Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

The VLT-FLAMES Survey of massive stars: Rotation and nitrogen enrichment as the key to understanding massive star evolution

Rotation has become an important element in evolutionary models of massive stars, specifically via the prediction of rotational mixing. Here we study a sample of stars, including rapid rotators, to constrain such models and use nitrogen enrichments as a probe of the mixing process. Chemical compositions (C, N, O, Mg, and Si) have been estimated for 135 early B-type stars in the Large Magellanic Cloud with projected rotational velocities up to similar to 300 km s(-1) using a non-LTE TLUSTY model atmosphere grid. Evolutionary models, including rotational mixing, have been generated attempting to reproduce these observations by adjusting the overshooting and rotational mixing parameters and produce reasonable agreement with 60% of our core hydrogen burning sample. We find (excluding known binaries) a significant population of highly nitrogen-enriched intrinsic slow rotators (nu sin i less than or similar to 50 km s(-1)) incompatible with our models (similar to 20% of the sample). Furthermore, while we find fast rotators with enrichments in agreement with the models, the observation of evolved (dex) fast rotators (log g < 3.7 dex) that are relatively unenriched (a further similar to 20% of the sample) challenges the concept of rotational mixing. We also find that 70% of our blue supergiant sample cannot have evolved directly from the hydrogen-burning main sequence. We are left with a picture where invoking binarity and perhaps fossil magnetic fields is required to understand the surface properties of a population of massive main- sequence stars.

Where are they now? Tracking the longitudinal evolution of environmental businesses from the 1990s

In 1991 Bennett published one of the first major publications on the concept of ecopreneurship, business opportunities resulting from the emerging environmental agenda of the late 1980s and early 1990s. Since then a body of literature has developed that explores the idea of the intersection of entrepreneurship with environmentally and socially responsible behaviour. Many of the business cases presented by Bennett represent early adopters of green products, services and emerging eco-markets. Given the current emphasis on the transformation of business practices towards a more sustainable paradigm it is timely to review these 94 early ecopreneurial examples and consider their status two decades on from the original publication. This paper explores the definitions of environmental and social enterprise, and considers the longitudinal survival of these companies and the emerging trends in consolidation and failure of the sampled companies.

Limbal stem cell deficiency:Concept, aetiology, clinical presentation, diagnosis and management

Defects in renewal and repair of ocular surface as a result of limbal stem cell deficiency are now known to cause varying ocular, surface morbidity including persistent photophobia, repeated and persistent surface breakdown and overt conjunctivalisation of the cornea. Ocular conditions with abnormalities of ocular surface repair include pterygium, limbal tumours, aniridia, severe scarring following burns, cicatricial pemphigoid and Stevens-Johnson Syndrome, sequelae of mustard gas exposure and Herpes simplex epithelial disease, radiation keratopathy, contact lens induced keratopathy, neuroparalytic keratitis and drug toxicity. Restoring ocular health in these eyes has traditionally been frustrating. An understanding of these intricate cell renewal and maintenance processes has spurred the evolution in recent years of new treatment methods for several blinding diseases of the anterior segment; many more exciting modalities are in the offing. However, there is inadequate awareness among ophthalmologists about the current principles of management of ocular surface disorders. The purpose of this article is to help elucidate the important principles and current treatment methods relevant to ocular surface disorders.