Product algebras for Galerkin discretisations of boundary integral operators and their applications

Operator products occur naturally in a range of regularized boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp we have implemented a complete operator algebra that depends on knowledge of the domain, range and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible while hiding the complexities of Galerkin discretisations. In this paper, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems

Product algebras for Galerkin discretisations of boundary integral operators and their applications

Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems

Histopathological variation in keratoconus

During examination of 131 penetrating keratoplasty specimens from patients with keratoconus obtained in an 11-year period, we observed two histopathologic variants based on the appearance of Bowman\u27s layer and the corneal epithelium. Typical keratoconus specimens had multiple breaks in Bowman\u27s layer and central epithelial thinning, whereas atypical corneas lacked breaks in Bowman\u27s layer and had less thinning of the central epithelium. Ninety-five corneas were from patients who underwent grafting in only one eye. Seventy-six (80%) of these corneas were typical and 19 corneas (20%) were atypical in appearance. Both variants had similar degrees of central stromal thinning. Patients with typical and atypical corneas differed demographically by race only; 49% of typical and 95% of atypical corneas were from white individuals. Thirty-six corneas were from 18 patients who underwent bilateral penetrating keratoplasty. The histologic appearance of these corneal pairs was concordant in 13 patients and discordant (one typical and one atypical cornea) in five patients. Statistical analysis indicated that this distribution is not significantly different from that predicted by chance and suggests that typical and atypical corneas are manifestations of the same disease process

Corneal argyrosis associated with silver soldering.

We report a patient who developed corneal argyrosis secondary to occupational silver soldering. Clinically, the cornea was notable for a green-brown discoloration localized to Descemet\u27s membrane by slit-lamp biomicroscopy. Silver particles were identified within the anterior three eighths of Descemet\u27s membrane by light and electron microscopy and energy-dispersive x-ray microanalysis. To our knowledge, the association between corneal argyrosis and silver soldering has not been previously reported