[see also: give, furnish, provide, present, yield, construct, create]
By induction, this process produces a sequence $(x_n)$ such that ......
This result shows that the mere existence of a nontrivial automorphism $j$ of $M$ produces the cut $I(j)$ of $M$ that satisfies (2).
The main new feature is the use of the face ring to produce lower bounds for the number of vertices.
Roughly speaking, we shall produce a synthesis of index theory with Fourier analysis.
This metric produces the usual topology of $X$.
Here, of course, the set $A$ produced is rather thin and certainly nowhere near the densities we are looking for.
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