Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and $\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and $x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$ the $\beta$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.Comment: 38 page

Multimode optical feedback dynamics in InAs/GaAs quantum dot lasers emitting exclusively on ground or excited states: transition from short- to long-delay regimes

© 2018 Optical Society of America. Users may use, reuse, and build upon the article, or use the article for text or data mining, so long as such uses are for non-commercial purposes and appropriate attribution is maintained. All other rights are reserved.The optical feedback dynamics of two multimode InAs/GaAs quantum dot lasers emitting exclusively on sole ground or excited lasing states is investigated. The transition from long- to short-delay regimes is analyzed, while the boundaries associated to the birth of periodic and chaotic oscillations are unveiled to be a function of the external cavity length. The results show that depending on the initial lasing state, different routes to chaos are observed. These results are of importance for the development of isolator-free transmitters in short-reach networks

American Twitter Users Revealed Social Determinants-related Oral Health Disparities amid the COVID-19 Pandemic

Objectives: To assess self-reported population oral health conditions amid COVID-19 pandemic using user reports on Twitter. Method and Material: We collected oral health-related tweets during the COVID-19 pandemic from 9,104 Twitter users across 26 states (with sufficient samples) in the United States between November 12, 2020 and June 14, 2021. We inferred user demographics by leveraging the visual information from the user profile images. Other characteristics including income, population density, poverty rate, health insurance coverage rate, community water fluoridation rate, and relative change in the number of daily confirmed COVID-19 cases were acquired or inferred based on retrieved information from user profiles. We performed logistic regression to examine whether discussions vary across user characteristics. Results: Overall, 26.70% of the Twitter users discuss wisdom tooth pain/jaw hurt, 23.86% tweet about dental service/cavity, 18.97% discuss chipped tooth/tooth break, 16.23% talk about dental pain, and the rest are about tooth decay/gum bleeding. Women and younger adults (19-29) are more likely to talk about oral health problems. Health insurance coverage rate is the most significant predictor in logistic regression for topic prediction. Conclusion: Tweets inform social disparities in oral health during the pandemic. For instance, people from counties at a higher risk of COVID-19 talk more about tooth decay/gum bleeding and chipped tooth/tooth break. Older adults, who are vulnerable to COVID-19, are more likely to discuss dental pain. Topics of interest vary across user characteristics. Through the lens of social media, our findings may provide insights for oral health practitioners and policy makers.Comment: Accepted for publication in Quintessence Internationa