Surface networks

© Copyright CASA, UCL. The desire to understand and exploit the structure of continuous surfaces is common to researchers in a range of disciplines. Few examples of the varied surfaces forming an integral part of modern subjects include terrain, population density, surface atmospheric pressure, physico-chemical surfaces, computer graphics, and metrological surfaces. The focus of the work here is a group of data structures called Surface Networks, which abstract 2-dimensional surfaces by storing only the most important (also called fundamental, critical or surface-specific) points and lines in the surfaces. Surface networks are intelligent and “natural ” data structures because they store a surface as a framework of “surface ” elements unlike the DEM or TIN data structures. This report presents an overview of the previous works and the ideas being developed by the authors of this report. The research on surface networks has fou

Hall conductivity as bulk signature of topological transitions in superconductors

Topological superconductors may undergo transitions between phases with different topological numbers which, like the case of topological insulators, are related to the presence of gapless (Majorana) edge states. In $\mathbb{Z}$ topological insulators the charge Hall conductivity is quantized, being proportional to the number of gapless states running at the edge. In a superconductor, however, charge is not conserved and, therefore, $\sigma_{xy}$ is not quantized, even in the case of a $\mathbb{Z}$ topological superconductor. Here it is shown that while the $\sigma_{xy}$ evolves continuously between different topological phases of a $\mathbb{Z}$ topological superconductor, its derivatives display sharp features signaling the topological transitions. We consider in detail the case of a triplet superconductor with p-wave symmetry in the presence of Rashba spin-orbit (SO) coupling and externally applied Zeeman spin splitting. Generalization to the cases where the pairing vector is not aligned with that of the SO coupling is given. We generalize also to the cases where the normal system is already topologically non-trivial.Comment: 10 pages, 10 figure

A variational algorithm for the detection of line segments

In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times

Robust oscillations in SIS epidemics on adaptive networks: Coarse-graining by automated moment closure

We investigate the dynamics of an epidemiological susceptible-infected-susceptible (SIS) model on an adaptive network. This model combines epidemic spreading (dynamics on the network) with rewiring of network connections (topological evolution of the network). We propose and implement a computational approach that enables us to study the dynamics of the network directly on an emergent, coarse-grained level. The approach sidesteps the derivation of closed low-dimensional approximations. Our investigations reveal that global coupling, which enters through the awareness of the population to the disease, can result in robust large-amplitude oscillations of the state and topology of the network.Comment: revised version 6 pages, 4 figure