Box-ball system: soliton and tree decomposition of excursions

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma in 1990. Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari, Nguyen, Rolla and Wang 2018 proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors have proposed a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In this paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall 2005. A ball configuration distributed as independent Bernoulli variables of parameter $\lambda<1/2$ is in correspondence with a simple random walk with negative drift $2\lambda-1$ and infinitely many excursions over the local minima. In this case the authors have proven that the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables. We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.Comment: 47 pages, 33 figures. This is the revised version after addressing referee reports. This version will be published in the special volume of the XIII Simposio de Probabilidad y Procesos Estoc\'asticos, UNAM Mexico, by Birkhause

A random tunnel number one 3-manifold does not fiber over the circle

We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December 200

Random division of an interval

The well-known relation between random division of an interval and the Poisson process is interpreted as a Laplace transformation. With the use of this interpretation a number of (in part known) results is derived very easily

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors

Random Neighbor Theory of the Olami-Feder-Christensen Earthquake Model

We derive the exact equations of motion for the random neighbor version of the Olami-Feder-Christensen earthquake model in the infinite-size limit. We solve them numerically, and compare with simulations of the model for large numbers of sites. We find perfect agreement. But we do not find any scaling or phase transitions, except in the conservative limit. This is in contradiction to claims by Lise & Jensen (Phys. Rev. Lett. 76, 2326 (1996)) based on approximate solutions of the same model. It indicates again that scaling in the Olami-Feder-Christensen model is only due to partial synchronization driven by spatial inhomogeneities. Finally, we point out that our method can be used also for other SOC models, and treat in detail the random neighbor version of the Feder-Feder model.Comment: 18 pages, 6 ps-figures included; minor correction in sec.

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure

Early Detection of Tuberculosis Outbreaks among the San Francisco Homeless: Trade-Offs Between Spatial Resolution and Temporal Scale

BACKGROUND: San Francisco has the highest rate of tuberculosis (TB) in the U.S. with recurrent outbreaks among the homeless and marginally housed. It has been shown for syndromic data that when exact geographic coordinates of individual patients are used as the spatial base for outbreak detection, higher detection rates and accuracy are achieved compared to when data are aggregated into administrative regions such as zip codes and census tracts. We examine the effect of varying the spatial resolution in the TB data within the San Francisco homeless population on detection sensitivity, timeliness, and the amount of historical data needed to achieve better performance measures. METHODS AND FINDINGS: We apply a variation of space-time permutation scan statistic to the TB data in which a patient's location is either represented by its exact coordinates or by the centroid of its census tract. We show that the detection sensitivity and timeliness of the method generally improve when exact locations are used to identify real TB outbreaks. When outbreaks are simulated, while the detection timeliness is consistently improved when exact coordinates are used, the detection sensitivity varies depending on the size of the spatial scanning window and the number of tracts in which cases are simulated. Finally, we show that when exact locations are used, smaller amount of historical data is required for training the model. CONCLUSION: Systematic characterization of the spatio-temporal distribution of TB cases can widely benefit real time surveillance and guide public health investigations of TB outbreaks as to what level of spatial resolution results in improved detection sensitivity and timeliness. Trading higher spatial resolution for better performance is ultimately a tradeoff between maintaining patient confidentiality and improving public health when sharing data. Understanding such tradeoffs is critical to managing the complex interplay between public policy and public health. This study is a step forward in this direction

Spatial variation and hot-spots of district level diarrhea incidences in Ghana: 2010–2014

