Character Values of Stanley Sequences

Stanley and Odlyzko proposed a method for greedily constructing sets with no 3-term arithmetic progressions. It is conjectured that there is a dichotomy between such sequences: those that have a periodic structure as the sequence satisfies certain recurrence relations while others appear to be chaotic. One large class of sequences that have these periodic behaviors are known as independent sequences that have two parameters, a character and a growth factor. It was conjectured by Rolnick that all but a finite set of integers can be achieved as characters of a independent sequences. Previously the only large class of integers known to be characters where those with base 3 representations consisting solely of the digits 0 and 2. This paper dramatically improves this result by demonstrating that all even integers not congruent to 244 mod 486 can be achieved as characters, therefore demonstrating that the set of all characters has a positive lower density.Comment: Java Code for Verification Adde

Towards an integrated understanding of neural networks

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 123-136).Neural networks underpin both biological intelligence and modern Al systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees. This thesis aims towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain.by David Rolnick.Ph. D