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Immersed finite element methods are designed to solve interface problems on interface- unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements with arbitrary interface-cutting configuration. Optimal a priori error estimates are rigorously proved in both energy and L2 norms, with the constant in the error bound independent of the interface location and its dependence on coefficient contrast explicitly specified. Numerical examples are provided not only to verify our theoretical results but also to demonstrate the applicability of this IFE method in tackling some real-world 3D interface models. 

As data are generated more and more from multiple disparate sources, multiview data sets, where each sample has features in distinct views, have grown in recent years. However, no comprehensive package exists that enables non-specialists to use these methods easily. mvlearn is a Python library which implements the leading multiview machine learning methods. Its simple API closely follows that of scikit-learn for increased ease-of-use. The package can be installed from Python Package Index (PyPI) and the conda package manager and is released under the MIT open-source license. The documentation, detailed examples, and all releases are available at https://mvlearn.github.io/. 

The effect of thermal treatment at 220 °C for 10 days on the structure and transport properties of a triptycene-based polybenzoxazole (TPBO) was investigated experimentally and theoretically. Gas and water vapor sorption in TPBO is virtually unaffected by thermal treatment, while diffusion and permeability coefficients decrease by just 20%. Remarkably, the CO2/CH4 selectivity exhibit a negligible change. Fluorescence spectroscopy, WAXD and FTIR analysis indicate that, in sharp contrast with typical behavior of glassy polymers, TPBO does not experience accelerated physical aging, and rule out formation of intermolecular charge transfer complexes upon thermal treatment. According with this physical picture, the diffusion coefficient of penetrant molecules sorbed in the Langmuir's mode, DH, does not change after treatment. Small molecule diffusivity and permeability decline is caused by a decrease in polymer chain mobility, which makes more difficult opening gaps to allow penetrant diffusion jumps. According to this picture, the Henry's mode diffusion coefficient, DD, substantially decreases upon thermal treatment. The higher stability exhibited by TPBO relative to other high Tg glassy polymers is ascribed to the presence of configurational free volume, which is not related to the non-equilibrium transient conformation, but to the molecular configuration and, as such, it is not relaxed upon protracted exposure to high temperatures.