When is negativity not a problem for the ultra-discrete limit?

The `ultra-discrete limit' has provided a link between integrable difference equations and cellular automata displaying soliton like solutions. In particular, this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the "Max" operator. Although it certainly is the case that dropping these positivity conditions creates potential difficulties, it is still possible for solutions to persist under the ultra-discrete limit even in their absence. To recognize when this will occur, one must consider whether a certain expression, involving a measure of the rates of convergence of different terms in the difference equation and their coefficients, is equal to zero. Applications discussed include the solution of elementary ordinary difference equations, a discretization of the Hirota Bilinear Difference Equation and the identification of integrals of motion for ultra-discrete equations

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).Comment: 14 pages, TEX FIL

Developing Age-Friendly Cities: An evidence-based evaluation tool

Recent years have seen a proliferation of initiatives aimed at enhancing the age-friendliness of urban settings. The World Health Organization's (WHO) global Age-Friendly Cities (AFC) programme has been central to these. Cities seeking to become more age-friendly need reliable ways of assessing their efforts. This article describes an evidence-based evaluation tool for age-friendly initiatives whose development was informed by fieldwork in Liverpool/UK. The tool complements existing assessment frameworks, including those provided by WHO, by paying particular attention to the structures and processes underlying age-friendly initiatives. It reflects the complexity of age-friendliness by reconciling a focus on breadth with detail and depth, and it allows for a highly accessible visual presentation of findings. Using selected examples from Liverpool, the article illustrates how the evaluation tool can be applied to guide policy and practice with an age-friendly focus in different urban contexts. Pilot testing in further settings is underway to refine the tool as a practical method for evaluation and for supporting city-level decision making. Key words: Age-Friendly City; evaluation tool; ageing; urbanisation; complex intervention

Kinematic characteristics of elite men's 50 km race walking.

Race walking is an endurance event which also requires great technical ability, particularly with respect to its two distinguishing rules. The 50 km race walk is the longest event in the athletics programme at the Olympic Games. The aims of this observational study were to identify the important kinematic variables in elite men's 50 km race walking, and to measure variation in those variables at different distances. Thirty men were analysed from video data recorded during a World Race Walking Cup competition. Video data were also recorded at four distances during the European Cup Race Walking and 12 men analysed from these data. Two camcorders (50 Hz) recorded at each race for 3D analysis. The results of this study showed that walking speed was associated with both step length (r=0.54,P=0.002) and cadence (r=0.58,P=0.001). While placing the foot further ahead of the body at heel strike was associated with greater step lengths (r=0.45,P=0.013), it was also negatively associated with cadence (r= -0.62,P<0.001). In the World Cup, knee angles ranged between 175 and 186° at initial contact and between 180 and 195° at midstance. During the European Cup, walking speed decreased significantly (F=9.35,P=0.002), mostly due to a decrease in step length between 38.5 and 48.5 km (t=8.59,P=0.014). From this study, it would appear that the key areas a 50 km race walker must develop and coordinate are step length and cadence, although it is also important to ensure legal walking technique is maintained with the onset of fatigue

Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations

Point Symmetries of Generalized Toda Field Theories

A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil

Point Symmetries of Generalized Toda Field Theories II Applications of the Symmetries

The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones.Comment: 26 pages, no figure

Optimised Traffic Flow at a Single Intersection: Traffic Responsive signalisation

We propose a stochastic model for the intersection of two urban streets. The vehicular traffic at the intersection is controlled by a set of traffic lights which can be operated subject to fix-time as well as traffic adaptive schemes. Vehicular dynamics is simulated within the framework of the probabilistic cellular automata and the delay experienced by the traffic at each individual street is evaluated for specified time intervals. Minimising the total delay of both streets gives rise to the optimum signalisation of traffic lights. We propose some traffic responsive signalisation algorithms which are based on the concept of cut-off queue length and cut-off density.Comment: 10 pages, 11 eps figs, to appear in J. Phys.

Bose-Hubbard model with occupation dependent parameters

We study the ground-state properties of ultracold bosons in an optical lattice in the regime of strong interactions. The system is described by a non-standard Bose-Hubbard model with both occupation-dependent tunneling and on-site interaction. We find that for sufficiently strong coupling the system features a phase-transition from a Mott insulator with one particle per site to a superfluid of spatially extended particle pairs living on top of the Mott background -- instead of the usual transition to a superfluid of single particles/holes. Increasing the interaction further, a superfluid of particle pairs localized on a single site (rather than being extended) on top of the Mott background appears. This happens at the same interaction strength where the Mott-insulator phase with 2 particles per site is destroyed completely by particle-hole fluctuations for arbitrarily small tunneling. In another regime, characterized by weak interaction, but high occupation numbers, we observe a dynamical instability in the superfluid excitation spectrum. The new ground state is a superfluid, forming a 2D slab, localized along one spatial direction that is spontaneously chosen.Comment: 16 pages, 4 figure