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.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

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.

Half-remembered quotations from mostly forgotten speeches: the limits of Labour’s European policy discourse

This article uses discourse analysis to study the continuities in British foreign policy thinking within the Labour party from the 1960s to the present day. Using representative extracts from speeches by Hugh Gaitskell, Harold Wilson, Tony Blair and Gordon Brown, it identifies the ideational consis- tencies in the leaders’ attitudes to: Empire; federalism in the EEC/EU; and laying down conditions that have to be met before any constructive engagement with ‘Europe’ can be countenanced. We argue that these consistencies, spanning a 50-year period, exemplify a certain stagnation both within Labour’s European discourses and within British foreign policy thinking more widely. We develop the idea that Labour party thinking has been crucially framed by both small ‘c’ conser- vative and upper-case Conservative ideology, popularised by Winston Churchill in his ‘three circles’ model of British foreign policy.

Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Sex and STEM occupation predict Autism-spectrum Quotient (AQ) scores in half a million people

This study assesses Autism-Spectrum Quotient (AQ) scores in a ‘big data’ sample collected through the UK Channel 4 television website, following the broadcasting of a medical education program. We examine correlations between the AQ and age, sex, occupation, and UK geographic region in 450,394 individuals. We predicted that age and geography would not be correlated with AQ, whilst sex and occupation would have a correlation. Mean AQ for the total sample score was m = 19.83 (SD = 8.71), slightly higher than a previous systematic review of 6,900 individuals in a non-clinical sample (mean of means = 16.94) This likely reflects that this big-data sample includes individuals with autism who in the systematic review score much higher (mean of means = 35.19). As predicted, sex and occupation differences were observed: on average, males (m = 21.55, SD = 8.82) scored higher than females (m = 18.95; SD = 8.52), and individuals working in a STEM career (m = 21.92, SD = 8.92) scored higher than individuals non-STEM careers (m = 18.92, SD = 8.48). Also as predicted, age and geographic region were not meaningfully correlated with AQ. These results support previous findings relating to sex and STEM careers in the largest set of individuals for which AQ scores have been reported and suggest the AQ is a useful self-report measure of autistic traits

Estrogen regulation of chicken riboflavin carrier protein gene is mediated by ERE half sites without direct binding of estrogen receptor

Estrogen is an important steroid hormone that mediates most of its effects on regulation of gene expression by binding to intracellular receptors. The consensus estrogen response element (ERE) is a 13 bp palindromic inverted repeat with a three nucleotide spacer. However, several reports suggest that many estrogen target genes are regulated by diverse elements, such as imperfect EREs and ERE half sites (ERE 1/2), which are either the proximal or the distal half of the palindrome. To gain more insight into ERE half site-mediated gene regulation, we used a region from the estrogen-regulated chicken riboflavin carrier protein (RCP) gene promoter that contains ERE half sites. Using moxestrol, an analogue of estrogen and transient transfection of deletion and mutation containing RCP promoter/reporter constructs in chicken hepatoma (LMH2A) cells, we identified an estrogen response unit (ERU) composed of two consensus ERE 1/2 sites and one non-consensus ERE 1/2 site. Mutation of any of these sites within this ERU abolishes moxestrol response. Further, the ERU is able to confer moxestrol responsiveness to a heterologous promoter. Interestingly, RCP promoter is regulated by moxestrol in estrogen responsive human MCF-7 cells, but not in other cell lines such as NIH3T3 and HepG2 despite estrogen receptor-alpha (ER-�) co transfection. Electrophoretic mobility shift assays (EMSAs) with promoter regions encompassing the half sites and nuclear extracts from LMH2A cells show the presence of a moxestrol-induced complex that is abolished by a polyclonal anti-ER� antibody. Surprisingly, estrogen receptor cannot bind to these promoter elements in isolation. Thus, there appears to be a definite requirement for some other factor(s) in addition to estrogen receptor, for the generation of a suitable response of this promoter to estrogen. Our studies therefore suggest a novel mechanism of gene regulation by estrogen, involving ERE half sites without direct binding of ER to the cognate elements.