Dipolar SLEs

We present basic properties of Dipolar SLEs, a new version of stochastic Loewner evolutions (SLE) in which the critical interfaces end randomly on an interval of the boundary of a planar domain. We present a general argument explaining why correlation functions of models of statistical mechanics are expected to be martingales and we give a relation between dipolar SLEs and CFTs. We compute SLE excursion and/or visiting probabilities, including the probability for a point to be on the left/right of the SLE trace or that to be inside the SLE hull. These functions, which turn out to be harmonic, have a simple CFT interpretation. We also present numerical simulations of the ferromagnetic Ising interface that confirm both the probabilistic approach and the CFT mapping.Comment: 22 pages, 4 figure

Vibration analysis of cracked aluminium plates

This research is concerned with analytical modelling of the effects of cracks in structural plates and panels within aerospace systems such as aeroplane fuselage, wing, and tail-plane structures, and, as such, is part of a larger body of research into damage detection methodologies in such systems. This study is based on generating a so-called reduced order analytical model of the behaviour of the plate panel, within which a crack with some arbitrary characteristics is present, and which is subjected to a force that causes it to vibrate. In practice such a scenario is potentially extremely dangerous as it can lead to failure, with obvious consequences. The equation that is obtained is in the form of the classical Duffing equation, in this case, the coefficients within the equation contain information about the geometrical and mass properties of the plate, the loading and boundary conditions, and the geometry, location, and potentially the orientation of the crack. This equation has been known for just over a century and has in the last few decades received very considerable attention from both the analytical dynamics community and also from the dynamical systems researchers, in particular the work of Ueda, Thompson, in the 1970s and 1980s, and Thomsen in the 1990s and beyond. An approximate analytical solution is obtained by means of the perturbation method of multiple scales. This powerful method was popularized in the 1970s by Ali H.Nayfeh, and discussed in his famous books, ‘Perturbation Methods’ (1974) and ‘Nonlinear Oscillations’ (1979, with D.T.Mook), and also by J.Murdock (1990), and M.P.Cartmell et al. (2003) and has been shown to be immensely useful for a wide range of nonlinear vibration problems. In this work it is shown that different boundary conditions can be admitted for the plate and that the modal natural frequencies are sensitive to the crack geometry. Bifurcatory behaviour of the cracked plate has then been examined numerically, for a range of parameters. The model has been tested against experimental work and against a Finite Element model, with good corroboration from both. In all events, this is a significant new result in the field and one that if implemented within a larger damage detection strategy, could be of considerable practical use

Hunter-gatherers in a howling wilderness: Neoliberal capitalism as a language that speaks itself

The 'self-referential' character of evolutionary process noted by Goldenfeld and Woese (2010) can be restated in the context of a generalized Darwinian theory applied to economic process through a 'language' model: The underlying inherited and learned culture of the firm, the short-time cognitive response of the firm to patterns of threat and opportunity that is sculpted by that culture, and the embedding socioeconomic environment, are represented as interacting information sources constrained by the asymptotic limit theorems of information theory. If unregulated, the larger, compound, source that characterizes high probability evolutionary paths of this composite then becomes, literally, a self-dynamic language that speaks itself. Such a structure is, for those enmeshed in it, more akin to a primitive hunter-gatherer society at the mercy of internal ecological dynamics than to, say, a neolithic agricultural community in which a highly ordered, deliberately adapted, ecosystem is consciously farmed so as to match its productivity to human needs

Umbral Vade Mecum

In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations --- discretized versions of the differential cornerstones appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.Comment: arXiv admin note: text overlap with arXiv:0710.230