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.

A Median Voter Theorem for Postelection Politics

We analyze a model of 'postelection politics', in which (unlike in the more common Downsian models of 'preelection politics') politicians cannot make binding commitments prior to elections. The game begins with an incumbent politician in office, and voters adopt reelection strategies that are contingent on the policies implemented by the incumbent. We generalize previous models of this type by introducing heterogeneity in voters' ideological preferences, and analyze how voters' reelection strategies constrain the policies chosen by a rent-maximizing incumbent. We first show that virtually any policy (and any feasible level of rent for the incumbent) can be sustained in a Nash equilibrium. Then, we derive a 'median voter theorem': the ideal point of the median voter, and the minimum feasible level of rent, are the unique outcomes in any strong Nash equilibrium. We then introduce alternative refinements that are less restrictive. In particular, Ideologically Loyal Coalition-proof equilibrium also leads uniquely to the median outcome.

Tribalism as a Minimax-Regret Strategy: Evidence from Voting in the 2007 Kenyan Elections

Although many studies find that voting in Africa approximates an ethnic census in that voting is primarily along ethnic lines, hardly any of the studies have sought to explain ethnic voting following a rational choice framework. Using data of voter opinions from a survey conducted two weeks before the December 2007 Kenyan elections, we find that the expected benefits associated with a win by each of the presidential candidates varied significantly across voters from different ethnic groups. We hypothesize that decision to participate in the elections was influenced by the expected benefits as per the minimax-regret voting model. We test the predictions of this model using data of voter turnout in the December 2007 elections and find that turnout across ethnic groups varied systematically with expected benefits. The results suggest that individuals participated in the elections primarily to avoid the maximum regret should a candidate from another ethnic group win. The results therefore offer credence to the minimax regret model as proposed by Ferejohn and Fiorina (1974) and refute the Downsian expected utility model.

DeMoivre's Theorem

DeMoivre's theorem is of great utility in some parts of physical chemistry and is re-introduced here.

Ehrenfest's Theorem

A one dimensional presentation of Ehrenfest's theorem is presented.