We consider the Schrödinger equation for the harmonic oscillator$i$$∂$_{$t$}$u$=$Hu$, where$H$=−Δ+|$x$|^{2}, with initial data in the Hermite–Sobolev space$H$^{−$s$/2}$L$^{2}(ℝ^{$n$}). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.

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 $p\in P\backslash (P_0\cup P_k)$.

In 1975, P. Erd\H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. The sequence $(c_1,c_2,\cdots,c_n)$ is the cycle length distribution of a graph $G$ of order $n$， where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1,a_2,\cdots,$ $a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i\leq a_i$ where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1,a_2,\cdots,a_n)，$ which extended the problem due to Erd\H os, it is clear that $f(n)=f(1,1,\cdots,1)$. Let $g(n,m)=f(a_1,a_2,\cdots,a_n),$ $a_i=1$ for all $i/m$ be integer, $a_i=0$ for all $i/m$ be not integer. It is clear that $f(n)=g(n,1)$. We prove that $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{40}{99}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2 + \frac{7654}{19071}}$ (Lai, 2017). We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}} > \sqrt {2.444},$ for all even integers $m$. We make the following conjecture: $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} > \sqrt {2.444}.$

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