderings skemas toe te pas, het Ninomiya-Victoir en Ninomiya- Ninomiya nuwe numeriese skemas van orde 2 ontwikkel, waar hulle die probleem omgeskakel het van een waar SDVs opgelos moet word, na een waar gewone differensiaalvergelykings (GDV) opgelos moet word. Hierdie twee skemas word in Hoofstuk 5 uiteengeset. Alhoewel die benaderings soortgelyk is, is daar ’n beduidende verskil in die algoritmes self. Hierdie hoofstuk begin met ’n algemene uiteensetting van die Ninomiya-Victoir algoritme waar ’n arbitrêre vaste tyd horison, T, gebruik word. Dié word toegepas op opsieprysvastelling en optimale portefeulje strategie probleme. Verder word numeriese simulasies uitgevoer, die prestasie van die Ninomiya-Victoir algoritme was bestudeer en vergelyk met die Euler-Maruyama metode. Ons maak die opmerking dat die Ninomiya-Victoir algoritme aansienlik vinniger is. Die belangrikste resultaat van die Ninomiya-Ninomiya benaderings skema word ook voorgestel. Deur die idee van ’n Lie algebra te gebruik, het Ninomiya en Ninomiya ’n stogastiese veranderlike met GDV-waardes gekonstrueer wat se gemiddeld die oplossing van ’n gegewe SDV benader. Die Runge-Kutta metode vir GDVs word dan toegepas op die GDV wat getrek is uit die stogastiese veranderlike en ’n lineêre operator gekonstrueer. ’n Veralgemeende uitdrukking vir die gekonstrueerde operator is afgelei en die algoritme is toegepas op die prysvasstelling van ’n Asiatiese opsie onder die Heston onbestendigheids model.

This dissertation demonstrates how Samuel Taylor Coleridge provides a unique vision of reality in which his evolving self-consciousness mirrors, contributes to, and is subsumed by a single universal consciousness. Utilizing the divine power of imagination, he is able to decipher the images from the material world as characters of God's symbolic language of self-revelation; subsequently, through the divine "attribute" of reason, he is able to transform them into a corresponding symbolic language of poetry. He realizes that his creativity is a finite repetition of God's infinite act of creation in which "spirit," God's consciousness in creation, comes to an awareness of itself through the human mind. This study argues that, according to Coleridge, these processes follow a divine intention, and the human faculties and the mind's structure have been molded precisely to achieve a particular understanding of reality that conforms to God's requirements and for spirit's self-actualization. Furthermore, the process by which Coleridge creates and derives knowledge from his poetic expressions follows an archetypal blueprint according to which all natural processes operate. This project illustrates not only how the theory of organicism lies at the foundation of the complex, reciprocal relationship between Coleridge's artistic expression and developing subjectivity, but also how there is an organic interrelationship between an individual's developing self-consciousness and spirit's growing awareness of its cosmic totality. Ultimately, Coleridge's writings reveal that the macrocosmic and microcosmic processes are organically interrelated, interdependent, and symbiotic and that this "truth" is gradually discovered through his experiences of the divine elements of love and beauty in creation.

This dissertation spans topics related to global trade, oil prices, optimum currency areas, the eurozone, monetary independence, capital controls and the international monetary policy trilemma. It consists of four chapters and three essays. Chapter one provides a brief summary of all three essays. Chapter two investigates the impact of oil prices on global trade. It is concluded that when oil prices increase, countries start trading relatively more with their neighbors. As an application this chapter provides a new estimate of the eurozone effect on trade. Chapter three continues to study the eurozone and asks whether it is an optimum currency area using the member countries’ desired monetary policies. It is concluded that Greece, Spain, and Ireland have desired policies that are the least compatible with the common euro policy and are therefore the least likely to have formed an optimum currency area with the euro. Chapter four provides a new methodology in testing the international trilemma hypothesis. It is concluded that the trilemma holds in the context of the Asian countries.

People have organized around gender equality in modern America for the last century. However, with the advent of new technology, people largely organize in around social movements in online spaces. This thesis explores the ways in which new media expands a popular understanding of the Feminist movement.

Electrospinning is a method of producing nano structured material from a polymer solution or melt using high strength electric fields. It is a process that has yet to find extensive industrial application yet shows promise if obstacles such as low rate of production overcome perhaps by more complete theoretical modelling. This work examines the effects of adding an ionic salt to a solution of poly(vinyl alcohol) in water. The direct effect was an increase the charge density and electric current. It was found that an increase in charge density decreases the mass deposition rate and forms a thinner initial jet. When the sign of the charge on the polymer solution was changed from positive to negative the charge density increased and the initial jet diameter and mass deposition rate also decreased. It was proposed that a smaller radius of curvature is formed by the Taylor cone at higher charge densities resulting in a smaller “virtual orifice”. The extent of the bending instability was explored and it was found that adding ionic salt results in a decrease in the bending instability resulting in thicker fibres. Changing the sign of the charge on the polymer solution from positive to negative resulted in an increase in the bending instability and resulted in thinner fibres. The charge transfer mechanisms used in different electrospinning models are explored and some assumptions not explicitly stated are discussed. From this discussion a generalized equation describing the charge transport mechanisms is proposed.

Land is a critical element in the formation of, maintenance and continuance of Native identity to tribes in North America. Since time immemorial, Native people have occupied these landscapes in a manner than can perhaps be best described as "persistent." Native views of the land can differ significantly from those of a Western, or Anglo-American tradition. And when managers of these lands come from a Western tradition, dissimilar views on how these lands should be used can become very problematic for Native people. This research examines how five tribes (Pueblo of Acoma, the Hopi Tribe, Pueblo of Laguna, Navajo Nation and Pueblo of Zuni) view their identity and future cultural continuity as their ancestral homelands are inundated by competing uranium mining interests that threaten to destroy the Mount Taylor landscape of northern New Mexico.

The Rayleigh-Taylor instability is a buoyancy driven instability that takes place in a stratified fluid system with a constant acceleration directed from the heavy fluid into the light fluid. In this study, both experimental data and numerical simulations are presented. Experiments are performed primarily using a lithium-tungstate aqueous solution as the heavy liquid, but sometimes a calcium nitrate aqueous solution is used for comparison purposes. Experimental data is obtained for both miscible and immiscible fluid combinations. For the miscible experiments the light liquid is either ethanol or isopropanol, and for the immiscible experiments either silicone oil or trans-anethole is used. The resulting Atwood number is either 0.5 when the lithium-tungstate solution is used or 0.2 when the calcium nitrate solution is used. These fluid combinations are either forced or left unforced. The forced experiments have an initial perturbation imposed by vertically oscillating the liquid containing tank to produce Faraday waves at the interface. The unforced experiments rely on random interfacial fluctuations, due to background noise, to seed the instability. The liquid combination is partially enclosed in a test section that is accelerated downward along a vertical rail system causing the Rayleigh-Taylor instability. Accelerations of approximately 1g (with a weight and pulley system) or 10g (with a linear induction motor system) are experienced by the liquids. The tank is backlit and digitally recorded with high speed video cameras. These experiments are then simulated with the incompressible, Navier-Stokes code Miranda. The main focus of this study is the growth parameter (ɑ) of the mixing region produced by the instability after it has become apparently self-similar and turbulent. The measured growth parameters are compared to determine the effects of miscibility and initial perturbations (of the small wavelength, finite bandwidth type used here). It is found that while initial perturbations do not affect the instability growth, miscibility does.