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In this paper, we tabulate the set of alternating oriented pretzel links. We derive a closed formula for the precise number of alternating oriented pretzel links with any given crossing number [Formula: see text]. Numerical computation suggests that this number grows approximately at the rate of [Formula: see text] 

There has been a growing number of datasets exhibiting an excess of zero values that cannot be adequately modeled using standard probability distributions. For example, microbiome data and single-cell RNA sequencing data consist of count measurements in which the proportion of zeros exceeds what can be captured by standard distributions such as the Poisson or negative binomial, while also requiring appropriate modeling of the nonzero counts. Several models have been proposed to address zero-inflated datasets including the zero-inflated negative binomial, hurdle negative binomial model, and the truncated latent Gaussian copula model. This study aims to compare various models and determine which one performs optimally under different conditions using both simulation studies and real data analyses. We are particularly interested in investigating how dependence among the variables, level of zeroinflation or deflation, and variance of the data affects model selection. KEYWORDS: Zero-InflatedModels; HurdleModels; Truncated Latent Gaussian CopulaModel; Microbiome Data; Gene-Sequencing Data; Zero-Inflation, Negative Binomial; Zero-Deflation 

We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the ``reduced dimensional method.'' Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples. 

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.