Tight upper bound on the maximum anti-forcing numbers of graphs

Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.Comment: 15 pages, 7 figure

Numerical simulation of Faraday waves

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier-Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The domain of calculation is periodic in the horizontal directions and bounded in the vertical direction by two rigid horizontal plates. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar and Tuckerman [J. Fluid Mech. 279, 49-68 (1994)]. The finite amplitude results are compared with the experiments of Kityk et al. [Phys. Rev. E 72, 036209 (2005)]. In particular we reproduce the detailed spatiotemporal spectrum of both square and hexagonal patterns within experimental uncertainty

Wave mechanics in media pinned at Bravais lattice points

The propagation of waves through microstructured media with periodically arranged inclusions has applications in many areas of physics and engineering, stretching from photonic crystals through to seismic metamaterials. In the high-frequency regime, modelling such behaviour is complicated by multiple scattering of the resulting short waves between the inclusions. Our aim is to develop an asymptotic theory for modelling systems with arbitrarily-shaped inclusions located on general Bravais lattices. We then consider the limit of point-like inclusions, the advantage being that exact solutions can be obtained using Fourier methods, and go on to derive effective medium equations using asymptotic analysis. This approach allows us to explore the underlying reasons for dynamic anisotropy, localisation of waves, and other properties typical of such systems, and in particular their dependence upon geometry. Solutions of the effective medium equations are compared with the exact solutions, shedding further light on the underlying physics. We focus on examples that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick up detailed qualitative and quantitative features

An aperiodic hexagonal tile

We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory Series A. Version 2 is a major revision. Parts of Version 1 have been expanded and parts have been moved to a separate article (arXiv:1003.4279