Background: Diarrhea is a public health menace, especially in developing countries. Knowledge of the biological and anthropogenic characteristics is abundant. However, little is known about its spatial patterns especially in developing countries like Ghana. This study aims to map and explore the spatial variation and hot-spots of district level diarrhea incidences in Ghana. Methods: Data on district level incidences of diarrhea from 2010 to 2014 were compiled together with population data. We mapped the relative risks using empirical Bayesian smoothing. The spatial scan statistics was used to detect and map spatial and space-Time clusters. Logistic regression was used to explore the relationship between space-Time clustering and urbanization strata, i.e. rural, peri-urban, and urban districts. Results: We observed substantial variation in the spatial distribution of the relative risk. There was evidence of significant spatial clusters with most of the excess incidences being long-Term with only a few being emerging clusters. Space-Time clustering was found to be more likely to occur in peri-urban districts than in rural and urban districts. Conclusion: This study has revealed that the excess incidences of diarrhea is spatially clustered with peri-urban districts showing the greatest risk of space-Time clustering. More attention should therefore be paid to diarrhea in peri-urban districts. These findings also prompt public health officials to integrate disease mapping and cluster analyses in developing location specific interventions for reducing diarrhea

A Spatial Cluster Analysis of Tractor Overturns in Kentucky from 1960 to 2002

Agricultural tractor overturns without rollover protective structures are the leading cause of farm fatalities in the United States. To our knowledge, no studies have incorporated the spatial scan statistic in identifying high-risk areas for tractor overturns. The aim of this study was to determine whether tractor overturns cluster in certain parts of Kentucky and identify factors associated with tractor overturns.A spatial statistical analysis using Kulldorff's spatial scan statistic was performed to identify county clusters at greatest risk for tractor overturns. A regression analysis was then performed to identify factors associated with tractor overturns.The spatial analysis revealed a cluster of higher than expected tractor overturns in four counties in northern Kentucky (RR = 2.55) and 10 counties in eastern Kentucky (RR = 1.97). Higher rates of tractor overturns were associated with steeper average percent slope of pasture land by county (p = 0.0002) and a greater percent of total tractors with less than 40 horsepower by county (p<0.0001).This study reveals that geographic hotspots of tractor overturns exist in Kentucky and identifies factors associated with overturns. This study provides policymakers a guide to targeted county-level interventions (e.g., roll-over protective structures promotion interventions) with the intention of reducing tractor overturns in the highest risk counties in Kentucky

Spatial variations in the incidence of breast cancer and potential risks associated with soil dioxin contamination in Midland, Saginaw, and Bay Counties, Michigan, USA

<p>Abstract</p> <p>Background</p> <p>High levels of dioxins in soil and higher-than-average body burdens of dioxins in local residents have been found in the city of Midland and the Tittabawassee River floodplain in Michigan. The objective of this study is threefold: (1) to evaluate dioxin levels in soils; (2) to evaluate the spatial variations in breast cancer incidence in Midland, Saginaw, and Bay Counties in Michigan; (3) to evaluate whether breast cancer rates are spatially associated with the dioxin contamination areas.</p> <p>Methods</p> <p>We acquired 532 published soil dioxin data samples collected from 1995 to 2003 and data pertaining to female breast cancer cases (<it>n </it>= 4,604) at ZIP code level in Midland, Saginaw, and Bay Counties for years 1985 through 2002. Descriptive statistics and self-organizing map algorithm were used to evaluate dioxin levels in soils. Geographic information systems techniques, the Kulldorff's spatial and space-time scan statistics, and genetic algorithms were used to explore the variation in the incidence of breast cancer in space and space-time. Odds ratio and their corresponding 95% confidence intervals, with adjustment for age, were used to investigate a spatial association between breast cancer incidence and soil dioxin contamination.</p> <p>Results</p> <p>High levels of dioxin in soils were observed in the city of Midland and the Tittabawassee River 100-year floodplain. After adjusting for age, we observed high breast cancer incidence rates and detected the presence of spatial clusters in the city of Midland, the confluence area of the Tittabawassee, and Saginaw Rivers. After accounting for spatiotemporal variations, we observed a spatial cluster of breast cancer incidence in Midland between 1985 and 1993. The odds ratio further suggests a statistically significant (<it>α </it>= 0.05) increased breast cancer rate as women get older, and a higher disease burden in Midland and the surrounding areas in close proximity to the dioxin contaminated areas.</p> <p>Conclusion</p> <p>These findings suggest that increased breast cancer incidences are spatially associated with soil dioxin contamination. Aging is a substantial factor in the development of breast cancer. Findings can be used for heightened surveillance and education, as well as formulating new study hypotheses for further research.</p