MITACS RESEARCH PROJECT CV --- April

MITACS RESEARCH PROJECT CV - April 15, 2000

1. TITLE of PROJECT: SYMBOLIC ANALYSIS

This project is within the theme of "The Mathematics of Information Technology".

1.a KEY WORDS

Symbolic algebra, interactive mathematics, mathematical algorithms, infinite precision computational mathematics

------------------------

2. PROJECT DESCRIPTION

The project addresses the central issues of putting mathematical analysis into the framework of symbolic algebra packages and other online resources. This involves the development and efficient implementation of algorithms for problems like exact definite integration and summation, identity and inequality verification, non-smooth differentiation and many others.

The principal problem is to be able to incorporate analytic objects into symbolic computation environments (specifically Maple) with the same computational fluency as is presently available for algebraic problems. How, for example, does one address continuity and all the geometric issues this entails? Any analysis of inequalities concerning functions has to address this issue. Current systems break down at this point. Some specific components are:

Analytic algorithms - including algorithms for exact integration, identity and inequality verification, automatic differentiation, differentiation of non-smooth function, exact series evaluation, asymptotics. This involves both design and implementation.

Analysis of functions represented by formulae, differential equations and programs - includes handling of domain information, relations and inequalities, and automatic differentiation of programs.

Reverse symbolic engineering - the problem of determining what calculation led to a specific answer. This is a fruitful approach to a large variety of problems including the definite integration problem and the exact summation problem.

Highly interactive mathematical tools available over the internet.

------------------------

3. PROJECT LEADER:

Name: Peter Borwein Fields: Classical and Computational Analysis and Number Theory Department: Mathematics and Statistics University: Simon Fraser University Phone Number: 604 291-4376 Fax Number: 604 291-4947 E-mail Address: pborwein@cecm.sfu.ca Web Page: www.cecm.sfu.ca/ pborwein/

------------------------

4. OTHER TEAM MEMBERS:

SIMON FRASER UNIVERSITY:

Name: Jonathan Borwein, Field: Modern Applied Mathematics Department: Mathematics Statistics University: Simon Fraser University Phone Number: (604) 291-3070 Fax Number: (604) 291-5614 E-mail Address: jborwein@cecm.sfu.ca Web Page: www.cecm.sfu.ca/ jborwein/

Name: Loki Jorgenson Field: Visualization, Digital Publishing Department: Mathematics Statistics University: Simon Fraser University Telephone: (604) 291-5616 Fax: (604) 291-5614 E-mail: loki@cecm.sfu.ca Web Page: www.cecm.sfu.ca/ loki/

Name: Michael Monagan Field: Computer Algebra Department: Mathematics Statistics University: Simon Fraser University Telephone: (604) 291-4279 Fax: (604) 291-4947 E-mail: michael_monagan@sfu.ca

UNIVERSITY OF WESTERN ONTARIO:

Name: Robert Corless Field: Applied Analysis Department: Applied Math University: University of Western Ontario Phone: (519) 679-2111 (ext 8794) Fax: (519) 661-3649 E-mail: rob.corless@uwo.ca

Name: David Jeffrey Field: Applied Analysis Department: Applied Math University: University of Western Ontario E-mail: DJJ@apmaths.uwo.ca

Name: Stephen Watt Field: Symbolic Mathematical Computation Department: Computer Science University: University of Western Ontario Phone: (519) 661-4244 E-mail: stephen.watt@uwo.ca.

UNIVERSITY of QUEBEC at MONTREAL

Name: Francois Bergeron Field: Combinatorics Department: Mathematics Phone: (514) 987-3000 Email: bergeron.francois@uqam.ca

------------------------

5. NON ACADEMIC PARTICIPANTS:

Company Name: Waterloo Maple Inc. Participant: Ian Suttie/Jacques Carette Phone Number: (519) 747-2373 Fax Number (519) 737-1338 E-mail Address: isuttie@maplesoft.com Web Page: Maplesoft.com

Company Name: MathResources Inc. Participant: R. Fitzgerald Phone: (902) 429-1323 Fax: (902) 492-7101 Email: rfitz@istar.ca URL: MathResources.com

------------------------ 6. SCIENCE

a) Project Objective

Symbolic algebra packages (Maple and Mathematica) have over the last fifteen years reached a remarkable degree of sophistication. Rather difficult problems, like exact integration of elementary functions have been significantly attacked. A number of the most important algorithms of the twentieth century like FFT's and LLL are centrally incorporated. The packages can now substantially deal with large parts of the standard mathematics curriculum (and can outperform most of our undergraduates).

