On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

The effect of calcium removal from milk on casein micelle stability and structure

The total calcium level of raw skimmed milk was reduced by 10, 19, 29, 40 and 51% using Duolite® ion-exchange resin. The products were examined for concentrations of ionic calcium, sodium and potassium and the pH, ethanol stability, micelle diameter and ζ-potential were also measured. Ionic calcium decreased with removal of calcium and pH increased. Calcium removal resulted in an increase in the ethanol stability from 88% to above 100%. Casein micelle diameter increased as calcium was removed. The ζ-potential of the skimmed bulk milk was -24.4 mV, gradually becoming more negative with calcium removal to -30.6 mV after 51% calcium removal. The milk became more translucent as calcium was removed. To investigate the reversibility of this process, calcium chloride was added back to the depleted samples to restore their original total calcium content. At 51% removal, restoration of the total calcium level resulted in formation of clots. At levels of 10 and 19% calcium removal, the ethanol stability remained above 100%, but at higher levels of calcium removal the alcohol stability was adversely affected when the calcium was added back. Adding back calcium resulted in partial restoration of the original casein micelle diameter.

Reactions of artemisinin and arteether with acid: Implications for stability and mode of antimalarial action

The currently accepted mechanism of trioxane antimalarial action involves generation of free radicals within or near susceptible sites probably arising from the production of distonic radical anions. An alternative mechanistic proposal involving the ionic scission of the peroxide group and consequent generation of a carbocation at C-4 has been suggested to account for antimalarial activity. We have investigated this latter mechanism using DFT (B3LYP/6-31+G* level) and established the preferred Lewis acid protonation sites (artemisinin O5a >> O4a approximate to O3a > O2a > O1a; arteether O4a >= O3a > O5b >> O2a > O1a; Figure 3) and the consequent decomposition pathways and hydrolysis sites. In neither molecule is protonation likely to occur on the peroxide bond O1-O2 and therefore lead to scission. Therefore, the alternative radical pathway remains the likeliest explanation for antimalarial action.

Intermediate ion stability and regio selectivity polymerization using neutral salicyladiminato in propene nickel(II) and palladium(II) complexes as catalysts

The mechanism of the Heck reaction has been studied with regard to transition metal catalysis of the addition of propene and the formation of unsaturated polymers. The reactivity of nickel and palladium complexes with five different bidentate ligands with O,N donor atoms has been investigated by computational methods involving density functional theory. Hence, it is possible to understand the electronic and steric factors affecting the reaction and their relative importance in determining the products formed in regard of their control of the regiochemistry of the products. Our results show that whether the initial addition of propene is trans to O or to N of the bidentate ligand is of crucial importance to the subsequent reactions. Thus when the propene is trans to 0, 1,2-insertion is favoured, but when the propene is trans to N, then 2,1-insertion is favoured. This difference in the preferred insertion pathway can be related to the charge distribution engendered in the propene moiety when the complex is formed. Indeed charge effects are important for catalytic activity but also for regioselectivity. Steric effects are shown to be of lesser importance even when t-butyl is introduced into the bidentate ligand as a substituent. (C) 2007 Elsevier B.V. All rights reserved.

The effect of free Ca2+ on the heat stability and other characteristics of low-heat skim milk powder

Low-heat skim milk powder (SMP), reconstituted to 25% total solids, was found to have poor heat stability. This could be improved by reducing the free Ca2+ concentration to 1.14 mm, or lower, by the addition of either Amberlite IR-120 ion-exchange resin in its sodium form or tri-sodium citrate in skim milk prior to evaporation and spray drying. Reduction in Ca2+ concentration was accompanied by increases in pH, particle size, and kinematic viscosity, and by a reduction in zeta-potential and changes in colour. In-container sterilisation of the reconstituted powder increased particle size, zeta-potential, kinematic viscosity and a* and b* values. However. Ca2+ concentration, pH and whiteness decreased. This study elucidated the importance of Ca2+ concentration and pH on heat stability of low-heat SMP, suggesting that Ca2+ concentration and pH in bulk milk are useful indicators for ensuring that spray dried milk powder has good heat stability. (C) 2009 Elsevier Ltd. All rights reserved.

