Personal Stories

EXPERIENCE WITH MAPLESIM

by: Lenin Araujo Castillo 1425 Product: MapleSim 2018

May 20 2019

5 2

It is a very good computational tool to perform modeling and simulation using our world as a reference. You can also teach math knowing how to choose the right icons.
I recommend this software to everyone who wants to simulate objects or multibodies. In any case, knowledge of physics and mathematics, especially vector mechanics, is necessary.
Very grateful to the Maplesoft company for sharing their projects through the MapleSim gallery.

From now on all projects will be with Maple and MapleSim.

Lenin AC

Ambassador of Maple

Introducing the Maple Quantum Chemistry Toolbox

by: David Mazziotti 80 Product: Quantum Chemistry Toolbox from RDMChem

May 06 2019

7 2

Maple 2019 has a new add-on package Maple Quantum Chemistry Toolbox from RDMChem for computing the energies and properties of molecules. As a member of the team at RDMChem that developed the package, I would like to tell the story of its origins and provide a brief demonstration of the package.

Thinking about Quantum Chemistry at Harvard

The story of the Maple Quantum Chemistry Toolbox begins with my graduate studies in Chemical Physics at Harvard University in the late 1990s. Even in 1998 programs for computing the energies and properties of molecules were extremely complicated and nonintuitive. Many of the existing programs had begun in the 1970s on computers whose programs would be recorded on punchcards.

Fig. 1: Used Punchcard by Pete Birkinshaw from Manchester, UK CC BY 2.0

Even today some of these programs have remnants of their early versions such as input files that must start on the second column to account for the margin of the now non-existent punchcards. As a student, I made a bound copy of one of these manuals at a local Kinkos photocopy shop and later found myself in Harvard Yard, thinking that there must be a better way to present quantum chemistry computations. The idea for a Maple-like package for quantum chemistry was born in that moment.

At the same time I was learning about something called the two-electron reduced density matrix (2-RDM). The basic variable in quantum chemistry is the wave function which is the probability amplitude for finding each of the electrons in a molecule. Because electrons are indistinguishable with pairwise interactions, the wave function contains much more information than is needed for computing the energies and electronic properties of molecules. The energies and properties of any molecule with any number of electrons can be expressed as a function of a 2 electron matrix, the 2-RDM [1-3]. A quantum chemistry based on the 2-RDM, it was known, would have potentially significant advantages over wave function calculations in terms of accuracy and computational cost, especially for molecules far from the mean-field limit. A 2-RDM approach to quantum chemistry became the focus of my Ph.D. thesis.

Representing Many Electrons with Only Two Electrons

The idea of using the 2-RDM in quantum chemistry can be attributed to four scientists: two physicists Kodi Husimi and Joseph Mayer, a chemist Per-Olov Lowdin, and a mathematician John Coleman [1-3]. In the early 1940s Husimi first published the idea in a Japanese physics journal, but in the midst of World War II the paper was not widely disseminated in the West. In the summer of 1951 John Coleman, which attending a physics conference at Chalk River, realized that the ground-state energy of any atom or molecule could be expressed as functional of the 2-RDM, and similar ideas later occurred to Per-Olov Lowdin and Joseph Mayer who published their ideas in Physical Review in 1955. It was soon recognized that computing the ground-state energy of an atom or molecule with the 2-RDM was potentially difficult because not every two-electron density matrix corresponds to an N-electron density matrix or wave function. The search for the appropriate constraints on the 2-RDM, known as N-representability conditions, became known as the N-representability problem [1-3].

Beginning in the late 1990s and early 2000s, Carmela Valdemoro and Diego Alcoba at the Consejo Superior de Investigaciones Científicas (Madrid, Spain), Hiroshi Nakatsuji, Koji Yasuda, and Maho Nakata at Kyoto University (Kyoto, Japan), Jerome Percus and Bastiaan Braams at the Courant Institute (New York, USA), John Coleman and Robert Erdahl at Queens University (Kingston, Canada), and my research group and I at The University of Chicago (Chicago, USA) began to make significant progress in the computation of the 2-RDM without computing the many-electron wave function [1-3]. Further contributions were made by Eric Cances and Claude Le Bris at CERMICS, Ecole Nationale des Ponts et Chaussées (Marne-la-Vallée, France), Paul Ayers at McMaster University (Hamilton, Canada), and Dimitri Van Neck at the University of Ghent (Ghent, Belgium) and their research groups. By 2014 several powerful 2-RDM methods had emerged for the computation of molecules. The Army Research Office (ARO) issued a proposal call for a company to develop a modern, built-from-scratch package for quantum chemistry that would contain two newly developed 2-RDM-based methods from our group: the parametric 2-RDM method [1] and the variational 2-RDM method with a fast algorithm for solving the semidefinite program [4,5,6]. The company RDMChem LLC was founded to work with the ARO to develop such a package built around RDMs, and hence, the name of the company RDMChem was selected as a hybrid of the RDM abbreviation for Reduced Density Matrices and the Chem colloquialism for Chemistry. To achieve a really new design for an electronic structure package with access to numeric and symbolic computations as well as advanced visualizations, the team at RDMChem and I developed a partnership with Maplesoft to build something new that became the Maple Quantum Chemistry Package (or Toolbox), which was released with Maple 2019 on Pi Day.

