Volume I, Issue 1
Cyclic Sieving Phenomenon
— The cyclic sieving phenomenon is an interesting phenomenon with connections to enumeration and representation theory. We will study the canonical example of multisets and present two proofs that illustrate these connections. We conclude by looking at a few other examples of the cyclic sieving phenomenon. Much of this paper is based on Sagan's survey.
Iterative Methods for Computing Eigenvalues and Eigenvectors
— We examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The five methods examined here range from the simple power iteration method to the more complicated QR iteration method. The derivations, procedure, and advantages of each method are briefly discussed.
De Broglie-Bohm and Feynman Path Integrals
— The de Broglie-Bohm theory offers what is arguably the clearest and most conceptually coherent formulation of nonrelativistic quantum mechanics known today. It not only renders entirely unnecessary all of the unresolved paradoxes at the heart of orthodox quantum theory, but moreover, it provides the simplest imaginable explanation for its entire (phenomenologically successful) mathematical formalism. All this, with only one modest requirement: the inclusion of precise particle positions as part of a complete quantum mechanical description. In this paper, we propose an alternative proof to a little known result—what we shall refer to as the de Broglie-Bohm path integral. Furthermore, we will show explicitly how the more famous Feynman path integral emerges and is, in fact, best understood as a consequence thereof.
The Wave Equation and Multi-Dimensional Time
— The classical wave equation initial value problem in single and multiple time dimensions is posed and subsequently, the physical and mathematical basis of it is discussed. The Theorem of Asgeirsson is proved and applied to study the wave equation with multiple time dimensions. Further, with the assembly of work by Courant and Hilbert, the well-posedness of such problems is determined in detail.