Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Este trabalho aborda uma investigação numérica-experimental sobre o comportamento à fratura por clivagem de um aço de alta resistência e baixa liga ASTM A572 Grau 50 usando corpos de prova SE(B) em configurações padronizadas e não padronizadas, incluindo a configuração PCVN não padronizada. O principal objetivo desse estudo é contribuir para o desenvolvimento de um procedimento de ensaio de tenacidade à fratura aplicável a geometrias SE(B) com distância entre apoios variável (S/W) e carregada sob configuração de flexão por 3 e 4 pontos. O objetivo secundário, mas não menos importante, é investigar os efeitos da geometria e do modo de carregamento sobre as distribuições dos dados de tenacidade à fratura medidas experimentalmente e suas implicações na caracterização da dependência da tenacidade em relação à temperatura baseada na metodologia da curva mestra. Com base nesses propósitos, é fornecido um novo e extenso conjunto de fatores plásticos eta (?p) aplicáveis às geometrias SE(B) não padronizadas para estimar os valores de tenacidade à fratura, incluindo a integral J e o parâmetro CTOD, medidos experimentalmente em termos de registros de carga e deslocamento. A fim de facilitar o contato com os demais protocolos de ensaios, é também fornecido um novo conjunto de fatores rotacionais plástico (rp) para determinar o CTOD baseado no modelo da rótula plástica. Os ensaios de tenacidade à fratura, realizados em diversas geometrias SE(B) com distância entre apoios variável (S/W) extraídas na direção transversal (T-L) de uma chapa de aço ASTM A572 Grau 50, fornecem os dados de tenacidade à fratura por clivagem em termos da integral-J medida no ponto de instabilidade, Jc. Os resultados experimentais mostram um potencial efeito da geometria do corpo de prova e do modo de carregamento sobre os valores de Jc, os quais podem ajudar a mitigar os efeitos da perda de restrição plástica frequentemente observada em corpos de prova de pequenas dimensões. A aplicação exploratória para determinar a temperatura de referência, T0, a partir da metodologia da curva mestra também fornece um suporte adicional para o uso de corpos de prova SE(B) não padronizados como uma alternativa geométrica nos procedimentos rotineiros de avaliação de tenacidade à fratura, incluindo o uso de corpos de prova de pequenas dimensões nas medições de tenacidade quando a disponibilidade limitada de material e a capacidade de carregamento da máquina são as principais preocupações. / This work addresses a numerical-experimental investigation on the cleavage fracture behavior of an ASTM A572 Grade 50 high strength, low alloy structural steel using standard and non-standard SE(B) specimens, including a non-standard PCVN configuration. The main purpose of this study is to contribute to the development of a fracture toughness test procedure applicable to bend geometries with varying specimen span over width ratio (S/W) and loaded under 3-point and 4-point flexural configuration. The secondary purpose, but none the less important, is to investigate the effects of geometry and loading mode on experimentally measured fracture toughness data distributions and implications for the characterization of the temperature dependence of toughness based on the Master Curve methodology. Based on these purposes, a large new set of plastic ?-factors applicable to these non-standard bend geometries which serve to estimate the experimentally measured toughness values in terms of load-displacement records, including the J-integral and the crack tip opening displacement (CTOD), is provided. In order to facilitate contact with other test protocols, a new set of rotational factors, rp, to determine the CTOD based on the plastic hinge model is also described. Fracture toughness testing conducted on various bend geometries with varying specimen span (S/W) extracted in the T-L orientation from an A572 Grade 50 steel plate provides the cleavage fracture resistance data in terms of the J-integral at cleavage instability, Jc. The experimental results show a potential effect of specimen geometry and loading mode on Jc-values which can help mitigating the effects of constraint loss often observed in smaller fracture specimens. An exploratory application to determine the reference temperature, T0, derived from the Master Curve methodology also provides additional support for using non-standard bend specimens as an alternative specimen geometry in routine fracture assessments, including the use of small-scale fracture specimens to measure fracture toughness properties when limited material availability and test machine capacity are of major concern are of major concern.