Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 86-92).

There is a growing appreciation for the insight mathematical models can yield on biological systems. In particular, due to the challenges inherent in experimental observation of disease progression, models describing the genesis, growth and evolution of cancer have been developed. Many of these models possess the common feature that one particular type of cellular population initiates a further, distinct population. This thesis explores two models containing this feature, which also employ branching processes to describe population growth. Firstly, we consider a deterministically growing wild type population which seeds stochastically developing mutant clones. This generalises the classic Luria- Delbruck model of bacterial evolution. We focus on how differing wild type growth manifests itself in the distribution of clone sizes. In our main result we prove that for a large class of wild type growth, the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. In the second model, we consider a fully stochastic system of cells in a growing population that can undergo birth, death and transitions. New cellular types appear via transitions, examples of which are genetic mutations or migrations bringing cells into a new environment. We concentrate on the scenario where the original cell type has the largest net growth rate, which is relevant for modelling drug resistance, due to fitness costs of resistance, or cells migrating into contact with a toxin. Two questions are considered in our main results. First, how long do we wait until a cell with a specific target type, an arbitrary number of transitions from the original population, exists. Second, which particular sequence of transitions initiated the target population. In the limit of small final transition rates, simple, explicit formulas are given to answer these questions.

Orientador: Jamil Viana Pereira / Banca: Rodrigo Martins / Banca: Renata Zotin Gomes de Oliveira / Resumo: O objetivo desse trabalho é estudar e apresentar detalhes da teoria de aproximação de funções e aplicá-la a problemas envolvendo prática esportiva, mais especificamente a encontrar limitantes e estimativas futuras para certos recordes, trazendo a interdisciplinariedade à tona. Dentre os conceitos matemáticos necessários para este fim, destacam-se o ajuste de curvas, a teoria de pontos de equilíbrio e critérios estabilidade, o Método dos Mínimos Quadrados e o Método de Ford - Walford. Por fim, utilizando estas ferramentas, poderemos analisar os recordes já estabelecidos e apresentar algumas estimativas futuras. Além do estudo teórico e das aplicações pretende-se fazer uma conexão com o ensino aprendizagem da matemática na educação básica, visando melhorar a dinâmica das aulas, podendo inclusive associar conceitos matemáticos e práticas esportivas / Abstract: The aim of this work is to study and present details about the approximation theory and to apply it in some problems involving sports practice, more specifically to find limitations and future estimates for certain records, bringing interdisciplinarity to the fore. The mathematical concepts required for this purpose are curve fitting, equilibrium point theory and stability criteria, Least Squares Method and Ford-Walford Method. Finally, using these tools, we will be able to analyze the al ready established records and present some future estimates. In addition to the theoretical and appplied studies, it is intended to make a connection with the teaching of mathematics in basic education, aiming to improve the dynamics of classes, and may even associate mathematical concepts and sports practices / Mestre

This thesis consists of two related parts. In the first part we conduct an empirical examination of the dynamics of Australian interest rates of six different maturities, covering the whole yield curve. This direct study of the long rates is quite novel. We use maximum likelihood estimation on a variety of models and find some results that are in stark contrast to previous studies. We estimate Poisson-jump diffusion (PJD) models and find very strong evidence for the existence of jumps in all daily interest rate series. We find that the PJD model fits short-rate data significantly better than a Bernoulli-jump diffusion model. We also estimate the CKLS model for our data and find that the only model not rejected for all six maturities is the CEV model in stark contrast to previous findings. Also, we find that the elasticity of variance estimate in the CKLS model is much higher for the short-rates than for the longer rates where the estimate is only about 0.25, indicating that different dynamics seem to be at work for different maturities. We also found that adding jumps to the simple diffusion model gives a larger improvement than comes from going from the simple diffusion to the CKLS model. In the second part of the thesis we examine the Flesaker and Hughston (FH) term structure model. We derive the dynamics of the short rate under both the original measure and the risk-neutral measure, and show that some criticisms of the bounds for the short rate may not be significant in actual applications. We also derive the dynamics of bond prices in the FH model and compare them to the HJM model. We also extend the FH model by allowing the martingale to follow a jump-diffusion process, rather than just a diffusion process. We derive the unique change of measure that guarantees the family of bond prices is arbitrage-free. We derive prices for caps and swaptions, and extend the results to include Bermudan swaptions and show how to price options with the jump-diffusion version of the FH model.

