Maple
シンプルな操作性の高度数学ソフトウェア
• 教育機関向け Maple • 学生向け Maple • Maple Learn • Maple Calculator App • 企業及び官公庁向け Maple • Maple Flow • 個人利用向け Maple
|
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Please use #mapleconference when sharing on social media!
AI is looming large over all aspects of our lives, and this is especially true when it comes to large language models. Their promise is to make our work easier, and to multiply the impact our efforts have a thousandfold. Yet, there is also an implied threat that they can replace humans altogether and make most of us redundant. When it comes to mathematics, both the promise and the threat seem to be more pronounced than in any other field. After all, math education and research have a long history of leveraging technology to great effect. At the same time, a field based on structure and logic is a prime candidate for AI to subsume.
In this presentation, we will investigate the potential and likely implications of the emergence of AI on mathematics. We will explore what AIs are good at, what they might become better at, and what their relationship with humans in the world of mathematics might ultimately turn out to be. We’ll discuss the implications on teaching, learning, doing, and leveraging mathematics, and what math tools could look like in a world where AI is ubiquitous.
Dr. Laurent Bernardin is President and CEO of Maplesoft. He has been with Maplesoft for over 25 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.
Whether you have been using Maple 2024 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 2024 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 Maple Development and long-time Maple user, finds particularly useful or interesting. You may even get a few hints of more good things to come.
This presentation will provide an overview of the AI features in Maplesoft products that you can use today, both as interactive features, and as tools for building your own applications. You may also get a glimpse of some new features we are actively working on.
Adam Gale (University of Western Ontario, Canada)
Partially Ordered Sets Package in Maple
Partially ordered sets (posets) are a specific type of directed graph that effectively models dependency relations. Examples of posets include the divisibility relation among integers and the set containment of polyhedron faces (vertices, edges, facets, etc.). Many operations on posets, such as finding maximal chains and antichains, computing minimal elements, and deciding isomorphism between posets, are algorithmic in nature and have numerous applications in areas like combinatorics, optimization of computer programs, social sciences, and polyhedral geometry.
Posets are often treated simply as directed graphs in software implementations, leading to missed opportunities for computational efficiency. Posets frequently possess additional properties, such as being lattices or graded posets, which can be leveraged to accelerate algorithms. Additionally, posets can be represented compactly, where many edges are implied by poset properties (e.g., transitivity) and thus do not need to be explicitly encoded. A notable example is the face lattice of a polyhedron, where not only edges but also nodes can be implicitly represented.
Motivated by these considerations, we developed a Maple package dedicated to posets and their most commonly used operations. Maple's programming language and extensive supporting libraries allowed us to capitalize on the aforementioned opportunities for efficiency improvements.
In this presentation, we will discuss the challenges and solutions involved in implementing posets in Maple, showcasing how our approach enhances computational performance. We will demonstrate the practical applications of our poset library in various domains, emphasizing the benefits of recognizing and exploiting poset properties in algorithmic computations.
Aishat Olagunju, Marc Moreno Maza, David Jeffrey (University of Western Ontario, Canada)
Comprehensive LU Decomposition and True Path
We explore the PLU decomposition of rectangular matrices with polynomial coefficients dependent on parameters and algebraic constraints. This process often requires splitting computations into multiple cases, leading to challenges such as generating unnecessary cases. We propose heuristic methods to minimize case numbers, avoiding splits when possible. Implemented in Maple, these methods were evaluated on various literature test problems.
Álvaro Pereira Albert, Tomás Recio, M. Pilar Vélez (Universidad Antonio de Nebrija, Spain)
Dealing with Non-Principal Locus in Automatic Reasoning in Geometry
The integration of the computer algebra system Giac into GeoGebra opened the possibility of algebraically modelling a geometric construction made in GeoGebra and manipulating it with computer algebra algorithms, thus allowing the development of mathematically rigorous and high-performance Automatic Reasoning Tools (ART) to deal with geometric statements. Hence, the GeoGebra Discovery version of GeoGebra includes some advanced tools offering the user a rich variety of ART to experiment, discover and assert in geometric contexts. In this presentation, we deal with the performance of one of these tools, the LocusEquation command. The purpose of this command is to automatically discover how to modify a given figure so that an incorrect or unfinished statement becomes true, returning where to place some point in a construction so that a given property is satisfied. We study some constructions where the use of this command gives interesting and unexpected results. As an example of this, we consider the problem of finding the geometric locus of a point C such that the Euler line of the triangle ABC is perpendicular to the side AB. A theoretic analysis suggests that this locus should be perpendicular bisector of the segment AB, and hence, a line. However, the output given by the LocusEquation command is not a principal ideal, and its zero set includes some additional points which correspond to degeneracy conditions of the construction. We propose a protocol for dealing with this kind of cases, using Maple to show how this protocol works, until these methods are implemented in Giac and in GeoGebra Discovery.
