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.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 0of operators, , which ensure that, if λ≠0 and λφ=Kkφ has only the trivial solution in X, for all k∈W, then, for 0⩽a⩽b, (λ−K)φ=ψ has exactly one solution φ∈Xa for every k∈W and ψ∈Xa. These conditions ensure further that is bounded uniformly in k∈W, for 0⩽a⩽b. As a particular application we consider the case when the kernel takes the form k(s,t)=κ(s−t)z(t), with , , and κ(s)=O(|s|−b) as |s|→∞, for some b>1. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

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.

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.

Applying robust control theory to solve problems in bio-medical sciences: study of an apoptotic model

Biological models of an apoptotic process are studied using models describing a system of differential equations derived from reaction kinetics information. The mathematical model is re-formulated in a state-space robust control theory framework where parametric and dynamic uncertainty can be modelled to account for variations naturally occurring in biological processes. We propose to handle the nonlinearities using neural networks.

The Impact of Vitamin A Supplementation on the Immune System of Vitamin A-deficient Children

Background & Aims: To investigate the effect of vitamin A supplementation on parameters of the immune system of vitamin A-deficient children. Methods: The study was carried out in four phases: 1) determination of serum retinol in 631 children from 36 to 83 months of age; 2) assessment of immunological markers [immunoglobulins and complement fractions, immunophenotyping of T and B lymphocytes, and natural killer (NK) cells], blood count, and serum ferritin of 52 vitamin A-deficient children (serum retinol <0.70 mu mol/L); 3) supplementation of the 52 deficient children with 200,000 IU of vitamin A; 4) determination of serum retinol and the immunological parameters 2 months after vitamin A supplementation. Results: Before vitamin A supplementation, 24.0% of the children were anemic and 4.3 %had reduced ferritin concentrations. There was no significant difference between mean values of retinol according to the presence/absence of anemia. The mean values of the humoral and cellular immunological parameters did not show a statistically significant difference before and after supplementation with vitamin A. Children with concomitant hypovitaminosis A and anemia presented a significant increase in absolute CD4 and CD8 T-cell counts after vitamin A supplementation (p < 0.05). Conclusion: Vitamin A had an effect on the recruitment of T and B lymphocytes to the circulation of children with hypovitaminosis A and anemia.

Effects of isoflavones on the coagulation and fibrinolytic system of postmenopausal women

Objective: We evaluated the effects of soy isoflavone supplementation on hemostasis in healthy postmenopausal women. Methods: In this double-blinded, placebo-controlled study, 47 postmenopausal women 47-66 y of age received 40 mg of soy isoflavone (n = 25) or 40 mg of casein placebo (n = 22) once a day for 6 mo. Levels of factors VII and X. fibrinogen, thrombin-antithrombin complex, prothrombin fragments I plus 2, antithrombin, protein C, total and free protein S, plasminogen, plasminogen activator inhibitor-1, and D-dimers were measured at baseline and 6 mo. Urinary isoflavone concentrations (genistein and daidzein) were measured as a marker of compliance and absorption using high-performance liquid chromatography. Baseline characteristics were compared by unpaired Student`s t test. Within-group changes and comparison between the isoflavone and casein placebo groups were determined by a mixed effects model. Results: The levels of hemostatic variables did not change significantly throughout the study in the isoflavone group; however, the isoflavone group showed a statistically significant reduction in plasma concentration of prothrombin fragments I plus 2; both groups showed a statistically significant reduction in antithrombin, protein C, and free protein S levels. A significant increase in D-dimers was observed only in the isoflavone group. Plasminogen activator inhibitor-l levels increased significantly in the placebo group. However, these changes were not statistically different between groups. Conclusion: The results of the present study do not support a biologically significant estrogenic effect of soy isoflavone on coagulation and fibrinolysis in postmenopausal women. However, further research will be necessary to definitively assess the safety and efficacy of isoflavone. (D 2008 Elsevier Inc. All rights reserved.

