Interview with Pam Pierce
Reuben Settergren

     How can you measure the variation of a function? One way to do it is to break it up into increasing or decreasing (monotonic) segments (a common calculus task), and add up the heights of all the pieces. For instance, you could call a (very boring) function that is all one monotonic piece of height 0. Or breaks up at every multiple of pi into an infinite number of monotonic pieces, all of height 2. Or how about a discontinuous function g, which is usually at the constant height of 0, but at blips up to . The increasing piece () has height 3, and so does the decreasing piece ( ). Since their total amount of variation is finite, functions fand gbelong to the class BV: functions of bounded variation (which might be familiar to students of Dr. Pierce's Real Analysis course). The function has an infinite amount of variation, and so does not belong to BV.

     Now one of the areas of mathematics in which Dr. Pierce is active involves generalizations and variations of the class BV. One such generalization that was recently introduced in the literature is called BVM, or functions of bounded variation in the mean. The number and heights of the monotonic pieces of function make a difference to BV, but (in a sense) not the widths. For example, the tent function

     also (like g) has one increasing and one decreasing piece, both of height 3 (check it out!). But in g, all the increasing and decreasing takes place instantaneously, while in h it happens over a wider interval. The definition of the class BVM has a more complicated measure of function variation, which is designed to yield different amounts of variation for functions such as g and h. (But since they would still have finite variation according to both measures, they would still both be in BVM as well as BV.)

     Now it has been proven that functions in the class BV have certain useful properties. This is why mathematicians are interested in the class BV, and in fact, the class BVM was created because of provable properties belonging to its functions. Therefore, it was a natural mathematical step for Dr. Pierce (and her collaborator, Daniel Waterman of Syracuse) to characterize the properties of this new class BVM---specifically, to determine how many known properties of BV have analogues for BVM?

     Their eventual result, however, was that each function in BVM is equal almost everywhere to a function in BV, which means the classes are essentially the same, so all known properties of BV apply to BVM. ("Almost everywhere" can mean a huge difference in variation, however... it gets complicated.) It just goes to show that mathematics is a process of discovery. For homeworks and tests, a student knows what has to be proven, so (except for typos), they can be sure it is true beforehand. But in the great unknown of mathematics, you can never be sure of where you're going until you get there.

     Dr. Pierce submitted her article for publication on Sept. 24, '98. It is interesting to note that on the same date, Dr. Pierce completed calculation of the next term in the well-known sequence P_n (a project she had been working on for approximately 9 months). This sequence has initial values Pam and Jim, and a recurrence relation has already produced P₃ = Brian. The most recent term will be referred to as Rebecca (which is a much nicer name than "P₄"). As for future terms, Dr. Pierce's only comment was "I would be quite alarmed if the recurrence yielded an infinite sequence."