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In this thesis a number of problems in harmonic analysis of a geometric flavour are discussed and, in particular, the Lebesgue space mapping properties of certain averaging and Fourier restriction operators are studied. The first three chapters focus on the perspective afforded by affine-geometrical considerations whilst the remaining chapter considers some discrete variants of these problems. In Chapter 1 there is an overview of the basic affine theory of the aforementioned operators and, in particular, the affine arc-length and surface measures are introduced. Chapter 2 presents work of the author, submitted for publication, concerning an operator which takes averages of functions on Euclidean space over both translates and dilates of a fixed polynomial curve. Moreover, the averages are taken with respect to the affine arc-length; this allows one to prove Lebesgue space estimates with a substantial degree of uniformity in the constants. The sharp range of uniform estimates is obtained in all dimensions except for an endpoint. Chapter 3 presents some work of the author, published in Mathematika, concerning a family of Fourier restriction operators closely related to the averaging operators discussed in Chapter 2. Specifically, a Fourier restriction estimate is obtained for a broad class of conic surfaces by introducing a certain measure which exhibits a special kind of affine invariance. Again, the sharp range of estimates is obtained, but the results are limited to the case of 2-dimensional cones. Finally, Chapter 4 discusses some recent joint work of the author and Jim Wright considering the restriction problem over rings of integers modulo a prime power. The sharp range of estimates is obtained for Fourier restriction to the moment curve in finitely-generated free modules over such rings. This is achieved by lifting the problem to the p-adics and applying a classical argument of Drury in this setting. This work aims to demonstrate that rings of integers offer a simplified model for the Euclidean restriction problem.

Este trabalho consiste em apresentar um método alternativo para a identificação de fontes de correntes harmônicas comumente encontradas em sistemas elétricos residenciais. Desta identificação, soluções viáveis poderão ser aplicadas com o intuito de mitigar os níveis de emissão das correntes harmônicas geradas, principalmente, por cargas com características não lineares. Para a identificação empregou-se redes neurais artificiais (RNAs), sendo esta técnica inteligente, apresentada como uma alternativa aos métodos convencionais. Os resultados reportados neste contexto procuram validar a proposta apresentada com dados experimentais obtidos em ensaios laboratoriais. / This work presents an alternative method for the identification of current harmonic sources commonly encountered in residential electrical systems. For this purpose, feasible solutions can be applied to minimize the levels of harmonic currents emission caused by nonlinear loads. Artificial neural networks are employed as alternative to conventional methods. The experimental results will be reported in order to validate the proposal presented with the experimental data obtained in laboratory.

Este trabalho consiste em apresentar um método alternativo para a identificação de fontes de correntes harmônicas comumente encontradas em sistemas elétricos residenciais. Desta identificação, soluções viáveis poderão ser aplicadas com o intuito de mitigar os níveis de emissão das correntes harmônicas geradas, principalmente, por cargas com características não lineares. Para a identificação empregou-se redes neurais artificiais (RNAs), sendo esta técnica inteligente, apresentada como uma alternativa aos métodos convencionais. Os resultados reportados neste contexto procuram validar a proposta apresentada com dados experimentais obtidos em ensaios laboratoriais. / This work presents an alternative method for the identification of current harmonic sources commonly encountered in residential electrical systems. For this purpose, feasible solutions can be applied to minimize the levels of harmonic currents emission caused by nonlinear loads. Artificial neural networks are employed as alternative to conventional methods. The experimental results will be reported in order to validate the proposal presented with the experimental data obtained in laboratory.

Two applications of second harmonic generation (SHG), a nonlinear optical technique, are studied. First, Fresnel factors are used with a bond model to describe SHG from vicinal silicon at five incidence angles: 7.5°, 22°, 30°, 45°, and 52°. Second, a prototype apparatus for applying SHG to enhance imaging capabilities of optical coherence tomography, a microscopy technique used in many biological fields, is briefly described. / text

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We study the Hausdorff dimension of a certain Borel measure associated to a positive weak solution of a certain quasilinear elliptic partial differential equation in a simply connected domain in the plane. We also assume that the solution vanishes on the boundary of the domain. Then it is shown that the Hausdorff dimension of this measure is less than one, equal to one, greater than one depending on the homogeneity of the certain function. This work generalizes the work of Makarov when the partial differential equation is the usual Laplace's equation and the work of Lewis and his coauthors when it is the p-Laplace's equation.

I propose and demonstrate a method by which barium titanate nanoparticle clusters can be used as exogenous contrast agents in Second Harmonic Optical Coherence Tomography imaging systems to localize and highlight desired regions of tissue. SH-OCT has previously been used to identify collagen within OCT images. However, SH-OCT signals from collagen are highly susceptible to inferior reflector artifacts because most of the second harmonic generated light is forward scattered. Second harmonic generating nanoparticle clusters exhibit high scattering properties, which can give them the advantage of backscattering a large quantity of second harmonic light while attenuating the forward scattered light. In this research project, a mathematical model is proposed in which the backward to forward scattering ratio of second harmonic generated light from nanoparticle layers is exponentially proportional to the thickness of the layer. This model was supported by measurements of the backward to forward scattering ratio of second harmonic light in barium titanate nanoparticles layers. This indicates that nanoparticle clusters can be designed and manufactured with the proper thickness so that they generate a large second harmonic signal without creating inferior reflector artifacts.

Line current distortion is an important issue to a high-power current source rectifier(CSR) system. There are two main challenges related to this issue. First, the CSR input LC resonance can be affected by the variation of the source inductance from the power system and the effects of the CSR DC side circuit, which may lead to a line current distortion higher than expected. Another challenge is that the traditional high-power CSR using Selective Harmonic Elimination (SHE) pulse-width modulation (PWM) technique attempts to eliminate certain harmonics in the PWM current, which limits its ability for line current harmonic control. To control the CSR line current harmonics, this thesis focuses on two aspects: 1) the analysis and design of CSR input filter to avoid unexpected input LC resonance, and 2) the development of a new PWM scheme that can compensate the effects of the grid voltage harmonics and DC link current ripples. The thesis work has been validated by simulations and on an experimental CSR prototype. / Power Engineering and Power Electronics