Maple
Powerful math software that is easy to use
• Maple for Academic • Maple for Students • Maple Learn • Maple Calculator App • Maple for Industry and Government • Maple Flow • Maple for Individuals
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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.
Dr. Laurent Bernardin is President and CEO of Maplesoft. He has been with Maplesoft for over 20 years and prior to his appointment to his current role, he held the positions of CTO and COO. Bernardin is a firm believer that mathematics matters. Under his leadership, Maple has grown from a research project in symbolic computing to a complete environment for mathematical calculations used by hundreds of thousands of engineers, scientists, researchers and students around the world.
Recording not available.
Whether you have been using Maple 2022 since the day it came out, or haven’t had a chance to try it yet, chances are good there are still new features in Maple 2022 that you haven’t explored yet. This talk will give you a closer look at some of the improvements that the presenter, the Senior Director of Research at Maplesoft and long-time Maple user, finds particularly useful or interesting. You may even get a few hints of more good things to come.
Recording not available.
Math anxiety is a complex problem, and no software is going to be able to wave its digital wand and make it go away. But the right technology can help reduce math anxiety, and dare we say it, even help indifferent students become interested in math. Maple Learn provides a flexible interactive environment for solving problems, a great platform for conceptual learning, and incredibly simple content development and deployment solutions. In this presentation, you’ll discover how Maple Learn can support your efforts to engage with your students, build their confidence, and maybe even get them excited about math.
Maple Flow is a math tool that reproduces the design metaphor of paper. You can place your calculations and text anywhere on a virtual whiteboard and move your work into position. Maple Flow updates your calculations automatically and rewards you with an environment that makes it easier to progressively refine and iterate your work. This talk introduces Maple Flow, and showcases many examples from different engineering domains. You’ll also get a glimpse of what we’re working on for the next release.
Rashid Barket (Coventry University)
Framework for Generating Integrable Expressions
Applications of machine learning are becoming more prominent in the field of computer algebra. Examples of such applications include selecting S-pairs in Buchberger’s algorithm or solving integrals and differential equations directly. With many of these applications, data must be generated to train a model. Methods such as generating binary trees representing mathematical expressions or created randomly in a recursive manner from a set of available function symbols, variables and constants have been discussed. However, these generated expressions do not represent a realistic dataset that draws from the typical Maple user’s experience.
I propose a framework for generating valid mathematical expressions. More precisely, the focus will be on integrable expressions. The difference from other methods lies in the fact that the data generation method will be based on a test suite of data generated from Maple users. Thus, the new synthetic data will have properties similar to integrable expressions that Maple users would typically try. This data generation method will be used to train machine learning models that make efficient choices algorithm selection problems.
Tereso del Rio (Coventry University)
Techniques to Find Relevant Features for Heuristics
In symbolic computation algorithms, it is very common to have to make a choice between multiple options. For example, a choice of two polynomials is needed in every step of Buchberger's algorithm or a variable ordering is needed in order to build a CAD. These choices have been traditionally made using human-made heuristics in which the features employed are selected relying on the expertise of the researcher designing the heuristics.
In this talk, it will be explored how these features can be obtained in a more systematic way. For that matter, it will be shown how through the examination of the complexity analysis of CAD, a relevant feature was identified and used to create a new heuristic to choose the variable ordering that outperformed existing ones. Similarly, it will also be shown how an explainability analysis of trained machine learning models can also identify relevant features to create heuristics.
AmirHosein Sadeghimanesh (Coventry University)
An SMT Solver for Non-linear Real Arithmetic Inside Maple
We report on work-in-progress to create an SMT-solver inside Maple for non-linear real arithmetic (NRA). We give background information on the algorithm being implemented: cylindrical algebraic covering as a theory solver in the lazy SMT paradigm. We then present some new work on the identification of minimal conflicting cores from the coverings.
Micaela Vancea, Victoria Quance and David Jeffrey (Western University)
Maple and the Cubic Formula
There are issues that inevitably arise when a standard mathematical solution is implemented in a computer algebra system. As a case study, we use the solution of cubic equations in Maple. Importance of branch selection in such a system will be discussed. To ensure the solutions are functional and ideal for each user, we consider the benefits and drawbacks of different commands in Maple when used in the cubic formula. Additionally, to correct some of the problems with the traditional cubic formula, we introduce a theorem for the purpose of implementation of solutions of the cubic into Maple.
Eugenio Roanes-Lozano (Universidad Complutense de Madrid)
A Bibliographic Study on the Computer Algebra System Maple
The computer algebra system DERIVE was discontinued, but more than 10 years after this decision it is still mentioned in scientific papers. Surprised by this fact, the authors carried out a bibliographic study on the evolution of the number of publications that mention DERIVE over time, which was published in 2019. In 2022 the authors conducted a similar study on the growing number of papers citing the dynamic geometry system GeoGebra, which was presented at the 8th European Seminar on Computing (ESCO 2022).
The authors have also wondered about the bibliographic impact of the computer algebra system Maple, compared to other computer algebra systems (such as Mathematica, Maxima, Macsyma, Sage/SageMath, REDUCE, Axiom, DERIVE, xCAS, Giac, GeoGebra and CoCoA). For this, they have carried out a new bibliographic study using the database Scopus, which considers computer algebra in general, as well as its educational applications. Searches are performed in the "Title-Abstract-Keywords" field, and "computer algebra" is included to refine the search as much as possible (for example, in this way references corresponding to the maple tree are excluded). As for educational applications they have added "education AND learning" to focus the search. The words REDUCE, Axiom and DERIVE introduce spurious results that pass the "computer algebra" filter, since they also have mathematical meanings (exact results would require manual screening of these references). However, Maple is the leader in both cases. Analyses by authors, countries and scientific areas are also included.
