Topic #5 ------------ OP-SF NET 5.5 ----------- September 15, 1998 ~~~~~~~~~~~~~ From: Martin Muldoon
Subject: Report on Minisymposium on Problems and Solutions in Special Functions On July 14, 1998, our Activity Group sponsored a Minisymposium "Problems and solutions in Special Functions" (Organizers: Willard Miller, Jr. and Martin E. Muldoon) at the SIAM Annual Meeting in Toronto. The organizers recognized that by providing concrete and significant problems, the problem sections in journals such as SIAM Review and the American Mathematical Monthly have been influential in advancing mathematical research and have played a role in attracting young people to the mathematical profession. At a time when the SIAM Review is phasing out its problem sections (see OP-SF NET 4-6, Topics #17, #18 and #19), it seemed appropriate to assess the history and impact of the problems sections and their future evolution. Cecil C. Rousseau, University of Memphis offered a retrospective on the 40-year history of the SIAM Review Problems and Solutions Section, based on his experience as a collaborating editor and then as an editor of the Section. We learned that of the 777 problems proposed, 329 were starred (no solution submitted by the proposer). The title most used was "A definite integral" while the keywords occurring most frequently were "integral" (131 times), "inequality" (47), "identity" (33), "series" (25) and "determinant" (24). The most frequent problem proposers were M. S. Klamkin (46), M. L. Glasser (38), D. J. Newman (24) and L. A. Shepp (20). Cecil chose a specific issue (April, 1972) and mentioned Problem 72-6 by Paul Erdos ("A solved and unsolved graph coloring problem") that provided the first contact between Erdos and the Memphis graph theory group (Faudree, Ordman, Rousseau, Schelp), and in that way led to more than 40 joint papers involving Erdos and the members of this group. He mentioned Problem 72-9 ("An extremum problem") by Richard Tapia, who, coincidentally, was honored on the same day as our Minisymposium by a Minisymposium for his 60th birthday. In the same issue, the solution to Problem 71-7 ("Special subsets of a finite group") was the very first publication by Doron Zeilberger. Rousseau himself had a solution of Problem 71-13 proposed by L. Carlitz, which called for a proof that a certain integral involving the product of Hermite polynomials was nonnegative. At the time, Rousseau looked for, but did not find, a combinatorial interpretation of the integral that would immediately imply its nonnegativity. That there is such an interpretation was shown by Foata and Zeilberger in 1988. Later in the discussion, Rousseau mentioned that problems sometimes get repeated in spite of the best efforts of the editors; for example, Problem 95-6 repeats part of Problem 75-12 but that he had found the relevant double integral later in Williamson's Calculus (6th ed), 1891! Otto G. Ruehr, Michigan Technological University, discussed the forty-year history of the Section with particular attention to the second half. He offered an anecdotal description of the trials, tribulations and satisfactions of being editor. Special attention was paid to problems in classical analysis, particularly those relating to orthogonal polynomials and special functions. He regretted that some problems he had proposed (73-12, 84-11) attracted only one solution other than that of the proposer. Sometimes, sheer luck played a role as in a solution of his which depended on the relatively sharp inequality 27e^2 < 200. In spite of the best editorial efforts, errors often crept in. In the very last issue which contained problems a complicated asymptotic expression (Problem 97-18) was correct except for an error in sign! Nevertheless, it led to collaboration between one of the proposers (D. H. Wood) and J. Boersma. Otto mentioned that, very appropriately, the last issue (December 1998) of the Section will be dedicated to its founding editor, Murray S. Klamkin. In some brief remarks, Murray discussed some highlights and problems such as "A network inequality" and (the very first) Problem 59-1 "The ballot problem", Proposed by Klamkin and Mary Johnson. This has not been solved in the general case. Willard Miller, Jr., University of Minnesota, spoke on "The Value of Problems Sections in Journals" He stressed their importance in getting young people interested in mathematics and as a place where a person not expert in an area can get their feet wet. He offered Doron Zeilberger as someone who exemplified the value of problem sections. Bill mentioned that by participation in problems sections you can get established researchers in other areas interested in what you have to say. People see that a problem is hard and when the solution comes out they are interested in it and are challenged to find a better proof. Richard Askey, University of Wisconsin, was unable to attend the Minisymposium but submitted a written statement, some of which was read by Bill Miller, and