Fractal Topology Foundations

In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topological space.Comment: 20 page

Curvature contraction of convex hypersurfaces by nonsmooth speeds

We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing parameter, we may pass to the limit to deduce that the flow by the nonsmooth speed converges to a point in finite time that, under a suitable rescaling, is round in the C² sense, with the convergence being exponential.The research of the second author was supported by an Australian Postgraduate Award. The research of the first, third, fourth and fifth authors was supported by Discovery Project grant DP120100097 of the Australian Research Council

Curvature contraction of convex hypersurfaces by nonsmooth speeds

We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing parameter, we may pass to the limit to deduce that the flow by the nonsmooth speed converges to a point in finite time that, under a suitable rescaling, is round in the C^2 sense, with the convergence being exponential

Natural selection maximizes Fisher information

In biology, information flows from the environment to the genome by the process of natural selection. But it has not been clear precisely what sort of information metric properly describes natural selection. Here, I show that Fisher information arises as the intrinsic metric of natural selection and evolutionary dynamics. Maximizing the amount of Fisher information about the environment captured by the population leads to Fisher's fundamental theorem of natural selection, the most profound statement about how natural selection influences evolutionary dynamics. I also show a relation between Fisher information and Shannon information (entropy) that may help to unify the correspondence between information and dynamics. Finally, I discuss possible connections between the fundamental role of Fisher information in statistics, biology, and other fields of science.Comment: Published version freely available at DOI listed her

Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater that $2^{17}\times 2^{17}$ elements. We examine the convergence properties of our algorithm. We also present two dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive

Poverty, Wealth and Place in Britain, 1968-2005

This paper examines long-term trends in the geography of poverty and wealth in Britain since 1968. To date, analysis of long-term trends in the spatial distribution of poverty in Britain have been frustrated by an absence of consistency in definitions, data sources and measures, as well as by changes over time in census and administrative geography. The research described here was commissioned by the Joseph Rowntree Foundation in order to further understanding of spatial inequalities in wealth and poverty in Britain since the 1960s (see Dorling et al., 2007). In particular, it draws upon a series of nationally representative poverty surveys conducted in 1968, 1983, 1990, and 1999 in order to derive methodologically consistent measures of ‘breadline poverty’ and ‘core poverty’. These results are then applied to UK Census data using longitudinally consistent boundary data (census tracts) in order to explore the changing geography of poverty in Britain. In comparison with poverty, much less is known about the geography of wealth in Britain, and establishing its distribution is essential for a more thorough understanding of the dynamics of social inequality in Britain. This study represents the first attempt to operationalise such a measure in order to produce longitudinally consistent small area measures of ‘asset wealth’ based on housing wealth data, and ‘exclusive wealth’ based upon analysis of Family Expenditure Survey data. These analyses suggest that not only is poverty widespread in Britain today, but that both poverty and wealth have become increasingly spatially concentrated since 1968. Rich and poor households are increasingly clustering together in different areas, and the ‘average’ group of households which are neither rich nor poor has gradually diminished in size during this period. As a result, poor, rich and ‘average’ households became progressively less likely to live next door to one another between 1971 and 2001