Symmetric Vertex Models on Planar Random Graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular Phi3 planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent -- a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models). The relations between the vertex weights and Ising model parameters in the 4-vertex model on Phi3 graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model. Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.Comment: Tidied up version accepted for publication in PL

Translated Chemical Reaction Networks

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analysed via a variety of techniques, including elementary flux mode analysis, algebraic techniques (e.g. Groebner bases), and deficiency theory. In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network's capacity to permit a particular class of steady states, called toric steady states, to topological properties of a related network called a translated chemical reaction network. These networks share their reaction stoichiometries with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature

Airtightness of UK dwellings

This paper presents the results and key messages that have been obtained from Phase 1 of a participatory action research project that was undertaken with 5 developers to investigate the practical design and construction issues that arise in making improvements to the airtightness of speculatively built mainstream housing. Two construction types were represented in the project, masonry cavity and light steel frame. Phase 1 of the project sought to assess in detail the design, construction and air permeability of 25 dwellings that were constructed to conform to the requirements of Approved Document Part L1 2002. While the total number of dwellings reported here is small, the results suggest that there is not a consistent approach to the way in which developers present information on air leakage to those on site, a mixture of approaches are utilised on site to achieve the same specification and there appears to be a lack of foresight in the detailed design stage, resulting in specifications that are practically very difficult to achieve. Despite this, the air permeability results suggest that dwellings constructed with a wet/mechanically plastered internal finish, can default to a reasonable standard of airtightness by UK standards, without much additional attention being given to airtightness