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.

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;

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.

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.

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.

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).

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.

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.

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

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.

Â•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.

“Exact solutions to Einstein’s equations” is one of those books that are difficult even to imagine: the authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, collecting, classifying, discarding repetitions in disguise, and organizing the whole material into chapters according to the physical properties of these solutions. The book is already in its second edition and it is a monumental piece of work.

As good as it is, however, the project resulted only in printed material, a textbook constituted of paper and ink. In 2006, when the DifferentialGeometry package was rewritten to enter the Maple library, one of the first things that passed through our minds was to bring the whole of “Exact solutions to Einstein’s equations” into Maple.

It took some time to start but in 2010, for Maple 14, we featured the first 26 solutions from this book. In Maple 15 this number jumped to 61. For Maple 17 we decided to emphasize the general relativity functionality of the DifferentialGeometry package, and Maple 18 added 50 more, featuring in total 225 of these solutions - great! but still far from the whole thing …

And this is when we decided to “step on the gas” - go for it, the whole book. One year later, working in collaboration with Denitsa Staicova from Bulgarian Academy of Sciences, Maple 2015 appeared with 330 solutions to Einstein’s equations. Today we have already implemented 492 solutions, and for the first time we can see the end of the tunnel: we are targeting finishing the whole book by the end of this year.

Wow^{2}! This is a terrific result. First, because these solutions are key in the area of general relativity, and at this point what we have in Maple is already the most thorough digitized database of solutions to Einstein’s equations in the world. Second, and not any less important, because within Maple this knowledge comes alive. The solutions are fully searcheable and are set by a simple call to the Physics:-g_ spacetime metric command, and that automatically sets the related coordinates, Christoffel symbols, Ricci and Riemann tensors, orthonormal and null tetrads, etc. All of this happens on the fly, and all the mathematics within the Maple library are ready to work with these solutions. Having everything come alive completely changes the game. The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords also makes the whole book concretely more useful.

And, not only are these solutions to Einstein’s equations brought to life in a full-featured way through the Physics package: they can also be reached through the DifferentialGeometry:-Library:-MetricSearch applet. Almost all of the mathematical operations one can perform on them are also implemented as commands in DifferentialGeometry.

Finally, in the Maple PDEtools package, we already have all the mathematical tools to start resolving the equivalence problem around these solutions. That is: to answer whether a new solution is or not new, or whether it can be obtained from an existing solution by transformations of coordinates of different kinds. And we are going for it.

What follows is a basic illustration of what has already been implemented. As usual, in order to reproduce these results, you need to update your Physics library from the Maplesoft R&D Physics webpage.

Load Physics, set the metric to (and everything else automatically) in one go

>

>

(1)

And that is all we do :) Although the strength in Physics is to compute with tensors using indicial notation, all of the tensor components and related properties of this metric are also derived on the fly (and no, they are not in any database). For instance these are the definition in terms of Christoffel symbols, and the covariant components of the Ricci tensor

>

(2)

>

(3)

These are the 16 Riemann invariants for Schwarzschild solution, using the formulas by Carminati and McLenaghan

These are the 2x2 matrix components of the Christoffel symbols of the second kind (that describe, in coordinates, the effects of parallel transport in curved surfaces), when the first of its three indices is equal to 1

>

(6)

In Physics, the Christoffel symbols of the first kind are represented by the same object (not two commands) just by taking the first index covariant, as we do when computing with paper and pencil

>

(7)

One could query the database, directly from the spacetime metrics, about the solutions (metrics) to Einstein's equations related to Levi-Civita, the Italian mathematician

>

(8)

These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in "Exact Solutions to Einstein's Equations"

>

(9)

Automatically, everything gets set accordingly; these are the contravariant components of the related Ricci tensor

>

(10)

One works with the Newman-Penrose formalism frequently using tetrads (local system of references); the Physics subpackage for this is Tetrads

>

(11)

This is the tetrad related to the book's metric with number 12.16.1

>

(12)

One can check these directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metric

>

(13)

This shows that, for the components given by (12), the definition holds

>

(14)

One frequently works with a different signature and null tetrads; set that, and everything gets automatically recomputed for the metric 12.16.1 accordingly

>

(15)

>

(16)

>

(17)

>

(18)

The related 16 Riemann invariant

>

(19)

The ability to query rapidly, set things in one go, change everything again etc. are at this point fantastic. For instance, these are the metrics by Kaigorodov; next are those published in 1962

>

(20)

>

(21)

The search can be done visually, by properties; this is the only solution in the database that is a Pure Ratiation solution, of Petrov Type "D", Plebanski-Petrov Type "O" and that has Isometry Dimension equal to 1:

>

Set the solution, and everything related to work with it, in one go

>

(22)

The related Riemann invariants:

>

(23)

To conclude, how many solutions from the book have we already implemented?