The identification of energy scales in the region of Isoscalar Giant Quadrupole Resonance (ISGQR) is motivated by their potential use in understanding how an ordered collective motion transforms into a disordered motion of intrinsic single-particle degrees-of-freedom in many-body quantum systems. High energy-resolution measurements of the ISGQR were obtained by proton inelastic scattering at Ep= 200 MeV using the K600 magnetic Spectrometer at iThemba LABS. The nuclei 58Ni, 90Zr, 120Sn and 208Pb, associated with closed shells, were investigated. Both the Fourier transform and Wavelet analysis were used to extract characteristic energy scales and were later compared with the results from the theoretical microscopic Quasi-particle Phonon Model (QPM), including contributions from collective and non-collective states. The scales found in the experimental data were in good agreement with the QPM. This provides a strong argument that the observed energy scales result from the decay of the collective modes into 2p-2h states. The different scale regions were tested directly by reconstruction of measured energy spectra using the Inverse Fourier Transform and the Continuous Wavelet Transform (CWT), together with a comparison to a previously available reconstruction using the Discrete Wavelet Transform (DWT).

The fast Fourier transform (FFT), even though it has been widely applicable in Physics and Engineering, it has become attractive in Finance as well for it’s enhancement of computational speed. Carr and Madan succeeded in implementing the FFT for pricing of an option. This project, inspired by Carr and Madan’s paper, attempts to elaborate and connect the various mathematical and theoretical concepts that are helpful in understanding of the derivation. Further, we derive the characteristic function of the risk neutral probability for the logarithmic terminal stock price. The Black-Scholes-Merton (BSM) model is also revised including derivation of the partial deferential equation and the formula. Finally, comparison of the BSM numerical implementation with and without the FFT method is done using MATLAB.

Neste trabalho são descritos e avaliados diferentes modelos de linhas aéreas de transmissão, considerando a variação dos parâmetros da linha com a frequência. Em geral os modelos apresentados são baseados no uso da matriz de admitâncias nodais, com algumas análises complementares empregando o método das características. São testados casos de circuito-aberto, curto-circuito, linha em carga e manobras com chaves. Os resultados de cálculo são comparados com as respostas simuladas nos programas ATP e PSCAD. Também são apresentadas as formulações para o cálculo de parâmetros da linha de transmissão, considerando o efeito pelicular e a resistividade do solo São avaliadas as respostas no domínio do tempo para diferentes modelos equivalentes de linha de transmissão, obtidos do modelo polo-resíduo com passividade da matriz de admitâncias nodais, sendo empregados: i) transformação do modelo racional para uma representação em espaço de estados, ii) construção de uma rede elétrica equivalente com resistências, indutores e capacitores e iii) representação por meio de um equivalente de Norton. Também é analisado o equivalente de Norton obtido dos modelos polo-resíduo com passividade de uma linha equivalente dobrada nas condições em curto-circuito e circuito-aberto. Foi analisada a solução de redes elétricas no domínio da frequência usando uma formulação nodal e sem a necessidade de ajustes intermediários da matriz de admitâncias nodais. As respostas de tensão e corrente são antitransformadas para o domínio do tempo por meio de diferentes formulações baseadas na transformada inversa de Fourier, levando em conta os métodos que mitigam erros de truncamento e discretização. As seguintes formulações foram avaliadas: i) transformada modificada de Fourier, ii) transformada inversa rápida de Fourier e iii) transformada inversa de Fourier com integração semianalítica. É também apresentado um modelo para linhas de transmissão em paralelo, baseado no domínio modal usando uma matriz de transformação constante, juntamente com a aplicação do método de ajuste Vector Fitting com o objetivo de modelar com precisão o efeito de acoplamento mútuo entre linhas. / In this work different overhead transmission lines models are described and evaluated, considering the frequency dependence of parameters. In general, the models are based on the nodal admittance matrix, with some complementary analysis using the method of characteristics. Cases of open-circuit, short-circuit, line connected to a load and switching maneuvers are tested. The calculation results are compared with the simulated responses obtained with the ATP and PSCAD. The formulations for the transmission line parameters are also presented, taking into account the skin effect and the ground return. The time-domain responses for different transmission line models are evaluated. These models are obtained from the passive pole-residue model of the nodal admittance matrix. The equivalent models are: i) state-space representation, ii) electrical network composed of resistors, inductors and capacitors, iii) the Norton\'s equivalent. In addition, an evaluation is performed for the Norton\'s equivalent obtained from the pole-residue models for the open-circuit and short-circuit conditions of a folded line equivalent. On the other hand, the network is solved completely in frequency-domain using a nodal formulation, without the fitting of the nodal admittance matrix. The voltage and current responses are transformed back to the time-domain by different formulations based on the inverse Fourier transform, taking into account methods to minimize errors by truncation and discretization. The formulations investigated are: i) modified Fourier transform, ii) fast Fourier transform, iii) inverse Fourier transform with semianalytic integration. The model for parallel transmission lines is also evaluated, based on modal domain using a constant transformation matrix and the application of the vector fitting method, with emphasis on accurate representation of mutual coupling effects.

The traditional Computed Tomography (CT) is based on the Radon Transform and its inversion. The Radon transform uses parallel beam geometry and its inversion is based on the Fourier slice theorem. In practice, it is very efficient to employ a back-projection algorithm in connection with the Fast Fourier Transform, and which can be interpreted as a 1-D filtering across the radial dimension of the 2-D Fourier plane of the transformed image. This approach can easily be adapted to windowing techniques in the frequency domain, giving the capability to reduce image noise. In this work we are investigating the capabilities of the so called Kaiser window (giving an optimal trade-off between the main lobe energy and the sidelobe suppression) to achieve a near optimal trade-off between the noise reduction and the image sharpness in the context of Radon inversion. Finally, we simulate our image reconstruction using MATLAB software and compare and estimate our results based on the normalized Least Square Error (LSE). We conclude that the Kaiser window can be used to achieve an optimal trade-off between noise reduction and sharpness in the image, and hence outperforms all the other classical window function in this regard.