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.

In this paper we study the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the L2-norm. We compare the arising errors for different window functions and show that the accuracy of the algorithm can be significantly improved by modifying the shape of the window function. Based on the considered error estimates for different window functions we are able to state an easy and efficient method to tune the involved parameters automatically. The numerical examples show that the optimal parameters depend on the given Fourier coefficients, which are assumed not to be of a random structure or roughly of the same magnitude but rather subject to a certain decrease.

[pt] Este trabalho estuda problemas geotécnicos e de interação solo-estrutura utilizando o método dos elementos finitos acoplado com a transformada de Fourier. Pela aplicação da transformada de Fourier, as equações diferenciais que governam o problema elástico linear, com as correspondentes condições de contorno, são reescritas no plano de Fourier, permitindo que um problema de natureza tridimensional possa ser numericamente analisado por uma discretização bidimensional. Esta técnica foi empregada neste trabalho para certos problemas de engenharia, como dutovias, túneis e fundações tipo radier, onde a geometria e os parâmetros dos materiais mantêm-se constantes ao longo do eixo longitudinal do corpo, porém admitindo-se variações espaciais no carregamento imposto ao sistema, gerando , assim, um estado tridimensional de tensões. Alguns elementos de interface, com formulação publicada na literatura, foram também considerados na implementação computacional, visto que em problemas de interação solo- estrutura o comportamento do sistema é bastante influenciado pelas propriedades e características mecânicas do solo imediatamente vizinho à estrutura. Os exemplos numéricos apresentados são comparados, sempre que possível, com os resultados obtidos por outra solução analítica ou numérica, procurando discutir as vantagens e limitações do acoplamento da transformada de Fourier com o método dos elementos finitos para a análise de determinada classe de problemas geotécnicos tridimensionais. / [en] In this work some geotechnical and soil-structure interaction problems are studied using the finite element method coupled with a Fourier transform technique. For linear elastic problems, Fourier transforms are applied to the governing field equations, thus enabling that some specific tridimensional problems can be analyzed using a 2D finite element mesh. In conventional finite element applications, a 3D discretization is usually required, but difficulties associated with the preparation of the finite element mesh and the involved computational efforts prevent, in general, the use of a true 3D model. The integral transform method is used in this research for the analysis of some very common problems in geotechnical engineering, such as piping systems, raft foundations and tunnels, where the geometry and the soil profile may be considered constant along a coordinate direction. The applied loading, however, can assume any possible surface distribution, which does not allow to treat the problem under the plane strain assumptions. Some special finite elements presented in the literature, called joint or interface elements, are also incorporated into the finite element computational program written in this research, given that for soil-structure interaction problems the material behavior at the common interface may greatly affect the entire system results. Some numerical examples are presented, and their numerical results are compared, whenever possible, with other solutions obtained using analytical or other numerical technique. Advantages and limitations of the integral transform method to solve tridimensional geomechanics problems are also discussed in this work. / [es] Este trabajo estudia problemas geotécnicos y de interacción suelo-extructura utilizando el método de los elementos finitos acoplado con la transformada de Fourier. Por la aplicación de la transformada de Fourier, las ecuaciones diferenciales que goviernan el problema elástico lineal, con las correspondentes condiciones de contorno, son reescritas en el plano de Fourier, permitiendo que un problema de naturaleza tridimensional pueda ser numericamente analizado por una discretización bidimensional. Esta técnica fue utilizada en este trabajo para ciertos problemas de Ingeniería, como canales, túneles y fundaciones tipo radier, donde la geometría y los parámetros de los materiales se mantienen constantes a lo largo del eje longitudinal del cuerpo, aunque se admiten variaciones espaciales en la carga impuesta al sistema, generando , así, un estado tridimensional de tensiones. En la implementación computacional fueron considerados algunos elementos de la interfaz, con formulación publicada en la literatura, ya que en problemas de interacción suelo-extructura, el comportamiento del sistema está bastante influenciado por las propiedades y características mecánicas del suelo imediatamente vecino a la extructura. Los ejemplos numéricos presentados se compararon, siempre que fue posible, con los resultados obtenidos por otra solución analítica o numérica, discutiendo las ventajas y limitaciones del acoplamiento de la transformada de Fourier con el método de los elementos finitos para el análisis de determinada clase de problemas geotécnicos tridimensionales.

