On a class of preconditioners for solving the Helmholtz equation

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−k²φ with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

Some non‐standard problems in viscous flow

Energy bounds are derived for Dirichlet type boundary value problems for the Navier–Stokes and Stokes equations when a combination of the solution values initially and at a later time is prescribed. The bounds are obtained by means of a differential inequality and imply uniqueness and continuous data dependence of the solutions for a range of values of the parameter in the non‐standard auxiliary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Non-protein amino acids in peptide design

An overview of the use of non-protein amino acids in the design of conformationally well-defined peptides, based on work from the author’s laboratory, is discussed. The crystal structures of several designed oligopeptides illustrate the useα-aminoisobutyric acid (Aib) in the construction of helices, D-amino acids in the design of helix termination segments and^DPro-Xxx segments for nucleating ofβ-hairpin structures.β- andγ-amino acid residues have been used to expand the range of designed polypeptide structures. Dedicated to Professor C N R Rao on his 70th birthday     

Electron transfer induced dissociations of 2- and 4-alkyl cyclohexadienones

Several 2- and 4-alkylcyclohexadienones were prepared and shown to accept electrons to produce ketyl radical anions that dissociated rapidly at room temperature to release carbon-centered radicals and an aromatic phenoxide type anion. In the PET process with benzyl-substituted cyclohexadienones, initiated with triethylamine, the benzyl radicals dimerised or abstracted an H-atom from solvent. In electrochemical reductions, and in reductions with alkali metals in liquid ammonia, the benzyl radicals were further reduced to anions.     

Microwave-accelerated conjugate addition of aldehydes to α,β-unsaturated ketones

Aldehydes undergo smooth conjugate addition to α,β-unsaturated ketones in the presence of 5-(2-hydroxyethyl)-1,3-thiazolium halides and DBU adsorbed onto the surface of basic alumina under microwave irradiation and solvent-free conditions to afford 1,4-diketones in enhanced yields and reduced reaction times compared to conventional methods.     

Addition of Bromotrichloromethane and Tetrachloromethane to cis-Cyclooctene,Cyclohexene, and Norbornadiene in the Presence of Palladium(II) Complexes

Addition of bromotrichloromethane and tetrachloromethane to alkenes in the presence of palladium(II) complexes begins with single-electron transfer from the palladium atom to polyhalomethane molecule, followed by fragmentation of the radical anion thus formed and generation of trichloromethyl radical. Halogen transfer to a carbon-centered radical arising from addition of trichloromethyl radical at the double bond occurs both from the polyhalomethane molecule and from halide palladium complex.     

Synthesis and Structure of Mixed Phosphonium-iodonium Ylide

A mixed phosphonium-iodonium ylide, phenyliodoniumethoxycarbonylmethylenetriphenylphosphorane borofluoride, was synthesized. Its structure was established by means of X-ray diffraction analysis. Temperature dependence of ¹H, ¹³C, and ³¹P spectra of the ylide synthesized was investigated. A dynamic equilibrium between Z and E-isomers was observed.