Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$ where $p$ is a prime number. Suppose that $G=\langle\sigma\rangle \simeq C_p$ acts on $k(x_0,\ldots,x_{p-1})$ by $k$-automorphisms defined as $\sigma:x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0$. Denote by $P$ the set of all prime numbers and define $P_0=\{p\in P:\bm{Q}(\zeta_{p-1})$ is of class number one$\}$ where $\zeta_n$ a primitive $n$-th root of unity in $\bm{C}$ for a positive integer $n$; $P_0$ is a finite set by \cite{MM}. Theorem. Let $k$ be an algebraic number field and $P_k=\{p\in P: p$ is ramified in $k\}$. Then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ for all $p\in P\backslash (P_0\cup P_k)$.

For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that we call essentially of finite type.

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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.

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