Visibility Graphs, Dismantlability, and the Cops and Robbers Game

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.Comment: 23 page

A 2-chain can interlock with a k-chain

One of the open problems posed in [3] is: what is the minimal number k such that an open, flexible k-chain can interlock with a flexible 2-chain? In this paper, we establish the assumption behind this problem, that there is indeed some k that achieves interlocking. We prove that a flexible 2-chain can interlock with a flexible, open 16-chain.Comment: 10 pages, 6 figure

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular face

Computing contour trees in all dimensions

AbstractWe show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al

Determination of the fate of polynuclear aromatic hydrocarbons in natural water systems

The polynuclear aromatic hydrocarbons, or PAH, are of current concern as water pollutants and potential health hazards. The presence of PAH in natural water systems was evaluated and an analytical technique for specific PAH was developed. It was found that the PAH are not soluble in water but they either are present as particulate material or as material adsorbed on solid surfaces in natural water systems. The photodecomposition of two PAH, 1,2 benzanthracene, or BA, and 3,4 benzpyrene, or BP, was examined. Both compounds decompose under ultraviolet light to form their quinones, which then further decompose. Both BP and BA decompose following first order kinetics in true solution in 20 percent acetone in water. Particulate BA also decomposes following first order reaction kinetics, a1 though particulate BP will decompose only to a depth of 0.2 pm before decomposition stops. This decomposition is relatively unaffected by water chemistry and will occur under solar radiation and in turbid waters.U.S. Department of the InteriorU.S. Geological SurveyOpe

Visibility graphs, dismantlability, and the cops and robbers game

We study versions of cop and robber pursuit–evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are , which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit–evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs

Separations in Solutions Based on Electric Field Induced Dielectric Polarization of Molecules

The electric field dielectric polarization-based separations mechanism represents a novel method for separating solutions at small length scales that is notably efficient. An applied electric field as low as $\mathrm{0.4~MV/m}$ across a $\mathrm{10~\mu m}$ channel increases the concentration inside the low electric field region by $\approx \mathrm{40}\%$ relative to the high electric field region. This concentration change is two orders of magnitude higher than the estimated change predicted using the classical equilibrium thermodynamics for the same electric field. The deviation between the predicted and the experimental results suggests that the change in volumetric electric field energy with solute concentration is insufficient to explain this phenomenon. The study also explores the effect of varying strength of electric field and frequency of supplied voltage on the polarization-based separation efficiency. While the increase in the former increases the separation efficiency, the increase in the latter reduces the degree of concentration change due to ineffective charging.Comment: Submitted to Langmui

Reduction of aqueous free chlorine with granular activated carbon

A surface reaction rate expression was developed to describe the heterogeneous reaction between aqueous free chlorine and granular activated carbon. This expression was then incorporated into a pore diffusion model and the relevant partial differential equations with the corresponding boundary conditions were solved for the case of (1) a constant concentration batch reactor and (2) a closed batch reactor. The solutions were then compared to similar batch data in order to evaluate the pore model constants. A packed bed reactor model was then solved using the rate information from the batch mathematical models and experimental data. The predicted results from the packed bed model were then compared to experimental results which were collected using applicable conditions. The effect of particle size on the rate of removal of free chlorine was investigated both in batch and packed bed column form. The effect of pH on the rate of reaction was studied in the pH range of 4-10. It was found that the pH only affects the rate insofar as it affects the distribution of free chlorine between OCl¯ and HOCl. Temperature effects were also studied in the range 2°-35°C. The effect of temperature on the surface dissociation rate constant was found to correspond to the Arrhenius law.U.S. Department of the InteriorU.S. Geological SurveyOpe