The Lattice Structure of Connection Preserving Deformations for q-Painlev\'e Equations I

We wish to explore a link between the Lax integrability of the $q$-Painlev\'e equations and the symmetries of the $q$-Painlev\'e equations. We shall demonstrate that the connection preserving deformations that give rise to the $q$-Painlev\'e equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice structure. Each Schlesinger transformation induces a B\"acklund transformation of the $q$-Painlev\'e equation. Each translational B\"acklund transformation may be lifted to the level of the associated linear problem, effectively showing that each translational B\"acklund transformation admits a Lax pair. We will demonstrate this framework for the $q$-Painlev\'e equations up to and including $q$-$\mathrm{P}_{\mathrm{VI}}$

Symmetries in Connection Preserving Deformations

We wish to show that the root lattice of B\"acklund transformations of the $q$-analogue of the third and fourth Painlev\'e equations, which is of type $(A_2+ A_1)^{(1)}$, may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we will show various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms

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

Evolution and turnover in scaling systems

Scaling has been discovered in the long tails of size distributions characterizing a variety of diverse systems, many of which evolve in terms of the size of their components through competition. Such time-invariant macro distributions, however, often obscure the micro-dynamics of change, such as continual turnover in the rank order of the constituents. Here we show how a model drawn from evolutionary theory can explain this change, such that the time spent in the top ranked constituents is finite and also characterized by longtailed distributions. To show the broad applicability of this model, we compare typical model runs to real-world examples including US boys’ names, UK Number One for pop albums, journal article keywords, and city sizes

Construction of a Lax Pair for the E6(1)E_6^{(1)} qq-Painlev\'e System

We construct a Lax pair for the $E^{(1)}_6$ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6$ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations

Accessing emergency rest centres in the UK - lesson learnt

Emergency rest centres (ERC) are premises that are used for the temporary accommodation of evacuees during an emergency situation. They form an important part of emergency response, by providing a focal point for receiving people and providing food, shelter, information and support. The Disability Discrimination Act 2005 creates a legislative right for ‘reasonable’ access to goods and services for disabled people. This legislation does not differentiate between emergency and non emergency situations which means that those with a responsibility for emergency planning need to consider the accessibility of ERCs. This article examines ERC provision and reviews access for disabled people. It focuses on a study of three ERCs that were established in different local authority areas within the Yorkshire and Humber region in the UK during a flooding event on 25th June 2007. While uncovering many instances of good practise, the results from the research also identified a number of lessons to be learnt, in particular it was noted that the main barriers to access were encountered with: • Facilities and elements that did not comprise part of the buildings normal operation, such as the provision of bedding, medical assistance and effective communication; and • Facilities that would not normally be expected to be used to the extent, or duration, whilst the emergency rest centre was in operation, such as the provision of adequate welfare facilities. The research also noted that Civil Protection Legislation within the UK contains limited instruction or guidance to those with responsibility for Emergency Rest Centre provision. This provides little impetus for Emergency Planners to consider the needs of disabled people. This research has broad implications for local authorities and national government representatives. It identifies a need for those with responsibility for emergency planning and response to strengthen their knowledge of disabled people, and to adopt a more holistic approach to the provision of emergency planning and response

Power Law Distribution of the Frequency of Demises of U.S Firms

Both theoretical and applied economics have a great deal to say about many aspects of the firm, but the literature on the extinctions, or demises, of firms is very sparse. We use a publicly available data base covering some 6 million firms in the US and show that the underlying statistical distribution which characterises the frequency of firm demises - the disappearances of firms as autonomous entities - is closely approximated by a power law. The exponent of the power law is, intriguingly, close to that reported in the literature on the extinction of biological species.Comment: 8 pages, 2 figure