A note on the γ\gamma-coefficients of the "tree Eulerian polynomial"

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these positive coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of the author and Wachs related to the poset of weighted partitions and the free multibracketed Lie algebra.Comment: 13 pages, 6 figures, Interpretations derived from results in arXiv:1309.5527 and arXiv:1408.541

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma(BA)$ where the deformation or twisting map $\sigma$ is a linear polynomial with a slope parameter that is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras whose deformation parameters are not roots of unity, and so we have a general solution for the Lie polynomial characterization problem for these algebras

Pop III GRBs: an estimative of the event rate for future surveys

We discuss the theoretical event rate of gamma-ray bursts (GRBs) from the collapse of massive primordial stars. We construct a theoretical model to calculate the rate and detectability of these GRBs taking into account all important feedback and recent results from numerical simulations of pristine gas. We expect to observe a maximum of N $\lesssim$ 0.2 GRBs per year integrated over at z > 6 with \textit{Swift} and N $\lesssim$ 10 GRBs per year integrated over at z > 6 with EXIST.Comment: 6 pages, 2 figures, published in Proceedings of the Gamma-Ray Bursts 2012 Conference (GRB 2012

A Lie algebra related to the universal Askey-Wilson algebra

Let $\mathbb{F}$ denote an algebraically closed field. Denote the three-element set by $\mathcal{X}=\{A,B,C\}$, and let \mathbb{F}\left denote the free unital associative $\mathbb{F}$-algebra on $\mathcal{X}$. Fix a nonzero $q\in\mathbb{F}$ such that $q^4\neq 1$. The universal Askey-Wilson algebra $\Delta$ is the quotient space \mathbb{F}\left/\mathbb{I}, where $\mathbb{I}$ is the two-sided ideal of \mathbb{F}\left generated by the nine elements $UV-VU$, where $U$ is one of $A,B,C$, and $V$ is one of \begin{equation} (q+q^{-1}) A+\frac{qBC-q^{-1}CB}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) B+\frac{qCA-q^{-1}AC}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) C+\frac{qAB-q^{-1}BA}{q-q^{-1}}.\nonumber \end{equation} Turn \mathbb{F}\left into a Lie algebra with Lie bracket $\left[ X,Y\right] = XY-YX$ for all X,Y\in\mathbb{F}\left. Let $\mathcal{L}$ denote the Lie subalgebra of \mathbb{F}\left generated by $\mathcal{X}$, which is also the free Lie algebra on $\mathcal{X}$. Let $L$ denote the Lie subalgebra of $\Delta$ generated by $A,B,C$. Since the given set of defining relations of $\Delta$ are not in $\mathcal{L}$, it is natural to conjecture that $L$ is freely generated by $A,B,C$. We give an answer in the negative by showing that the kernel of the canonical map \mathbb{F}\left\rightarrow\Delta has a nonzero intersection with $\mathcal{L}$. Denote the span of all Hall basis elements of $\mathcal{L}$ of length $n$ by $\mathcal{L}_n$, and denote the image of $\sum_{i=1}^n\mathcal{L}_i$ under the canonical map $\mathcal{L}\rightarrow L$ by $L_n$. We study some properties of $L_4$ and $L_5$

Applicability of Winthrop Score for the Diagnosis of Influenza A in the Emergency Department of Hospital Pablo Arturo Suárez, January to March of 2018

Introduction: In 2010, the Department of Infectious Diseases at Winthrop University Hospital designed a score system for the diagnosis of Legionella pneumonia. In this study, we applied the score to patients with acute respiratory symptoms suspected of having type A influenza. The identification of patients at medium to high risk of Influenza A allows for early initiation of treatment.Objective: To study the applicability of the Winthrop score for the diagnosis of Influenza A.Methodology: A prospective cohort study was performed in 2018 at Hospital Pablo Arturo Suárez, in Quito, Ecuador. Patients 0 to 100 years old presenting to the emergency department with influenza-like illness in January-March of 2018 were included in the study. Winthrop score results were then compared with the result of the reverse transcription polymerase chain reaction (RT-PCR) for influenza A, the gold standard for diagnosis. Sensitivity, specificity, positive and negative predictive values, and likelihood ratios were used to establish the diagnostic performance of this point system for influenza A within the sample at large and in subgroup analyses by age (<5 years, 5-65 years, and >65 years) and comorbidities.Results: 149 patients were enrolled in the study period. The study population included 81 males (54.4%) and the majority of patients were less than 5 years of age (N=85, 57.0%). Furthermore, almost one-third of the patients were less than one year old (N=38, 25.5%). According to the Winthrop point system, 68.5% of the cases had a low probability of having influenza (n = 102), 8.7% of cases had a medium probability (n = 13) and 22.8 % of cases had a high probability (n = 34). The RT-PCR test for influenza was positive for 26.2% of patients (n = 39). The Winthrop point system had a sensitivity of 97.4%, specificity of 91.8%, positive predictive value of 80.8%, negative predictive value of 99.0%, positive likelihood ratio of 11.9, and negative likelihood ratio of 35.8 in the total study population. For children under 5 years, a sensitivity of 100%, specificity of 96.3%, positive predictive value of 77.7%, negative predictive value of 100%, positive likelihood ratio of 27, and negative likelihood ratio of 0. In patients older than 6 years, a sensitivity of 96.9%, specificity of 89%, positive predictive value of 84.21%, negative predictive value of 98%, positive likelihood ratio of 8.8, and negative likelihood ratio of 29.4. Testing in patients over 65 years had a sensitivity of 100%, specificity of 90%, positive predictive value of 87.5%, negative predictive value of 100%, positive likelihood ratio of 10 and negative likelihood ratio of 0. Finally, patients with comorbidities had a sensitivity of 90%, specificity of 88.24%, positive predictive value of 81.82%, negative predictive value of 93.75%, positive likelihood ratio of 7.65, and negative likelihood ratio of 8.82.Conclusions: The Winthrop score performed well in predicting Influenza A in patients with acute respiratory symptoms. This score may be useful in settings were Influenza A PCR testing is unavailable

AMADA-Analysis of Multidimensional Astronomical Datasets

We present AMADA, an interactive web application to analyse multidimensional datasets. The user uploads a simple ASCII file and AMADA performs a number of exploratory analysis together with contemporary visualizations diagnostics. The package performs a hierarchical clustering in the parameter space, and the user can choose among linear, monotonic or non-linear correlation analysis. AMADA provides a number of clustering visualization diagnostics such as heatmaps, dendrograms, chord diagrams, and graphs. In addition, AMADA has the option to run a standard or robust principal components analysis, displaying the results as polar bar plots. The code is written in R and the web interface was created using the Shiny framework. AMADA source-code is freely available at https://goo.gl/KeSPue, and the shiny-app at http://goo.gl/UTnU7I.Comment: Accepted for publication in Astronomy & Computin