Valence bonds in planar and quasi-planar boron disks

Planar and Quasi Planar Simultaneous Geometric Embedding

k-Quasi-Planar Graphs

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most O ( n(logn) ) edges. This upper bound was recently improved (for la...

Quasi-Planar Transmission Lines

A newly proposed and tested full-wave mixed potential mode-matching method is presented for the analysis of planar and/or quasi-planar transmission lines. The transmission lines under investigation consist of layered (stratified) and nonlayered dielectric substrates and metal strips of finite thickness. The y-directed hybrid TE and TM Hertzian potentials, perpendicular to the interfaces between...

Visualizing Halogen Bonds in Planar Supramolecular Systems

Supramolecular interactions were studied in two planar model systems, 1,5and 2,6-dibromoanthraquinones, prepared on Au(111) using scanning tunneling microscopy. In both systems, we found rigid triangular structures that consisted of simultaneous halogen bonds and hydrogen bonds, as reported in protein-ligand complexes. We proposed molecular models that were well reproduced by first-principle st...

Valence of Complex-valued Planar Harmonic Functions

The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ C. Let S denote the critical set of f and C(f) the global cluster set of f . We show that f(S)∪C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f(S) ∪ C(f) has empty interior. We also...