A thermoelastic system of memory type in noncylindrical domains

In this paper, we consider a hyperbolic thermoelastic system of memory type in domains with moving boundary. The problem models vibrations of an elastic bar under thermal effects according to the heat conduction law of Gurtin and Pipkin. Global existence is proved by using the penalty method of Lions. (c) 2007 Elsevier Inc. All rights reserved.

On the dominance of roots of characteristic equations for neutral functional differential equations

We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail. (C) 2009 Elsevier Inc. All rights reserved.

Local solvability for a class of evolution equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider it class of evolution operators with real-analytic coefficients and study their local solvability both in L(2) and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg-Treves condition (psi) which is suitable to our study. (C) 2009 Published by Elsevier Inc.

Host-guest system of 4-nerolidylcatechol in 2-hydroxypropyl-beta-cyclodextrin: preparation, characterization and molecular modeling

The interaction of 4-nerolidylcatechol (4-NRC), a potent antioxidant agent, and 2-hydroxypropyl-beta-cyclodextrin (HP-beta-CD) was investigated by the solubility method using Fourier transform infrared (FTIR) methods in addition to UV-Vis, (1)H-nuclear magnetic resonance (NMR) spectroscopy and molecular modeling. The inclusion complexes were prepared using grinding, kneading and freeze-drying methods. According to phase solubility studies in water a B(S)-type diagram was found, displaying a stoichiometry complexation of 2:1 (drug:host) and stability constant of 6494 +/- A 837 M(-1). Stoichiometry was established by the UV spectrophotometer using Job`s plot method and, also confirmed by molecular modeling. Data from (1)H-NMR, and FTIR, experiments also provided formation evidence of an inclusion complex between 4-NRC and HP-beta-CD. 4-NRC complexation indeed led to higher drug solubility and stability which could probably be useful to improve its biological properties and make it available to oral administration and topical formulations.

The Pollination Biology and Mating System of a Peripheral Population of Witheringia solanacea (Solanaceae)

Pollinator visitation rates over the life of a flower are determined by pollinator abundance and floral longevity. If flowers are not visited frequently enough, pollen limitation may occur, favoring the evolution of self-compatibility (SC). In plant species with varying SC levels, central populations often are self-incompatible (SI) and peripheral populations are SC. Witheringia solanacea (Solanaceae) is a species that follows this trend with the exception of one population in the Monteverde Cloud Forest Reserve, which is peripheral yet SI. I investigated this population using multiple techniques including floral bagging, pollinator observations, microsatellite analysis, and floral longevity manipulations. My results confirmed the self-incompatibility of the Monteverde population and indicated low but perhaps adequate rates of pollinator visitation per flower per hour. I found reduced genetic diversity at Monteverde and gene flow occurring unidirectionally from San Luis (a central population) to Monteverde. In the greenhouse, there was more of an effect of male than female function on floral longevity, but the largest differences were environmental. Flowers stayed open substantially longer when cool, cloudy weather was simulated and shorter when conditions were hot and sunny. The results indicate that the Monteverde population of W. solanacea is SI because 1) it is unable to maximize its fitness due to gene flow from San Luis and its relatively recent colonization of the area and 2) pollen limitation may not be severe because of supplemental pollinator availability from other Witheringia species in the area and increased floral longevities due to cool and cloudy conditions.

The impact on the stationary state income of a pay-as-you-go system of social security

A model of overlapping generations in continuous time is composed. IndividuaIs pass through two distinct time periods during their life times. During the first period, they work, save and have a death probability equal to zero. During the second, from the periods T after birth, their probability of death changes to p and then they retire. Capital stock and the stationary state in come are calculated for two situations: in the first, people live from their accumulated capital after retirementj in the second, they live from a state transfer payment through income taxo To simplify matters, in this preliminary version, it is supposed that there is no population growth and that the instantaneous elasticity substitution of consumption is unitary.