We study the quadratic lower compensated convex transform C[l lambda]dist2(x, K) of the squared distance function to a nonempty, non-convex closed set K ⊂ R[n]. These transforms, introduced in [20], provide C[1,1]-smooth tight-approximations of the squared distance functions. We introduce a result to calculate explicit formulae for the lower transform of the squared distance functions. Our first main result is the geometric characterisation of critical points of the quadratic lower compensated convex transform Cllambda dist2(x,K), which shows that the geometric non-smooth critical points of the squared distance function are identical to the usual critical points for the smoothed squared distance function. We then consider K ⊂ Rn be finite and classify the Morse indices of critical points of the lower transform C[l lambda]dist2(x, K) in R2 and R3, that is, classify critical points into non-degenerate critical points and degenerate critical points. This classification of Morse indices of critical points cannot be fully justified without knowing the behaviour of the lower transform of the squared distance function to finite sets. Therefore, we study some local properties of the lower transform C[l lambda]dist2(x, K) for the squared distance functions to finite sets to understand its behaviour in a small neighbourhood of critical points. Ae show that the lower transform C[l lambda]dist2(x, K) has a semi-global property in that is, for lambda > 0 suffciently large, the lower transform of the squared distance function equals the squared distance function except on a small subset of a given bounded set. Under certain regularity assumptions of K, we establish that the lower transform of the squared distance function to finite sets has a semi-global representation in C(K) under triangulations, which helps us to understand the role of the lower transform C[l lambda]dist2(x, K) in surface reconstruction.

Orientador: Eduardo Kokubun / Banca: Fábio Yuzo Nakamura / Banca: Cláudio Alexandre Gobatto / Resumo: Manchado et al. (2002) determinaram os parâmetros do modelo da PCrit através dos valores submáximos da FC. Foi reportado a existência de uma relação linear da PCritFC e AWCFC submáxima com seus respectivos valores máximos. Diante da correlação entre a FC e o VO2, essas equações lineares foram re-inseridas no modelo hiperbólico da PCrit e deduzidas para predizer a cinética do VO2. O objetivo do presente estudo foi verificar a possibilidade de determinar os parâmetros da PCrit através da cinética do VO2 de apenas uma sessão de exercício supramáximo. Para isso, foram analisadas a resposta do VO2 de jovens universitários, 13 homens e 2 mulheres, submetidos a três ou quatro testes preditivos da PCrit e CTA. O VO2 foi coletado a cada 3 ciclos respiratórios, interpolado a cada um segundo e alisado com médias móveis de 15 segundos. Foram determinados os tempos para se atingir, 60, 65, 70, 75, 80, 85, 90, 95 e 100% do VO2max. Esses tempos apresentaram uma relação hiperbólica com a intensidade de trabalho e possibilitaram determinar os valores de PCrit e CTA correspondentes aos valores abaixo e no VO2max. Ancova demonstrou que esses valores de PCrit e CTA sub-VO2max se relacionavam de maneira linear com os valores máximos correspondentes. As equações lineares foram deduzidas ao modelo hiperbólico 2 2 2 1 * TVO k E P VO VO rep + = +, o qual apresentou bom ajuste à cinética do VO2 (0.90 < r2 <0.99). Os valores encontrados, para os termos da equação, possibilitaram predizer os valores de PCrit (PCrit_pred) e CTA (CTA_pred). Esses valores foram comparados, através do intervalo de confiança (IC), com seus respectivos parâmetros reais, mas não foi encontrada diferença para a PCrit. Entretanto, as grandes amplitudes do IC para a CTA e CTA_pred não permitiram realizar determinações acuradas para este parâmetro / Abstract: Manchado et al. (2002) determined CP model parameters through heart rate (HR) submaximal values. The authors reported a linear relationship between submaximal CPHR and AWCHR to its respectives maximal values. This work was based on the assumption that HR and VO2 have a qualitative correlation. Thus, those linear equations were re-inserted in CP hyperbolic model and deduced to predict VO2 kinetics. The main purpose of this study was test the use of VO2 kinetics, of one supramaximal exercise session, to determine the CP parameters. Were submitted to the experiment 13 men and 2 women. They realized 3 or 4 CP and AWC prediction tests and, during the trials, the VO2 response was collected every 3-breath cycle. It was interpolated to one-second values and smoothed by 15 s rolling average. The time to achieve 60, 65, 70, 75, 80, 85, 90, 95 and 100% of VO2max was recorded. Those times fitted to a hyperbolic relationship with exercise intensity, thus the CP and AWC related to sub- and VO2max were determined. ANCOVA revealed that each parameter sub- VO2max value was linear related to its maximal value. These two linear equations were deduced to the hyperbolic model 2 2 2 1 * TVO k E P VO VO rep + = + . The VO2 response was good fitted (0.90 < r2 < 0.99) and the equation term’s values were used to determine predicted values of CP (CP_pred) and AWC (AWC_pred). These values were compared, by confidence intervals (CI), with its respective measured real values and no differences were found to CP. However, AWC and AWC_pred presented elevated CI ranges and does not allowed an accurate prediction of this parameter. In Summary, the kinetics model adopted to VO2 makes possible determine CP values using just one rectangular test / Mestre

