On (αn)(\alpha_n)-regular sets

We define $(\alpha_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and in particular give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold in the real-line.Comment: 18 pages, 2 figure

A Subclass of Exceptional Parallel Self-similar G2 Cosmologies

We perform a qualitative and asymptotic analysis of a particular class of cosmological models, namely the exceptional G2 perfect fluid and vacuum models that are additionally self-similar with the fluid flow lying tangential to the H3 orbits. We show that for the values of the equation of state parameter in (1,3/2), there exist open sets of well-behaved vacuum models that are asymptotically spatially homogeneous, at large spatial distances. For the values of the equation of state parameter in the intervals (1,10/9) and (4/3,3/2), there exist open sets of well-behaved perfect fluid inhomogeneous cosmological models that are asymptotically spatially homogeneous, at large spatial distances, and we illustrate the spatial structure of their matter-energy density. In addition, the perfect fluid models exhibit only two possible asymptotic behaviours, namely they are well-behaved and asymptotically spatially homogeneous or badly-behaved

A Characterization of Signed Graphs with Generalized Perfect Elimination Orderings

An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs.Comment: 18 pages; (v2) Reference updated (v3) Major update including title change, shortening of proof of main theorem, addition of applications of main theorem to special cases, reference updat

Efficient simulations with electronic open boundaries

We present a reformulation of the Hairy Probe method for introducing electronic open boundaries that is appropriate for steady state calculations involving non-orthogonal atomic basis sets. As a check on the correctness of the method we investigate a perfect atomic wire of Cu atoms, and a perfect non-orthogonal chain of H atoms. For both atom chains we find that the conductance has a value of exactly one quantum unit, and that this is rather insensitive to the strength of coupling of the probes to the system, provided values of the coupling are of the same order as the mean inter-level spacing of the system without probes. For the Cu atom chain we find in addition that away from the regions with probes attached, the potential in the wire is uniform, while within them it follows a predicted exponential variation with position. We then apply the method to an initial investigation of the suitability of graphene as a contact material for molecular electronics. We perform calculations on a carbon nanoribbon to determine the correct coupling strength of the probes to the graphene, and obtain a conductance of about two quantum units corresponding to two bands crossing the Fermi surface. We then compute the current through a benzene molecule attached to two graphene contacts and find only a very weak current because of the disruption of the π-conjugation by the covalent bond between the benzene and the graphene. In all cases we find that very strong or weak probe couplings suppress the current