Linear-Size Approximations to the Vietoris-Rips Filtration

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees

Fulde-Ferrell-Larkin-Ovchinnikov state of two-dimensional imbalanced Fermi gases

The ground-state phase diagram of attractively-interacting Fermi gases in two dimensions with a population imbalance is investigated. We find the regime of stability for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, in which pairing occurs at finite wave vector, and determine the magnitude of the pairing amplitude $\Delta$ and FFLO wavevector $q$ in the ordered phase, finding that $\Delta$ can be of the order of the two-body binding energy. Our results rely on a careful analysis of the zero temperature gap equation for the FFLO state, which possesses nonanalyticities as a function of $\Delta$ and $q$, invalidating a Ginzburg-Landau expansion in small $\Delta$.Comment: 16 pages. 14 figure files. Submitted to Physical Review

Attitudes to Makaton in the ages on integration and inclusion

The Makaton Vocabulary was developed in the 1970’s and became, and has remained, one of most pervasive and influential pedagogical approaches for children with severe learning difficulties. This article looks at attitudes towards Makaton and compares findings from two studies, carried out in a sample of special schools in the south west of England during 1986 and 1995. Overall, the results suggest that attitudes towards the use of Makaton signs have become more positive. Makaton signs are now regarded, overall, as supporting and facilitating language development, and earlier concerns about stigmatisation have declined. There is some evidence to suggest that this latter change is influenced by changes in attitudes to British Sign Language. The 1986 study predicted that new technology would have a significant impact on attitudes to language and communication systems such as Makaton, but this prediction was not supported in the 2005 study. The article highlights also how different attitudes towards Makaton can exist within the same school, and how this situation can have a significant impact on the educational experiences and opportunities of children with severe learning difficulties. The article concludes that the apparent educational movements of integration or inclusion produce different attitudes towards Makaton and how it is used. However, although Makaton signing has become seen as a tool to create educational inclusion, the extent to which the system itself has actually changed is a contentious issu

Visualising Textual Knowledge about Risks to Aid Risk Communication

This paper demonstrates a potential application for latent semantic analysis and similar techniques in visualising the differences between two levels of knowledge about a risk issue. The HIV/AIDS risk issue will be examined and the semantic clusters of key words in a technical corpora derived from specific literature about HIV/AIDS will be compared with the semantic clusters of those in more general corpora. It is hoped that these comparisons will create a fast and efficient complementary approach to the articulation of mental models of risk issues that could be used to target possible inconsistencies between expert and lay mental models

Comment on "Superfluid stability in the BEC-BCS crossover"

We point out an error in recent work by Pao, Wu, and Yip [Phys. Rev.B {\bf 73}, 132506 (2006)], that stems from their use of a necessary but not sufficient condition [positive compressibility (magnetic susceptibility) and superfluid stiffness] for the stability of the ground state of a polarized Fermi gas. As a result, for a range of detunings their proposed ground-state solution is a local maximum rather than a minimum of the ground state energy, which thereby invalidates their proposed phase diagram for resonantly interacting fermions under an imposed population difference.Comment: 4 pages, 2 figures. Version in PR

Developing Point of Learning : an innovative approach to enhancing professional learning

In this paper we: * expand upon some of the issues with current approaches used in the development and assessment of professional practice, * outline the current emphasis upon a life of learning and why we need a new approach to assessment that enhances the development of professional practice, * introduce and provide a brief overview of Point of Learning (PoL) – a new approach that builds upon our growing understanding of learning and takes advantage of the power of digital technology, * explain how PoL overcomes problems with existing approaches, * develop the theoretical underpinning for this new approach and * present the overarching structure in which this new model can be developed. This is illustrated with an example from the schools sector – though it is important to remember that PoL is applicable to any professional development context

Mass Media and Polarisation Processes in the Bounded Confidence Model of Opinion Dynamics

This paper presents a social simulation in which we add an additional layer of mass media communication to the social network \'bounded confidence\' model of Deffuant et al (2000). A population of agents on a lattice with continuous opinions and bounded confidence adjust their opinions on the basis of binary social network interactions between neighbours or communication with a fixed opinion. There are two mechanisms for interaction. \'Social interaction\' occurs between neighbours on a lattice and \'mass communication,\' adjusts opinions based on an agent interacting with a fixed opinion. Two new variables are added, polarisation: the degree to which two mass media opinions differ, and broadcast ratio: the number of social interactions for each mass media communication. Four dynamical regimes are observed, fragmented, double extreme convergence, a state of persistent opinion exchange leading to single extreme convergence and a disordered state. Double extreme convergence is found where agents are less willing to change opinion and mass media communications are common or where there is moderate willingness to change opinion and a high frequency of mass media communications. Single extreme convergence is found where there is moderate willingness to change opinion and a lower frequency of mass media communication. A period of persistent opinion exchange precedes single extreme convergence, it is characterized by the formation of two opposing groups of opinion separated by a gradient of opinion exchange. With even very low frequencies of mass media communications this results in a move to central opinions followed by a global drift to one extreme as one of the opposing groups of opinion dominates. A similar pattern of findings is observed for Neumann and Moore neighbourhoods.Opinion Dynamics, Mass Media, Polarisation, Extremists, Consensus