On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities

We prove that all the eigenvalues of a certain highly non-self-adjoint Sturm–Liouville differential operator are real. The results presented are motivated by and extend those recently found by various authors (Benilov et al. (2003) [3], Davies (2007) [7] and Weir (2008) [18]) on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder.

Farming and food production and development in the Shropshire Hills Environmentally Sensitive Area, 1997-2008

European agricultural and environmental policy has evolved considerably over the last 15 years. In this paper the changes in farm businesses in an Environmentally Sensitive Area in England are evaluated based on two surveys with the same farmers at the start and end of this period. The rate of participation in the environmental scheme had increased significantly at a time when Government led goals in this area had developed and become more output focussed. A combination of policy, market and animal health status changes had encouraged a number to leave cattle production, and though remaining with stock and grass they had decided against any extensive development in the direction of pluriactivity – with or without Government encouragement. This left the future of this group in some uncertainty given that two significant forms of financial support, the environmental scheme and the Hill Farm Allowance, were due to close.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

Rapid topographic change measured by high-resolution satellite radar at Soufriere Hills Volcano, Montserrat, 2008-2010

High-resolution satellite radar observations of erupting volcanoes can yield valuable information on rapidly changing deposits and geomorphology. Using the TerraSAR-X (TSX) radar with a spatial resolution of about 2 m and a repeat interval of 11-days, we show how a variety of techniques were used to record some of the eruptive history of the Soufriere Hills Volcano, Montserrat between July 2008 and February 2010. After a 15-month pause in lava dome growth, a vulcanian explosion occurred on 28 July 2008 whose vent was hidden by dense cloud. We were able to show the civil authorities using TSX change difference images that this explosion had not disrupted the dome sufficient to warrant continued evacuation. Change difference images also proved to be valuable in mapping new pyroclastic flow deposits: the valley-occupying block-and-ash component tending to increase backscatter and the marginal surge deposits reducing it, with the pattern reversing after the event. By comparing east- and west-looking images acquired 12 hours apart, the deposition of some individual pyroclastic flows can be inferred from change differences. Some of the narrow upper sections of valleys draining the volcano received many tens of metres of rockfall and pyroclastic flow deposits over periods of a few weeks. By measuring the changing shadows cast by these valleys in TSX images the changing depth of infill by deposits could be estimated. In addition to using the amplitude data from the radar images we also used their phase information within the InSAR technique to calculate the topography during a period of no surface activity. This enabled areas of transient topography, crucial for directing future flows, to be captured.

Farmers' management of rice varietal diversity in the mid-hills of Nepal: implications for on-farm conservation and crop improvement

Season-long monitoring of on-farm rice (Oryza sativa, L.) plots in Nepal explored farmers' decision-making process on the deployment of varieties to agroecosystems, application of production inputs to varieties, agronomic practices and relationship between economic return and area planted per variety. Farmers deploy varieties [landraces (LRs) and modern varieties (MVs)] to agroecosystems based on their understanding of characteristics of varieties and agroecosystems, and the interaction between them. In marginal growing conditions, LRs can compete with MVs. Within an agroecosystem, economic return and area planted to varieties have positive relationship, but this is not so between agroecosystems. LRs are very diverse on agronomic and economic traits; therefore, they cannot be rejected a priori as inferior materials without proper evaluation. LRs have to be evaluated for useful traits and utilized in breeding programmes to generate farmer-preferred materials for marginal environments and for their conservation on-farm.

Adjoint methods in data assimilation for estimating model error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure.

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.

On the strong uniqueness of some finite dimensional Dirichlet operators

We prove essential self-adjointness of a class of Dirichlet operators in ℝn using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.

Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.