Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².

This thesis is an exposition of spectral theory for bounded operators on Hilbert space. Detailed proofs are given for the functional calculus, the multiplication operator, and the projection-valued measure versions of the spectral theorem for self-adjoint bounded operators. These theorems are then generalized to finite sequences of self-adjoint and commuting bounded operators. Finally, normal bounded operators are discussed, as a particular case of the generalization.

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This thesis is concerned with various aspects of stationary and nonstationary time series analysis. In the nonstationary case, we study estimation of the Wold-Cram'er evolutionary spectrum, which is a time-dependent analogue of the spectrum of a stationary process. Existing estimators of the Wold-Cram'er evolutionary spectrum suffer from several problems, including bias in boundary regions of the time-frequency plane, poor frequency resolution, and an inability to handle the presence of purely harmonic frequencies. We propose techniques to handle all three of these problems. We propose a new estimator of the Wold-Cram'er evolutionary spectrum (the BCMTFSE) which mitigates the first problem. Our estimator is based on an extrapolation of the Wold-Cram'er evolutionary spectrum in time, using an estimate of its time derivative. We apply our estimator to a set of simulated nonstationary processes with known Wold-Cram'er evolutionary spectra to demonstrate its performance. We also propose an estimator of the Wold-Cram'er evolutionary spectrum, valid for uniformly modulated processes (UMPs). This estimator mitigates the second problem, by exploiting the structure of UMPs to improve the frequency resolution of the BCMTFSE. We apply this estimator to a simulated UMP with known Wold-Cram'er evolutionary spectrum. To deal with the third problem, one can detect and remove purely harmonic frequencies before applying the BCMTFSE. Doing so requires a consideration of the aliasing problem. We propose a frequency-domain technique to detect and unalias aliased frequencies in bivariate time series, based on the observation that aliasing manifests as nonlinearity in the phase of the complex coherency between a stationary process and a time-delayed version of itself. To illustrate this ``unaliasing'' technique, we apply it to simulated data and a real-world example of solar noon flux data. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-05 10:18:13.476

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A complete stochastic theory of relaxation is developed in terms of a homogeneous equation for the averaged density matrix of a system immersed in a thermal bath. This theory is then used as the basis of a new stochastic approach to the phenomenon of collisional broadening of spectral line shapes. Single photon and multiphoton processes are studied. The features of a line shape are linked by simple expressions to the statistical properties of a stochastic hermitian Hamiltonian. The ordinary line shape predicted by Kubo's approach is generalized. The present approach predicts broadening as well as asymmetry and shift. A representation of line shapes in multiphoton processes by diagrams is also developed.

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The goal of this research was to demonstrate the potential of spectral CT for the discrimination of vulnerable atherosclerotic plaques. It was proposed that spectral CT has the potential to identify the presence of specific markers for vulnerable plaques: iron deposits and lipid core. A spectral micro-CT system incorporating the latest Medipix spectroscopic photon- counting detectors was commissioned for this purpose. Using spectroscopic methods developed with this system, it was possible to distinguish the presence of iron deposits and lipid core within ex vivo atherosclerotic plaques. Atherosclerosis or hardening of arteries is a systemic disease of the vessel wall that occurs in the aorta, carotid, coronary and peripheral arteries. It is characterised by the deposition of calcified plaques on the innermost layer of the artery wall. Vulnerable plaques are unstable, prone to rupture and put the person at risk of cardiovascular events and strokes. Factors that may lead to plaque instability are lipid content and iron deposits. This preclinical study is a precursor to the development of a clinical technique that will enable vulnerable atherosclerotic plaques to be identified in vivo prior to treatment or removal. Following a preliminary study on atherosclerotic plaques with a prototype system, the MARS-CT3 spectral micro-CT system incorporating Medipix3 was developed and commissioned for further plaque studies. The spectral CT data sets acquired by this system were assessed visually for morphology and analysed for material composition using a linear algebra method. The results were correlated with photography and histology (the histology is the current gold standard). The presence of iron and lipid can be differentiated from the background soft-tissue using a linear algebra method. However the quantification of iron in the presence of calcium is not currently possible without additional data or constraints. Nevertheless the presence of iron deposits within the plaques can be distinguished in the high resolution MARS-CT images and has been correlated with photographic and histological evidence. Thus, using the high spatial resolution spectral data from MARS-CT, the discrimination of lipid core and iron deposits within ex vivo advanced human atherosclerotic plaques is feasible. This may provide the basis for the development of a clinical technique that will identify vulnerable plaques in vivo by high resolution spectral CT.