Atharva Jamdade (Indian Institute of Technology-Madras), Mohammad Hadi Jalali, Mackenzie Savoy, and Kush Bubbar (University of New Brunswick)
High-Fidelity Modelling of a 4-Wheel Independent Drive and Steer Off-road Scaled Electric Vehicle
Electric vehicles are gradually gaining importance in the automotive industry due to their environmental benefits. The control, performance and safety of such vehicles in performing on-road and off-road manoeuvres play a crucial role during the design and development stages. In this paper, a high-fidelity vehicle dynamics model is developed for a 4-wheel independent drive/steer (4WID/4WIS) scaled electric vehicle using MapleSim™. The MapleSim™ CAD Toolbox is coupled with the MapleSim™ multibody dynamic toolbox and is utilized to import the CAD model of each vehicle subsystem into MapleSim™. The resulting system multibody dynamic model consists of all the mechanical subsystems of the vehicle, including double wishbone suspensions, independent steering mechanisms, independently driven powertrains, and the chassis. In order to identify parameters of some key subsystems of the vehicle model, experimental parameter identification is performed. The numerical computational features of MAPLE™ and MATLAB™ are utilized to determine the vehicular subsystem parameters such as vehicle frontal area, wheel inertia, vehicle CoG location and vehicle system inertia. The developed vehicle dynamics plant model is then imported into MATLAB™/SIMULINK™ using the Functional Mock-up Interface (FMI) and integrated with a control model developed in SIMULINK™ for analysis, leading to a co-simulation approach utilizing key features of each software package. The vehicle dynamics model and the electrical powertrain components are assembled in SIMULINK™ to create the complete electric vehicle system. Different drive cycles are used to verify that the developed electric vehicle system is able to accurately simulate different driving scenarios. It is observed that the MapleSim™ CAD Toolbox is an efficient tool to quickly assemble a multibody dynamics model of a complex system such as a vehicle, and it expedites the modelling process by reducing modelling errors. Future work includes parameter identification of the remaining vehicle subsystems and performing on-road tests to validate the electric vehicle model in different driving scenarios/manoeuvres. The high-fidelity electric vehicle model will be used in future work to develop more robust control models.
Marcin Kaminski (Lodz University of Technology)
Probabilistic Entropy Computations in Maple
This presentation is a part of a research project sponsored by the National Science Center in Poland. It includes some fundamental results concerning the usage of the system Maple in structural analysis, and also in some physical problems with uncertainty. The additional Maple procedures have been developed and programmed to determine both probabilistic moments of the system response as well as its probabilistic entropies. These entropies are available in a Python environment, while the system Maple has "Entropy" implemented but not in the context of the statistical library. This presentation documents both Shannon entropy implementation as well as Bhattacharyya relative entropy calculus.
Athanasios Tzemos (Research Center for Astronomy and Applied Mathematics of the Academy of Athens)
Critical Points of the Bohmian Quantum Flow: A Study with Maple
The highly nonlinear character of the Bohmian equations of motion allows, in general, the coexistence of ordered and chaotic Bohmian trajectories for a given quantum system. As we have shown in the past, chaos is introduced whenever a quantum particle comes close to a ‘nodal point-X-point complex’, a characteristic geometrical structure of the Bohmian flow in the close neighborhood of a nodal point of the wave function. The X-point is an unstable fixed point, in the frame of reference of the moving nodal point, which scatters the incoming trajectories. The cumulative effect of many such scattering events is the emergence of chaos. On the other hand, trajectories that do not come close to the NPXPCs are ordered. The research group of the RCAAM of the Academy of Athens has a long tradition in the study of Bohmian quantum dynamical systems. In this talk we are going to present our main results on the dynamics of these two critical points of the Bohmian quantum flow and its implications for the evolution of Bohmian trajectories, along with some new first results for the analytical detection of the X-point in a simple wave function. Our results will be supported by plots and animations made solely with Maple.
Jason Osborne (Appalachian State University)
Tensor Visualization and Computations for Differential Geometry in Anholonomic Frames
A collection of modules will be outlined illustrating an approach to tensor computations in an anholonomic frame using Maple. This approach closely mimics how one might work with tensors on paper by leveraging atomic variables and MathML available within Maple. A variety of visualization tools (e.g. matrix, vector, basis views) and computations (e.g. directional and covariant differentiation) in an anholonomic frame will be demonstrated as well as computational verification with a comparison of both Riemann and Cartan computations. For example, think of how one would write and work with a tensor (say the 4-index Riemann curvature tensor) in a by-hand, on-paper calculation. There are symmetries to consider, specifically anti-symmetry in at least one pair of indices if not two pairs of indices. There are also multiple ways to view this tensor, the most helpful of which might be as a matrix of anti-symmetric matrices. In a holonomic (or coordinated) frame, the Riemann curvature tensor of a parametrized surface is an anti-symmetric matrix of anti-symmetric matrices comprised of a single function, namely the Gauss curvature which is simple to compute. In contrast, curvature and other computations in an anholonomic and non-orthonormal frame curvature are more difficult and can be greatly accelerated with computer assistance using these Maple modules
Ajay Menon (Indian Institute of Technology Kharagpur), Ali Shahbaz Haider and Kush Bubbar (University of New Brunswick)
Development of an Ocean Engineering Toolbox in MapleSim
With the emergence of the blue economy, coupled with the motivation to drive down CO2 emissions, the importance of harnessing Marine Renewable Energy (MRE) has never been greater. Ocean wave energy is a form of MRE that is in its technological infancy and thus requires an engineering simulation toolset to advance the field. The following work explores the development of an Ocean Engineering Toolbox (OET) using the Modelica™ language that integrates with MapleSim™. In developing the OET, the primary focus is on solving Cummins equation for a floating, single-degree, heaving body subject to polychromatic wave excitation forces. Numerically, this is challenging due to the emergence of convolution terms in the time domain formulation that must be solved to calculate the instantaneous hydrodynamic radiation force. This work focuses on exploring several methods to solve for the radiation force including i) symbolic, ii) numerical integration, and iii) state space approximations. Results are as follows: Symbolic methods, while the most precise, are not feasible for many formulations. Numerical integration, while precise, is computationally intensive and challenging to implement in Modelica™ using a continuous function formulation. Finally, this work determined that by reformulating the convolution term using a state-space approach, MapleSim™ can solve Cummins’ equation accurately and efficiently for a single body when corroborated against results provided by the current standard WECSim™ using SimScape™. Having demonstrated the ability to solve the Cummins equation using MapleSim™, our future goals include extending this library to incorporate the MapleSim™ multibody dynamic library and a DAE solver. This research pioneers symbolic formulations of ocean engineering and MRE problems in MapleSim™.
