Maple in Mathematics Education and Research

“Correlation is not causation”, or so the old trope goes.This statement is true if we are talking about correlations revealed by familiar methods that are used to analyze data. In this talk, I want to put a new spin on this statement. I will argue that a new technology, quantum generated density maps, changes the way we look at complex data. The main point is that visualizing all the correlations hidden in a set of data, their shapes and their relationship to one another, can force us to ask questions that strongly suggest causal relationships. To make this point I will discuss the analysis of melanoma data provided to Quantum Insights Inc. by Genentech.

The present work is motivated by the problem of mathematical handwriting recognition where symbols are represented as parametric plane curves in a Legendre-Sobolev basis. An early work showed that approximating the coordinate functions as truncated series in a Legendre-Sobolev basis yields fast and effective recognition rates. Furthermore, this representation allows one to study the geometrical features of handwritten characters as a whole. These geometrical features are equivalent to baselines, bounding boxes, loops, and cusps appearing in handwritten characters. The study of these features becomes a crucial task when dealing with two-dimensional math formulas and the large set of math characters with different variations in style and size.In an early paper, we proposed methods for computing the derivatives, roots, and gcds of polynomials in Legendre-Sobolev bases to find such features without needing to convert the approximations to the monomial basis. Furthermore, in this paper, we propose a new formulation for the conversion matrix for constructing Legendre-Sobolev representation of the coordinate functions from their moment integrals.Our findings in employing parametrized Legendre-Sobolev approximations for representing handwritten characters and studying the geometrical features of such representation has led us to develop two Maple packages called LegendreSobolev and HandwritingRecognitionTesting. The methods in these packages rely on Maple’s linear algebra routines.

In this paper, we show how stabilizing controllers for 2D systems can effectively be computed based on computer algebra methods dedicated to polynomial systems, module theory and homological algebra. The complete chain of algorithms for the computation of stabilizing controllers, implemented in Maple, is illustrated with an explicit example.

We present the SIROPA Maple Library which has been designed to study serial and parallel manipulators at the conception level. We show how modern algorithms in Computer Algebra can be used to study the workspace, the joint space but also the existence of some physical capabilities w.r.t. to some design parameters left as degree of freedom for the designer of the robot.

We describe a Maple package that serves at least four purposes. First, one can use it to compute whether or not a given polyhedral structure is Zometool constructible. Second, one can use it to manipulate Zometool objects, for example to determine how to best build a given structure. Third, the package allows for an easy computation of the polytopes obtained by the kaleiodoscopic construction called the Wythoff construction. This feature provides a source of multiple examples. Fourth, the package allows the projection on Coxeter planes.

We adapt Victor Y. Pan’s root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of Guðbjörn Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process.

The Maple intersectplot command plots the intersection curve in three-dimensional space between a pair of two-dimensional surfaces. We will present the implementation in Maple of a new algorithm computing the intersection curve between two quadrics in 3D that improves the results produced by the intersectplot command.

Quadric surfaces play a very important role in solid geometric modeling and in the design and fabrication of mechanical and industrial parts. Solving the intersection curve between two quadrics is a fundamental problem in computer graphics and solid modeling. We present a new analytical method for parameterizing the intersection curve of two quadrics, which are represented by implicit quadratic equations in 3D. The method is based on the observation that the intersection curve of two quadrics comprises all the points that satisfy a parametric second order polynomial system. We show that the computation of the intersection problem of two general quadrics can be reduced to the solution of quartic polynomials. In particular, we show that the intersection problem of two quadric surfaces that are expressed in canonical forms can be reduced to the solution of quadratic polynomials. All the exact parametric solutions for the intersections of quadric surfaces are implemented in the Computer Algebra System MAPLE. Several previously published test problems of the intersection of quadric surfaces are presented and discussed.

Many systems in biology (as well as other physical and engineering systems) can be described by systems of ordinary differential equation containing large numbers of parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to identify and characterize the steady-state solutions as the model parameters vary, a technically challenging problem in a high-dimensional parameter landscape. Rather than simply determining the number and stability of steady-states at distinct points in parameter space, we decompose the parameter space into finitely many regions, the number and structure of the steady-state solutions being consistent within each distinct region. From a computational algebraic viewpoint, the boundary of these regions is contained in the discriminant locus. We develop global and local numerical algorithms for constructing the discriminant locus and classifying the parameter landscape. We showcase our numerical approaches by applying them to molecular and cell-network models.

