Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system in a very cost effective way. In this work we present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous evolution in a one-dimensional coupled map lattice but appears to be equally valid in a much wider range of cases. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system sizes are interleaved and we discuss the possible ways in which this may arise. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.Comment: 18 pages, double column, REVTeX, 27 embedded postscript figures with psfig. Submitted to Chao

Spacings and pair correlations for finite Bernoulli convolutions

We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.Comment: 17 pages, 6 figure

Don't bleach chaotic data

A common first step in time series signal analysis involves digitally filtering the data to remove linear correlations. The residual data is spectrally white (it is ``bleached''), but in principle retains the nonlinear structure of the original time series. It is well known that simple linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as fractal dimension and Lyapunov exponents. In theory, bleached data avoids these pitfalls. But in practice, bleaching obscures the underlying deterministic structure of a low-dimensional chaotic process. This appears to be a property of the chaos itself, since nonchaotic data are not similarly affected. The adverse effects of bleaching are demonstrated in a series of numerical experiments on known chaotic data. Some theoretical aspects are also discussed.Comment: 12 dense pages (82K) of ordinary LaTeX; uses macro psfig.tex for inclusion of figures in text; figures are uufile'd into a single file of size 306K; the final dvips'd postscript file is about 1.3mb Replaced 9/30/93 to incorporate final changes in the proofs and to make the LaTeX more portable; the paper will appear in CHAOS 4 (Dec, 1993

Phases of M2-brane Theories

We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the 'forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.Comment: 60 pages, 28 figures, 6 tables. v2: minor correction

A dynamical approach to the spatiotemporal aspects of the Portevin-Le Chatelier effect: Chaos,turbulence and band propagation

Experimental time series obtained from single and poly-crystals subjected to a constant strain rate tests report an intriguing dynamical crossover from a low dimensional chaotic state at medium strain rates to an infinite dimensional power law state of stress drops at high strain rates. We present results of an extensive study of all aspects of the PLC effect within the context a model that reproduces this crossover. A study of the distribution of the Lyapunov exponents as a function of strain rate shows that it changes from a small set of positive exponents in the chaotic regime to a dense set of null exponents in the scaling regime. As the latter feature is similar to the GOY shell model for turbulence, we compare our results with the GOY model. Interestingly, the null exponents in our model themselves obey a power law. The configuration of dislocations is visualized through the slow manifold analysis. This shows that while a large proportion of dislocations are in the pinned state in the chaotic regime, most of them are at the threshold of unpinning in the scaling regime. The model qualitatively reproduces the different types of deformation bands seen in experiments. At high strain rates where propagating bands are seen, the model equations are reduced to the Fisher-Kolmogorov equation for propagative fronts. This shows that the velocity of the bands varies linearly with the strain rate and inversely with the dislocation density, consistent with the known experimental results. Thus, this simple dynamical model captures the complex spatio-temporal features of the PLC effect.Comment: 17 pages, 18 figure

Neighbourhoods and oral health:Agent-based modelling of tooth decay

This research used proof of concept agent-based models to test various theoretical mechanisms by which neighbourhoods may influence tooth decay in adults. Theoretical pathways were constructed using existing literature and tested in two study areas in Sheffield, UK. The models found a pathway between shops and sugar consumption had the most influence on adult tooth decay scores, revealing that similar mechanisms influence this outcome in different populations. This highlighted the importance of the interactions between neighbourhood features and individual level variables in influencing outcomes in tooth decay. Further work is required to improve the accuracy and reliability of the models

M2-Branes and Fano 3-folds

A class of supersymmetric gauge theories arising from M2-branes probing Calabi-Yau 4-folds which are cones over smooth toric Fano 3-folds is investigated. For each model, the toric data of the mesonic moduli space is derived using the forward algorithm. The generators of the mesonic moduli space are determined using Hilbert series. The spectrum of scaling dimensions for chiral operators is computed.Comment: 128 pages, 39 figures, 42 table

Banking from Leeds, not London: regional strategy and structure at the Yorkshire Bank, 1859–1952

Industrial philanthropist Edward Akroyd created the Yorkshire Penny Savings Bank in 1859. Despite competition from the Post Office Savings Bank after 1861 and a serious reserve problem in 1911, it sustained his overall strategy to become a successful regional bank. Using archival and contemporary sources to build on recent scholarship illustrating how savings banks were integrated into local economies and the complementary roles of philanthropy and paternalism, we analyse an English regional bank's strategy, including an assessment of strategic innovation, ownership changes and management structure. This will demonstrate that the founder's vision continued, even though the 1911 crisis radically altered both strategy and structure