Wald for non-stopping times: The rewards of impatient prophets

Let X-1 , X-2 , ... be independent identically distributed nonnegative random variables. Wald's identity states that the random sum S-T := X-1 + ... + X-T has expectation ET . EX1 provided T is a stopping time. We prove here that for any 1 < alpha <= 2, if T is an arbitrary nonnegative random variable, then S-T has finite expectation provided that X-1 has finite alpha-moment and T has finite 1/(alpha - 1)-moment. We also prove a variant in which T is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence X-i violating them, there is a T satisfying the given condition for which S-T (and, in fact, X-T) has infinite expectation. An interpretation is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Bootstrap Percolation on Complex Networks

We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any $f>0$ and $p>0$, and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure

Phylogenetic relationships of the Wolbachia of nematodes and arthropods

Wolbachia are well known as bacterial symbionts of arthropods, where they are reproductive parasites, but have also been described from nematode hosts, where the symbiotic interaction has features of mutualism. The majority of arthropod Wolbachia belong to clades A and B, while nematode Wolbachia mostly belong to clades C and D, but these relationships have been based on analysis of a small number of genes. To investigate the evolution and relationships of Wolbachia symbionts we have sequenced over 70 kb of the genome of wOvo, a Wolbachia from the human-parasitic nematode Onchocerca volvulus, and compared the genes identified to orthologues in other sequenced Wolbachia genomes. In comparisons of conserved local synteny, we find that wBm, from the nematode Brugia malayi, and wMel, from Drosophila melanogaster, are more similar to each other than either is to wOvo. Phylogenetic analysis of the protein-coding and ribosomal RNA genes on the sequenced fragments supports reciprocal monophyly of nematode and arthropod Wolbachia. The nematode Wolbachia did not arise from within the A clade of arthropod Wolbachia, and the root of the Wolbachia clade lies between the nematode and arthropod symbionts. Using the wOvo sequence, we identified a lateral transfer event whereby segments of the Wolbachia genome were inserted into the Onchocerca nuclear genome. This event predated the separation of the human parasite O. volvulus from its cattle-parasitic sister species, O. ochengi. The long association between filarial nematodes and Wolbachia symbionts may permit more frequent genetic exchange between their genomes

Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks

We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-$k$-core process: the giant heterogeneous-$k$-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-$k$-core and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$ when the degree distribution tends to a power law $P(q) \sim q^{-\gamma}$ with $\gamma < 3$.Comment: 10 pages, 4 figure

Design for 'domestication': the decommercialisation of traditional crafts

This paper explores the contribution of design to the 'domestication' of traditional crafts: the reframing and support of such practices as amateur activities. Informed by twelve examples, six design strategies for the domestication of traditional crafts are identified and discussed. This issue emerges from a research project investigating the role of design in developing and revitalising culturally significant designs, products and associated practices. Within this paper, we focus on strategies that seek to revitalise traditional crafts by supporting domestic activity. This topic is introduced through a discussion of commercialisation, a more common approach to revitalisation. Two contemporary social trends support domestication: the strong interest – particularly in post-industrial countries – in provenance, local distinctiveness and authenticity; and the growth of maker culture and its ethos of amateur creativity. We gather twelve examples of various formats – such as books, kits, online communities, videos, workshops and holidays – which support amateur activity. The examples are analysed via a matrix, which considers their characteristics in terms of two variables: the way in which knowledge is exchanged, and the degree of experimentation facilitated by the activity. By categorising the examples, we identify six domestication strategies, each of which involves a different combination of design activities. Finally, we discuss domestication in terms of skill and innovation, arguing that amateur practice has much to offer in both respects

Stitching together: participatory textile making as an emerging methodological approach to research

Arising from a recently formed research network, Stitching Together, this article introduces a collection of case studies that critically examine participatory textile making as an emerging methodological approach to research. The twenty-first century resurgence of interest in textile processes such as knitting, sewing and weaving, whether as individual practice or community-based initiative, builds on a long and culturally diverse history of collaborative textile-making activity. This resurgence, combined with the familiarity, accessibility and flexibility of textile practices, has influenced a recent growth in the use of such activities as a means of inquiry within diverse research contexts. The article considers the ways in which collective textile making projects privilege social encounter as a format for learning skills, creating friendships and consolidating shared interests. It goes on to discuss how researchers are drawing on these characteristics when devising new projects, highlighting the quality of experience afforded by textile making, the diverse forms of data generated and the variety of ways in which these participatory activities can be set up. Recognising that this research approach is far from straightforward, three key methodological themes are then considered: the multifaceted nature of the researcher’s role and the complexities of relationships with participants and other stakeholders; the difficulties that can arise when using such familiar textile processes; and the opportunities, and complexities, of co-producing knowledge with participants through collaborative textile activity

Textiles as Material Gestalt: Cloth as a Catalyst in the Co-designing Process

Textiles is the common language within Emotional Fit, a collaborative research project investigating a person-centred, sustainable approach to fashion for an ageing female demographic (55+). Through the co-designing of a collection of research tools, textiles have acted as a material gestalt for exploring our research participants' identities by tracing their embodied knowledge of fashionable dress. The methodology merges Interpretative Phenomenological Analysis, co-design and a simultaneous approach to textile and garment design. Based on an enhanced understanding of our participants textile preferences, particular fabric qualities have catalysed silhouettes, through live draping and geometric pattern cutting to accommodate multiple body shapes and customisation. Printedtextiles have also been digitally crafted in response to the contours of the garment and body and personal narratives of wear. Sensorial and tactile interactions have informed the engineering and scaling of patterns within zero-waste volumes. The article considers the functional and aesthetic role of textiles

On the Sandpile group of the cone of a graph

In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d.Comment: 20 pages, 11 figures. The title was changed, other impruvements were made throughout the article. To appear in Linear Algebra and Its Application