It is well known that a classical solution of the initial value problem for a scalar conservation law may fail to exist on the whole domain of definition of the problem. For this reason, suitable generalized solutions of such problems, known as weak solutions, have been considered and studied extensively. However, weak solutions are not unique. In order to obtain a unique solution that is physically relevant, the vanishing viscosity method, amongst others, has been employed to single out a unique solution known as the entropy solution. In this thesis we present an alternative approach to the study of the entropy solution of conservation laws. The main novelty of our approach is that the theory of entropy solution of conservation law is presented in an operator theoretic setting. In this regard, the Order Completion Method for nonlinear PDEs, in the context of convergence vector spaces, is modified to obtain an operator equation which generalizes the initial value problem. This equation admits at most one solution, which may be represented as a Hausdorff continuous function. As a particular case, we apply our method to obtain the entropy solution of the Burger's equation. Copyright / Dissertation (MSc)--University of Pretoria, 2011. / Mathematics and Applied Mathematics / Unrestricted

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We begin by studying various topological properties of invariant sets of hyperbolic toral automorphisms in the linear case. Results related to cardinality, local maximality, entropy, and dimension are presented. Where possible, we extend the results to the case of hyperbolic toral automorphisms in higher dimensions, and further to general Anosov maps.

The subject of fractal geometry has exploded over the past 40 years with the availability of computer generated images. It was seen early on that there are many interesting questions at the intersection of probability and fractal geometry. In this dissertation we will introduce two random models for constructing fractals and prove various facts about them.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第22972号 / 理博第4649号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 藤原 耕二, 教授 山口 孝男, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Modeling the liver deformation forms the basis for the development of new clinical applications that improve the diagnosis, planning and guidance in liver surgery. However, the patient-specific modeling of this organ and its validation are still a challenge in Biomechanics. The reason is the difficulty to measure the mechanical response of the in vivo liver tissue. The current approach consist of performing minimally invasive or open surgery aimed at estimating the elastic constant of the proposed biomechanical models. This dissertation presents how the use of medical image analysis and evolutionary computation allows the characterization of the biomechanical behavior of the liver, avoiding the use of these minimally invasive techniques. In particular, the use of similarity coefficients commonly used in medical image analysis has permitted, on one hand, to estimate the patient-specific biomechanical model of the liver avoiding the invasive measurement of its mechanical response. On the other hand, these coefficients have also permitted to validate the proposed biomechanical models. Jaccard coefficient and Hausdorff distance have been used to validate the models proposed to simulate the behavior of ex vivo lamb livers, calculating the error between the volume of the experimentally deformed samples of the livers and the volume from biomechanical simulations of these deformations. These coefficients has provided information, such as the shape of the samples and the error distribution along their volume. For this reason, both coefficients have also been used to formulate a novel function, the Geometric Similarity Function (GSF). This function has permitted to establish a methodology to estimate the elastic constants of the models proposed for the human liver using evolutionary computation. Several optimization strategies, using GSF as cost function, have been developed aimed at estimating the patient-specific elastic constants of the biomechanical models proposed for the human liver. Finally, this methodology has been used to define and validate a biomechanical model proposed for an in vitro human liver. / Martínez Martínez, F. (2014). Determining the Biomechanical Behavior of the Liver Using Medical Image Analysis and Evolutionary Computation [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/39337