The saddle-point conditions are satisfied up to an error $o(n)$.
He used a new version of an algorithm for finding all normal subgroups of up to a given index in a finitely presented group.
Here $F$ is only defined up to an additive constant.
Therefore, $G$ is uniquely determined up to unitary equivalence.
If we know a covering space $E$ of $X$ then not only do we know that ......, but we can also recover $X$ (up to homeomorphism) as $E/G$.
from stage $A$ up to, but not including, stage $B$
The arguments from this point up to Theorem 2 do not depend on ......
Up to now, we have assumed that ......
The coefficients of $A$ add up to unity.
Other situations in dynamics where the $p$-adic numbers come up are surveyed in [W].
Arguing as before, we shall end up with a simple tree all of whose facets contain $V$.
Neighbourhoods of points in these spaces appear at first glance to have a nice regular structure, but upon closer scrutiny, one sees that many neighbourhoods contain collections of arcs hopelessly folded up.
These subsets join up to form a simple closed curve passing through $A$ and $B$.
The set $WF(u)$ is made up of bicharacteristic strips.
Each of the terms that make up $G(t)$ is well defined.
Now $F$ is defined to make $G$ and $H$ match up at the left end of $I$.
Write out the integers from 1 to $n$. Pair up the first and the last, the second and next to last, etc.
Values computed for the right side of (2) were rounded up in the fourth decimal place.
In Section 2 we set up notation and terminology. [= prepare]
On $TK$ we set up the symplectic structure induced by the metric. [= introduce]
This space of curves also shows up in the theorem of Meyer on ......
We shall split up $K$ as follows.
First we take up the trivial case $h=0$.
In closing this section we take up a result which will play a pivotal role in the characterization of ......
It should be no surprise that a condition like $a_i\ne b_i$ turns up in this theorem.
The lectures were written up by M. Stong.