The problem of singularity for planar grids

We consider simple graphs and their adjacency matrices. In [2], Rara (1996) gives methods of reducing graphs which simplify the procedure of computing the determinant of their adjacency matrices. We continue this subject matter and give a general method of reducing graphs. By the use of this method we define a formula for computing the determinant of any planar grid and in particular settle the problem of their singularity.     

Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

It is known that planar graphs without cycles of length 4, i, j, or 9 with 4<i<j<9, except that i=7 and j=8, are 3-choosable. This paper proves that planar graphs without cycles of length 4, 7, 8, or 9 are also 3-choosable.     

A structural theorem for planar graphs with some applications

In this note, we prove a structural theorem for planar graphs, namely that every planar graph has one of four possible configurations: (1) a vertex of degree 1, (2) intersecting triangles, (3) an edge xy with d(x)+d(y)≤9, (4) a 2-alternating cycle. Applying this theorem, new moderate results on edge choosability, total choosability, edge-partitions and linear arboricity of planar graphs are obtained.     

Interactions of amphiphilic block copolymers with lipid model membranes

Studies on interactions between amphiphilic block copolymers and lipid membranes have been focused traditionally on ABA triblock copolymers of poly(ethylene oxide)-b-poly(propylene oxide)-b-poly(ethylene oxide), widely due to their commercial availability. However, new architectures of amphiphilic block copolymer have been synthesized in recent years partially taking advantage of new polymerization techniques. This review focuses on amphiphilic block copolymers with potential biological activity and on model membrane systems used for studying interactions with such block copolymers. Experimental methods to study block copolymer–phospholipid interactions in Langmuir monolayers, liposomes, and planar bilayers are summarized. This work is intended to convey a better understanding of amphiphilic block copolymers used for in vitro and in vivo experiments in medicine and pharmacy. Recent developments and open questions are addressed.     

Bold graph drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r<w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r?w, a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time.     

Perturbations for Delaunay and weighted Delaunay 3D triangulations

The Delaunay triangulation and the weighted Delaunay triangulation are not uniquely defined when the input set is degenerate. We present a new symbolic perturbation that allows to always define these triangulations in a unique way, as soon as the points are not all coplanar. No flat tetrahedron exists in the defined triangulation. The perturbation scheme is easy to code. It is implemented in cgal, and guarantees that both vertex insertion and vertex removal are fully robust.     

不含4,5,6-圈的平面图的L(p,q)-标号

证明了若G为不含4,5,6-圈的平面图,则λp,q(G)≤(2q- 1)Δ(G)+6p+6q-6.这一结果暗含着对于△(G)≥12且不含4,5,6-圈的平面图G,x(G2)≤[3Δ(G)/2]+1成立.因此对于这样一类图部分地证实了Wegner猜想.     

Planar trivalent polygonal networks constructed from carbon nanotube Y-junctions

This work is the first step towards proving that general planar polygonal networks can be constructed as patterns of carbon nanotubes. A subset of the planar polygonal networks, namely the trivalent planar networks, are studied. These patterns can be constructed from carbon nanotubes such that every intersection point of the networks is replaced with a carbon nanotube Y-junction. Accordingly the basic task is to show the basic connections of the carbon nanotube Y-junctions. The basic set of connected Y-junctions is defined and models of the structures are shown. Nanorings with two, three or more branches are constructed from two, three or more Y-junctions such that two tubes of every Y-junction are joined to the neighbouring tubes with the third tube being free. Because of the known, exceptional electronic behaviour of carbon nanotubes, combinations of basic elements of planar nanotube networks could motivate new experimental and theoretical works having the goal of finding the basic electronic tools for nanocircuits.     

RF arbitrary waveform generation using tunable planar lightwave circuits

We demonstrate photonically-assisted generation of RF arbitrary waveforms using planar lightwave circuits (PLCs) fabricated on silica-on-silicon. We exploit thermo-optic effects in silica in order to tune the response of the PLC and hence reconfigure the generated waveform. We demonstrate the generation of pulse trains at 40 GHz and 80 GHz with flat-top, Gaussian, and apodized profiles. These results demonstrate the potential for RF arbitrary waveform generation using chip-scale photonic solutions.