There is a coherent argument that they are the most significant part in a paradigm shift in how mathematics is done and certainly they have become a central research tool in many subareas of mathematics both from an exploratory and from a formal point of view. (It is acceptable now to see a line in a proof that says something like "by a large calculation in Maple we see".) The first objective of symbolic algebra packages was to do as much exact mathematics as possible. A second objective is do it fast and to deal in an arbitrary precision environment with the more usual algorithms of analysis. Basically one would like to be able to incorporate the usual methods of numerical analysis into an exact environment or at least into an infinite precision environment.

The problems are obvious and hard. For example how does one do arbitrary precision numerical quadrature? When does one switch methods with precision required or with different analytic properties of the integrand? How does one deal with branch cuts of analytic functions? How does one even deal consistently with log (even this isn't completely worked out)? More ambitiously how does one do a similar analysis for differential equations? The goal is to marry the algorithms of analysis with symbolic and exact computation and to do this with as little loss of speed as possible.

Within this context a number of very interesting problems concerning the visualization of mathematics arise. How does one actually ßee" what one is doing. It has been argued that Cartesian graphing was the most important invention of the last millennium. Certainly it changed how we thought about mathematics - the subsequent development of differential calculus rested on it. More subtle and complicated graphics, like those of fractals, allow for a kind of exploration that was previously impossible. There are many issues to be worked out here that live at the interface of mathematics, pedagogy and even psychology but are very timely to get right. (Think of how one visualizes the human genome and its patterns - which is after all just a particular several billion digit number base four.)

The nature of our project is to attack those particular pieces of the puzzle that we see as important; where we have the appropriate skills; and which hold mathematical attraction. The speed with which new research can be incorporated is striking and in this sense the technology transfer is close to immediate. Scientific breakthroughs have a a variety of flavors. A thousand fold speed up in large sparse matrix multiplication is one type of breakthrough (as Maple has achieved between its last two versions). Brand new algorithms are another obvious type of breakthrough. A third type of significant achievement is the incremental building of large tools that in the end solve a large variety of problems. This is not very sexy but is very central and sometimes very effective (redolent of the successful treating of childhood leukemia which has been a lot of tuning of a lot of different therapies).

The specific places where we wish to make progress vary with the particular expertise of current post docs and students. To date they have centered on

- code relating to polynomial GCDs and resultant - simplex type method for sparse matrices - simplification of elementary functions (unwinding number problems) - infinite precision numerical quadrature - exact solutions of ODE's - an algebraic solver for differential equations - fast arbitrary precision multiplication - integer relation algorithms and inverse symbolic computation (PSLQ)

Projects we would like to get more involved with include

- finding all zeros of arbitrary analytic functions in arbitrary regions - visualization tools - design of specialized mathematical/engineering tool boxes

b) Methodology

Canada's position as a major participant in the development of mathematical software stems from the early eighties and the very successful development of Maple (at Waterloo). Tools such as Maple, and its main competitor Mathematica, and the related tool Matlab are now the primary research and development tools in mathematics. These are now substantial sized businesses.

The success of these tools relies on (literally) hundreds of individual developments of two varieties: mathematical development of new and faster algorithms and software design improvements. All of this is set against the vital issue of designing user compatible interfaces. Many of these issues are shared by the burgeoning industry involved with building on-line interactive (often Java based) mathematics interfaces. Initiatives in this direction are being undertaken at a number of Canadian Universities and by a number of recent high-tech companies and raise a host of interesting problems. An ancillary goal of this project is to support the development of these initiatives.

The great success of the symbolic algebra packages has been their mathematical generality and ease of use. These packages deal most successfully with algebraic problems while many (perhaps most) serious applications require analytic objects such as definite integrals, series and differential equations. All the elementary notions of analysis, like continuity and differentiability have to be given precise computational meaning. The first challenge involves mathematical algorithmic developments to allow the handling of a variety of these only partially handled problems - including the analysis of functions given by programs. Many of these relate to the difficult mathematical problems involved in automatic simplification of complicated analytic formulae and recognition of when two very different such expressions represent the same object. There is also an intrinsic need to mix numeric and symbolic (exact and inexact) methods.

