[see also: assumption, requirement]
We prove, under mild conditions on $f$, that ......
In this section we investigate under what conditions the converse holds.
They established the Hasse principle subject to a rank condition on the coefficients.
Take $ N$ to be a family of normal measures in $P(X)$ such that $ N$ is maximal subject to the condition that the supports of the measures in the family are pairwise disjoint.
This is a condition on how large $f$ is.
The next theorem provides conditions for the existence of ......
The corollary gives a necessary and sufficient condition on $p$ for $g$ to belong to $A_p$.
A necessary and sufficient condition for $A$ to be open is that $C$ be closed.
Then (3.5) is a necessary and sufficient condition for there to be a function $f$ such that ......
This condition also turns out to be necessary.
It should be no surprise that a condition like $a_i\ne b_i$ turns up [= appears, shows up] in this theorem.
Condition (c) is intended to give us firm control over ......
Finally, we must check that our $F$ satisfies condition (2) of Theorem 1. [Not: “verifies condition (2)”]
Let $A$ be the subset of $X$ where this coboundary condition obtains. [= is satisfied]
We can also appeal to Lemma 5 to see that the uniform continuity condition (5.3) is met.
When the rank condition fails to hold we use (3) instead.
We note that $H$ is in fact not Lipschitz continuous if this condition is violated.
a restrictive $\langle$stringent$\rangle$ condition
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