6,049 research outputs found

    Regular systems of paths and families of convex sets in convex position

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    In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes Toth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions f:[0,1]→Rf:[0,1] \to \mathbb{R}. On our way towards proving the Pach-Toth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdos and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest

    Dessins d'enfants and Hubbard Trees

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    We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane. We establish an explicit relationship between certain Hubbard trees and the trees known as ``dessins d'enfant'' introduced by Grothendieck.Comment: 27 pages, 8 PostScript figure

    Knowledge-based simulation using object-oriented programming

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    Simulations have become a powerful mechanism for understanding and modeling complex phenomena. Their results have had substantial impact on a broad range of decisions in the military, government, and industry. Because of this, new techniques are continually being explored and developed to make them even more useful, understandable, extendable, and efficient. One such area of research is the application of the knowledge-based methods of artificial intelligence (AI) to the computer simulation field. The goal of knowledge-based simulation is to facilitate building simulations of greatly increased power and comprehensibility by making use of deeper knowledge about the behavior of the simulated world. One technique for representing and manipulating knowledge that has been enhanced by the AI community is object-oriented programming. Using this technique, the entities of a discrete-event simulation can be viewed as objects in an object-oriented formulation. Knowledge can be factual (i.e., attributes of an entity) or behavioral (i.e., how the entity is to behave in certain circumstances). Rome Laboratory's Advanced Simulation Environment (RASE) was developed as a research vehicle to provide an enhanced simulation development environment for building more intelligent, interactive, flexible, and realistic simulations. This capability will support current and future battle management research and provide a test of the object-oriented paradigm for use in large scale military applications
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