The problems of interest are determined in conjunction with Waterloo Maple Inc. We suggest what interest us and they let us know what interest them. They are familiar with academic research and understand that they get the best work from people doing what they are most interested in. In this sense it is a very satisfactory partnership.

At SFU where the largest part of the project runs we have a weekly seminar and meeting. We also have occasional special days and have a variety of related courses. M. Monagan teaches a fourth year/graduate course in symbolic algebra which is a very useful training ground.

We house the participants together where possible to foster team research and indeed will house all our MITACS personell together in newly renovated space.

c) Sub-projects

Analysis of functions represented by formulae and programs - includes handling of domain information, relations and inequalities, and automatic differentiation of programs.

Simplification code relating to polynomial GCDs and resultants and some code for simplifying (combining) logarithms and exponentials.

Communication - having the various components talk to each other. LaTex to Maple to XML/MathML for example.

Building on-line interactive (often Java based) mathematics interfaces.

------------------------

7. PROGRESS MADE and RESULTS OBTAINED

a) Briefly summarize the Project's main achievements.

We have got most aspects of this project up and running. We have developed and incorporated significant Maple related code (as out lined below). This involves both theoretical developments and implementation improvements. We have substantially prototyped a reverse engineering tool "RevEng".

b) Describe the Project's progress towards the Project Plan.Explain any impediments to progress or changes in direction.

By Francois Bergeron (with Anouk Brlek):

1: Maple Prototype of: Algebraic Solver for Differential Equations.

By Peter Borwein (Matt Klassen, Steve Funk, Alan Meichsner).

1: The basic maple code for the reverse engineering tool RevEng has been written. This tool is designed to analyze numerical input and decide if it probably has a closed form. A prototype Java interface is available at "http://psg.cecm.sfu.ca/projects/revenge/client/RevEngClient.html". This includes an implementation of PSLQ.

By Mike Monagan

This lists code which we are installing which was not written under MITACS but which is being installed under MITACS.

1: A modular algorithm for polynomial resultants in Z[y1,y2,...,yn][x]. It turns out that the work on DEs being done by Edgardo Cheb-Terrab and also the work of David Boyd on A-polynomials both lead to large multivariate polynomial resultant computations which Maple cannot do but we now can do with this code, often in only a few minutes. The code was developed by Allan Wittkopf prior to MITACS, then improved and installed this summer by Michael Monagan under MITACS.The new code is automatically called by the resultant routine in Maple when the polynomials have sufficiently large degree that the determinant based methods would be too expensive.

2: Multi-polynomial resultants: an initial investigation. Given n+1 polynomials A1,A2,...,An+1 in Z[a1,...,am][y1,y2,...,yn][x], m>=0, one can use resultants to eliminate y1,y2,...,yn one variable at a time. Alternatively one can define a multi-polynomial resultant as the determinant of a square matrix which eliminates y1,...,yn in one step. Peter Lisonek's initial implementation follows the definition.It is ready to be installed and we propose to use the syntax for a multi-resultant resultant([A1,A2,...,An],[y1,y2,...,yn-1]); or resultant[y1,y2,...,yn](A1,A2,...,An); This computation leads us to investigate how to compute the determinant of a matrix A in Z[a1,...][x] efficiently. We are investigating how it compares with the approach of taking repeated resultants.

3: Simplifications for the constant logarithms and exponentials. Given an constant f in logarithms and exponentials, there exists a decision procedure due to Richardson, which assumes the generalized ERH, to determine whether f=0 or not.But it is very expensive. Peter Lisonek and Michael Monagan have developed their own "light-weight" approach to this problem and are presently installing the code as part of the combine command in Maple.

4: A modular implementation of Grobner bases. Michael Monagan has written a simple modular version of Grobner bases using Allan Meichsner's implementation over Q and has shown that this is much faster on more difficult problems than either the older Release 4 version in Maple or the newer Release 5 version in Maple. This investigation is an outcome of the course on Grobner bases that we held over the summer of 1999.

By E.S. Cheb-Terrab (with Mike Monagan):

1: Programs incorporated into the Maple system: -approximately 3,000 lines of ASCII code, mainly related to: 1. Solving systems of ordinary differential equations 2. Solving abelian type first order differential equations 3. Solving Lie groups in terms of derived algebras - this algorithm is now in use in dso 4. Interface for using differential elimination programs for DE systems 5. Enhancements all around in (throughout) the previously existing code

------------------------

9. RESULTS of the RESEARCH

a) Please create update the list of all publications directly arising from Network-funded research.