Ayoola Jinadu and Michael Monagan (Simon Fraser University)
Solving Parametric Polynomial Systems using Dixon resultants
Simon Fraser University, Canada. We present a new Dixon resultant algorithm and its implementation in Maple for solving parametric polynomial systems. Polynomial systems can be solved in Maple using either Gröbner bases or triangular sets or using Maple's solve command which uses an elimination method. But experiments have shown that these methods in Maple often fail on polynomial systems with many parameters; they can take a very long time to execute or run out of memory. The failure is due to the intermediate expression swell caused by the parameters.
The Dixon resultant R is a multiple of the unique generator of an elimination ideal of a polynomial system. It can be expressed as a determinant of a matrix M of polynomials called the Dixon matrix. So one way to compute R in Maple is to construct M and use Maple to compute det(M). However, the Dixon resultant R often has many repeated factors and a large polynomial content that is not needed so our goal is NOT to compute R in expanded form. Instead, using a black box representation for R=det(M), our new Dixon resultant algorithm probes the black box (evaluates M at integer points modulo primes) and interpolates the monic square-free factors Rj of R from monic univariate polynomial images in the main variable x1 using sparse multivariate rational function interpolation to interpolate the coefficients of the Rj's modulo primes and it uses Chinese remaindering and rational number reconstruction to recover the rational coefficients of Rj. The sparse rational function interpolation algorithm of Cuyt and Lee was modified for this purpose.
By not computing R in expanded form, our approach often dramatically reduces the number of polynomial terms to be interpolated and the number of primes needed. Consequently, our Dixon resultant algorithm is able to solve many parametric polynomial systems that Gröbner bases, triangular sets, and determinant algorithms (e.g. Gentleman and Johnson minor expansion) cannot solve. We have implemented our Dixon resultant algorithm in Maple with several parts coded in C for efficiency. We will present benchmarks that show that our algorithm significantly outperforms other approaches.
Garrett Paluck (Simon Fraser University)
New GCD Algorithm for Zp[x,y] with Cubic Cost
Suppose we are given two polynomials A and B in Z[x,y], that is, polynomials in two variables with integer coefficients. An important operation in Maple is to compute G = gcd(A,B), the greatest common divisor of A and B. Maple does this whenever it has to simplify the fraction A/B. To compute G, Maple computes G modulo primes p1, p2, ... and reconstructs the integer coefficients of A and B using the Chinese remainder theorem. For each prime p Maple uses evaluation and interpolation to compute gcd(A,B) mod p.
We present a new algorithm that uses bivariate Hensel lifting to compute gcd(A,B) mod p. Let dx = max(deg(A,x),deg(B,x)) and dy=max(deg(A,y),deg(B,y)). Our algorithm does O(dx2dy+dxdy2) arithmetic operations in Zp for some prime p >= dx. Our algorithm utilizes evaluation to compute a = A(x, α) and b = B(x, α) for some α in Zp, the modular gcd algorithm to compute g = GCD(a, b), and bivariate Hensel lifting to lift the univariate images g, a, b in Zp[x] and recover G in Zp[x,y]. Maple's modular gcd algorithm also does O(dx2dy + dxdy2) arithmetic operations in Zp. Experimental results show that our algorithm compares favorably to Maple’s gcd algorithm for a variety of input sizes.
Tian Chen and Michael Monagan (Simon Fraser University)
Factoring Multivariate Polynomials Represented by Black Boxes
The problem of factoring polynomials plays a critical role in symbolic computation. We present a new algorithm to factor multivariate polynomials with integer coefficients represented by black boxes. The black box representation of a polynomial a(x1,x2,...,xn) is a program which accepts input a prime p and an evaluation point (b1,b2,...,bn) in Zn and outputs a(b1,b2,...,bn) mod p. Given a polynomial a(x1,x2,...,xn) represented by a black box, we aim to compute its factors in their sparse representation. Three ways to do this include:
Method 1: first interpolates the sparse representation of a(x1,x2,...,xn) then factors it using a sparse Hensel lifting algorithm. However, this method is not space efficient as it must construct a(x1,x2,...,xn) which might be much larger than its factors.
Method 2: Kaltofen and Trager's method from 1990 first constructs black boxes for the factors then applies sparse interpolation to them.
Method 3: our new algorithm interpolates the factors in their sparse representation directly.
Method 3 requires less probes to the black box than methods 1 and 2.
In previous work we described Method 3 only for the monic and square-free case.
In this work, our black box factorization algorithm handles all inputs including non-monic, non-square-free and non-primitive polynomials.
One key design of our algorithm is that we transform each Hensel lifting step into many bivariate Hensel lifts. There are four major sub-steps for each Hensel lifting step, namely,
(1) probes to the black box to get the square-free part of bivariate images of a,
(2) evaluating the factors from the previous Hensel lifting step,
(3) bivariate Hensel lifts, and
(4) Solving Vandermonde systems.
We have made a hybrid Maple + C implementation of our new algorithm where sub-steps (1), (2), (3) and (4) are coded in C to speed up the computation and the main program is written in Maple. We shall show timing benchmarks for examples of computing the factors of the determinant of matrices with multivariate polynomial entries. We compare our algorithm with Maple and Magma's current best determinant and factorization algorithms.
Veronika Pillwein and Diego Dominici (RISC, JKU Linz)
Orthogonal Polynomial Solutions of Difference-Differential Equations
We consider a class of mixed first order linear difference-differential equation with polynomial coefficients. It is well known that all the classical orthogonal polynomials (Hermite, Laguerre, Jacobi, Bessel) satisfy such equations. In this talk we address the question: are these families the only ones? We use a combination of analytic and symbolic computation methods to find an answer.
