Natural clays have been used in ancient and modern medicine, but the mechanism(s) that make certain clays lethal against bacterial pathogens has not been identified. We have compared the depositional environments, mineralogies, and chemistries of clays that exhibit antibacterial effects on a broad spectrum of human pathogens including antibiotic resistant strains. Natural antibacterial clays contain nanoscale (<200 nm), illite-smectite and reduced iron phases. The role of clay minerals in the bactericidal process is to buffer the aqueous pH and oxidation state to conditions that promote Fe2+ solubility. Chemical analyses of E. coli killed by aqueous leachates of an antibacterial clay show that intracellular concentrations of Fe and P are elevated relative to controls. Phosphorus uptake by the cells supports a regulatory role of polyphosphate or phospholipids in controlling Fe2+. Fenton reaction products can degrade critical cell components, but we deduce that extracellular processes do not cause cell death. Rather, Fe2+ overwhelms outer membrane regulatory proteins and is oxidized when it enters the cell, precipitating Fe3+ and producing lethal hydroxyl radicals.

We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Carathéodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.

We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.1. Introduction. This paper is a contribution to the theory of stochastic flows in one dimension. The main result is Theorem 6.2. It establishes weak convergence of a certain class of discrete-time stochastic flows on the circle, which we call disturbance flows, to the coalescing Brownian flow. This is motivated by a surprising connection with a model of Hastings and Levitov [9] for planar aggregation, which is worked out in our companion paper [15]. In this model, the flow of harmonic measure on the cluster boundary is a disturbance flow, and our convergence theorem then shows that the random structure of fingers in the Hasting-Levitov cluster is well described in the small-particle limit by the coalescing Brownian flow.

Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distributiondriven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty.In this paper we propose an analytic approach to problem-driven scenario generation. This approach applies to stochastic programs where a tail risk measure, such as conditional value-at-risk, is applied to a loss function. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread their scenarios evenly across the support of the solution, struggle to adequately represent tail risk. Our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions. We provide conditions under which our approach is consistent with sampling, and as proof-of-concept demonstrate how our approach could be applied to two classes of problem. Numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard Monte Carlo sampling. arXiv:1511.03074v3 [math.OC] 25 Apr 2018 Tail risk measures and risk regionsIn this section we present the core theory to our scenario generation methodology. Specifically, in Section 3.1 we formally define tail-risk measures of random variables and in Section 3.2 we define risk regions and present some key results related to these.

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by |Φ ′ n | −η , where Φ n is the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for η > 1, aggregating particles attach to their immediate predecessors with high probability, while for η < 1 almost surely this does not happen. * sola@math.su.se † a.g.turner@lancaster.ac.uk