Made available in DSpace on 2014-06-11T19:22:51Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-11-27Bitstream added on 2014-06-13T19:28:18Z : No. of bitstreams: 1 moura_rf_me_rcla.pdf: 364424 bytes, checksum: 2ba8adc0206af1d0845327bf8bf6f7e8 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Manchado et al. (2002) determinaram os parâmetros do modelo da PCrit através dos valores submáximos da FC. Foi reportado a existência de uma relação linear da PCritFC e AWCFC submáxima com seus respectivos valores máximos. Diante da correlação entre a FC e o VO2, essas equações lineares foram re-inseridas no modelo hiperbólico da PCrit e deduzidas para predizer a cinética do VO2. O objetivo do presente estudo foi verificar a possibilidade de determinar os parâmetros da PCrit através da cinética do VO2 de apenas uma sessão de exercício supramáximo. Para isso, foram analisadas a resposta do VO2 de jovens universitários, 13 homens e 2 mulheres, submetidos a três ou quatro testes preditivos da PCrit e CTA. O VO2 foi coletado a cada 3 ciclos respiratórios, interpolado a cada um segundo e alisado com médias móveis de 15 segundos. Foram determinados os tempos para se atingir, 60, 65, 70, 75, 80, 85, 90, 95 e 100% do VO2max. Esses tempos apresentaram uma relação hiperbólica com a intensidade de trabalho e possibilitaram determinar os valores de PCrit e CTA correspondentes aos valores abaixo e no VO2max. Ancova demonstrou que esses valores de PCrit e CTA sub-VO2max se relacionavam de maneira linear com os valores máximos correspondentes. As equações lineares foram deduzidas ao modelo hiperbólico 2 2 2 1 * TVO k E P VO VO rep + = + , o qual apresentou bom ajuste à cinética do VO2 (0.90 < r2 <0.99). Os valores encontrados, para os termos da equação, possibilitaram predizer os valores de PCrit (PCrit_pred) e CTA (CTA_pred). Esses valores foram comparados, através do intervalo de confiança (IC), com seus respectivos parâmetros reais, mas não foi encontrada diferença para a PCrit. Entretanto, as grandes amplitudes do IC para a CTA e CTA_pred não permitiram realizar determinações acuradas para este parâmetro. / Manchado et al. (2002) determined CP model parameters through heart rate (HR) submaximal values. The authors reported a linear relationship between submaximal CPHR and AWCHR to its respectives maximal values. This work was based on the assumption that HR and VO2 have a qualitative correlation. Thus, those linear equations were re-inserted in CP hyperbolic model and deduced to predict VO2 kinetics. The main purpose of this study was test the use of VO2 kinetics, of one supramaximal exercise session, to determine the CP parameters. Were submitted to the experiment 13 men and 2 women. They realized 3 or 4 CP and AWC prediction tests and, during the trials, the VO2 response was collected every 3-breath cycle. It was interpolated to one-second values and smoothed by 15 s rolling average. The time to achieve 60, 65, 70, 75, 80, 85, 90, 95 and 100% of VO2max was recorded. Those times fitted to a hyperbolic relationship with exercise intensity, thus the CP and AWC related to sub- and VO2max were determined. ANCOVA revealed that each parameter sub- VO2max value was linear related to its maximal value. These two linear equations were deduced to the hyperbolic model 2 2 2 1 * TVO k E P VO VO rep + = + . The VO2 response was good fitted (0.90 < r2 < 0.99) and the equation term s values were used to determine predicted values of CP (CP_pred) and AWC (AWC_pred). These values were compared, by confidence intervals (CI), with its respective measured real values and no differences were found to CP. However, AWC and AWC_pred presented elevated CI ranges and does not allowed an accurate prediction of this parameter. In Summary, the kinetics model adopted to VO2 makes possible determine CP values using just one rectangular test.