David Mazziotti (University of Chicago)
Behind-the-Scenes Look at Maple's Quantum Chemistry Toolbox for Research and Education
Maple's Quantum Chemistry Toolbox provides a suite of tools for computing with molecules. In this session we interactively explore the key features of the Quantum Chemistry Toolbox for research and education. Mathematics and chemistry play critical roles in both research and education, and yet historically, the programs for facilitating our understanding of mathematics and chemistry have been separated. The Quantum Chemistry Add-on Package in Maple, we show, provides a unique and seamless integration of powerful algorithms for understanding the mathematics and chemistry of molecules within a single scientific environment. As a developer of the package, I will provide a behind-the-scenes look at the capabilities of the Toolbox as well as key design decisions by which Maple and the Add-on Package complement and reinforce each other's strengths. Participants will experience an up-close look at using the Toolbox to solve difficult research problems and to bring molecules to life in the Classroom.
Matteo Sacchet, Marina Marchisio, Fabio Roman, and Enrico Spinello (Università di Torino)
The Role of Context-based Problems with Maple for the Training of Military Officers
At the European level, the DEAP, Digital Education Action Plan 2021-2027 (European Education Area, 2020) outlines the strategic value of digital competences for educators, while the DigComp 2.2 (Vuorikari et al., 2022) outlines digital knowledge, skills and competences for all citizens. In this setting, the importance of digital technologies in Security and Defense education is increasing due to the numerous disciplines involved and the growing number of issues: logistics, cybersecurity, drone defense and many others. Military officers need problem-solving skills to face the threats in a continuously changing scenario, to evaluate the best strategies, to implement proper solutions to reach the target and to take proper decisions. Officers’ basic education is the perfect time to develop skills, and modules in Mathematics provide a suitable ground for this purpose. At the Bachelor’s and Master’s Degrees in Strategic Sciences at the University of Turin, instructors use innovative digital technologies, like the Advanced Computing Environment Maple, paired with problem-solving methodology to facilitate the learning of Mathematics and the development of problem solving skills. Students learn how to use Maple through dedicated and participative activities to solve context-based problems related to situations they will face in their future careers. Moreover, as part of their educational activities, students have to analyze, solve, present, and discuss - using Maple - a science-based problem in the military context.
In this paper, we investigate the use of Maple from the students' side, analyzing their work connected to different military tasks and across different dimensions: comprehension, resolution strategy, representation, use of Maple, in order to evaluate the effectiveness of the adopted approach. Works done by students will be discussed to highlight the presence of indicators of good and bad performance from the point of view of problem-solving skills. In conclusion, context-based problem-solving activities with Maple increase students’ engagement, allowing the development of those skills that are essential and specific for military officers. More instructional resources should be invested by educational stakeholders to face such an important need.
Donatella Merlini (Dipartimento di Statistica, Informatica, Applicazioni, Università di Firenze)
Analysis of Algorithms as a Teaching Experience
Teaching analysis of algorithms to students in Computer Science degrees, using the approach popularized by Knuth in his series of books "The Art of Computer Programming" and later by Sedgewick and Flajolet in the book "An Introduction to the Analysis of Algorithms", is not a simple task since, in general, these students are more interested in the implementation of an algorithm than in the corresponding theoretical aspects. This approach concentrates on precisely characterizing the performance of algorithms, by determining their best, worst and average case performance using a methodology based on symbolic tools such as recurrence relations and generating functions. The most difficult aspect is to understand the average case since this corresponds to studying the algorithm as its possible inputs vary: this represents the most important goal since generally students have no difficulty in understanding the best and worst cases corresponding to particular input configurations. A compromise that has been successful over the years consists in teaching students the analytical aspects of the problem and then organizing a simulation of the algorithm with a system of symbolic computation in order to precisely check the theoretical results.
We illustrate this problem by analyzing the well-known Quicksort algorithm and some of its variants, performing with Maple an average case analysis in terms of comparisons and exchanges executed to sort an array containing a random permutation of the values 1,..,n. For small dimensions of the array it is possible to execute the algorithm on all possible inputs while for larger dimensions it is necessary to execute it on a sufficiently large sample of inputs. In addition, the recurrence relations describing the problem and the corresponding generating functions can be easily handled in Maple.
Ovidiu Bagdasar and Uchenna Dial (University of Derby)
Using Maple at the University of Derby, for Teaching and Research
This paper presents the usage of Maple at the University of Derby (UoD).
As a powerful tool for symbolic computations, this software package has been used extensively by the authors, particularly for calculations with special polynomials and recurrences (Andrica and Bagdasar, 2020), or in the analysis and design of nonlinear systems like energy harvesters or vibration isolation systems (Diala et al., 2022).
In the classroom, the Maple platform provided the students with hands-on experience in working with complicated formulae across various modules in Mathematics and Electrical Engineering, being notably used for Renewable Energy systems modeling.
At the University of Derby, mathematics-based engineering modules were often characterized by high failure rates and low average module marks. Working together across disciplines (mathematics, psychology, technology, engineering) with a focus on math pedagogy and math anxiety (Hunt et al, 2019; Mahoney et al, 2012), we managed to centralize the mathematics support provided to our learners in a Maths Hub.
Maple has now become a key product within the Maths Hub project, developed to provide support for mathematics education across the University, its academic partners, Small and Medium Enterprises (SMEs) or learning communities in the region.
Want to know more about what goes on behind the scenes at Maplesoft? This is your opportunity ask questions of members of the Maplesoft R&D team. The panel will include people who are highly involved with the development of various aspects of Maple, the Maple Calculator app, and Maple Learn. Between them, this panel has many(!!) years of experience developing products for doing, learning, and teaching math. This is meant to be an interactive session, so come with lots of questions!
Panelists:
This meeting is for Maple Ambassadors. Attendance is by invitation only. Check your email for instructions on how to join the meeting.
Erik Hoy (Rowan University)
Using Maple+QuantumChemistry Toolbox for Nanoelectronics
Nanoscale electronic devices such as switches, resistors, and transistors built from single molecules have attracted significant experimental and theoretical interest as potential replacements for/supplements to silicon-based electronics. Compared to traditional electronic components, these devices offer both additional miniaturization potential and unique charge transport properties. Designing effective nanoscale devices, however, requires an accurate theoretical description of charge transport at the quantum level. Using the quantum mechanical tools provided by Maple's QuantumChemistry Toolbox, we have designed a series of worksheets for both designing nanoscale electronics and calculating their transport properties. We will demonstrate how to use Maple worksheets to construct a series of molecular resistors to investigate non-classical conductance patterns.
