A checkerboard constraint is a bounded measurable set 2, containing the origin. A binary labeling of the 2 lattice satisfies the checkerboard constraint if whenever 2 is labeled 1, all of the other 2-lattice points in the translate + are labeled 0. Two-dimensional channels that only allow labelings of 2 satisfying checkerboard constraints are studied. Let ( ) be the area of , and let ( ) mean t...

Phase 1, with electrical conductivity σ1, occupies the “bowtie” region of angle ±α, while phase 2, with conductivity σ2 σ1, occupies the remaining region. Deduce the approximate form of lines of current density J when a background electric field is applied along the symmetry axis of phase 1. What is the effective conductivity σ of the system, defined by the relation I = σΔφ between the total cu...

Since the introduction of the P300 brain-computer interface (BCI) speller by Farwell and Donchin in 1988, the speed and accuracy of the system has been significantly improved. Larger electrode montages and various signal processing techniques are responsible for most of the improvement in performance. New presentation paradigms have also led to improvements in bit rate and accuracy (e.g. Townse...

Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize chec...

Scanning tunneling microscopy (STM) studies of the high-T superconductors have led to a number of important discoveries. In particular, STM has revealed spatial patterns in electronic density due to phenomena such as checkerboard order and quasiparticle interference. This thesis presents two studies of these patterns and their implications. In the first, I present a doping and temperature depen...