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
|
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Maple is powerful, versatile, and efficient. But how did it get that way? It’s not just about adding more packages – over the years there have been key design decisions that have had a profound impact on what Maple can do and how it does it. In this talk, Dr. Laurent Bernardin, CEO and President of Maplesoft, will provide insight into some of these choices, the reasons behind them, and how they affect users of Maplesoft products today. Along the way, he’ll share his own memories of some of these turning points and the people who made them happen.
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.
Whether you have been using Maple 2023 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 2023 that you haven’t explored yet. This talk will give you a closer look at some of the improvements that our Maple Product Manager finds particularly useful or interesting. You may even get a few hints of more good things to come.
Students have to do math to learn math, but there are only so many problems with solutions in the textbook. What can you do for your students when that’s not enough? In this session, you will learn about the various tools available in Maple and other products in the Maplesoft Mathematics Suite that will tell students if they answered a problem correctly, give them new problems to try, help them understand where they went wrong, and show them how to get back on track.
Maple 2023 introduced the Canvas Example Gallery, which provides template applications that illustrate the use of a wide variety of features such as clickable plots, interactive visualizations, quizzes, examples that provide solution steps, and more. The Maple code used to create these applications can be easily viewed, modified, and copied, so you can customize them or use them as a starting point for your own work. In this session, we will show you how to leverage the examples in the Gallery to create engaging and enlightening interactive applications designed specifically for your own students, without having to start from scratch. In many cases, it only takes a small tweak to the code to create very different applications, so you can create interactive experiences for your class quickly and with only minimal programming experience.
Meet up with other conference attendees for conversation, quick demos, and fun activities. If there is a specific topic you wish to explore in more depth, you can ask the moderator to set up a breakout room for a smaller group discussion.
AmirHosein Sadeghimanesh (Coventry University)
Using CDCAC for SMT Inquiries with Special Constraints
In the previous year, we presented a Maple package with the implementation of a lazy SMT solver using the Conflict Driven Cylindrical Algebraic Covering (CDCAC) algorithm [1]. This is capable of verifying the satisfiability of systems of real polynomial equations and inequalities. Moreover, in cases where a real solution exists, it also provides a solution by assigning values to the variables that satisfy the input formula.
In this year’s work, we present two further implementations of the CDCAC that address specific requirements in an SMT inquiry. The first involves finding a solution to the input system that is as close as possible to a given point. This scenario arises when a numeric solution to an SMT problem is already available, and there is a need to derive an exact symbolic solution based on this numeric solution. We demonstrate the flexibility of the CDCAC structure in accommodating such requests. The second is to obtain solutions from specific components of the solution set of a given dimension, such as isolated points or points on a solution curve. This is particularly relevant in scenarios involving the motion of a robotic arm. Once again, the CDCAC algorithm’s structure proves to be capable of easily accommodating such requests. A comparison here can be drawn with a layered sub-CAD, as defined in [2].
References:
[1] E. Ábrahám, J. H. Davenport, M. England, and G. Kremer. Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings. Journal of Logical and Algebraic Methods in Programming, 119:100633, 2021. doi:10.1016/j.jlamp.2020.100633.
[2] D. J. Wilson, R. J. Bradford, J. H. Davenport, and M. England. Cylindrical algebraic sub-decompositions. Mathematics in Computer Science, 8(2):263 – 288, 2014. doi:10.1007/s11786-014-0191-z.
Brandilyn Stigler (Southern Methodist University)
Applications of Algebra, Geometry, and Combinatorics to Infer Biological Networks
Biological data science is a field replete with many substantial data sets from laboratory experiments and myriad diverse methods for modeling, simulation, and analysis. As a data set can have a large number of associated models, model selection is often required as a post-processing step. In parallel experimental design can be utilized as a preprocessing step to minimize the number of resulting models, many of which may be biologically irrelevant.
In this talk we focus on the problem of inferring polynomial models of biological networks from data. We will outline theoretical results and computational algorithms related to model construction and model selection. This work draws from algebraic geometry and algebraic combinatorics over finite fields, and has been implemented as web-based applications mostly using the computer algebra system Macaulay2. It has also been used to model a variety of biological processes including tissue development and tumor progression.
Gregory Akulov (Luther College Regina)
Arc Midpoint Computation: New Symbolic and Numeric Algorithms for Maple, CS and Math
Arc midpoint computation locates midpoint of circular arc on Cartesian plane and in Euclidean space. Its mathematical formula and computer algorithm are discovered/invented in Canada. Arc midpoint algorithm is invariant for all variables, it works with no irregularities for any length and location of the arc, and outperforms other algorithms with similar type of output. Furthermore, unlike other algorithms for locating midpoint of an arc, it allows the software implementations that output result in both symbolic and numeric forms.
Presentation discusses formula, diagram, and flow chart the arc midpoint algorithm is based on, and gives examples of code that demonstrate the work of different versions of the algorithm. It also includes selected applied and theoretic problems for students and engineers that are easier to solve using arc midpoint algorithm and harder to solve otherwise. Presentation concludes with the brief remarks about the history of the arc midpoint algorithm invention.
Tereso del Rio Armajano (Coventry University)
Explainable AI Insights for Symbolic Computation: A Case Study on Selecting the Variable Ordering for Cylindrical Algebraic Decomposition
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.
Tomas Recio (Coventry University)
Measuring with GeoGebra Discovery and Maple the Difficulty of Geometric Theorems
GeoGebra Discovery is an experimental version of GeoGebra focusing on the development of automatic proving and discovering of elementary geometry statement. For example, it has a “Discover” command that automatically finds all properties of a certain kind that hold over some element of a construction. Or the recent “StepwiseDiscovery" command, that discovers automatically all statements involving each of the new elements that the user is adding in every construction step. Or the more general command in the web version called the "Automated Geometer" that finds all the statements (quite often, hundreds of them) involving the different elements of a figure. Now, it happens that a great number of such discovered statements are just obvious, like finding that the midpoint of a segment (defined as a point with coordinates half of the sum of the coordinates of the extreme points of the segment) is equidistant from both ends of the segment!