Marcin Kamiński (Lodz University of Technology, Poland)
On Integration of the Relative Bhattacharyya Entropy in Maple
The main aim of this presentation is to demonstrate how one can derive the Bhattacharyya relative entropy in the system Maple. This type of relative entropy (probabilistic distance) as the geometric mean of two probability distributions is applicable in all the problems of applied sciences and engineering where two different probability distributions need to be contrasted. It may serve for the quantification of a diversity of two given populations in biology and demography, and also - to measure the reliability of structures, systems or machines. The key problem is that even for the Gaussian distributions the additional integral defining this entropy cannot be directly determined using symbolic computing and needs rather complex algebraic transformations. Therefore, an additional computational implementation in the system Maple has been prepared with additional parametric visualization, and would be used to make some engineering illustrations.
Athanasios Tzemos (Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Greece)
Partial Ergodicity in 2d Bohmian trajectories
In this talk, we will present our new results on the chaotic dynamics of the Bohmian trajectories of a 2D harmonic oscillator with non-commensurable frequencies. In particular, we will show an example of a wavefunction with a single nodal point that produces partially ergodic chaotic trajectories, in sharp contrast to the usual case where all the chaotic trajectories cover the entire support of the wavefunction in a similar way. In the case of partial ergodicity, the chaotic trajectories cover specific parts of the phase space. We find that this phenomenon stems from the existence of an unstable point in the Bohmian flow, which we call the Y-point.
The Y-point completes the so-called nodal point-X-point complex mechanism, which describes the production of chaos in 2D systems and provides us with a complete understanding of the behavior of the maximum Lyapunov exponent. Finally, the partial ergodicity shows that there are cases where chaos itself prevents convergence to Born's rule by an arbitrary initial distribution of Bohmian particles. All our calculations have been carried out using Maple and Python.
Daulet Nurakhmetov, Serik Jumabayev (Institute of Mathematics and Mathematical Modelling, Kazakhstan)
On Volterra three-point problems for the Sturm-Liouville operator related to potential symmetry
In this talk, we consider on L_2 (0;π) the three-point problem for the second order ordinary differential equation. We show a connection of the three-point problem with one-point Cauchy problem if a potential is an integrable function on [0, π] posing symmetry with respect to the points π⁄2 and π⁄4. Symbolic calculations were carried out in the Maple computer algebra system.
Olga Yaremchuk, Gennady Chuiko (Petro Mohyla Black Sea National University, Ukraine)
How does Maple support Data Feature Ranking and Selection?
We presented a successful case of integrating two software products, WEKA and Maple, in the Machine Learning section, often referred to as "Feature Engineering."
Mª Pilar Vélez Melón (Escuela Politécnica Superior, Universidad Antonio de Nebrija, Spain)
Francisco Botana (Universidad de Vigo, Spain)
Tomás Recio (Escuela Politécnica Superior, Universidad Antonio de Nebrija, Spain)
Using GeoGebra Discovery and ChatGPT for dealing with geometric statements
Our presentation explores the performance of ChatGPT and GeoGebra Discovery, when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence, over sixty years ago, and a non-trivial inequality concerning triangles, from the American Mathematical Monthly. In two of these statements both mentioned systems succeed –but in the case of ChatGPT, not systematically, as the answers change quite often, for the same question, along the time-- on proving these two examples. On the other hand, ChatGPT fails on the third one.
Our thesis, that will be developed with the concourse of Maple to describe the algebraic geometry-related difficulties, and the involved symbolic computations, is that both GeoGebra Discovery and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery could cooperate to get sound and—more relevant— interesting results.
Join Dr. Laurent Bernardin’s virtual lunch table for an informal conversation about anything that comes up. Many Maple developers will also be there. It’s a great chance to ask questions, share experiences, and meet the people behind the product.
Breakout rooms are also available for smaller group discussions. If you want to continue a discussion from one of the sessions, catch up with colleagues, or talk about a subject that isn’t in the main room, just ask and we’ll create a room for you.
Yuzhuo Lei (University of Western Ontario, Canada)
Rui-Juan Jing (Jiangsu University, China)
Christopher F.S. Maligec, Marc Moreno Maza (University of Western Ontario, Canada)
Erik Postma (Maplesoft, Canada)
Counting the Number of Integer Points of Non-Parametric and Parametric Polytopes with the PolyhedralSets Library in Maple
When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints this task may simply become an undecidable problem. However, studying the integer solutions of systems of equations and inequalities is of practical importance in areas like combinatorial optimization and compiler construction. Over the past 10 years, a series of projects has equipped the computer algebra system Maple with a number of tools for dealing with the integer points of systems of linear equations and inequalities, even in the presence of parameters. With these tools, one can either decide whether integer point solutions exist, or count them, or describe them in a compact way, or enumerate them. In this talk, we will give a tour of these facilities and illustrate their usage with a number of applications. In particular, we will discuss the adaptation of Barvinok’s algorithm for integer point counting from non-parametric polytopes to parametric polytopes. We report on a Maple implementation of this adaptation. This includes a general framework called ValueUnderConstraints in support of parametric system solving. Additionally, in order to represent the parametric results of the integer point counting of parametric systems, a new Maple data structure called QuasiPolynomials was built.
