论文标题
组代数的统一子组的顺序
The Order of the Unitary Subgroups of Group Algebras
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If $p$ is an odd prime, then the order of the $\cd$-unitary subgroup of $FG$ is established. For the case $p=2$ we generalize a result obtained for finite abelian $2$-groups. It is proved that the order of the $*$-unitary subgroup of $FG$ of a non-abelian $2$-group is always divisible by a number which depends only on the size of $F$, the order of $G$ and the number of elements of order two in $G$. Moreover, we show that the order of the $*$-unitary subgroup of $FG$ determines the order of the finite $p$-group $G$.