Maple Quantum Chemistry Toolbox

The Maple Quantum Chemistry Toolbox provides a powerful, parallel platform for quantum chemistry calculations that is directly integrated into the Maple 2019 environment. It is optimized for both cutting-edge research as well as chemistry education. The Toolbox can be used from the worksheet, document, or command-line interfaces. Plus there is a Maplet interface for rapid exploration of molecules and their properties. Figure 2 shows the Maplet interface being applied to compute the ground-state energy of 1,3-dibromobenzene by density functional theory (DFT) in a 6-31g basis set.

Fig. 2: Maplet interface to the Quantum Chemistry Toolbox 2019, showing a density functional theory (DFT) calculation

After entering a name into the text box labeled Name, the user can click on: (1) the button Web to import the geometry from an online database containing more than 96 million molecules, (2) the button File to read the geometry from a standard XYZ file, or (3) the button Input to enter the geometry. As soon the geometry is entered, the Maplet displays a 3D picture of the molecule in the window on the right of the options. Dropdown menus allow the user to select the basis set, the electronic structure method, and a boolean for geometry optimization. The user can click on the Compute button to perform the computation. When the quantum computation completes, the total energy appears in the box labeled Total Energy. The dropdown menu Analyze contains a list of data tables, plots, and animations that can be selected and then displayed by clicking the Analyze button. The Maplet interface contains nearly all of the options available in the worksheet interface. The Help Pages of the Toolbox include extensive curricula and lessons that can be used in undergraduate, graduate, and even high school chemistry courses. Next we look at some sample calculations in the worksheet interface.

Reproducing an Early 2-RDM Calculation

One of the earliest variational calculations of the 2-RDM was performed in 1975 by Garrod, Mihailović, and Rosina [1-3]. They minimized the electronic ground state of the 4-electron atom beryllium as a functional of only two electrons, the 2-RDM. They imposed semidefinite constraints on the particle-particle (D), hole-hole (Q), and particle-hole (G) metric matrices. They solved the resulting optimization problem of minimizing the energy as a linear function of the 2-RDM subject to the semidefinite constraints, known as a semidefinite program, by a cutting-plane algorithm. Due to limitations of the cutting-plane algorithm and computers circa 1975, the calculation was a difficult one, likely taking a significant amount of computer time and memory.

With the Quantum Chemistry Toolbox we can use the command Variational2RDM to reproduce the calculation on a Windows laptop. First, in a Maple 2019 worksheet we load the commands of the Add-on Quantum Chemistry Toolbox:

>	with(QuantumChemistry);

(1.1)

Then we define the atom (or molecule) using a Maple list of lists that we assign to the variable atom:

>	atom := [["Be",0,0,0]];

(1.2)

We can then perform the variational 2-RDM method with the Variational2RDM command to compute the ground-state energy and properties of beryllium in a minimal basis set like the one used by Rosina and his collaborators. By default the method uses the D, Q, and G N-representability conditions and the minimal "sto-3g" basis set. The calculation, which completes in seconds, contains a wealth of information in the form of a convenient Maple table that we assign to the variable data.

>	data := Variational2RDM(atom);

table(%id = 18446744313704784158)

(1.3)

The table contains the total ground-state energy of the beryllium atom in the atomic unit of energy (hartrees)

>	data[e_tot];

(1.4)

We also have the atomic orbitals (AOs) employed in the calculation

>	data[aolabels];

Vector(5, {(1) = "0 Be 1s", (2) = "0 Be 2s", (3) = "0 Be 2px", (4) = "0 Be 2py", (5) = "0 Be 2pz"})

(1.5)

as well as the Mulliken populations of these orbitals

>	data[populations];

Vector(5, {(1) = 1.9995807710723152, (2) = 1.7913484714571852, (3) = 0.6969023822632789e-1, (4) = 0.6969026475511847e-1, (5) = 0.6969029119010149e-1})

(1.6)

We see that 2 electrons are located in the 1s orbital, 1.8 electrons in the 2s orbital, and about 0.2 electrons in the 2p orbitals. By default the calculation also returns the 1-RDM

>	data[rdm1];

(1.7)

The eigenvalues of the 1-RDM are the natural orbital occupations

>	LinearAlgebra:-Eigenvalues(data[rdm1]);

Vector(5, {(1) = 1.9999941387490443+0.*I, (2) = 1.7909351037804568+0.*I, (3) = 0.6969023822632789e-1+0.*I, (4) = 0.6969026475511847e-1+0.*I, (5) = 0.6969029119010149e-1+0.*I})

(1.8)

We can display the density of the 2s-like 2^nd natural orbital using the DensityPlot3D command providing the atom, the data, and the orbitalindex keyword

>	DensityPlot3D(atom,data,orbitalindex=2);

Similarly, using the DensityPlot3D command, we can readily display the 2p-like 3^rd natural orbital

>	DensityPlot3D(atom,data,orbitalindex=3);

By using Maple keyword arguments in the Variational2RDM command, we can readily change the basis set, use point-group symmetry, add active orbitals with or without self-consistent-field, change the N-representability conditions, as well as explore many other options. Having reenacted one of the first variational 2-RDM calculations ever, let's examine a more complicated molecule.

Explosive TNT

We consider the molecule TNT that is used as an explosive. Using the command MolecularGeometry, we can import the experimental geometry of TNT from the online PubChem database.

>	mol := MolecularGeometry("TNT");

(1.9)

The command PlotMolecule generates a 3D ball-and-stick plot of the molecule

>	PlotMolecule(mol);

We perform a variational calculation of the 2-RDM of TNT in an active space of 10 electrons and 10 orbitals by setting the keyword active to the list [10,10]. The keyword casscf is set to true to optimize the active orbitals during the calculation. The keyword basis is used to set the basis set to a minimal basis set sto-3g for illustration.

