# LETTERS

A Hierarchy for Electronic Publishing
Keith Tognetti
On Fermat's Little Theorem and Integer Matrices
Geoff C. Smith

## A Hierarchy for Electronic Publishing

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.

1. Local Pre-print. This gives an opportunity for anybody to get into print and be given an imprimatur by the Society which at least certifies the date of publication of the manuscript together with security on the content at that date. Although these pre-prints are not formally refereed perhaps they could be at least vetted informally within the Department as a check against obvious plagiarism and for fear of releasing material of too low a standard. These pre-prints should contain much more connective material than the final paper: that is they should be transparent. Perhaps in this way we might eliminate the all too common practice of including the expression it easily follows'' wherein the author has withheld that small amount of explanation that would make all the difference (- in too many cases the author has taken a great deal of effort and time to develop that small amount of explanation). Such pre-prints should also be accompanied by a comprehensive abstract and perhaps even some attempt at an expository version could be encouraged.

2. Local Abstracts. These are simply user provided abstracts of the pre-prints. The Society should keep a central register of these which can be accessed and copied by Site ftp. The Society would accept no responsibility for the quality or accuracy of the associated pre-print: this is entirely the responsibility of the author and perhaps the author's Department.

3. National Abstracts. These, and the International publications, are in the form of extended abstracts based on fully refereed papers. At this stage (which is simply a more elegant form of the pre-print updated by referees' comments) should be kept on a data base at the Site and, at least for the first few years, a certified hard copy should also be centrally stored so that hard copies can be sent on request to colleagues with no access to the Internet.

4. International Abstracts. These are abstracts of papers of the highest calibre designed to demonstrate that Australia is capable of truly outstanding work - whether the paper has in fact been published in an existing reputable journal outside the Society is incidental. In the case of a paper which has not already been published elsewhere, it might be appropriate to consider only those papers that have been accepted in National Abstracts. Then the referee can take account of subsequent commentaries from the community.

1. It is claimed that it is quite unnecessary to publish the full paper: the point is that the abstract in the Journal should allow the user to be able to make a decision about asking for the full paper. To get a copy, perhaps the reader should be required to contact the author directly: this could be a condition of acceptance by the Journal and would help to disseminate the electronic load within the community.

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.

2. The real-politik now is that we will become more and more under centralised bureaucratic control so it is only a matter of time before we will be forced to allocate Brownie points to each publication. Perhaps the Society should take the initiative and address this problem and at the same time the related problem of Salami Science - breaking up into LTUs (Least Publishable Units). After all, our publications do vary considerably in quality and in the effort required for preparation, so at least some attempt should be made to address this problem by the referee. Yes, I do realise the dangers of this, as on too many occasions, the referee has not realised the significance of the result.

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.

3. Local Abstracts should start emerging as soon as possible at the beginning of the study so as to encourage colleagues to interact and work collaboratively. The big question here is whether the author is brave enough to reveal only part of the picture and have someone else finish it off.

4. Finally, the question of costs. With the proposed new system, no university libraries will keep copies of post electronic journal publications. Thus, the university itself should subsidise the Society for the saving in the cost of the Journal and of course this should include both the direct cost and indirect costs (e.g. the cost of cataloguing and storing). So it is reasonable to expect that the nett contribution to the Society from a university may be substantially more than the former subscription costs. Of course, such an arrangement could be implemented only if the AVCC strongly supported the concept. And of course this will happen if and only if the Society is prepared to lobby vigorously with the AVCC on this matter.

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.

## References

1. Groves, J.R.J. and Newman, M.F. Electronic Publishing' Australian Mathematical Society Gazette , v.~21, no.~3 (1994) pp.72-80.
2. Mulvaney, John and Steele, Colin (eds) Changes in Scholarly Communication Patterns: Australia and the Electronic Library. Canberra: Australian Academy of Humanities, Occasional Paper No. 15, 1993.
Keith Tognetti
University of Woolongong
k.tognetti@uow.edu.au

## On Fermat's Little Theorem and Integer Matrices

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 $p.$

## References

1. Garry J. Tee, `Fermat's little theorem generalized to algebraic integers and to integer matrices' Australian Mathematical Society Gazette 21 (5) (1994) 160-164.

Geoff C. Smith
University of Melbourne and University of Bath

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