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.     

A stereoselective synthesis of 1,2-diols from α-hydroxyaldehydes

The addition of lithium acetylides to (20R)-20-hydroxypregnane-22-carboxaldehydes in the absence and in the presence of BF₃ afforded predominantly 20R,22R- or 20R,22s-diols, respectively, characteristic of ecdysones.     

Synthesis of functionalised cyclopentanes by intramolecular radical-mediated cyclisations of terminal allenic ketones

Electrolysis of a series of terminal allenic ketones e.g. (7), (10), (17) and (21) is shown to result in intramolecular reductive cyclisation, through the $\text{exo}$-mode, producing five-membered rings, $\text{viz}$ (8), (9), (20) and (22) respectively, incorporating a bridgehead hydroxyl group. The unsaturated alcohols (8), (20) and (22), are also obtained when the allenic ketones (7), (17) and (21) are treated with sodium naphthalenide. By contrast treatment of (10) in the sodium naphthalenide gives a low yield ($\text{ca}$ 7%) of the isomeric indenol (11) by cyclisation through the $\text{endo}$-mode.Reductive cyclisations of the terminal acetylenic ketones (14) and (30) produce the corresponding unsaturated alcohols (26) and (31) respectively. Treatment of the alcohols (26) and (31) with catalytic selenium dioxide then provides the enediols (27) and (32) respectively, which make up the B/C ring systems of the novel marine natural products capnellenediol (1) and isoamijiol (2).