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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.

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the rst moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, suf cient conditions for the second moment to be bounded or unbounded are proposed.

Analyzing phylogenetic relationships using mathematical methods has always been of importance in bioinformatics. Quantitative research may interpret the raw biological data in a precise way. Multiple Sequence Alignment (MSA) is used frequently to analyze biological evolutions, but is very time-consuming. When the scale of data is large, alignment methods cannot finish calculation in reasonable time. Therefore, we present a new method using moments of cumulative Fourier power spectrum in clustering the DNA sequences. Each sequence is translated into a vector in Euclidean space. Distances between the vectors can reflect the relationships between sequences. The mapping between the spectra and moment vector is one-to-one, which means that no information is lost in the power spectra during the calculation. We cluster and classify several datasets including Influenza A, primates, and human rhinovirus (HRV) datasets to build up the phylogenetic trees. Results show that the new proposed cumulative Fourier power spectrum is much faster and more accurately than MSA and another alignment-free method known as k-mer. The research provides us new insights in the study of phylogeny, evolution, and efficient DNA comparison algorithms for large genomes. The computer programs of the cumulative Fourier power spectrum are available at GitHub (https://github.com/YaulabTsinghua/cumulative-Fourier-power-spectrum).

We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^ c case. In particular, we prove a K-theory version of the main results of [{\bf H}],[{\bf Liu1}, Theorem B] for a family of almost complex manifolds.