A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

Theoretical perspectives on the development of a meta-methodology for value management

This paper provides a new set of theoretical perspectives on the topic of value management in building procurement. On the evidence of the current literature it is possible to identify two distinct methodologies which are based on different epistemological positions. An argument is developed which sees these two methodologies to be complementary. A tentative meta-methodology is then outlined for matching methodologies to different problem situations. It is contended however that such a meta-methodology could never provide a prescriptive guide. Its usefulness lies in the way in which it provides the basis for reflective practice. Of central importance is the need to understand the problem context within which value management is to be applied. The distinctions between unitary, pluralistic and coercive situations are seen to be especially significant.

Should we be utopophobes about democracy in particular?

In his book Democratic Authority, David Estlund puts forward a case for democracy, which he labels epistemic proceduralism, that relies on democracy's ability to produce good – that is, substantively just – results. Alongside this case for democracy Estlund attacks what he labels ‘utopophobia’, an aversion to idealistic political theory. In this article I make two points. The first is a general point about what the correct level of ‘idealisation’ is in political theory. Various debates are emerging on this question and, to the extent that they are focused on ‘political theory’ as a whole, I argue, they are flawed. This is because there are different kinds of political concept, and they require different kinds of ideal. My second point is about democracy in particular. If we understand democracy as Estlund does, then we should see it as a problem-solving concept – the problem being that we need coercive institutions and rules, but we do not know what justice requires. As democracy is a response to a problem, we should not allow our theories of it, even at the ideal level, to be too idealised – they must be embedded in the nature of the problem they are to solve, and the beings that have it.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Participation in and responsibility for state injustices

This paper discusses the criteria for acceptably holding citizens partly responsible for wrongs their state or its agents commit. Some proposed criteria are not, it argues, appropriately sensitive to the particular coercive relation between state and citizen. Others, which are, conceive of it wrongly and fail to match our judgments about a range of cases. Alternative criteria of breadth and joint authorship, built around Christopher Kutz's account of participation, better match these considered judgments as well as linking them to a more powerful theoretical framework. Understanding citizens' responsibility will mean understanding these criteria more fully.