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / O processo de recuperação secundária de petróleo é comumente realizado com a injeção de água ou gás no reservatório a fim de manter a pressão necessária para sua extração. Para que o investimento seja viável, os gastos com a extração precisam ser menores do que o retorno financeiro obtido com a produção de petróleo. Objetivando-se estudar possíveis cenários para o processo de exploração, costuma-se utilizar simulações dos processos de extração. As equações que modelam esse processo de recuperação são de caráter hiperbólico e não lineares, as quais podem ser interpretadas como Leis de Conservação, cujas soluções são complexas por suas naturezas descontínuas. Essas descontinuidades ou saltos são conhecidas como ondas de choque. Neste trabalho foi abordada uma análise matemática para os fenômenos oriundos de leis de conservação, para em seguida utilizá-la no entendimento do referido problema. Foram estudadas soluções fracas que, fisicamente, podem ser interpretadas como ondas de choque ou rarefação, então, para que fossem distinguidas as fisicamente admissíveis, foi considerado o princípio de entropia, nas suas diversas formas. As simulações foram realizadas nos âmbitos dos escoamentos bifásicos e trifásicos, em que os fluidos são imiscíveis e os efeitos gravitacionais e difusivos, devido à pressão capilar, foram desprezados. Inicialmente, foi feito um estudo comparativo de resoluções numéricas na captura de ondas de choque em escoamento bifásico água-óleo. Neste estudo destacam-se o método Composto LWLF-k, o esquema NonStandard e a introdução da nova função de renormalização para o esquema NonStandard, onde obteve resultados satisfatórios, principalmente em regiões onde a viscosidade do óleo é muito maior do que a viscosidade da água. No escoamento bidimensional, um novo método é proposto, partindo de uma generalização do esquema NonStandard unidimensional. Por fim, é feita uma adaptação dos métodos LWLF-4 e NonStandard para a simulação em escoamentos trifásicos em domínios unidimensional. O esquema NonStandard foi considerado mais eficiente nos problemas abordados, uma vez que sua versão bidimensional mostrou-se satisfatória na captura de ondas de choque em escoamentos bifásicos em meios porosos. / The process of secondary oil recovery is commonly accomplished by injecting water or gas into the reservoir to maintain the necessary pressure for their extraction. So that the investment is viable spending extraction must be smaller than the financial return to oil production. Aiming to study possible scenarios for the exploration process, it is customary to use simulations of extraction processes. The equations that model this process of recovery are hyperbolic and nonlinear, which can be interpreted as Conservation Laws , whose solutions are complex by their discontinuous nature . These discontinuities or jumps are known as shock waves. Due to this importance, this work will be discussed a mathematical analysis of the phenomena arising from conservation laws, to then use it in the understanding of this problem. Weak solutions that physically can be interpreted as shock waves or rarefaction, so that they might be distinguished physically admissible were studied, was considered the principle of entropy, in its various forms. The simulations were performed in the fields of two-phase and three-phase flow, in which the fluids are immiscible and gravitational and diffusive effects due to capillary pressure, were discarded. Initially a comparative study of numerical resolutions in the capture of shock waves in water-oil two-phase flow was made. This study highlights LWLF k Composite method and Nonstandard. Was also presented a new renormalization function for nonstandard scheme with satisfactory results, especially in regions where the oil viscosity is much higher than the viscosity of the water. In twodimensional flow, a new method will be presented. The same is a generalization of onedimensional nonstandard schema. Finally, the adaptation of nonstandard and LWLF-4 methods for simulating in three-phase one-dimensional flows. In general, the nonstandard scheme was considered the most efficient method in problems addressed, since its twodimensional version was satisfactory in capturing shock waves in two-phase flow in porous media.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / O processo de recuperação secundária de petróleo é comumente realizado com a injeção de água ou gás no reservatório a fim de manter a pressão necessária para sua extração. Para que o investimento seja viável, os gastos com a extração precisam ser menores do que o retorno financeiro obtido com a produção de petróleo. Objetivando-se estudar possíveis cenários para o processo de exploração, costuma-se utilizar simulações dos processos de extração. As equações que modelam esse processo de recuperação são de caráter hiperbólico e não lineares, as quais podem ser interpretadas como Leis de Conservação, cujas soluções são complexas por suas naturezas descontínuas. Essas descontinuidades ou saltos são conhecidas como ondas de choque. Neste trabalho foi abordada uma análise matemática para os fenômenos oriundos de leis de conservação, para em seguida utilizá-la no entendimento do referido problema. Foram estudadas