Approximately Counting Embeddings into Random Graphs

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme. We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put the first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n-1 queues in tandem with n-1 priority-types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multi-line process and its invariant measure the transformation described above of the product measure.Comment: 21 pages, 6 figure

The Interpretation of Near-Infrared Star Counts at the South Galactic Pole

We present new deep $K'$ counts of stars at the South Galactic Pole (SGP) taken with the NAOJ PICNIC camera to $K'=17.25$. Star-galaxy separation to $K'=17.5$ was accomplished effectively using image profiles because the pixel size we used is 0.509 arcsec. We interpret these counts using the SKY (Cohen 1994) model of the Galactic point source sky and determine the relative normalization of halo-to-disk populations, and the location of the Sun relative to the Galactic plane, within the context of this model. The observed star counts constrain these parameters to be: halo/disk $\sim$ 1/900 and z$_\odot$=16.5$\pm$2.5 pc. These values have been used to correct our SGP galaxy counts for contamination by the point source Galactic foreground.Comment: accepted for publication in AJ, 15 pages with 2 figure

Modeling Star counts in the Monoceros stream and the Galactic anti-centre

There is a continued debate as to the form of the outer disc of the Milky Way galaxy, which has important implications for its formation. Stars are known to exist at a galacto-centric distance of at least 20 kpc. However, there is much debate as to whether these stars can be explained as being part of the disc or whether another extra galactic structure, the so called Monoceros ring/stream, is required. To examine the outer disc of the Galaxy toward the anti-centre to determine whether the star counts can be explained by the thin and thick discs alone. Using Sloan star counts and extracting the late F and early G dwarfs it is possible to directly determine the density of stars out to a galacto-centric distance of about 25 kpc. These are then compared with a simple flared disc model. A flared disc model is shown to reproduce the counts along the line of sights examined, if the thick disc does not have a sharp cut off. The flare starts at a Galacto-centric radius of 16 kpc and has a scale length of 4.5+/-1.5 kpc. Whilst the interpretation of the counts in terms of a ring/stream cannot be definitely discounted, it does not appear to be necessary, at least along the lines of sight examined towards the anti centre.Comment: 11 pages, 4 figures, accepted to be published in A&

What is research with children and young people?

Research with children and young people has a long history, going back at least to the end of the nineteenth century (de Landsheere 1988). However, its character has changed over time, and become more diverse. This reflects the range of different areas and disciplines involved, such as the study of health and education, of psychology, anthropology and sociology. In this chapter we will look, first of all, at the nature of research: the aim will be to give you an initial sense of the different approaches to inquiry that you will find in the literature about children and young people, the ideas underpinning them, and the debates that surround them. You will be introduced to these in more detail in later chapters. In the second half of the chapter we will look at what it means to do research with children and young people. In particular, since it has become very influential, we will pay close attention to what is referred to as ‘participatory research’. As will become clear, this kind of work highlights some important methodological, ethical and political issues

Localization criteria for Anderson models on locally finite graphs

We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on \ZZ^d. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

A study of the social and physical environment in catering kitchens and the role of the chef in promoting positive health and safety behaviour

This is the account of a mixed method study of chefs and their kitchens in order to identify the nature of their workplace and how this affects their ability to manage health and safety in the kitchen. It included extended periods of observation, monitoring of physical parameters, analysis of records of reported accidents, and a series of reflexive interviews. The findings were integrated and then fed back in a smaller number of second interviews in order to test whether the findings fitted in with the chefs' understanding of their world. Major factors identified included survival in a market environment, the status of the chef (and the kitchen) within organisations, marked autocracy of chefs, and an increasing tempo building up to service time with commensurate heat, noise, and activity. In particular during the crescendo, a threshold shift in risk tolerance was identified. The factors, their interplay, and their implications for health and safety in the catering kitchen are discussed

On the effect of random errors in gridded bathymetric compilations

We address the problem of compiling bathymetric data sets with heterogeneous coverage and a range of data measurement accuracies. To generate a regularly spaced grid, we are obliged to interpolate sparse data; our objective here is to augment this product with an estimate of confidence in the interpolated bathymetry based on our knowledge of the component of random error in the bathymetric source data. Using a direct simulation Monte Carlo method, we utilize data from the International Bathymetric Chart of the Arctic Ocean database to develop a suitable methodology for assessment of the standard deviations of depths in the interpolated grid. Our assessment of random errors in each data set are heuristic but realistic and are based on available metadata from the data providers. We show that a confidence grid can be built using this method and that this product can be used to assess reliability of the final compilation. The methodology as developed here is applied to bathymetric data but is equally applicable to other interpolated data sets, such as gravity and magnetic data

Religion as practices of attachment and materiality: the making of Buddhism in contemporary London

This article aims to explore Buddhism’s often-overlooked presence on London’s urban landscape, showing how its quietness and subtlety of approach has allowed the faith to grow largely beneath the radar. It argues that Buddhism makes claims to urban space in much the same way as it produces its faith, being as much about the practices performed and the spaces where they are enacted as it is about faith or beliefs. The research across a number of Buddhist sites in London reveals that number of people declaring themselves as Buddhists has indeed risen in recent years, following the rise of other non-traditional religions in the UK; however, this research suggests that Buddhism differs from these in several ways. Drawing on Baumann’s (2002) distinction between traditionalist and modernist approaches to Buddhism, our research reveals a growth in each of these. Nevertheless, Buddhism remains largely invisible in the urban and suburban landscape of London, adapting buildings that are already in place, with little material impact on the built environment, and has thus been less subject to contestation than other religious movements and traditions. This research contributes to a growing literature which foregrounds the importance of religion in making contemporary urban and social worlds