Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Root pruning as a means to encourage root growth in two ornamental shrubs, Buddleja davidii ‘Summer Beauty’ and Cistus ‘Snow Fire'

The inability of a plant to grow roots rapidly upon transplanting is one of the main factors contributing to poor establishment. In bare-rooted trees, treatments such as root pruning or application of the plant hormone auxin [e.g., indole butyric acid (IBA)] can promote root growth and aid long-term establishment. There is little information on ornamental containerised plants, however, other than the anecdotal notion that 'teasing' the roots out of the rootsoil mass before transplanting can be beneficial. In the present study we tested the ability of various root-pruning treatments and application of IBA to encourage new root and shoot growth in two shrub species, commonly produced in containers - Buddleja davidii 'Summer Beauty' and Cistus 'Snow Fire'. In a number of experiments, young plants were exposed to root manipulation (teasing, light pruning, or two types of heavy pruning) and/or treatment with IBA (at 500 or 1,000 mg l-1) before being transplanted into larger containers containing a medium of 1:1:1 (v/v/v) fine bark, sand and loam. Leaf stomatal conductance (gl) was measured 20 min, and 1, 2, 4 and 6 h after root manipulation. Net leaf CO2 assimilation (A) was measured frequently during the first week after transplanting, then at regular intervals up to 8 weeks after transplanting. Plants were harvested 8 weeks after transplanting, and root and shoot weights were measured. In both species, light root pruning alone, or in combination with 500 mg l-1 IBA, was most effective in stimulating root growth. In contrast, teasing, which is commonly used, showed no positive effect on root growth in Buddleja, and decreased new root growth in Cistus. The requirement for exogenous auxin to encourage new root growth varied between experiments and appeared to be influenced by the age and developmental stage of the plants. There were no consistent responses between root treatments and net CO2 assimilation rates, and changes in root weight were not closely correlated with changes in assimilation. The mechanisms whereby new root growth is sustained are discussed.

The influence of lodgepole pine (Pinus contorta) provenance on the development and survival of larvae of the pine beauty moth Panolis flammea (Lepidoptera: Noctuidae)

Larvae of the pine beauty moth Panolis flammae (Denis & Schiffermuller) were reared in sleeve cages on five different seed origins (provenances) of pole stage Pinus contorta in the field in each of four years from 1985 to 1988. Survival varied significantly between the years. In those years when survival was high, significant differences between tree provenance were not found. However, between provenance significant differences were found in larval weight and stage of development. In the years when survival was low, the results seen in good years were reversed. Significant differences attributable to provenance were found but these were not reflected in significant differences between larval weight or development. In addition, there was a significant correlation between the proportion surviving and larval weight, which was not the case in those years where larval survival was high. The results are discussed in light of the pest status of P, flammea in Britain and in view of current silvicultural policies. The use of trees resistant to insect attack as part of an integrated pest management programme is highlighted and the need to coordinate laboratory and field studies so as to control for environmental variation discussed.

Are differences in life history parameters of the pine beauty moth Panolis flammea modified by host plant quality or gender?

Pine beauty moth (Panolis flammea D&S, Lepidoptera: Noctuidae) were reared individually from egg hatch to pupation on one of three host plants, Pinus sylvestris (native host plant), Pinus contorta (Central Interior seed origin - good quality introduced host) and P. contorta (Alaskan seed origin - poor quality introduced host). After emerging from the pupae the adult moths were confined to a Skeena River seed origin of P. contorta. Female pupal weight and adult life span were significantly higher on P. sylvestris than on the two lodgepole pine seed origins. Development time was, however, not significantly different between treatments, but larval mean relative growth rate was found to be negatively correlated with birth weight and positively correlated with pupal weight. The time to emerge from the pupa was also not significantly different between treatments. However, there were marked differences between the genders. Male moths lost a significantly greater proportion of their weight over the pupal stage but lived significantly longer as adults than the females. Female moths emerged from the pupal stage significantly sooner than male moths. There was no apparent advantage of lai-ge birth size when looked at in terms of subsequent performance. These results are discussed in light of current life history theory.

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.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

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.