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.

Translation of Longus: 'Daphnis and Chloe'

A translation into English of 'Daphnis and Chloe', the ancient Greek novel by Longus.

Barbara Torelli Benedetti, Partenia, a pastoral play. A bilingual edition edited with notes and translation

This edition presents for the first time in print what is probably the earliest known secular, 'regular' play by a woman in Italy. As suggested in the introduction to the work, Torelli's play offers a fascinating example of female dramaturgy and creative adaptation of the pastoral genre in the context of late sixteenth-century Parma. Critical textual study is combined with new biographical material on the author and her literary milieu. The edition provides the first detailed palaeographical study of the complex Cremona manuscript of the play, which unusually includes emendations thought to be in the hand of the author herself (besides others). The transcription also describes for the first time in detail a newly discovered second manuscript of the play in Rome. The play-text is presented in the original Italian with a new facing English translation. Also included is the first transcription of the paratextual verse from the Cremona manuscript and the first transcribed collection of the author’s extant verse (both with translation).

Translating Lucretia: word, image and 'ethical non-indifference' in Simon de Hesdin's translation of Valerius Maximus's 'Facta et dicta memorabilia'

An analysis of how illustrations functioned as a distinctive and important aspect of the translation of Latin versions of the story of the rape and suicide of Lucretia into Middle French texts, especially the 'Faits et dits memorables' (a translation-adaptation of Valerius Maximus's 'Facta et dicta memorabilia'). The study focuses on a selection of 14th- and 15th- century illuminations, and proposes also that the early modern 'Lucretia' portrait tradition should be viewed in the context of these images.

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.