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CECM Home > Events > CECM Annual Summer Meeting 2005 > Talk Abstracts

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Talk Abstracts

Rational functions, Hilbert series, and forbidden subwords

Jason Bell · Simon Fraser University

We give an overview of the theory of commutative and noncommutative Grobner bases, giving some unexpected applications along the way. We also discuss the connections between Grobner bases, counting words on an alphabet with certain forbidden subwords, and the computation of the Hilbert series of an algebra. We end with some applications of these ideas.

Taming Apparent Singularities via Ore Closure

Marni Mishna · Simon Fraser University

It is a classical result that the coefficients of power series solutions of a linear differential equation obey a linear recurrence. This recurrence can be computed from simple ring morphism from linear differential operators to linear difference operators. However, the recurrence might not be minimal if the initial differential equations possesses apparent singularities, that is, if there are points where the differential equation is singular, but its solutions are not. The problem is to find a suitable multiple of the original equation, which may be an equation of higher order, which is free of apparent singularities.

In our approach to this problem, and in order to simultaneously consider the inverse problem, q-analogues and other related operators, we consider an equivalent problem in the framework of skew polynomial rings.

This talk will describe algorithms to remove apparent singularites and compute polynomial torsion modules in algebras of skew polynomials. In the differential case, we revisit algorithms by Tsai with a goal of greater efficiency. In the case of recurrences, $q$-recurrences, and Mahler equations, our algorithmic results seem new, and require a more involved machinery.

Joint work with F. Chyzak, Ph. Dumas, H. Le, J. Martins, and B. Salvy.

The Monic Integer Chebyshev Problem

Kevin Hare · Department of Pure Mathematics, University of Waterloo

We study the problem of finding monic integer polynomials $P$ with small supremum norm $||P||_I$ on an interval $I$. The monic integer transfinite diameter $t_{\mathrm{M}}(I)$ is defined as \[ t_{\mathrm{M}}(I) := {\inf_P ||P||_I^{1/\deg(P)}, \] where the infimum is taken over all non-constant monic polynomials with integer coefficients. We show that if $I$ has length $1$ then $t_{\mathrm{M}}(I) = 1/2$.

We also consider the problem of determining $t_{\mathrm{M}}(I)$ when $I$ is a Farey interval. We give some partial results, which support a conjecture of Borwein, Pinner and Pritsker concerning this value.

Maple 10: New GUI Features and Math Capabililites

Allan Wittkopf · Simon Fraser University

We will demonstrate the new interface and math features of Maple 10.

Generating Numerical Algorithms Using Computer Algebra

Walter Gander · Institute for Computational Science, ETH Zurich

We show how numerical algorithms can be derived in a simple way using computer algebra. Examples are numerical differentiation, quadrature and multi-step methods for ODE. It is also shown how the discretization error of a method can be computed automatically. This approach not only makes formularies obsolete (in fact some errors were found in Abramowitz/Stegun) but is also useful in teaching since principles and fundamentals are emphasized and we can leave the sometimes tedious derivation of the specific algorithm to the machine.

Theory & application of space-filling curves

Richard Crandall · Apple Distinguished Scientist, Advanced Computation Group

Space-filling curves (or "space-fills") provide an elegant way to transform an N-dimensional problem into an M-dimensional one, for M <> N. Examples of this expedient include: "sorting" in N-dimensions, fractal measures derived from spacefill trajectories, data compression, many-body physics, and so on. The key to such applications is the transition from the theoretical (analytic) picture to the discrete (digital) picture. Indeed, discrete spacefills are demonstrably powerful software tools that apply across a wide spectrum, from graphics to information theory to hard science.