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Structures and functions of proteins play various essential roles in biological processes. The functions of newly discovered proteins can be predicted by comparing their structures with that of known functional proteins. Many approaches have been proposed for measuring the protein structure similarity, such as the template-modeling (TM)-score method, GRaphlet (GR)-Align method as well as the commonly used root-mean-square deviation (RMSD) measures. However, the alignment comparisons between the similarity of protein structure cost much time on large dataset, and the accuracy still have room to improve. In this study, we introduce a new three-dimensional (3D) Yau–Hausdorff distance between any two 3D objects. The (3D) Yau–Hausdorff distance can be used in particular to measure the similarity/dissimilarity of two proteins of any size and does not need aligning and super- imposing two structures. We apply structural similarity to study function similarity and perform phylogenetic analysis on several datasets. The results show that (3D) Yau–Hausdorff distance could serve as a more precise and effective method to discover biological relationships between proteins than other methods on structure comparison.

It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.

With the advance in virtual reality applications, garment industry has strived for new developments. This paper reviews state-of-the-art CAD methods in 3D garment design. A large range of techniques are selected and organized into several key modules which form the core of a 3D garment design technology platform. In each module, basic techniques are presented first. Then advanced developments are systematically discussed and commented. The selected key modules – digital human modeling, 3D garment design and modification, numerical integration of draping, 2D pattern generation, geometric details modeling, parallel computation and GPU acceleration – are discussed in turn. Major challenges and solutions that have been addressed over the years are discussed. Finally, some of the ensuing challenges in 3D garment CAD technologies are outlined.

We apply our model of quantum gravity to an AdS black hole resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur.