## Computing with Thermophysical Data

by: Maple 2016

When Maple 2016 hit the road, I finally relegated my printed Mollier charts and steam tables to a filing cabinet, and moved my carefully-curated spreadsheets of refrigerant properties to a distant part of my hard drive. The new thermophysical data engine rendered those obsolete.

Other than making my desk tidier, what I find exciting is that I can compute with fluid properties in a tool that has numerical integrators, ODE solvers, optimizers, programmatic visualisation and more.

Here are several small examples that demonstrate how you can use fluid properties with Maple’s math and visualization tools (this worksheet contains the complete examples).

## Work Done in Compressing a Gas

The work done (per unit mass) in compressing a fluid at constant temperature is

where V1 and V2 are specific volumes and p is pressure.

You need a relationship between pressure and specific volume (either theoretical or experimental) to calculate the work done.

Assuming the ideal gas law, the work done becomes

where R is the ideal gas constant, T is the temperature (in K) and M is the molecular mass (in kg mol-1), and V is the volume.

Ideal gas constant

Molecular mass of propane

Hence the work done predicted by the Ideal Gas Law is

Let’s now use real fluid properties instead and numerical integrators to compute the work done.

Here, the work done predicted with the Ideal Gas Law and real fluid properties is similar. This isn’t, however, always the case for all gases (try experimenting with ammonia – its strong intermolecular forces result in non-ideal behavior).

## Minimum Specific Heat Capacity of Water

The specific heat capacity of water varies with temperature like so.

Let's find the temperature at which the specific heat capacity of water is the lowest.

The lowest specific heat capacity occurs at 309.4 K; this is the temperature at which water requires the least energy to raise or lower its temperature.

Incidentally, this isn’t that far from the standard human body temperature of 310.1 K (given that the human body is largely water, one might hazard a guess why we have evolved to maintain this temperature).

## Temperature-Entropy Plot for Water

Maple 2016 generates pressure-enthalpy-temperature charts and psychrometric charts out of the box. However, you can create your own customized thermodynamic visualizations.

This, for example, is a temperature-entropy chart for water, together with the two-phase vapor dome (the worksheet contains the code to generate this plot).

I'm also working on a lumped-parameter heat exchanger model with fluid properties (and hence heat transfer coefficients) that change with temperature. That'll be more complex than these simple examples, and will use Maple's numeric ODE solver.

## Maple T.A. 2016 is here!

by: Maple T.A.

I’m pleased to announce the release of Maple T.A. 2016, our online assessment system.

For this release, we put a lot of effort into streamlining the authoring experience. We worked closely with customers to find out how they authored content, the places where they found the interface awkward, the tasks that took longer than they should have, and what they’d like to see changed. Then we made it better.

Right away you’ll notice that questions and assignments are no longer in separate places in Maple T.A. All your content is stored in a convenient location that makes it simple to browse your content. Contextual navigation, filtering options, sorting tools, question details, drag and drop organization, combined import feature, and more make it easier than ever to find and organize your content. The Maple T.A. Cloud also sees improvements. Not only can questions be shared, but assignments and entire course modules can be as well.

For question creation, we consolidated all question authoring into the question designer, so you have a single starting point no matter what kind of question you want to create. We have also refined the text editor to help authors find the tools they need to modify their questions. This includes embedding external content, importing questions from the repository, text formatting options, and more.

Of course, once you have questions, you’ll want to put them into an assignment, and assignment creation is now easier than ever. A key change is that you can now create and modify questions while you are creating an assignment, without having to leave the assignment editor. There are also changes to how you preview questions, set properties, and even save your assignments, all of which contribute to making assignment creation simpler and faster.

Of course, there’s more than just a significantly improved author workflow. Here are some highlights:

• Assignment groups for efficient organization, both in the content repository and on the class homepage.
• Easy-to-create sorting questions – no coding required!
• HTML questions, which can be authored directly in the question designer.
• Clickable image questions are Java-free and easier to author.
• Maximum word counts and other improvements to the essay question type.
• A new scanned document feature lets instructors upload and even grade scanned documents.
• Officially certified LTI integration for connectivity with a wide range of course management systems

See What’s New in Maple T.A. 2016 for more information on these and other new features.

Jonny Zivku
Maplesoft Product Manager, Online Education Products

## Transforming Functions to Names: Further explaining...

by: Maple

A wealth of knowledge is on display in MaplePrimes as our contributors share their expertise and step up to answer others’ queries. This post picks out one such response and further elucidates the answers to the posted question. I hope these explanations appeal to those of our readers who might not be familiar with the techniques embedded in the original responses.

The Question: Transforming functions to names

Bendesarts wanted to know how to make programmatic changes to characters in a list. He wrote:

I have this list :

T:=[alpha(t),beta(t)]

I would like to create this list automatically:

Tmod:=[alpha_,beta_]

In other words, how can I remove the 3 characters "(t)" and replace it by "_"

Do you have ideas to do so ?

