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This chapter is devoted to the spectral analysis of birth–death processes on nonnegative integers, which are the most basic and important continuous-time Markov chains. These processes will be characterized by an infinitesimal operator which is a tridiagonal matrix whose spectrum is always contained in the negative real line (including 0). The Karlin–McGregor integral representation formula of the transition probability functions of the process is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside a positive real interval. Although many of the results are similar or equivalent to those of discrete-time birth–death chains, the methods and techniques are quite different. The chapter gives an extensive collection of examples related to orthogonal polynomials, including the M/M/k queue for any k servers, the continuous-time Ehrenfest and Bernoulli–Laplace urn models, a genetics model of Moran and linear birth–death processes. As in the case of discrete-time birth–death chains, the Karlin–McGregor formula is applied to the probabilistic aspects of birth–death processes, such as processes with killing, recurrence, absorption, the strong ratio limit property, the limiting conditional distribution, the decay parameter, quasi-stationary distributions and bilateral birth–death processes on the integers.
The differences in the clinical features and outcomes of respiratory adenovirus infection (RAI) between immunocompetent and immunocompromised adult patients remain unclear. Thirty-nine adult RAI patients, including 28 (71.8%) immunocompetent patients and 11 (28.2%) immunocompromised patients were enrolled in this retrospective study. Demographic characteristics, symptoms, laboratory tests, radiographic findings, therapies and clinical outcomes were compared between the two groups. We found fever (94.9%), cough (66.7%) and sputum production (56.4%) were the most frequent symptoms. Compared with immunocompetent RAI patients, the immunocompromised RAI patients were more likely to experience anaemia (g/l; 90.8 ± 24.0 vs 134.3 ± 14.6, P < 0.001), thrombocytopaenia ( × 109/l; 116.9 ± 92.7 vs 178.4 ± 74.6, P = 0.037), hypoalbuminaemia (g/l; 29.6 ± 5.5 vs 36.9 ± 5.2, P < 0.001), hyponatraemia (mmol/l; 134.8 ± 5.6 vs 138.5 ± 3.9, P = 0.026) and low levels of cholinesterase (U/l; 2650.5 ± 1467.4 vs 5892.8 ± 1875.1, P < 0.001). Chest computed tomography (CT) scans indicated that lung infiltrate was the most common finding (64.1%). Immunocompromised patients had a higher likelihood of bilateral lung involvement (72.7%) and lower lobe involvement (81.8%) of both lungs. The hospitalized mortality rate was 27.3% in immunocompromised RAI patients, but no death occurred among immunocompetent RAI patients (P = 0.018). Our data suggested immunocompromised RAI patients had worse laboratory test results, more bilateral lung and lower lobe involvement and higher in-hospital mortality compared with immunocompetent RAI patients.
In a Nicaraguan population-based cohort, SARS-CoV-2 seroprevalence reached 28% in the first 6 months of the country's epidemic and reached 35% 6 months later. Immune waning was uncommon. Individuals with a seropositive household member were over three times as likely to be seropositive themselves, suggesting the importance of household transmission.
Accurate prediction of laminar-turbulent transition is a critical element of computational fluid dynamics simulations for aerodynamic design across multiple flow regimes. Traditional methods of transition prediction cannot be easily extended to flow configurations where the transition process depends on a large set of parameters. In comparison, neural network methods allow higher dimensional input features to be considered without compromising the efficiency and accuracy of the traditional data-driven models. Neural network methods proposed earlier follow a cumbersome methodology of predicting instability growth rates over a broad range of frequencies, which are then processed to obtain the N-factor envelope, and then, the transition location based on the correlating N-factor. This paper presents an end-to-end transition model based on a recurrent neural network, which sequentially processes the mean boundary-layer profiles along the surface of the aerodynamic body to directly predict the N-factor envelope and the transition locations over a two-dimensional airfoil. The proposed transition model has been developed and assessed using a large database of 53 airfoils over a wide range of chord Reynolds numbers and angles of attack. The large universe of airfoils encountered in various applications causes additional difficulties. As such, we provide further insights on selecting training datasets from large amounts of available data. Although the proposed model has been analyzed for two-dimensional boundary layers in this paper, it can be easily generalized to other flows due to embedded feature extraction capability of convolutional neural network in the model.
We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.