Product Tips & Techniques

Tips and Tricks on how to get the most about Maple and MapleSim

by: Maple 2019

Yesterday, user @lcz , while responding as a third party to one of my Answers regarding GraphTheory, asked about breadth-first search. So, I decided to write a more-interesting example of it than the relatively simple example that was used in that Answer. I think that this is general enough to be worthy of a Post.

This application generates all maximal paths in a graph that begin with a given vertex. (I'm calling a path maximal if it cannot be extended and remain a path.) This code requires Maple 2019 or later and 1D input. This works for both directed and undirected graphs. Weights, if present. are ignored.

restart:

AllMaximalPaths:= proc(G::GRAPHLN, v)
description
"All maximal paths of G starting at v by breadth-first search"
;
option Author: Carl Love <carl.j.love@gmail.com> 2021-Mar-17;
uses GT= GraphTheory;
local
P:= [rtable([v])], R:= rtable(1..0),
VL:= GT:-Vertices(G), V:= table(VL=~ [$1..nops(VL)]), Departures:= {op}~(GT:-Departures(G)) ; while nops(P) <> 0 do P:= [ for local p in P do local New:= Departures[V[p[-1]]] minus {seq}(p); if New={} then R,= [seq](p); next fi; ( for local u in New do local p1:= rtable(p); p1,= u od ) od ] od; {seq}(R) end proc : #large example: GT:= GraphTheory: K9:= GT:-CompleteGraph(9): Pa:= CodeTools:-Usage(AllMaximalPaths(K9,1)): memory used=212.56MiB, alloc change=32.00MiB, cpu time=937.00ms, real time=804.00ms, gc time=312.50ms nops(Pa); 40320 #fun example: P:= GT:-SpecialGraphs:-PetersenGraph(): Pa:= CodeTools:-Usage(AllMaximalPaths(P,1)): memory used=0.52MiB, alloc change=0 bytes, cpu time=0ns, real time=3.00ms, gc time=0ns nops(Pa); 72 Pa[..9]; #sample paths {[1, 2, 3, 4, 10, 9, 8, 5], [1, 2, 3, 7, 8, 9, 10, 6], [1, 2, 9, 8, 7, 3, 4, 5], [1, 2, 9, 10, 4, 3, 7, 6], [1, 5, 4, 3, 7, 8, 9, 2], [1, 5, 4, 10, 9, 8, 7, 6], [1, 5, 8, 7, 3, 4, 10, 6], [1, 5, 8, 9, 10, 4, 3, 2], [1, 6, 7, 3, 4, 10, 9, 2]}  Notes on the procedure: The two dynamic data structures are • P: a list of vectors of vertices. Each vector contains a path which we'll attempt to extend. • R: a vector of lists of vertices. Each list is a maximal path to be returned. The static data structures are • V: a table mapping vertices (which may be named) to their index numbers. • Departures: a list of sets of vertices whose kth set is the possible next vertices from vertex number k. On each iteration of the outer loop, P is completely reconstructed because each of its entries, a path p, is either determined to be maximal or it's extended. The set New is the vertices that can be appended to the (connected to vertex p[-1]). If New is empty, then p is maximal, and it gets moved to R The following code constructs an array plot of all the maximal paths in the Petersen graph. I can't post the array plot, but you can see it in the attached worksheet: BreadthFirst.mw #Do an array plot of each path embedded in the graph: n:= nops(Pa): c:= 9: plots:-display( (PA:= rtable( (1..ceil(n/c), 1..c), (i,j)-> if (local k:= (i-1)*ceil(n/c) + j) > n then plot(axes= none) else GT:-DrawGraph( GT:-HighlightTrail(P, Pa[k], inplace= false), stylesheet= "legacy", title= typeset(Pa[k]) ) fi )), titlefont= [Times, Bold, 12] ); #And recast that as an animation so that I can post it: plots:-display( [seq]($~(plots:-display~(PA), 5)),
insequence
);



Wirtinger derivatives in Maple 2021

by: Maple

Wirtinger Derivatives in Maple 2021

Generally speaking, there are two contexts for differentiating complex functions with respect to complex variables. In the first context, called the classical complex analysis, the derivatives of the complex components ( abs , argument , conjugate , Im , Re , signum ) with respect to complex variables do not exist (do not satisfy the Cauchy-Riemann conditions), with the exception of when they are holomorphic functions. All computer algebra systems implement the complex components in this context, and computationally represent all of  as functions of z . Then, viewed as functions of z, none of them are analytic, so differentiability becomes an issue.