In this environment, very prone to mechanically produce large amounts of geometric information, it is very relevant to develop instruments that allow humans to assess the relevance of the obtained results and, eventually, to filter them. In our presentation we will introduce and discuss the proposal of a specific measure of the complexity (or interest?) of a geometric statement, supported by the commutative algebra notion of “syzygy”, and we will exemplify its behavior, using Maple to perform the involved computations, through a variety of examples: from well-known theorems from the traditional school curricula, such as the theorems of Pythagoras, or the statement that declares the common intersection of the Medians of a triangle, etc. to problems from the realm of the Mathematics Olympiads.
This is joint work of Zoltán Kovács (The Private University College of Education of the Diocese of Linz, Linz, Austria) Tomás Recio and M.Pilar Vélez (University Antonio de Nebrija, Madrid, Spain).
Alexandre Cavalcante (University of Toronto)
Deepening the Connections Between Financial Literacy and Mathematics Through Maple Learn
The goal of this presentation is to share a package of Maple Learn documents created within a collaborative research project between Maplesoft and the University of Toronto. The package focuses on financial literacy, a topic which has been receiving significant attention in mathematics curricula worldwide. Yet, the connections between financial concepts and mathematics have yet to be explored in depth in research and practice. The package of documents was created to emphasize such connections beyond mere applications of formulas. Through Maple Learn, we constructed learning situations that involve a wider spectrum of mathematical ideas and representations (graphs, modelling, tables, functions, etc.). These learning situations also leverage technical functionalities available on the platform that can benefit both students and teachers. Furthermore, they reflect particular concerns with equity and social justice in financial matters. During the presentation, we will present a theoretical framework to understand financial literacy in mathematics education. We will also share examples of documents included in the package on topics such as interest, budgeting, exponential growth, credit cards. All documents are currently available on the Maple Learn gallery.
Vladimir Martinusi (Technion - Israel Institute of Technology)
A Poisson Brackets Formulation of Modern Astrodynamics
One of the most efficient formulations of Astrodynamics lies within the framework of Hamiltonian Mechanics. The main challenge of using a Poisson brackets processor in this context emanates from the implicit nature of the coordinates that are involved. This drawback can be overcome by using non-standard orbital elements that were recently introduced by the author. The formulation is already being used within the course of Astrodynamics in the Faculty of Aerospace Engineering of the Technion, with very efficient results.
Patrick Mills (Texas A&M University-Kingsville)
Incorporation of Maple into Chemical Engineering Core Graduate Courses and MS Thesis Research
Mathematical models for chemical engineering process systems are based upon macroscopic or microscopic forms of the conservation laws for total mass, fluid momentum, energy and species transport. The model equations are single or systems of linear or nonlinear ODE's, PDE's, DAE's, or algebraic equations of various degrees of complexity. Undergraduate students in chemical engineering take courses on fluid mechanics, heat transfer, mass transfer, reaction engineering, process controls, and biomolecular process systems, during their second, third and third year of study. These courses employ a building-block approach to the derivation and application of the conservation laws starting from general forms and then moving to application-specific cases. Undergraduate applications are often limited to situations where the process system model is described by a single or system of linear or nonlinear ODE’s or algebraic equations. Advanced study of the above-cited topics and others, such as rheology, materials science, sustainable process systems, pharmaceutical engineering, and energy systems, is performed in graduate-level courses as well as in graduate thesis research. The equation forms in this case are more advanced and require a strong knowledge of mathematical methods to develop solutions.
A survey of recent textbooks on the above subjects shows that the use of symbolic computation, hybrid symbolic computation with numerical analysis, or purely numerical solution approaches as supporting teaching instruments is limited. The text by White and Subramanian (2010) is the only one that focuses upon the use of Maple and how it can be utilized to solve various types of model equations that describe chemical engineering systems. A more recent text by Foley (2021) describes the use of Mathematica to solve basic chemical engineering system models. Other mathematical software for generating numerical solutions, such as Matlab, is often incorporated into textbooks as a means of generating numerical results (Tesser and Russo, 2020). However, by not using the full potential of computer-aided solution methods, the scope of coverage of the subject matter as well as student exposure and insight into approaches for solving real-world problems is notably limited. Student learning is negatively impacted as a result.
This presentation will first provide a high-level overview of the author’s approach and experience over the past decade in incorporating Maple into teaching of various graduate-level core courses in the chemical engineering curriculum, including applied mathematics, transport phenomena, chemical and catalytic reaction engineering, and process control. It is shown that one successful approach has been to use selected content from the classical course texts as the starting point, (e.g., Amundson, 1966; Bird et al., 2006; Froment and Bischoff, 1990) so that students develop knowledge on the fundamentals and the use of manual methods. Selected aspects are then repeated using Maple as the symbolic or symbolic-numerical engine, which provides students with a computer-based approach while also reinforcing the concepts. This allows parametric studies to be represented in graphical form and defines the basis for development of more advanced models by relaxing the initial assumptions, e.g., 1-D vs 2-D models.
In addition to the above, several examples will be summarized on how graduate students have used their knowledge of Maple gained in these courses as an integral component of their MS thesis research. These works collectively show that previous approaches based on either manual or numerical computations can often be replaced by a more modern approach where Maple is used to generate closed-form symbolic or numerical solutions. One important end result from an educational perspective is a notable increase in student enthusiasm for learning the subject matter and also expansion of their knowledge to solve research-level problems.
References:
Amundson, N. R., “Mathematical Methods in Chemical Engineering: Matrices and Their Application,” pp.160-164, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.
Bird, R. Byron, Stewart, W. E., and Lightfoot, E. N., “Transport Phenomena,” Revised Second Edition, John Wiley & Sons, New York, 2006.
Froment, G. F., Bischoff, K. B., and De Wilde, J, “Chemical Reactor Analysis and Design,” Vol. 2. New York: Wiley, 1990.
Tesser, R. and Russo, V., “Advanced Reactor Modeling with Matlab: Case Studies with Solved Examples,” Walter de Gruyter GmbH & Co KG, 2020.
