In this paper, we generalize the Cao-Yau'’s gradient estimate for the sum of squares of vector …elds up to higher step under ssumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel. 1.

For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with coordinates in the topological closure of this subgroup in the product of the multiplicative group of those local completions of this function field over all but finitely many places.

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Protein universe is a complex system with critical problem of protein evolution to be analyzed. Early studies have used geometric distances and polygenetic-trees to solve this problem. However, the traditional methods are bivariate, whose taxonomy classification relies on bivariate branching. This is not sufficient to describe the complex nature of protein universe. Therefore, we propose a novel approach on multivariate protein classification. The new method bases on the theory of information and network, can be used to analyze multivariate relationships of proteins. The new method is alignment-free and have wide-applications to both sequences and 3D structures. We demonstrate the new method on six protein examples, results show that the new method is efficient and can potentially be used for future protein classifications.

For the purpose of isogeometric analysis, one of the most common ways is to construct structured hexahedral meshes, which have regular tensor product structure, and fit them by volumetric T-Splines. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces. The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on surfaces. This trinity theorem lays down the theoretic foundation for quadrilateral/hexahedral mesh generation, and leads to practical, automatic algorithms. The work proposes the following algorithm: the user inputs a set of disjoint, simple loops on a high genus surface, and specifies a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user’s input; the Strebel differential assigns a flat metric on the surface and decomposes the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to produce a solid cylindrical decomposition of the volume; the hexadhedral meshing is generated for each volumetric cylinder and then glued together to form a globally consistent hex-mesh. The method is rigorous, geometric, automatic and conformal to the geometry. This work focuses on the theoretic aspects of the framework, the algorithmic details and practical evaluations will be given in the future expositions.