Bob Piziak and Megan Schneck
Megan Schneck, a current graduate student in our program, was a University Scholar while an undergraduate at Baylor. Even so, she was essentially one of our mathematics majors and took the usual B.S. courses including the Matrix Theory course. She came to me to discuss the requirement of the University Scholar's program to write a thesis. She was interested in continuing her study of matrix theory. I was working on Hermitian matrices at the time to I suggested that as a topic. She agreed. Her thesis looked at a variety of aspects of Hermitian matrices including their trace properties, their purely algebraic properties, the connection with the Gram matrix (where she noticed that the Cauchy-Schwarz inequality could be extended to more than two vectors), and eigenvalue properties including Rayleigh-Ritz, Courant-Fischer and Cauchy Interlacing Theorems.
Tim Sheng and Myles Baker
Under the guidance of Professor Tim Sheng, Mr. Myles Baker has been working on an exploration of optimized finite difference approximations of two-dimensional Black- Scholes differential equations on nonuniform grids. The differential equation, under proper initial and boundary conditions, is fundamental to modern financial and banking industries and has numerous important applications in the national economy.
It is not easy to compute numerical solutions of a multidimensional Black-Scholes equation, especially when the investment environments are turbulent, or market prices change too quickly. Highly reliable adaptive methods are often required and nonuniform grids become essential. With the support in computer simulations from a local high school team, Dr. Sheng and Mr. Baker will extend the diamond-alpha dynamic derivative theory and strategy to the challenging computational issues. Highly accurate, reliable and applicable algorithms are anticipated. Rigorous numerical analysis will be implemented.