In one of the paper I am reading, I came across two mathematical notations which I cannot understand. The author also did not give any explanation. In both, note that $N$ is a positive integer.
Term 1: $x \in \{ 0,1 \}^N$
Term 2: $h\in \mathbb{R}^N_{>0}$
Thanks
On
In this case $x \in S^N$ where $S$ is some set means that $x$ is a tuple of length $N$ with entries from the set $S$, that is
$$x = (s_1, s_2, \ldots, s_N)$$
where all the $s_i$ are elements of $S$. So in the case of $S = \{0, 1\}$ (the set containing the elements $0$ and $1$) and $N=2$ we have $$S^N= \{(0,0),(0,1),(1,0),(1,1)\}.$$
In the case of $S = \mathbb R_{>0}$, the entries are positive numbers.
In general, given two sets $A$ and $B, A^B$ represents set of all functions $f$ such that $f:B\to A$.
The first one represents the set of all functions $f$ such that $f:\mathbb N \to \{0,1\}$. Recall set of all sequences of $0$s and $1$s, you'll notice that both these sets are the same.
So if $x\in \{0,1\}^\mathbb N$, then $x$ is a function from $\mathbb N$ to $\{0,1\}$.
The second one represents set of all functions $f$ such that $f:\mathbb N\to \mathbb R^+$. Recall set of all sequences of positive numbers, you'll notice that both these sets are the same.
So if $h\in \mathbb R_{>0}^\mathbb N$, then $h$ is a function from $\mathbb N$ to $\mathbb R_{>0}$, where $\mathbb R_{>0}=\{x\in \mathbb R: x\gt 0\}$