The Current Support Theorem in Context

This work builds up the theory surrounding a recent result of Erlandsson, Leininger, and Sadanand: the Current Support Theorem. This theorem states precisely when a hyperbolic cone metric on a surface is determined by the support of its Liouville current. To provide background for this theorem, we will cover hyperbolic geometry and hyperbolic surfaces more generally, cone surfaces, covering spaces of surfaces, the notion of an orbifold, and geodesic currents. A corollary to this theorem found in the original paper is discussed which asserts that a surface with more than $32(g-1)$ cone points must be rigid. We extend this result to the case that there are more than $3(g-1)$ cone points. An infinite family of cone surfaces which are not rigid and which have precisely $3(g-1)$ cone points is also provided, hence demonstrating tightness

A MapReduce-Based Big Spatial Data Framework for Solving the Problem of Covering a Polygon with Orthogonal Rectangles

The polygon covering problem is an important class of problems in the area of computational geometry. There are slightly different versions of this problem depending on the types of polygons to be addressed. In this paper, we focus on finding an answer to a question of whether an orthogonal rectangle, or spatial query window, is fully covered by a set of orthogonal rectangles which are in smaller sizes. This problem is encountered in many application domains including object recognition/extraction/trace, spatial analyses, topological analyses, and augmented reality applications. In many real-world applications, in the cases of using traditional central computation techniques, working with real world data results in a performance bottlenecks. The work presented in this paper proposes a high performance MapReduce-based big data framework to solve the polygon covering problem in the cases of using a spatial query window and data are represented as a set of orthogonal rectangles. Orthogonal rectangular polygons are represented in the form of minimum bounding boxes. The spatial query windows are also called as range queries. The proposed spatial big data framework is evaluated in terms of horizontal scalability. In addition, efficiency and speed-up performance metrics for the proposed two algorithms are measured

Equivalent String Networks and Uniqueness of BPS States

We analyze string networks in 7-brane configurations in IIB string theory. We introduce a complex parameter M characterizing equivalence classes of networks on a fixed 7-brane background and specifying the BPS mass of the network as M_{BPS} = | M |. We show that M can be calculated without knowing the particular representative of the BPS state. Based on detailed examination of backgrounds with three and four 7-branes we argue that equivalent networks may not be simultaneously BPS, an essential requirement of consistency.Comment: 28 pages, LaTeX, 18 eps figure