In the second context, first introduced by Poincare (also called Wirtinger calculus), in brief z and its conjugate  are taken as independent variables, and all the six derivatives of the complex components become computable, also with respect to . Technically speaking, Wirtinger calculus permits extending complex differentiation to non-holomorphic functions provided that they are ℝ-differentiable (i.e. differentiable functions of real and imaginary parts, taking  as a mapping ).

In simpler terms, this subject is relevant because, in mathematical-physics formulations using paper and pencil, we frequently use Wirtinger calculus automatically. We take z and its conjugate  as independent variables, with that , , and we compute with the operators ,  as partial differential operators that behave as ordinary derivatives. With that, all of , become differentiable, since they are all expressible as functions of z and .

Wirtinger derivatives were implemented in Maple 18 , years ago, in the context of the Physics package. There is a setting, , that when set to true - an that is the default value when Physics is loaded - redefines the differentiation rules turning on Wirtinger calculus. The implementation, however, was incomplete, and the subject escaped through the cracks till recently mentioned in this Mapleprimes post.

Long intro. This post is to present the completion of Wirtinger calculus in Maple, distributed for everybody using Maple 2021 within the Maplesoft Physics Updates v.929 or newer. Load Physics and set the imaginary unit to be represented by

The complex components are represented by the computer algebra functions

 (1)

They can all be expressed in terms of  and

 (2)

The main differentiation rules in the context of Wirtinger derivatives, that is, taking  and  as independent variables, are

 (3)

Since in this context  is taken as - say - a mathematically-atomic variable (the computational representation is still the function ) we can differentiate all the complex components also with respect to

 (4)

For example, consider the following algebraic expression, starting with conjugate

 (5)

Differentiating this expression with respect to z and  taking them as independent variables, is new, and in this example trivial

 (6)

 (7)

Switch to something less trivial, replace  by the real part

 (8)

To verify results further below, also express  in terms of

 (9)

New: differentiate  with respect to  and

 (10)

 (11)

Note these results (10) and (11) are expressed in terms of , not . Let's compare with the derivative of  where everything is expressed in terms of z and . Take for instance the derivative with respect to z

 (12)

To verify this result is mathematically equal to (10) expressed in terms of  take the difference of the right-hand sides

 (13)

One quick way to verify the value of expressions like this one is to replace  and simplify   and are real

 (14)

 (15)

The equivalent differentiation, this time replacing in   by ; construct also the equivalent expression in terms of  and   for verifying results

 (16)

 (17)

Since these two expressions are mathematically equal, their derivatives should be too, and the derivatives of  can be verified by eye since  and   are taken as independent variables

 (18)

 (19)

Eq (18) is expressed in terms of  while (19) is in terms of . Comparing as done in (14)

 (20)

 (21)

 (22)

To mention but one not so famliar case, consider the derivative of the sign of a complex number, represented in Maple by . So our testing expression is

 (23)

This expression can also be rewritten in terms of  and

 (24)

This time differentiate with respect to ,

 (25)

Here again, the differentiation of , that is expressed entirely in terms of  and  , can be computed by eye

 (26)

Eq (25) is expressed in terms of  while (26) is in terms of . Comparing as done in (14),

 (27)

 (28)

 (29)

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

Unveiling Maple 2021

by: Maple 2021

I’ll admit it. There are times when I don't fully understand every mathematical advancement each release of Maple brings. Given the breadth of what Maple does, I guess that isn't surprising.

In development meetings, I make the pretence of keeping up by looking serious, nodding knowingly and occasionally asking to go back to the previous slide “for a minute”. I’ve been doing this since 2008 and no one’s caught on yet.

But I do understand

• the joy on a user’s (Zoom) face when they finally solve a complex problem with a new version of Maple
• the smiley emojis that students send us when they understand a tricky math concept with the help of an improved Maple tutor
• and the wry smile on a developer’s face when they get to work on a project they really want to work on, and the bigger smile when that project gets positive feedback

These are all moments that give me that magic dopamine hit.

The job that Karishma and I have is to make users happy. We don’t have to be top-flight mathematicians, engineers or computer scientists to do that. We just have to know what itch to scratch.

Here’s some things I think might give you that dopamine hit when you get your hands on Maple 2021. You can also explore the new release yourself at What’s New in Maple 2021.

Worksheet mode has been my go-to interface for when I just want to get stuff done. This is mostly because worksheet mode always felt like a more structured environment for developing math when I didn’t have all the steps planned out in advance, and I found that structure helpful. I’d use Document mode when I needed to use the Context Panel for math operations and didn’t want to see the commands, or I needed to create a nice looking document without input carets. And this was fine – each mode has its own strengths and uses – but I what I really wanted was the best of both worlds in a single environment.