Thanks a lot for your help

Joe Riel provided a complete answer that had three different approaches. He wrote:

Map, over the list, a procedure that extracts the name of the function and catenates an underscore to it. The function name is extracted via the op procedure, e.g. op(0,f(a,b,c)) evaluates to f. Thus

map(f->cat(op(0,f),_),T);

Note that this fails if the name of a function is indexed, e.g. f[2](a). No error is generated but the catenation doesn't evaluate to a symbol. Presumably that isn't an issue here.  One way to handle that case is to first convert the indexed name to a symbol, then catenate the underscore.  So a more robust version is

map(f->cat(convert(op(0,f),'symbol'),_),T);

However, if you are actually dealing with indexed names you might want a different result. Another way to do the conversion, and combine it with the catenation, is to use nprintf, which generates a name (symbol). Thus

map(f -> nprintf("%a_", op(0,f)),T);

Let’s discuss each approach by understanding the definitions and functionalities of the commands used.

The map command, map(fcn, expr, arg1, ..., argN) applies a procedure or name, fcn, to the operands or elements of an expression, expr. The result of a call to map is a copy of expr with the ith operand of expr replaced by the result of applying fcn to the ith operand.  This concept is easier to grasp by looking at a few examples related to the usage of map in this question.

Example 1.  map(x-> x2,x+y)         returns     x2+y2

Example 2. map(a -> a-b, sin(x))    returns     sin(x-b)

The cat function, cat(a,b,c,…), is commonly used to concatenate (or join) string and names together. This function’s parameters: a,b,c…, can be any expressions.

Example 1. cat(a,2)                      returns     a2

Example 2.  cat(“a”,3,sin(x))          returns    “a3sin(x)”

The op function, op(i..j,e), extracts operands from an expression. The parameters i and j are the integers indicating positions of the operands and e is the expression. For functions, as in this example, op(0,e) is the name of the function.

Example 1.  op(0,alpha(t))            returns   the symbol alpha

Example 2.  op(0, sin(x))              returns    sin

Now analyzing Joe Riel's code will be easier.

1. map(f->cat(op(0,f),_),T);

In this approach Joe is extracting the name of the functions, alpha and beta, and then concatenating it to the underscore symbol. Then using the mapping function he applies the previous procedure to the list T.

1. map(f->cat(convert(op(0,f),'symbol'),_),T);

This approach is a lot similar to the previous one, but he added the convert function in case the function inside of map was indexed. Convert(expr, form, arg3,..), is used to change an expression from one form to another. In this example op(0,f) has been changed from type name to type symbol.

1. map(f -> nprintf("%a_", op(0,f)),T);

Again this is a similar approach but it uses nprintf. This command, nprintf(fmt,x1,..xn), is based on a C standard library command of the same name. It uses the format specifications in the fmt string to format and writes the expression into a Maple symbol, which is returned. In this example the format specified is the algebraic format “%a”.

This blog was written by Maplesoft’s intern Pia under the supervision of Dr. Robert Lopez. We both hope that you find this useful. If there is a particular question on MaplePrimes that you would like further explained, please let us know.

## New and improved integration results in Maple...

by: Maple 2016

The attached worksheet shows a small selection of new and improved results in integration for Maple 2016. Note that integration is a vast topic, so there will always be more improvements that can be made, but be sure that we are working on them.

A selection of new and improved integration results for Maple 2016

Indefinite integrals:

 > int(sqrt(1+sqrt(z-1)), z);
 (1.1)
 > int(arctan((-1+sec(x))^(1/2))*sin(x), x);
 (1.2)
 > int(((1+exp(I*x))^2+(1+exp(-I*x))^2)/(1-2*c*cos(x)+c^2), x);
 (1.3)
 > int(x^4/arccos(x)^(3/2),x);
 (1.4)

Definite integrals:

 > int(arcsin(sin(z)), z=0..1);
 (1.5)
 > int(sqrt(1 - sqrt(1+z)), z=0..1);
 (1.6)
 > int(z/(exp(2*z)+4*exp(z)+10),z = 0 .. infinity);
 (1.7)
 > simplify(int(sinh(a*abs(x-y)), y=0..c, 'method'='FTOC'));
 (1.8)
 > int(ln(x+y)/(x^2+y), [x=0..infinity, y=0..infinity]);
 (1.9)

Definite integrals with assumptions on the parameters:

 > int(x^(-ln(x)),x=0..b) assuming b > 0;
 (1.10)
 > int(exp(-z)*exp(-I*n*z)*cos(n*z),z = -infinity .. infinity) assuming n::integer;
 (1.11)

Integral of symbolic integer powers of sin(x) or cos(x):

