Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.     

Dynamics of Relative Phases: Generalised Multibreathers

For small Hamiltonian perturbation of a Hamiltonian systemof arbitrary number of degrees of freedom with anormally non-degenerate submanifold of periodic orbits we construct a nearbysubmanifold and an `effective Hamiltonian' on it such that the differencebetween the two Hamiltonian vector fields is small. The effective Hamiltonianis independent of one coordinate, the `overall phase', and hence thecorresponding action is preserved. Unlike standard averaging approaches,critical points of our effective Hamiltonian subject to given actioncorrespond to exact periodic solutions. We prove there has to be at least acertain number of these critical points given by global topological principles.The linearisation of the effective Hamiltonian about critical points is provedto give the linearised dynamics for the full system to leading order in theperturbation. Hence in the case of distinct eigenvalues which move at non-zerospeed with ,the linear stability type of the periodic orbit can be read offfrom the effective Hamiltonian. Our principal application is to networks ofoscillators or rotors where many such submanifolds of periodic orbits occurat the uncoupled limit – simply excite a number N 2 of the units inrational frequency ratio and put the others on equilibria, subject to anon-resonance condition. The resulting exact periodic solutions for weakcoupling are known as multibreathers. We call the approximate solutions givenby the effective Hamiltonian dynamics, `generalised multibreathers'. Theycorrespond to solutions which look periodic on a short time scale but therelative phases of the excited units may evolve slowly. Extensions aresketched to travelling breathers and energy exchange between degrees offreedom.     

Edge states at the interface between monolayer and bilayer graphene

The electronic properties for monolayer-bilayer hybrid graphene with zigzag interface are studied by both the Dirac equation and numerical calculation in zero field and in a magnetic field. Basically there are two types of zigzag interface dependent on the way of lattice stacking at the edge. Our study shows they have different locations of the localized edge states. Accordingly, the energy-momentum dispersion and local density of states behave quit differently along the interface near the Fermi energy EF=0.     

Simulation of Lamb wave reflections at plate edges using the semi-analytical finite element method

In typical Lamb wave simulation practices, effects of plate edge reflections are often not considered in order to simplify the wave signal interpretations. Methods that are based on infinite plates such as the semi-analytical finite element method is effective in simulating Lamb waves as it excludes the effect of plate edges. However, the inclusion of plate edges in a finite plate could render this method inapplicable, especially for transient response simulations. Here, by applying the ratio of Lamb mode reflections at plate edges, and representing the reflection at plate edges using infinite plate solutions, the semi-analytical finite element method can be applied for transient response simulation, even when the plate is no longer infinite.     

Selective depiction of susceptibility transitions using Laplace-filtered phase maps

In this work, we aim to demonstrate the ability of Laplace-filtered three-dimensional (3D) phase maps to selectively depict the susceptibility transitions in an object. To realize this goal, it is first shown that both the Laplace derivative of the z component of the static magnetic field in an object and the Laplacian of the corresponding phase distribution may be expected to be zero in regions of constant or linearly varying susceptibility and to be nonzero when there is an abrupt change in susceptibility, for instance, at a single point, a ridge, an interface, an edge or a boundary. Next, a method is presented by which the Laplace derivative of a 3D phase map can be directly extracted from the complex data, without the need for phase unwrapping or subtraction of a reference image. The validity of this approach and of the theory behind it is subsequently demonstrated by simulations and phantom experiments with exactly known susceptibility distributions. Finally, the potential of the Laplace derivative analysis is illustrated by simulations with a Shepp-Logan digital brain phantom and experiments with a gel phantom containing positive and negative focal susceptibility deviations.     

Sizable band gap in organometallic topological insulator

Based on first principle calculation when Ceperley–Alder and Perdew–Burke–Ernzerh type exchange-correlation energy functional were adopted to LSDA and GGA calculation, electronic properties of organometallic honeycomb lattice as a two-dimensional topological insulator was calculated. In the presence of spin–orbit interaction bulk band gap of organometallic lattice with heavy metals such as Au, Hg, Pt and Tl atoms were investigated. Our results show that the organometallic topological insulator which is made of Mercury atom shows the wide bulk band gap of about ∼120 meV. Moreover, by fitting the conduction and valence bands to the band-structure which are produced by Density Functional Theory, spin–orbit interaction parameters were extracted. Based on calculated parameters, gapless edge states within bulk insulating gap are indeed found for finite width strip of two-dimensional organometallic topological insulators.     

Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W_1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.     

On the localization of the antisymmetric flexural edge waves for obtuse angles

It is well known that the existence of edge waves is directly related to the localization of the acoustic field in the wedge. In this paper, it is shown experimentally and numerically that, for wedge angles smaller than about 100° (this angle may vary from one material to an other), the edge modes are confined in the tip of the wedge and may be considered as localized. For higher wedge angles, the analysis of the results shows a delocalization of the guided waves, which induces a new repartition of the acoustical energy in the wedge and a decrease in the amplitude of the wedge wave. This observation is numerically verified via an analysis in the time domain. Experiments realized on obtuse wedges demonstrate that the first ASF mode may be detected for wedge angles up to about 110°.     

Demonstration of the stability or instability of multibreathers at low coupling

Whereas there exists a mathematical proof for one-site breathers stability, and an unpublished one for two-site breathers, the methods for determining the stability properties of multibreathers rely on numerical computation of the Floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear Schrödinger equations. Here we present a set of multibreather stability theorems (MST) that provides a simple method to determine multibreathers stability in Klein–Gordon systems. These theorems are based in the application of degenerate perturbation theory to Aubry’s band theory. We illustrate them with several examples.