The Motion of a Point Vortex in Multiply Connected Polygonal Domains

We study the motion of a single point vortex in simply and multiply connected polygonal domains. In case of multiply connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion

Algorithmic and technical improvements: Optimal solutions to the (Generalized) Multi-Weber Problem

Rosing has recently demonstrated a new method for obtaining optimal solutions to the (Generalized) Multi-Weber Problem and proved the optimality of the results. The method develops all convex hulls and then covers the destinations with disjoint convex hulls. This paper seeks to improve implementation of the algorithm to make such solutions economically attractive. Four areas are considered: sharper decision rules to eliminate unnecessary searching, bit pattern matching as a method of recording a history and eliminating duplication, vector intrinsic functions to speed up comparisons, and profiling a program to maximize operating efficiency. Computational experience is also presented

Approximation algorithms for multi-facility location

This thesis deals with the development and implementation of efficient algorithms to obtain acceptable solutions for the location of several facilities to serve customer sites. The general version of facility location problem is known to be NP-hard; For locating multiple facilities we use Voronoi diagram of initial facility locations to partition the customer sites into k clusters. On each Voronoi region, solutions for single facility problem is obtained by using both Weizfield\u27s algorithm and Center of Gravity. The customer space is again partitioned by using the newly computed locations. This iteration is continued to obtain a better solution for multi-facility location problem. We call the resulting algorithm: Voronoi driven k-median algorithm ; We report experimental results on several test data that include randomly distributed customers and distinctly clustered customers. The observed results show that the proposed approximation algorithm produces good results

Active shape models with focus on overlapping problems applied to plant detection and soil pore analysis

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Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$