Analisa-se numericamente o comportamento das soluções da equação X(t) + X(t) = f(X(t-)) para f(X) = A X (l-X), em função dos parâmetros , A. Constroem-se as curvas de duplicação de período no espaço de parâmetros para uma determinada condição inicial, que assegura um determinado tipo de solução assintótica (pertencente ao \"ramo 1\"). Verifica-se a conjectura de que a \"rota para o caos\" neste ramo 1\", possa ser a rota de Feigenbaum. Realça-se o fato de que esta equação, para alguns valores de , A, possui diversos atratores. Estuda-se a organização das soluções globais e limitadas da equação acima em \"ramos\" (certos domínios de soluções), e faz-se uma análise das relações entre as soluções dos diversos \"ramos\". Constata-se que uma cascata de duplicação de período no ramo 1, implica em cascatas de duplicação, ao menos parciais, em outros ramos. Para a equação acima com f(X) = A X (l-X), apresentam-se algumas soluções sob a forma de série, parcialmente computáveis sobre a reta, e faz-se uma aplicação de um resultado acerca da estabilidade do ramo 1 no caso f(X) = A sen(X-C), que corresponde a uma equação da ótica. / Numerical analysis are made of the behavior of the solutions of the equation X(t) + X(t) = f(X(t-)) for f(X) = A X (1 - X), as function of the parameters , A. Period-doubling bifurcation curves are constructed in the parameter space for some particular initial conditions, that insures a certain asymptotic behavior of the solutions (it belongs to \"branch 1\"). It is verified the conjecture that the \"route to chaos\" in the \"branch 1\" may be the Feigenbaum\'s route. The organization of the global and bounded solutions of the above equation in branches (certain domains of solutions) is studied. An analysis is made of the relations between solutions belonging to different branches. It is verified that the existence of a full period-doubling cascade in the branch 1 implies the existence, at least partially, of period-doubling cascade in other branches. It is noted that, for some values of (, A), the equation has many attractors. Some series expansions of solutions of the above equation are presented. These series expantions may be partially computed on the set R. An application of a result about the \"stability\" of branch 1 is made for the case f(X) = A sin(X-C), used to describe an optical system.

No description available.

Over recent years, we have witnessed a rapid development in the body of economic theory with applications to finance. It has had great success in finding theoretical explanations to economic phenomena. Typically, theories are employed that are defined by mathematical models. Finance in particular has drawn upon and developed the theory of stochastic differential equations. These produce elegant and tractable frameworks which help us to better understand the world. To directly apply such theories, the models must be assessed and their parameters estimated. Implementation requires the estimation of the model's elements using statistical techniques. These fit the model to the observed data. Unfortunately, existing statistical methods do not work satisfactorily when applied to many financial models. These methods, when applied to complex models often yield inaccurate results. Consequently, simpler analytical models are often preferred, but these are typically unrealistic representations of the underlying process, given the stylised facts reported in the literature. In practical applications, data is observed at discrete intervals and a discretisation is typically used to approximate the continuous-time model. This can lead to biased estimates, since the true underlying model is assumed continuous. This thesis develops new methods to estimate these types of models, with the objective of obtaining more accurate estimates of the underlying parameters present. The methods are applicable to general models. As the solution to the true continuous process is rarely known for these applications, the methods developed rely on building an Euler-Maruyama approximate model and using simulation techniques to obtain the distribution of the unknown quantities of interest. We propose to simulate the missing paths between the observed data points to reduce the bias from the approximate model. Alternatively, one could use a more sophisticated scheme to discretise the process. Unfortunately, their implementation with simulation methods require us to simulate from the density and evaluate the density at any given point. This has until now only been possible for the Euler-Maruyama scheme. One contribution of the thesis is to show the existence of a closed form solution from use of the higher order Milstein scheme. The likelihood based method is implemented within the Bayesian paradigm, as in the context of these models, Bayesian methods are often analytically easier. Concerning the estimation methodology, emphasis is placed on simulation efficiency; design and implementation of the method directly affects the accuracy and stability of the results. In conjunction with estimation, it is important to provide inference and diagnostic procedures. Meaningful information from simulation results must be extracted and summarised. This necessitates developing techniques to evaluate the plausibility and hence the fit of a particular model for a given dataset. An important aspect of model evaluation concerns the ability to compare model fit across a range of possible alternatives. The advantage with the Bayesian framework is that it allows comparison across non-nested models. The aim of the thesis is thus to provide an efficient estimation method for these continuous-time models, that can be used to conduct meaningful inference, with their performance being assessed through the use of diagnostic tools.