 > int(sin(x)^n,x) assuming n::integer;
 (1.12)
 > int(cos(x)^n,x) assuming n::negint;
 (1.13)
 > int(cos(x)^n,x) assuming n::posint;
 (1.14)

 > int(sqrt(1+sqrt(x)), x);
 (2.1)
 > int(sqrt(1+sqrt(1+z)), z= 0..1);
 (2.2)
 > int(signum(z^k)*exp(-z^2), z=-infinity..infinity) assuming k::real;
 (2.3)
 > int(2*abs(sin(x*p)*sin(x)), x = 0 .. Pi) assuming p> 1;
 (2.4)
 > int(1/(x^4-x+1), x = 0 .. infinity);
 (2.5)

In Maple 2016, this multiple integral is computed over 3 times faster than it was in Maple 2015.

 > int(exp(abs(x1-x2))*exp(abs(x1-x3))*exp(abs(x3-x4))*exp(abs(x4-x2)), [x1=0..R, x2=0..R, x3=0..R, x4=0..R], AllSolutions) assuming R>0;
 (2.6)

Austin Roche
Mathematical Software, Maplesoft

## Announcing Maple 2016

by: Maple 2016

You, I, and others like us, are the beneficiaries of decades of software evolution.

From its genesis as a research project at the University of Waterloo in the early 80s, Maple has continually evolved to meet the challenges of technical computing.

## Can you grade conceptual understanding with Maple...

by: Maple T.A.

Disclaimer: This blog post has been contributed by Dr. Nicola Wilkin, Head of Teaching Innovation (Science), College of Engineering and Physical Sciences and Jonathan Watkins from the University of Birmingham Maple T.A. user group*.

We all know the problem. During the course of a degree, students become experts at solving problems when they are given the sets of equations that they need to solve. As anyone will tell you, the skill they often lack is the ability to produce these sets of equations in the first place. With Maple T.A. it is a fairly trivial task to ask a student to enter the solution to a system of equations and have the system check if they have entered it correctly. I speak with many lecturers who tell me they want to be able to challenge their students, to think further about the concepts. They want them to be able to test if they can provide the governing equations and boundary conditions to a specific problem.

With Maple T.A. we now have access to a math engine that enables us to test whether a student is able to form this system of equations for themselves as well as solve it.

In this post we are going to explore how we can use Maple T.A. to set up this type of question. The example I have chosen is 2D Couette flow. For those of you unfamiliar with this, have a look at this wikipedia page explaining the important details.

In most cases I prefer to use the question designer to create questions. This gives a uniform interface for question design and the most flexibility over layout of the question text presented to the student.

1. On the Questions tab, click New question link and then choose the question designer.
2. For the question title enter "System of equations for Couette Flow".
3. For the question text enter the text

The image below shows laminar flow of a viscous incompressible liquid between two parallel plates.

What is the system of equations that specifies this system. You can enter them as a comma separated list.

e.g. diff(u(y),y,y)+diff(u(y),y)=0,u(-1)=U,u(h)=0

You then want to insert a Maple graded answer box but we'll do that in a minute after we have discussed the algorithm.

When using the questions designer, you often find answers are longer than width of the answer box. One work around is to change the width of all input boxes in a question using a style tag. Click the source button on the editor and enter the following at the start of the question

` <style id="previewTextHidden" type="text/css">input[type="text"] {width:300px !important}</style> `

Pressing source again will show the result of this change. The input box should now be significantly wider. You may find it useful to know the default width is 186px.
4. Next, we need to add the algorithm for this question. The teacher's answer for this question is the system of equations for the flow in the picture.

` \$TA="diff(u(y),y,y) = 0, u(0) = 0, u(h) = U";\$sol=maple("dsolve({\$TA})");`

I always set this to `\$TA` for consitency across my questions. To check there is a solution to this I use a maple call to the dsolve function in Maple, this returns the solution to the provided system of equations. Pressing refresh on next to the algorithm performs these operations and checks the teacher's answer.

The key part of this question is the grading code in the Maple graded answer box. Let's go ahead and add the answer box to the question text. I add it at the end of the text we added in step 3. Click Insert Response area and choose the Maple-graded answer box in the left hand menu. For the answer enter the `\$TA` variable that we defined in the algorithm. For the grading code enter
`a:=dsolve({\$RESPONSE}):evalb({\$sol}={a}) `

This code checks that the students system of equations produces the same solution as the teachers. Asking the question in this way allows a more open ended response for the student.

To finish off make sure the expression type is Maple syntax and Text entry only is selected.
5. Press OK and then Finish on the Question designer screen.

That is the question completed. To preview a working copy of the question, have a look here at the live preview of this question. Enter the system of equations and click How did I do?

I have included a downloadable version of the question that contains the .xml file and image for this question. Click this link to download the file. The question can also be found on the Maple T.A. cloud under "System of equations for Couette Flow".

