3,528 research outputs found

    Maximizing Maximal Angles for Plane Straight-Line Graphs

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    Let G=(S,E)G=(S, E) be a plane straight-line graph on a finite point set S⊂R2S\subset\R^2 in general position. The incident angles of a vertex p∈Sp \in S of GG are the angles between any two edges of GG that appear consecutively in the circular order of the edges incident to pp. A plane straight-line graph is called ϕ\phi-open if each vertex has an incident angle of size at least ϕ\phi. In this paper we study the following type of question: What is the maximum angle ϕ\phi such that for any finite set S⊂R2S\subset\R^2 of points in general position we can find a graph from a certain class of graphs on SS that is ϕ\phi-open? In particular, we consider the classes of triangulations, spanning trees, and paths on SS and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

    On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space R31

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    In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper’s surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space R3Ministerio de Ciencia y Tecnología MTM2004-00160Ministerio de Ciencia y Tecnología MTM2007-61775Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32

    Recognizing and Drawing IC-planar Graphs

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    IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph GG with nn vertices, we present an O(n)O(n)-time algorithm that computes a straight-line drawing of GG in quadratic area, and an O(n3)O(n^3)-time algorithm that computes a straight-line drawing of GG with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

    Quasiconvex Programming

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    We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

    Maximal surfaces in anti-de Sitter 3-manifolds with particles

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    We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than π\pi. We interpret this result in terms of Teichm\"uller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than π\pi.Comment: Accepted for publication at "Annales de l'institut Fourier
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