In recent decades, the rapid development of optical spectroscopy for tissue diagnosis has been indicative of its high clinical value. The goal of this research is to prove the feasibility of using diffuse reflectance spectroscopy and fluorescence spectroscopy to assess myocardial infarction (MI) in vivo. The proposed optical technique was designed to be an intra-operative guidance tool that can provide useful information about the condition of an infarct for surgeons and researchers. In order to gain insight into the pathophysiological characteristics of an infarct, two novel spectral analysis algorithms were developed to interpret diffuse reflectance spectra. The algorithms were developed based on the unique absorption properties of hemoglobin for the purpose of retrieving regional hemoglobin oxygenation saturation and concentration data in tissue from diffuse reflectance spectra. The algorithms were evaluated and validated using simulated data and actual experimental data. Finally, the hypothesis of the study was validated using a rabbit model of MI. The mechanism by which the MI was induced was the ligation of a major coronary artery of the left ventricle. Three to four weeks after the MI was induced, the extent of myocardial tissue injury and the evolution of the wound healing process were investigated using the proposed spectroscopic methodology as well as histology. The correlations between spectral alterations and histopathological features of the MI were analyzed statistically. The results of this PhD study demonstrate the applicability of the proposed optical methodology for assessing myocardial tissue damage induced by MI in vivo. The results of the spectral analysis suggest that connective tissue proliferation induced by MI significantly alter the characteristics of diffuse reflectance and fluorescence spectra. The magnitudes of the alterations could be quantitatively related to the severity and extensiveness of connective tissue proliferation.

The Hilbert-P\'olya conjecture asserts that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ correspond (in a certain canonical way) to the eigenvalues of some positive operator. R. Meyer constructed a differential operator $D_-$ acting on a function space $\H$ and showed that the eigenvalues of the adjoint of $D_-$ are exactly the nontrivial zeros of $\zeta(s)$ with multiplicity correspondence. We follow Meyer's construction with a slight modification. Specifically, we define two function spaces $\H_\cap$ and $\H_-$ on $(0,\infty)$ and characterize them via the Mellin transform. This allows us to show that $Z\H_\cap\subseteq\H_-$ where $Zf(x)=\sum_{n=1}^\infty f(nx)$. Also, the differential operator $D$ given by $Df(x)=-xf'(x)$ induces an operator $D_-$ on the quotient space $\H=\H_-/Z\H_\cap$. We show that the eigenvalues of $D_-$ on $\H$ are exactly the nontrivial zeros of $\zeta(s)$. Moreover, the geometric multiplicity of each eigenvalue is one and the algebraic multiplicity of each eigenvalue is its vanishing order as a nontrivial zero of $\zeta(s)$. We generalize our construction on the Riemann zeta function to some $L$-functions, including the Dirichlet $L$-functions and $L$-functions associated with newforms in $\mathcal S_k(\Gamma_0(M))$ with $M\ge1$ and $k$ being a positive even integer. We give spectral interpretations for these $L$-functions in a similar fashion.