>	data := Variational2RDM(mol, active=[10,10], casscf=true, basis="sto-3g");

table(%id = 18446744493271367454)

(1.10)

The ground-state energy of TNT in hartrees is

>	data[e_tot];

(1.11)

Unlike beryllium, the electric dipole moment of TNT in debyes is nonzero

>	data[dipole];

Vector(3, {(1) = .5158925019252739, (2) = -0.5985274393363119e-1, (3) = .1277528280025474})

(1.12)

We can easily visualize the dipole moment relative to the molecule's ball-and-stick model with the DipolePlot command

>	DipolePlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

The 1-RDM is returned by default

>	data[rdm1];

_rtable[18446744313709602566]

(1.13)

The natural molecular-orbital (MO) occupations are the eigenvalues of the 1-RDM

>	data[mo_occ];

_rtable[18446744313709600150]

(1.14)

All of the occupations can be viewed at once by converting the Vector to a list

>	convert(data[mo_occ], list);

(1.15)

We can visualize these occupations with the MOOccupationsPlot command

>	MOOccupationsPlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

The occupations, we observe, show significant deviations from 0 and 2, indicating that the electrons have substantial correlation beyond the mean-field (Hartree-Fock) limit. The blue lines indicate the first N/2 spatial orbitals where N is the total number of electrons while the red lines indicate the remaining spatial orbitals. We can visualize the highest "occupied" molecular orbital (58) with the DensityPlot3D command

>	DensityPlot3D(mol,data, orbitalindex=58);

Similarly, we can visualize the lowest "unoccupied" molecular orbital (59) with the DensityPlot3D command

>	DensityPlot3D(mol,data, orbitalindex=59);

>

Comparison of orbitals 58 and 59 reveals an increase in the number of nodes (changes in the phase of the orbitals denoted by green and purple), which reflects an increase in the energy of the orbital.

Looking Ahead

The Maple Quantum Chemistry Toolbox 2019, an new Add-on for Maple 2019 from RDMChem, provides a easy-to-use, research-grade environment for the computation of the energies and properties of atoms and molecules. In this blog we discussed its origins in graduate research at Harvard, its reproduction of an early 2-RDM calculation of beryllium, and its application to the explosive molecule TNT. We have illustrated only some of the many features and electronic structure methods of the Maple Quantum Chemistry package. There is much more chemistry and physics to explore. Enjoy!

Selected References

[1] D. A. Mazziotti, Chem. Rev. 112, 244 (2012). "Two-electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics"

[2] Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (Adv. Chem. Phys.) ; D. A. Mazziotti, Ed.; Wiley: New York, 2007; Vol. 134.

[3] A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag, New York, 2000).

[4] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). "Large-scale Semidefinite Programming for Many-electron Quantum Mechanics"

[5] A. W. Schlimgen, C. W. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 7, 627-631 (2016). "Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex"

[6] J. M. Montgomery and D. A. Mazziotti, J. Phys. Chem. A 122, 4988-4996 (2018). "Strong Electron Correlation in Nitrogenase Cofactor, FeMoco"

Download QCT2019_PrimesV17_05.05.19.mw

Experiences introducing Maple in a high school class ...

by: nielsle 20

April 17 2019

4 4

I am a highschool teacher and we just started using Maple. It is a great program, but the students had a few problems with the worksheet-mode in the editor. Here is a list of small problems that are hopefully easy to fix.

In text editors such as Word you can highlight a piece of text and select "insert->equation" to turn the text into an equation. Maple doesn't supprt this, so if the students have written an equation in textmode, then they have trouble turning it into math-mode. (the answer is: highlight, ctrl-X, F5, ctrl-v but that is difficult to guess for beginners)

Once in a while the students place two equations next to eachother with no space in between. When this happens they cannot place the cursor between the two eqations, and therefore they cannot place a newline between the equations. In other words the two equations are glued together. (I think that you can solve this by pressing f5 in the right place, but it would be better to mirror the behavior of word)

It would be great to allow the user to delete the pink error messages by pressing delete. Sometimes the students use math mode to write text inside a large paragraph and the result is an error message a few lines below. This error message cannot be deleted unless you trace down the math field and delete it. In one situation a student had deleted all the letters in the math field, but the error message could not be deleted until the empty mathfield had been found and deleted. (One solution is to highlight all math fields on the page whenever the user is editing text in math mode. That would make them easier to find and understand)

If a student pressed enter or alt-enter inside an equation in worksheet mode, then the student stays in worksheet mode after executing the equation. I would prefer to switch to text-mode after pressing enter. This would mirror the behavior of word. Sometimes the students end up writing a long paragraph because they expect Maple to mirror the behavior of word. Once the text has been written in math mode then it is difficult to convert it into textmode again (This requires highlight, ctrl-X, F5, ctrl-v)

It would be great to write an equation that isn't meant to be executed by the kernel. In other words the equation should only be for display. You can solve this by writing ":" at theb end of the equation, but that makes the math look weird. It would be nice to have a way to do this.

If mac users havent installed a printer then they cannot export to pdf. It would be great to solve this problem (or at least write a helpful error message)

All of these problems are beginner problems, but if I you want to sell Maple in highschools then I think that you can gain from focusing on making Maple approachable so that the students have an early succes-story with the program. In my class we switched to Maple from another math program, and it was a hard sell because the other math program had an easier editor.

