几乎所有方程组解的鲁棒性
Robustness of solutions of almost every system of equations
摘要：
在数学建模中，通常有一个方程$F(p)=c$，其中$F$的确切形式未知。这篇文章表明，有大量的$F$，其中几乎所有的$F$共享相同的属性。我们研究的类是$C^1$函数的向量空间$\mathcal{F}$ $F:\mathbb{R}^N \to \mathbb{R}^M$，满足以下条件：如果存在一个恒定的整数$ρ(\mathcal{F}) \geq 0$，使得对$(DF(p))=ρ(\mathcal{F})$进行排名，则$\mathcal{F}$具有“几乎恒定的排名”（ACR），以便对“几乎所有”$F\in \mathcal{F}$和几乎所有$p\in\mathbb{R}^N$进行排名。如果向量空间$\mathcal{F}$是有限维的，则“几乎每一个”是关于$\mathcal{F}$上的勒贝格测度的，否则，它意味着几乎每一个在流行的意义上，如本文所述。通常用于建模目的的大多数函数空间都是ACR。特别是，我们表明，如果所有的功能在$\mathcal{F}$是线性或多项式或真正的分析，或者如果$\mathcal{F}$是一个“结构化系统”的所有功能的集合，那么$\mathcal{F}$是ACR。对于每个$F$和$p$，$p \in \mathbb{R}^N$的解集是SolSet $(p):= \{x: F(x)=F(p)\}.$如果$F(p)=c$的解集在$F$和$c$中存在小的变化时仍然存在，则称为稳健的。对ACR向量空间$\mathcal{F}$中的几乎所有$F$，证明了以下两个整体结果：（1）要么解集SolSet $(p)$对几乎所有$p\in \mathbb{R}^N$都是鲁棒的，要么没有解集是鲁棒的. (2)解集SolSet $(p)$是维数为$d = N-ρ(\mathcal{F})$的$C^\infty$ -流形。特别是，几乎每种$F \in \mathcal{F}$的$d$均相同。
英文摘要：
In mathematical modeling, it is common to have an equation $F(p)=c$ where the exact form of $F$ is not known. This article shows that there are large classes of $F$ where almost all $F$ share the same properties. The classes we investigate are vector spaces $\mathcal{F}$ of $C^1$ functions $F:\mathbb{R}^N \to \mathbb{R}^M$ that satisfy the following condition: $\mathcal{F}$ has ``almost constant rank'' (ACR) if there is a constant integer $ρ(\mathcal{F}) \geq 0$ such that rank$(DF(p))=ρ(\mathcal{F})$ for ``almost every'' $F\in \mathcal{F}$ and almost every $p\in\mathbb{R}^N$. If the vector space $\mathcal{F}$ is finite-dimensional, then ``almost every'' is with respect to Lebesgue measure on $\mathcal{F}$, and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in $\mathcal{F}$ are linear or polynomial or real analytic, or if $\mathcal{F}$ is the set of all functions in a ``structured system'', then $\mathcal{F}$ is ACR. For each $F$ and $p$, the solution set of $p \in \mathbb{R}^N$ is SolSet$(p):= \{x: F(x)=F(p)\}.$ A solution set of $F(p)=c$ is called robust if it persists despite small changes in $F$ and $c$. The following two global results are proved for almost every $F$ in an ACR vector space $\mathcal{F}$: (1) Either the solution set SolSet$(p)$ is robust for almost every $p\in \mathbb{R}^N$, or none of the solution sets are robust. (2) The solution set SolSet$(p)$ is a $C^\infty$-manifold of dimension $d = N-ρ(\mathcal{F})$. In particular, $d$ is the same for almost every $F \in \mathcal{F}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
