A bound on the treewidth of planar even-hole-free graphs

The class of planar graphs has unbounded treewidth, since the k×k grid, ∀k∈N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.     

A graph-theoretic approach to a partial order of knots and links

We say that a link L₁ is an s-major of a link L₂ if any diagram of L₁ can be transformed into a diagram of L₂ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods.     

In situ 2-D piezoelectric wafer active sensors arrays for guided wave damage detection

This paper presented development work of an in situ method for damage detection in thin-wall structures using embedded two-dimensional ultrasonic phased arrays. Piezoelectric wafer active sensors were used to generate and receive guided Lamb waves propagating in the plate-like structure. The development of a generic beamforming algorithm that does not require parallel ray assumption through using full wave propagation paths is described. A virtual beam steering method and device, the embedded ultrasonic structural radar, was implemented as a signal post-processing procedure. Several two-dimensional configurations were investigated and compared with beamforming simulation. Finally, rectangular shape arrays were developed for verifying the generic formulas and omnidirectionality. The rectangular arrays yield good directionality within the 360° full range and are able to detect damage anywhere in the entire plate.     

Gallai’s path decomposition conjecture for triangle-free planar graphs

A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most $?(n+1)∕2?$. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai’s Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai’s Conjecture for triangle-free planar graphs.     

Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.     

Studying the Electrically Driven Switching of the Planar Light Guide

Liquid crystalline materials can be used as an active media for the new generation of planar light guides. The main characteristics which governs light-guiding and switching abilities of such devices are the spatial distributions of the refraction indices (defined via the distribution of nematic director) for the liquid crystal confined within a light-guiding pore. We aim to obtain these distributions from the molecular dynamics simulation of the liquid crystalline cell with the homeotropic boundary conditions being applied. We discuss the reorientation kinetics of the homeotropic-planar transition and obtain the equilibrium director profile upon application of the planar reorienting field.     

Existence and stability of spatially localized patterns

Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at $\lambda =0$ and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results.     

Emergence of a non-Fowler-Nordheim-type behavior for a general planar tunneling barrier

In this work we investigate a generalized tunneling barrier for planar emitters at zero-temperature. We present the evidence of the emergence of a non-Fowler-Nordheim-type general behavior for the field emission current density in the case that the Fermi energy (μ) is comparable with or smaller that the decay width ($d_F$). Therefore, for some non-metals or materials that have very small Fermi energy the standard Fowler-Nordheim-type theory may require a correction. In the opposite regime, i.e., for μ much larger that $d_F$, we confirm that the conventional theory is suitable for metals.     

Simplified fabrication of high areal capacitance all-solid-state micro-supercapacitors based on graphene and MnO2 nanosheets

A universal simplified strategy was developed to fabricate all-solid-state planar micro-supercapacitors with high areal capacitance (~355 mF/cm²), based on interdigital patterned films of 2D pseudocapacitive MnO₂ nanosheets and electrochemically exfoliated graphene.