Unknown operations in differential equations paper

Question

I'm reading this (https://www.sciencedirect.com/science/article/pii/S0893965901001380) paper at the moment and came across a few operations that are either ambiguous to me, or for which I can't find a defintion.

The first is they give a definition for what they mean by "$<$" when comparing two vectors in $\mathbb{R}^n$. They say "for $x,y\in \mathbb{R}^n$, $x<y$ means that there exists an $i\in\Lambda=\{ 1,\dots,n \}$ such that $x_i<y_i$." And "$x<<y$ denotes $x_i<y_i$ for $i\in\Lambda$." So the second definition is clear to me in that every element of the $x$ vector has to be less than the corresponding element of the $y$ vector. But the first seems to lead to some ambiguity. For example, let $x,y\in\mathbb{R}^2$ and let $x=\left[1,2\right]^T$ and $y=\left[2,1\right]^T$. So clearly we have $x_1<y_1$ and $y_2<x_2$. So wouldn't the first definition imply simultaneously that $x<y$ and $y<x$?

The second point of confusion is a definition. They say for $y\in \mathbb{R}^n$, we define $\left[y\right]^+=\text{col}\{|{y_i}|\}$ and for $\phi \in C$, $C$ being the space of continuous functions mapping the interval $\left[-r,0\right]$ to $\mathbb{R}^n$, $\left[\phi\right]^+=\text{col}\{\|{\phi_i}\|_r\}$ where the norm on the function $\phi$ is the sup norm over the interval $\left[-r,0\right]$.

My issue is I'm not sure what the function "col" is. I did some searching and the most similar looking thing I've found is that this sometimes denotes the column space of a matrix. However the inputs in this context aren't matrices, but look to me to be numbers. So the column space of those would be trivial.

I'm probably missing something obvious here, so apologies in advance if I did. But any clarification you can offer on either point would be great.

Answer 1

$\mathrm{col}$ forms a column vector from the elements. So

$$\mathrm{col}\{\|y_i\|\} = \begin{bmatrix}\|y_1\| \\ \vdots \\ \|y_n\|\end{bmatrix}$$

Regarding the inequality, from the paper we have

For $x,$ $y \in \mathbb{R}^n,$ $x \dot{<} y$ means that there at least exists an $i\in\Lambda = \{1,\ldots,n\}$ such that $x_i < y_i.$

Notice that $\dot{\phantom{<}}$ over the $<$; I do not think this is a typo. This is not defining $<$ but is defining $\dot{<}.$ You can verify this isn't a typo by observing its (only) use in (H3).(i). Unfortunately, (H3).(i) is not explicitly called so it is hard to say what the real purpose of this additional notation is without getting into the details.

Answer 2

$1)$ Yes, under this definition you could indeed have $x < y$ and $y < x$.

$2)$ "col" seems to mean "column vector"