## do

We shall do this by showing that ......

This is done to simplify the notation.

As the space of Example 3 shows, complete regularity of $X$ is not enough to allow us to do that.

For binary strings, the algorithm does not do quite as well.

Recent improvements in the HL-method enable us to do better than this.

In fact, we can do even better, and prescribe finitely many derivatives at each point of $M$.

A geodesic which meets $bM$ does so either transversally or ......

We have not required $f$ to be compact, and we shall not do so except when explicitly stated.

This will hold for $n>1$ if it does for $n=1$.

Consequently, $A$ has all geodesics closed if and only if $B$ does.

We can do a heuristic calculation to see what the generator of $x_t$ must be.

In contrast to the previous example, membership of $D(A)$ does impose some restrictions on $f$.

We may (and do) assume that ......

The statement does appear in [3] but there is a simple gap in the sketch of proof supplied.

Only for $x=1$ does the limit exist.

In particular, for only finitely many $k$ do we have $F(a_k)>1$. [Note the inversion after only.]

In doing this we will also benefit from having the following notation.

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