Planar cubic hypohamiltonian and hypotraceable graphs

Infinite families of planar cubic hypohamiltonian and hypotraceable graphs are described and these are used to prove that the maximum degree and the maximum number of edges in a hypohamiltonian graph with n vertices are approximately $\text{n}\text{2}$ and $\text{n}{}^2\text{4}$, respectively. Also, the possible order of a cubic hypohamiltonian graph is determined.     

Saturation of the all-optical Kerr effect in solids

We discuss the influence of the higher-order Kerr effect (HOKE) in wide bandgap solids at extreme intensities below the onset of optically induced damage. Using different theoretical models, we employ multiphoton absorption rates to compute the nonlinear refractive index by a Kramers-Kronig transform. Within this theoretical framework we provide an estimate for the appearance of significant deviations from the standard optical Kerr effect predicting a linear index change with intensity. We discuss the role of the observed saturation behavior in practically relevant situations, including Kerr lens mode-locking and supercontinuum generation in photonic crystal fibers. Furthermore, we present experimental data from a multiwave mixing experiment in BaF2, which can be explained by the appearance of the HOKE.     

Planar and infinite hypohamiltonian and hypotraceable graphs

Chvátal raised the question whether or not planar hypohamiltonian graphs exist and Grünbaum conjectured the nonexistence of such graphs. We shall describe an infinite class of planar hypohamiltonian graphs and infinite classes of planar hypotraceable graphs of connectivity two (resp. three). Infinite hypohamiltonian (resp. htpotraceable) graphs are also described. It is shown how the study of infinite hypotraceable graphs leads to a new infinite family of finite hypotraceable graphs.     

Staggered flux vortices and the superconducting transition in the layered cuprates

We propose an effective model for the superconducting transition in the high-T(c) cuprates motivated by the SU(2) gauge theory approach. In addition to variations of the superconducting phase we allow for local admixture of staggered flux order. This leads to an unbinding transition of vortices with a staggered flux core that are energetically preferable to conventional vortices. Based on parameter estimates for the two-dimensional t-J model we argue that the staggered flux vortices provide a way to understand a phase with a moderate density of mobile vortices over a large temperature range above T(c) that yet exhibits otherwise normal transport properties. This picture is consistent with the large Nernst signal observed in this region.     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.