On global aspects of exact WKB analysis

A case-study of the Stokes geometry in the large is given for some concrete examples of WKB type operators that admit infinitely many phases. In all the examples discussed in this article virtual turning points and new Stokes curves emanating from them neatly resolve the trouble we encounter. Some computerassisted study of the configuration of the steepest descent paths is employed to show that the introduction of new Stokes curves is reasonable and appropriate to obtain the complete Stokes geometry

On the Number of Permutations

Let m be a positive integer, and p n (m) the proportion of permutations of the symmetric group S n that admit an m-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula where ' is the Euler function and m an explicit constant