论文标题
部分可观测时空混沌系统的无模型预测
Semialgebraic and Continuous Solution of Linear Equations with Semialgebraic Coefficients
论文作者
论文摘要
从Charles Fefferman和Janoskollár的结果开始，在\ desxit {线性方程的连续解决方案} [1]中，我们采用了一种基于Fefferman Glaeser改进技术的新方法，以显示比使用Algebraicemeetry的技术证明的结果比Kollár证明的一种更一般的结果。考虑一个线性方程式系统具有半gebraic的线性方程式（不仅像[1]中的多项式一样），在$ \ mathbb {r}^{n} $上，我们在$ \ mathbb {r}^n} $上存在连续和半的溶液。这与Fefferman和Luli在\ textit {semialgebraic pection the Plane}中获得的不同之处在于他们在平面$ \ mathbb {r}^2 $上陈述了规律性解决方案$ c^m $的结果。更深入地，我们证明存在$ \ mathbb {r}^{n} $的连续和半gebraic解决方案，并且仅当存在连续的解决方案时，即与系统相关的Glaeser稳定捆绑包没有空纤维。
Starting from the results of Charles Fefferman and Janos Kollár in \texit{Continuous Solutions of Linear Equations} [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the one proved by Kollár by using techniques from algebraic geometry. Considering a system of linear equations with semialgebraic (not only polynomial as in [1]) coefficients on $\mathbb{R}^{n}$, we get a necessary and sufficient condition for the existence of a continuous and semialgebraic solution on $\mathbb{R}^{n}$. This is different from what Fefferman and Luli obtained in \textit{Semialgebraic Sections Over the Plane} since they stated their result for solutions of regularity $C^m$ on the plane $\mathbb{R}^2$. More in depth, we prove that a continuous and semialgebraic solution on $\mathbb{R}^{n}$ exists if and only if there is a continuous solution i.e., if the Glaeser-stable bundle associated to the system has no empty fiber.
