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In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented tropical polynomials and tropical polynomials in compact forms to quickly find roots of given tropical polynomials. We also prove the existence and uniqueness of tropical theorems for meromorphic functions with prescribed roots and poles. Moreover, we explain the relations between classical and tropical meromorphic functions. Different definitions and applications of tropical meromorphic functions are discussed. Finally, we point out the properties of tropical meromorphic functions are very similar to complex ones and prove some tropical analogues of theorems in complex analysis.2010 Mathematics Subject Classification. 14A10,52B20,30C15.

Cell culture remains as the golden standard for primary isolation of viruses in clinical specimens. In the current practice, researchers have to recognize the cytopathic effects (CPE) induced by virus infection and subsequently use virus-specific monoclonal antibody to confirm the presence of virus. Considering the broad applications of neural network in various fields, we aimed to utilize convolutional neural networks (CNN) to shorten the timing required for CPE identification and to improve the assay sensitivity. Based on the characteristics of influenza-induced CPE, a CNN model with larger sizes of filters and max-pooling kernels was constructed in the absence of transfer learning. A total of 601 images from mock-infected and influenza-infected MDCK cells were used to train the model. The performance of the model was tested by using extra 400 images and the percentage of correct recognition was 99.75%. To further examine the limit of our model in evaluating the changes of CPE overtime, additional 1190 images from a new experiment were used and the recognition rates at 16 hour (hr), 28 hr, and 40 hr post virus infection were 71.80%, 98.25%, and 87.46%, respectively. The specificity of our model, examined by images of MDCK cells infected by six other non-influenza viruses, was 100%. Hence, a simple CNN model was established to enhance the identification of influenza virus in clinical practice.

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This paper studies the properties of the parabolic free-boundary problem arising from pricing of American volatility options in mean-reverting volatility processes. When the volatility index follows the mean-reverting square root process (MRSRP), we derive a closed-form pricing formula for the perpetual American power volatility option. Moreover, an artificial neural network (ANN) approach is extended to find an approximate solution of the free boundary problem arising from pricing the perpetual American option. The comparison results demonstrates that the ANN provides an accurate approach to approximate solution for the free boundary problem.