In this thesis, the structure of pure measures is investigated. These are elements of the dual of the space of essentially bounded functions. A more precise representation of the dual space of the space of essentially bounded functions is given, leading to the definition and analysis of density measurs which constitute a new large class and yield numerous new examples of pure measures which are well-suited for applications in very general Divergence Theorems. The existence of pure normal measures for sets of finite perimeter is demonstrated. These yield Gauß formulas for essentially bounded vector fields having divergence measure. Furthermore, a result of Silhavy is extended. In particular, it is shown that a Gauß-Green Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field.

In this thesis, the structure of pure measures is investigated. These are elements of the dual of the space of essentially bounded functions. A more precise representation of the dual space of the space of essentially bounded functions is given, leading to the definition and analysis of density measurs which constitute a new large class and yield numerous new examples of pure measures which are well-suited for applications in very general Divergence Theorems. The existence of pure normal measures for sets of finite perimeter is demonstrated. These yield Gauß formulas for essentially bounded vector fields having divergence measure. Furthermore, a result of Silhavy is extended. In particular, it is shown that a Gauß-Green Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field.

Made available in DSpace on 2016-03-15T19:37:58Z (GMT). No. of bitstreams: 1 RODRIGO FREITAS LIMA.pdf: 13768169 bytes, checksum: 153d5257eed9a0961aaeaac94e224f89 (MD5) Previous issue date: 2015-08-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Paranasal sinus are important objects of study to rhinosinusitis diagnostic, having some papers related incidence between asthma and allergic rhinitis.Many applications can calculate to various parts of the human body, getting a CT scan or MRI input, and returning information about the region of interest observed as volume and area. The accumulated mucus in the sinuses is one of the areas of interest that have not yet been implemented methods for the calculation of volume and area. In the present scenario, the patient monitoring is done visually, depending largely on perception of the evaluator. Therefore, we seek to implement more accurate metrics to facilitate medical care to the patient and it can help prevent the worsening of rhinitis in a given patient, developing mechanisms of visual and numerical comparison, where it is possible observe the progress of treatment. This work contains a detailed study of how certain existing techniques, combined into one methodology can segment and calculate the accumulated mucus in the maxillary sinus. In addition to techniques such as Thresholding, Gaussian filter, Mathematical Morphology, Metallic Artifacts Reduction during processing and segmentation, MUNC and DTA to calculate the volume and area, and visualization techniques as the Marching Cubes, it was also necessary some adjustments in the algorithm for limit the region of interest where the thresholding combined with the gaussian filter has not been effective of retaining edges. The application will use two open source platforms, one for processing, ITK, and another for visualization, VTK. The results demonstrated that it is possible to perform segmentation and the calculation with the use of platforms as well as the methodology used is adequate to solve this problem. / Os seios paranasais são importantes objetos de estudo para o diagnóstico de rinossinusites, tendo alguns estudos relacionado a incidência de asma na fase adulta a quadros de rinite alérgica na infância. Muitas aplicações atendem a diversas partes do corpo humano, obtendo de entrada uma tomografia computadorizada ou ressonância magnética, e devolvendo, muitas vezes, números que dizem respeito ao objeto de interesse observado, como volume e área. O muco acumulado nos seios paranasais é uma das regiões de interesse que ainda não tiveram métodos implementados para o cálculo do volume e área. No cenário atual, o acompanhamento do paciente é feito de forma visual, dependendo muito da percepção do avaliador. Portanto, busca-se a implementação de métricas mais precisas para facilitar o acompanhamento médico ao paciente e ajudar na prevenção do agravamento de um quadro de rinite em um determinado paciente, criando mecanismos de comparação visual e numérica, onde é possível observar a evolução do tratamento. Este trabalho contém um estudo detalhado de como determinadas técnicas existentes, combinadas em uma metodologia, podem segmentar e calcular o muco acumulado nos seios paranasais maxilares. Além de técnicas como a Binarizacão, Filtro Gaussiano, Morfologia Matemática, Redução de Ruídos Metálico durante o processamento e segmentação, MUNC e DTA para o cálculo do volume e área, e técnicas de visualização como o Marching Cubes, foram necessários também ajustes no algoritmo para limitar a área segmentada onde a binarizacão combinada ao filtro não foi capaz de manter as bordas da região de interesse. A aplicação fará uso de duas plataformas de código livre, sendo uma para o processamento, ITK, e outra para visualização de imagens, VTK. Os resultados demonstraram que é possível realizar a segmentação e o cálculo com o uso das plataformas, bem como a metodologia empregada é adequada a resolução deste problema.

Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1