Danish primary schools are using Maple to revoluti...

by: sshamma 40 Product: Maple

March 26 2019

8 1

I recently had a wonderful and valuable opportunity to meet with some primary school students and teachers at Holbaek by Skole in Denmark to discuss the use of technology in the classroom. The Danish education system has long been an advocate of using technology and digital learning solutions to augment learning for its students. One of the technology solutions they are using is Maple, Maplesoft’s comprehensive mathematics software tool designed to meet the unique and complex needs of STEM courses. It is rare to find Maple being used at the primary school level, so it was fascinating to see first-hand how Maple is being incorporated at the school.

In speaking with some of the students, I asked them what their education was like before Maple was incorporated into their course. They told me that before they had access to Maple, the teacher would put an example problem on the whiteboard and they would have to take notes and work through the solution in their notebooks. They definitely prefer the way the course is taught using Maple. They love the fact that they have a tool that let them work through the solution and provide context to the answer, as opposed to just giving them the solution. It forces them to think about how to solve the problem. The students expressed to me that Maple has transformed their learning and they cannot imagine going back to taking lectures using a whiteboard and notebook.

Here, I am speaking with some students about how they have adapted Maple to meet their needs ... and about football. Their team had just won 12-1.

Mathematics courses, and on a broader level, STEM courses, deal with a lot of complex materials and can be incredibly challenging. If we are able to start laying the groundwork for competency and understanding at a younger age, students will be better positioned for not only higher education, but their careers as well. This creates the potential for stronger ideas and greater innovation, which has far-reaching benefits for society as a whole.

Jesper Estrup and Gitte Christiansen, two passionate primary school teachers, were responsible for introducing Maple at Holbaek by Skole. It was a pleasure to meet with them and discuss their vision for improving mathematics education at the school. They wanted to provide their students experience with a technology tool so they would be better equipped to handle learning in the future. With the use of Maple, the students achieved the highest grades in their school system. As a result of this success, Jesper and Gitte decided to develop primary school level content for a learning package to further enhance the way their students learn and understand mathematics, and to benefit other institutions seeking to do the same. Their efforts resulted in the development of Maple-Skole, a new educational tool, based on Maple, that supports mathematics teaching for primary schools in Denmark.

Maplesoft has a long-standing relationship with the Danish education system. Maple is already used in high schools throughout Denmark, supported by the Maple Gym package. This package is an add-on to Maple that contains a number of routines to make working with Maple more convenient within various topics. These routines are made available to students and teachers with a single command that simplifies learning. Maple-Skole is the next step in the country’s vision of utilizing technology tools to enhance learning for its students. And having the opportunity to work with one tool all the way through their schooling will provide even greater benefit to students.

(L-R) Henrik and Carolyn from Maplesoft meeting with Jesper and Gitte from Holbaek by Skole

It helps foster greater knowledge and competency in primary school students by developing a passion for mathematics early on. This is a big step and one that we hope will revolutionize mathematics education in the country. It is exciting to see both the great potential for the Maple-Skole package and the fact that young students are already embracing Maple in such a positive way.

For us at Maplesoft, this exciting new package provides a great opportunity to not only improve upon our relationships with educational institutions in Denmark, but also to be a part of something significant, enhancing the way students learn mathematics. We strongly believe in the benefits of Maple-Skole, which is why it will be offered to schools at no charge until July 2020. I truly believe this new tool has the potential to revolutionize mathematics education at a young age, which will make them better prepared as they move forward in their education.

Announcing Maple 2019

by: Samir Khan 1119 Product: Maple 2019

March 14 2019

14 8

Last year, I read a fascinating paper that presented evidence of an exoplanet, inferred through the “wobble” (or radial velocity) of the star it orbits, HD 3651. A periodogram of the radial velocity revealed the orbital period of the exoplanet – about 62.2 days.

I found the experimental data and attempted to reproduce the periodogram. However, the data was irregularly sampled, as is most astronomical data. This meant I couldn’t use the standard Fourier-based tools from the signal processing package.

I started hunting for the techniques used in the spectral analysis of irregularly sampled data, and found that the Lomb Scargle approach was often used for astronomical data. I threw together some simple prototype code and successfully reproduced the periodogram in the paper.

After some (not so) gentle prodding, Erik Postma’s team wrote their own, far faster and far more robust, implementation.

This new functionality makes its debut in Maple 2019 (and the final worksheet is here.)

From a simple germ of an idea, to a finished, robust, fully documented product that we can put in front of our users – that, for me, is incredibly satisfying.

That’s a minor story about a niche I’m interested in, but these stories are repeated time and time again. Ideas spring from users and from those that work at Maplesoft. They’re filtered to a manageable set that we can work on. Some projects reach completion in under a year, while other, more ambitious, projects take longer.

The result is software developed by passionate people invested in their work, and used by passionate people in universities, industry and at home.

We always pack a lot into each release. Maple 2019 contains improvements for the most commonly used Maple functions that nearly everyone uses – such as solve, simplify and int – as well features that target specific groups (such as those that share my interest in signal processing!)

I’d like to to highlight a few new of the new features that I find particularly impressive, or have just caught my eye because they’re cool.

Of course, this is only a small selection of the shiny new stuff – everything is described in detail on the Maplesoft website.

Edgardo, research fellow at Maplesoft, recently sent me a recent independent comparison of Maple’s PDE solver versus those in Mathematica (in case you’re not aware, he’s the senior developer for that function). He was excited – this test suite demonstrated that Maple was far ahead of its closest competitor, both in the number of PDEs solved, and the time taken to return those solutions.

He’s spent another release cycle working on pdsolve – it’s now more powerful than before. Here’s a PDE that Maple now successfully solves.

