The Hasse-Minkowski Theorem

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a finite constant field of odd characteristic. Our treatments of quadratic forms and local fields, though, are more general than what is strictly necessary for our proofs of the Hasse-Minkowski theorem over Q and its analogue over rational function fields (of odd characteristic). Our discussion concludes with some applications of the Hasse-Minkowski theorem.

Alternative Formulations for Angular Momentum Operators, Cartesian and Spherical Polar Forms

The conversion between representations of angular momentum in spherical polar and cartesian form is discussed.

Analysis of the Phosphorylated Forms of Protein Kinase R

Protein Kinase R (PKR) is induced by interferon and activated by dsRNA. Subsequent autophosphorylation and phosphorylation of eIF2alpha inhibits viral replication. In the latent state PKR exists as an unphosphorylated monomer. Work in the Cole laboratory has shown two additional states, a phosphorylated monomeric state (pPKRm) and a phosphorylated dimeric state (pPKRd). RNA serves as a scaffold bringing two PKRs together allowing dimerization and autophosphorylation to occur. The contribution of each state to the function of PKR remains unclear. Western blots were performed to examine the phosphorylation states of the essential residues, T446 and T451. Activity assays have shown activation of pPKRm at a level comparable to pPKRd in its ability to phosphorylate eIF2alpha. Phosphorylation of eIF2alpha by both pPKRm and pPKRd was shown to be RNA independent. Despite reaching similar terminal levels of eIF2alpha phosphorylation, kinetic measurements revealed a faster reaction from pPKRd. Therefore, pPKRm and pPKRd may both contribute to the activity of PKR.