Teaching and learning about math, Maple and MapleSim

--- Prolog.ue ---

The best things in life come free of charge.

Happiness, love, and wisdom of expertise are first few that hit my mind.

As a business economist, I keep my eyes keenly open to opportunities for growth; such as Maple 2017 training session.

It was a Saturday afternoon in Waterloo, ON, this chilly Feburary which was blessed by snowstorm warning.


--- Encountering with Maple ---

I was aware of Maple for many years back when my academic career began.

In fact, Maple was available in the lab computers at university. 

But I did not know what to do with it.

Nor did I use any mathematics softwares until recently, but I had this thought : one day I could learn.

The motivation for this `learn how to use it' did not occur to me for a long time (14 years!!).

Things changed this year when I enrolled to an Electrical Engineering program at Lassonde.

Mind you, I have already been using various types of languages and tools such as: Python, C, Java, OpenOfficeSuites, Stata, SAS, Latex just to mention a few.

These stuffs also run on multiple platforms which I am sure you have heard of if you're reading this post; Windows, OSX and Linux. And Maple supports all these major operating systems.


--- Why do I like Maple ---

During the first week of school, Dr. Smith would ask us to purchase and practice using MATLAB because it had a relatively easy learning curve for beginners like python and we were going to use it for labs.

Furthermore, students get a huge discount (i.e. $1500 to $50) for these softwares.

Then, the professor also added; "Maple is also a great tool to use, but we won't use it for this class".

ME: ' Why not ? '

The curiosity inside me gave in and I decided to try both!

After all, my laziness in taking derivatives by hand or the possibility of making mistake would disappear if I can verify results using software.

That's it...!

Being able to check correct answer was already worth more than $50.

I can not emphasize this point enough; 

For people in the industry being paid for their time, or students like me who got a busy schedule can not afford to waste any time. (i.e. need to minimize homework effort & frustration, while maximizing the educational attainment & final grades)

Right? Time is money.

Don't we all just want more spare time for things we care?

Googling through many ambiguous Yahoo Answers or online forums like Stackoverflow replies are often misleading and time consuming. 

I have spent years (estimated 3000+ hours) going through those wildly inaccurate webpages hoping for some clearly written information with sub-optimal outcome.

Diverting many hours of study time is not something a first year S.T.E.M. students can afford.


--- Maple Training ---

Now you know about my relationships with Maple; Let me describe how the training session went.

I will begin with the sad news first, =(

First of all, there was no coffee available when I arrived. It arrived only after lunch.

Although it was a free event aside other best things in life, this was only a material factor I didn't enjoy at the site. 

Still a large portion of Canadians start their work with a zolt of caffeine in my defence.

Secondly, there was a kind of assumption which expected attendee were familiar with software behavior.

A handful of people were having trouble opening example file, perhaps because of their browser setting or link to preferred software by OS.

Not being able to follow the tutorials as the presenter demonstrated various facets of software substantially diminished the  efficacy of training session for those who could not be on the same page.

These minor annoyances were the only drawbacks I experinced from the event.


Here comes the happy side, =)

1. The staffs were considerate enough to provide vegetarion options for inclusive lunch as well as answering all my curious, at times orthogonal questions regarding Maplesoft company.

2. Highly respectable professionals were presenting themselves; 

That is, Prof. Illias Kotsireas, Dr. Erik Postma and Dr. Jürgen Gerhard.

I can not appreciate enough of their contribution for the training in an eloquent and humble manners.

To put it other way, leading of the presentation was well structured and planned out.

In the beginning, Prof. Kotsireas presented `Introduction to Maple' which included terminology and basic behaviors of Maple (i.e. commands and features) with simple examples you can quickly digest. Furthermore, Maple has internal function to interface with Latex! No more typing hours of $$s and many frac{}{}, \delta_{} to publish. In order for me to study all this would have been two-weeks kind of commitment in which he summarized in a couple of hours time. Short-cut keys that are often used by his project was pretty interesting, which will improve work efficiency.

