Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Rensch's rule in large herbivorous mammals derived from metabolic scaling

Rensch’s rule, which states that the magnitude of sexual size dimorphism tends to increase with increasing body size, has evolved independently in three lineages of large herbivorous mammals: bovids (antelopes), cervids (deer), and macropodids (kangaroos). This pattern can be explained by a model that combines allometry,life-history theory, and energetics. The key features are thatfemale group size increases with increasing body size and that males have evolved under sexual selection to grow large enough to control these groups of females. The model predicts relationships among body size and female group size, male and female age at first breeding,death and growth rates, and energy allocation of males to produce body mass and weapons. Model predictions are well supported by data for these megaherbivores. The model suggests hypotheses for why some other sexually dimorphic taxa, such as primates and pinnipeds(seals and sea lions), do or do not conform to Rensh’s rule.

Robotic multiwell planar patch-clamp for native and primary mammalian cells

Robotic multiwell planar patch-clamp has become common in drug development and safety programs because it enables efficient and systematic testing of compounds against ion channels during voltage-clamp. It has not, however, been adopted significantly in other important areas of ion channel research, where conventional patch-clamp remains the favored method. Here, we show the wider potential of the multiwell approach with the ability for efficient intracellular solution exchange, describing protocols and success rates for recording from a range of native and primary mammalian cells derived from blood vessels, arthritic joints and the immune and central nervous systems. The protocol involves preparing a suspension of single cells to be dispensed robotically into 4-8 microfluidic chambers each containing a glass chip with a small aperture. Under automated control, giga-seals and whole-cell access are achieved followed by preprogrammed routines of voltage paradigms and fast extracellular or intracellular solution exchange. Recording from 48 chambers usually takes 1-6 h depending on the experimental design and yields 16-33 cell recordings.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Dissolution or disillusion: the unravelling of civil partnerships

This chapter considers the following questions: • Why did more men register a civil partnership initially than women? • Why has the gender ratio equalised? • Why have women dissolved their civil partnership at a consistently higher rate than men? • What has happened to the argument that same-sex unions will be different from, or better than (with its assumption of longer-lasting), heterosexual unions?

Variance in male reproductive success and sexual size dimorphism in pinnipeds: testing an assumption of sexual selection theory

The theory of evolution by sexual selection for sexual size dimorphism (SSD) postulates that SSD primarily reflects the adaptation of males and females to their different reproductive roles. For example, competition among males for access to females increases male body size because larger males are better able to maintain dominant status than smaller males. Larger dominant males sire most offspring while smaller subordinate males are unsuccessful, leading to skew in reproductive success. Therefore, species with male-biased SSD are predicted to have greater variance in male reproductive success than those in which both sexes are similar in size. We tested this prediction among the Pinnipedia, a mammalian group with a great variation in SSD. From a literature review, we identified genetic estimates of male reproductive success for 10 pinniped taxa (eight unique species and two subspecies of a ninth species) that range from seals with similarly sized males and females to species in which males are more than four times as large as females. We found no support for a positive relationship between variance in reproductive success and SSD among pinnipeds after excluding the elephant seals Mirounga leonina and Mirounga angustirostris, which we discuss as distinctive cases. Several explanations for these results are presented, including the revival of one of Darwin's original ideas. Darwin proposed that natural selection may explain SSD based on differences in energetic requirements between sexes and the potential for sexual niche segregation. Males may develop larger bodies to exploit resources that remain unavailable to females due to the energetic constraints imposed on female mammals by gestation and lactation. The importance of this alternative explanation remains to be tested.