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

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

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

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

Humic substances removal by activated carbon

Humic substances are a major concern to the water treatment industry because they can cause odor and color problems and are potentially toxic and/or cancinogenic. A commercial humic acid and fulvic acid extracted from peat were used to evaluate the removal or humic substances by various activated carbons. Different analytical methods were investigated to measure the concentration of humic substances and their chloroform formation potential. Total organic carbon measurement and ultraviolet/visible absorbance correlate well with the chloroform formation potential of humic substances. Pore volumes of activated carbon in pores between 100 and 500 Å radius were correlated with adsorption capacity. A decrease of pH in solution, lower molecular weight fractions of humic substances, smaller carbon particles, and presence of soluble alum were found to increase the carbon adsorption. Chemical pretreatment with alum increased the adsorption capacity of carbon almost threefold. This increase of adsorption capacity is probably because of the removal of weakly- or non-adsorbable humic substances by alum coagulation. The presence of soluble alum in the solution also enhances the adsorbability of humic substances. The application of a developed mathematical model to predict the performance of the carbon bed for the adsorption of humic substances was successful. Model parameter estimation techniques and model verification were evaluated and the applicability of the model was verified. In general, this model is sensitive to values of adsorption capacity of activated carbon and the surface diffusion coefficient was also correlated with the adsorption capacity. The higher the adsorption capacity, the lower the diffusion coefficient. Based on the model predictions, the empty bed contact time of the carbon bed and the influent concentrations were the two most important factors in designing a carbon bed for humic substances removal.U.S. Department of the InteriorU.S. Geological SurveyOpe

Modeling of activated carbon and coal gasification char absorbents in single-solute and bisolute systems

A mathematical model of fixed-bed adsorption was used to predict the bed response to a sustained step change in influent concentration. The model was employed to compare the performance of different adsorbents in the removal of organics from water and to analyze factors that affect desorption owing to a decrease in influent concentration and to competitive adsorption. Model equations, which considered that film transfer and surface diffusion controlled the adsorption rate, were solved with the technique of orthogonal collocation. Three species, 3,5-dimethylphenol (DMP), 3,5-dichlorophenol (DCP) and rhodamine 6G (R6G), were the single solutes studied, and the two phenols were also examined as a mixture. Four activated carbons and a coal gasification char were the adsorbents studied. The model was used to compare the adsorbents in the removal of DMP, R6G and the bisolute mixtures and equilibrium capacity was found to have a greater influence than kinetics on fixed-bed performance. It was observed that, under conditions approximating a drinking water plant, the time during which the effluent concentration of a desorbed species was higher than the influent concentration was significant (on the order of weeks) whether a reduced influent concentration or competition was responsible for the desorption.U.S. Department of the InteriorU.S. Geological SurveyOpe

Improved Implementation of Point Location in General Two-Dimensional Subdivisions

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in CGAL, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph G is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(log n) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.Comment: 21 page

Spanning trees crossing few barriers

We consider the problem of finding low-cost spanning trees for sets of n points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of m barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost O(min(m2,m SQRT(n))); (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost O(m); (iii) if the barriers are disjoint convex objects, then there is always a spanning tree of cost O(n + m). All our bounds are worst-case optimal