论文标题
$ \ mathfrak {c} $ - 单一理想的本地共同体学模块的分级组件
Graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals
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 $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1, \ldots, a_c U_c)\subseteq R$ an ideal, where $a_j \in A$ (not necessarily units) and $U_j$'s are monomials in $X_1, \ldots, X_n$. We call such an ideal $I$ as a $\mathfrak{C}$-monomial ideal. Consider the standard multigrading on $R$. We produce a structure theorem for the multigraded components of the local cohomology modules $H^i_I(R)$ for $i \geq 0$. We further analyze the torsion part and the torsion-free part of these components. We show that if $A$ is a PID then each component can be written as a direct sum of its torsion part and torsion-free part. As a consequence, we obtain that their Bass numbers are finite.
