Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

Vitrectomy vs. Spontaneous Closure for Traumatic Macular Hole: A Systematic Review and Meta-Analysis

Objective: This systematic review and meta-analysis aimed to determine the traumatic macular hole (TMH) closure rate and visual acuity (VA) improvement rate by comparing two treatment methods for TMH: vitrectomy and observation for spontaneous closure.Methods: PubMed, Cochrane, Web of Science Library, Embase, CNKI, Wanfang, VIP, and Sino Med were systematically searched from their inception to June 10, 2021. Studies in the surgery group (n = 32) and studies in the observation group (n = 12) were meta-analyzed. The primary outcomes were the TMH closure and VA improvement rates in the surgery and observation groups. The secondary outcomes were best-corrected visual acuity (BCVA) improvement in the surgery group. Stata software (version 15.1) was used for the analyses.Results: Thirty-six studies that included 1,009 eyes were selected for this meta-analysis, among which 33 were retrospective studies and 3 were prospective studies. The meta-analysis showed that the random-model pooled event rate for TMH closure was 0.37 (95% confidence interval [CI], 0.26–0.48) in the observation group, while it was 0.9 (95% CI, 0.85–0.94) in the surgery group. The fixed-model pooled event rate for VA improvement was 0.39 (95% CI, 0.33–0.45) in the observation group, while the random-model pooled event rate of VA improvement for the surgery group was 0.72 (95% CI, 0.63–0.80). The pooled event rate for BCVA improvement in the surgery group was 0.39 (95% CI, 0.33–0.46).Conclusions: This meta-analysis suggests that TMH hole closure and VA improvement rates in the surgery group were significantly higher than those in the observation group. Vitrectomy is an effective method for treating TMH. However, further randomized controlled trials (RCTs) are required to evaluate the efficacy and safety of surgery and observation for TMH treatment.Systematic Review Registration:https://www.crd.york.ac.uk/PROSPERO/#recordDetails, identifier: CRD42021276684.

Randomness in classes of matroids

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

Randomness in classes of matroids

<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

Min-Entropy Estimation for Semiconductor Superlattice True Random Number Generators

Semiconductor superlattice true random number generator (SSL-TRNG) has an outstanding practical property to serve as high-throughput and high-security cryptographic applications. Security in random number generators is closely related to the min-entropy of the raw output because feeding cryptographic applications with insufficient entropy leads to poor security and vulnerability to malicious attacks. However, no research has focused on the minimum entropy estimation based on the random model for SSL-TRNG, which is a highly recommended method for evaluating the security of a specific TRNG structure. This paper proposes a min-entropy estimation method for the SSL-TRNG by extending the Markov stochastic model derived from the memory effects. By calculating the boundary of the transition matrix, the min-entropy result is that the average value of each sample (1 bit) is 0.2487. Moreover, we demonstrate that the estimator is accurate enough to adjust compression rate dynamically in post-processing to reach the required security level, estimating entropy on the fly rather than off-line.

On the Temperature of SAT Formulas

The remarkable advances in SAT solving achieved in the last years have allowed to use this technology in many real-world applications of Artificial Intelligence, such as planning, formal verification, and scheduling, among others. Interestingly, these industrial SAT problems are commonly believed to be easier than classical random SAT formulas, but estimating their actual hardness is still a very challenging question, which in some cases even requires to solve them. In this context, realistic pseudo-industrial random SAT generators have emerged with the aim of reproducing the main features shared by the majority of these application problems. The study of these models may help to better understand the success of those SAT solving techniques and possibly improve them. In this work, we present a model to estimate the temperature of real-world SAT instances. This temperature represents the degree of distortion into the expected structure of the formula, from highly structured benchmarks (more similar to real-world SAT instances) to the complete absence of structure (observed in the classical random SAT model). Our solution is based on the Popularity-Similarity (PS) random model for SAT, which has been recently presented to reproduce two crucial features of application SAT benchmarks: scale-free and community structures. The PS model is able to control the hardness of the generated formula by introducing some randomizations in the expected structure. Our solution is a first step towards a hardness oracle based on the temperature of SAT formulas, which may be able to estimate the cost of solving real-world SAT instances without solving them.

A systematic and meta-analysis study on the prevalence of tuberculosis and relative risk factors for prisoners in Iran

Introduction: This study aimed to evaluate the incidence of Tuberculosis (TB) among prisoners in Iran, by performing a systematic and meta-analysis study on the related articles. Methodology: Scopus, Iran doc, Cochrane, Pubmed, Medline, Embase and Iran Medex, Magiran, SID, Google Scholar, and EBSCO were searched. After quality assessment of the articles, a fixed or random model, as appropriate, was used to pool the results in a meta-analysis. Heterogeneity between the studies was assessed using I-square and Q-test. Results: The overall sample size of included studies was 19562 that 64 of them were with TB. The highest prevalence of tuberculosis was related to the study of Rasht, 517 in 100,000 but the lowest rate was related to the study of Sought Khorasan, 25 in 100,000. The ES of the random effect model is 0.003 (95% CI, 0.001-0.005) and p-value <0.0001. The Higgins’ I2 of all studies is 86.55%, and the p-value of the Cochrane Q statistics is <0.001, indicating that there is heterogeneity. Based on the Egger regression plot (t=2.18, p = 0.08, CI 95%: -0.001, 0.005) no publication bias existed. Conclusion: The frequency of TB among the prisoners in Iran was low. Due to important limitations in this study, it is not possible to indicate the exact prevalence of TB among prisoners in Iran and compare this with the general population. More studies are needed to assess the related risk factor for designing health interventions plan to decrease the incidence rate of TB among prisoners.