This thesis considers three problems in the field of elastodynamics. The first concerns small-amplitude elastic disturbances in an infinite cylinder, a problem first investigated by Pochhammer and Chree. Our approach extends the results of Pochhammer and Chree by utilising a method of successive approximation through which the governing equations are solved to produce dispersion relations. The second investigation, recently considered by Eringen and Suhubi, is of the propagation of elastic waves in a prestressed body, with particular reference to the circular cylinder and the half-space. The governing equations are again solved via successive approximation to give new and detailed results describing the wave motion. The final investigation is of a compressible strain-energy function which is an extension of the Ko model. The model is examined in the light of various a priori inequalities, and is then used to obtain solutions to the problem of vibrations in a stressed plate.

University of Macau / Faculty of Science and Technology / Department of Mathematics

In a large class of modern applications, we have to estimate the initial (or final) state of an infinite-dimensional system (typically a system governed by a Partial Differential Equation) from its partial measurement over some finite time interval. This kind of identification problems arises in medical imaging. For instance, the detection of sick cells (tumor) by thermo-acoustic tomography can be viewed as an initial data reconstruction problem. Some other methods need the identification of a source term, which can be rewritten, under some assumptions, under the form of an initial data reconstruction problem. In this thesis, we are dealing with the reconstruction of the initial state of a system of evolution, working as much as possible on the infinite-dimensional system, using the new algorithm developed by Ramdani, Tucsnak and Weiss (Automatica 2010). We perform in particular the numerical analysis of the algorithm in the case of Schrödinger and wave equations, with internal observation. We study the suitable functional spaces for its use in Maxwell’s equations, with internal and boundary observation. In the last chapter, we try to extend the framework of this algorithm when the initial system is perturbed or when the inverse problem is ill-posed, with application to thermoacoustic tomography.

M.Sc. / A class of numerical methods for solving nonstiff initial value problems in ordinary differential equations has been developed. These methods, designated RKrGLn, are based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration over n+ 1 nodes. The interval of integration for the initial value problem is subdivided into an integer number of subintervals. On each of these n + 1 nodes are defined in accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta method is used to find an approximate solution at each of these nodes; Gauss-Legendre integration is used to find the solution at the endpoint of the subinterval. The process then carries over to the next subinterval. We find that for a suitable choice of n, the order of the local error of the Runge- Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor choice of n can actually limit the order of RKrGLn, irrespective of the choice of r. What is more, the inclusion of Gauss-Legendre integration slightly reduces the number of arithmetical operations required to find a solution, in comparison with RKr at the same number of nodes. These two factors combine to ensure that RKrGLn is considerably more efficient than RKr, particularly when very accurate solutions are sought. Attempts to control the error in RKrGLn have been made. The local error has been successfully controlled using a variable stepsize strategy, similar to that generally used in RK methods. The difference lies in that it is the size of each subinterval that is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local error has been successfully controlled for relative tolerances ranging from 10 -4 to 10-10 . We have also developed algorithms for estimating and controlling the global error. These algorithms require that a complete solution be obtained for a specified distribution of nodes, after which the global error is estimated and then, if necessary, a new node distribution is determined and another solution obtained. The algorithms are based on Richardson extrapolation and the use of low-order and high-order pairs. The algorithms have successfully achieved desired relative global errors as small as 10-1° . We have briefly studied how RKrGLn may be used to solve stiff systems. We have determined the intervals of stability for several RKrGLn methods on the real line, and used this to develop an algorithm to solve a stiff problem. The algorithm is based on the idea of stepsize/subinterval adjustment, and has been used to successfully solve the van der Pol system. Lagrange interpolation on each subinterval has been implemented to obtain a piecewise continuous polynomial approximation to the numerical solution, with same order error, which can be used to find the solution at arbitrary nodes.