Maplesoft tracks visits to our online help pages - simplify is well-inside the top-ten most visited pages. It’s one of those core functions that nearly everyone uses.

For this release, R&D has made many improvements to simplify. For example, Maple 2019 better simplifies expressions that contain powers, exponentials and trig functions.

Everyone who touches Maple uses the same programming language. You could be an engineer that’s batch processing some data, or a mathematical researcher prototyping a new algorithm – everyone codes in the same language.

Maple now supports C-style increment, decrement, and assignment operators, giving you more concise code.

We’ve made a number of improvements to the interface, including a redesigned start page. My favorite is the display of large data structures (or rtables).

You now see the header (that is, the top-left) of the data structure.

For an audio file, you see useful information about its contents.

I enjoy creating new and different types of visualizations using Maple's sandbox of flexible plots and plotting primitives.

Here’s a new feature that I’ll use regularly: given a name (and optionally a modifier), polygonbyname draws a variety of shapes.

In other breaking news, I now know what a Reuleaux hexagon looks like.

Since I can’t resist talking about another signal processing feature, FindPeakPoints locates the local peaks or valleys of a 1D data set. Several options let you filter out spurious peaks or valleys

I’ve used this new function to find the fundamental frequencies and harmonics of a violin note from its periodogram.

Speaking of passionate developers who are devoted to their work, Edgardo has written a new e-book that teaches you how to use tensor computations using Physics. You get this e-book when you install Maple 2019.

The new LeastTrimmedSquares command fits data to an equation while not being signficantly influenced by outliers.

In this example, we:

Artifically generate a noisy data set with a few outliers, but with the underlying trend Y =5 X + 50
Fit straight lines using CurveFitting:-LeastSquares and Statistics:-LeastTrimmedSquares

LeastTrimmedSquares function correctly predicts the underlying trend.

We try to make every release faster and more efficient. We sometimes target key changes in the core infrastructure that benefit all users (such as the parallel garbage collector in Maple 17). Other times, we focus on specific functions.

For this release, I’m particularly impressed by this improved benchmark for factor, in which we’re factoring a sparse multivariate polynomial.

On my laptop, Maple 2018 takes 4.2 seconds to compute and consumes 0.92 GiB of memory.

Maple 2019 takes a mere 0.27 seconds, and only needs 45 MiB of memory!

I’m a visualization nut, and I always get a vicarious thrill when I see a shiny new plot, or a well-presented application.

I was immediately drawn to this new Maple 2019 app – it illustrates the transition between day and night on a world map. You can even change the projection used to generate the map. Shiny!

So that’s my pick of the top new features in Maple 2019. Everyone here at Maplesoft would love to hear your comments!

How MapleSim Helped Canada’s Paralympic Curling...

by: John McPhee 25 Products: Maple MapleSim

January 23 2019

5 0

Recently, my research team at the University of Waterloo was approached by Mark Ideson, the skip for the Canadian Paralympic men’s curling team, about developing a curling end-effector, a device to give wheelchair curlers greater control over their shots. A gold medalist and multi-medal winner at the Paralympics, Mark has a passion to see wheelchair curling performance improve and entrusted us to assist him in this objective. We previously worked with Mark and his team on a research project to model the wheelchair curling shot and help optimize their performance on the ice. The end-effector project was the next step in our partnership.

The use of technology in the sports world is increasing rapidly, allowing us to better understand athletic performance. We are able to gather new types of data that, when coupled with advanced engineering tools, allow us to perform more in-depth analysis of the human body as it pertains to specific movements and tasks. As a result, we can refine motions and improve equipment to help athletes maximize their abilities and performance. As a professor of Systems Design Engineering at the University of Waterloo, I have overseen several studies on the motor function of Paralympic athletes. My team focuses on modelling the interactions between athletes and their equipment to maximize athletic performance, and we rely heavily on Maple and MapleSim in our research and project development.

The end-effector project was led by my UW students Borna Ghannadi and Conor Jansen. The objective was to design a device that attaches to the end of the curler’s stick and provides greater command over the stone by pulling it back prior to release. Our team modeled the end effector in Maple and built an initial prototype, which has undergone several trials and adjustments since then. The device is now on its 7^th iteration, which we felt appropriate to name the Mark 7, in recognition of Mark’s inspiration for the project. The device has been a challenge, but we have steadily made improvements with Mark’s input and it is close to being a finished product.

Currently, wheelchair curlers use a device that keeps the stone static before it’s thrown. Having the ability to pull back on the stone and break the friction prior to release will provide great benefit to the curlers. As a curler, if you can only push forward and the ice conditions aren’t perfect, you’re throwing at a different speed every time. If you can pull the stone back and then go forward, you’ve broken that friction and your shot is far more repeatable. This should make the game much more interesting.

For our team, the objective was to design a mechanism that not only allowed curlers to pull back on the stone, but also had a release option with no triggers on the curler’s hand. The device we developed screws on to the end of the curler’s stick, and is designed to rest firmly on the curling handle. Once the curler selects their shot, they can position the stone accordingly, slide the stone backward and then forward, and watch the device gently separate from the stone.

For our research, the increased speed and accuracy of MapleSim’s multibody dynamic simulations, made possible by the underlying symbolic modelling engine, Maple, allowed us to spend more time on system design and optimization. MapleSim combines principles of mechanics with linear graph theory to produce unified representations of the system topology and modelling coordinates. The system equations are automatically generated symbolically, which enables us to view and share the equations prior to a numerical solution of the highly-optimized simulation code.

