论文标题
几乎每个方程系统的解决方案的鲁棒性
Robustness of solutions of almost every system of equations
论文作者
论文摘要
在数学建模中，通常拥有一个等式$ f（p）= c $，其中$ f $的确切形式尚不清楚。 本文表明，几乎所有$ f $共享相同的属性，都有大量的$ f $。我们调查的类是向量空间$ \ Mathcal {f} $ of $ c^1 $ functions $ f：\ mathbb {r}^n \ to \ mathbb {r}^m $，满足以下条件：$ \ nathcal {f} \ geq 0 $这样，即rank $（df（p））=ρ（\ mathcal {f}）$ for``几乎每个'''''''''''in \ mathcal {f} $，几乎每个$ p \ in \ mathbb {r}^n $。如果矢量空间$ \ MATHCAL {f} $是有限维度的，那么``几乎每个''是关于$ \ Mathcal {f} $的Lebesgue Measure，而否则，如本文所述，它几乎在流行意义上的意思。大多数用于建模目的的功能空间是ACR。特别是，我们表明，如果$ \ MATHCAL {F} $中的所有功能都是线性或多项式或真实分析性的，或者如果$ \ Mathcal {f} $是```ofstructed system''''的所有功能的集合，则$ \ nathcal {f} $是ACR。对于每个$ f $和$ p $，$ p \ in \ mathbb {r}^n $的解决方案集为 解决方案$（p）：= \ {x：f（x）= f（p）\}。$，如果$ f（p）= c $的解决方案集，尽管尽管$ f $和$ c $的变化很小，但它仍然称为稳定性。在ACR矢量空间$ \ MATHCAL {f} $中，几乎每个$ f $的全局结果都被证明：（1）对于几乎每个$ p \ in \ mathbb {r}^n $中的每个$ p \，或者没有解决方案集都不强大。 （2）解决方案集解决方案$（p）$是$ c^\ infty $ - dimension $ d = n-ρ（\ Mathcal {f}）$。特别是，对于\ Mathcal {f} $中的几乎每一个$ f \，$ d $都是相同的。
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 $F \in \mathcal{F}$.