The process of calibrating a numerical model is examined in this thesis with an application to the flow of groundwater in the Willunga Basin in South Australia. The calibration process involves estimating unknown parameters of the numerical model so that the output obtained from the model is comparable with data that is observed in the field. Three methods for calibrating numerical models are discussed, these being the steepest descent method, the nonlinear least squares method, and a new method called the response function method. The response function method uses the functional relationship between the model's output and the unknown parameters to determine improved estimates for the unknown parameters. The functional relationships are based on analytic solutions to simplifed model problems or from previous experience. The three calibration methods are compared using a simple function involving one parameter, an idealised steady state model of groundwater flow and an idealised transient model of groundwater flow. The comparison shows that the response function method produces accurate estimates in the least amount of iterations. A numerical model of groundwater flow in the Willunga Basin in South Australia has been developed and the response function method used to estimated the unknown parameters for this model. The model of the Willunga Basin has been used to examine the sustainable yield of groundwater from the basin. The effect on groundwater levels in the basin using current and estimated extraction rates from the literature for sustainable yield has been examined. The response function method has also been used to estimate the rate of extraction to return the groundwater levels at a specific location to a desirable level. / Thesis (M.Sc.)--Department of Applied Mathematics, 2001.

This thesis consists of two related parts. In the first part we conduct an empirical examination of the dynamics of Australian interest rates of six different maturities, covering the whole yield curve. This direct study of the long rates is quite novel. We use maximum likelihood estimation on a variety of models and find some results that are in stark contrast to previous studies. We estimate Poisson-jump diffusion (PJD) models and find very strong evidence for the existence of jumps in all daily interest rate series. We find that the PJD model fits short-rate data significantly better than a Bernoulli-jump diffusion model. We also estimate the CKLS model for our data and find that the only model not rejected for all six maturities is the CEV model in stark contrast to previous findings. Also, we find that the elasticity of variance estimate in the CKLS model is much higher for the short-rates than for the longer rates where the estimate is only about 0.25, indicating that different dynamics seem to be at work for different maturities. We also found that adding jumps to the simple diffusion model gives a larger improvement than comes from going from the simple diffusion to the CKLS model. In the second part of the thesis we examine the Flesaker and Hughston (FH) term structure model. We derive the dynamics of the short rate under both the original measure and the risk-neutral measure, and show that some criticisms of the bounds for the short rate may not be significant in actual applications. We also derive the dynamics of bond prices in the FH model and compare them to the HJM model. We also extend the FH model by allowing the martingale to follow a jump-diffusion process, rather than just a diffusion process. We derive the unique change of measure that guarantees the family of bond prices is arbitrage-free. We derive prices for caps and swaptions, and extend the results to include Bermudan swaptions and show how to price options with the jump-diffusion version of the FH model.

Thesis (Ph. D.)--University of California, San Diego, 2006. / Title from first page of PDF file (viewed April 6, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 201-206).

