Quadrature domains and kernel function zipping

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be zipped down to a strikingly small data set. It is also proved that the kernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati

Conditions for negative specific heat in systems of attracting classical particles

We identify conditions for the presence of negative specific heat in non-relativistic self-gravitating systems and similar systems of attracting particles. The method used, is to analyse the Virial theorem and two soluble models of systems of attracting particles, and to map the sign of the specific heat for different combinations of the number of spatial dimensions of the system, $D$($\geq 2$), and the exponent, $\nu$($\neq 0$), in the force potential, $\phi=Cr^\nu$. Negative specific heat in such systems is found to be present exactly for $\nu=-1$, at least for $D \geq 3$. For many combinations of $D$ and $\nu$ representing long-range forces, the specific heat is positive or zero, for both models and the Virial theorem. Hence negative specific heat is not caused by long-range forces as such. We also find that negative specific heat appears when $\nu$ is negative, and there is no singular point in a certain density distribution. A possible mechanism behind this is suggested.Comment: 11 pages, 1 figure, Published version (including correlation between positive specific heat and singular points

Exact General Solutions to Extraordinary N-body Problems

We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If the potential is supplemented by any function that scales as the inverse square of the radius there is still no damping of the fundamental breathing mode. For such systems a remarkable new statistical equilibrium is found for the other coordinates and momenta, which persists even as the radius changes continually.Comment: 15 pages, LaTeX. Accepted for publication in Proc. Roy. Soc.

Quasi-invariant measures on the path space of a diffusion

The author has previously constructed a class of admissible vector fields on the path space of an elliptic diffusion process $x$ taking values in a closed compact manifold. In this Note the existence of flows for this class of vector fields is established and it is shown that the law of $x$ is quasi-invariant under these flows

Theorising teaching

Theorising of teaching is not limited to the major theorists, as researchers and teachers also theorise teaching. This article explores theorising by researchers and a teacher in two ways: firstly, an overview or framework from the research literature on teaching as a sociocultural practice is developed, and secondly, the framework is tested by analysing the transcripts of a beginning teacher talking about her teaching

Comment on "Separability of quantum states and the violation of Bell-type inequalities"

The statement of E.R. Loubenets, Phys. Rev. A 69, 042102 (2004), that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell's concept of local hidden variables. In a Bell type setting the notion of classicality used by Loubenets corresponds to the assumption of perfect correlations if the same observable is measured on both sides. While it is obvious that most separable states do not satisfy this assumption, this does not constitute "non-classical" behaviour in any usual sense of the word.Comment: 1 page, accepted by Phys. Rev.

LSE centre for economic performance: bankers’ bonuses

A new series of Election Analyses is now available from the LSE’s Centre for Economic Performance (CEP). The series will discuss the research evidence on some of the key policy battlegrounds of the 2010 General Election, including macroeconomic policy, immigration, health, education, crime, poverty and inequality, labour market policy, regional policy, energy and the environment, financial regulation and bankers’ bonuses, and foreign aid. The latest CEP Election Analysis, by Brian Bell, gives an overview of bankers’ bonuses and extreme wage inequality, one of the key battlegrounds of the 2010 General Election. The publication is summarised below and can be found in full on the CEP Election Analysis Site

Connectivism: Its place in theory-informed research and innovation in technology-enabled learning

The sociotechnical context for learning and education is dynamic and makes great demands on those trying to seize the opportunities presented by emerging technologies. The goal of this paper is to explore certain theories for our plans and actions in technology-enabled learning. Although presented as a successor to previous learning theories, connectivism alone is insufficient to inform learning and its support by technology in an internetworked world. However, because of its presence in massive open online courses (MOOCs), connectivism is influential in the practice of those who take these courses and who wish to apply it in teaching and learning. Thus connectivism is perceived as relevant by its practitioners but as lacking in rigour by its critics. Five scenarios of change are presented with frameworks of different theories to explore the variety of approaches educators can take in the contexts for change and their associated research/evaluation. I argue that the choice of which theories to use depends on the scope and purposes of the intervention, the funding available to resource the research/evaluation, and the experience and philosophical stances of the researchers/practitioners