Hari Sitaula (Montana Technological University, USA)
Algorithms in Maple to Compute Discrete Residues of a Rational Function
The concept of discrete residues for rational functions was introduced by Chen and Singer in 2012.
Discrete residues have numerous important applications beyond their primary role as obstructions to summability: they are useful in creative telescoping problems, determining (differential-)algebraic relations among hypergeometric sequences, and computing (differential) Galois groups of difference equations.
However, discrete residues are initially defined through the complete partial fraction decomposition of a rational function. This makes direct computation impractical due to the complexity of factoring arbitrary denominator polynomials into linear factors. We have developed a practical and efficient algorithm using Maple to compute discrete residues of rational functions using only gcd computations and linear algebra.
Juan-Pablo Gonzalez Trochez, Manuel Romero (University of Western Ontario, Canada)
Matthew Calder, Erik Postma (Maplesoft, Canada)
Marc Moreno Maza (University of Western Ontario, Canada)
Power series solutions of algebraic and ordinary differential equations with the MultivariatePowerSeries in Maple
The MultivariatePowerSeries (MPS) package in Maple is based on the lazy evaluation scheme, that is, power series terms are computed on demand and cached for possible later reuse. In this presentation, we report on two features of the existing MultivariatePowerSeries package. Namely, users are now able to compute Power series solutions of algebraic and ordinary differential equations (ODEs).
One of the key features of the MPS package is the ability to increase the accuracy of its members (power series, polynomials over power series) without restarting computations from scratch. Another distinctive property of this package, compared with other Maple packages dealing with power series, is that power series are encoded using functions generating their coefficients. This allows users to handle a larger class of power series, beyond the usual examples coming from elementary functions or solutions of algebraic and differential equations.
Nowak's approach to the Puiseux theorem allows MPS users to solve univariate equations with arbitrary Puiseux series coefficients (that MPS can construct). On the other hand, with the Extended Hensel Constructions (EHC), the coefficients of univariate equations are restricted to be formal multivariate power series. EHC, however, allows for higher computational performance. Another application of EHC is to compute the Puiseux series solutions of one-dimensional regular chains, and thus space curves, from which one can deduce real or complex branches of such curves around a point.
As new features of the MPS package, it is now possible to create power series using linear ODEs with polynomial coefficients as an input. It is also possible to find solutions of algebraic equations by means of the Extended Hensel Constructions, as well as Nowak's formulation of Puiseux's theorem.
Mahsa Ansari (Simon Fraser University, Canada)
Computing the Resultant and GCDs of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions
Let f1 and f2 be two multivariate polynomials over an algebraic number field Q(α1, . . ., αn). We designed two modular algorithms; Algorithm MRES for computing the resultant of f1 and f2 and Algorithm MGCD for computing the monic gcd of f1 and f2. To enhance the efficiency, our algorithms convert f1 and f2 to their corresponding polynomials over Q(γ) where γ is a primitive element of Q(α1, . . ., αn). This conversion is executed modulo a prime to prevent the coefficient growth. Next, our algorithms employ evaluation and dense interpolation to reduce the problem to the computation of the resultant and monic gcd of two univariate polynomials where we apply the monic Euclidean algorithm. Moreover, employing the monic Euclidean algorithm, we present a new formula for computing the resultant of univariate polynomials. Finally, our modular algorithms apply the Chinese remaindering (CRT) and the rational number reconstruction (RNR) to recover the rational coefficients of the resultant and monic gcd. We have implemented our algorithms in Maple. We computed the expected time complexity of both algorithms and a partial failure probability analysis. Additionally, we designed two benchmarks for our resultant algorithm and one benchmark for our monic gcd algorithm.
Sepideh Bahrami, Taylor Brysiewicz, David Jeffrey, Marc Moreno Maza (University of Western Ontario, Canada)
Computing Osculating Curves of Power Series in Maple
Given a complex analytic curve X ⊂ ℂ² and a point p ∈ X, an interesting problem is to approximate X geometrically at p. For instance, in calculus, students learn how to find the tangent line to the graph of a function y = f(x) at a point p = (x₀, f(x₀)) using the derivative of f at x₀. Locally, we assume the curve X is represented as a power series expansion, with the point p translated to the origin (0, 0) ∈ ℂ²:
f(x) = Σ cᵢ xⁱ. (1)
This assumption that X is locally given in this manner follows from the implicit function theorem: near x = 0, this series converges, and the graph Γ of x ↦ f(x) coincides with X near p = (0, 0).
Another way to define a plane curve is as the zero set of an implicit polynomial function F(x, y) ∈ ℂ[x, y]. We say that F(x, y) defines the variety
V(F(x, y)) = {(x, y) ∈ ℂ² | F(x, y) = 0}.