White, R. E. and Subramanian, V. R., “Computational Methods in Chemical Engineering with Maple,” Springer Verlag, Berlin and Heidelberg, 2010.
Scot Gould (Claremont McKenna, Pitzer, Scripps)
The Maturing of Learning Introductory Undergraduate Physics Via Maple Immersion
This talk reports on the outcomes of incorporating Maple into an experimental first-semester undergraduate introductory physics for the second time, several years after the first attempt and after working with it in upper-division physics courses. The talk covers what worked and failed the first time and my attempts to improve on the integration of Maple into the course experience.
In my first attempt to introduce Maple into this course, Maple was taught during class time and heavily at the beginning of the semester. For the second attempt, the students learned the basic skillsets of Maple through a collection of ten-minute videos and associated problem sets. Each skillset was introduced at the time it was needed to solve the problems for the physics principle being studied. All presentations and all homework submissions were in the form of a Maple worksheet using the 2D input style. Maple allowed the students to 1) learn new mathematics, such as solving sets of linked differential equations, and 2) practice presenting results through publishable graphs, animations, and simple "what if" applications.
By the end of the course, the protocol to solve a standard physics problem by nearly every student was to use Maple to input and manipulate the fundamental equations. In addition, most students stated Maple allowed them a greater opportunity to concentrate on understanding physics principles while spending less time performing the mathematical minutia that typically exists within the course.
Ibrahim Larchie (Humber College)
Maple in Up-Skilling in Trades Technology
The current push by all levels of government to get as many people as possible in the Skilled Trades has necessitated the need for an efficient and cost-effective way of helping many of the candidates who require an upgrade of key basic skills in Mathematics (Numeracy, Foundation Algebra, and Trigonometry. For most of these learners, the need for a platform that offers an all-inclusive set of tools is critical for getting up to speed on these topics, if they are to succeed in their chosen field. Maple provides a comprehensive platform to make this process both convenient and affordable. This is the focus of the presentation.
Cyrille Chenavier (University of Limoges, XLIM)
Computation of Koszul Homology and Application to Involutivity of Partial Differential Systems
The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.
Kaze Atsi (Federal University Gashua)
A Treatment of a Multistep Collocation Method for the Direct Solution of Second-Order ODEs Using a Class of Modified BDF-Type
This research has been conducted at the Department of Mathematics, Faculty of Science, Federal University, Gashua, Yobe State to study multistep collocation technique for the direct solution of second order ordinary differential equations using a class of modified BDF-type with one super-future point. Using the technique as it has been adopted in many literatures, the D matrix is obtained for a step number, k=3. Following a number of manipulations, the values of the continuous coefficients of the new method are obtained from the inverse of D. The continuous form of the new method is obtained by substituting the continuous coefficients into the three-step modified BDF-type with one super-future point. The continuous form is interpolated at some points and collocated its first and second derivative at some points also, to get a system of eight equations which are used in block for the direct solution of second order ordinary differential equations.
The stability properties of the newly constructed methods are investigated using Maple software. The order, error constant, consistency and zero stability properties of the new block method are presented. Numerical efficiency of the method has been tested on some second-order initial valued problems, in order to ascertain its suitability. The solutions of the problems are compared with the corresponding exact solutions and the associated absolute errors are presented. Tables and graph have been adopted in the presentation of results. Conclusion and recommendation are made for further investigations.
Rashid Barket (Coventry University)
Machine Learning for Symbolic Integration Algorithm Selection
Within Maple, there are many different sub-algorithm choices when calculating the integral of a given expression (refer to the int/methods documentation in Maple). Selecting the wrong algorithm can result in wasted computation time and/or a more complex answer than required. Thus, there exists a need to produce a ranked list of which algorithms to try first when the int function is called. We first show how to generate data for this problem using the Risch Algorithm, which has several benefits over currently existing data generation methods. Then, several different deep learning architectures are tried to attempt to select the best sub-algorithm for a given expression to integrate. Results from the ML models are compared to Maple's meta-heuristic algorithm and show improvements on the output for several different types of expressions.
Behzad Djafari Rouhani (University of Texas at El Paso)
Algorithms for Approximating Zeros of Some Nonlinear Operators
We design an algorithm to approximate the zeros of a pseudo-monotone operator in a real Hilbert space and prove the strong convergence of the generated sequence by the algorithm to a zero of the operator. We provide also some examples of applications of our result to optimization problems, as well as some numerical experiments showing the efficacy of our algorithm.
Sarah Hamdi (University of Toronto Schools)
Assessment of Airborne Infection Risks of COVID-19 in Confined Indoor Spaces using Maple
The transmission by small airborne particles (aerosols) has been recognized as a predominant pathway for the spread of SARS-Cov-2, the virus that causes the novel coronavirus disease COVID-19. Confined indoor environments, poor ventilation, and high occupant density were major factors for the spread of the disease, early in the pandemic, in enclosed spaces such as nursing homes, hospitals, and schools.
The COVID-19 pandemic showed the need for tools for quantifying the risk associated with airborne transmission of the respiratory virus SARS-Cov-2.
We present a MAPLE interactive tool for computing airborne infection risks of COVID-19 in confined indoor spaces. The well-known Wells-Riley model equation is used for the calculation of the probability risk of infection, which is computed as a function of quanta (viruses produced by infector), number of infectors, time of exposure, room ventilation rate with clear air, room volume and other factors. This MAPLE application calculates the effects of room ventilation rates, and filtration efficiency as well as the relative risk reductions of transmission by wearing masks. We model several scenarios representing a range of ventilation rates and air filtration strategies for reducing the spread of COVID-19.
Interactive plots and visualization of the probability risk of infection for various values of the model parameters are presented using MAPLE explore command.