soluções fracas que, fisicamente, podem ser interpretadas como ondas de choque ou rarefação, então, para que fossem distinguidas as fisicamente admissíveis, foi considerado o princípio de entropia, nas suas diversas formas. As simulações foram realizadas nos âmbitos dos escoamentos bifásicos e trifásicos, em que os fluidos são imiscíveis e os efeitos gravitacionais e difusivos, devido à pressão capilar, foram desprezados. Inicialmente, foi feito um estudo comparativo de resoluções numéricas na captura de ondas de choque em escoamento bifásico água-óleo. Neste estudo destacam-se o método Composto LWLF-k, o esquema NonStandard e a introdução da nova função de renormalização para o esquema NonStandard, onde obteve resultados satisfatórios, principalmente em regiões onde a viscosidade do óleo é muito maior do que a viscosidade da água. No escoamento bidimensional, um novo método é proposto, partindo de uma generalização do esquema NonStandard unidimensional. Por fim, é feita uma adaptação dos métodos LWLF-4 e NonStandard para a simulação em escoamentos trifásicos em domínios unidimensional. O esquema NonStandard foi considerado mais eficiente nos problemas abordados, uma vez que sua versão bidimensional mostrou-se satisfatória na captura de ondas de choque em escoamentos bifásicos em meios porosos. / The process of secondary oil recovery is commonly accomplished by injecting water or gas into the reservoir to maintain the necessary pressure for their extraction. So that the investment is viable spending extraction must be smaller than the financial return to oil production. Aiming to study possible scenarios for the exploration process, it is customary to use simulations of extraction processes. The equations that model this process of recovery are hyperbolic and nonlinear, which can be interpreted as Conservation Laws , whose solutions are complex by their discontinuous nature . These discontinuities or jumps are known as shock waves. Due to this importance, this work will be discussed a mathematical analysis of the phenomena arising from conservation laws, to then use it in the understanding of this problem. Weak solutions that physically can be interpreted as shock waves or rarefaction, so that they might be distinguished physically admissible were studied, was considered the principle of entropy, in its various forms. The simulations were performed in the fields of two-phase and three-phase flow, in which the fluids are immiscible and gravitational and diffusive effects due to capillary pressure, were discarded. Initially a comparative study of numerical resolutions in the capture of shock waves in water-oil two-phase flow was made. This study highlights LWLF k Composite method and Nonstandard. Was also presented a new renormalization function for nonstandard scheme with satisfactory results, especially in regions where the oil viscosity is much higher than the viscosity of the water. In twodimensional flow, a new method will be presented. The same is a generalization of onedimensional nonstandard schema. Finally, the adaptation of nonstandard and LWLF-4 methods for simulating in three-phase one-dimensional flows. In general, the nonstandard scheme was considered the most efficient method in problems addressed, since its twodimensional version was satisfactory in capturing shock waves in two-phase flow in porous media.

Le Cyberknife (Accuray, Sunnyvale, CA) est un appareil de radiochirurgie stéréotaxique sans cadre. Il a été développé pour administrer de fortes doses dans des volumes restreints. Aussi, pour obtenir une conformation optimale de traitement, des champs circulaires de petites dimensions sont utilisés (\phi = 0,5 à 6 cm). L'étude dosimétrique de ces petits champs doit être menée selon de nouveaux standards puisque ceux-ci échappent aux définitions du TG-51. L'objectif de ce projet est d'implanter une plateforme de calcul de dose de type Monte Carlo pour le CyberKnife en clinique. Il s'articule autour de deux réalisations principales. Tout d'abord, une caractérisation dosimétrique du modèle Monte Carlo de l'accélérateur linéaire du CyberKnife a été menée à travers des simulations Monte Carlo générées par le moteur de EGSnrc. Cette étude est basée sur la caractérisation de la réponse d'un détecteur à un champ de type CK à partir de simulations EGS_chamber. Cette approche permet de prendre en compte l'impact du détecteur sur les mesures expérimentales. Cet aspect est d'autant plus important que le modèle Monte Carlo de l'accélérateur est validé à partir de mesures expérimentales. Les résultats obtenus montrent une bonne concordance, <1% ou 1 mm, entre les mesures expérimentales et les données de simulations pour les grands champs. Pour les champs de diamètre < 12,5 mm, le modèle est moins exact et une correction est appliquée pour atteindre une différence de <1% ou 1 mm. Deuxièmement, ce modèle validé du CK a été implanté dans un cadre de calcul Monte Carlo complet. Une plateforme de calcul dédiée aux calculs Monte