An ultradiscrete matrix version of the fourth Painleve equation

We establish a matrix generalization of the ultradiscrete fourth Painlev\'e equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-PIV. The dynamics, irreducibility, and integrability of the matrix valued ultradiscrete systems are studied.Comment: 12 pages, 12 figures, Latex2e, Submitted to J. Phys. A, corrections mad

Thick-film materials for silicon photovoltaic cell manufacture

Thick film technology is applicable to three areas of silicon solar cell fabrication; metallization, junction formation, and coating for protection of screened ohmic contacts, particularly wrap around contacts, interconnection and environmental protection. Both material and process parameters were investigated. Printed ohmic contacts on n- and p-type silicon are very sensitive to the processing parameters of firing time, temperature, and atmosphere. Wrap around contacts are easily achieved by first printing and firing a dielectric over the edge and subsequently applying a low firing temperature conductor. Interconnection of cells into arrays can be achieved by printing and cofiring thick film metal pastes, soldering, or with heat curing conductive epoxies on low cost substrates. Printed (thick) film vitreous protection coatings do not yet offer sufficient optical uniformity and transparency for use on silicon. A sprayed, heat curable SiO2 based resin shows promise of providing both optical matching and environmental protection

Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity

We discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments.Comment: 16 pages plus one figure (plain TeX), MIT-CTP #2291. (Extraordinarily minor corrections.

On the zero set of G-equivariant maps

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(\Sigma)$ for isotropy subgroups $\Sigma$ of $G$ which is the difference of the dimension of the fixed point subspace of $\Sigma$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $\Sigma$ satisfying $s(\Sigma)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $\Sigma$ of dimension $s(\Sigma)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. This paper introduces and discusses an instructive example of an ODE where one can observe and analyse robust cycling behaviour. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rossler system), and/or saddle equilibria. For this model, we distinguish between cycling that include phase resetting connections (where there is only one connecting trajectory) and more general non-phase resetting cases where there may be an infinite number (even a continuum) of connections. In the non-phase resetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability whereas more general cases may give rise to `stuck on' cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase-resetting and connection selection

Student wellbeing through teacher wellbeing: A Study with law teachers in the UK and Australia

Research confirms law students and lawyers in the US, Australia and more recently in the UK are prone to symptoms related to stress and anxiety disproportionately to other professions. In response, the legal profession and legal academy in Australia and the UK have created Wellness Networks to encourage and facilitate research and disseminate ideas and strategies that might help law students and lawyers to thrive. This project builds on that research through a series of surveys of law teachers in the UK and Australia on the presumption that law teachers are in a strong position to influence their students not only about legal matters, but on developing attitudes and practices that will help them to survive and thrive as lawyers. The comparative analysis reveals several differences, but also many similarities with law teachers in both countries reporting negative effects from neoliberal pressures on legal education programs that impact their wellbeing, performance as teachers and ability to adequately respond to student concerns