Minors for alternating dimaps

We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.Comment: 51 pages, 7 figure

Fractal space frames and metamaterials for high mechanical efficiency

A solid slender beam of length $L$, made from a material of Young's modulus $Y$ and subject to a gentle compressive force $F$, requires a volume of material proportional to $L^{3}f^{1/2}$ [where $f\equiv F/(YL^{2})\ll 1$] in order to be stable against Euler buckling. By constructing a hierarchical space frame, we are able to systematically change the scaling of required material with $f$ so that it is proportional to $L^{3}f^{(G+1)/(G+2)}$, through changing the number of hierarchical levels $G$ present in the structure. Based on simple choices for the geometry of the space frames, we provide expressions specifying in detail the optimal structures (in this class) for different values of the loading parameter $f$. These structures may then be used to create effective materials which are elastically isotropic and have the combination of low density and high crush strength. Such a material could be used to make light-weight components of arbitrary shape.Comment: 6 pages, 4 figure

Estimate for the fractal dimension of the Apollonian gasket in d dimensions

We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. {\bf 131} 244104 (2009)] in order to predict the Hausdorff dimension $d_{A}$ of the Apollonian gasket in dimensions 2 and above. Our approximate results agree with published values in $2$ and $3$ dimensions to within $0.05$ and $0.6$ respectively, and we provide predictions for dimensions $4$ to $8$.Comment: 4 pages, 4 figure

Variable Factor Utilization and International Business Cycles

When an economic boom produces high output, employment, and investment in the United States, there is usually a simultaneous boom in other industrialized countries. But, why? Answering this question is a central goal of international macroeconomics. However, multi-country dynamic equilibrium models have struggled with two major problems. The first difficulty is that the productivity shocks required by the model are implausibly large and volatile. Second, these models have difficulty explaining why factor inputs move together so closely across countries: realistic international comovement of business cycles requires implausibly high cross-country correlations of productivity shocks. This paper builds a model in which the utilization rates of capital and labor can be varied in response to shocks. We find that variable factor utilization is quite successful in (i) reducing the required size of productivity shocks; and (ii) increasing international comovement of factor inputs, with most of the improvement stemming from variable capital utilization.

A rock or a hard place? Teaching assistants supporting physically disabled pupils in mainstream secondary school physical education: the tensions of professionalising the role

As a Physical Education (PE) teacher in both special and mainstream schools over a 15 year period, I witnessed the use of the teaching assistant (or Learning Support Assistant as they were known) for purposes which might be deemed to be related to a medical/welfare/care-giver role. In addition, previous small-scale research into the experiences of secondary-age disabled pupils in mainstream as opposed to special school PE showed that their experiences in an inclusive setting were restricted and that the presence of a TA did little to rectify this situation (Farr, 2005). Recently, the professionalisation of the role of the TA may have created a ‘teacher-in-waiting’ (Neill, 2002) and thus the nature of the TA’s role in PE, and the ability of the specialist teacher to work collaboratively with them is complex. This mixed methods study, inspired by critical ethnography (Thomas, 1993, 2003) incorporated five techniques of enquiry initially based on the work of Giangreco and Broer (2005). In keeping with a constructionist paradigm and integrating what I have termed a distorical theoretical perspective, I counted the interaction between people and the social structure in which they operated as important (Crotty, 1998, Broido, 2002) and drew on dominant participant voices (Lincoln and Guba, 2003). Adopting a theoretical perspective grounded in disability studies, I explored the perceptions of the role of the TA in inclusive PE through qualitative and quantitative data and presented a role definition which combines the humanistic with the instructional (or professional) after Reiter, 2000. I argued whether responsibility for the child’s learning should be devolved through the TA. Do we use the TA to make the teacher’s life easier or to support, collaboratively, the inclusion of the disabled pupil? The impact of this study on professional practice relates to the clarity of role definition for TAs generally and for TAs specifically who work in PE; the collaborative nature (or otherwise) of the TA/teacher relationship and the implications of these findings for the future training and deployment of teaching assistants in PE with a physically disabled pupil in a mainstream secondary school. This study found that TAs in PE share many traits or characteristics with those TAs working in other subject disciplines, or across subjects. However, in PE they were inclined to rate a willingness and ability to ‘join in’ and participate in practical activities alongside pupils above pedagogical knowledge. Training either reinforces an instructional or coaching role, or it focuses on the caring or medical aspects. The reality for the TA in this study however, is that they neither define themselves as one or the other but see themselves as drawing on their own skills, empathy and initiative to facilitate a positive, inclusive environment, with or without the input of the PE teacher. They deem themselves to be both care-givers where appropriate as well as supporters of autonomous participation (as opposed to learning). That the professionalization of their role moves them towards the pedagogical places the TA between a rock and a hard place