Given d and a sequence c = (c1, c2, . . .) defining (1), our goal is to find the best approximation of Γ by a curve of the form V(F(x, y)) where F(x, y) has degree at most d. Such a curve is referred to as the degree d osculating curve of (1), or equivalently, the degree d osculating curve of X at p. Such a curve, by definition, belongs to the osculating space Vd,k. This complex vector space depends only on the first k − 1 derivatives of f(x).
For example, the degree 1 osculating curve of X at p is simply the tangent line of X at p. Given (1), a formula for the tangent line is
F(x, y) = c₁ x − y.
Running our Maple code for computing osculating curves with the c’s left as
indeterminates reveals the formula for the osculating conic of (1):
F(x, y) =(c₁ c₂³) x + (- c₂³) y + (c₂⁴ - c₁ c₂² c₃ - c₁² c₃² + c₁² c₂ c₄) x² +
(c₂² c₃ + 2 c₁ c₃² - 2 c₁ c₂ c₄) x y + (- c₃² + c₂ c₄) y².
Rafal Ablamowicz (formerly Tennessee Technological University, USA)
Jane Liu (Tennessee Technological University, USA)
A MaterialConstants Package ver. 03 for Maple
In this presentation consisting of two parts we wish to describe a Maple 2020.2 package called MaterialConstants ver. 03 (June 30, 2024), whose main goal is to display a material constant matrix (also called the stiffness matrix) C for materials with various types of symmetries. In particular, the main procedure of the package called TransformedMaterialConstants (aliased as TCM) applies desired material symmetries to a generic material constant matrix C which is a 6 x 6 symmetric real matrix. It then returns the material constant matrix C for the material with the specified symmetry(ies).
This presentation will consist of two parts: Part 1 - A theoretical background by Dr. Jane Liu, and Part 2 - A Maple 2020.2 worksheet with examples by Dr. Rafal Ablamowicz.
Matthew Calder (Maplesoft, Canada)
A Maple baseball scorekeeping application to generate defences, optimize lineups, keep score, generate score sheets, and compute and display statistics
Calling up all baseball/Maple/mathematics lovers! A baseball scorekeeping application written entirely in Maple will be presented.
The main aspects of the scorekeeper are:
| 1. | a defence table generator and optimizer that can quickly assign fielders to positions for every inning based on preferences and subject to constraints, such as minimizing the difference in the number of innings played between fielders, ensuring any requirements for girls or guys on the field is met, and preventing successive innings on the bench for fielders; |
| 2. | a lineup optimizer, which orders the hitters in the lineup to maximize the expected number of runs scored compared with other permutations; |
| 3. | a scorekeeper to quickly record the play-by-play of the game and generate a more traditional score sheet; and finally |
| 4. | a statistics generator to view performance data based on all the recorded games. |
Willem Andriessen (HAN University of Applied Sciences), Ans Koster
Model based control of Heating, Ventilation, and Air Conditioning (HVAC) components
I am a retired professor at HAN University of Applied Sciences and also for one year at San Diego State University in California. I am still working with students to accelerate the energy transition with sustainable home energy for all and to help 3 million households transition by 2030. Our students are highly motivated, ambitious and not afraid to make their hands dirty to make a real dent in the climate change challenge that we are already facing.
We train our students in electrical and control engineering by developing gray-box/black-box models of HVAC (Heating, Ventilation, and Air Conditioning) components like water tanks and heat pumps. They design and refine control systems for domestic heating applications, ensuring energy efficiency and reliability objectives are met. Also, low-level controls are implemented that enable rapid prototyping of control algorithms and develop/update simulation environments to test and compare control designs.
The future job of our students is that of a Control Systems Engineer that can play a crucial role in designing and implementing the control algorithms for domestic heating products.
During my lectures I have used Maple 2022.2 and its packages LinearAlgebra and DynamicSystems to implement models of several HVAC components and to simulate the model-based control of these components, both analog and digitally. Usually this is done using MATLAB and Simulink, but I have succeeded to do it by using only Maple, even without the help of MapleSim.
In this talk, I will zoom in to the so-called 4 +1 approach to state space modeling of thermodynamical systems we use at HAN. With such a state-space model, the next steps are to design a state-space controller that meets specified control requirements, to simulate the behavior and to convert to a standard PID controller when control goals are met. I’m planning to use an electric tap water boiler and a domestic kitchen oven for this modeling and control exercise using Maple, with the opportunity to explain what the advantage of Maple is above using Matlab/Simulink that also often is used.
Tomás Recio (Escuela Politécnica Superior, Universidad Antonio de Nebrija, Spain)
Zoltán Kovács (The Private University College of Education of the Diocese of Linz, Austria)
Mª Pilar Vélez Melón (Escuela Politécnica Superior, Universidad Antonio de Nebrija, Spain)
Describing, through Maple, GeoGebra Discovery’s new features: proof certificates, complexity estimation
We describe some on-going improvements concerning the Automated Reasoning Tools developed in GeoGebra Discovery, providing different examples of the performance of these new features. Thus, we describe the new ShowProof command, a first attempt to enhance the visibility of the proofs achieved by GeoGebra Discovery, by showing the result of the different steps performed by GeoGebra Discovery to confirm a certain statement: algebraic translation of the geometric input construction, numerical specialization of the coordinates of some free points, the automatic inclusion of non-degeneracy conditions, and the writing the expression of 1 as a combination of the hypotheses and the negation of the thesis (thus proving the statement by reductio ad absurdum).