which offered some thoughts about problems and the role that a problem section can play in a scientific journal. Askey's first example was on the generalization to Jacobi polynomials of an inequality for trigonometric functions. Rather than writing a one page paper, he decided to submit it as a problem to have people work on it. Unfortunately his plan failed. Nobody else submitted a solution because he had not been explicit enough about a limiting case which would be more familiar to readers. Askey also described some of the history (including an incorrect published solution) of a problem where it was required to show that the sum from 1 to n of (-1)^(k+1)[sin(kx)/ksin(x)]^(2m) is positive for all real x, m = 1,2,... Askey described his favorite SIAM Review problem as Problem 74-6 ("Three multiple integrals") submitted by a physicist, M.L. Mehta. It called for the evaluation of a multidimensional normal integral. "I spent many hours on this problem, unsuccessfully. Eventually, a multidimensional beta integral which Atle Selberg had evaluated about 1940, and published a derivation of in 1944, came to light. Then it was easy to prove the Mehta-Dyson conjecture, as Dyson realized once Bombieri told him of Selberg's result. I heard about this from George Andrews, who was in Australia at the time, and he heard of it from Kumar, a physicist there. I worked out what should happen in a q-case, and published my conjectures in SIAM Journal of Mathematical Analysis. All of these conjectures have now been proven. Ian Macdonald heard about Selberg's result from someone in Israel, and he came up with some very significant conjectures about other q-beta integrals. He had been working on questions like this for root systems, and his conjecture for a constant term identity for BC(n) was equivalent to Selberg's result. Some of this would have been done exactly as it was without Mehta's problem in SIAM Review, but I doubt that I would have appreciated the importance of Selberg's result as rapidly if I had not spent so much time on the Mehta-Dyson conjecture." In the general discussion which followed there was mention of "opsftalk" the discussion forum for this Activity Group. It was generally agreed that it could not replace Problem Sections of the kind being discussed both because of the limited readership and the fact that it is restricted to orthogonal polynomials and special functions. Dick Askey had cautioned: "I am afraid that having a problem section only on line will lead to a restricted group of readers, those with a love of problems for their own sake, and not reach the wider group of mathematicians, applied mathematicians, and scientists who could use some of the results in these problem sections." A wide-ranging discussion continued informally between those attending. Some of the points raised in these discussions follow: It was felt that it was very important to stress that any web initiative for a Problem Section should cover all areas. It would be unsatisfactory to have separate operations for say, the various SIAM activity groups. There was some skepticism about the web proposal. In particular, the importance of careful editing was stressed. It is a commonplace that much material on the web is sloppy and done in a hurried manner. It will be very important to make sure that the present proposal is carefully monitored. There was also a sense that "putting it on the web" is sometimes offered as a panacea for all sorts of information distribution without a realistic understanding of the work involved. Nevertheless, the advantages of speed and access which are provided by a web site are eagerly anticipated by those interested in preserving and enhancing the SIAM Review Problem Section. There should be a part of the web initiative devoted to problems suitable for high school students. This has the potential to greatly broaden the audience for the problem sections and to attract more young people to mathematics research. The web pages should be divided into two parts. Part A would contain the problems and refereed solutions, and would be comparable to what appears in the SIAM Review now (but with hyperlinks and other bells and whistles). Part B would be more informal. It would contain proposed solutions (before they have been fully refereed), comments on the solutions and other comments and background information related to the problems. Part B would be more timely. The editor would still control what is posted in Part B but wouldn't vouch for the accuracy of all proofs. Part B would be more lively, and give a better indication of how mathematics research is actually carried on. Part A would be more polished. There should be some way to archive in print the problems and solutions of Part A. Perhaps a volume could be produced every few years. SIAM should refer routinely to the website in the Review, say a paragraph in each issue. Once the website is well launched, there should be an article about the project in the SIAM Newsletter.