Solving eigenvalue problems on curved surfaces using the Closest Point Method

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach

Algorithms for Closest Point Problems: Practice and Theory

This paper describes and evaluates know sequential algorithms for constructing planar Voronoi diagrams and Delaunay triangulations. In addition, it describes a new incremental algorithm which is simple to understand and implement, but whose performance is competitive with all known methods. The experiments in this paper are more than just simple benchmarks, they evaluate the expected performance of the algorithms in a precise and machine independent fashion. Thus, the paper also illustrates how to use experimental tools to both understand the behaviour of different algorithms and to guide the algorithm design process

On the Ranking of Bilateral Bargaining Opponents

We fix the status quo (Q) and one of the bilateral bargaining agents to examine how shifting the opponent.s ideal point (type) away from Q in a unidimensional space affects the Nash and Kalai-Smorodinsky bargaining solutions when opponents differ only in their ideal points. The results are similar for both solutions. As anticipated, the bargainer whose ideal point is farthest from Q prefers a opponent whose ideal is closest to her own. A similar intuitive ranking emerges for the player closest to Q when opponent\'s preferences exhibit increasing absolute risk aversion. However, if the opponent\'s preferences exhibit decreasing absolute risk aversion (DARA), the player closest to Q prefers a more extreme opponent. This unintuitive result arises for opponents with DARA preferences because the farther their ideal point is from Q, the easier they are to satisfy.Game Theory; Nash bargaining problems; bargaining solutions, rankings

Proximity problems on line segments spanned by points

AbstractFinding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S∖{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space

Efficient Algorithms for the Closest Pair Problem and Applications

The closest pair problem (CPP) is one of the well studied and fundamental problems in computing. Given a set of points in a metric space, the problem is to identify the pair of closest points. Another closely related problem is the fixed radius nearest neighbors problem (FRNNP). Given a set of points and a radius $R$, the problem is, for every input point $p$, to identify all the other input points that are within a distance of $R$ from $p$. A naive deterministic algorithm can solve these problems in quadratic time. CPP as well as FRNNP play a vital role in computational biology, computational finance, share market analysis, weather prediction, entomology, electro cardiograph, N-body simulations, molecular simulations, etc. As a result, any improvements made in solving CPP and FRNNP will have immediate implications for the solution of numerous problems in these domains. We live in an era of big data and processing these data take large amounts of time. Speeding up data processing algorithms is thus much more essential now than ever before. In this paper we present algorithms for CPP and FRNNP that improve (in theory and/or practice) the best-known algorithms reported in the literature for CPP and FRNNP. These algorithms also improve the best-known algorithms for related applications including time series motif mining and the two locus problem in Genome Wide Association Studies (GWAS)