Samir Hamdi (University of Toronto)
New Gauss Hypergeometric Solutions of the Quartic Equation and Summation Identities of Reciprocal of First Derivatives of Polynomials
We present new fundamental solutions for the quartic polynomial equation in terms of hypergeometric function. The solution method is based on reducing the cubic resolvent to a trinomial that can be solved using the classical Gauss hypergeometric function using Maple. The new closed form formula is compact compared to existing solutions for the quartic. We also extend the Newton's and Vieta's formulas by introducing new identities for the sums that involve the reciprocals of the first derivatives evaluated at the roots of the polynomial. These identities are general and valid for any polynomials P(x) of any degree with nonzero distinct roots and with nonzero first derivatives at roots. In this work, we present only the summation identities for the quartic equation, which can be derived using a computer algebra system such as Maple. Several applications of the new hypergeometric solutions of the quartic and the polynomial summation identities are presented. We derive travelling solitary pulse solutions for a high order boundary value equation governing the FitzHugh-Nagumo model for an excitable reaction- diffusion neuron system. The quartic solution is applied for the analytical inversion of the specific energy–depth relationship of water flow in open channels with parabolic cross sections. We present applications for computing solutions for water flow with smooth change in channel width, and flow under sluice gate.
Jean-François Hermant (ESIEE-IT)
Using Maple to Compute a Tight Upper Bound on Worst-case Searches for a t-leaf Balanced m-ary Tree
In this paper, we compute a tight upper bound on worst-case searches for a t-leaf balanced m-ary tree. More specifically, we compute Xitk, the worst-case search time for isolating k leaves in a t-leaf balanced m-ary tree, which satisfies the following recursive equation: Xitk = 1 + max{Xit/mk_1 + ... + Xit/mk_m}, with: k_1 + ... + k_m = k and (k_1, ..., k_m) in {0, ..., t/m}m, if k in { 2, ..., t }, Xitk = 0 if k = 1, and Xitk = 1 if k = 0, where: t=mn, m in N*\{1}, n in N*. We use Maple to help us solve the previous recursive equation, which is far from being trivial.
Luis Felipe Tabera (Universidad de Cantabria), Rafael Losada-Liste (Sociedad Asturiana de Educación Matemática Agustín de Pedrayes), Tomás Recio (Universidad Antonio de Nebrija), and Carlos Ueno (CEAD-Las Palmas)
Handling a cube through Maple and GeoGebra
In our contribution we will reflect on the issues involved in the modeling of linkages through GeoGebra, a widespread program that combines dynamic geometry (DGS) and Computer Algebra systems (CAS) features. As it is standard in the DGS context, the performance of the graphic model (i.e. the positions of the other vertices when dragging a given one) must correspond to a mathematically rigorous, symbolic computation driven output. We will exhibit the different, quite challenging programming, geometric and algebraic problems arising in this context, by focusing on a specific, apparently simple, structure: the cubic linkage, i.e. the spatial linkage formed by the vertices and rigid edges that constitute the frame of a regular cube. The mathematical model for our treatment of linkages embedded in a Euclidean space has a strong algebraic flavour, since we associate to a linkage an algebraic (non-necessarily irreducible) variety (sometimes over R^n, sometimes over C^n). Among the obtained results, we will describe the seemingly evident, but hard to prove, complete determination of the dimension of the cubic linkage from an algebraic geometry perspective, achieved through the indispensable concourse of Maple. Moreover, we will exhibit different GeoGebra books allowing advanced (and mathematically sound) 3D visualizations of this structure.
Bennett Palmer (Idaho State University)
Construction of Equilibria with Symmetry for Anisotropic Surface Functionals
The surfaces we see in the real world occur as interfaces between immiscible materials or between distinct phases of a single material. Thermodynamics tells us that the interface forms so as to attempt to minimize a particular potential energy. When one of the constituent materials is in an ordered phase (e.g. crystalline, or liquid crystalline), the potential energy is anisotropic, i.e. it may depend on the direction of the surface. This accounts for the obvious difference between the shape of an equilibrium drop of liquid and an equilibrium crystal. In this talk, we discuss algorithms for the generation of equilibrium shapes for certain types of anisotropic surface energies. We present Maple graphics of the resulting surfaces.
Yagub Aliyev (ADA University)
Cayley’s Centro-Surface: Old and New Attempts to Draw this Elusive Surface and Some New Ideas Around It
In the talk we will discuss a little less known chapter in the history of mathematics: Cayley’s attempt to find the shape of the surface containing all the centers of the principal curvatures of an ellipsoid. Cayley called it the centro-surface of an ellipsoid. Nowadays, this surface is known as focal surface of an ellipsoid, Cayley’s Astroid or by more popular name (because of applications in Optics and related fields) caustic of an ellipsoid. Arthur Cayley’s paper from 1873 contains sophisticated algebraic calculations followed by an attempt to draw this surface. He says: "I constructed on a large scale a drawing of the centro-surface for the values a2= 50,b2=25,c2 = 15. (These were chosen so that a,b,c should have approximately the integer values 7, 5, 4, and that a2 + c2 should be well greater than 2b2 ; they give a good form of surface,…"
After A. Cayley there were many attempts to recreate this surface both as drawings and as 3-D models. With the dawn of computer graphics era the quality and accuracy of these recreations increased and we can now find many interesting and colorful renderings of this surface in books, articles and websites. The author of the current abstract, after spending more than a year of online Calculus teaching, tried to find some new motivating examples for his students of the use of modern computer technology in tandem with the methods of vector calculus to create colorful images of surfaces which are not easy to describe using equations. I used Maple for this purpose. The talk contains colorful images from history of mathematics and more modern computer graphics as comparison for how mathematics evolved over the centuries.
