Other Mathematics Commons^™

Harmony Amid Chaos, Drew Schaffner

Pence-Boyce STEM Student Scholarship

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as …

Perturbation Of Burkholder's Martingale Transform And Monge-Ampère Equation, Nicholas Boros, Prabhu Janakiraman, Alexander Volberg

Faculty Scholarship – Mathematics

Given a sequence of martingale differences, Burkholder found the

sharp constant for the Lp-norm of the corresponding martingale transform. We

are able to determine the sharp Lp-norm of a small "quadratic perturbations"

of the martingale transform in Lp. By "quadratic perturbation" of the martin-

gale transform we mean the Lp norm of the square root of the squares of the

martingale transform and the original martingale (with small constant). The

problem of perturbation of martingale transform appears naturally if one wants

to estimate the linear combination of Riesz transforms (as, for example, in the

case of Ahlfors{Beurling operator).