Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space

Associated to any finite simple graph $\Gamma$ is the chromatic polynomial $P_\Gamma(q)$ whose complex zeroes are called the chromatic zeros of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the limiting measure of chromatic zeros associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added considerably more background on activity currents and especially on the Dujardin-Favre classification of the passive locus. Exposition in the proof of the main theorem was improved. Comments welcome

Gambling Participation and Prevalence Estimates of Pathological Gambling in a Far-East Gambling City: Macao

This research is the first scientific study of gambling participation and pathological gambling in Macao. In 2003, a computer-assisted random digit dialing method was used to conduct 1,121 telephone interviews. Two-thirds of respondents expressed that they have participated in at least one of the fourteen forms of gambling in the past year. The three most popular forms of gambling include social gambling, Mark Six lottery, and soccer/basketball betting. As for the prevalence of pathological gambling, the Chinese DSM-IV Gambling Behavior Index was used as the screening tool and results indicate that 1.78% of respondents are classified as probable pathological gamblers and 2.5% are classified as probable problem gamblers. Logistic Regression test indicates that male respondents with a monthly personal income of less than MOP 8,000 (Macau currency is the Pataca) tend to be more vulnerable to problem and pathological gambling when they participate in casino gambling, soccer matches betting and mahjong house gambling

The Appearance of the Rural in China's Tourism

State-led programs for rural development through tourism serve to reaffirm and reinstate rural spaces as the ideal periphery, a desirable and attractive decorative edge. to the modern, contemporary Chinese nation. By tracing the ways in which tourism development both centralizes the necessity of modernizing rural regions for the nation as a whole while simultaneously emphasizing the otherness. of rural communities in order to promote them as tourist attractions, in this essay I seek to understand how the rural. appears in Chinese tourism as a vital concern of the state by characterizing what is rural as increasingly different and distant in order to satisfy perceived tourist desires. In particular, the Chinese state represents village-based tourism projects through discourses of distance that render the rural. both absolutely critical to national processes of development and fundamentally peripheral to modern conditions

Some Connections Between Complex Dynamics and Statistical Mechanics

Indiana University-Purdue University Indianapolis (IUPUI)Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree

Statistical Analyses of Historical Pipeline Incident Data with Application to the Risk Assessment of Onshore Natural Gas Transmission Pipelines

Statistical analyses of the pipe-related incident data for onshore gas transmission pipelines between 2002 and 2013 collected by the Pipeline and Hazardous Material Safety Administration (PHMSA) of the United States Department of Transportation (DOT) are conducted. It is found that the total length of the onshore gas transmission pipelines in the US is approximately 480,000 km as of 2013. The third-party interference, external corrosion, material failure and internal corrosion are the leading causes for the pipe-related incidents, responsible for over 75% of the total incidents between 2002 and 2013. Based on the pipeline mileage and incident data, the average rate of rupture incidents over the 12-year period between 2002 and 2013 is calculated to be 3.1 × 10-5 per km per year. Furthermore, external corrosion is found to be the leading cause for rupture incidents, with a corresponding rupture rate of 1.0 × 10-5 per km per year A log-logistic model is developed to evaluate the probability of ignition (POI) given a rupture of an onshore gas transmission pipeline using the maximum likelihood method based on a total of 188 rupture incidents between 2002 and 2014 collected from the PHMSA pipeline incident database. The product of the pipeline internal pressure at the time of incident and outside diameter squared is observed to be strongly correlated to POI while the location class of the pipeline is not, and thus the former is adopted as the sole predictor in the model. The 95% confidence interval is evaluated, and for practical engineering use, the 95% upper confidence bound is tabulated in a look-up table. The proposed model is further validated using an independent dataset reported in the literature. The quantitative risk assessment of a hypothetical onshore gas transmission pipelines is illustrated by incorporating the statistics of the pipeline rupture incidents and POI model obtained in the present study. The thermal radiation hazards resulting from an ignited rupture of the pipeline are quantified using the well-known C-FER model. The heat intensity thresholds leading to fatality and injury for both the outdoor and indoor exposure conditions are selected from the literature. The societal risk is then evaluated in terms of the expected number of casualties and F-N curve for the population located in the vicinity of the pipeline, whereas the individual risk is calculated as the annual probability of casualty of a specific individual located in the vicinity of the pipeline. The F-N curve is evaluated for each one kilometer section of the pipeline such that the section corresponding to the most critical F-N curve is identified