After a brief lunch, which was supplied more than enough for all, Dr. Erik Postma delivered a critical component of simluation. That is, `Random Number Generation'. Again, he showed us some software-related tricks such as `Text mode' vs. `Math mode'.  The default RNG embedded in the software allows reproducible results unless we set seed and randomize further. Main part of the presentation was regarding `Optimization of solution through simulation'. He iteratively improved efficiency of test model, which I will not go in depth here. However, visually and quantitatively showing the output was engaging the attendees and Maple has modularized this process (method available for all the users!!).

Finally, we got some coffee break that allowed to me to push through all the way to the end. I believe if we had some coffee earlier less attendees would have left.

The last part of the training was presented by Dr. Jürgen Gerhard. In this part, we were using various applications of Maple in solving different types of problems. We tackled combinatorics of Fibonacci sequence by formula manipulation. In this particular example he showed us how to optimize logic of a function that made a huge impact in processing time and memory usage. Followed by graph theory example, damped harmonic oscillator, 2 DOF chaotic system, optimization and lastly proof of orthocentre by coding. I will save the examples for you to enjoy in future sessions. 

The way they went through examples were super easy to follow. This can only be done with profound understanding of the subject and a lot of prior effort in preparing the presentation.

I appreciate much efforts put together by whom organized this event, allocating their own precious weekend time and allowing many to gain opportunity to learn directly from the person in the house.


--- Epilogue ---

My hope for Maple usage lies in enhancing education outcome for first year students, especially in the field of Science and Economics. This is a free opportunity for economic empowerment which is uncaptured.

Engineering students are already pretty good at problem solving, and will figure things out as I witnessed my colleagues have.

However, students of natural sciences and B.A. programs tend to skimp on utilizing tools due to lack of exposure.

Furthermore, I am supporting their development of SaaS, software as service, which delivers modules like gRPC does.

Also, I hope the optimization package from prior version written by Dr. Postma will become available to public sometime.

Here's a BIG thank you to staffs once again, and forgive me for any grammatical errors from rushed writing. I tried to incorporate as much observation as possible gathered from the event.

To contact me, my email is hyonwoo.kee (at);


Minimize the number of tensor components according to its symmetries
(and relabel, redefine or count the number of independent tensor components)



The nice development described below is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, Université Lille 1, France, used in the Mapleprimes post Magnetic traps in cold-atom physics


A new keyword in Define  and Setup : minimizetensorcomponents, allows for automatically minimizing the number of tensor components taking into account the tensor symmetries. For example, if a tensor with two indices in a 4D spacetime is defined as antisymmetric using Define with this new keyword, the number of different tensor components will be exactly 6, and the elements of the diagonal are automatically set equal to 0. After setting this keyword to true with Setup , all subsequent definitions of tensors automatically minimize the number of components while using this keyword with Define  makes this minimization only happen with the tensors being defined in the call to Define .


Related to this new functionality, 4 new Library routines were added: MinimizeTensorComponents, NumberOfIndependentTensorComponents, RelabelTensorComponents and RedefineTensorComponents



restart; with(Physics)


Define an antisymmetric tensor with two indices

Define(F[mu, nu], antisymmetric)

`Defined objects with tensor properties`


{Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}


Although the system knows that F[mu, nu] is antisymmetric, you need to use Simplify to apply the (anti)symmetry

F[mu, nu]+F[nu, mu]

F[mu, nu]+F[nu, mu]



Simplify(F[mu, nu]+F[nu, mu])



so by default the components of F[mu, nu] do not automatically reflect the (anti)symmetry; likewise

F[1, 2]+F[2, 1]

F[1, 2]+F[2, 1]


Simplify(F[1, 2]+F[2, 1])



and computing the array form of F[mu, nu]we do not see the elements of the diagonal equal to zero nor the lower-left triangle equal to the upper-right triangle but for a different sign:

TensorArray(F[mu, nu])

Matrix(%id = 18446744078270093062)



On the other hand, this new functionality, here called minimizetensorcomponents, makes the symmetries of the tensor be explicitly reflected in its components.


