Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Analysis of the sequence and expression of a second putative acetylcholinesterase cDNA from organophosphate-susceptible and organophosphate-resistant cattle ticks

The cattle tick, Boophilus microplus, is a major pest of cattle in Australia, Central and South America, and parts of Africa and Asia. Control of ticks with organophosphates (OPs) and carbamates, which target acetylcholinesterases (AChE), led to evolution of resistance to these pesticides. Alleles at the locus studied here, AChE2, from OP-susceptible female ticks from Australia and Mexico differed at 46 of 1689 nucleotide positions (20 putative amino acid differences) whereas alleles from three strains of OP-resistant ticks from Australia differed with the allele from the Australian susceptible ticks at six to 13 nucleotide positions (three to six putative amino acid differences). However, the role, if any, of these polymorphisms in the OP-resistance phenotype is unknown. Certainly none of the polymorphisms correspond to sites in ACK that are involved in catalysis or binding of acetylcholine in other organisms. Both of the AChE loci of B. microplus, AChE1 and AChE2, are apparently expressed in synganglia; AChE1 is also expressed in salivary glands and ovaries, in OP-susceptible and OP-resistant ticks. This seems to contradict studies of enzyme kinetics, which indicated that only one form of AChE was present in the synganglia, the site of the action of OPs, in this species of tick. (C) 2002 Elsevier Science Ltd. All rights reserved.

On the overabundance of light rare earth elements in terrestrial zircons and its implication for Earth's earliest magmatic differentiation

We present whole-rock and zircon rare earth element (REE) data from two early Archaean gneisses (3.81 Ga and 3.64 Ga) from the Itsaq gneiss complex, south-west Greenland. Both gneisses represent extremely rare examples of unaltered, fresh and relatively undeformed igneous rocks of such antiquity. Cathodoluminescence imaging of their zircons indicates a single crystallisation episode with no evidence for either later metamorphic and/or anatectic reworking or inheritance of earlier grains. Uniform, single-population U/Pb age data confirm the structural simplicity of these zircons. One sample, a 3.64 Ga granodioritic gneiss from the Gothabsfjord, yields a chondrite-normalised REE pattern with a positive slope from La to Lu as well as substantial positive Ce and slight negative Eu anomalies, features generally considered to be typical of igneous zircon. In contrast, the second sample, a 3.81 Ga tonalite from south of the Isua Greenstone Belt, has variable but generally much higher light REE abundances, with similar middle to heavy REE. Calculation of zircon/melt distribution coefficients (D-REE(zircon/melt)) from each sample yields markedly different values for the trivalent REE (i.e. Ce and Eu omitted) and simple application of one set of D-REE(zircon/melt) to model the melt composition for the other sample yields concentrations that are in error by up to two orders of magnitude for the light REE (La-Nd). The observed light REE overabundance in the 3.81 Ga tonalite is a commonly observed feature in terrestrial zircons for which a number of explanations ranging from lattice strain to disequilibrium crystallisation have been proposed and are further investigated herein. Regardless of the cause of light REE overabundance, our study shows that simple application of zircon/melt distribution coefficients is not an unambiguous method for ascertaining original melt composition. In this context, recent studies that use REE data to claim that > 4.3 Ga Hadean detrital zircons originally crystallised from an evolved magma, in turn suggesting the operation of geological processes in the early Earth analogous to those of the present day (e.g. subduction and melting of hydrated oceanic crust), must be regarded with caution. Indeed, comparison of terrestrial Hadean and > 3.9 Ga lunar highland zircons shows remarkable similarities in the light REE, even though subduction processes that have been used to explain the terrestrial zircons have never operated on the Moon. (C) 2002 Elsevier Science B.V. All rights reserved.

Second primary oesophageal cancer following radiation for breast cancer

The management of 12 women who presented with a second primary oesophageal cancer following radiotherapy for breast cancer was reviewed. It was concluded that nine cases fitted the classical description of a radiation-induced malignancy. Most cases were successfully managed with combined modality therapy in spite of their previous radiotherapy. (C) 2002 Elsevier Science Ireland Ltd. All rights reserved.

Was there a second adaptive radiation of giant tortoises in the Indian Ocean? Using mitochondrial DNA to investigate speciation and biogeography of Aldabrachelys (Reptilia, Testudinidae)

A radiation of five species of giant tortoises (Cylindraspis ) existed in the southwest Indian Ocean, on the Mascarene islands, and another (of Aldabrachelys ) has been postulated on small islands north of Madagascar, from where at least eight nominal species have been named and up to five have been recently recognized. Of 37 specimens of Madagascan and small-island Aldabrachelys investigated by us, 23 yielded significant portions of a 428-base-pair (bp) fragment of mitochondrial (cytochrome b and tRNA-Glu), including type material of seven nominal species (A. arnoldi, A. dussumieri, A. hololissa, A. daudinii, A. sumierei, A. ponderosa and A. gouffei ). These and nearly all the remaining specimens, including 15 additional captive individuals sequenced previously, show little variation. Thirty-three exhibit no differences and the remainder diverge by only 1-4 bp (0.23-0.93%). This contrasts with more widely accepted tortoise species which show much greater inter- and intraspecific differences. The non-Madagascan material examined may therefore only represent a single species and all specimens may come from Aldabra where the common haplotype is known to occur. The present study provides no evidence against the Madagascan origin for Aldabra tortoises suggested by a previous molecular phylogenetic analysis, the direction of marine currents and phylogeography of other reptiles in the area. Ancient mitochondrial DNA from the extinct subfossil A. grandidieri of Madagascar differs at 25 sites (5.8%) from all other Aldabrachelys samples examined here.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.