arXiv2022
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 $\rho(\mathcal{F}) \geq 0$ such that rank$(DF(p))=\rho(\mathcal{F})$ 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 $d = N-\rho(\mathcal{F})$. In particular, $d$ is the same for almost every $F \in \mathcal{F}$.