We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2...

Background: It was reported in a literature that approximately one third of the 4-cell stage embryos did not exhibit a tetrahedral shape. Non-tetrahedral embryos showed a lower in vitro developmental potential than tetrahedral embryos. Recently vitrification technology has been widely employed for embryo cryopreservation. The objective of this study was to prove our hypothesis that vitrified - ...

The anomalous dispersion of iron at its K-absorption edge in small-angle scattering of an aqueous solution of hemoglobin has been used to establish the geometrical arrangement of the four iron atoms in this protein. Although the anomalous contributions are 0.001-0.01 of the total scattering, experiments with synchrotron radiation from the storage ring DORIS have shown that these effects can be ...

Density maps are commonly used for data presentation, and their symmetry is a central characteristic. Quite often near symmetry can be detected but not the exact one. Here we present a method for estimating the symmetry content of density maps. We thus extend the application of the continuous symmetry measure (CSM) methodology to many-points problems. We demonstrate the applicability of the met...

To date there is a general consensus on the structure of the first coordination shells of liquid water, namely tetrahedral short range order of molecules. In contrast, little is known about the structure at longer distances and the influence of the tetrahedral molecular arrangement of the first shells on the order at these length scales. An expansion of the distance dependent excess entropy is ...

To a set of n points in the plane, one can associate a graph that has less than n2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2k−1 on the size of a planar point set for which the graph has chromatic number k, matching the bound conjectured by Szekeres for the clique number. Constructions of Erdős...

The Strong Dirac conjecture, open in some form since 1951 [5], is that every set of n points in R includes a member incident to at least n/2 − c lines spanned by the set, for some universal constant c. The less frequently stated dual of this conjecture is that every arrangement of n lines includes a line incident to at least n/2 − c vertices of the arrangement. It is known that an arrangement o...

A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppstein’s notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric sym...

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n). No known input causes our algorithm to use area Ω(n ) for any > 0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puz...