Moreover, the ShowProof command also outputs a number that we consider could grade the difficulty or interest of the given geometric assertion. The proposal of this assessment measure involves a formal comparison of the expression of the thesis (or of number 1) as a combination of the hypotheses (respectively, of the hypotheses and the negation of the thesis). Roughly speaking, we will consider that a statement is more complex or difficult if the polynomial describing the thesis (or 1) is a sum of products of the hypothesis’s polynomials (respectively, of the hypotheses and the negation of the thesis) multiplied by polynomials of higher degree. Thus, we could label as “complexity zero” the implication {x+y=0, x-y=0} => {x=0}, as x=1/2*(x+y)+1/2(x-y), meaning that we only have to multiply the hypotheses by constant polynomials. On the other hand, the implication {x-y^2=0, y=0} => {x=0} could be labeled as of complexity 1, since x=1*(x-y^2)+y*(y), requiring polynomials of degree 1 to express the thesis in terms of the hypotheses.
This notion of complexity will be developed and illustrated through some handmade examples implemented in Maple worksheets, as the current execution in GeoGebra Discovery of the complexity computation remains in a kind of black box, making impossible to visualize the computation of this complexity grade. Let us remark that the use of Maple in this context has been facilitated through a new feature of GeoGebra Discovery’s Edition menu, allowing to copy GeoGebra’s CAS view as Maple (or Mathematica, or Giac) to the clipboard.
Finally, we will sketch some on-going steps addressing the potential relevance in the educational world of these two new features (showing proof steps, estimating difficulty), e.g. to adequately propose and rank mathematic problems for students with high capacities, to evaluate the performance of some A.I. tools in problems of different levels or to assess if human performance (say, math Olympiad teams) reasonably agrees (or can be predicted) with our complexity measure.
Bahia Si Lakhal (University of Science and Technology Houari Boumediene (USTHB), Algeria)
The solutions of Lane-Emden Equation
The Lane–Emden equation is a classical equation of mathematical physics. First, it was introduced to describe the mass density distribution inside a polytropic star in hydrostatic equilibrium, that depends on a polytropic index n. Given the value of n, we can get solutions of the equation which can be exact or approximate. Up to now, there exist analytic solutions only for the polytropic indices n=0, 1, and 5. By using Maple software, we find the solutions of the Lane-Emden equation for n=0, 1, 2, 3, 4, 5, and 6. When an analytic solution is not possible, we will use some commands of Maple software such as DEtools, to get different numerical solutions. We then use the package odeplot to plot the solutions found.
Samir Hamdi (University of Toronto, Canada)
Maple simulation for travelling wave solutions for FitzHugh-Nagumo model equations for nerve conduction in an excitable reaction-diffusion system
We present Maple interactive tools for studying travelling wave pulses in the Bonhoeffer-van der Pol type reaction-diffusion system, which is governed by a coupled set of FitzHugh-Nagumo equations for an activator and inhibitor and exhibits excitability. The FitzHugh-Nagumo coupled reaction-diffusion equations model the propagation of electrical signals in nerve axons and other biological tissues. A piecewise linear inhibition is considered based on McKean and Rinzel-Keller models and using the Heaviside function for approximating the cubic nonlinearity. Solitary wave solutions are derived by solving a fourth order boundary value problem and the associated quartic polynomial using Maple.
Maple Interactive plots and visualization tools are developed for the animation of the analytical relations between the propagating pulse celerity and width. Numerical simulations of different solitary wave profiles for various values of the diffusion coefficient and model parameters are presented using MAPLE explore command.
Alexander Rusnak (Microchip, USA)
A Stochastic & Predictive Scheduling Technique for Mega Projects
In the fast-paced world of semiconductor design, where FPGA projects are characterized by their complexity and size, traditional scheduling techniques often fall short. This presentation will explore a novel approach to project management, specifically tailored to address the challenges faced in the semiconductor industry. The speaker, a project management specialist, was tasked with developing a method to predict project completion for an FPGA semiconductor company, amidst an ongoing project with no time to create a detailed schedule. The solution involved leveraging projected milestone data.
Utilizing Maplesoft's powerful computational tools, the speaker applied several different curve-fitting methods to the milestone data. Then, the speaker selected a particular curve. This curve, once defined, was used to analyze the randomness of the project's progress was visualized using Maplesoft's graphing capabilities, providing clear insights into the main trend, statistical confidence intervals, and projected completion. Furthermore, this technique proved beneficial in detecting significant changes in the project's schedule, allowing for earlier intervention and adjustments by management. Over the course of a year, this innovative approach not only facilitated more accurate schedule predictions but also enhanced the company's ability to proactively manage project timelines.