Yagub Aliyev (ADA University)
Digits of Powers of 2 in Odd Based Numeral System
We study the digits of the powers of 2 in the odd based number system. First we discuss an algorithm for writing doubles of numbers in 2k+1 based number system. Using this algorithm, we explain the appearance of “stairs” formed by the digits (2k)’s and 0’s when the number 2^(n+1) is written below 2^n (n=0, 1, 2, …) in a natural way so that for example the last digits are forming one column, the pre-last digits are forming another column, etc. We also look at the patterns formed by the first digits, the patterns formed by the last digits and use this to prove that the sizes of these “stairs” as blocks of 0’s and 2k’s are unbounded. We also discuss how this regularity changes when the digits move from left end of the numbers to the right end. Maple tools such as base conversion are aggressively used to detect and explain these patterns. Special case of ternary representations was featured in Maple Art and Creative Works Exhibit of Maple Conference Art Gallery in November 2-3, 2022. The topic has many interesting connections with The Euler Phi-Function, primitive roots, Benford's law, Erdős’s conjecture about the powers of 2. As an example, look at the following list created by Maple:
1
2
4
31
13
211
422
3001
1102
2204
44031
34113
143211
232032
4101101
3302202
1214404
24234131
43024313
321432311
103320232
2012014101
4024023302
3143141214
13323334231
21202224023
424044431411
304134423332
1133144022201
2212334144402
4424124334414
34043432244331
14132420044223
233104010340021
412303120141042
3341113402330301
1243221414121112
2432003333342224
43201012222400041
32012024444310033
101240434442300121
202431324440210242
4043231044414204301
3132023034433013212
1310141114422121034
21302332234003420141
42114120024101401233
303233400433023124121
111022410322141343342
2220443301003331422401
4440342212001223300412
3441140034002441210334
14432310141043434201241
23420230233032424012433
Patrick Mills (Texas A&M University-Kingsville)
Analysis of Transient Behavior in a Staged Absorber and Multiple Steady-states in a Non-isothermal One-dimensional Tubular Reactor Model Using Maple
Mathematical models for chemical reactors (Froment and Bischoff, 1990), separation process equipment (Seader et al., 2016), and other process equipment (Turton et al., 2018) have received considerable attention in the chemical engineering literature since they are employed during the various stages of process scale-up (Chaouki and Sotudeh-Gharebagh, 2021). These models sometimes assume the form of either linear or nonlinear systems of ODE’s or DAE’s where either time or a spatial coordinate is the independent variable, and temperature, concentration, and total pressure are dependent variables (Bird et al., 2006). Development of computer codes for solving these model equations is a key component of process design, analysis and optimization packages.
Maple is particularly suited as a mathematical platform for solving the above class of equations and exploring their solution behavior due to its extensive array of symbolic and numerical tools for analysis of this class of equations. A modern approach would benefit by the use of symbolic computation or a hybrid symbolic computation – numerical approach versus a purely numerical one (Tesser and Russo, 2020). However, literature that illustrates the use of Maple in chemical kinetics and chemical engineering is very limited (Korobov and Ochkov, 2011; Mackenzie and Allen, 1998; White and Subramanian, 2010). A need clearly exists for Maple to be more broadly introduced to the chemical engineering community and for it to be utilized as a useful and powerful tool for analysis and design using model-based approaches.
This presentation will provide an overview from two specific research-level chemical engineering applications where Maple was used as an alternative to manual and purely numerical approaches. These applications are part of a broader effort to employ Maple as a tool for teaching graduate-level subjects in chemical engineering and also for its use in graduate research. In the first example, the linear system of ODE’s that describe the transient behavior of a staged countercurrent gas-liquid absorber (Amundson, 1966) are solved in closed form for an arbitrary number of stages. The use of Maple to implement Sylvester’s theorem and the Cayley-Hamilton theorem as an integral part of the solution is described. The second example identifies the multiple steady-states that occur in a one-dimensional nonlinear dispersion model of a tubular nonadiabatic reactor (Kubíček et al., 1979). The governing equations are two coupled, second-order nonlinear ODE’s whose dimensionless forms for the concentration and temperature profiles can exhibit slow to rapidly changing responses over particular ranges for the reactor spatial coordinate. The parametric dependence of the solution on three dimensionless numbers, namely, the Damkohler number Da, the heat-of-reaction parameter B, and the Peclet number Pe, is performed by implementing a generalized parametric mapping (GPM) technique (Kubíček and Hlaváček, 1972) in Maple. It is shown that multiple solutions exist and that the results are consistent with those based upon purely numerical approaches. Suggestions will be provided on how Maple can be more effectively utilized as a tool for modeling and analysis in the modern practice of chemical engineering.
References:
Amundson, N. R., Mathematical Methods in Chemical Engineering: Matrices and Their Application, pp.160-164, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.
Bird, R. Byron, Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, Revised Second Edition, John Wiley & Sons, New York, 2006.
Chaouki, Jamal, and Sotudeh-Gharebagh, R. (editors), Scale-Up Processes: Iterative Methods for the Chemical, Mineral and Biological Industries. Walter de Gruyter GmbH & Co KG, 2021.
Froment, G. F., Bischoff, K. B., and De Wilde, J, Chemical Reactor Analysis and Design. Vol. 2. New York: Wiley, 1990.
Korobov, V. and Ochkov, V., Chemical Kinetics with Mathcad and Maple. Springer Science & Business Media, 2011.
Kubíček, M. and Hlaváček, V., Solution of Nonlinear Boundary value Problems—Va and Vb. A Novel Method: General Parameter Mapping (GPM) and Predictor-Corrector GPM Method, Chem. Eng. Sci., 27, pp. 743-750, 1972.
Kubíček, M., Hofmann, H., and Hlaváček, V., Modeling of Chemical Reactors – XXXII: Non-isothermal Nonadiabatic Tubular Reactor. One Dimensional Model-Detailed Analysis, Chem. Eng. Sci., 34, pp. 595-600, 1979.
Mackenzie, J. G. and Allen, M., Mathematical Power Tools: Maple, Mathematica, MATLAB, and Excel. Chemical Engineering Education 32(2), pp. 156-160, 1998.
Maplesoft, https://www.maplesoft.com/books/index.aspx, June 28, 2023.
Seader, J. D., Henley, E. J. and Roper, D. K., Separation process principles: With applications using process simulators. John Wiley & Sons, 2016.