Carlo, WebTPS, a été adaptée aux calculs de dose CK. Cette plateforme reçoit les données relatives au plan de traitement et lance des calcul EGSnrc sur un superordinateur. Cette approche tend à réduire les approximations lors de l'évaluation dosimétrique de plans de traitements cliniques. Une incertitude inférieure à 1% peut être atteinte en deux heures de calcul. Ce projet a donc pour objectif de développer une référence clinique pour le calcul de dose dans le cadre de la radiochirurgie stéréotaxique. L'outil WebTPS pourrait être particulièrement utile en clinique, l'algorithme de calcul de dose du CK étant limité dans plusieurs situations de traitement. / Purpose: The scope of this study is to implement a clinical Monte Carlo dose calculation system based on the EGSnrc engine. This web-based tool will be mostly used to evaluate clinical treatment plans in highly heterogeneous phantoms. Methods: The Monte Carlo calculation tool is based on the DOSXYZnrc user code. The platform automatically converts CyberKnife clinical plan to the user code input files. Phantoms can be created from HU to ED curves or by manually assigning material using medical contours. Parallel computation is made on a Compute Canada high-performance cluster to reduce simulation time. A Monte Carlo CyberKnife model is built on BEAMnrc user code using the manufacturer specifications. Simulated and experimental data is compared to estimate the electron beam parameters. The beam energy estimation is based on percent depth dose (PDD) comparison while the full width at half max (FWHM) is validated by output factor (OF) and off-axis ratio (OAR). An EGS_chamber model of the PTW60012 diode is used in the OF calculation. A set of phase-spaces is generated from the optimal model and for each collimator to calculate dose contribution from each incident beam. Results: The linac model optimisation yielded a 0.5% PDD agreement between experimental and simulation data, and a 0.5% or 1 mm for OAR. DOSxyz simulation of full treatment plan, based on the preliminary CyberKnife model, were achieved. Total Monte Carlo dose calculation have been achieved for heterogeneous phantoms. Uncertainty under 1% can be achieved for less than 2 hour of computing time. However, computing time estimation is nontrivial due to its dependence on cluster availability. Conclusion: This work aims to develop a suitable tool for reference plan dose calculation. This web-based tool would be used in several clinical and research applications where the CyberKnife embedded ray-tracing algorithm would show significant limitations. Because it is destined to a clinical use, the whole dose calculation system will be rigorously validated. / Le travail de modélisation a été réalisé à travers EGSnrc, un logiciel développé par le Conseil National de Recherche Canada.

Le travail de modélisation a été réalisé à travers EGSnrc, un logiciel développé par le Conseil National de Recherche Canada. / Le Cyberknife (Accuray, Sunnyvale, CA) est un appareil de radiochirurgie stéréotaxique sans cadre. Il a été développé pour administrer de fortes doses dans des volumes restreints. Aussi, pour obtenir une conformation optimale de traitement, des champs circulaires de petites dimensions sont utilisés (\phi = 0,5 à 6 cm). L'étude dosimétrique de ces petits champs doit être menée selon de nouveaux standards puisque ceux-ci échappent aux définitions du TG-51. L'objectif de ce projet est d'implanter une plateforme de calcul de dose de type Monte Carlo pour le CyberKnife en clinique. Il s'articule autour de deux réalisations principales. Tout d'abord, une caractérisation dosimétrique du modèle Monte Carlo de l'accélérateur linéaire du CyberKnife a été menée à travers des simulations Monte Carlo générées par le moteur de EGSnrc. Cette étude est basée sur la caractérisation de la réponse d'un détecteur à un champ de type CK à partir de simulations EGS_chamber. Cette approche permet de prendre en compte l'impact du détecteur sur les mesures expérimentales. Cet aspect est d'autant plus important que le modèle Monte Carlo de l'accélérateur est validé à partir de mesures expérimentales. Les résultats obtenus montrent une bonne concordance, <1% ou 1 mm, entre les mesures expérimentales et les données de simulations pour les grands champs. Pour les champs de diamètre < 12,5 mm, le modèle est moins exact et une correction est appliquée pour atteindre une différence de <1% ou 1 mm. Deuxièmement, ce modèle validé du CK a été implanté dans un cadre de calcul Monte Carlo complet. Une plateforme de calcul dédiée aux calculs Monte Carlo, WebTPS, a été adaptée aux calculs de dose CK. Cette plateforme reçoit les données relatives au plan de traitement et lance des calcul EGSnrc sur un superordinateur. Cette approche tend à réduire les approximations lors de l'évaluation dosimétrique de plans de traitements cliniques. Une incertitude inférieure à 1% peut être atteinte en deux heures de calcul. Ce projet a donc pour objectif de développer une référence clinique