We will present Maple's functionality for computing truncated series expansions and limits, mainly in the univariate case, but also multivariate. We will discuss best practices for getting the most out of the series, limit and asympt commands.
Learn how to build Jupyter notebooks that use Maple for computation. You will also discover the different ways you can run Python code and functions from within Maple, and how you can perform computations with Maple directly from Python using the OpenMaple API.
Yagub Aliyev (ADA University, Azerbaijan)
The Number of Inscribed and Circumscribed Graphs of a Convex Polyhedron
Let A be a graph (vertices and edges) also called 1-skeleton of a convex polyhedron. On each edge of A choose a point which does not coincide with the vertices of A. If the points on each polygonal face of A are connected to form a convex polygon, then these polygons together form a graph, which we denote by B. We say B is inscribed into A and similarly, A is circumscribed around B. Repeating this process for B, we obtain its inscribed graph C. The vertices of C are chosen on the edges of B one on each. Suppose now that graphs A and C are fixed. Is it possible to find graphs different from B which are also inscribed into A and circumscribed around B, and if yes, then what is the maximum of their number? We will answer the second question by proving that this number cannot exceed 4 and by using Maple to give examples of convex polyhedra for which 4 such graphs exist. Along the way we also solved the 2-dimensional variant of the problem for polygons. There are many open questions which will be discussed in the presentation.
Shaoxuan Huang (Chinese University of Hong Kong, China)
Finding Reduction Transformation that Transforms a Rational First Order ODE to Linearizable Equation
We propose a method for finding, if possible, the local reduction transformation that transforms a general first order rational ODE to a solvable first order linear, Riccati, or Abel equation, and its generalized type. We have tested the algorithm on a number of cases and improved its computation speed so that it is already able to handle equations of considerable size.
Flóra Hajdu (Széchenyi István University, Hungary)
Parallelization of the One-at-a-Time Sensitivity Study of Nonlinear Systems using Maple
In this presentation the parallelization of the One-at-a-Time (OAT) sensitivity study of nonlinear systems is presented using Maple’s Grid package. The parallelization is carried out with an Intel® Core™ i7-7700HQ CPU @ 2.80GHz processor with 8 nodes. First the parallelization opportunities using Maple are discussed, then the OAT sensitivity study using Fuzzy-logic with Maple’s FuzzySets[RealDomain] package is described. Then the parallelization of the sensitivity study of 2 systems is presented, which are a Duffing-type nonlinear vibration system using 5 parameters and a nonlinear fire truck suspension system using the mass of the superstructure. Then the results of parallelization are described using speedup and efficiency. A scalability study is also carried out at the largest number of parameters. The presentation concludes with further research tasks.
Falade Kazeem Iyanda (Aliko Dangote University of Science and Technology, Nigeria)
Abd’gafar A.T. (Chinese University of Hong Kong, China)
Adeyemo K.A. (Nigeria Police Academy, Nigeria)
Bello K.A., Babatunde V.O. (University of Ilorin, Nigeria)
Tanko A. (Federal College of Forestry, Nigeria)
Algorithm Approach for Solving Time-Fractional Coupled Systems of Partial Differential Equations
In this paper, we introduce and apply an efficient computational algorithm utilizing the capabilities of the Maple 18 software, incorporating coded fractional derivatives in the Riemann-Liouville sense. Our focus is on solving coupled systems of partial differential equations commonly encountered in engineering and mathematical physics domains, including fluid dynamics, viscoelastic materials, viscous damping, polymer physics, and seismic analysis. Our algorithm leverages various mathematical commands within the Maple software package. We present four illustrative examples of both linear and nonlinear time-fractional coupled systems of partial differential equations. The obtained results are systematically compared with analytical solutions to assess their accuracy. These findings are presented in both tabular format and as 2D and 3D graphical representations. The proposed algorithm is characterized by its ease of use, reliability, and efficiency. It holds the potential to serve as a valuable mathematical tool for addressing a wide range of systems of partial differential equations in applied mathematics.
John Pais (Ladue Horton Watkins High School)
Comparing Flavell and Guralnick-Tracey Criteria for Subgroup Containment in a Proper Normal Subgroup
The not “unreasonable effectiveness” and in fact necessity of computation in understanding, comparing and evaluating certain closure theorems in group theory is illustrated by comparing Flavell and Guralnick-Tracey criteria for containment of a subgroup H of G in a proper normal subgroup of G. In order to find useful examples one must first consider non-nilpotent groups with some but relatively few normal subgroups and with several layers in their lattice of subgroups. This precludes an effective approach by hand calculation, so instead we may find suitable groups by considering the Shephard-Todd finite unitary reflection groups of rank 2 contained in U(2,C), which we represent using the Maple built-in interactive matrix algebra over the complex numbers. As a somewhat challenging side-project, we have captured and identified all nineteen of these Shephard-Todd groups (order 24-3600) within one Maple document using the generators provided in Chapter 6 of the classic book “Unitary Reflection Groups” by G. I. Lehrer and D. E. Taylor.
