Exterior cloaking with active sources in two dimensional acoustics

We cloak a region from a known incident wave by surrounding the region with three or more devices that cancel out the field in the cloaked region without significantly radiating waves. Since very little waves reach scatterers within the cloaked region, the scattered field is small and the scatterers are for all practical purposes undetectable. The devices are multipolar point sources that can be determined from Green's formula and an addition theorem for Hankel functions. The cloaking devices are exterior to the cloaked region.     

[1l‐3,4‐Bis‐O‐(diphenylphosphino)‐1,2,5,6‐tetra‐O‐methyl‐chiro‐inositol‐κ2P,P′][η4‐(Z,Z)‐cycloocta‐1,5‐diene]rhodium(I) tetrafluoridoborate chloroform solvate

The title compound, [Rh(C₈H₁₂)(C₃₄H₃₈O₆P₂)]BF₄·CHCl₃, a novel asymmetric hydrogenation catalyst, crystallizes with two independent almost identical cations and anions. Cell‐packing interactions are provided by nonclassical hydrogen bonding between phenyl and chloroform H atoms and fluoro and chloro donors of the BF₄⁻ anion and the chloroform solvent molecule.     

Tin(II) aminoalkoxides and heterobimetallic derivatives: the structures of Sn6(O)4(dmae)4, Sn6(O)4(OEt)4 and [Sn(dmae)2Cd(acac)2]2

Tin(II) methoxide reacts with N,N′‐dimethylaminoethanol (dmaeH) to yield Sn(dmae)₂ ( 1 ) along with small amounts of the hydrolysis product Sn₆(O)₄(dmae)₄ ( 2 ). The geometrically more regular iso‐structural cage Sn₆(O)₄(OEt)₄ ( 3 ) was obtained as the only tractable product isolated from reaction of 2 and Sb(OEt)₃, while 1 reacted with CdX₂ (X = acac, I) to afford Sn(dmae)₂Cd(acac)₂ ( 4 ) and Sn(dmae)₂CdI₂ ( 5 ). The X‐ray structures of 2, 3 and 4 are reported. Decomposition of 4 under aerosol‐assisted chemical vapour deposition conditions leads to amorphous tin oxide films with no detectable cadmium (i.e. ca < 2% cadmium), rather than a stoichiometric Sn:Cd oxide. Copyright © 2006 John Wiley & Sons, Ltd.     

The synthesis of a pyrrole-functionalized cyclobis(paraquat-p-phenylene) derivative and its corresponding [2]rotaxane and [2]catenane and their subsequent deposition onto an electrode surface

We report the convenient synthesis of a pyrrole-functionalized tetracationic cyclophane, [2]rotaxane, and [2]catenane. X-ray crystallography has confirmed the interlocked structure of the catenane. We have investigated the solution properties of these systems using solution electrochemistry, NMR, and UV-vis spectroscopy. We have also demonstrated that it is possible to immobilize these systems onto a platinum working electrode surface. We have shown that films of the cyclophane have the ability to undergo complexation with a dialkyloxynaphthalene derivative.     

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

An orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the H^l–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the quantisation of one-dimensional bags

We discuss the quantisation of one-dimensional MIT bags by expanding the fields as a sum of classical modes and truncating the series after the first term. We obtain the lowest states of a bag in a world containing two scalar quark fields. Problems associated with the zero-point oscillations of the field are discussed.     

The simple layer potential method of fundamental solutions for certain biharmonic problems

A novel formulation of the method of fundamental solutions for the numerical solution of plane biharmonic problems, based on the simple layer potential representation of Fichera, is presented. The applicability and accuracy of the method are demonstrated by examining its performance on a set of practical problems arising in Stokes fluid flow.     

Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture

Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform. He then conjectured that the converse is true, that is, that if the field inside an inclusion is uniform for all uniform loadings, then the inclusion is of elliptic or ellipsoidal shape. We call this the weak Eshelby conjecture. In this paper we prove this conjecture in three dimensions. In two dimensions, a stronger conjecture, which we call the strong Eshelby conjecture, has been proved: if the field inside an inclusion is uniform for a single uniform loading, then the inclusion is of elliptic shape. We give an alternative proof of Eshelby’s conjecture in two dimensions using a hodographic transformation. As a consequence of the weak Eshelby’s conjecture, we prove in two and three dimensions a conjecture of Pólya and Szegö on the isoperimetric inequalities for the polarization tensors (PTs). The Pólya–Szegö conjecture asserts that the inclusion whose electrical PT has the minimal trace takes the shape of a disk or a ball.