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.

Space physics for graduate students: an activities-based approach

The geospace environment is controlled largely by events on the Sun, such as solar flares and coronal mass ejections, which generate significant geomagnetic and upper atmospheric disturbances. The study of this Sun-Earth system, which has become known as space weather, has both intrinsic scientific interest and practical applications. Adverse conditions in space can damage satellites and disrupt communications, navigation, and electric power grids, as well as endanger astronauts. The Center for Integrated Space Weather Modeling (CISM), a Science and Technology Center (STC) funded by the U.S. National Science Foundation (see http://www.bu.edu/cism/), is developing a suite of integrated physics-based computer models that describe the space environment from the Sun to the Earth for use in both research and operations [Hughes and Hudson, 2004, p. 1241]. To further this mission, advanced education and training programs sponsored by CISM encourage students to view space weather as a system that encompasses the Sun, the solar wind, the magnetosphere, and the ionosphere/thermosphere. This holds especially true for participants in the CISM space weather summer school [Simpson, 2004].

Atypical neural self-representation in autism

The 'self' is a complex multidimensional construct deeply embedded and in many ways defined by our relations with the social world. Individuals with autism are impaired in both self-referential and other-referential social cognitive processing. Atypical neural representation of the self may be a key to understanding the nature of such impairments. Using functional magnetic resonance imaging we scanned adult males with an autism spectrum condition and age and IQ-matched neurotypical males while they made reflective mentalizing or physical judgements about themselves or the British Queen. Neurotypical individuals preferentially recruit the middle cingulate cortex and ventromedial prefrontal cortex in response to self compared with other-referential processing. In autism, ventromedial prefrontal cortex responded equally to self and other, while middle cingulate cortex responded more to other-mentalizing than self-mentalizing. These atypical responses occur only in areas where self-information is preferentially processed and does not affect areas that preferentially respond to other-referential information. In autism, atypical neural self-representation was also apparent via reduced functional connectivity between ventromedial prefrontal cortex and areas associated with lower level embodied representations, such as ventral premotor and somatosensory cortex. Furthermore, the magnitude of neural self-other distinction in ventromedial prefrontal cortex was strongly related to the magnitude of early childhood social impairments in autism. Individuals whose ventromedial prefrontal cortex made the largest distinction between mentalizing about self and other were least socially impaired in early childhood, while those whose ventromedial prefrontal cortex made little to no distinction between mentalizing about self and other were the most socially impaired in early childhood. These observations reveal that the atypical organization of neural circuitry preferentially coding for self-information is a key mechanism at the heart of both self-referential and social impairments in autism.