The $$\mathbb Z$$-Polyhedra is a library written in Maple and dedicated to solving problems dealing with the integer points of polyhedral sets. Those problems include decomposing the integer points of polyhedral sets, solving parametric integer programs, performing dependence analysis in for-loop nests and determining the validity of certain Presburger formulas. This article discusses the design of the $$\mathbb Z$$-Polyhedra library and provides numerous illustrations of its usage.

Local bifurcation analysis of singular smooth maps plays a fundamental role in understanding the dynamics of real world problems. This analysis is accomplished in two steps: first performing the Lyapunov-Schmidt reduction to reduce the dimension of the state variables in the original smooth map and then applying singularity theory techniques to the resulting reduced smooth map. In this paper, we address an important application of the so-called Extended Hensel Construction (EHC) for computing the aforementioned reduced smooth map, which, consequently, leads to detecting the type of singularities of the original smooth map. Our approach is illustrated via two examples displaying pitchfork and winged cusp bifurcations.

The study of Drazin inverses is an active research area that is developed, among others, in three directions: theory, applications and computation. This paper is framed in the computational part.Many authors have addressed the problem of computing Drazin inverses of matrices whose entries belong to different domains: complex numbers, polynomial entries, rational functions, formal Laurent series, meromorphic functions. Furthermore, symbolic techniques have proven to be a suitable tools for this goal.In general terms, the main contribution of this paper is the implementation, in a package, of the algorithmic ideas presented in [10, 11]. Therefore, the package computes Drazin inverses of matrices whose entries are elements of a finite transcendental field extension of a computable field. The computation strategy consists in reducing the problem to the computation of Drazin inverses, via Gröbner bases, of matrices with rational functions entries.More precisely, this paper presents a Maple computer algebra package, named DrazinInverse, that computes Drazin inverses of matrices whose entries are elements of a finite transcendental field extension of a computable field. In particular, the implemented algorithm can be applied to matrices over the field of meromorphic functions, in several complex variables, on a connected domain.

The problem of Quantifier Elimination (QE) in Computer Algebra is that of eliminating all quantifiers from a statement featuring polynomial constraints. This problem is known to be worst case time complexity worst case doubly exponential in the number of variables. As such implementations are sometimes seen as undesirable to use, despite problems arising in algebraic geometry and even economics lending themselves to formulations as QE problems. This paper largely concerns discussion of current progress of a package QuantifierElimination written using Maple that uses a poly-algorithm between two well known algorithms to solve QE: Virtual Term Substitution (VTS), and Cylindrical Algebraic Decomposition (CAD). While mitigation of efficiency concerns is the main aim of the implementation, said implementation being built in Maple reconciles with an aim of providing rich output to users to make use of algorithms to solve QE valuable. We explore the challenges and scope such an implementation gives in terms of the desires of the Satisfiability Modulo Theory (SMT) community, and other frequent uses of QE, noting Maple’s status as a Mathematical toolbox.

Problem solving and computational thinking are the key competences that all individuals need for professional fulfillment, personal development, active citizenship, social inclusion and employment. In mathematics, during contextualized problem solving using Maple, the differences between these two skills become thinner. A very important feature of Maple for problem solving is the programming of animated graphs: an animation obtained by generalizing a static graph, choosing the parameter to be varied and its interval of variation. The first objective of this research is to analyze the computational thinking processes behind the creation of animated graphs for the resolution of a contextualized problem. To this end, we selected and analyzed some resolutions of problems carried out by fourth-grade students of upper secondary schools in Italy (grade 12). The paper shows some examples in which different processes of computational thinking have emerged, which reflect resolutive strategies and different generalization processes. From the analysis it emerged that all the processes underlying the mental strategies of the computational thought useful for solving problems are activated in the creation of animated graphs. In the second part of the article we discuss examples of animations created during training activities with secondary school teachers, and how animations can support the learning of scientific concepts. It is very important to train the teachers in this regard, both to understand the processes that the students would activate during the creation of animated graphics and to enrich the theoretical or practical explanations with animated representations.

In this article we demonstrate how to solve a variety of problems and puzzles using the built-in SAT solver of the computer algebra system Maple. Once the problems have been encoded into Boolean logic, solutions can be found (or shown to not exist) automatically, without the need to implement any search algorithm. In particular, we describe how to solve the n-queens problem, how to generate and solve Sudoku puzzles, how to solve logic puzzles like the Einstein riddle, how to solve the 15-puzzle, how to solve the maximum clique problem, and finding Graeco-Latin squares.