* Any views or opinions presented are solely those of the author(s) and do not necessarily represent those of the University of Birmingham unless explicitly stated otherwise.

## Why use Maple T.A. for STEM courses?

by: Maple T.A.

Disclaimer: This blog post has been contributed by Dr. Nicola Wilkin, Head of Teaching Innovation (Science), College of Engineering and Physical Sciences and Jonathan Watkins from the University of Birmingham Maple T.A. user group*.

If you have arrived at this post you are likely to have a STEM background. You may have heard of or had experience with Maple T.A or similar products in the past. For the uninitiated, Maple T.A. is a powerful system for learning and assessment designed for STEM courses, backed by the power of the Maple computer algebra engine. If that sounds interesting enough to continue reading let us introduce this series of blog posts for the mapleprimes website contributed by the Maple T.A. user group from the University of Birmingham(UoB), UK.

These posts mirror conversations we have had amongst the development team and with colleagues at UoB and as such are likely of interest to the wider Maple T.A. community and potential adopters. The implementation of Maple T.A. over the last couple of years at UoB has resulted in a strong and enthusiastic knowledge base which spans the STEM subjects and includes academics, postgraduates, undergraduates both as users and developers, and the essential IT support in embedding it within our Virtual Learning Environment (VLE), CANVAS at UoB.

By effectively extending our VLE such that it is able to understand mathematics we are able to deliver much wider and more robust learning and assessment in mathematics based courses. This first post demonstrates that by comparing the learning experience between a standard multiple choice question, and the same material delivered in a Maple TA context.

To answer this lets compare how we might test if a student can solve a quadratic equation, and what we can actually test for if we are not restricted to multiple choice. So we all have a good understanding of the solution method, let's run through a typical paper-based example and see the steps to solving this sort of problem.

Here is an example of a quadratic

To find the roots of this quadratic means to find what values of x make this equation equal to zero. Clearly we can just guess the values. For example, guessing 0 would give

So 0 is not a root but -1 is.

There are a few standard methods that can be used to find the roots. The point though is the answer to this sort of question takes the form of a list of numbers. i.e. the above example has the roots -1, 5. For quadratics there are always two roots. In some cases two roots could be the same number and they are called repeated roots. So a student may want to answer this question as a pair of different numbers 3, -5, the same number repeated 2, 2 or a single number 2. In the last case they may only list a repeated roots once or maybe they could only find one root from a pair of roots. Either way there is quite a range of answer forms for this type of question.

With the basics covered let us see how we might tackle this question in a standard VLE. Most are not designed to deal with lists of variable length and so we would have to ask this as a multiple choice question. Fig. 1, shows how this might look.

Fig 1: Multiple choice question from a standard VLE

Unfortunately asking the question in this way gives the student a lot of implicit help with the answer and students are able to play a process of elimination game to solve this problem rather than understand or use the key concepts.

They can just put the numbers in and see which work...

Let's now see how we may ask this question in Maple T.A.. Fig. 2 shows how the question would look in Maple T.A. Clearly this is not multiple choice and the student is encouraged to answer the question using a simple list of numbers separated by commas. The students are not helped by a list of possible answers and are left to genuinely evaluate the problem. They are able to provide a single root or both if they can find them, and moreover the question is not fussy about the way students provide repeated roots. After a student has attempted the question, in the formative mode, a student is able to review their answer and the teacher's answer as well as question specific feedback, Fig. 3. We'll return to the power of the feedback that can be incorporated in a later post.

Fig. 2: Free response question in Maple T.A.

Fig. 3: Grading response from Maple T.A.

The demo of this question and others presented in this blog, are available as live previews through the UoB Maple T.A. user group site.

The question can be downloaded from here and imported as a course module to your Maple T.A. instance. It can also be found on the Maple TA cloud by searching for "Find the roots of a quadratic". Simply click on the Clone into my class button to get your own version of the question to explore and modify.

* Any views or opinions presented are solely those of the author(s) and do not necessarily represent those of the University of Birmingham unless explicitly stated otherwise.

## Bring Learning to Life (LIVE!)

by: Maple Maple T.A.

This January 28th, we will be hosting another full-production, live streaming webinar featuring an all-star cast of Maplesoft employees: Andrew Rourke (Director of Teaching Solutions), Jonny Zivku (Maple T.A. Product Manager), and Daniel Skoog (Maple Product Manager). Attend the webinar to learn how educators all around the world are using Maple and Maple T.A. in their own classrooms.

Any STEM educator, administrator, or curriculum coordinator who is interested in learning how Maple and Maple T.A. can help improve student grades, reduce drop-out rates, and save money on administration costs will benefit from attending this webinar.