Philip Yasskin (Texas A & M University)
You Too Can Put Manipulatable 3D Graphics in Your Webpage
Maple can export 3D plots in the x3doc format. I will show you how to incorporate these into your webpage. Here is an example from my second semester calculus class:
https://www.math.tamu.edu/~yasskin/prevclas/172H.22a/Projects/
Here are two samples from my online calculus textbook, MY Math Apps Calculus:
https://mymathapps.com/mymacalc-sample/MYMACalc3/Part I - Geometry & Vectors/CurveProps/TNB.html
https://mymathapps.com/mymacalc-sample/MYMACalc3/Part%20I%20-%20Geometry%20&%20Vectors/CurveProps/Cubic.html
You can rotate these by dragging your mouse, zoom using the mouse wheel or pan these using Ctrl-mouse drag. I will also discuss some hurdles I overcame in getting these to work properly and where we go from here.
Beata Hebda and Piotr Hebda (University of North Georgia)
Making Typical Problems on the Trapezoidal Rule More Interesting and More Relevant to Students by Using Technology
The presentation will show an approach to Trapezoidal Rule problems that combines the use of a calculator with the use of Maple. As a result, students are more interested in solving typical problems, because they understand connections between the steps. The approach can be used in other situations that require approximating function values and calculating errors of such approximations.
We both use this approach to teaching the topic very successfully in our calculus classes. Students seem to be much more interested in solving typical approximation problems when they see a connection between by-hand, calculator, and Maple methods. The same idea can be used when teaching other approximation topics, for example, given by the Taylor or Mclaurin polynomials.
Serkan Dag (Middle East Technical University)
Maple-Aided Teaching in Mechanical Engineering Undergraduate Courses
This presentation outlines and details Maple-aided teaching activities in mechanical engineering undergraduate level courses Numerical Methods and Composite Structures. Maple worksheets are used in both courses in the teaching of the main approaches to solution methodologies. These worksheets illustrate how parametric algorithms are constructed, implemented, and executed to generate the complete set of numerical results pertaining to a given mathematical or engineering problem. The undergraduate level course Numerical Methods covers fundamental solution techniques regarding error analysis, root finding, linear system of equations, optimization, curve fitting, numerical integration and differentiation, and ordinary differential equations. Sample Maple worksheets for numerical methods will be presented with a discussion on the impact of Maple based instruction on student performance. The second course that is supported by Maple-aided instruction is the fourth year technical elective Composite Structures. The ultimate objective in this course is to impart the know-how and skills necessary for stress and failure analysis of general fiber-reinforced composite laminates. For this purpose, throughout the semester the students are acquainted with Maple scripts written for angle lamina analysis, stiffness matrix computation, thermal stress and failure analysis, and laminate design. Sample worksheets will be shown and their role in improving the programming skills of the enrolled students will be discussed. Lastly, Maple-aided instruction in online courses will be considered and web links for complete sets of related Maple worksheets will be provided.
Michael Monagan (Simon Fraser University)
What integration methods should we teach for a first integral calculus course?
I am presently teaching integral calculus using Stewart's text. For calculating antiderivatives, Stewart covers simple substitutions, integration by parts, trigonometric substitutions, and partial fractions in sections 5.5, 7.1, 7.3 and 7.4 respectively. To enable trigonometric substitutions, in section 7.2 Stewart also covers trigonometric integrals where the integrand is either of the form sin^m x cos^n x or tan^m sec^n x. There is also a review section 7.5 Strategies for Integration, so, 6 one hour lectures in total focused on calculating antiderivatives.
Given that we have computer algebra systems like Maple, many of us believe that the students would be better served if we replace one or two of those topics, perhaps 7.2 and 7.3, by a lecture on how to use Maple, or perhaps another application of integration. Indeed, for our integral calculus course for social science students we do not cover 7.2 and 7.3.
Dropping trigonometric integrals is problematic, however. The application Section 8.1 Arc Length generates integrals which need trigonometric substitution. There is also the desire to cover a second type of substitution so students are more skilled at manipulating integrals. We have dropped integrals with integrands of the form tan^m x sec^n x and have included instead the tan half angle substitution. The tan half angle substitution covers a wider class of functions (rational functions in sin x and cos x). For example it includes csc x which is useful. But it generates rational functions of rather high degree for which computing partial fractions is error prone. Another difficulty is inverting the tan half angle substitution. In the talk I will share the difficulties I ran into, how to program this in Maple and what to do so that students can more easily apply the tan half angle substitution.
Camille Pinto, Alban Quadrat (Sorbonne Université & Inria Paris), and Thomas Cluzeau (University of Limoges, XLIM)
Towards an Effective Integro-Differential Elimination Theory
In spite of the importance of calculus in mathematics and mathematical physics, the study of integro-differential equations does not seem to have attracted much attention from the computer algebra community. In particular, an effective elimination theory for integro-differential equations has not yet been developed. For linear systems of integro-differential equations, such an elimination theory is equivalent to the "coherence property" of rings of integro-differential operators. In 2013, Bavula gave a non-constructive proof of this property for the ring I1 of integro-differential operators with polynomial coefficients.
To develop an effective integro-differential elimination for I1, we shall first recall that a ring is coherent if the (left/right) annihilator of every element of I1 is a finitely generated (left/right) ideal and the intersection of two (left/right) finitely generated ideals of I1 is also finitely generated. In this talk, using the normal forms of elements of I1, effective computations over the ring A1 of differential operators with polynomial coefficients and linear algebra, we shall focus on the explicit characterization of the annihilators of the elements of I1. The different results will be illustrated with explicit examples computed with the Maple packages IntDiffOp developed by Korporal-Regensburger and OreModules (built upon Ore_algebra) by Chyzak-Quadrat-Robertz.
