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Abstract We define an antipode of a point p as a farthest point from p . In this paper we describe, on the surface of a convex polyhedron endowed with its intrinsic metric, points admitting at least two antipodes.
Abstract We prove that every flock of a finite-dimensional locally compact connected circle plane is homeomorphic to ℝ or 1 and that every flock of a real Miquelian circle plane defines a compact 4-dimensional translation plane. Furthermore we investigate (topological) properties of...
Abstract Let B be a class of point-line geometries. Given Γ i ∈ B with subspace i for i = 1, 2, does any isomorphism Γ 1 − 1 → Γ 2 − 2 extend to a unique isomorphism Γ 1 → Γ 2 ? It is known to be true if B is the class of almost all projective spaces or the class of almost all...
Abstract The paper is a continuation of work initiated by the first two authors in (S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem. Trans. Amer. Math. Soc. 354 (2002), 4285–4301). Section 1 is introductory. In Section 2 we prove a basic lemma,...
Abstract In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen–Macaulay curves in ℙ 3 . We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen–Macaulay curves...
Abstract We study different, non-compact groups acting on unitals in compact projective planes in such a way that the geometry can be reconstructed from the group action. These unitals have many ambient automorphisms, but are not induced by polarities. In passing, we construct polarities for a...
Abstract We show that each of the Artin groups of type B n and D n can be presented as a semidirect product F ⋊ ℬ n , where F is a free group and ℬ n is the n -string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in...
Abstract An involutory automorphism of an inversive plane with exactly two fixed points is called an harmonic involution. In this paper we study inversive planes with sufficiently many harmonic involutions satisfying an analog of the theorem about the three reflections.
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