Both the Flavell and Guralnick-Tracey criteria for containment of a subgroup H of G in a proper normal subgroup of G, require H to be contained in at least two maximal subgroups of G. The Flavell criteria also require that H is subnormal in all the maximal subgroups containing it, except possibly one. The Guralnick-Tracey criteria introduces a descending normal closure operator which is applied to H with respect to each one of its containing maximal subgroups yielding a set S of such descending normal closures, which must have a unique maximal element (subgroup).
In addition, we implement several relevant closure operators and use them to explore these different criteria with some interesting and illuminating results, which we illustrate using Maple tables and plottools.
Philip Yasskin (Texas A&M University)
Maple Plots in MY Math Apps Calculus
MY Math Apps Calculus (MYMACalc) is an online calculus course. A sample (about half the chapters) is available at https://mymathapps.com/mymacalc-sample/. Most of the plots have been made using Maple.
These are 2D and 3D, static and animated. Some can even be rotated or zoomed with a mouse. For example, there are:
| * | animations demonstrating the proof of the triangle inequality. |
| * | a sequence of animations showing the limits of a function from the left and right. |
| * | animations showing the convergence of secant lines (and vectors) to tangent lines (and vectors). |
| * | plots showing how the derivative of an inverse function is related to that of the function. |
| * | a sequence of plots showing the convergence of Newton's method. |
| * | many beautiful plots for related rates and max/min problems for functions of 1, 2 and 3 variables. |
| * | animations showing the convergence of Riemann sums for functions of 1 and 2 variables. |
| * | an animated proof of the Fundamental Theorem of Calculus. |
| * | many animations for arc length, surface area, volume by slicing and revolution, work and fluid force. |
| * | animations showing the convergence of Taylor series. |
| * | plots showing parallel, intersecting and skew lines which can be rotated with a mouse. |
| * | plots of quadratic surfaces which can be rotated and zoomed with a mouse. |
| * | animations showing T, N and B along a curve which can be rotated. |
| * | an animation showing the osculating circle on a curve. |
| * | animations explaining how torsion distinguished between left and right handed curves. |
| * | animations explaining linear approximations for functions of 1 and 2 variables. |
| * | animated proofs of the properties of gradients. |
| * | animations deriving the Lagrange multiplier method. |
| * | a sequence of plots showing how to reverse the order of integration in multiple integrals. |
| * | plots for polar, cylindrical, spherical and other curvilinear coordinate systems deriving the Jacobian. |
| * | animations to understand orientations in Green's, Stokes' and Gauss' Theorems. |
Scot Gould (Scripps, Pitzer, Claremont McKenna)
Like and Subscribe: Using YouTube to Teach and Promote Maple
This talk covers the experiences of creating the YouTube channel - Learning Maple. The aim of the channel is to educate new Maple users or remind infrequent users how to solve problems using Maple. The target audience is undergraduate-educated students who use Maple in an applied mathematics discipline or profession. The videos are not designed to teach mathematics.
The less-than-year-old channel contains over forty 8-minute to 12-minute videos organized into five playlists: Maple Fundamentals, Mathematics for Undergraduates, Advanced Mathematics, Applications and Learning Physics Using Maple. The videos are ordered in each playlist like chapters in a book. The format of the majority of the videos is identical. I walk through the process of typing out 2D input into a worksheet. This choice of presentation allows the user to engage in active learning. Each video is accompanied by a PDF of a Maple document akin to what one would find in a textbook.
Because of YouTube’s extensive array of tools to analyze the channel and individual videos, I know the content from the channel is most frequently viewed by people aged 18-24 or 25-34. However, people 65 and older tend to watch a larger fraction of each video. Over the lifetime of the channel, the percentage of female viewers has been under 10%. More recently, it has increased to 16%. The most popular video is on how to solve symbolic problems, followed by one on vector products and one on generating and manipulating matrices. Finally, with each video, I can see what fraction of the video and at what period of the video it was viewed. Not surprisingly, most viewers stop watching a video early on. But, should a viewer complete at least a minute of viewing, they tend to watch until the end. Since the channel is not a social media hit, YouTube algorithms rarely promote the videos. Consequently, the most likely way a video is discovered is through a search. Most viewers come from the United States. However, it is also viewed in Malaysia and India. I have discovered it is essential to over-tag videos to increase the likelihood that a search recommends them.
The channel is at: YouTube.com/@MapleProf. Or find information at: gould.prof/learning-maple.