## Maplesoft at Joint Math 2016 in Seattle

by: Maple

The Joint Mathematics Meetings are taking place this week (January 6 – 9) in Seattle, Washington, U.S.A. This will be the 99th annual winter meeting of the Mathematical Association of America (MAA) and the 122nd annual meeting of the American Mathematical Society (AMS).

Maplesoft will be exhibiting at booth #203 as well as in the networking area. Please stop by our booth or the networking area to chat with me and other members of the Maplesoft team, as well as to pick up some free Maplesoft swag or win some prizes.

Given the size of the Joint Math Meetings, it can be challenging to pick which events to attend. Hopefully we can help by suggesting a few Maple-related talks and events:

Maplesoft is hosting a catered reception and presentation ‘Challenges of Modern Education: Bringing Math Instruction Online’ on Thursday, January 7th at 18:00 in the Cedar Room at the Seattle Sheraton. You can find more details and registration information here: www.maplesoft.com/jmm

Another not to miss Maple event is “30 Years of Digitizing Mathematical Knowledge with Maple”, presented by Edgardo Cheb-Terrab, on Thursday, January 7 at 10:00 in Room 603 of the Convention Center.

Here’s a list of Maple-related events and talks:

Wednesday, Jan 6, 10:20, Room 2B, Convention Center

Lina Wu

30 Years of Digitizing Mathematical Knowledge with Maple

Thursday, Jan 7, 10:00, Room 603, Convention Center

Edgardo Cheb-Terrab

MAA Poster Session – Collaborative Research: Maplets for Calculus

Thursday, Jan 7, 14:00, Hall 4F, 4th Floor, Convention Center

Challenges of Modern Education: Bringing Math Instruction Online

Thursday, Jan 7, 18:00, Cedar Room, 2nd Floor, Sheraton Center

Using Maple to Promote Modelling in Differential Equations

Friday, Jan 8, 10:40, Room 617, Convention Center

Patrice G Tiffany; Rosemary C Farley

If you are presenting at Joint Math and would like to advertise your Maple-related talk, please feel free to comment below, or send me a message with your event and I’ll add it to the list above.

See you in Seattle!

Daniel

Maple Product Manager

## Would Brunel Have Used a Spreadsheet?

This is a post that I wrote for the Altair Innovation Intelligence blog.

I have a grudging respect for Victorian engineers. Isambard Kingdom Brunel, for example, designed bridges, steam ships and railway stations with nothing but intellectual flair, hand-calculations and painstakingly crafted schematics. His notebooks are digitally preserved, and make for fascinating reading for anyone with an interest in the history of engineering.

His notebooks have several characteristics.

• Equations are written in natural math notation
• Text and diagrams are freely mixed with calculations
• Calculation flow is clear and well-structured

Hand calculations mix equations, text and diagrams.

Engineers still use paper for quick calculations and analyses, but how would Brunel have calculated the shape of the Clifton Suspension Bridge or the dimensions of its chain links if he worked today?

If computational support is needed, engineers often choose spreadsheets. They’re ubiquitous, and the barrier to entry is low. It’s just too easy to fire-up a spreadsheet and do a few simple design calculations.

Spreadsheets are difficult to debug, validate and extend.

Spreadsheets are great at manipulating tabular data. I use them for tracking expenses and budgeting.

However, the very design of spreadsheets encourages the propagation of errors in equation-oriented engineering calculations

• Results are difficult to validate because equations are hidden and written in programming notation
• You’re often jumping about from one cell to another in a different part of the worksheet, with no clear visual roadmap to signpost the flow of a calculation

For these limitations alone, I doubt if Brunel would have used a spreadsheet.

Technology has now evolved to the point where an engineer can reproduce the design metaphor of Brunel’s paper notebooks in software – a freeform mix of calculations, text, drawings and equations in an electronic notebook. A number of these tools are available (including Maple, available via the APA website).

Modern calculation tools reproduce the design metaphor of hand calculations.

Additionally, these modern software tools can do math that is improbably difficult to do by hand (for example, FFTs, matrix computation and optimization) and connect to CAD packages.

For example, Brunel could have designed the chain links on the Clifton Suspension Bridge, and updated the dimensions of a CAD diagram, while still maintaining the readability of hand calculations, all from the same electronic notebook.

That seems like a smarter choice.

Would I go back to the physical notebooks that Brunel diligently filled with hand calculations? Given the scrawl that I call my handwriting, probably not.

## Top 5 Webinars in 2015

by: Maple

Since we’re almost at the end of the year, I thought it would be interesting to look back at our most popular webinars for academics in 2015. I found that they fell into one of two categories: live streaming webinars featuring Dr. Robert Lopez and Maple how-to tutorials.  (If you missed the live presentation, you can watch the recordings of all these webinars below.)