Tesser, R. and Russo, V., Advanced Reactor Modeling with Matlab: Case Studies with Solved Examples. Walter de Gruyter GmbH & Co KG, 2020.
Turton, R., Shaeiwitz, J. A., Bhattacharyya, D., and Whiting, W. B., Analysis, Synthesis, and Design of Chemical Processes, 5th Edition, Pearson Education, 2018.
White, R. E. and Subramanian, V. R., Computational Methods in Chemical Engineering with Maple, Springer Verlag, Berlin and Heidelberg, 2010.
Samir Hamdi (University of Toronto)
Travelling Wave Solutions for FitzHugh-Nagumo Model Equations for Nerve Conduction in an Excitable Reaction-Diffusion System using Maple
We study 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 a 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. Analytical relations between the propagating pulse celerity and width are obtained. Interactive plots and animations are presented for different solitary wave profiles for various values of the diffusion coefficient and model parameters using Maple explore command. Method of lines simulations of travelling wave solutions are performed for FitzHugh-Nagumo coupled reaction-diffusion system of equations.
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:
Louis Roussel (CRIStAL, Université de Lille)
Parameter Estimation Using Integral Equations
Many models, such as the SIR (Susceptible, Infectious, Recovered) epidemiology model, consist of non-linear differential equations, involving unknown numerical parameters (e.g rate of propagation of a virus).
Under conditions, it is possible to compute Input/Output (I/O) differential equations involving only the inputs (known quantity), the output (known measured quantity with noise) and the parameters. Those I/O equations can then be used to retrieve the values of the parameters from experimental curves.
Instead of using differential equations, we consider integral equations. Indeed, in some examples, the introduction of integral equations increases the expressiveness of the models, improves the estimation of parameter values from error-prone measurements and reduces the size of the intermediate equations.
Differential I/O equations can be automatically computed using differential elimination, with the DifferentialAlgebra in Maple. Integral I/O equations are more difficult to obtain. For the moment, we either integrate differential I/O equations (which sometimes require the computation of integrating factors) or perform an ad-hoc "integral" elimination.
In my talk, I will first describe the computation of integral I/O equations using academic examples and present some partial results of the computation of integrating factors using deep learning. Then I will compare I/O equations of an academic example in terms of the quality of parameter estimation.
Samuel Feldman, Meagan James and Daniel Quan (University of Western Ontario)
The Matroids and Hypergraphs Packages in Maple
Matroids are mathematical objects whose construction abstracts the concept of independence. They were invented by Hassler Whitney in his paper "On the abstract properties of linear dependence". A matroid may, indeed, codify the linear dependencies of a finite collection of vectors within a vector space. But it may encode other types of dependencies as well, like the dependent subsets of edges in a graph, or the intersections amongst a collection of hyperplanes. Matroids show up whenever a notion of independence is present. Algebraic dependence is another example. Matroid theory has been applied to a large variety of subjects ranging from combinatorial optimization to mechanical engineering through coding theory and electricity networks. This motivated us to implement matroids in the environment that Maple provides with its unique collection of software libraries dedicated to science and engineering.
The versatility of matroids creates a number of challenges towards their software implementation. Not only matroids may arise from diverse mathematical entities (graphs, matrices, polynomial ideals, groups, etc.), but also, matroids can be defined using several distinct (yet equivalent) characterizations. These characterizations are stated in terms of hypergraphs satisfying certain axioms: bases, circuits, and hyperplanes are some examples. Moreover, some matroids can only arise from these axiomatic hypergraph definitions. All These facts motivated us in supporting our Matroids package with a Maple package implementing Hypergraphs. Hypergraph theory is a natural generalization of graph theory where (hyper)edges can be any non-empty subset of the set of vertices. Hypergraph operations supporting algorithms on matroids generally require a number of bit operations which is exponential in the number vertices. This is yet another implementation challenge.
In the proposed presentation, we would explain how our implementation of the Matroids and Hypergraphs packages in Maple address the above mentioned challenges. We would also demonstrate these packages in action with problems from applications, namely combinatorial optimization and electricity networks.
Venkat Subramanian (University of Texas at Austin)
A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple
In this talk, efficient numerical methods are implemented in Maple to solve index-1 nonlinear Differential Equations (DAEs) and stiff systems of Ordinary Differential Equations (ODEs). Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs). Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and the analytic Jacobian automatically. The algorithm and implementation were found to be robust and efficient for index-1 DAE problems and scales well for finite difference/finite element discretization of two-dimensional models with system size up to 100,000 nonlinear DAEs and solves the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs which is one of the commonly used sparse DAE solvers in the industry.
A Maple Transactions paper has been submitted. Relevant codes and discussion can be found at https://sites.utexas.edu/maple/sparsedaesolver/. MaplePrimes discussion at https://www.mapleprimes.com/questions/235335-Will-Anyone-Be-Able-To-Speed-Up-This
Stephen Watt (University of Waterloo)
An Implementation of Generic Skew Polynomials
Skew polynomials are a generalization of polynomials where variables do not commute with coefficients. They may be used to represent differential, difference and shift operators and their q-generalizations as well as other algebraic systems. Skew polynomials have been studied by various authors, notably including Ore, for almost a century. The have become a standard component of computer algebra systems and Maple provides the Ore_Algebra package for their use.
We are interested in the case of skew polynomials where the coefficients lie in a non-commutative ring, for example differential operators with matrix coefficients. For this purpose we have implemented a generic version of skew polynomials over arbitrary rings using the Maple Domains package. We present this SkewPolynomial package, the algorithms used and the lessons learned in its development.
Examples are given of differential and difference operators with coefficients in various rings.
