Land, rights and the gift: the Countryside and Rights of Way Act 2000 and the negotiation of citizenship

The Countryside and Rights of Way Act came into force at the end of 2000 with,as part of its content, new provisions relating to public access to the English and Welsh countryside. In this paper we review the main elements of the Act and assess its meaning in relation to citizenship, territoriality and the place of land in English law and society. We invoke Mauss’s (1954)concept of Gift to explain the process of brokerage being made over access and rights in the countryside. In conclusion we reflect on the Act as being indicative of a wider move towards Bromley’s (1998)post-feudal scenario for land and its governance.

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods applied to boundary value problems for the positive definite Helmholtz-type problem -DeltaU + alpha U-2 = 0 in a bounded or unbounded domain, with the parameter alpha real and possibly large. Applications arise in the implementation of space-time boundary integral methods for the heat equation, where alpha is proportional to 1/root deltat, and deltat is the time step. The corresponding layer potentials arising from this problem depend nonlinearly on the parameter alpha and have kernels which become highly peaked as alpha --> infinity, causing standard discretization schemes to fail. We propose a new collocation method with a robust convergence rate as alpha --> infinity. Numerical experiments on a model problem verify the theoretical results.

Leisure, consumerism and active citizenship in the UK

Since the first election victory of the Thatcher administration in 1979, Britain has witnessed a cultural transformation from the municipal socialism enshrined in the post-World War 2 development of the Welfare State to a form of post-industrial entrepreneurialism based largely on market rationality. This has had a profound effect on all aspects of civil life, not least the redefinition of the role of active leisure. Since the late 1950s the dominant policy for active leisure has been 'Sport For All', an assertion of a social right too important to be left to the market. The transformation has, therefore, signalled a shift from government support for active leisure as an element of citizen rights to the use of leisure to promote the government's interest in legitimating a new social order based not on rights but on means. Thus access to active living is no longer a societal goal for all, but a discretionary consumer good, the consumption of which signifies 'active' citizenship. It furthermore signifies differentiation from the growing mass of 'deviants' who are unwilling or unable to embrace this new construction of citizenship and are, therefore, increasingly denied access to active living and, hence, active citizenship.

Rights, citizenship and access to the countryside

This paper presents a reading of current UK Government policy on recreational access to the countryside of England, in terms of its citizenship and rights agenda. Given the continuity of traditional forms of land tenure and occupation, it is argued that the policy is less of recognition of the changing needs of a tranisitory society than it is a revisionist menifesto for resisting external influence and change. This is particularly so in terms of recreation, where the underlying organisation of the physical environment has been appropriated to reproduce a reflection of the social order which increasingly descriminates between culturally legitimate and illegitimate uses of rural space.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

On the entrainment assumption in Schatzmann's integral plume model

The behaviour of stationary, non-passive plumes can be simulated in a reasonably simple and accurate way by integral models. One of the key requirements of these models, but also one of their less well-founded aspects, is the entrainment assumption, which parameterizes turbulent mixing between the plume and the environment. The entrainment assumption developed by Schatzmann and adjusted to a set of experimental results requires four constants and an ad hoc hypothesis to eliminate undesirable terms. With this assumption, Schatzmann’s model exhibits numerical instability for certain cases of plumes with small velocity excesses, due to very fast radius growth. The purpose of this paper is to present an alternative entrainment assumption based on a first-order turbulence closure, which only requires two adjustable constants and seems to solve this problem. The asymptotic behaviour of the new formulation is studied and compared to previous ones. The validation tests presented by Schatzmann are repeated and it is found that the new formulation not only eliminates numerical instability but also predicts more plausible growth rates for jets in co-flowing streams.

ELMs disease? Corporatism, stewardship and citizenship in the English countryside

Varying concepts of citizenship, implicit within policy providing countryside access opportunities in England and the sometimes contrasting political rhetoric concerning citizenship, are evaluated here. The focus for this paper surrounds the Countryside Stewardship Scheme and, generically, the access elements of Environmental Land Management schemes (ELMs) and the implications of the 1994 Criminal Justice and Public Order Act in this context. Policy formulation in respect of countryside access may not be prepared considering the philosophical implications for citizens rights or property rights constructions. However, it is hypothesized that particular modes of regulation and commodification (of certain countryside goods) are imbued with certain values which reflect a neo-Liberal political philosophy. This view is contextualized within present theoretical debates concerning rural society.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat “patchy” connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a “lattice-directed” traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

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.

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

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.