Given positive integers$n$_{1}<$n$_{2}<... we show that the Hardy-type inequality $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ holds true for all$f$∈$H$^{1}, provided that the$n$_{$k$}'s, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of$H$^{1}for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.

We construct a virus database called VirusDB (http://yaulab.math.tsinghua.edu.cn/VirusDB/) and an online inquiry system to serve people who are interested in viral classification and prediction. The database stores all viral genomes, their corresponding natural vectors, and the classification information of the single/multiple-segmented viral reference sequences downloaded from National Center for Biotechnology Information. The online inquiry system serves the purpose of computing natural vectors and their distances based on submitted genomes, providing an online interface for accessing and using the database for viral classification and prediction, and back-end processes for automatic and manual updating of database content to synchronize with GenBank. Submitted genomes data in FASTA format will be carried out and the prediction results with 5 closest neighbors and their classifications will be returned by email. Considering the one-to-one correspondence between sequence and natural vector, time efficiency, and high accuracy, natural vector is a significant advance compared with alignment methods, which makes VirusDB a useful database in further research.

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX ) with curvature bounded above by a constant κ (κ 􏰣 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

Using the degeneration formula for Donaldson-Thomas invariants [L-W,MNOP2], we proved formulae for blowing up a point, simple flops, and extremal transitions.

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