The first and second most popular webinar were, unsurprisingly, both of the live streaming webinars that featured Dr. Robert Lopez (Emeritus Professor at Rose Hulman Institute of Technology and Maple Fellow at Maplesoft). These webinars were streamed live to an audience and allowed many people to get their first glimpse of the man behind the Clickable Calculus series and Teaching Concepts with Maple:

1.       Eigenpairs Enlivened

In this webinar, Dr. Robert Lopez demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms by which students are expected to practice finding eigenpairs.

2.       Resequencing Concepts and Skills via Maple's Clickable

In this webinar, Dr. Lopez presents examples of what "resequencing" looks like when implemented with Maple's point-and-click syntax-free paradigm. Not only can Maple be used to elucidate the concept, but in addition, it can be used to illustrate and implement the manipulations that ultimately the student must master.

The next three were all brief webinars on how to complete specific tasks in Maple 2015. Just under a dozen of these were created in 2015 and they were all quite popular, but these three stood out above the rest:

3.       Working with Data Sets in Maple

This video walks through examples of working with several types of data in Maple, including visualizing stock and commodity data, forecasting future temperatures using weather data, and analyzing macroeconomic data, such as employment statistics, GDP and other economic indicators.

4.       Custom Color Schemes in Maple

This webinar provides an overview of the colorscheme option for coloring surfaces, curves and collections of points in Maple, including how to color with gradients, according to function value or point position. Examples of how the colorscheme option is used with various commands from the Maple library are also demonstrated.

5.       Working with Units in Maple

Maple 2015 allows for more fluid and natural interaction with units. This webinar provides an overview of the new unit formatting controls and new Temperature object, and demonstrates how to compute with units and tolerances.

Are there any topics you’d like to see Robert cover in upcoming webinars? Or, any Maple how-to videos you think would be a helpful addition to our library? Let us know in the comments below!

Kim

## An Arts Co-op Student in a Math Software Company

As an Arts major at the University of Waterloo, my first day as a co-op student in the Maplesoft marketing department was a bit of a blur. I was hearing a lot of mathematical jargon that I did not understand. Other than a mandatory statistics class in my second year at university, I haven’t taken a math course since high school, over two years ago. I spent my first week as the marketing assistant educating myself about the basics of marketing complex math software. My favourite method for doing this was to read through the Maplesoft user stories. As I read, I was amazed by the variety of customers and the endless applications that Maplesoft products had contributed to. It became apparent that math is a part of every industry and it is in the design of many products. There were a few stories from the robotics industry in particular that really sparked my interest in the software that I now market.

We’ve all seen the futuristic movies where robots gradually get smarter and smarter, developing enough intelligence to control the human race, and eventually, take over the world. As it turns out, Engineered Arts, a UK robotics company, is bringing us one step closer to that reality. Well… they’re maybe not ready for world domination just yet, but they are working on one of the most advanced and human-like robots that the world has seen outside of a Hollywood production, and they are doing this using MapleSim. The first generation of the biologically inspired robot was named RoboThespian. With his ability to speak and sing, he was used to educate, entertain, and investigate new developments in robotics. However, he was largely static. That’s when the engineers began work on generation two of their robot, named Byrun, who has the ability to walk, run, jump, and hop as well as speak and sing. Byrun can even express thousands of different facial features thanks to his projective head display. This makes him even more human-like; scary or cool? I’m thinking a bit of both. If you’re interested in the story, click here to continue reading about it.

Another unexpected use of MapleSim was adopted as a joint research project between Ryerson University and McMaster University. I never would have guessed that math software could be applied to the process of human birth. Nevertheless, a group of researchers used MapleSim to simulate induced labour with a Foley Catheter. In short, this is when a small balloon is inserted through the opening of the cervix creating a downward pressure that effectively tricks the cervix into opening for labour to begin. Though the application of this story surprised me, it makes a lot of sense to use modelling software for a research project like this. It’s more efficient to get all of the kinks out of the virtual model in a simulation program before building a physical model that could end up being dysfunctional. According to Dr. James Andrew Smith, a Biomedical Engineering researcher and Assistant Professor in Electrical and Computer Engineering, who is the lead researcher on the project, “Modern engineering has a lot to offer the medical world,” especially when it saves on time and cost. Click here to read more about this story and to watch a video of the finished model.

After two months at Maplesoft, I have noticed that I don’t look at things in the same way that I used to. I find myself staring at a toaster and imagining how it was designed. Did the engineers use advanced physical simulation and modeling software to make the most efficient toaster possible? Well, if it can still only toast on one side then, my guess is no! Maplesoft has many more user stories that I haven’t had the chance to read yet. With customers ranging from BMW to Pixar, Maplesoft continues to expand its customer base and adapt its software to support more and more unique applications. I can’t wait to hear what new and unexpected things will be done with the software next!