This year, we’ve made one change that has let me transition far more of my work into Document mode.

In Document Mode, pressing Enter in a document block (math input) now always moves the cursor to the next math input (in previous releases, the cursor may have moved to the start of the next line of text).

This means you can now quickly update parameters and see the downstream effects with just the Enter key – previously, a key benefit of worksheet mode only.

There’s another small change we’ve made - inserting new math inputs.  In previous releases of Maple, you could only insert new document blocks above the in-focus block using a menu item or a three-key shortcut.

In Maple 2021, if you move the insertion point to the left of a document block (Home position), the cursor is now bold, as illustrated here:

Now, if you press Enter, the in-focus prompt is moved down and a new empty math input is created.

Once you get used to this change, Ctrl+Shift+K seems like a distance memory!

@Scot Gould logged a request that Maple numerically solve a group of differential equations collected together in a vector. And now you can!

Before Maple 2021, this expression was unchanged after evaluation. Now, it is satisfyingly simpler.

We’ve dramatically increased the scope of the signal processing package.

My favorite addition is the MUSIC function. With some careful tuning, you can generate a pseudo power spectrum at frequencies smaller than one sample.

First generate a noisy data set with three frequencies (two frequencies are closer than one DFT bin).

with(SignalProcessing):
num_points:= 2^8:
sample_rate := 100.0:
T := Vector( num_points, k -> 2 * Pi * (k-1) / sample_rate, 'datatype' = 'float[8]' ):
noisy_signal:=Vector( num_points, k -> 5 * sin( 10.25 * T[k] ) + 3 * sin( 10.40 * T[k] ) - 7 * sin( 20.35 * T[k] )) + LinearAlgebra:-RandomVector(num_points, generator=-10..10):
dataplot(noisy_signal, size = [ 800, 400 ], style = line)

Now generate a standard periodogram

Periodogram( noisy_signal, samplerate = sample_rate, size = [800, 400] )

This approach can’t discriminate between the two closely spaced frequencies.

And now the MUSIC pseudo spectrum

MUSIC( noisy_signal, samplerate = sample_rate, dimension = 6, output = plot );

The Maple Quantum Chemistry Toolbox from RDMChem, a separate add-on product to Maple, is a powerful environment for the computation and visualization of the electronic structure of molecules. I don’t pretend to understand most of what it does (more knowing nods are required). But I did get a kick out of its new molecular dictionary. Did you know that caffeine binds to adenosine receptors in the central nervous system (CNS), which inhibits adenosine binding? Want to know more about the antiviral drug remdesivir? Apparently it looks like this:

We put a lot of work into resources for students and educators in this release, including incorporating study guides for Calculus, Precalculus, and Multivariate Calculus, a new student package for ODEs, and the ability to obtain step-by-step solutions to even more problems.  But my favourite thing out of all this work is the new SolvePractice command in the Grading Tools package.  Because it lets you build an application that does this:

I like this for three main reasons:

1. It lets students practise solving equations in a way that actually helps them figure out what they’ve done wrong, saving them from a spiral of frustration and despair
2. The same application can be shared via Maple Learn for students to use in that environment if they don’t have Maple
3. The work we did to create that “new math entry box” can also be used to create other Maple applications with unknown numbers of inputs (see DocumentTools). I’m definitely planning on using this feature in my own applications.

Okay, yes, we know. Up until recently, our LaTeX export has been sadly lacking. It definitely got better last year, but we knew it still wasn’t good enough. This year, it’s good. It’s easy. It works.  And it’s not just me saying this. The feedback we got during the beta period on this feature was overwhelmingly positive.

That’s just the tip of the Maple 2021 iceberg of course. You can find out more at What’s New in Maple 2021.  Enjoy!

Feynman Integrals

by: Maple

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

Pythagorean Triples Ternary Tree

Maple 18

I make a maple worksheet for generating Pythagorean Triples Ternary Tree :

Around 10,000 records in the matrix currently !

You can set your desire size or export the Matrix as text ...

But yet ! I wish to understand from you better techniques If you have some suggestion ?

the mapleprimes Don't load my worksheet for preview so i put a screenshot !

Pythagoras_ternary.mw

Pythagoras_ternary_data.mw

Pythagoras_ternary_maple.mw

Document attachments using links in MaplePrimes...

by: MaplePrimes

The above doc/worksheet can be clicked and viewed.

Youtube URL is given below. This also can be clicked and viewed. Thanks to Carl Love's suggestion, I modified using the chain link in the editor.

https://youtu.be/0pDa4FWMSQo

Maple Evaluation Order Example

by: Maple 2020

The order in which expressions evaluate in Maple is something that occasionally causes even advanced users to make syntax errors.

