A modified version of John May's modification of Bruce Char's animated Christmas tree, using some Maple 2015 features to automatically scale the 3D plot and automatically play the animation.

I went with the viewpoint option, and removed the blinking lights. In the worksheet the tree shows as scaled larger. Perhaps it could be a submission to walkingrandomly.

bchartree1.mw

The CIELAB perceptual model of human vision can be used to predict the color of a wavelength in sRGB. I found the CIEDE2000 corrections to hue, chroma, and lighness were required to get the right values in the blue to violet region. The CIEDE2000 results seem pretty close to what I see looking through a diffraction grating considering I have no way of knowing how my eyes are adapted.

Use the new, faster  version. It can run in under 1 minute;

nprocWavelength_Color_CIEDE2000.mw   and  WtoC_data.xlsx

You will need the data for the color matching functions in the WtoC Excel File. The data directory can now be controlled in the mw file.

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.

I am learning to use maple for my notes preparation for the subject Finite Element Analysis. It is interesting to know that how often we blame maple or computer for the silly mistakes we made in our commands and expect the exact answers. I have used a small file and find it easy to analyse my mistakes fatser. If we make a small mistake in a big file, it not only gives us problem finding our mistakes, it leads to more mistakes in other parts as well. A command working in one document need not necessarily work the same way in other document.

I have made my first document and people will come with suggestions to make appropriate modifications in the various sections to improve my knowledge on maple as well as the subject.

Download FINITE_ELEMENT_ANALYSIS.mw

Ramakrishnan V

rukmini_ramki@hotmail.com

Um den Studierenden zu helfen, deren Mathematikkenntnisse nicht auf dem von Studienanfängern erwarteten Niveau waren, hat die TU Wien einen Auffrischungskurs mit Maple T.A. entwickelt.  Die vom Team der TU Wien ausgearbeiteten Fragen zu mathematischen Themen wie der Integralrechnung, linearen Funktionen, der Vektoranalysis, der Differentialrechnung und der Trigonometrie, sind in die Maple T.A. Cloud übernommen worden.  Außerdem haben wir diesen Inhalt als Kursmodul zur Verfügung gestellt.

Laden Sie das Kursmodul der TU Wien herunter.

Bei Interesse können Sie mehr über das Projekt der TU Wien in diesem Anwenderbericht lesen: Erfolgreiches Auffrischen von Mathematikkenntnissen an der Technischen Universität Wien mit Maple T.A.

Jonny
Maplesoft Product Manager, Maple T.A.

Here's a simple package for drawing knot diagrams and computing the Alexander polynomial. A typical usage case for the AlexanderPolynomial function is when a knot needs to be identified and only a visual representation of the knot is available. Then it's trivial to write down the Dowker sequence by hand and then the sequence can be used as an input for this package. The KnotDiagram function also takes the Dowker sequence as an input.

TorusKnot(p, q) and PretzelKnot(p, q, r) are accepted as an input as well and can also be passed to the DowkerNotation function.

The algorithm is fairly simple, it works as follows: represent each double point as a quadrilateral (two 'in' vertices and two 'out' vertices); connect the quads according to the Dowker specification; draw the result as a planar graph; erase the sides of each quad and draw its diagonals instead. This draws the intersections corresponding to the double points and guarantees that there are no other intersections. The knot polynomial is then computed from the diagram.

The diagrams work fairly well for pretzel knots, but for certain knots they can be difficult to read because some of the quads around the double points can become too small or too skewed. Also, the code doesn't check that the generated quadrilaterals are convex (which is an implicit assumption in the algorithm).

knot.txt

knot.mw

Download knot.mw

knot.txt

ABSTRACT. In this paper we demonstrate how the simulation of dynamic systems engineering has been implemented with graphics software algorithms using maple and MapleSim. Today, many of our researchers the computational modeling performed by inserting a piece of code from static work; with these packages we have implemented through the automation components of kinematics and dynamics of solids simple to complex.

It is very important to note that once developed equations study; recently we can move to the simulation; to thereby start the physical construction of the system. We will use mathematical and computational methods using the embedded buttons which lie in the dynamics leaves and viewing platform cloud of Maplesoft and power MapleNet for online evaluation of specialists in the area. Finally they will see some work done; which integrate various mechanical and computational concepts implemented for companies in real time and pattern of credibility.

Selasi_2015.pdf

(in spanish)

Lenin Araujo Castillo

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.

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.

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.

Download JordanForm_blog.mw

Some time ago, @marc005 asked how he could send an email from the Maple command line.

Why would you want to do this? Using Maple's functionality, you could programatically construct an email - perhaps with the results of a computation - and email it yourself or someone else.

I originally posted a solution that involved communicating with a locally-installed SMTP server using the Sockets package. But of course, you need to set up an SMTP server and ensure the appropriate ports are open.

I recently found a better solution. Mailgun (http://mailgun.com) is a free email delivery service with an web-based API. You can communicate with this API via the URL package; simply send Mailgun a URL:-Post() message that contains account-specific information, and the text of your email.

The general steps and Maple commands are given below, and you can download the worksheet here.

Note: Maplesoft have no affiliation with Mailgun.

Step 1:
Sign up for a free Mailgun account.

Step 2:
In your Mailgun account, go to the Domains section - it should look like the screengrab below (account-specific information has been blanked).

Note down the API Base URL and the API key.

• the API Base URL looks like https://api.mailgun.net/v3/sandboxXXXXXXXXXXXXXXXXXXXXXXXXXXX.mailgun.org.  
• •the API Key looks like key-XXXXXXXXXXXXXXXXXXXXXXXXXX

Step 3:

In Maple, define strings containing your own API Base URL and API Key. Also, define the recipient's email address, the email you want the recipient to reply to, the email subject and email body.

>restart:

>APIBaseURL := "https://api.mailgun.net/v3/sandboxXXXXXXXXXXXXXXXXXXXXXXXX.mailgun.org":

>APIKey:="key-XXXXXXXXXXXXXXXXXXXXXXXX":

>toEmail := "xxxxx@xxxxxx.com":

>fromEmail:="First Last <FirstLast@Domain.com>":

>subject:= "Email Subject Goes Here":

>emailBody := "I'd rather have a bottle in front of me than a frontal lobotomy":

Step 4:
Run the following code

> URL:-Post(cat(APIBaseURL,"/messages"),[

"from"=fromEmail,"to"=toEmail,

"subject"=subject,

"text"= emailBody],

user="api",password=APIKey);

If you've successfully sent the email, you should see something like this (account-specific information is blanked out)

You can also send HTML emails by replacing the "text" line with "html" = str, where str is a string with your HTML code.