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.