45,304 research outputs found
On sets defining few ordinary planes
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)
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
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
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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