MITACS Day -- Symbolic Analysis

(Tuesday) August 3, 1999--Room K9509, Burnaby Campus, Simon Fraser University

Schedule of Talks
9:00 - 9:15	Peter Borwein, Simon Fraser University Opening
9:15 - 10:00	David Bailey, National Energy Research Scientific Computing Center, Berkeley, CA, USA Parallel Integer Relation Detection: Techniques and Applications Let {x_1, x_2, ..., x_n} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the $n$ integers a_k, if they exist, such that a_1 x_1 + a_2 x_2 + ... + a_n x_n= 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This talk will present a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Performance results on large-scale parallel computer systems will be given. Several applications of these programs will be described, including some new results in number theory, quantum field theory and chaos theory.
10:00 - 10:30	François Bergeron, Université du Québec à Montréal Guessing Liouvillian Solutions to Linear Differential Equations We will show how to guess Liouvillian solutions to linear differential equations. We can then prove that they are indeed solutions by constructing a new linear differential equation which is shown to have the same solution as the original equation.
10:30 - 11:00	Robert Corless, University of Western Ontario What Symbolic Analysis means at Western Computer Algebra, as a tool for analysis, has been useful for many years but less so than it should be. This is a direct consequence of the (necessary) focus on algebraic ideas and techniques, many of which are mathematically very well developed. On the analytical side, many extremely elementary questions (such as how to automatically simplify complex-valued functions correctly and efficiently) are nowhere near as well developed (aside, of course, from some algebraically-derived non-computability results). This has hampered efforts to implement a class of useful tools. This talk will discuss some basic problems and approaches considered at Western.
11:00 - 11:30	Coffee Break
11:30 - 11:50	Jonathan Borwein, Simon Fraser University Interactive Network Mathematics I will describe the basic goals of our research and how it fits with the new Advanced Publishing Research Lab and with the JaMaLabs (Java Math Labs) we are designing within the Telelearning Research Network.
11:50 - 12:10	Matt Klassen, Simon Fraser University Number Recognition in Maple Number Recognition has been implemented in such projects as the Inverse Symbolic Calculator primarily as a table look-up tool. We will discuss the algorithmic approach to Number Recognition in Maple beginning with simple classes of numbers and functions.
12:10 - 12:30	Edgardo S. Cheb-Terrab, Simon Fraser University Abel ODEs, Their Equivalence, Classification and New Integrable Classes A classification, based on invariant theory, of integrable Abel ODEs is presented. A Computer Algebra implementation for solving the related equivalence problem is also shown. The work included collecting these integrable classes having {\it non-constant} invariant (corresponding canonical representatives and solutions), as well as the insertion of this data and the programs for the related equivalence problem into the ODEtools (Maple) package. Classifying the integrable Abel families presented in the literature (19$^{th}$ century works by Abel, Liouville, Appell and Halphen), four classes depending on one parameter plus other four without parameters were collected. Different approaches yielded three new integrable classes without parameters plus one depending on four parameters. The idea used to obtain this new 4-parameter class also leads to a wider 6-parameter non-constant invariant Abel class all of whose members can be systematically mapped into Riccati ODEs, as well as a recipe to generate new integrable Abel classes from other known classes when the latter satisfy certain conditions. Finally, an alternative Lie symmetry approach for these non-constant invariant classes was developed, leading to the same aforementioned new 4-parameter class as well as to a different way of resolving its related equivalence problem.
12:30 - 12:50	Petr Lisonek, Simon Fraser University Simplification of Elementary Constants and Elementary Functions in a Computer Algebra System Elementary constants/functions are obtained from the field of rational numbers/functions by a finite number of logarithmic, exponential or algebraic extensions. We discuss existing and newly proposed algorithms for simplification of elementary constants/functions, as well as their efficient implementations in the environment of a computer algebra system. Particular attention is paid to the zero recognition problem.
12:50 - 2:00	Lunch Break
2:00 - 2:30	David Boyd, University of British Columbia Computing A-polynomials The A-polynomial A(x,y) of a knot is a recently defined invariant. In contrast with most other polynomial invariants, the computation of A(x,y) for most knots presents considerable practical difficulties. We will describe some classes of knots for which the computation is now feasible and present some of the techniques that have been used to compute many previous intractible A-polynomials.
2:30 - 2:50	Coffee Break
2:50 - 3:20	Peter Borwein, Simon Fraser University Overview of the MITACS Project
3:20	Open Stage