Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

The Effect of Package Geometry on Moisture Driven Degradation of Polymer Aluminum Capacitors

Polymer aluminum electrolytic capacitors were introduced to provide an alternative to liquid electrolytic capacitors. Polymer electrolytic capacitor electric parameters of capacitance and ESR are less temperature dependent than those of liquid aluminum electrolytic capacitors. Furthermore, the electrical conductivity of the polymer used in these capacitors (poly-3,4ethylenedioxithiophene) is orders of magnitude higher than the electrolytes used in liquid aluminum electrolytic capacitors, resulting in capacitors with much lower equivalent series resistance which are suitable for use in high ripple-current applications. The presence of the moisture-sensitive polymer PEDOT introduces concerns on the reliability of polymer aluminum capacitors in high humidity conditions. Highly accelerated stress testing (or HAST) (110ºC, 85% relative humidity) of polymer aluminum capacitors in which the parts were subjected to unbiased HAST conditions for 700 hours was done to understand the design factors that contribute to the susceptibility to degradation of a polymer aluminum electrolytic capacitor exposed to HAST conditions. A large scale study involving capacitors of different electrical ratings (2.5V – 16V, 100µF – 470 µF), mounting types (surface-mount and through-hole) and manufacturers (6 different manufacturers) was done to determine a relationship between package geometry and reliability in high temperature-humidity conditions. A Geometry-Based HAST test in which the part selection limited variations between capacitor samples to geometric differences only was done to analyze the effect of package geometry on humidity-driven degradation more closely. Raman spectroscopy, x-ray imaging, environmental scanning electron microscopy, and destructive analysis of the capacitors after HAST exposure was done to determine the failure mechanisms of polymer aluminum capacitors under high temperature-humidity conditions.

Visual Insight in Geometry

According to a traditional rationalist proposal, it is possible to attain knowledge of certain necessary truths by means of insight—an epistemic mental act that combines the 'presentational' character of perception with the a priori status usually reserved for discursive reasoning. In this dissertation, I defend the insight proposal in relation to a specific subject matter: elementary Euclidean plane geometry, as set out in Book I of Euclid's Elements. In particular, I argue that visualizations and visual experiences of diagrams allow human subjects to grasp truths of geometry by means of visual insight. In the first two chapters, I provide an initial defense of the geometrical insight proposal, drawing on a novel interpretation of Plato's Meno to motivate the view and to reply to some objections. In the remaining three chapters, I provide an account of the psychological underpinnings of geometrical insight, a task that requires considering the psychology of visual imagery alongside the details of Euclid's geometrical system. One important challenge is to explain how basic features of human visual representations can serve to ground our intuitive grasp of Euclid's postulates and other initial assumptions. A second challenge is to explain how we are able to grasp general theorems by considering diagrams that depict only special cases. I argue that both of these challenges can be met by an account that regards geometrical insight as based in visual experiences involving the combined deployment of two varieties of 'dynamic' visual imagery: one that allows the subject to visually rehearse spatial transformations of a figure's parts, and another that allows the subject to entertain alternative ways of structurally integrating the figure as a whole. It is the interplay between these two forms of dynamic imagery that enables a visual experience of a diagram, suitably animated in visual imagination, to justify belief in the propositions of Euclid’s geometry. The upshot is a novel dynamic imagery account that explains how intuitive knowledge of elementary Euclidean plane geometry can be understood as grounded in visual insight.