45,304 research outputs found

    On sets defining few ordinary planes

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    Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than (Formula presented.) then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four.Postprint (updated version

    Extended and Reshetikhin Twists for sl(3)

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    The properties of the set {L} of extended jordanian twists for algebra sl(3) are studied. Starting from the simplest algebraic construction --- the peripheric Hopf algebra U_ P'(0,1)(sl(3)) --- we construct explicitly the complete family of extended twisted algebras {U_ E(\theta)(sl(3))} corresponding to the set of 4-dimensional Frobenius subalgebras {L(\theta)} in sl(3). It is proved that the extended twisted algebras with different values of the parameter \theta are connected by a special kind of Reshetikhin twist. We study the relations between the family {U_E(\theta)(sl(3))} and the one-dimensional set {U_DJR(\lambda)(sl(3))} produced by the standard Reshetikhin twist from the Drinfeld--Jimbo quantization U_DJ(sl(3)). These sets of deformations are in one-to-one correspondence: each element of {U_E(\theta)(sl(3))} can be obtained by a limiting procedure from the unique point in the set {U_DJR(\lambda)(sl(3))}.Comment: 14 pages, LaTeX 20

    Tropical Convexity

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    The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications to phylogenetic trees are discussed. Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to injective hulls, are incorrect. See the erratum at http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure

    Ree geometries

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    We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines
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