PAPERS:

1) Peer-reviewed publications.

J. M. Borwein, D. M. Bradley, D. J. Broadhurst, P. Lisonek, "Special values of multiple polylogarithms." Transactions of the American Mathematical Society, accepted.

Jonathan M. Borwein and Robert Corless Ëmerging tools for experimental mathematics" MAA Monthly, December 1999.

These papers take advantage of EZFace Java tool we developed, which is at the URL http://wayback.cecm.sfu.ca/projects/ezface+

2) Refereed/invited conference proceedings

Robert M. Corless, Mark W. Giesbrecht, David J. Jeffrey, Stephen M. Watt, Äpproximate Polynomial Decomposition" Proc. ISSAC 99

3) Other publications (including submissions).

Francois Bergeron and Francis Gacon "Counting Young Tableaux of Bounded Height" submitted to Journal of Integer Sequences.

Jonathan M. Borwein, David J. Broadhurst and Joel Kamnitzer "Central binomial sums and multiple Clausen values" preprint, August 1999

J. M. Borwein and T. Stanway "Numerical and computational mathematics (at the undergraduate level)," Proceedings of the Technology in Mathematics Education at the Secondary and Tertiary Levels Conference, June 1999 at Brock University, August 1999. (Also to appear in Cubo.)

Peter Borwein and Loki Jorgenson "Visible Structures in Number Theory" preprint, August 1999

Erich Kaltofen "My favourite Open Problems in Computer Algebra", with an appendix by Robert M. Corless and David J. Jeffrey, submitted

Petr Lisonek and Robert Israel "Metric invariants of tetrahedra via polynomial elimination". Accepted conference paper, ISSAC 2000, St. Andrews (August 2000)

E.S. Cheb-Terrab and A.D. Roche Än Abel ODE Class Generalizing Known Integrable Classes" submitted to European Journal of Applied Mathematics, 1999

Robert M. Corless and Jacques Carrette (WMI) Are preparing a paper Änalysis is not Algebra" for submission to ISSAC 2000

Kevin Hare "Some applications of the LLL algorithm" submitted

Colin Percival "FFT Multiplication" to be submitted

b) Describe the Intellectual Property (IP), including software, developed as a result of this research.Describe the state of this IP and its readiness for use by other institutions or industry, if applicable. Indicate whether the IP has or will be licensed.Are patents or other protection being sought?

Much of the software developed is outlined above. The IP issues are still being finalized with Maple. None of the software will be "checked in" with Maple till this is done.

c) List presentations (lectures, workshops, conferences, poster sessions) of Network supported research.

LECTURES:

Francois Begeron: August, 1999 - MITACS Day at SFU "Guessing Liouvillian Solutions to Linear Differential Equations"

Jonathan Borwein: - January 1999 - MAA Invited Address, Combined Math Meetings, San Antonio Ëxperimental Mathematics: Insight from Computation" - January 1999 - Institute of Advanced Research (IAS), Technion, Haifa, Israel Ëxperimental Mathematics: Insight from Computation, II" - February 1999- Burnaby Rotary Club "Publishing on the Web" - March 1999 - Invited Address, MAA Pacific Northwest Section Meeting, Salem Ëxperimental Mathematics: Insight from Computation, II" - June 1999 - Technology in Mathematics Education, Plenary, Brock University "Numerical and computational Mathematics at the undergraduate level" - July 1999 - Plenary Lecture (ISSAC), Vancouver Ëxperimental mathematics and exact computation" - August, 1999 - MITACS Day at SFU Ïnteractive Network Mathematics" - September 1999 - Physics Department, University of Bologna, Colloquium Ëxperimental mathematics and exact computation" - October 1999 - Miami University of Ohio Colloquium "Doing Math in the Presence of Technology" Colloquium - October 1999 - Ëxperimental Mathematics" Miami University Ëxperimental Mathematics: Insight from Computation" - March 2000 - ``Parallel Symbolic Computation: Methods and Issues," Haifa-Technion Workshop on `Inherently parallel algorithms in optimization and feasibility and their applications'

Peter Borwein: - April, 1999 ECCAD, Raleigh "Three Problems in Computational Number Theory" - August, 1999 ISSAC Panel, Vancouver - August, 1999 ISSAC, Vancouver Demo of RevEng - September, 1999 University of Western Ontario "Seeing Math" - September, 1999 Simon Fraser University "Seeing Math" - January, 2000 Cognos Lecture, Carleton University - April, 2000 Changing the Culture Conference Panel on Mathematical Visualization

