The Jahn-Teller effect in icosahedral symmetry: extension of Ham factors in strongly coupled systems

We extend the standard definition of reduction factors (Ham factors) in strongly coupled Jahn-Teller (JT) systems. Our aim is to cover linear JT systems in which the vibronic ground state at strong coupling is in close proximity in energy to low-lying excited states belonging to singlet and non-trivial irreducible representations of the JT centre. Such a structure of low-lying vibronic states is present in the linear JT systems of the icosahedral orbital quarter and quintet, G and H. We calculate all the standard reduction factors as well as extended matrix elements, for the icosahedral systems G(X)g, G(X)h and H(X)g. We calculate the matrix of Ham factors needed to handle the extra multiplicity of an H operator in an H state. A direct group-theoretical approach which explains the origins of various features of our analysis is included

The pedagogical power of context: extending the Epidemiology of Eyam

This paper describes the further development of the Epidemiology of Eyam project at Winchester College since the publication of our previous paper (French et al 2019 Phys. Educ. 54 045008). A much wider cadre of students have been involved with this recent phase of the study, and we have also benefitted from a sabbatical collaboration with Oxford University Mathematical Institute. It is hoped that the research presented here can be used as a case study for an Extended Project Qualification (EPQ) or equivalent in a range of schools around the UK and worldwide. We feel that a project such as this, with a strong humanities and public health rooted context, could incentivise students of mathematics and sciences to participate in inter-disciplinary teams over an educationally significant period, and offer an opportunity to develop vital independent research skills that are required at University, but are often difficult to experience in a School context. The main educational goal of our previous paper was to provide an suitable context to incentivise the introduction of Calculus ideas. In this paper we assume a slightly higher level of mathematical skill, and aim to present a more comprehensive analysis of the epidemiological model that we will refer to as the 'Eyam Equations.' We describe a 'semi-analytic' solution, and make the connection to the approximation of Kermack and McKendrick (Kermack and McKendrick 1927 Proc. R. Soc. Lond. A 115 700-21) which held sway for much of the early twentieth century. We revisit the Eyam 1660s plague data of William Mompesson, and also apply the model to Ebola data collected in Liberia during the 2014-16 epidemic in West Africa. Motivated by uncertainty in the size of the 'at-risk' Susceptible population, we re-parameterize the model in terms of alternative inputs which enable a curve-fitting mechanism to be conducted more efficiently, with a much more tightly bounded range of possible parameter values. In addition to a spreadsheet model, we have created a software application in the MATLAB environment which has dynamic tools that could potentially enable a sensible curve fit to be calculated very rapidly in response to new data. From a time series describing the Infective population of the 2014-16 Liberia Ebola epidemic, we can predict Infective I(t), Susceptible S(t) and Dead D(t) populations, calculate the associated total population size N, calculate the Kendall Susceptible threshold ρ, and hence the Basic Reproduction Number R0 = N/ρ. For the Liberia Ebola data, these are: N = 2542, ρ = 1373, R0 = 1.85. Note this suggests that out of an 'at risk' population of N = 2542, about 75% may ultimately have been infected by the Ebola virus. In the WHO Ebola Response Team report (WHO Ebola Response Roadmap Situation Report), R0 = 1.83 ± 0.11 for the 2014-16 Liberia outbreak