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Option Pricing using the Fast Fourier Transform MethodBerta, Abaynesh January 2020 (has links)
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.
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Automated Parameter Tuning based on RMS Errors for nonequispaced FFTsNestler, Franziska 16 February 2015 (has links)
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.
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[pt] ANÁLISE DE PROBLEMAS TRIDIMENSIONAIS SOLO-ESTRUTURA PELO MÉTODO DOS ELEMENTOS FINITOS NO DOMÍNIO DE FOURIER / [es] ANÁLISIS DE PROBLEMAS TRIDIMENSIONALES SUELO-EXTRUCTURA POR EL MÉTODO DE LOS ELEMENTOS FINITOS EN EL DOMINIO DE FOURIER / [en] THREE-DIMENSIONAL ANALYSIS OF SOIL-STRUCTURE PROBLEMS USING THE FINITE ELEMENT METHOD IN THE FOURIER DOMAINJANAINA VEIGA CARVALHO 03 August 2001 (has links)
[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.
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Matrix Approximation And Image CompressionPadavana, Isabella R 01 June 2024 (has links) (PDF)
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.
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Isotopic analysis of nitrogen in ammonia by Fourier transform infrared spectroscopyAbuzwida, Mohamed Ali January 2011 (has links)
Typescript (photocopy). / Digitized by Kansas Correctional Industries
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Resolution of composite radioactive decay curves by Fourier decay analysisJowzani-Moghaddam, Ali. January 1978 (has links)
Call number: LD2668 .T4 1978 J69 / Master of Science
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Probability theory, fourier transform and central limit theoremSorokin, Yegor January 1900 (has links)
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.
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Laser and Fourier transform spectroscopy of gas phase molecules.O'Brien, Leah Christine. January 1987 (has links)
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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Molecular spectroscopic techniques in the characterisation and structural determination of novel systemsGrose, Richard Ian January 1993 (has links)
No description available.
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The reactivity of magnesium oxide surfaces towards methane and other moleculesAulton, Simon Philip January 1999 (has links)
No description available.
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