Almir Aniyarov (Astana International University)
Methods of solving differential equations
Language is used as a means of describing phenomena and processes in the environment. In order to observe phenomena and processes from the outside and understand them from a scientific point of view, there is a need for their mathematical modeling. This involves representing them with special symbols and their arithmetic combinations. This branch of mathematics is called differential equations. Differential equations have an incredible ability to predict the world around us. They are used in biology, physics, economics, and various other fields. Mathematical models make it easier to predict the results of experiments carried out in real systems and allow us to study the phenomenon as a whole, predicting its development and the changes that will occur over time.
In this article, as a concrete example, we consider the solution of second-order differential equations with a variable coefficient by the variational method and the solution of the Green’s function by the method of construction..
The availability of tools like Maple enables us to introduce an experimental approach into the research we do in mathematics, when appropriate. In this talk we present four concrete examples where the use of Maple has been essential to discover, conjecture, and in some cases, demonstrate new properties of the mathematical objects that we were considering - mostly polynomials and matrices.
In the first example we will show how the use of Maple allowed us to provide the first non-trivial results on the Casas-Alvero Conjecture, which asks if every polynomial that has a root in common with each of its derivatives (not always the same root) is necessarily a power of a linear factor. The second example will show how useful is Maple to determine the structure of the polynomials that describe the projection of the intersection of a torus and a quadric.
The third example will illustrate how Maple helps generate and characterize correlation matrices when their entries are -1, 0 and 1. The last example will be devoted to introducing the use of Maple for trying to prove (or disprove) a conjecture about the spread of a symmetric matrix (i.e. the maximum absolute value of the difference between any two eigenvalues) with entries in the closed interval [a,b].
Dr. Laureano González Vega is the Director of the Department of Quantitative Methods at CUNEF University and Professor of Algebra at University of Cantabria (on leave). He is one of the co-founders of the EACA conferences (Meetings on Computational Algebra and its Applications), and the editor of The Computational Mathematics Column in the Gazette of the Royal Spanish Mathematical Society. His research activity is concentrated on topics related with Computer Algebra, Symbolic Computation and Computer Aided Geometric Design. A long-time Maple enthusiast, Dr. González Vega has made many important contributions to the field of matrices and polynomials in computer algebra, and been a strong advocate for the use of Maple in research promoting the experimental approach when appropriate, and in teaching to increase students’ understanding of and interest in mathematics.
Maple Transactions is an open-access journal that publishes expositions on topics of interest to the Maple community, including researchers, educators, and students. Containing both peer-reviewed research articles and general interest content, the journal is free to read, and free to publish in. In this session, you’ll explore highlights of past issues, learn about recent changes to the format as the journal grows and evolves, and get a chance to ask questions of the Editor-in-Chief.
In a world of ever-tightening budgets and ever-increasing class sizes, student retention has become both more important and more difficult than ever. How can universities and other higher education institutions leverage both existing and emerging technology to help students succeed in their math courses, so that they can, and wish to, continue their studies? In this talk, we’ll explore some new ideas Maplesoft is working on to support student success. We’d love to know what you think!
If you use Maple in your teaching, or are thinking of doing so, come share experiences, best practices, and concerns with your fellow Maple enthusiasts and Maplesoft developers.
Breakout rooms are also available for smaller group discussions. If you want to continue a discussion from one of the sessions, catch up with colleagues, or talk about a subject that isn’t in the main room, just ask and we’ll create a room for you.
Visualization: Going beyond plot and plot3d
Dave Linder - Software Architect
This session will explore some of Maple’s visualization commands to take you beyond the standard graphs you get from plot and plot3d. Examples will include parametric plotting, density plots, plotting a 3-D surface from data, and plotting implicitly defined curves and surfaces.
Better than Loops: Language Features Every Maple User Should Know
Dave Linder - Software Architect
Many operations that would require nested for- or while-loops in other languages can be done with a single command in Maple. See why these fundamental commands, such as map and zip, are incredibly useful as soon as you begin writing small scripts in Maple, no matter what your application.
Useful Interface Tips (or “Oh, I didn’t know you could do that!”)
Karishma Punwani — Director, Product Management – Academic Market
Here’s your chance to look over the shoulder of a Maple expert and learn about some small but useful features of the Maple interface that you may have never encountered, but once you know, you’ll use all the time.
Making Numeric Computations Faster
Matthew Calder - Intermediate Developer
By default, Maple assumes that you might need its symbolic power and flexibility. But if you have an application that requires crunching numbers, you can speed up your code significantly by using features in Maple designed to optimize numeric computation speed. In this session, you’ll learn how to take advantage of compiled procedures and hardware floats and integers procedures, to make your numeric problem solving faster.
The Power of Structured Types
Erik Postma - Manager, Mathematical Software Group
Learn how you can use define your own datatypes in Maple, using the structured types mechanism, to simplify your code, make your type checking more robust, and take advantage of powerful utility functions like select/remove to perform complex manipulations in a single command.
Whether you’ve written a few lines or a complex procedure, sometimes your code just doesn’t do what you want. In this training session, you will learn about tools and techniques to help you debug your Maple code, as well as some best practices that will increase the chances of your code working right the first time!
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.
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. He 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.
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.
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.
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.
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!