The Mark 7 is an invention that could have significant ramifications in the curling world. Shooting accuracy across wheelchair curling is currently around 60-62%, and if new technology like the Mark 7 is adopted, that number could grow to 70 or 75%. Improved accuracy will make the game more enjoyable and competitive. Having the ability to pull back on the stone prior to release will eliminate some instability for the curlers, which can help level the playing field for everyone involved. Given the work we have been doing with Mark’s team on performance improvements, it was extremely satisfying for us to see them win the bronze medal in South Korea. We hope that our research and partnership with the team can produce gold medals in the years to come.

Improve Your Chess Rating with Math

by: Joe Riel 7667

January 05 2019

9 4

Over the holidays I reconnected with an old friend and occasional
chess partner who, upon hearing I was getting soundly thrashed by run
of the mill engines, recommended checking out the ChessTempo site. It
has online tools for training your chess tactics. As you attempt to
solve chess problems your rating is computed depending on how well you
do. The chess problems, too, are rated and adjusted as visitors
attempt them. This should be familar to any chess or table-tennis
player. Rather than the Elo rating system, the Glicko rating system is
used.

You have a choice of the relative difficulty of the problems.
After attempting a number of easy puzzles and seeing my rating slowly
climb, I wondered what was the most effective technique to raise my
rating (the classical blunder). Attempting higher rated problems would lower my
solving rate, but this would be compensated by a smaller loss and
larger gain. Assuming my actual playing strength is greater than my
current rating (a misconception common to us patzers), there should be a
rating that maximizes the rating gain per problem.

The following Maple module computes the expected rating change
using the Glicko system.

Glicko := module()

export DeltaRating
    ,  ExpectedDelta
    ,  Pwin
    ;

    # Return the change in rating for a loss and a win
    # for player 1 against player2
    DeltaRating := proc(r1,rd1,r2,rd2)
    local E, K, g, g2, idd, q;

        q := ln(10)/400;
        g := rd -> 1/sqrt(1 + 3*q^2*rd^2/Pi^2);
        g2 := g(rd2);
        E := 1/(1+10^(-g2*(r1-r2)/400));
        idd := q^2*(g2^2*E*(1-E));

        K := q/(1/rd1^2+idd)*g2;

        (K*(0-E), K*(1-E));

    end proc:

    # Compute the probability of a win
    # for a player with strength s1
    # vs a player with strength s2.

    Pwin := proc(s1, s2)
    local p;
        p := 10^((s1-s2)/400);
        p/(1+p);
    end proc:

    # Compute the expected rating change for
    # player with strength s1, rating r1 vs a player with true rating r2.
    # The optional rating deviations are rd1 and rd2.

    ExpectedDelta := proc(s1,r1,r2,rd1 := 35, rd2 := 35)
    local P, l, w;
        P := Pwin(s1,r2);
        (l,w) := DeltaRating(r1,rd1,r2,rd2);
        P*w + (1-P)*l;
    end proc:

end module:

Assume a player has a rating of 1500 but an actual playing strength of 1700. Compute the expected rating change for a given puzzle rating, then plot it. As expected the graph has a peak.

Ept := Glicko:-ExpectedDelta(1700,1500,r2):
plot(Ept,r2 = 1000...2000);

Compute the optimum problem rating

fsolve(diff(Ept,r2));

                     {r2 = 1599.350691}

As your rating improves, you'll want to adjust the rating of the problems (the site doesn't allow that fine tuning). Here we plot the optimum puzzle rating (r2) for a given player rating (r1), assuming the player's strength remains at 1700.

Ept := Glicko:-ExpectedDelta(1700, r1, r2):
dEpt := diff(Ept,r2):
r2vsr1 := r -> fsolve(eval(dEpt,r1=r)):
plot(r2vsr1, 1000..1680);

Here is a Maple worksheet with the code and computations.

Glicko.mw

Later

After pondering this, I realized there is a more useful way to present the results. The shape of the optimal curve is independent of the user's actual strength. Showing that is trivial, just substitute a symbolic value for the player's strength, offset the ratings from it, and verify that the result does not depend on the strength.

Ept := Glicko:-ExpectedDelta(S, S+r1, S+r2):
has(Ept, S);
                    false

Here's the general curve, shifted so the player's strength is 0, r1 and r2 are relative to that.

r2_r1 := r -> rhs(Optimization:-Maximize(eval(Ept,r1=r), r2=-500..0)[2][]):
p1 := plot(r2_r1, -500..0, 'numpoints'=30);

Compute and plot the expected points gained when playing the optimal partner and your rating is r-points higher than your strength.

EptMax := r -> eval(Ept, [r1=r, r2=r2_r1(r)]):
plot(EptMax, -200..200, 'numpoints'=30, 'labels' = ["r","Ept"]);

When your playing strength matches your rating, the optimal opponent has a relative rating of

r2_r1(0);
                       -269.86

The expected points you win is

evalf(EptMax(0));
                       0.00956

Making Math Fun – the Teenage Way!

by: Tina 80 Product: Maple

September 14 2018

11 0

Fourteen year old Lazar Paroski is an exceptional student. Not only is he an overachiever academically, but he has a passion to help struggling students, and to think of innovative ways to help them learn. Lazar is particularly fond of Math, and in his interactions with other students, he noticed how students have a hard time with Math.

Putting on his creative cap, Lazar came up with the idea of an easily accessible “Math Wall” that explains simple math concepts for students in a very visual way.

“The Music Wall on Pinterest was my inspiration,” says Lazar. “I thought I can use the same idea for Math, and why not a Math Wall”?