David Boyd: August, 1999 - MITACS Day at SFU "Computing A-polynomials"

Edgardo S. Cheb-Terrab: -July 1999, Oxford, Foundations of Computational Mathematics (FoCM) Äbel ODEs, their equivalence, classification and new integrable classes" -Maple Day at CECM, August, 1999 "Computer Algebra approach for solving analytic systems of DEs" -Four talks for the SFU Computer Algebra Group

Robert Corless: - June, 1999 - invited talk at U. Cantabria, Santander, Spain Ëmerging Tools for Experimental Mathematics" - July, 1999 - tutorial, with Larry Shampine "Numerical Solution of IVP in a PSE" - August, 1999 - MITACS Day at SFU "What Symbolic Analysis means at Western" - August, 1999 - SciCADE 99, Fraser Island, Australia, "Numerical Solution of IVP in a PSE" - September, 1999 - invited talk at the 5th Spanish Computer Algebra Meeting (EACA 99) Santa Cruz de Tenerife Öpen Problems in Computer Algebra: Converting the Risch Integration Algorithm from an Algebraic Algorithm to an Analytic Algorithm" - October, 1999, Dept. Mathematics, University of Pisa "Dynamics of the Lambert W function" - November, 1999, Dagstuhl Seminar No. 99471, Report No. 260 Änalysis of Approximate Polynomials"

Loki Jorgenson: -March 1999 - Departamento de thematica Universidad de Chile "Distribuido el aprender: El cuer de matematica para la red" -June 1999 - Conference on Technology in Mathematics Education, Brock "Future of Mathematics on the Web"

Matt Klassen: -August, 1999 - MITACS Day at SFU "Number Recognition in Maple"

Petr Lisonek: -April, 1999 - A Day of Maple "Simplification of elementary functions and elementary constants in Maple" August, 1999 - MITACS Day at SFU "Simplification of elementary constants and elementary functions in a computer algebra system" August, 1999 - ISAAC 1999, Vancouver EZFace demo -Four talks for the SFU Computer Algebra Group -November 20, 1999, ITACS Information Technology Meeting, Toronto Överview of the Simplification Project" -January 18, 2000, CECM/MITACS Afternoon of Number Theory "Mahler Measure of Fewnomials"

Michael Lamoureux - August, 1999 - First China-Canada Congress, Beijing "Computation in Modern Analysis"

Alan Meichsner - February 2000 - ASI Exchange, Vancouver Poster and Demo of Revenge

------------------------

10. INTERACTION WITH INDUSTRY - period: April-October, 1999

Ä Day of Maple at Simon Fraser" - presentations focusing on the issues that this project addresses were held on April 20, 1999.Jacques Carette, the head of the mathematics research group at Maple, which is the main industrial partner on this project, had two presentations titled "What is new in the Maple kernel" and "What's new in the Maple library". Project investigators, Edgardo Cheb-Terrab, Petr Lisonek, and Matt Klassen had presentations also. Peter Borwein has visited Maple in Waterloo as have Rob Corless and Stephen Watt.

------------------------

11. OTHER RESEARCH PERSONNEL

a) Visiting Academics:

Name: David Bailey University: Livermore Labs (Berkeley) Visiting SFU from Livermore Labs (Berkeley) August, 1999

Lecture: "Parallel Integer Relation Detection: Techniques and Applications"

Name: Francois Bergeron University: University of Quebec Visiting SFU July-August, 1999

Name: Frederic Chyzak University: INRIA Visiting UQAM from October 25, 99 - November 5, 99

Name: Michael Lamoureux University: University of Calgary Visiting SFU July-August, 1999

Name: Kieth Geddes University: University of Waterloo Visiting SFU Jan-April, 2000

c) POST-DOCTORAL FELLOWS:

Name: Petr Lisonek University: SFU Period of involvement: January 1999 -

Name: Edgardo Cheb-Terrab University: SFU Period of involvement: February 1999 -

Name: Matt Klassen University: SFU Period of involvement: January 1999 - September, 1999

d) RESEARCH STAFF

Name: Stephen Funk University: SFU Period of involvement: July 1999 - April 2000

e) GRADUATE STUDENTS

Name: Hans Bauck (M.Sc. Candidate) University: SFU Period of involvement: Feb 1999 -

