We put $b$ in $R$ unless $a$ is already in.

This equation has a solution in integers for $N>7$.

Expand $f$ in powers of $x$.

It is this point of view which is close to that used in $C^*$-algebras.

The maps $f_t$ are homotopic, so $f_1$ and $f_0$ induce the same maps in homology.

These intervals are disjoint from all those used in defining $J_1$.

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

This also resolves the ambiguity introduced earlier in choosing an order of the lifts of $U$.

Each $A_i$ meets $A$ in a finite set. [= $A_i\cap A$ is finite]

Then one $Y_i$ can intersect another only in one point.

Values computed for the right side of (2) were rounded up in the fourth decimal place.

Then $F$ varies smoothly in $t$.

Thus $F_n(x,y)$ converges to $F(x,y)$ uniformly in $x$.

Clearly, $A_j$ is increasing in $j$.

The word ends in $a$.

in diagonal form

in geometric language

In less precise language, the requirement is that the two angles are the same in size and in orientation.

Then $P$ is the product of several integer factors of about $x^n$ in size.

The set $A$ is roughly triangular in shape.

The proof proper [= The actual proof] will consist of establishing the following statements in sequence.

convergence in probability $\langle$in distribution$\rangle$

Less than 1 in $p$ of its points will result in a quartic with ideal class number $p$.

Theorem 3 is remarkable in that considerably fewer conditions than in the previous theorems ensure universality.

As $M$ is ordered, we have no difficulty in assigning a meaning to $(a,b)$.

The prime 2 is anomalous in this respect, in that the only edge from 2 passes through 3.

This is where the notion of an upper gradient comes in.

The set $A$ is obtained from $B$ by removing a neighbourhood of $C$ and gluing in a copy of $D$.

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