I recently saw a single line of Maple code that provided a good example of a command not evaluating in the order the user desired.

The command in question (after calling with(plots):) was

animate(display, [arrow(<cos(t), sin(t)>)], t = 0 .. 2*Pi)

This resulted in the error:

Error, (in plots/arrow) invalid input: plottools:-arrow expects its 3rd argument, pv, to be of type {Vector, list, vector, complexcons, realcons}, but received 0.5000000000e-1*(cos(t)^2+sin(t)^2)^(1/2)

This error indicates that the issue in the animation is the arrow command

arrow(<cos(t), sin(t)>)

on its own, the above command would give the same error. However, the animate command takes values of t from 0 to 2*Pi and substitutes them in, so at first glance, you wouldn't expect the same error to occur.

What is happening is that the command

arrow(<cos(t), sin(t)>)

in the animate expression is evaluating fully, BEFORE the call to animate is happening. This is due to Maple's automatic simplification (since if this expression contained large expressions, or the values were calculated from other function calls, that could be simplified down, you'd want that to happen first to prevent unneeded calculation time at each step).

So the question is how do we stop it evaluating ahead of time since that isn't what we want to happen in this case?

In order to do this we can use uneval quotes (the single quotes on the keyboard - not to be confused with the backticks).

animate(display, ['arrow'(<cos(t), sin(t)>)], t = 0 .. 2*Pi)

By surrounding the arrow function call in the uneval quotes, the evaluation is delayed by a level, allowing the animate call to happen first so that each value of t is then substituted before the arrow command is called.

Maple_Evaluation_Order_Example.mw

One more way of inverse kinematics

by:

As a continuation of the posts:
https://www.mapleprimes.com/posts/208958-Determination-Of-The-Angles-Of-The-Manipulator
https://www.mapleprimes.com/posts/209255-The-Use-Of-Manipulators-As-Multiaxis
https://www.mapleprimes.com/posts/210003-Manipulator-With-Variable-Length-Of
But this time without Draghilev's method.
Motion along straight lines can replace motion along any spatial path (with any practical precision), which means that solving the inverse problem of the manipulator's kinematics can be reduced to solving the movement along a sequential set of segments. Thus, another general method for solving the manipulator inverse problem is proposed.
An example of a three-link manipulator with 5 degrees of freedom. Its last link, like the first link, geometrically corresponds to the radius of the sphere. We calculate the coordinates of the ends of its links when passing a straight line segment. We do this in a loop for interior points of the segment using the procedure for finding real roots of polynomial systems of equations RootFinding [Isolate]. First, we “remove” two “extra” degrees of freedom by adding two equations to the system. There can be an infinite set of options for additional equations - if only they correspond to the technical capabilities of the device. In this case, two maximally easy conditions were taken: one equation corresponds to the perpendicularity of the last (third) link directly to the segment of the trajectory itself, and the second equation corresponds to the perpendicularity to the vector with coordinates <1,1,1>. As a result, we got four ways to move the manipulator for the same segment. All of these ways are selected as one of the RootFinding [Isolate] solutions (in text  jj=1,2,3,4).
In this text jj=4
without_Draghilev_method.mw

As you can see, everything is very simple, there is practically no programming and is performed exclusively by Maple procedures.

The Physics Examples and LaTeX

by: Maple

One of the most interesting help page about the use of the Physics package is Physics,Examples. This page received some additions recently. It is also an excellent example of the File -> Export -> LaTeX capabilities under development.

Below you see the sections and subsections of this page. At the bottom, you have links to the updated PhysicsExample.mw worksheet, together with PhysicsExamples.PDF.

The PDF file has 74 pages and is obtained by going File -> Export -> LaTeX (FEL) on this worksheet to get a .tex version of it using an experimental version of Maple under development. The .tex file that results from FEL (used to get the PDF using TexShop on a Mac) has no manual editing. This illustrates new automatic line-breakingequation labels, colours, plots, and the new LaTeX translation of sophisticated mathematical physics notation used in the Physics package (command Latex in the Maplesoft Physics Updates, to be renamed as latex in the upcoming Maple release).

In brief, this LaTeX project aims at writing entire course lessons or scientific papers directly in the Maple worksheet that combines what-you-see-is-what-you-get editing capabilities with the Maple computational engine to produce mathematical results. And from there get a LaTeX version of the work in two clicks, optionally hiding all the input (View -> Show/Hide -> Input).

