On point sets with many unit distances in few directions

We study the problem of the maximum number of unit distances among n points in the plane under the additional restriction that we count only those unit distances that occur in a xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and suciently large n n k the extremal sets are essentially sections of lattices bounded by edges parallel to the k directions and of equal lengt

Extracellular NAD and ATP: Partners in immune cell modulation

Extracellular NAD and ATP exert multiple, partially overlapping effects on immune cells. Catabolism of both nucleotides by extracellular enzymes keeps extracellular concentrations low under steady-state conditions and generates metabolites that are themselves signal transducers. ATP and its metabolites signal through purinergic P2 and P1 receptors, whereas extracellular NAD exerts its effects by serving as a substrate for ADP-ribosyltransferases (ARTs) and NAD glycohydrolases/ADPR cyclases like CD38 and CD157. Both nucleotides activate the P2X7 purinoceptor, although by different mechanisms and with different characteristics. While ATP activates P2X7 directly as a soluble ligand, activation via NAD occurs by ART-dependent ADP-ribosylation of cell surface proteins, providing an immobilised ligand. P2X7 activation by either route leads to phosphatidylserine exposure, shedding of CD62L, and ultimately to cell death. Activation by ATP requires high micromolar concentrations of nucleotide and is readily reversible, whereas NAD-dependent stimulation begins at low micromolar concentrations and is more stable. Under conditions of cell stress or inflammation, ATP and NAD are released into the extracellular space from intracellular stores by lytic and non-lytic mechanisms, and may serve as ‘danger signals–to alert the immune response to tissue damage. Since ART expression is limited to naïve/resting T cells, P2X7-mediated NAD-induced cell death (NICD) specifically targets this cell population. In inflamed tissue, NICD may inhibit bystander activation of unprimed T cells, reducing the risk of autoimmunity. In draining lymph nodes, NICD may eliminate regulatory T cells or provide space for the preferential expansion of primed cells, and thus help to augment an immune response

On Equilateral Simplices in Normed Spaces

: It is the aim of this note to improve the lower bound for the problem of Petty on the existence of equilateral simplices in normed spaces. We show that for each k there is a d(k) such that each normed space of dimension d d(k) contains k points at pairwise distance one, and that if the norm is sufficiently near to the euclidean norm, the maximal equilateral sets behave like their euclidean counterparts. 1. Introduction The question whether each d-dimensional normed space contains d+ 1 points at pairwise distance one, i.e. an equilateral simplex, was first raised by Petty in 1971 [6]. This seems obvious at first, especially in the equivalent packing version: each convex body K admits a packing (K + t i ) d+1 i=1 of d + 1 pairwise touching translates. But it turned out much more difficult, as illustrated by the following near-counterexample constructed by Petty: define a norm on IR d by fl fl fl(x 1 ; : : : ; x d ) fl fl fl : = jx 1 j + q x 2 2 + \Delta \Delta \Delta + x 2 d..

Isoperimetric Inequalities for Densities of Lattice-periodic Sets

The minimum boundary length density of a lattice-periodic set with given period-lattice and area density is determined, together with the extremal sets, and a conjecture on the higher-dimensional analogue is made. This improves previous results of Hadwiger for d-dimensional sets with integer period lattice and of Schnell and Wills for twodimensional sets with arbitrary period lattice. * Institut fur Informatik, Freie Universitat Berlin, Takustraße 9, D-14195 Berlin, Germany email: [email protected] Isoperimetric Inequalities for Densities of Lattice-periodic Sets by Peter Brass, Berlin Abstract. The minimum boundary length density of a lattice-periodic set with given period lattice and area density is determined, together with the extremal sets, and a conjecture on the higher-dimensional analogue is made. This improves previous results of Hadwiger for Z d -periodic d-dimensional sets and of Schnell and Wills on twodimensional sets with arbitrary period-lattice. 1. Introductio..

Combinatorial Geometry Problems in Pattern Recognition

In the following I consider combinatorial geometry problems motivated by point pattern matching algorithms, and discuss the classical exact matching situation and several variants

Triangles of extremal area or perimeter in a finite planar point seit

We show the following two results on a set on "n" points in the plane, tus answering questions posed by Erdős and Purdy (1971). 1\. The maximum number of triangles of maximum area (or of maximum perimeter) in a set of "n" points in the plane is exactly "n". 2\. The maximum possible number of triangles of minimum positive area in a set of "n" points in the plane is (-)(n²)

On the Number of Cylinders Touching a Ball

: In this note we prove an upper bound of seven for the maximum number of unit cylinders touching a unit ball in a packing. This improves a previous bound of eight by Heppers and Szabo. The value conjectured by Kuperberg in 1990 is six. 1. Introduction The problem to determine the maximum number of unit-radius innite cylinders touching a unit-radius ball was rst raised by W. Kuperberg [3] in 1990. This problem is analogous to the classical `kissing number': the maximum number of unit balls touching a unit ball in a packing, which is six in two dimensions and twelve in three dimensions; and one would expect that the innite cylinders essentially reduce the problem by one dimension. Packings of innite cylinders, however, have produced already some surprises, like the packing of positive density with no two cylinders parallel [4], and similarly this problem turned out to be more dicult than expected. There are several known constructions of six cylinders touching a central ball: one b..