Name: Mhenni Benghorpal (Ph.D. candidate) University: Western Ontario Period of involvement: September 1999 -

Name: Jennifer de Kleine (M.Sc. Candidate) University: SFU Period of involvement: September 1999 -

Name: Gurjeet Litt (M.Sc. Candidate, soon to be Ph.D. candidate) University: Western Ontario Period of involvement: September 1997-August 2003

Name: Alan Meichsner (M.Sc. Candidate) University: SFU Period of involvement: September 1999 -

Name: Francis Gascon University: UQAM Period of involvement: June 99 December 99

Name: Kevin Hare (Ph.D. candidate) University: SFU Period of involvement: Feb 1999 -

Name: Terry Stanway (M.Sc. Candidate) University: SFU Period of involvement: Feb 1999 -

Name:Allan Wittkopf (Ph.D. candidate) University: SFU Period of involvement: Feb 1999 -

f) OTHERS (e.g. co-op student, summer student, consultant, etc.)

Name: Anouk Brlek University: UQAM Period of involvement: June 99 September 99

Name: Colin Percival (Summer NSERC) University: SFU Period of involvement: Summer, 1999

Name: Simon Plouffe University: UQAM Period of involvement: September 99 -

------------------------

12.(a) STUDENTS WORKING ON NETWORK RESEARCH

PhDs 3 Male (two Canadian) PhDs 1 Female (Canadian) Masters 3 Male (all Canadian) Masters 2 Female (both Canadian)

(b) POST NETWORK EMPLOYMENT OF GRADUATE STUDENT/Postdocs who left the Network during the past year

Fiscal Year 1999-2000 Matt Klassen (C): PDF January - September, 1999 Now employed by Digipen (digital university) in Seattle

Stephen Funk RA July 1999 - April 2000 Now employed by Tryllian an Amsterdam

------------------------

14. TRAINING, WORKSHOPS, ETC.

We ran several one day conferences:

A DAY of MAPLE at SIMON FRASER: Tuesday April 20. 1999. This had eight speakers including Jacques Carette from Waterloo Maple.

MITACS Day - Symbolic Analysis:Tuesday August 3, 1999 This had all the principal investigators for our project together for a day of lectures and discussion.

PROJECT DAY - October 12, 1999 This was a morning of presentations on our project to the theme leader and ourselves.

MITACS/MAPLE Day - March 10, 2000 A day of presentation to and discussion with Maple Inc.

We were heavily involved in running ISAAC'99 the main international symbolic algebra meeting - particularly Mike Monagan who was on the organizing committee.

We took a long term view of our MITACS project on symbolic analysis and felt that in the, hopefully, likely event that the project will run many years, we should provide some more formal way to (re-)educate ourselves in computer algebra because several of the people on the project have little formal training in computer algebra. Michael Monagan in January-April 1999 was teaching MATH 814, an introductory course on polynomial factorization and indefinite integration. Peter Borwein and Matt Klassen sat in and Hans Bauck and Alan Meischner took the course. Another important area in computer algebra, which is directly relevant to two of our sub projects is Grobner bases.We ran MATH 895, a seminar type course in the summer of 1999 based on the Cox/Little/O'Shea text Ïdeals Varieties and Algorithms: An Introduction to Computation Commutative Algebra and Algebraic Geometry".This provided us with a solid background on Grobner bases. Four students Allan Wittkopf, Kevin Hare, Alan Meichsner, and Hans Bauck took this course and Peter Borwein, Michael Monagan, Matthew Klassen, Greg Fee, attended the course. During the period we have also run a weekly meeting at which one of us presents some results or presents a problem. In addition to this, we hosted ISSAC '99 in Vancouver in late July 1999.This provided an inexpensive (no travel expenses) opportunity for everyone to attend the main international event in computer algebra.

------------------------ 14. AWARDS

J. Borwein received an Honorary Doctorate from Limoges in September, 1999.

J. Borwein and P. Borwein received a University of Wetern Ontario National Alumni Merit Award in October, 1999.

P. Borwein gave the Cognos Lecture in Carleton in January 1999

S. M. Watt received a Premier's Research Excellence Award, 1999

R. M. Corless was awarded the Distinguished Research Professorship (Faculty of Science) 1999-2000

File translated from T_EX by T_TH, version 2.20.
On 21 May 2000, 10:36.