## Sponsoring Innovators from All Over the Globe

Here at Maplesoft, we like to foster innovation in technological development. Whether that is finding solutions to global warming, making medical discoveries that save millions, or introducing society to very advanced functional robots, Maplesoft is happy to contribute, support and encourage innovative people and organizations researching these complex topics. This year, we are delighted to have sponsored two contests in the robotics field that provide opportunities to think big and make an impact: Create the Future Design Contest and the International Space Apps Challenge.

Create the Future Design Contest

Established in 2002, and organized by TechBriefs, the goal of the Create the Future Design Contest is to help engineers bring their product design ideas to life. The overall ‘mission of the contest is to benefit humanity, the environment, and the economy.’ This year, there were a record 1,159 new product ideas submitted by students, engineers, and entrepreneurs from all over the globe. In the machinery/automation/robotics category, which Maplesoft sponsored, the project with the top votes was designed by two engineers who chose to name their innovation CAP Exoskeleton, a type of assistive robotic machine designed to aid the user in walking, squatting, and carrying heavy loads over considerable distances. It can either be used to enhance physical endurance for military purposes or to help the physically impaired perform daily tasks. A contest like Create the Future is a perfect opportunity, for engineers in particular, to learn, explore, and create.

The CAP Exoskeleton - ©2015 Create the Future Design Contest

International Space Apps Challenge

The exploration of space has always been unique in its search for knowledge. The International Space Apps Challenge, a NASA incubator innovation program, is an ‘international mass collaboration focused on space exploration that takes place over 48-hours in cities around the world’. It is a unique global competition where people rally together to find solutions to real world problems, bringing humanity closer to understanding the Earth, the universe, the human race, and robotics. These goals, the organizers believe, can be reached much faster if we combine the power of the seven billion or so brains that occupy the planet, not forgetting the six that are currently orbiting above us aboard the International Space Station. The competition is open to people of all ages and in all fields, including engineers, technologists, scientists, designers, artists, educators, students, entrepreneurs, and so on. With an astounding 13,846 participants from all over the world, several highly innovative solutions were presented.

Maplesoft sponsored the University of York location in the UK where the winning team of five modeled an app called CropOp, a communication tool that connects the government to local farmers with the goal of providing instantaneous, crucial information regarding pest breakout warnings, extreme weather, and other important updates. This UK-based team believes the quality and quantity of food produced will be improved, especially benefiting the undernourished communities in Africa. Maplesoft supports the Space Apps Challenge because it proves that collaboration makes for bigger and better discoveries that can save millions of people.

Donating Maplesoft software for contestants to use is part of the sponsorship. The real delight is to wait and see what innovative concepts they come up with. When we sponsor contests like these, we find it benefits our software as much as it does the participants. Plus, if the contestants can provide solutions to real world issues, well, that benefits everyone!

## Transitioning to Jordan Form

by: Maple 2015

I have two linear algebra texts [1, 2]  with examples of the process of constructing the transition matrix  that brings a matrix  to its Jordan form . In each, the authors make what seems to be arbitrary selections of basis vectors via processes that do not seem algorithmic. So recently, while looking at some other calculations in linear algebra, I decided to revisit these calculations in as orderly a way as possible.

First, I needed a matrix  with a prescribed Jordan form. Actually, I started with a Jordan form, and then constructed  via a similarity transform on . To avoid introducing fractions, I sought transition matrices  with determinant 1.

Let's begin with , obtained with Maple's JordanBlockMatrix command.

The eigenvalue  has algebraic multiplicity 6. There are sub-blocks of size 3×3, 2×2, and 1×1. Consequently, there will be three eigenvectors, supporting chains of generalized eigenvectors having total lengths 3, 2, and 1. Before delving further into structural theory, we next find a transition matrix  with which to fabricate .

The following code generates random 6×6 matrices of determinant 1, and with integer entries in the interval . For each, the matrix  is computed. From these candidates, one  is then chosen.

After several such trials, the matrix  was chosen as

for which the characteristic and minimal polynomials are

So, if we had started with just , we'd now know that the algebraic multiplicity of its one eigenvalue  is 6, and there is at least one 3×3 sub-block in the Jordan form. We would not know if the other sub-blocks were all 1×1, or a 1×1 and a 2×2, or another 3×3. Here is where some additional theory must be invoked.

The null spaces  of the matrices  are nested: , as depicted in Figure 1, where the vectors , are basis vectors.

 Figure 1   The nesting of the null spaces

The vectors  are eigenvectors, and form a basis for the eigenspace . The vectors , form a basis for the subspace , and the vectors , for a basis for the space , but the vectors  are not yet the generalized eigenvectors. The vector  must be replaced with a vector  that lies in  but is not in . Once such a vector is found, then  can be replaced with the generalized eigenvector , and  can be replaced with . The vectors  are then said to form a chain, with  being the eigenvector, and  and  being the generalized eigenvectors.

