Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have ...

We generalize the Hadwiger theorem on line transversals to collections of compact convex sets in the plane to the case where the sets are connected and the transversals form an arrangement of pseudolines. The proof uses the embeddability of pseudoline arrangements in topological affine planes. In 1940 Santaló showed [8], by an example, that Vincensini’s proof [9] of an extension of Helly’s theo...

Let G be a graph embedded in the plane and let A be an arrangement of pseudolines intersecting the drawing of G. An aligned drawing of G and A is a planar polyline drawing Γ of G with an arrangement A of lines so that Γ and A are homeomorphic to G and A. We show that if A is stretchable and every edge e either entirely lies on a pseudoline or intersects at most one pseudoline, then G and A have...

The cover picture shows sketches of the two novel structures emphasizing manifold possibilities tetrahedral building blocks. Na10AlTaP6 and Na3GaP2 comprise discrete edge-sharing [(Al/Ta)P4] linear chains 1∞[GaP4/2] tetrahedra, respectively. overlap experimental calculated Raman spectrum show for crystallographically mixed Al/Ta site exclusively heteroatomic dimers. Thus swinging units unfold a...

A molecular-mechanics computational study is performed in order to analyze the arrangement of (1R,2S)-(-)-ephedrine molecules within the 12-MR channels of the AFI aluminophosphate microporous framework and the influence on the spatial distribution of dopants embedded in the tetrahedral network. Results showed that ephedrine molecules arrange exclusively as dimers by π-π stacking of the aromatic...

Refer to Figure 1. A monotone matching is a set of n segments, each the portion of a unique pseudoline, and each spanning a unique slab, such that the left endpoint of each segment is above the right endpoint of the segment in the previous slab. In addition, the point on the first vertical line is below the left endpoint of the first segment, and the point on the last vertical line is above the...

This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitri...

The relationship between structural order and water-like anomalies in tetrahedral liquids is still open. Here, first-principle molecular dynamics are performed to study it in metastable liquid Si. It is found that in T-P phase diagram, there indeed exists a structural anomaly region, which encloses density anomaly but not diffusivity anomaly. This is consistent with that of SW Si and BKS SiO2 b...

Spinels are compounds that have the general formula AB2O4 where A and B are metallic cations. The formal charges of A and B are strictly specifi ed and they are a criterion of the spinel type. There are three types of spinel compounds, the so-called 2-3, 4-2 and 6-1 spinels, where the numbers indicate the formal charges of the cations. The fi rst cation of each pair occupies a tetrahedral site ...