Marcin Kamiński (Politechnika Łódzka)
Integrating Probabilistic Entropy in the System Maple
The main issue of this presentation would be to demonstrate an application coded in the system Maple, whose main objective is to demonstrate analytical and numerical integration of more popular probabilistic entropies. These entropies formulas are based upon different algebraic combinations of more popular probability density functions like Gaussian, Weibull, or Gumbel, but no solid mathematical background can be found in the literature. The system Maple uses an 'Entropy' internal function to evaluate some data strings and also some graphs, but this function has no strict connection with probability theory and its integrals. Therefore, the algorithm proposed, based on analytical integrals and also on the Monte-Carlo simulation scheme and histograms processing would enable to determine numerically and parametrically probabilistic entropies in several cases, where the use has or may generate some random inputs. Although the main area of interest is civil engineering, this application may be useful for any statistical and probabilistic studies with large populations with unknown uncertainty propagation, and particularly in fluid mechanics, aeronautical and mechanical engineering as well as in financial mathematics.
Camille Pinto (Inria & Sorbonne Université)
Towards an Effective Integro-Differential Elimination Theory with Maple
Algebraic analysis is a mathematical theory which studies linear systems of ordinary or partial differential equations using module theory, homological algebra, sheaf theory, etc. Within this approach, a linear differential system yields a finitely presented left module over a non-commutative polynomial ring of ordinary or partial differential operators. The solvability of inhomogeneous linear partial differential systems, the structural properties, and equivalence problems of linear partial systems can then be intrinsically reformulated and studied within algebraic analysis.
Our goal is to extend the above approach to handle linear systems of ordinary integro-differential equations by hopefully developing an "effective integro-differential elimination theory". In this context, the ring of ordinary integro-differential operators is a non-commutative polynomial ring not only containing ordinary differential operators but also integral operators defined as compositions of an indefinite integral operator, and evaluation operators defined by compositions of differential operators with an evaluation at a fixed point (the lower bound of the integral operator).
In the case of polynomial coefficients, the corresponding ring is the so-called I_1(k) algebra that has recently been studied by Bavula (see Bavula 2013). In particular, he proved that I_1(k) was not noetherian but a coherent ring. That means that all finitely generated sub-modules of I_1(k) are finitely presented, namely, the modules of relations among finite sets of generators are always finitely generated.
Yet Bavula gave a theorical proof of the coherence property of I_1(k), we would like to develop an effective version and an implementation in Maple. To do that, there are two conditions that must be made effective: the first one concerns the fact that the annihilators of elements of I_1(k) are finitely generated. This point has recently been made algorithmic (see Cluzeau, Pinto, Quadrat, 2023 & Quadrat, Regensburger, 2021); the second condition concerns the fact that the intersection of two finitely generated ideals of I_1(k) is finitely generated. The goal of the talk is to explain recent progress towards an algorithmic version of this last result. Finally, our study uses the OreModules and IntDiffOp Maple packages to handle effective computations.
Athanasios Tzemos (Research Center for Astronomy and Applied Mathematics of the Academy of Athens)
Approximate Integrals of Motion in Time Periodic Hamiltonian Systems: A Study with Maple
Hamiltonian Mechanics is the backbone of modern Theoretical Physics and describes many physical systems at all scales, from atoms up to galaxies. Hamiltonian systems are isolated and conservative, i.e. their energy is conserved in time. However, their dynamics may be also subject to other complete integrals of motion (e.g. conservation of angular momentum etc.) as well as secondary approximate integrals of motion (the so called `formal integrals’), which are usually written as power series of perturbation parameter.
Hamiltonian dynamics is governed by the so called Hamilton’s equations of motion which lead, in general, to the appearance of both ordered and chaotic trajectories in the phase space of a given system. Hamiltonian chaos is one of the most important and well studied fields of Mathematical Physics.
In the present talk we are going to present a computational algorithm in Maple for the construction of approximate integrals of motion in the case of time periodic Hamiltonian systems. We will show the corresponding stroboscopic Poincare surfaces of section and discuss both chaos and order through the prism of our approximate integrals. Finally, we will give examples of their applications in Classical Mechanics, Astronomy and Quantum Mechanics.
Selden Crary
Experimental NuMath/Physics Using Maple
NuMath/Physics is a research programme exploring the remarkable mathematics and applications of finite-sized, totally-positive*, covariance matrices. We commence by applying NuMath/Physics to optimal design and analysis of computer experiments (DACE) and then show how the lessons learned in this DACE application provide a framework for quantum-gravity (QG).
DACE: We focus on the following simple-to-complex uses of Maple in our exploration of this maturing, applied-mathematics subject: (1) evaluation of integrals involving simple-exponential, Gaussian-exponential, and Matérn functions; (2) symbolic evaluation of determinants; (3) advanced, high-numerical-precision search techniques for finding optimal sets of input points (designs) for evaluation of the computationally-intensive, computer-experiments simulator being used; (4) identification of “miraculous” cancellations of algebraic terms that lead to optimal designs with limit-zero-separated points; (5) confirmation of optimal-design searches, via well-documented alternate optimizers; (6) identification of sequences of integers (e.g., those that may appear in the OEIS), as well as rational numbers (or ratios of integers augmented by radicals of integers, e.g., sqrt(3), sqrt(5), etc.) in the objective function; (7) summation of thousands of discrete terms in required summations; (8) identification of the objective function as a low-degree-truncated, rational, generalized function of the inputs; (9) evaluation of Nu-approximants, which are generalizations of Padé approximants; (10) identification of the topologies of the objective functions near aggregated design points (e.g., as representations of the real projective plane); (11) visualization of the mathematical objects, via 3D plots, contour plots, and rainbow plots; and (12) investigation of the dynamics of optimal designs, as functions of changes in the parameters of the underlying simulation model. QG: We demonstrate how Nu Math/Physics begins to fulfill Neil Gershenfeld’s call for physical theories based on information and computation, rather than on partial differential equations.
*N.B.: Neither “strictly totally positive” nor “positive-definite”
Arxiv: 1406.6326, 1510.01685, 1604.05278, 1704.06250, 1707.00705, & 1709.09599
Authorea: 156277706.69664177 & 156587648.87399449
Olga Dvornik (Petro Mohyla Black Sea National University)
Maple in Practicum on Digital Signal Processing for Computer Engineering Students
Authors: Gennady Chuiko, Olga Dvornik, Yevhen Darnapuk.
