A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Metalloestrogens: an emerging class of inorganic xenoestrogens with potential to add to the oestrogenic burden of the human breast

Many compounds in the environment have been shown capable of binding to cellular oestrogen receptors and then mimicking the actions of physiological oestrogens. The widespread origin and diversity in chemical structure of these environmental oestrogens is extensive but to date such compounds have been organic and in particular phenolic or carbon ring structures of varying structural complexity. Recent reports of the ability of certain metal ions to also bind to oestrogen receptors and to give rise to oestrogen agonist responses in vitro and in vivo has resulted in the realisation that environmental oestrogens can also be inorganic and such xenoestrogens have been termed metalloestrogens. This report highlights studies which show metalloestrogens to include aluminium, antimony, arsenite, barium, cadmium, chromium (Cr(II)), cobalt, copper, lead, mercury, nickel, selenite, tin and vanadate. The potential for these metal ions to add to the burden of aberrant oestrogen signalling within the human breast is discussed. Copyright (c) 2006 John Wiley & Sons, Ltd.

Molybdenum eta(3)-allyl dicarbonyl complexes as a new class of precursors for highly reactive epoxidation catalysts with tert-butyl hydroperoxide

New Mo(II) diimine derivatives of [Mo(q (3)allyl)X(CO)(2)(CH3CN)(2)] (allyl = C3H5 and C5H5O; X = Cl, Br) were prepared, and [MO(eta(3)-C3H5)Cl(CO)(2)(BIAN)] (BIAN = 1,4-(4-chloro)phenyl-2,3-naphthalene-diazabutadiene) (7) was structurally characterized by single-crystal X-ray diffraction. This complex adopted an equatorial-axial arrangement of the bidentate ligand (axial isomer), in contrast with the precursors, found as the equatorial isomer in the solid and fluxional in solution. The new complexes of the type [Mo(eta(3)-allyl)X(CO)(2)(N-N)l (N-N is a bidentate chelating dinitrogen ligand) were tested for the catalytic epoxidation of cyclooctene using tert-butyl hydroperoxide as oxidant. All catalytic systems were 100% selective toward epoxide formation. While their turnover frequencies paralleled those of related Mo(eta) carbonyl compounds or Mo(VI) compounds bearing similar N-donor ligands, they exhibited similar olefin conversions in consecutive catalytic runs. The acetonitrile precursors were generally more active than the diimine complexes, and the chloro derivatives more active than the bromo ones. Combined vibrational and NMR spectroscopy and computational studies (DFT) were used to investigate the nature of the molybdenum species formed in the catalytic system with [Mo(eta(3)-C3H5)Cl(CO)(2){1,4-(2,6-dimethyl)phenyl-2.3-dimethyldiazabuta diene}] (4) and to propose that the resulting species may be dimeric bearing oxide bridges.

Dendrimers: a new class of nanoscopic containers and delivery devices

Dendrimers and hyperbranched polymers are a relatively new class of materials with unique molecular architectures and dimensions in comparison to traditional linear polymers. This review details recent notable advances in the application of these new polymers in terms of the development of new polymeric delivery systems. Although comparatively young, the developing field of hyperbranched drug delivery devices is a rapidly maturing area and the key discoveries in drug-conjugate systems amongst others are highlighted. As a consequence of their ideal hyperbranched architectures, the utilisation of host-guest chemistries in dendrimers has been included within the scope of this review. (C) 2003 Elsevier Ltd. All rights reserved.

6,6 '-bis-(5,6-diethyl-[1,2,4]triazin-3-yl)-2,2 '-bipyridyl the first example of a new class of quadridentate hetercyclic extraction reagents for the separation of americium(III) and europium(III)

The synthesis of the first example of a new class of tetradentate reagents for the efficient separation of americium(Ill) and europium(111) is reported together with the structure of the complex formed with europium(III), (C) 2004 Elsevier B.V. All rights reserved.

A new class of ammonium ylid for [2,3]-sigmatropic rearrangement reactions: ene-endo-spiro ylids

The first examples of sigmatropic rearrangements of ene-endo-spirocyclic, tetrahydropyridine-derived ammonium ylids are reported. Thus, spiro[6.7]-ylids rearrange primarily by a [2,3]-pathway, whereas the analogous [6.6]-ylids rearrange by [1,2]- and [2,3]-mechanisms in roughly equal proportions. This method serves as a rapid entry to the core of a range of alkaloids bearing a pyrrolo[1,2-a]azepine or octahydroindolizidine nucleus.

Synthesis of a novel class of chiral polyaromatic amide dendrimers bearing an amino acid derived C-3-symmetric core

Chiral polyaromatic amide dendrimers incorporating a C-3-core have been prepared as potential catalysts for asymmetric reactions. (C) 2002 Elsevier Science Ltd. All rights reserved.

Diheteroarylmethyl substituted titanocenes: a novel class of possible anti-cancer drugs

From the reaction of tert-butyl lithium or n-butyl lithium with N-methylpyrrole (1a), furan (1b) or 2-bromo-thiophen (1c), 2-N-methylpyrrolyl lithium (2a), 2-furyl lithium (2b) or 2-thiophenyl lithium (2c), respectively, was obtained. When reacted with 6-(2-N-methylpyrrolyl) fulvene (3a), 6-(2-furyl) fulvene (3b) or 6-(2-thiophenyl) fulvene (3c), the corresponding lithiated intermediates were formed (4a-c). Titanocenes (5a-c) were obtained through transmetallation with titanium tetrachloride. When these titanocenes were tested against pig kidney epithelial (LLC-PK) cells, inhibitory concentrations (IC50) of 32 mu M, 140 mu M, and 240 mu M, respectively, were observed. These values represent improved cytotoxicity against LLC-PK, compared to their ansa-analogues. (c) 2006 Elsevier B.V. All rights reserved.