We just published a paper showing that the properties of the shift invariant spaces, f , generated by the translates by Z n of an f in L 2 (R n ) correspond to the properties of the spaces L 2 (T n , p), where the weight p equals [f ,f ]. This correspondence helps us produce many new properties of the spaces f . In this paper we extend this method to the case where the role of Z n is taken over by locally compact abelian groups G, L 2 (R n ) is replaced by a separable Hilbert space on which a unitary representation of G acts, and the role of L 2 (T n , p) is assumed by a weighted space L 2 ( b G, w), where b G is the dual group of G. This provides many different extensions of the theory of wavelets and related methods for carrying out signal analysis.2010 Mathematics Subject Classification: 42C40, 43A65, 43A70.

The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at the surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.

The Penny Pusher, a simple stochastic model, offers a useful conceptual framework for the investigation of lens growth mechanisms and provides a plausible alternative to growth models that postulate the existence of lens stem cells.

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Lévy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.