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COUNTEREXAMPLES TO THE COLORFUL TVERBERG CONJECTURE FOR HYPERPLANES
Karasev [16] conjectured that for every set of r blue lines, r green lines, and r red lines in the plane, there exists a partition of them into r ...
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A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions
An m -free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m -free imply ( m + 1)-free...
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Calculating the Fundamental Group of Galois Cover of the (2,3)-embedding of ℂℙ1 × T
As known to all, it is quite difficult to compute the fundamental group of a surface of general type. In this paper, applying Moishezon-Teicher’s...
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Counting chambers in restricted Coxeter arrangements
Solomon showed that the Poincaré polynomial of a Coxeter group W satisfies a product decomposition depending on the exponents of W . This polynomial...
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The axiomatization of affine oriented matroids reassessed
In an unpublished manuscript of 1992, Johan Karlander has given an axiomatization of affine oriented matroids, which can be thought of as oriented...
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A generalisation of Sylvester’s problem to higher dimensions
In this article we consider S to be a set of points in d -space with the property that any d points of S span a hyperplane and not all the points of S ...
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Inductively Free Multiderivations of Braid Arrangements
The reflection arrangement of a Coxeter group is a well-known instance of a free hyperplane arrangement. In
2002 , Terao showed that equipped with a... -
Optimal configurations of lines and a statistical application
Motivated by the construction of confidence intervals in statistics, we study optimal configurations of 2 d − 1 lines in real projective space ℝℙ ...
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Rational realization of the minimum ranks of nonnegative sign pattern matrices
A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum...
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On the angle sum of lines
What is the maximum of the sum of the pairwise (non-obtuse) angles formed by n lines in the Euclidean 3-space? This question was posed by Fejes Tóth...
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Another Ham Sandwich in the Plane
We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from any k -fan when k is odd and in...
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The Szemerédi-Trotter theorem in the complex plane
It is shown that n points and e lines in the complex Euclidean plane ℂ 2 determine O ( n 2/3 e 2/3 + n + e ) point-line incidences. This bound is the...