Aspects of quark mass generation on a torus

In this talk we report on recent results for the quark propagator on a compact manifold. The corresponding Dyson-Schwinger equations on a torus are solved on volumes similar to the ones used in lattice calculations. The quark-gluon interaction is fixed such that the lattice results are reproduced. We discuss both the effects in the infinite volume/continuum limit as well as effects when the volume is small.Comment: 3 pages, 3 figures; talk given by CF at QNP06, Madrid, June 200

Finite volume effects in a quenched lattice-QCD quark propagator

We investigate finite volume effects in the pattern of chiral symmetry breaking. To this end we employ a formulation of the Schwinger-Dyson equations on a torus which reproduces results from the corresponding lattice simulations of staggered quarks and from the overlap action. Studying the volume dependence of the quark propagator we find quantitative differences with the infinite volume result at small momenta and small quark masses. We estimate the minimal box length L below which chiral perturbation theory cannot be applied to be L \simeq 1.6 fm. In the infinite volume limit we find a chiral condensate of ||_{\bar{MS}}^{2 GeV} = (253 \pm 5.0 MeV)^3, an up/down quark mass of m_{\bar{MS}}^{2 GeV} = 4.1 \pm 0.3 MeV and a pion decay constant which is only ten percent smaller than the experimental value.Comment: 19 pages, 8 figures. v2: minor clarifications added, version published in PR

Quark Condensates: Flavour Dependence

We determine the q-bar q condensate for quark masses from zero up to that of the strange quark within a phenomenologically successful modelling of continuum QCD by solving the quark Schwinger-Dyson equation. The existence of multiple solutions to this equation is the key to an accurate and reliable extraction of this condensate using the operator product expansion. We explain why alternative definitions fail to give the physical condensate.Comment: 9 pages, 7 figures, uses appolb.cls, LaTeX. Talk presented by R. Williams at the EURIDICE Final Meeting, August 24-27th, 2006, Kazimierz, Polan

An Introduction to Conformal Ricci Flow

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur

Rayleigh-Benard Convection with a Radial Ramp in Plate Separation

Pattern formation in Rayleigh-Benard convection in a large-aspect-ratio cylinder with a radial ramp in the plate separation is studied analytically and numerically by performing numerical simulations of the Boussinesq equations. A horizontal mean flow and a vertical large scale counterflow are quantified and used to understand the pattern wavenumber. Our results suggest that the mean flow, generated by amplitude gradients, plays an important role in the roll compression observed as the control parameter is increased. Near threshold the mean flow has a quadrupole dependence with a single vortex in each quadrant while away from threshold the mean flow exhibits an octupole dependence with a counter-rotating pair of vortices in each quadrant. This is confirmed analytically using the amplitude equation and Cross-Newell mean flow equation. By performing numerical experiments the large scale counterflow is also found to aid in the roll compression away from threshold but to a much lesser degree. Our results yield an understanding of the pattern wavenumbers observed in experiment away from threshold and suggest that near threshold the mean flow and large scale counterflow are not responsible for the observed shift to smaller than critical wavenumbers.Comment: 10 pages, 13 figure

Speed Limits in General Relativity

Some standard results on the initial value problem of general relativity in matter are reviewed. These results are applied first to show that in a well defined sense, finite perturbations in the gravitational field travel no faster than light, and second to show that it is impossible to construct a warp drive as considered by Alcubierre (1994) in the absence of exotic matter.Comment: 7 pages; AMS-LaTeX; accepted for publication by Classical and Quantum Gravit

Red blood cells and other non-spherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition

We consider the motion of red blood cells and other non-spherical microcapsules dilutely suspended in a simple shear flow. Our analysis indicates that depending on the viscosity, membrane elasticity, geometry and shear rate, the particle exhibits either tumbling, tank-treading of the membrane about the viscous interior with periodic oscillations of the orientation angle, or intermittent behavior in which the two modes occur alternately. For red blood cells, we compute the complete phase diagram and identify a novel tank-treading-to-tumbling transition at low shear rates. Observations of such motions coupled with our theoretical framework may provide a sensitive means of assessing capsule properties.Comment: 11 pages, 4 figure

A general review of concepts for reducing skin friction, including recommendations for future studies

Four main concepts which have significantly reduced skin friction in experimental studies are discussed; suction, gaseous injection, particle additives, and compliant wall. It is considered possible that each of these concepts could be developed and applied in viable skin friction reduction systems for aircraft application. Problem areas with each concept are discussed, and recommendations for future studies are made