The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

RNAG: a new Gibbs sampler for predicting RNA secondary structure for unaligned sequences

Motivation: RNA secondary structure plays an important role in the function of many RNAs, and structural features are often key to their interaction with other cellular components. Thus, there has been considerable interest in the prediction of secondary structures for RNA families. In this article, we present a new global structural alignment algorithm, RNAG, to predict consensus secondary structures for unaligned sequences. It uses a blocked Gibbs sampling algorithm, which has a theoretical advantage in convergence time. This algorithm iteratively samples from the conditional probability distributions P(Structure | Alignment) and P(Alignment | Structure). Not surprisingly, there is considerable uncertainly in the high-dimensional space of this difficult problem, which has so far received limited attention in this field. We show how the samples drawn from this algorithm can be used to more fully characterize the posterior space and to assess the uncertainty of predictions

Landholder Typologies Used in the Development of Natural Resource Management Programs in Australia - A Review

This article reviews the literature on the identification of landholder typologies that can be used to assist the design and delivery of natural resource management (NRM) programs. Australian researchers have developed typologies of landholders based on a variety of criteria. The rationale for developing landholder typologies is first discussed before reviewing the various approaches that have been used by Australian researchers and comparing their findings. The methods employed have differed according to the theories used to guide the research and the 'clients' or 'sponsors' of the research. The landholder types they describe, however, have a number of similarities. These similarities suggest that the studies have identified the same fundamental divisions in the rural community, and that it may be possible to integrate landholder typologies for a variety of NRM and non-NRM applications. It is concluded that further research could usefully investigate whether concepts of social class or sub-cultures may be appropriate to define and describe the variations in landholder types

Trihydrophobin 1 Phosphorylation by c-Src Regulates MAPK/ERK Signaling and Cell Migration

c-Src activates Ras-MAPK/ERK signaling pathway and regulates cell migration, while trihydrophobin 1 (TH1) inhibits MAPK/ERK activation and cell migration through interaction with A-Raf and PAK1 and inhibiting their kinase activities. Here we show that c-Src interacts with TH1 by GST-pull down assay, coimmunoprecipitation and confocal microscopy assay. The interaction leads to phosphorylation of TH1 at Tyr-6 in vivo and in vitro. Phosphorylation of TH1 decreases its association with A-Raf and PAK1. Further study reveals that Tyr-6 phosphorylation of TH1 reduces its inhibition on MAPK/ERK signaling, enhances c-Src mediated cell migration. Moreover, induced tyrosine phosphorylation of TH1 has been found by EGF and estrogen treatments. Taken together, our findings demonstrate a novel mechanism for the comprehensive regulation of Ras/Raf/MEK/ERK signaling and cell migration involving tyrosine phosphorylation of TH1 by c-Src

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Social capital and natural resources management: an application to landcare

Social capital has been increasingly recognised as a fundamental ingredient to the success of natural resource management (NRM) within rural and regional communities. Landcare, with its extensive local group structure and community-state partnership, provides a notable case of the importance of social capital to NRM. However identifying processes and practices to enhance social capital have been problematic given the theoretical and empirical ambiguities surrounding the concept. In this paper we apply an analytical framework recognising the sources, characteristics and consequences of social capital to Landcare. The framework recognises the importance of both community and state/institutional sources of social capital and suggests a series of measures of social capital appropriate to Landcare