Theorem 2, at the end of Section 2, was not originally obtained in the manner indicated there.

That approach was used earlier in [2]. There, however, it was applied in simply connected regions only.

This shows that there is $r\ge 0$ such that ...... [Or: there is an $r\ge 0$]

Now, there are $a$ and $b$ such that ......

There exists a function $f$ and a constant $c$ such that ...... [Or: There exist a function $f$ and a constant $c$]

However, there are a large number of examples showing that ...... [Not: “There is a large number”]

If $p=0$ then there are an additional $m$ arcs. [Note the article an.]

We have to show that the property of there being $x$ and $y$ such that $x<y$ uniquely determines $P$ up to isomorphism.

Then (3.5) is a necessary and sufficient condition for there to be a function $f$ such that ......

How many such expressions are there?

How many entries are there in this section?

In general, we have $a\le b$; there is equality if ......

There has recently been increasing interest in the theory of ......

There is not space to enumerate them all here.

From the viewpoint of the Fox theorem, there is not an exact parallel between the odds and the evens.

For general linear operators, there is not such an extensive functional calculus as there is for self-adjoint operators.

Hence there can be no condition on the norms which guarantees (7).

In representation theory, there can never be a $B$-map whose domain is finite-dimensional.

There cannot be two edges between one pair of vertices.

With each $D$ there is associated a region $V_D$.

There remain four intervals of length 1/4 each.

There remains one further case to consider. [ But: It remains to consider one further case.]

In [6] there occur the following formulas.

There seems to be no simple formula for ......

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