6,049 research outputs found
Regular systems of paths and families of convex sets in convex position
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 . 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
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
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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