Juan Pablo Gonzales Trochez (Western University)
Laurent Series and Puiseux Series in Maple
Let K be an algebraically closed field of characteristic zero. The field of fractions of the ring of formal multivariate power series over K, is called the field of formal multivariate Laurent series. In this document, we follow the ideas introduced by Monforte and Kauers in their paper Formal Laurent Series in Several Variables. Our objective is to report on a first implementation of formal multivariate Laurent series inside of Maple, and explain the challenges we had to overcome. In order to accomplish this goal, we make use of the already existing MultitivariatePowerSeries package, and its lazy evaluation scheme. In particular, we expose our ideas for adding and multiplying Laurent series with support inside different cones, where the support of a Laurent series is the set of all exponents of all non-zero monomials of our series. We also describe our biggest challenge, how to invert a Laurent series. Unfortunately, this problem cannot be completely solved in a lazy evaluation context. We describe some situations where we can solve the problem completely; our approach for the cases that fall outside of these situations; and how we let the user customize this approach, trading off between speed and the likelihood of an incorrect result.
The algebraic closure of the field of formal multivariate Laurent series is called the field of formal multivariate Puiseux series. As an extension of our current work, we also present our ideas for an implementation of a multivariate Puiseux series object inside of Maple.
Bertrand Teguia Tabuguia (Max Planck Institute for Mathematics in the Sciences)
Symbolic Powers of Functions Defined by Second-Order Linear ODEs
By symbolic power of a given function f, we shall mean fn with a non-valued n so that fn follows all the basic rules for exponentiation. An ordinary differential equation (ODE) satisfied by the symbolic power of f gathers ODEs fulfilled by any valued power of f. Therefore such an ODE encodes an uncountable set of ODEs.
We focus on ODEs with polynomial coefficients over a number field. Although D-finite functions form an algebra, it is generally impossible to automatically compute ODEs for their symbolic powers because of a linear dependency linking the order and the symbolic power. We prove that degree-two ODEs overcome this situation for second-order D-finite functions and some second-order D-algebraic functions. Our result emphasizes an advantage for guessing with quadratic ODEs, which we also demonstrate with many examples. Indeed, degree-two lower-order ODEs exist for big integer powers of the underlying generating functions.
Alexandre Lê (Sorbonne Université, Paris Université / Safran / Inria / CNRS)
On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators
This presentation deals with the study of a Spherical Parallel Manipulator (SPM) with coaxial input shafts. Such a parallel robot provides three degrees of freedom (DOF) in orientation and is capable of unlimited rolling, in the context of stabilizing a set of cameras on a moving carrier. A special focus is made on the kinematics of this mechanism, especially taking into account uncertainties (e.g. on conception parameters, measures) and their propagation. Such considerations are crucial if we want to make sure to control our robot correctly without any undesirable behaviour in its workspace. One of the biggest issues of robots is singularities. These are configurations where the robot does not behave properly: either it loses at least one DOF or it gains at least one uncontrollable DOF. Such configurations should therefore be avoided. By studying the kinematics and singularities of this robot using Algebraic Geometry and Numerical Analysis techniques, we can compute a certified singularity-free zone in the work- and joint spaces, considering given uncertainties in the parameters. Problems related to the workspace will be solved using exact symbolic computation whereas the ones related to the joint space will be treated using a certified semi-numerical approach. This work of certification will allow us to use any control law to stabilize the upper platform of the robot, providing that the robot remains in the singularity-free zone.
Tomás Recio (Universidad Antonio de Nebrija) and M. Pilar Vélez (Universidad Antonio de Nebrija)
Dissecting Clough's Conjecture with Maple
Michael De Villiers is a well-known emeritus professor from South Africa, with world-leading research contributions on mathematics education and Dynamic Geometry. In different papers (De Villiers 2004, 2012) he worked on what he called “Clough’s conjecture”, namely that the sum of distances APc + BPa + CPb (see figure below, left) is constant. Trying to prove this result with the help of GeoGebra Discovery (see Kóvacs et. al. 2021) leads to a series of unexpected issues (figure above, right, showing how GeoGebra declares the result is “true on parts”), due to the complicated interlacing of real and complex algebraic geometry algorithms. In our contribution we will show how the use of different performing Maple packages and tools (PolynomialIdeals, algcurves, RegularChains, ConstructibleSetTools, SemialgebraicSetTools), allows to cleanly dissect the anatomy of Clough’s conjecture, yielding to a very coherent panorama of the geometry behind this nice result.
Cecilia Fissore, Francesco Floris and Marina Marchisio (Università di Torino)
Maple for the Development of Problem Solving Skills in Upper Secondary School
Among the eight key skills for lifelong learning to achieve success are mathematical competence, defined as the ability to develop and apply mathematical thinking to solve a variety of problems in everyday situations, and digital competence, which presupposes an interest in technologies and their use with critical thinking. Solving contextualized mathematical problems using Maple allows students to develop mathematical skills and digital skills at the same time. In particular, problem solving skills include several dimensions: understanding (analyzing the problematic situation, representing and interpreting data); identify (model the situation and identify the most suitable solution strategy); develop the solution process (resolve the situation problematic in a coherent, complete and correct way); and argue (comment and appropriately justify the applied process). The development of these skills, combined with digital skills, is particularly important for upper secondary school students who can be facilitated in university study or in the world of work. Our research question is: during a problem solving activity how much does the knowledge and effective use of Maple by students contribute to the development of problem solving skills? To answer this research question, the resolution of contextualized mathematical problems with Maple of about 100 secondary school students will be analyzed during a three-month online training. During this training the students learned how to use Maple and solved a mathematical problem every ten days, collaborating with the other students. Student resolutions were assessed by university tutors using an assessment grid that takes into account all dimensions of problem solving (understanding, identifying, developing, arguing, using maple). The research methodology consists of a quantitative analysis (of the evaluations of the students' resolutions during the training) and qualitative (the satisfaction questionnaires filled in by the students). The results of the analysis show how the use of Maple for problem solving activities positively influences the development of all problem solving skills. Explanatory examples are also given to show how Maple supports the development of problem solving skills and in particular argumentation skills. These results are useful for understanding the impact of this learning methodology and the use of technologies in the teaching and learning of Mathematics.