PS: MANY THANKS to all of you who provided so-valuable feedback on the new Latex here in Mapleprimes.

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

Polar Point Plot and Zooming

by:

A customer wondered if they could create a plot of points on a polar axis, and if they could zoom a polar plot.

We suggested a couple of options for making these plots, and (although zooming is not supported in polar coordinates in Maple 2020) an alternative to manually scale a polar plot.

Hope it's a useful demonstration.

polar-point-plot-with-zoom.mw

2-line tickmarks on Maple plots

by: Maple

A customer wondered if it was possible to create 2-line tickmarks on Maple plots like so

We achieved this using typeset, fractions, and backticks. Worksheet follows.

2line-tickmarks-mprimes.mw

To export a worksheet as pdf on a Mac, select...

by: Maple

'Export as pdf' is not one of the  'Export as' options on  a Mac. Instead, you should follow these instructions, from

?pdf:

1. Open the worksheet that you want to export.
2. From the File menu, select Print.
3. Specify options for export to PDF as described in Section Options for Export to PDF below.
4. From the PDF drop-down list, select Save as PDF...
5. In the Save As dialog, enter a filename.
Click Save.

small note about the new Maple 2020.2 and Latex...

by: Maple 2020

I am testing Maple 2020.2 with new Latex with Physics version latest 879.

The latex generated now issues \! between the symbol and the () next to it to improve the spacing. This post is just to let anyone using the package mleftright in Latex, that this will cause a problem. So it is better to remove this package if you are allready using it.

Here is an example

eq:=y(x)=exp(x/2);
Latex(eq)

y \! \left(x \right) = {\mathrm e}^{\frac{x}{2}}

In earlier version of Physics:-Latex (now it is just Latex), the above generated this

             y  \left(x \right) = {\mathrm e}^{\frac{x}{2}}

Notice, no \! in earlier version.

If you happen to be using \usepackage{mleftright} to improve the spacing for \left and \right, which I was using, you'll get negative side effect. A solution is to remove this package. Here is an example showing the above Latex compiled with this package added, and without it, so you can see the differerence.

\documentclass[12pt]{book}
\usepackage{amsmath}
\usepackage{mleftright}
\mleftright
\begin{document}

which gives using latest Latex V 879. Maple 2020.2
$y \! \left(x \right) = {\mathrm e}^{\frac{x}{2}}$

And which gives using earlier Physics Latex. Using Maple 2020.1
$y \left(x \right) = {\mathrm e}^{\frac{x}{2}}$
\end{document}

This is the output without using this package. by removing the inlcude command in the above Latex code and not calling mlfright. Now the problem is gone:

I like the effect added from \! , which is a manual way to improve the space, which this package was doing.

just be careful not to use mleftright package now, which is a somewhat popular package in latex and It was recommended to use sometime ago to improve the spacing, as it will over correct the spacing, and looks like not needed any more with latest Maple Latex.

Maple 2020.2 includes corrections and improvements to printing and export to PDF, support for macOS 11.0, more MATLAB connectivity, resolves issues with the installation of the Maplesoft Physics Updates, and more.  We recommend that all Maple 2020 users install these updates.

This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2020.2 download page, where you can also find more details.

Something about one degree of freedom for testing...

by: Maple 17

One forum had a topic related to such a platform. You can download a video of the movement of this platform from the picture at this link. The manufacturer calls the three-degrees platform, that is, having three degrees of freedom. Three cranks rotate, and the platform is connected to them by connecting rods through ball joints. The movable beam (rocker arm) has torsion springs.  I counted 4 degrees of freedom, because when all three cranks are locked, the platform remains mobile, which is camouflaged by the springs of the rocker arm. Actually, the topic on the forum arose due to problems with the work of this platform. Neither the designers nor those who operate the platform take into account this additional fourth, so-called parasitic degree of freedom. Obviously, if we will to move the rocker with the locked  cranks , the platform will move.
Based on this parasitic movement and a similar platform design, a very simple device is proposed that has one degree of freedom and is, in fact, a spatial linkage mechanism. We remove 3 cranks, keep the connecting rods, convert the rocker arm into a crank and get such movements that will not be worse (will not yield) to the movements of the platform with 6 degrees of freedom. And by changing the length of the crank, the plane of its rotation, etc., we can create simple structures with the required design trajectories of movement and one degree of freedom.
Two examples (two pictures for each example). The crank rotates in the vertical plane (side view and top view)
PLAT_1.mw

and the crank rotates in the horizontal plane (side view and top view).

The program consists of three parts. 1 choice of starting position, 2 calculation of the trajectory, 3 design of the picture.  Similar to the programm  in this topic.

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