A Rectangular Area Filling Display System Architecture

A display system architecture which has rectangular area filling as its primitive operation is presented. It is shown that lines can be drawn significantly faster while rendition of filled boxes shows an O(n^2) speed improvement. Furthermore filled polygons can be rendered with an O(n) speed improvement. Implementation of this rectangular area filling architecture is discussed and refined. A custom VLSI integrated circuit is currently being designed to implement this rectangular area filling architecture and at the same time reduce the display memory system video refresh bandwidth requirements

Lenstra-Hurwitz Cliques In Real Quadratic Fields

Let $K$ be a number field and let \OO_K denote its ring of integers. We can define a graph whose vertices are the elements of \OO_K such that an edge exists between two algebraic integers if their difference is in the units \OO_K^{\times}. Lenstra showed that the existence of a sufficiently large clique (complete subgraph) will imply that the ring \OO_K is Euclidean with respect to the field norm. A recent generalization of this work tells us that if we draw more edges in the graph, then a sufficiently large clique will imply the weaker (but still very interesting) conclusion that $K$ has class number one. This thesis aims to understand this new result and produce further examples of cliques in rings of integers. Lenstra, Long, and Thistlethwaite analyzed cliques and gave us class number one through a prime element. We were able to extend and generalize their result to larger cliques through prime power elements while still preserving our desired property of class number one. Our generalization gave us that class number one is preserved if the number field $K$ contained a clique that is generated by a prime power