We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of n points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Ha...

The use of null hypotheses (in a statistical sense) is common in hard sciences but not in theoretical linguistics. Here the null hypothesis that the low frequency of syntactic dependency crossings is expected by an arbitrary ordering of words is rejected. It is shown that this would require star dependency structures, which are both unrealistic and too restrictive. The hypothesis of the limited...

In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n) time and O(n) ...

Given a set of k-colored points in the plane, we consider the problem of finding k trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k = 1, this is the well-known Euclidean Steiner tree problem. For general k, a kρ-approximation algorithm is known, where ρ ≤ 1.21 is the Steiner ratio. We present a PTAS ...

We analyze extremal statistics in non-crossing configurations on the n vertices of a convex polygon.We prove that themaximumdegree and the largest component are of logarithmic order, and that, suitably scaled, they converge to a well-defined constant. We also prove that the diameter is of order √ n. The proofs are based on singularity analysis, an application of the first and second moment meth...

By the application of an ultra fast magnetic field sweeping (2.0 × 105 T/sec) and the zero-field magnetic resonance (ZFMR)/FID techniques, we are able to probe the dynamics at the level anti-crossing region of the photo-excited triplet states of pentacene-h14 in p-terphenyl (PHPT) and pentacene-d14 in p-terphenyl (PDPT) systems. The crossing is a non-adiabatic passage at such a high jumping rat...