Preconditioners for the spectral multigrid method

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problems preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented

Minimal Bending Energies of Bilayer Polyhedra

Motivated by recent experiments on bilayer polyhedra composed of amphiphilic molecules, we study the elastic bending energies of bilayer vesicles forming polyhedral shapes. Allowing for segregation of excess amphiphiles along the ridges of polyhedra, we find that bilayer polyhedra can indeed have lower bending energies than spherical bilayer vesicles. However, our analysis also implies that, contrary to what has been suggested on the basis of experiments, the snub dodecahedron, rather than the icosahedron, generally represents the energetically favorable shape of bilayer polyhedra

An atomic clock with 10−1810^{-18} instability

Atomic clocks have been transformational in science and technology, leading to innovations such as global positioning, advanced communications, and tests of fundamental constant variation. Next-generation optical atomic clocks can extend the capability of these timekeepers, where researchers have long aspired toward measurement precision at 1 part in $\bm{10^{18}}$. This milestone will enable a second revolution of new timing applications such as relativistic geodesy, enhanced Earth- and space-based navigation and telescopy, and new tests on physics beyond the Standard Model. Here, we describe the development and operation of two optical lattice clocks, both utilizing spin-polarized, ultracold atomic ytterbium. A measurement comparing these systems demonstrates an unprecedented atomic clock instability of $\bm{1.6\times 10^{-18}}$ after only $\bm{7}$ hours of averaging

Experimental investigation of some aspects of insect-like flapping flight aerodynamics for application to micro air vehicles

Insect-like flapping flight offers a power-efficient and highly manoeuvrable basis for micro air vehicles for indoor applications. Some aspects of the aerodynamics associated with the sweeping phase of insect wing kinematics are examined by making particle image velocimetry measurements on a rotating wing immersed in a tank of seeded water. The work is motivated by the paucity of data with quantified error on insect-like flapping flight, and aims to fill this gap by providing a detailed description of the experimental setup, quantifying the uncertainties in the measurements and explaining the results. The experiments are carried out at two Reynolds numbers-500 and 15,000-accounting for scales pertaining to many insects and future flapping-wing micro air vehicles, respectively. The results from the experiments are used to describe prominent flow features, and Reynolds number-related differences are highlighted. In particular, the behaviour of the leading-edge vortex at these Reynolds numbers is studied and the presence of Kelvin-Helmholtz instability observed at the higher Reynolds number in computational fluid dynamics calculations is also verified

Field ordering and energy density in texture cosmology

We use numerical simulations of the time evolution of global textures to investigate the relationship between ordering dynamics and energy density in an expanding universe. Events in which individual textures become fully wound are rare. The energy density is dominated by the more numerous partially wound configurations, with median topological charge alpha ~ 0.44. This verifies the recent supposition (Borrill et al. 1994) that such partially wound configurations should dominate the cosmic microwave background

Possible co-existence of local itinerancy and global localization in a quasi-one-dimensional conductor

In the chain compound PrBa$_2$Cu$_4$O$_8$ localization appears simultaneously with a dimensional crossover in the electronic ground state when the scattering rate in the chains exceeds the hopping rate between the chains. Here we report the discovery of a large, transverse magnetoresistance in PrBa$_2$Cu$_4$O$_8$ in the localized regime. This result suggests a novel form of localization whereby electrons retain their metallic (quasi-one-dimensional) character over a microscopic length scale despite the fact that macroscopically, they exhibit localized (one-dimensional) behavior.Comment: 4 pages, 4 Figure