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In this work, we use the immersed boundary method with four extensions to simulate a moving liquid–gas interface on a solid surface. We first define a moving contact line model and implements a static-dynamic friction condition at the immersed solid boundary. The dynamic contact angle is endogenous instead of prescribed, and the solid boundary can be non-stationary with respect to time. Second, we simulate both a surface tension force and a Young's force with one general equation that does not involve estimating local curvature. In the third extension, we splice liquid–gas interfaces to handle topological changes, such as the coalescence and separation of liquid droplets or gas bubbles. Finally, we re-sample liquid–gas interface markers to ensure a near-uniform distribution without exerting artificial forces. We demonstrate empirical convergence of our methods on non-trivial examples and apply them to several benchmark cases, including a slipping droplet on a wall and a rising bubble. 

We introduce Video Transformer (VidTr) with separableattention for video classification. Comparing with commonly used 3D networks, VidTr is able to aggregate spatiotemporal information via stacked attentions and provide better performance with higher efficiency. We first introduce the vanilla video transformer and show that transformer module is able to perform spatio-temporal modeling from raw pixels, but with heavy memory usage. We then present VidTr which reduces the memory cost by 3.3× while keeping the same performance. To further optimize the model, we propose the standard deviation based topK pooling for attention (pooltopK std), which reduces the computation by dropping non-informative features along temporal dimension. VidTr achieves state-of-the-art performance on five commonly used datasets with lower computational requirement, showing both the efficiency and effectiveness of our design. Finally, error analysis and visualization show that VidTr is especially good at predicting actions that require long-term temporal reasoning. 

In this paper, a discrete Markov chain model is developed to describe the inventory at a bike share station. The uniqueness of solutions is first studied. Then the model calibration is considered by investigating a constrained optimization problem. Numerical simulations involving real data are conducted to demonstrate the model effectiveness as well.