"The math wall is basically all the tools you'll have on the wall in your classroom outside," said Lazar. Making the Math Wall and getting it set up, took time and effort. But he had help along the way, which, fueled by his passion and enthusiasm, helped turn his creative dream into reality. Lazar received a grant of $6000 from the local government to implement the project; his teachers, principal and family helped promote it; and the community of parents provided encouragement.

The Math Wall covers fundamental math concepts learnt in grades 1 to 6. Lazar engaged with over 450 students in the community to understand what would really be helpful for students to see in this Math Wall, and then he carefully picked the top themes he wanted to focus on.

The three meter Math Wall is located in the Morrison community park, and was officially inaugurated by the Mayor of Kitchener in July 2018. Many students have already found it to be useful and educative. Parents who bring their children to the park stop by to give their kids a quick math lesson.

At Maplesoft, we love a math story like this! And that too in our backyard! We wanted to appreciate and encourage Lazar and his efforts in making math fun and easy and accessible to students. So we invited Lazar to our offices, gifted him a copy of Maple, and heard more about his passion and future plans. “In many ways, Lazar embodies the same qualities that Maplesoft stands for – making it easy for students to understand and grasp complex STEM concepts,” said Laurent Bernardin, Maplesoft’s Chief Operating Officer. “We try to impress upon students that math matters, even in everyday life, and they can now use advanced, sophisticated technology tools to make math learning fun and efficient.”

We wish Lazar all the very best as he thinks up new and innovative ways to spread his love for math to other kids. Well done, Lazar!

Great Hall Floor and Maple

by: vv 5877 Product: Maple

September 01 2018

13 5

I have recently visited the Queen's House at Greenwich (see wiki), an important building in British architectural history (17th century).
I was impressed by the Great Hall Floor, whose central geometric decoration seems to be generated by a Maple program :-)

Here is my code for this. I hope you will like it.

restart;
with(plots): with(plottools):
n:=32: m:=3:#    n:=64: m:=7:

a[0], b[0] := exp(-Pi*I/n), exp(Pi*I/n):
c[0]:=b[0]+(a[0]-b[0])/sqrt(2)*exp(I*Pi/4):  
for k to m+1 do  
  c[k]:=a[k-1]+b[k-1]-c[k-1];
  b[k]:=c[k]*(1+exp(2*Pi*I/n))-b[k-1];
  a[k]:=conjugate(b[k])  od:
b[-1]:=c[0]*(1+exp(2*Pi*I/n))-b[0]:
a[-1]:=conjugate(b[-1]):
c[-1]:=a[-1]+b[-1]-c[0]:
seq( map[inplace](evalf@[Re,Im], w), w=[a,b,c] ):
Q:=polygonplot([seq([a[k],c[k],b[k],c[k+1]],k=0..m-1), [c[m],a[m],b[m]], [a[-1],b[-1],c[0]]]):
display(seq(rotate(Q, 2*k*Pi/n, [0,0]),k=0..n-1), disk([0,0],c[m][1]/3), axes=none, size=[800,800], color=black);

Meet Your Developers - Margaret Hinchcliffe

by: RochelleAngyal 270

July 23 2018

8 0

We’re excited to bring you another Meet Your Developers post. This one comes from Senior Developer, Margaret Hinchcliffe.

Enjoy!

What do you do at Maplesoft?
I work on the software team that develops the user interface for Maplesoft products. In my time at Maple, I’ve worked on Maple, MapleSim, and MapleNet.

What did you study in school?
I studied Computer Science at the University of Waterloo, in the Co-op program.

What area(s) of Maple or MapleSim are you currently focusing on in your development?
I’m currently working on MapleNet and on ways to bring Maple functionality to the web.

What’s the coolest feature of Maple or MapleSim that you’ve had a hand in developing?
I helped develop the feature that lets you embed videos in a Maple worksheet. I thought that was pretty cool.

What do you like most about working at Maplesoft? How long have you worked here?
I celebrated my twentieth anniversary at Maplesoft this spring. Obviously, I like working here. What I like most is the opportunity to learn new skills in a supportive environment. Our company gym is pretty awesome, too.

Favourite hobby?
I took up boxing a couple of years ago and I really enjoy it. It’s a great workout and there’s always something new to learn.

What do you like on your pizza?
Pepperoni, mushrooms, and fresh basil.

What’s your favourite movie?
The Wizard of Oz. When people say “The book is always better”, I point to this movie as a counterexample.

What skill would you love to learn? Why?
I’d like to try archery. If nothing else, it would come in handy in a zombie apocalypse.

Who’s your favourite mathematician?
Alan Turing. He made important contributions to computer science and he helped fight the Nazis.

Thanks Margaret!

What makes an expert?

by: Ian Thompson 50 Product: Maple

June 12 2018

9 5

Who should be considered an 'expert'? How does one achieve expert status? In this guest MaplePrimes blog post, 'Understanding Maple' author Ian Thompson discusses his view of what makes an expert, his journey of becoming an expert in Maple, and the process of putting together and perfecting this resource for Maple users.

In days of 8-bit computers, one would sometimes encounter individuals who knew everything about a particular device or piece of software. Single programmers wrote entire applications or games, and some could debug their work by looking directly at a core dump (a printout of the numbers stored in the computer’s memory). Some even managed to take computers beyond their specifications by exploiting design loopholes that the manufacturers hadn’t foreseen or intended. It would be fair to classify such individuals as ‘experts’.