Fabio Roman, Alice Barana, Marina Marchisio and Valeria Fradiante (Università di Torino)
Developing Civic Education and Digital Citizenship Skills by Solving Mathematical Problems with Maple
In recent years, European reference frameworks have stressed the importance of developing Civic Education and Digital Citizenship skills in education, providing young people with means to live as active citizens in today's digital era. Civic Education, or Civics, means the provision of information and learning experiences to empower citizens to participate in society processes. Digital Citizenship is related to the positive engagement with digital technologies and data, so that people can participate actively and responsibly in today's digital society. Since 2020, Civic Education has become compulsory in all Italian upper secondary schools: teachers of all disciplines must dedicate some of their lessons to Civics, therefore adopting an interdisciplinary approach.
The objectives of this study are to understand how to design mathematical problems with Maple to develop Civics and Digital Citizenship skills, and how the use of Maple can enable upper secondary school students to develop this kind of skills. This study concerns a set of 40 contextualized mathematical problems to be solved with Maple on Civics and Digital Citizenship topics. They were conceived to help students develop these kinds of competences, besides Mathematics and problem-solving skills. A selected group of 542 upper secondary school students worked collaboratively in solving these problems with Maple through the Digital Learning Environment of the Digital Math Training project of the University of Turin funded by the Fondazione CRT. At the same time, the 108 Mathematics teachers of the students involved in the project were trained in a six-hour course aimed at showing how to use these problems in their teaching to engage students in Digital and Civic Citizenship Education. For this study, we have analyzed the task design of the 40 problems and how Maple was used in task design. We also studied how students and teachers reacted to the use of this type of problems through questionnaires submitted at the end of the project. From the analysis, we draw a model for the design of interdisciplinary mathematical problems to be solved with Maple. Moreover, the results show that students were quite aware of having developed digital and problem-solving skills, and teachers really appreciated how the problems helped students build Civics and Digital Citizenship competences. These results could help Mathematics teachers develop Civic Education activities within their teachings and encourage them to use an interdisciplinary problem-solving approach to make students develop different kinds of competences.
Alexandre Cavalcante, Sarah Lu and Sisi (Xiao) Feng (University of Toronto / OISE)
Exploring Open-Ended Tasks in Secondary Mathematics with Maple Learn - the Case of Functions
Teaching mathematics with a focus on mathematical processes (such as problem solving) often leads teachers to approach situations with single solutions. In doing so, students are presented with narrow ideas of what these processes can be. Problem solving, for example, can lead to open-ended explorations when students investigate a mathematical situation and need to make assumptions about aspects of the situation. In this presentation, we will introduce examples of Maple Learn activities that promote an open-ended perspective in mathematics explorations. The platform offers multiple features that allow students to identify their own assumptions about mathematical situations and work with them to create unique and individual solutions to problems, all the while still providing automated support to student learning. The presentation will focus particularly on the concept of functions from grade 9 to 12, starting with linear and quadratic functions. The presentation is part of an ongoing collaboration between Maplesoft and OISE (University of Toronto) to create Maple Learn resources that reflect contemporary mathematics education research.
John Pais (Ladue Horton Watkins High School)
Qubit Tensor Product Circuits Modeling Boolean Functions
This Maple interactive text develops the standard reversible extension (SRE) of an n-ary Boolean function by encoding each such function using an (n+1)-ary tensor product of qubit state spaces, with each n-tuple function argument encoded in the first n tensor product components and the corresponding value of the function encoded using the (n+1)st tensor product component. Each SRE of an n-ary Boolean function is represented by a unitary permutation matrix acting on the (n+1)-ary tensor product space, which when viewed as a quantum gate is reversible in the sense that it has a matrix inverse. This is in contrast to classical computing n-ary gates which are not reversible or invertible as functions.
In addition, for some small values of n, the matrix group of all such SREs that represent an n-ary Boolean function is generated and interactively explored, including its subgroups. Furthermore, the matrix tensor product of these and various other quantum gates, including the Hadamard gate, are developed and used to present a matrix group approach to several quantum computing algorithms, including the Deutsch Problem, the Deutsch-Jozsa Problem, Bernstein-Vazirani Problem and the Simon Problem.
This is the fifth in a sequence of Maple interactive texts that are being developed in order to introduce quantum computing to high school students. The target audience for this course is comprised of students that have already taken Calculus III (vector calculus) and/or AP Computer Science A. It is not assumed that the students have studied quantum mechanics or that they are familiar with Dirac notation, which is not used.
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.
Dr. Robert M. Corless is Emeritus Distinguished University Professor at Western University, a member of the Rotman Institute of Philosophy and of The Ontario Research Center for Computer Algebra, and Adjunct Professor at the Cheriton School of Computer Science, the University of Waterloo. His is also Editor-in-Chief of Maple Transactions. His primary research interests are computational linear and polynomial algebra, computational dynamical systems, and computational special functions. His underlying principles are Computational Discovery and Computational Epistemology, and the Ethics of AI, especially in teaching. His current focus is the new field of Bohemian Matrices. He has collaborated and published widely, and is the winner of a Halmos-Ford prize for mathematical exposition.
Each of the products in the Maple Math Suite include tools that encourage highly visual point-and-click style explorations. While appropriately similar in some ways, each product offers its own unique advantages. In this session, you’ll discover some of the ways Maple Calculator, Maple Learn, and Maple offer students and educators a highly interactive approach to conceptual learning and problem solving, the particular strengths of each approach, different methods for sharing interactive content, and how these tools can be used together to further enhance the student experience.
Everyone is familiar with using packages in Maple, from the widely useful plots package to specialized packages like AudioTools, DifferentialGeometry, and PolyhedralSets. But have you ever considered creating your own? Packages provide structure and organization to your code, and they make your work easier to reuse and share. This session will reveal some of the secrets used by algorithm developers at Maplesoft to write packages. Along the way we will touch on some essential programming topics such as modules, code-edit regions, debugging, revision control, and sharing.
Come learn more about the artwork in the Maple Art and Creative Works Exhibit and the Maple Learn Showcase, and meet some of the artists. Make sure you visit the Art Gallery ahead of time and vote for your favorites for the People’s Choice Awards.
** CANCELLED **
This meeting is for the associate editors of the Maple Transactions open-access journal (mapletransactions.org). Attendance is by invitation only. Check your email for instructions on how to join the meeting.
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