We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].

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We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the K¨unneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes.

Due to vast sequence divergence among different viral groups, sequence alignment is not directly applicable to genome-wide comparative analysis of viruses. More and more attention has been paid to alignment-free methods for whole genome comparison and phylogenetic tree reconstruction. Among alignment-free methods, the recently proposed ‘‘Natural Vector (NV) representation” has successfully been used to study the phylogeny of multi-segmented viruses based on a 12-dimensional genome space derived from the nucleotide sequence structure. But the preference of proteomes over genomes for the determination of viral phylogeny was not deeply investigated. As the translated products of genes, proteins directly form the shape of viral structure and are vital for all metabolic pathways. In this study, using the NV representation of a protein sequence along with the Hausdorff distance suitable to compare point sets, we construct a 60-dimensional protein space to analyze the evolutionary relationships of 4021 viruses by whole-proteomes in the current NCBI Reference Sequence Database (RefSeq). We also take advantage of the previously developed natural graphical representation to recover viral phylogeny. Our results demonstrate that the proposed method is efficient and accurate for classifying viruses. The accuracy rates of our predictions such as for Baltimore II viruses are as high as 95.9% for family labels, 95.7% for subfamily labels and 96.5% for genus labels. Finally, we discover that proteomes lead to better viral classification when reliable protein sequences are abundant. In other cases, the accuracy rates using proteomes are still comparable to that of genomes.

In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller--Segel (KS) chemotaxis model. We improve the results upon (Y. Epshteyn and A. Kurganov, SIAM Journal on Numerical Analysis, 47 (2008), 368-408) and give optimal rate of convergence under special finite element spaces before the blow-up occurs (the exact solutions are smooth). Moreover, to construct physically relevant numerical approximations, we consider $P^1$ LDG scheme and develop a positivity-preserving limiter to the scheme, extending the idea in (Y. Zhang, X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). With this limiter, we can prove the $L^1$-stability of the numerical scheme. Numerical experiments are performed to demonstrate the good performance of the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model will yield blow-up solutions under certain initial conditions. We numerically demonstrate how to find the approximate blow-up time by using the $L^2$-norm of the $L^1$-stable numerical solution.