On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Relationship between 24-hour urinary-free cortisol excretion and salivary cortisol levels sampled from awakening to bedtime in healthy subjects

The utility of repeated salivary cortisol sampling as a substitute for 24-hour urinary-free cortisol (UFC) assessment was examined. Forty-four participants completed both 24-hour collections and 6 salivary collections at wake-up, 08:00, 12:00, 16:00, 20:00 and bedtime, during the same 24-hour period. The results demonstrated that mean, maximum, and amplitude (maximum minus minimum) for salivary cortisol all correlated positively with urinary cortisol, but the associations of these variables with urinary-free cortisol excretion were relatively small. Furthermore, a single salivary sample taken at wake-up was as good an indicator of overall cortisol production as the measures derived from multiple salivary samples. An examination of subject compliance indicated that many subjects failed to collect the timed salivary collections as instructed. The authors conclude that diurnal salivary cortisol sampling versus 24-hour urinary cortisol collections are likely to provide different information about ambient hypothalamic-pituitary-adrenal productivity, and therefore these measures should not be used interchangeably. In addition, subject compliance is a serious consideration in designing studies that employ home salivary collections. Published by Elsevier Science Inc.

Local gratings due to angular hole burning in a photorefractive polymer

We investigate the processes involved in writing real-time holographic gratings in a photorefractive polymer (PRP) that incorporates an azo-dye. In such systems there may be gratings due to mechanisms associated with trans–cis isomerization (angular hole burning (AHB) and/or angular redistribution), which appear in addition to those arising from the photorefractive (PR) effect. The work presented here helps to understand the interactions which may occur between these different gratings. The formation of local gratings due to mechanisms associated with photoisomerization is studied, in a new PRP based on the photoconductor poly(N-vinylcarbazole):2, 4, 7-trinitro-9-fluorenone, plasticized with N-ethylcarbazole. The polymer includes the azo-dye 4-nitro-4'-pentyloxy-azobenzene and we observe both PR and photoisomerization gratings. The gratings are shown to be both polarization-sensitive and reversible. The presence of the photoisomerization gratings (which diffract almost as strongly as the PR gratings) significantly affects the field-dependent diffractive behaviour of the composite. A measurement of the lifetime of the cis state is made (τcis = 38 s) using photoinduced dichroism. This is close to the decay time constant of the local gratings (τdecay = 42 s), and it is suggested that the local grating mechanism is AHB of the azo-dye. This is the first time (to the knowledge of the authors) that a local grating due to AHB has been demonstrated in a PRP.

A method for under-sampled ecological network data analysis: plant-pollination as case study

In this paper, we develop a method, termed the Interaction Distribution (ID) method, for analysis of quantitative ecological network data. In many cases, quantitative network data sets are under-sampled, i.e. many interactions are poorly sampled or remain unobserved. Hence, the output of statistical analyses may fail to differentiate between patterns that are statistical artefacts and those which are real characteristics of ecological networks. The ID method can support assessment and inference of under-sampled ecological network data. In the current paper, we illustrate and discuss the ID method based on the properties of plant-animal pollination data sets of flower visitation frequencies. However, the ID method may be applied to other types of ecological networks. The method can supplement existing network analyses based on two definitions of the underlying probabilities for each combination of pollinator and plant species: (1), pi,j: the probability for a visit made by the i’th pollinator species to take place on the j’th plant species; (2), qi,j: the probability for a visit received by the j’th plant species to be made by the i’th pollinator. The method applies the Dirichlet distribution to estimate these two probabilities, based on a given empirical data set. The estimated mean values for pi,j and qi,j reflect the relative differences between recorded numbers of visits for different pollinator and plant species, and the estimated uncertainty of pi,j and qi,j decreases with higher numbers of recorded visits.

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.