论文标题
具有身份的交换环的某些子环的频谱研究
Study of spectrum of certain subrings of a commutative ring with identity
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。
By a ring we always mean a commutative ring with identity. It is well known that maximal spectrum of $C(X)$, $C^*(X)$ and any intermediate subrings between $C(X)$ and $C^* (X)$ are homeomorphic and homeomorphic with $βX$, the Stone-$\check{C}$ech compactification of $X$. In this paper we generalized these results to an arbitrary ring by introducing a notion of dense subring. We proved that if $A$ is completely normal and dense subring of $B$, then maximal spectrum of $A$ and $B$ are homeomorphic and hence maximal spectrum of all intermediate subrings between $A$ and $B$ where $A$ is dense, are homeomorphic. We also proved that $A$ is dense subring of $B$ if and only if spectrum of $B$ is densely embedded in spectrum of $A$ and have further shown that if $A$ is dense subring of $B$, any minimal prime ideal of $A$ is precisely of the form $Q\cap A$ for some unique minimal prime ideal $Q$ of $B$. As a consequence, we concluded that if $A$ is dense in $B$, minimal spectrum of $A$ and that of $B$ are homeomorphic. We also studied different properties of dense subrings of a ring.
