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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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A análise harmônica e o ramo da matemática que estuda a representação de funções ou sinais como a sobreposição de ondas base. Ela investiga e generaliza as noções das séries de Fourier e da transformação de Fourier. Neste trabalho, investigou-se um teorema de restrição da transformada de Fourier devido a Mitsis e Mockenhaupt (uma generalização do teorema de Stein-Tomas). Foram realizados estudos analíticos sobre o método para operadores integrais oscilatórios, baseado na fase estacionária. Os resultados permitem deduzir o teorema de restrição no plano (em seu caso geral) e o teorema de Carleson-Sjölin. / Harmonic analysis is the mathematical branch that studies the function or signals representation as a base wave overlay. It investigates and generalizes the notions of Fourier series and of the Fourier transform. In this work, was investigated a restriction theorem of the Fourier transform due to Mitsis and Mockenhaupt (a generalization of Stein-Tomas theorem) . Were performed analytic studies on the method for oscillating integral operators, based in the stationary phase. The results allow deducing the restriction theorem on the plane (in the general case) and the Carleson-Sjölin theorem.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Physics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / by David Hopkins Shoemaker. / M.S.

Typescript. / Digitized by Kansas Correctional Industries

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant under automorphisms of a binary tree. We present a technique by which a slightly relaxed form of the generalized Fourier transform in this case can eventually be computed using only simple tools from linear algebra, which has possible advantages in computational efficiency.