What’s New in the World of Maplesoft

Math matters. Maplesoft’s mission is to provide powerful technology to help students, researchers, engineers, and scientists take advantage of the power of math so they in turn can enrich the world we live in. Since technology evolves, research advances, and needs change, Maplesoft is continuously looking for new ways to improve, experiment, and innovate, in order to fulfill that mission. In this talk, Dr. Laurent Bernardin, CEO and President of Maplesoft, will give you a tour of some new and coming things at Maplesoft that he is personally excited about, and divulge some of his thoughts on the future of math technology.

The Mathieu Functions: Computational and Historical Perspectives

The Mathieu functions, which are also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu in order to help understand the vibrations of an elastic membrane set within a fixed elliptical hoop. These functions still occur frequently in applications today. Our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This talk surveys the historical development of both the theory of Mathieu functions and the methods used to compute them, with a particular focus on some of the interesting people who did the major work: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch. Time permitting, we will discuss some gaps in current software capability involving double eigenvalues of the Mathieu equation, and some possible ways to fill those gaps using methods developed by Blanch.

An Integral View on Dimensional Analysis: Scaling Invariants for Parameter Reductions in Dynamical Systems

Dimensional analysis, also known as parameter reduction, is a recommended practice before analyzing a dynamical system, such as a physical system or biological model. The Buckingham Pi Theorem shows how linear algebra can be used to bring out dimensionless variables, as power products of the original variables, which simplifies the analysis. One issue that arises, however, is that the powers provided by the Pi Theorem can be fractional, resulting in roots, and thus they require some care when determining the regions of positivity of the variables.

In this talk, I will present an algorithm involving scaling invariants that performs a similar transformation into dimensionless variables, but the results only involve integer powers and so are much easier to work with. I will also provide a simple rewriting algorithm, in the form of substitutions, that can be used to find the induced equations in the dimensionless variables.

This talk is based on: E. Hubert & G. Labahn. Scaling Invariants and Symmetry Reduction of Dynamical Systems. Foundations Computational Mathematics. 13:4 pages 479-516 (2013)

Two-Eyed Seeing: Mathematics and Indigenous Traditions and Cultures

Elder Albert Marshal of the Mi’kmaw Nation describes “two-eyed seeing” as the ability to see with the strength of Indigenous knowledge from one eye while seeing with the strength of Western knowledge from the other. This dual perspective can be applied to many aspects of life, including mathematics.

In this presentation, I will explore the concept of “two-eyed seeing” and the field of ethnomathematics, the study of the relationship between mathematics and culture first introduced by Brazilian educator and mathematician Ubiratan D'Ambrosio. I will address some of the dynamics between these two concepts and illustrate them with several examples. These examples will include a brief analysis of the geometry evident in a traditional Haida Nation hat, as well as the work of contemporary Salish artist Dylan Thomas.

In addition, I will discuss a project that used mathematical modeling of a traditional Tla’amin Nation stone fish trap to communicate cultural, engineering, environmental, and mathematical ideas. This project was a collaboration with the Tla’amin Nation and the Callysto Program, an online education tool that helps students in elementary and high school learn about and apply data science skills.