The Digital Signal Processing Practicum is a basic course designed for computer engineering undergraduates. The course includes nine practical workshops offering students theoretical foundations, a step-by-step guide to actions, implementation in Maple code, and analysis of the results.
The course covers a wide range of topics related to digital signal processing, including the sampling and processing of analog signals and the analysis of non-periodic datasets. The course also simulates analog and digital frequency filters, using FIR filters as an example. Furthermore, workshops dedicated to digital image processing help students gain a holistic understanding of the subject.
To help students in their projects, the lecturers have used examples from the extensive Maple help system and their research, which they have presented at the Maplesoft Application Center. All projects showcase the impressive capabilities of Maple, which have been effectively utilized in our research and education for over two decades.
As a final project, students will need to digitally process a short recording of their voice, enabling them to apply the knowledge and skills they have acquired throughout the course. This Maple Practicum provides a comprehensive learning experience for students interested in digital signal processing.
John Pais (Ladue Horton Watkins High School)
Eclectic Interactive Group Theory
These group theory notes are eclectic in several different ways. They are not organized as in the usual encyclopedic textbook, but rather in terms of certain interesting eclectic topics, including primarily: (1) the use of group actions in proving theorems, in combinatorics, and in representing symmetries of various mathematical (geometrical) structures; (2) the representation of finite unitary reflection groups as subgroups of the multiplicative group of the real quaternion algebra; (3) the proofs of simplicity of classical groups using Iwasawa criteria with a detailed discussion comparing and contrasting various versions of these criteria.
Another sense in which these notes are eclectic is that they both include complete proofs as a Maplesoft™ mathematical word processing document, and also leverage the Maplesoft™ interactive matrix algebra over the complex numbers, as a “naturally occurring” representation for group theoretic objects and computations. Furthermore, interactive code is developed to aid the reader in understanding proofs, performing calculations, and creating examples. Many complex proofs are more easily elucidated by the ability to interactively work with specific concrete vectors and matrices representing the main ideas. For example, code is created to seamlessly move back and forth between quaternion representations as four dimensional vectors and two by two matrices, and to use the latter to explicitly generate unitary reflection groups. In addition, transvection matrices, which are essential in proving the simplicity of the projective special linear group, are represented and used to elucidate several proofs and algorithms describing their important properties. These notes may be viewed as a more gentle introduction to interesting advanced topics using only “pure group theory” and sets the stage for the more advanced group representation theory via group characters and/or Lie algebras.
Philip Yasskin (Texas A&M University)
MY Math Apps Calculus - Maple Plots
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. |
We will discuss a selection of these plots and how they are made.
Bahia Si Lakhal (Université des Sciences et Technologie)
Application of Lagrangian Formalism: The Frahm Damper
Abstract:
Using the Lagrangian formalism, we aim to study the Frahm damper, called also the tuned mass damper (TMD).
1) We give the two equations of motion in matrix form.
2) We study the effect of a TMD on a simple spring–mass system, excited by a sinusoidal force.
3) We study the optimization of the TMD by defining some particular parameters.
Víctor Medina (Universidad de Carabobo)
Using Maple and GRTensorIII in Relativistic Spherical Models
The use of algebraic manipulation software, applied to general relativity, to study the gravitational field equations in material distributions with spherical symmetry. The symbolic computational study has been growing rapidly as the processing and memory capacity of computer systems has increased and its use is now not only intended for numerical relativity, but also for symbolic manipulation. In addition to the fact that they become easier to use, more researchers have moved to use a wide variety of algebraic manipulators. The growing popularity of graphical platforms has made it possible to approach the problem of the simplifications of many expressions from another point of view. Here we will try to present some algebraic programming procedures, in Maple with the GRTensorIII package, to obtain and study the Einstein field equations and their Conservation Theorems for material distributions with spherical symmetry. The primary purpose is to show how useful a computer algebra system can be for application to a particular problem or to teaching. All calculations were performed using the GRTensorIII computer algebra package, which runs on top of Maple 2017, along with various routines that we have used specifically for the simplification of many of the algebraic expressions that are very common in this type of problem.
Grace Younes (Sorbonne University Abu Dhabi)
Computation of the L∞-norm of Finite-Dimensional Linear Systems Depending on Parameters
Authors: A. Quadrat(*), F. Rouillier(*), G. Younes(**)
(*) OURAGAN Project team, Sorbonne Université, Paris Université, CNRS, Inria Paris,
(**) Department of Science and Engineering, Sorbonne University Abu Dhabi
A major issue in control theory aims to design a system, called a controller, which stabilizes a given unstable system, called a plant, and then optimizes the performances of the closed-loop system obtained by adding the controller to the plant in a feedback loop. Within the frequency domain approach to linear systems, in a series of seminal papers, Zames shows that the standard concept of stability corresponds to a system defined by a transfer matrix (which defines the dynamics between the inputs and the outputs of the system) whose entries belong to the Hardy algebra H^∞ formed by the holomorphic functions in the right half complex plane which are bounded for the supremum norm. By the maximum modulus principle, it means that the H^∞-norm corresponds to the L∞-norm of the restriction of the transfer matrix to the pure imaginary axis. In the case of finite-dimensional systems, i.e., systems defined by linear ordinary differential equations, the coefficients of the corresponding transfer matrix are rational functions of a complex variable. Zames' approach was the starting point for the development of the so-called robust control theory or H^∞-control theory (see [4] and the references therein), a major achievement in control theory, which is nowadays commonly used by companies working in the area of control theory. In this approach, a property of a system is said to be robust if it is valid for all systems defined in a ball around the original system with a small radius for the H^∞-norm. This concept of robustness is crucial in control theory since it can be used to mathematically model the different uncertainties and errors naturally occurring in any mathematical model of a physical system.
The computation of the H^∞-norm of finite-dimensional systems, i.e., the L∞-norm of the restriction of a matrix with complex univariate rational function entries to the purely imaginary axis, is thus a fundamental problem in robust control theory. Unfortunately and contrary to the L^2-norm (see, e.g., [4]), it is not possible to simply characterize this L∞-norm using simple closed-form expressions.