This paper describes an application of Maple in the teaching of linear algebra. The topic is the construction of an orthogonal basis for a set of vectors or a matrix using Householder transformations. We present a method for generating matrices which, when subject to using Householder transformations, require only rational computations and give rational results. The pedagogical problem addressed is that numerical examples in this topic will usually contain unsimplified square roots, which add an extra layer of difficulty for students working examples.

A senior undergraduate course on cryptography for computer science majors that combines the use of conventional materials with Maple worksheets and Möbius modules is in development. The design intent and impact of the Maple-based materials on course conduct is discussed. Pedagogical and practical considerations are discussed along and initial impressions given regarding benefits and difficulties in using the tools.

What role does faculty confidence and skills play in the use of a Computer Algebra System (CAS) play in increasing student success in college calculus courses? Studies show that improving students’ spatial abilities in math is a key indicator to improving their success in calculus Sorby et al. (2013). This proposal discusses challenges faced while attempting to implement the use of Maple within the classroom in the Calculus sequences, Linear Algebra and Differential Equations at Johnston Community College as well as across North Carolina Community Colleges.

Dirac-notation based upper division undergraduate quantum mechanics was taught in the Spring semester of 2019 using Maple to present the mathematics of the course and to solve all mathematical and computational problems. In addition to step-by-step presentations on using Maple, students were provided with numerous examples of solving quantum mechanical problems using Maple. Students were required to submit all homework and “take-home” exam solutions as PDF documents, primarily generated from a Maple worksheet. However, students were not required to solve all problems using Maple. Through external evaluation and student survey, it was determined that by the end of the semester, all students used Maple for solving over half the problems; nearly three-quarters of the students developed sophisticated Maple skill sets; and a third of the students used Maple to solve every type of problem – completing assignments in a single worksheet. Maple was most frequently used to solve problems involving single variable continuous functions, vectors and matrices. Maple was least frequently used to solve problems involving Dirac-notation based algebra. Maple was nearly universally appreciated by the students.

The Fermat-Torricelli problem of triangles on the sphere under Euclidean metric asks to find the optimal point P on the sphere $$S^2$$ for three given points A, B, C on $$S^2$$, so that the sum of the Euclidean distances $$L=PA+PB+PC$$ from that point P to the three vertices is minimal (or maximal). In this paper we introduce a solution to this problem done with help of the symbolic computation software Maple and interpolation of implicit function, where the minimal and the maximal sum of the distances are expressed by same polynomial f(L, a, b, c) of degree 12 with $$a=BC, b=CA, c=AB$$.

Leslie matrices may be used to model the age distribution of a population as well as population growth. The dominant eigenvalue tells us the long term population growth and the corresponding eigenvector tells us the long term age distribution. Because the model is so simple, and it does not require any knowledge of physics or chemistry or biology, it’s ideal for presenting in a first course on Linear Algebra as the main application of eigenvalues and eigenvectors.In this paper we present the Leslie age distribution model and provide accompanying exercises suitable for students. We use Maple for both numerical calculations and symbolic calculations. We include some data for real populations that instructors may use for classroom presentation or for assignments.

This article describes an intelligent, model-tracing system for tutoring expansion and factoring of algebraic expressions. The system is implemented as a set of procedures in a Maple document that tutor a breadth of 18 top-level mathematical skills (algebraic operations). Twelve (12) skills for expansion (monomial multiplication, monomial division and power of monomial, monomial-polynomial and polynomial-polynomial multiplication, parentheses elimination, collection of like terms, identities square of sum and difference, product of sum by difference, cube of sum and difference) and six (6) skills for factoring (common factor, identities square of sum and difference, product of sum by difference, quadratic form by sum and product, quadratic form by roots). These skills are further decomposed in simpler ones giving a deep domain expertise model of 68 primitive skills. The tutor has two novel features: (a) it exhibits intelligent task recognition by identifying all skills present in any expression through intelligent parsing, and (b) for each identified skill, the tutor traces all the sub-skills, a feature called deep model tracing. Furthermore, based on these features, the tutor achieves broad knowledge monitoring by recording student performance for all skills present in any expression.

