Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

Integrable cluster dynamics of directed networks and pentagram maps

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. This paper expands on our research announcement arXiv:1110.0472Comment: 46 pages, 22 figure

Compactifying Exchange Graphs I: Annuli and Tubes

We introduce the notion of an \emph{asymptotic triangulation} of the annulus. We show that asymptotic triangulations can be mutated as the usual triangulations and describe their exchange graph. Viewing asymptotic triangulations as limits of triangulations under the action of the mapping class group, we compactify the exchange graph of the triangulations of the annulus. The cases of tubes are also considered.Comment: 14 page

Population status and habitat preferences of critically endangered Dipterocarpus littoralis in West Nusakambangan, Indonesia

The conservation of the endemic tree species Dipterocarpus littoralis (Bl.) Kurz. is hampered by the paucity of information on its population biology and ecology. Consequently, a targeted survey was carried out in the West Nusakambangan Nature Reserve to assess its population size and structure as well as habitat preferences. In total, 676 individuals of D. littoralis were located at 52 locations, with an extent of occurrence of 3.66 km2 and an area of occupancy of 1.71 km2. The population had an inverse-J-shaped distribution of diameter at breast height (DBH), with 63% of individuals in the 0-5 cm class and another 21% in the 5-10 cm class; only 11 (1.6%) mature individuals (DBH≥30) were found. D. littoralis was associated with steep, low, southwest-facing sites and sites that had high litter cover and thickness. Illegal logging and fuel-wood chopping were the main threats to D. littoralis and its habitat. In addition, an invasive shrub, Langkap (Arenga obtusifolia, Arecaceae), was a potential competitor with the seedlings throughout the reserve. In view of its endemism, narrow range and localized distribution, small population, environmental preferences, and the severe threats from anthropogenic activities and invasive species, D. littoralis appears to more than justify its conservation status of Critically Endangered