Maple Transactions−The Early Years

A vision statement for Maple Transactions giving our goals and plans for the first few years.

An Interview with the Authors: Log-Lightning Computation, and Some Instructive Mathematical Errors

An interview with the authors of "Log-lightning computation of capacity and Green's function", Maple Trans. 1, 1, Article 14124 (July 2021), and the author of "Some Instructive Mathematical Errors" Maple Trans. 1, 1, Article 14069 (July 2021). This interview was conducted by Annie Cuyt, with authors Peter Baddoo & Nick Trefethen and with Richard Brent, on Wednesday Sep 22, 2021 7am – 8am (EDT) via Zoom.

Problems, Puzzles, and Challenges

We give here some problems and puzzles that need a combination of thought and computation to solve. Please submit your solutions to the journal at mapletransactions.org. Include the problem number with your solution.

The Theodorus Variation

The Spiral of Theodorus, also known as the "root snail" from its connection with square roots, can be constructed by hand from triangles made with from paper with scissors, ruler, and protractor. See the Video Abstract. Once the triangles are made, two different but similar spirals can be made. This paper proves some things about the second spiral; in particular that the open curve generated by the inner vertices monotonically approaches a circle, and that the vertices are ultimately equidistributed around that inner circle.

Experimenting with Apéry Limits and WZ pairs

This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.

Viète's formula in "A Fractal Eigenvector"

This paper explores a relationship of the asymptotic behavior ofthe leading element of eigenvectors belonging to the dominant eigenvalueof a recursively-constructed family of Mandelbrot matrices toViète's formula, helping to explain the appearance of π in thiselement.

Jonathan Michael Borwein 1951 − 2016: Life and Legacy

Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.

Skew-symmetric tridiagonal Bohemian matrices

Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous "numbers" dress Abstract: Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent. This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment. I also give a terrible pun. Don't say you weren't warned.

Welcome to Maple Transactions

Maple was conceived over forty years ago as a general purpose system for mathematical calculations. Its strength, however, has always been its community. The work of hundreds of researchers from around the world has produced a mathematical engine unique in its depth, breath and efficiency. Forward thinking educators have used Maple to transform the way mathematics is taught, all the way supporting each other with advice, examples and myriads of Maple worksheets. Scientists and engineers have been taking advantage of the power and ease of use of the Maple system to help them in their discovery and the development of new products. Together we have tackled environmental issues, taken on disease and reached for the stars. At Maplesoft, we are firm believers that Math Matters and our mission is to provide technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. This mission, we share with hundreds of thousands of Maple users from all over the world and indeed we rely on that community’s constant stream of feedback and support. With Maple Transactions, our community is gaining a new place to come together. A place to exchange ideas, share experiences and discoveries. A place to welcome newcomers and discuss possibilities. The drive, vision and energy of editor in chief Prof Rob Corless together with the fantastic editorial board that he assembled, have given me a glimpse into a bright future for the journal and this first issue bears witness to the high quality of contributions we can expect.

How to use Fibonacci numbers to teach recursive programming

There are too many examples and programming guides (which, e.g., an internet search for "recursive procedure Fibonacci" will turn up) to count that use Fibonacci numbers as an example to illustrate recursive programming. The motivation for this article is to show why the naive way of doing this is a bad idea, as it is horrendously inefficient. We will exhibit much more efficient ways of computing Fibonacci numbers, both iterative and recursive, and analyze and compare worst case running times and memory usages. Using some mathematical properties of Fibonacci numbers leads to the most efficient method for their computation. For illustration and benchmarking, we will use Maple and its programming language, however, similar behaviour can be demonstrated in almost any other programming language. This exposition combines and explores the mathematical properties of Fibonacci numbers, notions of algorithmic complexity, and efficient Maple programming and profiling techniques, and may be used as an introduction to any of these three subjects. The techniques described can be readily generalized to more general types of linear recurrences with constant coefficients.