Measure solutions for the Smoluchowski coagulation-diffusion equation

A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. Sufficient conditions are identified on the diffusivity, coagulation rates and initial data for existence, uniqueness and mass conservation of solutions. These conditions impose no form of monotonicity on the coagulation kernel, which may depend on complex characteristics of the particles. They also allow singular behaviour in both diffusivity and coagulation rates for small particles. The general results apply to the Einstein-Smoluchowski model for colloidal particles suspended in a fluid

Essential edges in Poisson random hypergraphs

Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process "identifiable". Also any hyperedge all of whose vertices are removed is called "identifiable". We say that a hyperedge is "essential" if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal sub-graph with minimal vertex degree 2.Comment: 12 pages, 3 figures. Revised version with minor corrections/clarifications and slightly expanded introductio

Weak convergence of the localized disturbance flow to the coalescing Brownian flow

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.Comment: Published at http://dx.doi.org/10.1214/13-AOP845 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:0810.021

Surprise probabilities in Markov chains

In a Markov chain started at a state $x$, the hitting time $\tau(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(\tau(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with $n$ states, $\mathbf{P}_x(\tau(y) = t) \le \frac{n}{t}$. 2) In a reversible chain with $n$ states, $\mathbf{P}_x(\tau(y) = t) \le \frac{\sqrt{2n}}{t}$ for $t \ge 4n + 4$. 3) For random walk on a simple graph with $n \ge 2$ vertices, $\mathbf{P}_x(\tau(y) = t) \le \frac{4e \log n}{t}$. We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain. To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): For random walk on an $n$-vertex graph, for every initial vertex $x$, \[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \

Large deviations for the Yang-Mills measure on a compact surface

We prove the first mathematical result relating the Yang-Mills measure on a compact surface and the Yang-Mills energy. We show that, at the small volume limit, the Yang-Mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the Yang-Mills energy

Structural Identification of an Elevated Water Tower

Elevated water tanks are generally located at higher elevations within a particular geographic area. The location and height of these structures often makes them desirable for installing wireless and cellular communications antennas. The impact of this practice on the long-term serviceability performance of the primary structure is not clear and could be an important consideration deserving further analysis. This paper presents a case study of an elevated water tank that was motivated by this particular consideration. The structure had been retrofitted with cellular antennas during its service and later experienced fatigue cracking at the fill pipe-tank interface. The owner wished to determine if the addition of the cellular antennas had contributed to this damage. This paper presents a structural identification program that was implemented to determine how wind, water level, and antenna modifications affect the water tank