In the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of A. Bukhgeim. We then uses the range characterization of the Radon transform of functions to characterize the range of the attenuated Radon transform of vector fields as they appear in the medical diagnostic techniques of Doppler tomography. As an application we determine necessary and sufficient conditions for the Doppler and X-ray data to be mistaken for each other. We also characterize the range of real symmetric second order tensor field using the range characterization of the Radon transform of zero tensor field.

The master's thesis is concerned with digital image denoising methods. The theoretical part explains some elementary terms related to image processing, image noise, categorization of noise and quality determining criteria of denoising process. There are also particular denoising methods described, mentioning their advantages and disadvantages in this paper. The practical part deals with an implementation of the selected denoising methods in a Java, in the environment of application RapidMiner. In conclusion, the results obtained by different methods are compared.

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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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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 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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