In this paper, we present a proof of the property that for any convex nonagon $$P_1P_2\ldots P_9$$ in the plane, the smallest area of a triangle $$P_{i}P_{j}P_{k} (1\le i< j < k \le 9)$$ is at most a fraction of $$4\cdot \sin ^2(\pi /9)/9= 0.05199\ldots $$ of the area of the nonagon. The problems is transformed into an optimization problem with bilinear constraints and solved by symbolic computation with Maple.

Pseudo-linear systems constitute a large class of linear functional systems including the usual differential, difference and q-difference systems. The Maple package PseudoLinearSystems is dedicated to the study of this class of linear systems. It contains a generic procedure for computing a so-called simple form of a pseudo-linear system as well as local data useful for the local analysis: k-simple forms, super-irreducible forms, integer slopes of the Newton polygon, indicial equations, etc. It is also devoted to the computation of rational solutions (using simple forms) of a single linear differential, difference or q-difference system, as well as rational solutions of a system of mixed linear partial differential, difference and q-difference equations. In this software presentation, we demonstrate the use of several procedures of the package that are all based on the simple form procedure.

Many algorithms in computer algebra systems can have their performance improved through the careful selection of options that do not affect the correctness of the end result. Machine Learning (ML) is suited for making such choices: the challenge is to select an appropriate ML model, training dataset, and scheme to identify features of the input. In this extended abstract we survey our recent work to use ML to select the variable ordering for Cylindrical Algebraic Decomposition (CAD) in Maple: experimentation with a variety of models, and a new flexible framework for generating ML features from polynomial systems. We report that ML allows for significantly faster CAD than with the default Maple ordering, and discuss some initial results on adaptability.

This presentation gives an overview of ball arithmetic as a tool for computing with real numbers in the context of computer algebra, and discusses recent development to the Arb library.

Lie symmetry groups of transformations (mappings) of differential equations leave them invariant, and are most conveniently studied through their Lie algebra of vector fields (essentially the linearization of the mappings around the identity transformation). Maple makes powerful and frequent use of such Lie algebras, mostly through routines that are dependent on Maple’s powerful exact integration routines, that essentially automate traditional hand-calculation strategies. However these routines are usually heuristic, and algorithmic approaches require a deeper integration of differential elimination (differential algebraic) approaches in applications to differential equations. This is the underlying motivation of the LieAlgebrasOfVectorFields (LAVF) package of Huang and Lisle. The LAVF package introduces a powerful algorithmic calculus for doing calculations with differential equations without the heuristics of integration to calculate efficiently many properties of such systems.We use LAVF in the development of our MapDE package, which determines the existence of analytic invertible mappings of an input DE to target DE. Theory, algorithms, and examples of MapDE can be found in [5, 6]. Here we present a brief summary, through examples, of the application of LAVF to MapDE.

Maple 2019 has a new multivariate polynomial factorization algorithm for factoring polynomials in $$\mathbb {Z}[x_1,x_2,...,x_n]$$, that is, polynomials in n variables with integer coefficients. The new algorithm, which we call MTSHL, was developed by the authors at Simon Fraser University. The algorithm and its sub-algorithms have been published in a sequence of papers [3–5]. It was integrated into the Maple library in early 2018 by Baris Tuncer under a MITACS internship with Maplesoft. MTSHL is now the default factoring algorithm in Maple 2019.

We apply computer algebra, especially linear algebra over polynomial rings and Gröbner bases, to classify inhomogeneous distributive laws between the operads for Lie algebras and commutative associative algebras.

Combinatorial power series are formal power series of the form $$\sum c_{n,H}X^n/H $$ where, for each n, H runs through subgroups of the symmetric group $$S_n$$ and the coefficients $$c_{n,H}$$ are complex numbers (or ordinary power series involving some “weight variables”). Such series conveniently encode species of combinatorial (possibly weighted) structures according to their stabilizers (up to conjugacy). We give general lines for expressing these kinds of series – as well as the main operations $$(+,\cdot ,\times ,\circ ,d/dX)$$ between them – by making use of the GroupTheory package and give suggestions for possible extensions of that package and some other specific procedures such as collect, expand, series, etc. An analysis of multivariable combinatorial power series is also presented.

In this talk I will provide a quick tour through some of the different ways in which I have used Maple to improve student understanding of traditional topics in an introductory differential equations course, such as direction fields, the phenomenon of beats, and developing an understanding of solutions to first-order systems in phase space.