There are three ways to use it. First, one can minimize the number of tensor components of a tensor previously defined. For example



Matrix(%id = 18446744078270064630)


After this, both (1.2) and (1.3) are automatically equal to 0 without having to use Simplify

F[mu, nu]+F[nu, mu]






And the output of TensorArray  in (1.6) becomes equal to (1.7).


NOTE: in addition, after using minimizetensorcomponents in the definition of a tensor, say F, all the keywords implemented for Physics tensors are available for F:



F[mu, nu] = Matrix(%id = 18446744078247910206)






F[mu, nu] = {(1, 2) = F[1, 2], (1, 3) = F[1, 3], (1, 4) = F[1, 4], (2, 1) = -F[1, 2], (2, 3) = F[2, 3], (2, 4) = F[2, 4], (3, 1) = -F[1, 3], (3, 2) = -F[2, 3], (3, 4) = F[3, 4], (4, 1) = -F[1, 4], (4, 2) = -F[2, 4], (4, 3) = -F[3, 4]}



F[`~1`, mu] = Vector[row](%id = 18446744078247885990)


Alternatively, one can define a tensor, specifying that the symmetries should be taken into account to minimize the number of its components passing the keyword minimizetensorcomponents to Define .




Define a tensor with the symmetries of the Riemann  tensor, that is, a tensor of 4 indices that is symmetric with respect to interchanging the positions of the 1st and 2nd pair of indices and antisymmetric with respect to interchanging the position of its 1st and 2nd indices, or 3rd and 4th indices, and define it minimizing the number of tensor components


Define(R[alpha, beta, mu, nu], symmetric = {[[1, 2], [3, 4]]}, antisymmetric = {[1, 2], [3, 4]}, minimizetensorcomponents)

`Defined objects with tensor properties`


{Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}


We now have

R[1, 2, 3, 4]+R[2, 1, 3, 4]



R[alpha, beta, mu, nu]-R[mu, nu, alpha, beta]



One can always retrieve the symmetry properties in the abstract notation used by the Define command using the new Library:-GetTensorSymmetryProperties, its output is ordered, first the symmetric then the antisymmetric properties



{[[1, 2], [3, 4]]}, {[1, 2], [3, 4]}


After making the symmetries explicit (and also before that), it is frequently useful to know the number of independent components of a given tensor. For this purpose you can use the new Library:-NumberOfIndependentTensorComponents





and besides taking into account the symmetries, in the case of the Riemann  tensor, after taking into account the first Bianchi identity this number of components is further reduced to 20.


A third way of using the new minimizetensorcomponents functionality is using Setup , so that, automatically, every subsequent definition of tensors with symmetries is performed minimizing the number of its components using the indicated symmetries



Setup(minimizetensorcomponents = true)

[minimizetensorcomponents = true]


So from hereafter you can define tensors taking into account their symmetries explicitly and without having to include the keyword minimizetensorcomponents at each definition


Define(C[alpha, beta], antisymmetric)

`Defined objects with tensor properties`


{C[mu, nu], Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}




C[mu, nu] = Matrix(%id = 18446744078408747598)


Two new related functionalities are provided via Library:-RelabelTensorComponents and Library:-RedefineTensorComponent, the first one to have the number of tensor components directly reflected in the names of the components, the second one to redefine only one of these components


Matrix(%id = 18446744078408729774)



Suppose now we want to make one of these components equal to 1, say C__2

Library:-RedefineTensorComponent(C[1, 2] = 1)

C[mu, nu] = Matrix(%id = 18446744078270104390)


This nice development is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France.





Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft



I study mainly subjects that fall under umbrella of number theory, but i have specified a little further in the worksheet. This is really a request for assistance, because in as much as i have met so many brilliant people online via social media etc,  I would always love to meet more, and especially ones who are more experienced in this field. 


Basically i am too cheap and old to think about going to a good university, so I am trying to get free advice from the people who have probably completed doctorates in the relevant field. Got to be honest I say.


Anyway my contact email is at the top of the attached worksheet.


First thing that stood out to me about the distributions produced in this worksheet is how sparse the number of points is for N=17 relative to all the other values of N.


Edit: Another example worksheet added.