[see also: give, afford, furnish, offer, produce, supply, yield
Our last example was kindly provided by B. Johnson.
This itself does not produce a solution of (1), but an additional hypothesis such as the Palais--Smale condition does provide such a solution.
This provides an effective means for computing the index.
In addition to illustrating how our formulas work in practice, it provides a counterexample to Brown's conjecture.
The method falls short of providing an explicit formula for the index.
This will provide us with a way of getting some information about...... without relying on general theory.