Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n \to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.Comment: 12 pages, 2 figures, acroread recommended for figure displa

Dairy Product Manufacturing Costs at Cooperative Plants

Cost data are summarized for 14 plants manufacturing cheese, butter, and powder and average costs are presented for each product. Average cost curves are estimated for each plant. The scale of plant for least-cost operations is identified for plants of each product type. Plant capacity utilization and seasonal volume variation and their impacts on manufacturing cost are delineated.Cooperatives, dairy, average cost curve, productivity, capacity utilization, seasonal variation, economies of scale, Agribusiness,

Identifying Keystone Species in the Human Gut Microbiome from Metagenomic Timeseries using Sparse Linear Regression

Human associated microbial communities exert tremendous influence over human health and disease. With modern metagenomic sequencing methods it is possible to follow the relative abundance of microbes in a community over time. These microbial communities exhibit rich ecological dynamics and an important goal of microbial ecology is to infer the interactions between species from sequence data. Any algorithm for inferring species interactions must overcome three obstacles: 1) a correlation between the abundances of two species does not imply that those species are interacting, 2) the sum constraint on the relative abundances obtained from metagenomic studies makes it difficult to infer the parameters in timeseries models, and 3) errors due to experimental uncertainty, or mis-assignment of sequencing reads into operational taxonomic units, bias inferences of species interactions. Here we introduce an approach, Learning Interactions from MIcrobial Time Series (LIMITS), that overcomes these obstacles. LIMITS uses sparse linear regression with boostrap aggregation to infer a discrete-time Lotka-Volterra model for microbial dynamics. We tested LIMITS on synthetic data and showed that it could reliably infer the topology of the inter-species ecological interactions. We then used LIMITS to characterize the species interactions in the gut microbiomes of two individuals and found that the interaction networks varied significantly between individuals. Furthermore, we found that the interaction networks of the two individuals are dominated by distinct "keystone species", Bacteroides fragilis and Bacteroided stercosis, that have a disproportionate influence on the structure of the gut microbiome even though they are only found in moderate abundance. Based on our results, we hypothesize that the abundances of certain keystone species may be responsible for individuality in the human gut microbiome