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Comparing DNA or protein sequences plays an important role in the functional analysis of genomes. Despite many methods available for sequences comparison, few methods retain the information content of sequences. We propose a new approach, the Yau-Hausdorff method, which considers all translations and rotations when seeking the best match of graphical curves of DNA or protein sequences. The complexity of this method is lower than that of any other two dimensional minimum Hausdorff algorithm. The Yau-Hausdorff method can be used for measuring the similarity of DNA sequences based on two important tools: the Yau-Hausdorff distance and graphical representation of DNA sequences. The graphical representations of DNA sequences conserve all sequence information and the Yau-Hausdorff distance is mathematically proved as a true metric. Therefore, the proposed distance can preciously measure the similarity of DNA sequences. The phylogenetic analyses of DNA sequences by the Yau-Hausdorff distance show the accuracy and stability of our approach in similarity comparison of DNA or protein sequences. This study demonstrates that Yau-Hausdorff distance is a natural metric for DNA and protein sequences with high level of stability. The approach can be also applied to similarity analysis of protein sequences by graphic representations, as well as general two dimensional shape matching.

Geometric compression plays a fundamental rolein virtual reality and augmented reality (VR/AR) applications. Dense meshesare re-sampled and re-tessellatedtoreduce thecomplexity. This process is called remeshing. In this work,we propose a novel remeshing algorithm based on both angle-preservingparameterization, and measurecontrollable parameterization. The conformal parameterization iscarried out by discrete surface Ricci flow method, the measurecontrollable parameterization is obtaind by an optimal mass transportation map. The sampling is performed on the measurecontrollable parameterization domain, the triangulation is computed on the conformal parameterization domain using Delaunay refinement algorithm. This method gives the user full control of sampling distribution, and produces mesheswith curvature measure convergence. The meshing resultcan emphsizethe region of interests, is curvature sensitive. Experimental results demonstrate the efficiency and efficacy of the proposed method.

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magneto-hydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

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