Summary

My thesis developed a two-parameter Melnikov framework for nearly integrable Hamiltonian systems with both fast periodic forcing and slow drift. The work decomposes the Melnikov function into a fast component measuring invariant-manifold splitting and a slow component measuring accumulated drift in the slow variable. Using averaging theory, I showed that the slow-drift term is phase-independent to leading order, giving a rigorous first-order separation between fast oscillations and slow modulation. The thesis also connects periodic and homoclinic Melnikov theory by studying the large-period limit and clarifies the O(ϵ)-neighborhood in which the perturbative approximation remains valid. A key outcome is that the perturbed unstable manifold reconnects with the deformed family of saddle points, laying the groundwork for a full homoclinic existence criterion. You can download the full paper here.

Abstract

The way all things in the world have their building blocks the same goes for groups. In this paper we will be talking about finite simple groups, the building blocks of all finite groups. Akin to the way prime numbers are the building blocks of natural numbers, we will see the parallel runs very deep when it comes to finite simple groups and finite groups. The classification of all finite groups to this day remains the longest ever proof in mathematics, approximately 15,000 pages in length. We will be focusing on the specific proof of the simplicity of PSL2(F), the induced action of the special linear group on the associated projective space. It should be noted that the group is simple if and only if the degree of the field F is larger than 3. We will only concern ourselves with the case of the field F having an order 4; larger orders go beyond the scope of this paper. You can download the full paper here.

Abstract

In this paper we will be focusing on proving a case of measure of concentration on the sphere. The proof relies on Euclidean geometry as well as a corollary of the Brunn-Minkowski Theorem along with utilizing the Isoperimetry Inequality for the sphere. The former is employed for proving the case when the measurable subset is a spherical cap while the latter is used for proving the general case when the measurable subset is arbitrarily chosen. You can download the full paper here.

Abstract

In this paper I explore mass hysteria as a social contagion phenomenon utilizing Social Identification Theory (SIT) and Social Cognitive Theory (SCT) as the theoretical framework. SIT is viewed as a plausible explanation for the spread of mass hysteria on group basis, while SCT offers useful insights for the spread of mass hysteria behavior on individual basis. I analyze two cases: he June Bug Epidemic of 1962 and the Tanganyika Laughter Epidemic in order to examine the explanatory power of the proposed theories. You can download the full paper here.