This thesis concerns the mathematics and application of various methods for approximating matrices, with a particular eye towards the role that such methods play in image compression. An image is stored as a matrix of values with each entry containing a value recording the intensity of a corresponding pixel, so image compression is essentially equivalent to matrix approximation. First, we look at the singular value decomposition, one of the central tools for analyzing a matrix. We show that, in a sense, the singular value decomposition is the best low-rank approximation of any matrix. However, the singular value decomposition has some serious shortcomings as an approximation method in the context of digital images. The second method we consider is the discrete Fourier transform, which does not require the storage of basis vectors (unlike the SVD). We describe the fast Fourier transform, which is a remarkably efficient method for computing the discrete cosine transform, and how we can use this method to reduce the information in a matrix. Finally, we look at the discrete cosine transform, which reduces the complexity of the calculation further by restricting to a real basis. We also look at how we can apply a filter to adjust the relative importance of the data encoded by the discrete cosine transform prior to compression. In addition, we developed code implementing the ideas explored in the thesis and demonstrating examples.

Typescript (photocopy). / Digitized by Kansas Correctional Industries

Call number: LD2668 .T4 1978 J69 / Master of Science

Master of Science / Department of Mathematics / David R. Auckly / In this report we present the main concepts of probability theory: sample spaces, events, random variables, distributions, independence, central limit theorem. Most of the material may be found in the notes of Bass. The work is motivated by wide range of applications of probability theory in quantitative finance.

Laser and Fourier transform spectroscopy were used to study a variety of gas phase molecules. A series of calcium and strontium monoalkoxides, CaOR and SrOR [R=H, CH₃, CH₂CH₃, CH(CH₃)₂, CH₂CH₂CH₃, CH₂(CH₂)₂CH₃, CH₂(CH₃)CH₂CH₃ and C(CH₃)₃] was studied by laser induced fluorescence and laser excitation techniques. The A - X and B - X electronic transitions were recorded. Calcium and strontium monocyclopentadienide, CaCp and SrCp, were produced in a Broida oven and the spectra were recorded at low resolution by laser techniques. The geometry of these molecules is like an "open-faced sandwich", with C₅ᵥ symmetry. The A²Π - X²Σ⁺ and B²Σ⁺ - X²Σ⁺ transitions of calcium and strontium monoisocyanate, CaNCO and SrNCO, were recorded for the first time. The spectra are interpreted as arising from a molecule with linear, ionic Sr⁺ - ⁻NCO structure. The 0-0 band of the A - X transition of SrNCO was recorded at high resolution and rotationally analyzed. The use of a 0.64 m monochromator, acting as a narrow bandpass filter, was necessary for recording the spectra and enabled the correct assignment of the rotational lines. By comparing the molecular constants of SrNCO with related molecules, the NCO⁻ ligand was found to be nitrogen bonding to the strontium atom. The Sr-N bond length was estimated to be 2.26 A in the ground state. The rotational analysis of the 0-0 band of the A²E₁/₂ - X²A₁ transition of the SrOCH₃ free radical has been carried out by laser excitation spectroscopy. The SrOCH₃ molecule was found to have C₃ᵥ symmetry with a Sr-O bond length of 2.12 A. There was no evidence of a Jahn-Teller effect in the A²E state. This work is the first high-resolution analysis of a metal alkoxide molecule. The rotationally cool (30 K) spectrum of the 0-0 vibrational band of the A²Δ - X²Π electronic transition for the diatomic free radical CCl was recorded using a Fourier transform spectrometer. The first rotational lines of the six strongest branches were observed, and transitions in three other branches were also detected. The CCl radical was produced in a corona-excited supersonic expansion source.

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