Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3

Generalised dimensions of measures on almost self-affine sets

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q>1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of `multienergy' integrals which we then bound using induction on families of trees

A multifractal zeta function for cookie cutter sets

Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectrum for a class of measures

The horizon problem for prevalent surfaces

We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2$. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most \alpha, for \alpha $\in$ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions

On the arithmetic sums of Cantor sets

Let C_\la and C_\ga be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions d_\la and d_\ga, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having \Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0 where C_\la + C_\ga denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form C_\la \times C_\ga. We prove that for Lebesgue almost all directions $\theta$ for which the projection is not one-to-one, the projection has zero (d_\la + d_\ga)-dimensional Hausdorff measure. We demonstrate the results on the case when C_\la and C_\ga are the middle-(1-2\la) and middle-(1-2\ga) sets

Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions

Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\nu$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a $Z_2$ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents which we show (numerically) to depend on whether the insulating state is topologically trivial or not.Comment: 12 pages, 13 figures, selected for an Editors' Suggestion in PR