pour le calcul de dose dans le cadre de la radiochirurgie stéréotaxique. L'outil WebTPS pourrait être particulièrement utile en clinique, l'algorithme de calcul de dose du CK étant limité dans plusieurs situations de traitement. / Purpose: The scope of this study is to implement a clinical Monte Carlo dose calculation system based on the EGSnrc engine. This web-based tool will be mostly used to evaluate clinical treatment plans in highly heterogeneous phantoms. Methods: The Monte Carlo calculation tool is based on the DOSXYZnrc user code. The platform automatically converts CyberKnife clinical plan to the user code input files. Phantoms can be created from HU to ED curves or by manually assigning material using medical contours. Parallel computation is made on a Compute Canada high-performance cluster to reduce simulation time. A Monte Carlo CyberKnife model is built on BEAMnrc user code using the manufacturer specifications. Simulated and experimental data is compared to estimate the electron beam parameters. The beam energy estimation is based on percent depth dose (PDD) comparison while the full width at half max (FWHM) is validated by output factor (OF) and off-axis ratio (OAR). An EGS_chamber model of the PTW60012 diode is used in the OF calculation. A set of phase-spaces is generated from the optimal model and for each collimator to calculate dose contribution from each incident beam. Results: The linac model optimisation yielded a 0.5% PDD agreement between experimental and simulation data, and a 0.5% or 1 mm for OAR. DOSxyz simulation of full treatment plan, based on the preliminary CyberKnife model, were achieved. Total Monte Carlo dose calculation have been achieved for heterogeneous phantoms. Uncertainty under 1% can be achieved for less than 2 hour of computing time. However, computing time estimation is nontrivial due to its dependence on cluster availability. Conclusion: This work aims to develop a suitable tool for reference plan dose calculation. This web-based tool would be used in several clinical and research applications where the CyberKnife embedded ray-tracing algorithm would show significant limitations. Because it is destined to a clinical use, the whole dose calculation system will be rigorously validated.

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The dissertation analyses the stratification of Czech in terms of teaching Czech as a foreign language. The general theoretical part relates to teaching Czech as a foreign language or other foreign languages as close as possible. This part also includes the analysis of recordings of authentic communication between Czech speakers and foreigners. The findings of the analysis are subsequently applied to the sample plans of teaching units. The introductory part deals with the history and development of the theory of the standard Czech language and it also presents the attitudes held by the members of the Prague Linguistic Circle, which were published in Spisovná čeština a jazyková kultura (Czech Literary Language and Culture) in 1932. The theoretical part reflects the issue of defining the standard, mentions diglossia and presents a variety of contemporary opinions on future trends in the development of codification. In accordance with its aim, the paper lists the recommendations of benchmark descriptions for Czech as a foreign language and also describes so called Foreigner Talk, including the way foreigners are spoken to by the speakers of Czech. Furthermore, the extent to which spoken Czech is reflected upon in the instruction of Czech language for foreigners and/or teaching materials were studied....

This master thesis deals with middle school pupils and their comprehension of Part-whole theory within the area of Mathematics. The theory section is divided into five parts. The first part presents sets of knowledge and skills of lower-secondary school pupils that are essential for completing a particular time period in selected thematic areas of mathematics at primary schools. In the second part, individual stages and mechanisms of cognitive processes needed in Mathematics are determined, and the third part focuses on teaching methods of Mathematics. Among the issues mentioned in this part are for example the difference between formal and informal knowledge or transmissive and constructivist way of teaching. Lastly, the fourth and fifth parts specify various expressions of wholes and their parts. The research part of this paper provides an analysis of interviews with certain teachers that are employing different teaching methods, testing applied on their pupils and also an analysis of the textbooks that are being used by these teachers. The other part of the thesis is the evaluation of the interviews with the selected teachers, what gives us the information about the relationship of the teachers to particular didactic approaches, and the evaluation of the tests taken by pupils including standard...

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