In response to a request by the Council of the Australian Mathematical Society, Groves and Newman [1] submitted a report which addressed the problems associated with the introduction of electronic publishing by the Society. Although their report was most impressive and addressed in particular the impact of various options on the Society, I believe that it did not go far enough in opening up discussion on the full suite of scenarios that now present themselves in the emerging electronic environment.
However, no matter what scenario is accepted by the Society, it must be agreed by all that their recommendation to set up an electronic site is a necessary preliminary.
I think that we should take this as an opportunity to review the very nature of journal publications and try to relate this to the other machine environment which is being forced upon us, namely the way in which we are allocated Brownie points for promotion (and now perhaps even survival) by the Unified National System.
I offer the following scenario, in the hope that it might provoke some spirited discussions. First of all, I wish to suggest that although the Society keeps a master copy of each accepted paper, it should quickly move to a policy of not actually publishing the paper. Instead the main concern of the Society should be to publish only abstracts. Eventually, perhaps even the abstracts should be published only electronically. The papers can be accessed as described below.
Of course with such a system, papers will continue to be refereed: indeed perhaps they will be refereed more thoroughly than previously.
In what follows, `Site' means the elctronic site run by the Society as proposed in the report [1]. It is proposed that we have two broad types of publications, namely A. Research Publications and B. Teaching Modules.
A. Research Publications. These should fit into a hierarchy as follows.
Comments.
It is emphasised that, even if such a recommendation were accepted, the Site should store a secure authenticated copy of the entire paper, probably at some nominated library which could be accessed if there is any dispute about precedence or if the author slips off the system.
When I start to do research in a new area, it is very time consuming to do a full literature search in an unfamiliar area. So, it would be nice at this early stage to make contact with a kind person who is familiar with the area and has the time and patience to give me some early guidance. This is why I would value local abstracts as I would be alerted to emerging work that was not done some years ago. Such a service would also increase the probability of finding that kind person.
B. Teaching Modules. One of the invariants that characterise teaching in every discipline and every department is the complaint that each institution appears to be continually reinventing the wheel and preparing lessons as though none of the others existed. Not only that, but in many cases, when a staff member leaves, even with a reputation as a fine teacher, there appears to be immediate obliteration of that member's teaching material.
What is suggested here is that members of the Society are challenged to present lessons for classifiation as modules. After some vetting, but please not nit picking refereeing at this stage, the lesson is accepted as a module and made available to the community for scrutiny and commentaries: that is, the module is tried out in the classroom and commentaries are reported back to be appended to the module. These commentaries might be included into the module at a later stage. In this way, the development of a lesson becomes an evolutionary process but one that depends more on co-operation rather than competition.
At this stage I must add a caveat. Even if you are convinced that the module proposal has some merit, staff will be encouraged to make the necessary committment to produce these very time consuming publications only if such publications are formally acknowledged for promotion.
I should point out that the Australian Academy of Humanities organised a conference on electronic publications (see Mulvaney [2]) which had several very valuable presentations, in particular those from Joe Gani and Robin Derricoat are excellent: the first explores the general problem of journal publication in the widest context including commercial as well as society publishers, and the second really comes to grips with the problems facing librarians in conversion to an electronic library.
In particular Derricoat asserts that hard copy will not be phased out until computer technologies ``freeze'' into a a more permanent format than we have been offered so far. How long has any computer storage medium lasted before the next generation replaced it? ``Will a 1995 CD-ROM be as useful to a mathematician in 25 years time as a 1995 book?''. The key question is not only for whom are we writing but ``for when are we writing''.
I finish by emphasising that this note is a discussion paper. If you have any comments, you might like to send them by e-mail to me at the address below. I would then like to make the gesture of including your comments in a summary which I will send to the panel.
This note addresses the implicit challenge in the final sentence of [1]. In that paper it is shown that if $A$ is a square matrix with integer entries and $p$ is a prime number then $tr(A) \equiv tr(A^p) \hbox{ mod } p.$ Other related results are also proved.
We work with polynomials in a variable $X$ and coefficients in the integers modulo $p.$ Observe that if $f = f(X), g$ are any such polynomials then both $(f + g)^p = f^p + g^p$ and $f(X)^p = f(X^p).$ Now let $\chi = \hbox{det}(X\cdot I - A)$ be the characteristic polynomial of $A$ with coefficients in the integers modulo $p.$ Let $\phi$ be the corresponding characteristic polynomial for $A^p$ then $$\phi(X^p) = \hbox{det}(X^p I - A^p) = \chi(X)^p = \chi(X^p).$$ Thus $\phi = \chi$ and so all elementary symmetric polynomials in the eigenvalues of $A$ and $A^p$ coincide modulo $p.$ In particular, the trace of $A$ and the trace of $A^p$ are the same modulo $p.$ If $A$ is invertible modulo $p$ then of course $tr(A^{-1})$ and $tr(A^{-p})$ also coincide modulo $p.$
Geoff C. Smith
University of Melbourne and University of Bath