The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Effect of dietary supplementation of different oils during the first or second half of pregnancy on the glucose tolerance of the sow

Poor glucose tolerance may be an under-researched contributory factor in the high (10% to 20%) pre-weaning mortality rate observed in pigs. Insulin resistance commences at around week 12 of gestation in the sow, although there are conflicting reports in the literature about the extent to which insulin resistance is modulated by maternal diet. The aim of the study was to determine the effects of supplementing the maternal diet with different dietary oils during either the first half or the second half of gestation on the glucose tolerance of the sow. Sows were offered the control (C: n = 5) diet as pellets or the C diet plus 10% extra energy (h = 16 per group) derived from either. (i) extra pellets; (ii) palm oil; (iii) olive oil; (iv) sunflower oil; or (v) fish oil. Experimental diets were fed during either the first (G1) or second (G2) half of gestation. A glucose tolerance test (GTT) was conducted on day 108 of gestation by administering 0.5g/kg glucose i.v. Blood samples were taken every 5 to 10 min for 90 min post administration. The change in body weight and backfat thickness during gestation was similar but both type and timing of dietary supplementation influenced litter size and weight. With the exception of the sunflower oil group, supplementing the maternal diet in G1 resulted in larger and heavier litters, particularly in mothers offered palm oil. Basal blood glucose concentrations tended to be more elevated in G1 than G2 groups, whilst plasma insulin concentrations were similar Following a GTT, the adjusted area under the curve was greater in G1 compared to G2 sows, despite no differences in glucose clearance. Maternal diet appeared to influence the relationship between glucose curve characteristics following a GTT and litter outcome. In conclusion, the degree of insulin sensitivity can be altered by both the period during which maternal nutritional supplementation is offered and the fatty acid profile of the diet.

Spectroscopic properties and electronic structures of 17-electron half-sandwich ruthenium acetylide complexes, [Ru(CCAr)(L2)Cp′]+ (Ar=phenyl, p-tolyl, 1-naphthyl, 9-anthryl; L2=(PPh3)2, Cp′=Cp; L2=dppe; Cp′=Cp∗)

A series of half-sandwich bis(phosphine) ruthenium acetylide complexes [Ru(C CAr)(L-2)Cp'] (Ar = phenyl, p-tolyl, 1-naphthyl, 9-anthryl; L2 = (PPh3)(2), Cp' = Cp; L-2 = dppe; Cp' = Cp*) have been examined using electrochemical and spectroelectrochemical methods. One-electron oxidation of these complexes gave the corresponding radical cations [Ru(C CAr)(L2)Cp'](+). Those cations based on Ru(dppe)Cp*, or which feature a para-tolyl acetylide substituent, are more chemically robust than examples featuring the Ru(PPh3)(2)Cp moiety, permitting good quality UV-Vis-NIR and IR spectroscopic data to be obtained using spectroelectrochemical methods. On the basis of TD DFT calculations, the low energy (NIR) absorption bands in the experimental electronic spectra for most of these radical cations are assigned to transitions between the beta-HOSO and beta-LUSO, both of which have appreciable metal d and ethynyl pi character. However, the large contribution from the anthryl moiety to the frontier orbitals of [Ru(C CC14H9)(L2)CP'](+) suggests compounds containing this moiety should be described as metal-stabilised anthryl radical cations.

Sleeping with half open eyes: dreams and realities in The Cry of the Owl

This article is a close analysis of The Cry of the Owl (Thraves, 2009). It is also part of larger project to bring together traditions of detailed criticism with those of production history, which culminates in second article on the film due to be published in 2011. The detail of the argument concerns analysing a range of the film’s key signifying systems, with a particular interest in the way the film explores the gap between images / impressions and characters’ realities; engages in a complex way with generic traditions and modes of address; establishes complex patterns of connection and contrast through blocking, camera strategies and narrative structure.

‘Anything can happen in the next half-hour’: Gerry Anderson’s transnational science fiction

Gerry Anderson’s 1960s puppet series have hybrid identities in relation to their medial, geographical, and production histories. This chapter ranges over his science fiction series from Supercar (1961) to Joe 90 (1968), arguing that Anderson’s television science fiction in that period crossed many kinds of boundary and border. Anderson’s television series were a compromise between his desire to make films for adults versus an available market for children’s television puppet programs, and aimed to appeal to a cross-generational family audience. They were made on film, using novel effects, for a UK television production culture that still relied largely on live and videotaped production. While commissioned by British ITV companies, the programs had notable success in the USA, achieving national networked screening as well as syndication, and they were designed to be transatlantic products. The transnational hero teams and security organisations featured in the series supported this internationalism, and simultaneously negotiated between the cultural meanings of Britishness and Americanness. By discussing their means of production, the aesthetic and narrative features of the programs, their institutional contexts, and their international distribution, this chapter argues that Anderson’s series suggest ways of rethinking the boundaries of British science fiction television in the 1960s.

New half-sandwich heterobimetallic CpMPt (M=Rh, Ir) dithiolato bridged complexes

Cationic heterobimetallic complexes 5–7 [(PPh3)2Pt(μ-edt)MClCp′)]BF4 (edt=−S(CH2)2S−; 5: M=Rh and Cp′=η5-C5H5; 6: M=Rh and Cp′=η5-C5Me5 and 7: M=Ir and Cp′=η5-C5Me5) were prepared by reaction of [Pt(edt)(PPh3)2] with [Cp′ClM(μ-Cl)2MClCp′] in THF in the presence of two equivalents of AgBF4. The crystalline structure of 5 was determined by X-ray diffraction methods. Cationic heterobimetallic complexes [(PPh3)2Pt(μ-S(CH2)2S)MClCp′)]BF4 (M=Rh, Ir) were prepared. The crystalline structure of [(PPh3)2Pt(μ-edt)RhClCp)]BF4 was determined by X-ray diffraction methods.

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.