Brownian Motion and the Airy Function

Deriving the location of the maximum of a Brownian Motion with downward quadratic drift. Proving it is welldefined then finding an algorithmic method to simplify expressions of the moments.

The Borsuk-Ulam Theorem in Combinatorics

We examine and prove the Borsuk-Ulam theorem and its combinatorial equivalent Fan's lemma. The theory of simplicial complexes and triangulations plays an important role in this. The Borsuk-Ulam theorem and Fan's lemma will be used to provide proofs for the Brouwer fixed point theorem and the combinatorial Sperner's lemma. Lastly the Borsuk-Ulam...

The Theorem of Reicht and Network Matrices

In 1969, the theorem of Reichert was formulated. Since then, two proofs have been found, both by Z. Jiang and Malcolm C. Smith, one of which in collaboration with Sara Y. Zhang. These proofs are based on the transfer functions of the electrical networks. <br/>The aim of this paper is to work toward a proof based on graph theory, linear algebra...

Synchronize with noise: Synchronization in a mean-field model of interacting oscillators in a random environment

A variation on the random Kuramoto model was analyzed, a mean-field model which describes the behavior of coupled oscillators in a random environment. The analytic results are based on the behavior of the system in the infinite volume limit. In the analysis critical values were found for the parameters in the model which can be used to determine...