|Kuratowski's Closure-Complement Cornucopia|
The motivation to build this site
can be traced back to early 1983, when
Cantor’s ternary function was introduced in
undergraduate analysis sequence (MATH 424, 425, 426) at the
University of Washington in Seattle (it is interesting to note that to this day the UW still uses the
which I prefer). An inquisitive classmate named Peter asked about the values of its derivative on the
Claiming to not know the answer, Lind posed it as a fun problem to investigate.
He sometimes gave us “lollipop” problems to solve (an actual
lollipop went to the author of the winning solution), but this time he left us with
a motivation more powerful than sugar:
Applying knowledge gained from a number theory course (MATH 414) taught by Ralph Greenberg during Winter Quarter 1982, I went home and proved that the derivative is positive infinity on a certain subset of the Cantor set. After reading my proof, Professor Lind posed a follow‑up question:
With respect to the standard coin-flipping measure, does the Cantor function have infinite derivative almost everywhere on the Cantor set?
This new question appeared to call for knowledge acquired from the UW’s undergraduate probability sequence (MATH 394, 395, 396) taught by Ronald Pyke during 1981‑82, except we never went into any measure theory in that course. Lebesgue integration was still a ways off in analysis, too. So I went to Professor Pyke’s office. After reading my proof and taking a few seconds to think about the follow‑up problem, he concluded the answer is “yes” by the strong law of large numbers. He sketched out an argument and suggested I write up a full proof to send to the American Mathematical Monthly as a problem proposal.
The idea of publishing mathematical work had never occurred to me, so this was a very exciting suggestion! However, when I told Professor Lind about it, his response was uncharacteristically muted. He didn’t explicitly dissuade me from submitting the proposal, but he seemed to have some sort of misgiving about it. The following problem might provide a clue as to why:
This was Lind’s first Monthly problem. He was still in high school when he proposed it. After receiving 79 solutions, the editors gave the problem its unflattering title.
If Professor Lind suspected (or knew) that his follow-up problem was actually just another major theorem “thinly disguised,” his suspicion was correct!
In this case there really wasn’t even any “disguise” to speak of.
Here is my second Monthly problem [1997 BRR PMark Bowron, Stanley Rabinowitz, Closure, Complement, and Arbitrary Union, Problem 10577, Amer. Math. Monthly, v. 104 no. 2, 1997, p. 169, Solution: John Rickard, v. 105 no. 3, 1998, pp. 282‑283.], co-proposed with Stanley Rabinowitz in 1997:
When we proposed it, neither of us was aware that multiple solutions had already appeared in print (a list is posted here).
The unoriginality of these first two problem proposals of mine played a key role in my decision to post the large list of references and English translations found here. Indeed, the primary goal of this website is to help future researchers (and editors) avoid republishing old results.
4 Jul 2017