A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem

Improving convergence in smoothed particle hydrodynamics simulations without pairing instability

The numerical convergence of smoothed particle hydrodynamics (SPH) can be severely restricted by random force errors induced by particle disorder, especially in shear flows, which are ubiquitous in astrophysics. The increase in the number NH of neighbours when switching to more extended smoothing kernels at fixed resolution (using an appropriate definition for the SPH resolution scale) is insufficient to combat these errors. Consequently, trading resolution for better convergence is necessary, but for traditional smoothing kernels this option is limited by the pairing (or clumping) instability. Therefore, we investigate the suitability of the Wendland functions as smoothing kernels and compare them with the traditional B-splines. Linear stability analysis in three dimensions and test simulations demonstrate that the Wendland kernels avoid the pairing instability for all NH, despite having vanishing derivative at the origin (disproving traditional ideas about the origin of this instability; instead, we uncover a relation with the kernel Fourier transform and give an explanation in terms of the SPH density estimator). The Wendland kernels are computationally more convenient than the higher-order B-splines, allowing large NH and hence better numerical convergence (note that computational costs rise sub-linear with NH). Our analysis also shows that at low NH the quartic spline kernel with NH ~= 60 obtains much better convergence then the standard cubic spline.Comment: substantially revised version, accepted for publication in MNRAS, 15 pages, 13 figure

Unforgetting Hillsborough: researching memorialisation

This poster was presented at the 'X-Scapes: 10th Linguistic Landscapes Workshop, 02-04 May, 2018, University of Bern, Switzerland', It presented ongoing research into the memorialisation of the Hillsborough Football Tragedy (15 April, 1989) in which 96 Liverpool Football fans were unlawfully killed. It explores the notion of 'unforgetting', that is replacing a false narrative with a true account, and investighates this through and as “a nexus of trajectories shaped by the ongoing interventions of inter alia individuals, institutions, activist groups, artists, passers-by etc. producing their accounts, artefacts, transgressive emplacements and acts of unforgettting across different places, media and timescales"

Requiem banana man: banana farming in the commonwealth of Dominica

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Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH

In this paper we discuss the treatment of discontinuities in Smoothed Particle Hydrodynamics (SPH) simulations. In particular we discuss the difference between integral and differential representations of the fluid equations in an SPH context and how this relates to the formulation of dissip ative terms for the capture of shocks and other discontinuities. This has important implications for many problems, in particular related to recently highlighted problems in treating Kelvin-Helmholtz instabilities across entropy gradients in SPH. The specific problems pointed out by Agertz et al. (2007) are shown to be related in particular to the (lack of) treatment of contact discontinuities in standard SPH formulations which can be cured by the simple application of an artificial thermal conductivity term. We propose a new formulation of artificial thermal conductivity in SPH which minimises dissipation away from discontinuities and can therefore be applied quite generally in SPH calculations.Comment: 31 pages, 10 figures, submitted to J. Comp. Phys. Movies + hires version available at http://www.astro.ex.ac.uk/people/dprice/pubs/kh/ . v3: modified as per referee's comments - comparison with Ritchie & Thomas formulation added, quite a few typos fixed. No major change in metho

Smoothed Particle Magnetohydrodynamics II. Variational principles and variable smoothing length terms

In this paper we show how a Lagrangian variational principle can be used to derive the SPMHD (smoothed particle magnetohydrodynamics) equations for ideal MHD. We also consider the effect of a variable smoothing length in the SPH kernels after which we demonstrate by numerical tests that the consistent treatment of terms relating to the gradient of the smoothing length in the SPMHD equations significantly improves the accuracy of the algorithm. Our results complement those obtained in a companion paper (Price and Monaghan 2003a, paper I) for non ideal MHD where artificial dissipative terms were included to handle shocks.Comment: 14 pages, 4 figures, accepted to MNRA

A turbulence model for smoothed particle hydrodynamics

The aim of this paper is to devise a turbulence model for the particle method Smoothed Particle Hydrodynamics (SPH) which makes few assumptions, conserves linear and angular momentum, satisfies a discrete version of Kelvin's circulation theorem, and is computationally efficient. These aims are achieved. Furthermore, the results from the model are in good agreement with the experimental and computational results of Clercx and Heijst for two dimensional turbulence inside a box with no-slip walls. The model is based on a Lagrangian similar to that used for the Lagrangian averaged Navier Stokes (LANS) turbulence model, but with a different smoothed velocity. The smoothed velocity preserves the shape of the spectrum of the unsmoothed velocity, but reduces the magnitude for short length scales by an amount which depends on a parameter $\epsilon$. We call this the SPH-$\epsilon$ model. The effectiveness of the model is indicated by the fact that the second order velocity correlation function calculated using the smoothed velocity and a coarse resolution, is in good agreement with a calculation using a resolution which is finer by a factor 2, and therefore requires 8 times as much work to integrate to the same time.Comment: 34 pages, 11 figure

A Density Independent Formulation of Smoothed Particle Hydrodynamics

The standard formulation of the smoothed particle hydrodynamics (SPH) assumes that the local density distribution is differentiable. This assumption is used to derive the spatial derivatives of other quantities. However, this assumption breaks down at the contact discontinuity. At the contact discontinuity, the density of the low-density side is overestimated while that of the high-density side is underestimated. As a result, the pressure of the low (high) density side is over (under) estimated. Thus, unphysical repulsive force appears at the contact discontinuity, resulting in the effective surface tension. This tension suppresses fluid instabilities. In this paper, we present a new formulation of SPH, which does not require the differentiability of density. Instead of the mass density, we adopt the internal energy density (pressure), and its arbitrary function, which are smoothed quantities at the contact discontinuity, as the volume element used for the kernel integration. We call this new formulation density independent SPH (DISPH). It handles the contact discontinuity without numerical problems. The results of standard tests such as the shock tube, Kelvin-Helmholtz and Rayleigh-Taylor instabilities, point like explosion, and blob tests are all very favorable to DISPH. We conclude that DISPH solved most of known difficulties of the standard SPH, without introducing additional numerical diffusion or breaking the exact force symmetry or energy conservation. Our new SPH includes the formulation proposed by Ritchie & Thomas (2001) as a special case. Our formulation can be extended to handle a non-ideal gas easily.Comment: 24 pages, 21 figures. Movies and high resolution figures are available at http://v1.jmlab.jp/~saitoh/sph/index.htm

Controlling Artificial Viscosity in SPH simulations of accretion disks

We test the operation of two methods for selective application of Artificial Viscosity (AV) in SPH simulations of Keplerian Accretion Disks, using a ring spreading test to quantify effective viscosity, and a correlation coefficient technique to measure the formation of unwanted prograde alignments of particles. Neither the Balsara Switch nor Time Dependent Viscosity work effectively, as they leave AV active in areas of smooth shearing flow, and do not eliminate the accumulation of alignments of particles in the prograde direction. The effect of both switches is periodic, the periodicity dependent on radius and unaffected by the density of particles. We demonstrate that a very simple algorithm activates AV only when truly convergent flow is detected and reduces the unwanted formation of prograde alignments. The new switch works by testing whether all the neighbours of a particle are in Keplerian orbit around the same point, rather than calculating the divergence of the velocity field, which is very strongly affected by Poisson noise in the positions of the SPH particles.Comment: 8 pages, 5 figure