[on sth; see also: base, rely, rest]
However, the space $Z$ depends crucially on the particular ambient algebra $A$ chosen.
Note that $O(g)$ depends on $g$ only through its differential $dg$.
We have to keep track of how the constant $K$ depends on the domain $D$.
We do not know how $V$ depends on the various choices made.
The answer depends on how broadly or narrowly the term “matrix method” is defined.
Up to equivalence it only depends on the pair $(f,g)$.
We divide them into three classes depending on the width of $k$.
The proof of (2) is a matter of straightforward computation, and depends on the relation $ab-cd=1$.
These functions are sometimes called elementary factors. Their utility depends on the fact that ......
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