where $z(i):=z_i$, $i=1,\ldots,N$.

\[defextr\] Let $z\in{\mathbb{R}}^N$. A *local minimum* of $z$ is a sequence of indices $\left\{i,{i+1},\dots,{i+n}\right\}$ such that $z(i) = z(i+1) = \dots = z(i+n)$, where $0 \leq n < N-1$, and it holds that

-   if $i-1\geq 1$, then $z(i-1) > m$,

-   if $i+n+1\leq N$, then $z(i+n+1) > m$.

A *local maximum* of $z$ is a sequence of indices $\left\{i,{i+1},\dots,{i+n}\right\}$ such that $z(i) = z(i+1) = \dots = z(i+n)$, where $0 \leq n < N-1$, and it holds that

-   if $i-1\geq 1$, then $z(i-1) < m$,

-   if $i+n+1\leq N$, then $z(i+n+1) < m$.

A *local extremum* of $z$ is either a local minimum of $z$ or a local maximum of $z$. A *boundary extremum* is a *local extremum* where $i=1$ or $i+n=N$. A local extremum that is not a boundary extremum is an *interior extremum*.

We say that $z$ is *monotone increasing* (*monotone decreasing*) over the set of indices ${\left\{ {k,k+1,\ldots, k+j}\right\}}$ if $z(k)\leq z(k+1) \leq \cdots \leq z(k+j)$ ($z(k)\geq z(k+1) \geq \cdots \geq z(k+j))$.

We make the following assumption for the remainder of this paper

A

:   The values of all local extrema of $z$ are distinct.

The important consequence of assumption [**A**]{} is 