provide

[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.



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