Exact and approximate results for deposition and annihilation processes on graphs

We consider random sequential adsorption processes where the initially empty sites of a graph are irreversibly occupied, in random order, either by monomers which block neighboring sites, or by dimers. We also consider a process where initially occupied sites annihilate their neighbors at random times. We verify that these processes are well defined on infinite graphs, and derive forward equations governing joint vacancy/occupation probabilities. Using these, we derive exact formulae for occupation probabilities and pair correlations in Bethe lattices. For the blocking and annihilation processes we also prove positive correlations between sites an even distance apart, and for blocking we derive rigorous lower bounds for the site occupation probability in lattices, including a lower bound of 1/3 for Z^2. We also give normal approximation results for the number of occupied sites in a large finite graph.Comment: Published at http://dx.doi.org/10.1214/105051604000000765 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimising Matrix Product State Simulations of Shor's Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on $r$, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.Comment: 8 pages, 2 figures, 2 tables. v2 using PDFLaTeX compiler. v3 to include extra references. v4 for publication in Quantu

A storage-based model of heterocyst commitment and patterning in cyanobacteria

When deprived of fixed nitrogen (fN), certain filamentous cyanobacteria differentiate nitrogen-fixing heterocysts. There is a large and dynamic fraction of stored fN in cyanobacterial cells, but its role in directing heterocyst commitment has not been identified. We present an integrated computational model of fN transport, cellular growth, and heterocyst commitment for filamentous cyanobacteria. By including fN storage proportional to cell length, but without any explicit cell-cycle effect, we are able to recover a broad and late range of heterocyst commitment times and we observe a strong indirect cell-cycle effect. We propose that fN storage is an important component of heterocyst commitment and patterning in filamentous cyanobacteria. The model allows us to explore both initial and steady-state heterocyst patterns. The developmental model is hierarchical after initial commitment: our only source of stochasticity is observed growth rate variability. Explicit lateral inhibition allows us to examine $\Delta$patS, $\Delta$hetN, and $\Delta$patN phenotypes. We find that $\Delta$patS leads to adjacent heterocysts of the same generation, while $\Delta$hetN leads to adjacent heterocysts only of different generations. With a shortened inhibition range, heterocyst spacing distributions are similar to those in experimental $\Delta$patN systems. Step-down to non-zero external fixed nitrogen concentrations is also investigated.Comment: This is an author-created, un-copyedited version of an article accepted for publication in Physical Biology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version will be available onlin

Bayesian regression discontinuity designs: Incorporating clinical knowledge in the causal analysis of primary care data

The regression discontinuity (RD) design is a quasi-experimental design that estimates the causal effects of a treatment by exploiting naturally occurring treatment rules. It can be applied in any context where a particular treatment or intervention is administered according to a pre-specified rule linked to a continuous variable. Such thresholds are common in primary care drug prescription where the RD design can be used to estimate the causal effect of medication in the general population. Such results can then be contrasted to those obtained from randomised controlled trials (RCTs) and inform prescription policy and guidelines based on a more realistic and less expensive context. In this paper we focus on statins, a class of cholesterol-lowering drugs, however, the methodology can be applied to many other drugs provided these are prescribed in accordance to pre-determined guidelines. NHS guidelines state that statins should be prescribed to patients with 10 year cardiovascular disease risk scores in excess of 20%. If we consider patients whose scores are close to this threshold we find that there is an element of random variation in both the risk score itself and its measurement. We can thus consider the threshold a randomising device assigning the prescription to units just above the threshold and withholds it from those just below. Thus we are effectively replicating the conditions of an RCT in the area around the threshold, removing or at least mitigating confounding. We frame the RD design in the language of conditional independence which clarifies the assumptions necessary to apply it to data, and which makes the links with instrumental variables clear. We also have context specific knowledge about the expected sizes of the effects of statin prescription and are thus able to incorporate this into Bayesian models by formulating informative priors on our causal parameters.Comment: 21 pages, 5 figures, 2 table

Rectangular Hierarchical Cartograms for Socio-Economic Data

We present rectangular hierarchical cartograms for mapping socio-economic data. These density-normalising cartograms size spatial units by population, increasing the ease with which data for densely populated areas can be visually resolved compared to more conventional cartographic projections. Their hierarchical nature enables the study of spatial granularity in spatial hierarchies, hierarchical categorical data and multivariate data through false hierarchies. They are space-filling representations that make efficient use of space and their rectangular nature (which aims to be as square as possible) improves the ability to compare the sizes (therefore population) of geographical units. We demonstrate these cartograms by mapping the Office for National Statistics Output Area Classification (OAC) by unit postcode (1.52 million in Great Britain) through the postcode hierarchy, using these to explore spatial variation. We provide rich and detailed spatial summaries of socio-economic characteristics of population as types of treemap, exploring the effects of reconfiguring them to study spatial and non-spatial aspects of the OAC classification

Franch-Canadian Communities in the American Upper Midwest during the Nineteenth Century

La distribution des établissements canadiens-français dans l'Ouest américain au XIXe siècle n'avait aucune relation avec la distribution des postes de traite de fourrures du siècle précédent. Les Canadiens français du XIXe siècle furent attirés par les opportunités d'emploi le long des frontières agricole, minière et forestière. Détroit, Chicago et Minneapolis-St-Paul ont vu naître des paroisses canadiennes-françaises qui entretenaient des liens avec les communautés rurales. Le maintien de la langue française, l'héritage culturel et l'affiliation à l'église catholique variaient à travers la région. L'américanisation des Canadiens français fut directement reliée à leur réussite économique. Une identité canadienne-française a été maintenue dans les régions rurales les plus pauvres et marginales du nord des états du Michigan, du Winsconsin et du Minnesota.The pattern of nineteenth-century French-Canadian settlements in the American Midwest bore no relation to the pattern of fur-trading posts of the eighteenth century. French-Canadians of the nine-teenth century were attracted by employment opportunities along the farming, lumbering, and mining frontiers. Detroit, Chicago, and Minneapolis-St. Paul developed French-Canadian parishes which maintained links with rural communities. Survival of the French language, cultural heritage, and affiliation with the Catholic Church varied throughout the region. Americanization of French-Canadians went hand in hand with their commercial success. A French-Canadian identity survived in the poorest, marginal, rural areas of northern Michigan, Wisconsin, and Minnesota