Orientador: Sidineia Barrozo / Banca: Erika Capelato / Banca: Érica Regina Filletti Nascimento / Resumo: Inspirada na Modelagem Matemática proposta por Bassanezi, mas entusiasmada com a possibilidade de aplicação proposta por Burak, neste trabalho busquei entender como o estudo das Recorrências Lineares de Primeira Ordem poderia contribuir para a construção de modelos matemáticos, em especial, fórmulas inerentes à Matemática Financeira, com alunos da 1ª série do Ensino Médio, utilizando, para isso, Matemática acessível aos mesmos e inteiramente imersa em um tema único e em suas ramificações, envolvendo no processo, simultaneamente, aprendizagem matemática - símbolos, algoritmos e técnicas de resolução -, interação entre a Matemática desenvolvida e a realidade do aluno - contexto sociocultural -, auxílio da tecnologia - Excel e Geo- gebra para observação de regularidades e compreensão dos processos e interpretação dos dados, além de Word e Power Point para escrita e apresentacão da atividade, respectivamente -, o desenvolvimento do aluno como protagonista de seu processo de aprendizagem e, por fim, a interdisciplinaridade. Os resultados foram tão positivos que repercutiram na comunidade escolar e culminaram no convite feito pela Diretoria de Ensino de Piracicaba para que o trabalho fosse adaptado para exposição em uma competição de pesquisas nos moldes de Iniciacão Científica na Universidade Metodista de Piracicaba - Unimep / Abstract: Inspired by the mathematical modeling proposed by Bassanezi, but enthusiastic about the possibility of application proposed by Burak, in this work I tried to understand how the study of Linear Recurrences of First Order could contribute to the construction of mathematical models, especially formulas inherent to Financia lMathematics, with students of the 1st grade of the High School, using, for this, Mathematics accessible to them and entirely immersed in a single theme and its ramifications, involving in the process simultaneously mathematical learning - symbols, algorithms and resolution techniques - interaction between developed mathematics and student reality - sociocultural context -, technology assistance - Excel and Geogebra for observation of regularities and understanding of processes and interpretation of data, in addition to Word and Power Point for writing and presentation of the activity, respectively -, the development of the student as the protagonist of their learning process and, finally, interdisciplinarity. The results were so positive that they reverberated in the school community and culminated in the invitation made by the Teaching Board of Piracicaba so that the work was adapted for exhibition in a competition of research in the form of Scientific Initiation at the Methodist University of Piracicaba - Unimep / Mestre

The main goal of my studies has been to search for the optimal control strategy of controlling epidemics when taking into account both economical and social costs of the disease. Three control scenarios emerge with treating the whole population (global strategy, GS), treating a small number of individuals in a well-defined neighbourhood of a detected case (local strategy, LS) and allowing the disease to spread unchecked (null strategy, NS). The choice of the optimal strategy is governed mainly by a relative cost of palliative and preventive treatments. Although the properties of the pathogen might not be known in advance for emerging diseases, the prediction of the optimal strategy can be made based on economic analysis only. The details of the local strategy and in particular the size of the optimal treatment neighbourhood weakly depends on disease infectivity but strongly depends on other epidemiological factors (rate of occurring the symptoms, spontaneously recovery). The required extent of prevention is proportional to the size of the infection neighbourhood, but this relationship depends on time till detection and time till treatment in a non-nonlinear (power) law. The spontaneous recovery also affects the choice of the control strategy. I have extended my results to two contrasting and yet complementary models, in which individuals that have been through the disease can either be treated or not. Whether the removed individuals (i.e., those who have been through the disease but then spontaneously recover or die) are part of the treatment plan depends on the type of the disease agent. The key factor in choosing the right model is whether it is possible - and desirable - to distinguish such individuals from those who are susceptible. If the removed class is identified with dead individuals, the distinction is very clear. However, if the removal means recovery and immunity, it might not be possible to identify those who are immune. The models are similar in their epidemiological part, but differ in how the removed/recovered individuals are treated. The differences in models affect choice of the strategy only for very cheap treatment and slow spreading disease. However for the combinations of parameters that are important from the epidemiological perspective (high infectiousness and expensive treatment) the models give similar results. Moreover, even where the choice of the strategy is different, the total cost spent on controlling the epidemic is very similar for both models. Although regular and small-world networks capture some aspects of the structure of real networks of contacts between people, animals or plants, they do not include the effect of clustering noted in many real-life applications. The use of random clustered networks in epidemiological modelling takes an impor- tant step towards application of the modelling framework to realistic systems. Network topology and in particular clustering also affects the applicability of the control strategy.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Economics, 2008. / Includes bibliographical references.