Fast forward twenty five years, and the picture is far less clear. The complexity of computers and software has grown to such an extent that even relatively small smartphone applications are created by teams of developers, and nobody understands every aspect of a CPU chip, much less an entire PC or tablet. Who now should be classified as an expert? One possibility is that an expert is a person who may sometimes need to look up the details of a rarely used command or feature, but who is never confused or frustrated by the behavior of the system or software in question (except where there is a bug), and never needs help from anyone, except perhaps on rare occasions from its creators.

This rather stringent definition makes me an expert in only two areas of computing: the Fortran programming language, and the mathematical computation system Maple. An argument could be made for the typesetting system LATEX, but whilst this has a large number of expert users, there is also a much smaller group of more exalted experts, who maintain the system and develop new packages and extensions. It would be fair to say that I fall into the first category, but not the second.*

How does one achieve expert status? Some software actively prevents this, by hiding its workings to such an extent that fully understanding its behavior is impossible. Where it is possible to gain expert status, I have experienced two very different routes, both starting during my time as a research student, when it became clear that Fortran and Maple would be useful in my work. There were several parallels. I knew a little about both, having used them for basic tasks as an undergraduate. However, working out why things went wrong and how to fix them was time-consuming and unrewarding, since it often relied on magic recipes obtained from unreliable sources, and in many cases I didn’t really understand why these worked, any more than I understood why my own attempts had not. I realized then that knowing a little was at the root of these problems. Partial knowledge, supplemented by contradictory, outdated and even downright bad advice from websites and well-meaning individuals (some of whom invariably labor under false pretences of their own expert status) is not an efficient way to approach scientific computing. In fact it’s just a recipe for frustration. In the case of Fortran, fixing this turned out to be easy, because there are lots of good books on the subject. Reading one of these eliminated all of my problems with the language at a stroke. I can’t claim that I remembered every command and its syntax, nor do I know them all now. This is hardly surprising — the Fortran Language Standard (a very terse document that sets out everything the language provides) now extends to more than 600 pages. Instead, the book provided a general picture of how things work in Fortran, and showed the right way to go about tackling a problem. This investment in time has since paid itself back hundreds of times over.

The route to expert status in Maple was far more challenging. Its own help pages give a very comprehensive description of individual commands, but they are intended as a reference guide, and if it’s possible to become an expert using these alone, then I never discovered the correct order in which to read them. I found a number of books on Maple in the university library, but most were too basic to be useful, and others focused on particular applications. None seemed likely to give me the general picture — the feel for how things work — that would make Maple into the time-saving resource it was intended to be.

The picture became clearer after I taught Maple to students in three different courses. Nothing encourages learning better than the necessity to teach someone else! Investigating the problems that students experienced gave me new opportunities to properly understand Maple, and eventually the few remaining gaps were filled in by the Programming Guide. This is a complex document, similar in length to the Fortran Language Standard, but with more examples. Personally I would only recommend it to readers with experience of programming language specifications. Students now started to ask how I came to know so much about Maple, and whether there was a book that would teach them the same. Since no such book existed, I decided to write one myself. As the old adage goes, if you want something doing properly, do it yourself. The project soon began to evolve as I tried to set down everything that the majority of Maple users need to know. I’ve always hated books that skirt around important but difficult topics, so where before I might have used a dirty trick to circumnavigate a problem, now I felt compelled to research exactly what was going on, and to try to explain it in a simple, concise way. When the first draft was complete, I approached Cambridge University Press (CUP). The editor arranged for reviews by four anonymous referees**, and by Maplesoft’s own programming team. This led to several major improvements. My colleague, Dr Martyn Hughes, also deserves a mention for his efforts in reading and commenting on four different drafts. Meanwhile, Maplesoft continued to release new editions of their software, and the drafts had to be revised to keep up with these. The cover was created by one of CUP’s designers, with instructions that it should not look too ‘treeish’ — one might be surprised by the number of books that have been written about Maple syrup, and it would be a shame for Understanding Maple to be mixed up with these by potential readers browsing the internet. Then there were the minor details: how wide should the pages be? What font should be used? Should disk be spelled with a ‘c’ or a ‘k’? Could quotes from other sources be used without the threat of legal action over copyright infringement? One rights holder laughably tried to charge $200 for a fragment of text from one of their books. Needless to say, no greenbacks were forthcoming.

The resulting book is concise, with all the key concepts needed to gain an understanding of Maple, alongside numerous examples, packed into a mere 228 pages. It gives new users a solid introduction, and doesn’t avoid difficult topics. It isn’t perfect (in fact I have already started to list revisions that will be made if a second edition is published in the future) but I’ve seen very few problems that can’t be solved with the material it contains. Only time will tell if Understanding Maple will it create new experts. At the very least, I would certainly like to think it will make Maple far easier to grasp, and help new users to avoid some of the traps that caught me out many years ago.

Learn more about Understanding Maple, which is published by Cambridge University Press.

Induction is just a solid gamble

by: Adam Ledger 225 Product: Maple

May 24 2018

0 14

I just feel that if at least one person less experienced than me reads this it will be a worth while post, because it will help them avoid things that eluded me when I was younger.

The omitted function definitions are not relevant to the reason for which I decided to post about this. I would like the maple user to simply observe how many variables are involved in the relation's (R) three equalities in the consideration of the output.

The reason I believe this is important, is that it is sometimes very easy to believe induction is sufficient proof of the truth value of a relation over the superset of a subset that has been enumerated, much like the example of the coefficients of the
"105^(th) cyclotomic polynomial if one were to inductively reason statements about the coeffiecents of the previous 104 polynomials."