If we could carry out these steps, we'd be in the state depicted in Figure 2.

 Figure 2   The null spaces  with the longest chain determined

Next, basis vector  is to be replaced with , a vector in  but not in , and linearly independent of . If such a  is found, then  is replaced with the generalized eigenvector . The vectors  and  would form a second chain, with  as the eigenvector, and  as the generalized eigenvector.

Define the matrix  by the Maple calculation

and note

The dimension of  is 3, and of , 5. However, the basis vectors Maple has chosen for  do not include the exact basis vectors chosen for .

We now come to the crucial step, finding , a vector in  that is not in  (and consequently, not in  either). The examples in  are simple enough that the authors can "guess" at the vector to be taken as . What we will do is take an arbitrary vector in  and project it onto the 5-dimensional subspace , and take the orthogonal complement as .

A general vector in  is

A matrix that projects onto  is

The orthogonal complement of the projection of Z onto  is then . This vector can be simplified by choosing the parameters in Z appropriately. The result is taken as .

The other two members of this chain are then

A general vector in  is a linear combination of the five vectors that span the null space of , namely, the vectors in the list . We obtain this vector as

A vector in  that is not in  is the orthogonal complement of the projection of ZZ onto the space spanned by the eigenvectors spanning  and the vector . This projection matrix is

The orthogonal complement of ZZ, taken as , is then

Replace the vector  with , obtained as

The columns of the transition matrix  can be taken as the vectors , and the eigenvector . Hence,  is the matrix

Proof that this matrix  indeed sends  to its Jordan form consists in the calculation

 =

The bases for , are not unique. The columns of the matrix  provide one set of basis vectors, but the columns of the transition matrix generated by Maple, shown below, provide another.

I've therefore added to my to-do list the investigation into Maple's algorithm for determining an appropriate set of basis vectors that will support the Jordan form of a matrix.

References

 [1] Linear Algebra and Matrix Theory, Evar Nering, John Wiley and Sons, Inc., 1963 [2] Matrix Methods: An Introduction, Richard Bronson, Academic Press, 1969

## They're watching...

by: MaplePrimes

Like most companies today, Maplesoft monitors its website traffic, including the traffic coming to MaplePrimes. This allows us to view statistical data such as how many total visits MaplePrimes gets, how many unique visitors it gets, what countries these visitors come from, how many questions are asked and answered, how many people read but never respond to posts, etc.

Recently one of our regular MaplePrimes users made the comment that MaplePrimes does not reach new Maple users. We found this comment interesting because our data and traffic numbers show a different trend. MaplePrimes gets unique visitors in the hundreds of thousands each year, and since its inception, it has welcomed unique visitors in the many millions. Based on these unique visitor numbers and the thousands of common searches specifically about Maple that people are doing, we can see that many of these unique visitors are in fact new Maple users looking for resources and support as they begin using Maple. Other visitors to MaplePrimes include people who use Google (or other search engines) to find an answer to a particular mathematics or engineering question, regardless of what mathematics software they choose to use, and Google points them to MaplePrimes. There are some popular posts that were written months, even years ago, that are still getting high visitor views today, showing the longevity of the information on MaplePrimes.

MaplePrimes gets the majority of its visible activity from a small number of extremely active members. In public user forums around the world, these types of members are given many names – power users, friendlies, evangelists. Every active public user forum has them. On MaplePrimes, it’s this small number of active members that are highly visible. But, what our traffic data reveals is the silent majority. These people, many of them repeat visitors, are quietly reviewing the questions and answers that our evangelists are posting. The silent majority of MaplePrimes visitors are the readers; they are the quiet consumers of information. For every person that writes, comments on, or likes a post, there are thousands more that read it.

Here are a few more MaplePrimes traffic data points for your reference:

• MaplePrimes is very international and draws people from all around the world. Here are the top 10 countries where the most MaplePrimes visitors come from:
1. USA
2. India
4. Germany
5. China
6. United Kingdom
7. Brazil
8. Australia
9. France
10. Denmark
• Here are the top 5 keywords people are using in their searches on MaplePrimes:
1. Data from plot
2. Physics
3. Sprintf
4. Size of plot
5. Fractal
• MaplePrimes is growing at a very fast rate: Traffic (visitors to the site) and membership size is growing at nearly double the pace it was last year. The total number of posts and questions this year is also much higher compared to the same timeframe last year.
• Our top 5 MaplePrimes members have each visited MaplePrimes more than 1200 times and viewed a combined total of more than 10,000 pages (that is total page views, not unique page views). Our top 25 MaplePrimes members have visited at least 250 times each (many of them nearly 1000 times each) and our top 50 MaplePrimes members have visited a combined total of over 23,000 times, visiting nearly 200,000 pages. Thank you! We’re glad you like it. :-)

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