Effects of chronic consumption of fruit and vegetable puree-based drinks on vasodilation, plasma oxidative stability and antioxidant status

Background: Fruit and vegetable-rich diets are associated with a reduced cardiovascular disease (CVD) risk. This protective effect may be a result of the phytochemicals present within fruits and vegetables (F&V). However, there can be considerable variation in the content of phytochemical composition of whole F&V depending on growing location, cultivar, season and agricultural practices, etc. Therefore, the present study investigated the effects of consuming fruits and vegetables as puree-based drinks (FVPD) daily on vasodilation, phytochemical bioavailability, antioxidant status and other CVD risk factors. FVPD was chosen to provide a standardised source of F&V material that could be delivered from the same batch to all subjects during each treatment arm of the study. Methods: Thirty-nine subjects completed the randomised, controlled, cross-over dietary intervention. Subjects were randomised to consume 200 mL of FVPD (or fruit-flavoured control), daily for 6 weeks with an 8-week washout period between treatments. Dietary intake was measured using two 5-day diet records during each cross-over arm of the study. Blood and urine samples were collected before and after each intervention and vasodilation assessed in 19 subjects using laser Doppler imaging with iontophoresis. Results: FVPD significantly increased dietary vitamin C and carotenoids (P < 0.001), and concomitantly increased plasma α- and β-carotene (P < 0.001) with a near-significant increase in endothelium-dependent vasodilation (P = 0.060). Conclusions: Overall, the findings obtained in the present study showed that FVPD were a useful vehicle to increase fruit and vegetable intake, significantly increasing dietary and plasma phytochemical concentrations with a trend towards increased endothelium-dependent vasodilation.

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

On nonlinear symmetric stability and the nonlinear saturation of symmetric instability

A nonlinear symmetric stability theorem is derived in the context of the f-plane Boussinesq equations, recovering an earlier result of Xu within a more general framework. The theorem applies to symmetric disturbances to a baroclinic basic flow, the disturbances having arbitrary structure and magnitude. The criteria for nonlinear stability are virtually identical to those for linear stability. As in Xu, the nonlinear stability theorem can be used to obtain rigorous upper bounds on the saturation amplitude of symmetric instabilities. In a simple example, the bounds are found to compare favorably with heuristic parcel-based estimates in both the hydrostatic and non-hydrostatic limits.

Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

Disparity-driven vs blur-driven models of accommodation and convergence in binocular vision and intermittent strabismus

Background. Current models of concomitant, intermittent strabismus, heterophoria, convergence and accommodation anomalies are either theoretically complex or incomplete. We propose an alternative and more practical way to conceptualize clinical patterns. Methods. In each of three hypothetical scenarios (normal; high AC/A and low CA/C ratios; low AC/A and high CA/C ratios) there can be a disparity-biased or blur-biased “style”, despite identical ratios. We calculated a disparity bias index (DBI) to reflect these biases. We suggest how clinical patterns fit these scenarios and provide early objective data from small illustrative clinical groups. Results. Normal adults and children showed disparity bias (adult DBI 0.43 (95%CI 0.50-0.36), child DBI 0.20 (95%CI 0.31-0.07) (p=0.001). Accommodative esotropes showed less disparity-bias (DBI 0.03). In the high AC/A and low CA/C scenario, early presbyopes had mean DBI of 0.17 (95%CI 0.28-0.06), compared to DBI of -0.31 in convergence excess esotropes. In the low AC/A and high CA/C scenario near exotropes had mean DBI of 0.27, while we predict that non-strabismic, non-amblyopic hyperopes with good vision without spectacles will show lower DBIs. Disparity bias ranged between 1.25 and -1.67. Conclusions. Establishing disparity or blur bias, together with knowing whether convergence to target demand exceeds accommodation or vice versa explains clinical patterns more effectively than AC/A and CA/C ratios alone. Excessive bias or inflexibility in near-cue use increases risk of clinical problems. We suggest clinicians look carefully at details of accommodation and convergence changes induced by lenses, dissociation and prisms and use these to plan treatment in relation to the model.