Previous studies (see [1, 2] and the references therein) investigated this problem by reformulating it as the search for the maximum y-projection of real solutions (x,y) for a system of two polynomial equations Σ = {P =0, ∂P/∂x = 0}, where P ∈ Z[x,y]. To solve this problem in a certified manner, standard computer algebra methods (such as RUR, Sturm-Habich sequences, and certified root isolation of univariate polynomials) were used, and three different algorithms were proposed with their complexity analysis. These results were presented at the Maple 2020 Conference [1] and published in [2].
The goal of this paper is to generalize our approach to the case of linear time-invariant systems with parameter-dependent coefficients by studying the representation of their H∞-norm.
In this contribution, we extend the approach developed in [1, 2] to address the parametric case assuming that P now belongs to Z[u_1,..,u_k][x,y], the indeterminates u_1,..,u_k being the parameters. In this setting, the “maximal” y-projection of real solutions of Σ can now be viewed as a semi-algebraic function of the given parameters. We show that it can be computed as part of a “cylindrical algebraic decomposition” of the parameter's space adapted to polynomials computed with an ad-hoc projection strategy.
We finally propose an experimental part showing how this parametric version allows us to certify and complement the study of simple gyrostabilized sight models whose transfer matrices depend on a few physical parameters [3].
References:
[1] Bouzidi, Quadrat, Rouillier, and Younes, Computation of the L∞-norm of finite-dimensional linear systems, Maple Conference 2020, Waterloo, Canada.
[2] Bouzidi, Quadrat, Rouillier, Younes, G. Computation of the L∞ -norm of finite-dimensional linear systems, in Maple in Mathematics Education and Research, Communications in Computer and Information Science, vol. 1414, Springer, 2021, 119-136.
[3] Guillaume Rance. “Commande H-infini paramétrique et application aux viseurs gyrostabilisés”. PhD thesis. Université Paris-Saclay, 2018.
[4] Zhou, Doyle, Glover, Robust and Optimal Control, Prentice Hall, 1996.
Valerii Dryuma (Institute of Mathematics and Computer Science R. Moldova, Chishinau)
The Ricci-flat Metrics and Their Applications to the Navier-Stokes Equations
The examples of 14D and 6D Riemann metrics are applied to the study of properties of compressible viscous flows of liquids. The properties of geodesics in multidimensional Riemann spaces are introduced to obtain exact solutions for the Navier-Stokes system of equations. The use of symbolic calculations in Maple programs with GRTensor package are important to investigating this problem.
Albert Wang (Calallen High School)
Computation of Nash Equilibria of Two-Person m x n Games
Game theory has found its applications in many areas including economics, biological evolution, and social science. Game theory shows that mathematics can not only be used in natural science and engineering, but can also be used in economics and social science. Nash equilibrium is the key to solving a game, but the proof of the existence of Nash equilibrium is nonconstructive. Computing Nash equilibria is a fundamental computational problem in game theory. In fact, it is one of the actively investigated problems in theoretical computer science in recent times. The numeric and symbolic computation of Nash equilibrium of two-person 2 x 2 games is completely solved. In this paper, we discuss the computation of Nash equilibria of two-person m x n games with Maple, where m and n are positive integers greater than 1.
A finite two-person game in strategic form can be represented as a matrix of ordered pairs, called a bimatrix. The first component of the pair represents Player I's payoff and the second component represents Player II's payoff. The players’ mixed strategies are represented by vectors. Naturally, Matrix computations are involved when computing Nash equilibria. Maple has a powerful LinearAlgebra package, which is one of the reasons why we use Maple to compute Nash equilibria. The problem of finding Nash equilibria is closely related to nonlinear programming problems. Maple has a very powerful nonlinear programming package NLPSolve, which is the second reason we use Maple to compute Nash equilibria. We have written a Maple procedure to solve a nonlinear programming problem to find Nash equilibrium of m x n games. The input of the procedure is an m x n bimatrix. The output of the procedure gives Nash equilibrium. The output is a list, with the first entry of the list being the expected payoff for player 1 and the second entry being the expected payoff for player 2. The remaining entries of the list are player 1's mixed strategy and player 2's mixed strategy. The Maple procedure was tested with 3 examples. The test shows that the Maple procedure gives correct Nash Equilibria. Both the Maple procedure and examples are listed in this paper.
Maple Transactions is an open-access journal that publishes expositions on topics of interest to the Maple community. The journal is intended for a broad audience of researchers, educators, students, and anyone else with an interest in Maple, and includes 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 learn about the diversity of topics, content, and formats accepted by the journal, explore highlights of past issues, and get a chance to ask questions of the Editor-in-Chief.
Meet up with other conference attendees for conversation, quick demos, and fun activities. If there is a specific topic you wish to explore in more depth, you can ask the moderator to set up a breakout room for a smaller group discussion.
How many Pikachus does it take to power a lightbulb? How many calories would a Charizard consume? And what’s the probability of catching a Pokémon? Once posed, these are questions your students will want to know the answer to. In this talk Dr Tom Crawford, aka Tom Rocks Maths, will show you how you too can incorporate video games into your lessons, and how technology can help to bring the topics to life.
Dr. Tom Crawford holds the position of Early Career Teaching and Outreach Fellow in Mathematics at St Edmund Hall, University of Oxford, with a mission to share his love of math with the world. His award-winning website and associated social media profiles feature videos, podcasts, articles, and puzzles designed to make math more entertaining, exciting and enthralling for all. Crawford works with several partners including the BBC and the Numberphile YouTube channel - the largest math channel on the platform with over pi‑million subscribers.
In this presentation we will discuss a number of techniques, tips, and tricks to speed up your Maple code. These include choosing memory-efficient data structures, taking advantage of highly efficient commands such as map and seq to replace much slower for-loops, caching previous results to avoid re-computations, and more.
Come learn more about the artwork in the Maple Art Gallery and Creative Works Showcase, and meet some of the artists. Make sure you visit the Art Gallery ahead of time and vote for your favorite for the People’s Choice Award.
Please use #mapleconference when sharing on social media!
