## Bible of Mathematics

by: Maple

Advanced Engineering Mathematics with Maple (AEM) by Dr. Lopez is such an art.

Mathematics and Control Theory talks easily in Maple...

Thanks Prof. Lopez. You are the MAN !!

Dr. Ali GÜZEL

## Inert arithmetic operators %*, %+, ... in 2D-Math...

by: Maple

Hi
It's been years since expressions like A %* B %+ C involving inert arithmetic operators used in infix form are correctly understood (parsed) when written on a 1D-Math input line. The idea is simple: have the operators %. %*, %+, %-, %^, %/ work on input as infix operators the same way their active forms: ., *, +, -, ^, / do. This useful functionality, however, remained elusive when using 2D-Math input notation, so one would have to resort to using the functional form of the operators. E.g., input the above as `%+`(`%*`(A, B) ,C), which for me is really ugly. Besides being a bit demoralizing: we do all this fuzz about how great computer algebra and 2D-Math input notation is, and then input things in that way …

So this is to mention that this elusive functionality of inert arithmetic operators used in infix form within a 2D-Math input line now works. The novelty is present in the latest Maplesoft Physics Updates for Maple 2023, which is version 1490. As usual, to install the Updates open a Maple worksheet and input Physics:-Version(latest).

Here is an image (worksheet at the end) showing the new thing The implementation is pretty new; reports of anything related to these inert operators not displayed/working as you'd expect are much appreciated.

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

## An existential question

by: MaplePrimes

What do you think is the acceptable limit to the effort required to answer a question?

At what point does the question-and-answer game between two contributors become unreasonable?

How do you, the most highly ranked, deal with situations that last for days?

## Kudos for Maple 2023!

by: Maple 2023

Just installing Maple 2023 on my office machine (a mac); installed it on my travel computer (a Surface Pro running Windows) yesterday.

Configured Jupyter notebooks to use the 2023 Maple Kernel and it all went smoothly.  I was *delighted* to notice that plotting Lambert W in Jupyter with the command

plot( [W(x), W(-1,x)], x=-1..4, view=[-1..4, -3.5..1.5], colour=[red,blue], scaling=constrained, labels=[x,W(x)] );

produced a *better* plot near the branch point.  This is hard to do automatically!  It turns out this is a side effect of the better/faster/more memory efficient adaptive plotting software, which I gather from "What's New" was written for efficiency not for quality.  But the quality is better, too!  Nice!

I am working my. way through the "What's New" and I'm really pleased to learn about the new univariate polynomial rootfinder, *not least because it cites the paper describing the algorithm*.  Lots of other goodies too; the new methods of integration look like serious improvements.  Well done. (One thing there: "parallel Risch" is a term of art, and may lead people to believe that Maple is doing something with parallel computing there.  I don't think so.  Could a reference be supplied?)

The new colour schemes and plotting features in 3d and contour plotting look fabulous.

Direct Python language support from a code edit region is not at all what I expected to see---I wonder if it will work in a Jupyter notebook?  I'm going to have to try it...

I'm quite impressed.  The folks at Maplesoft have been working very hard indeed.  Congratulations on a fine release!

## Maple 2023: The colorbar option for contour plots...

by: Maple 2023

Maple 2023: The colorbar option for contour plots does not work when used with the Explore command. See the example below.

## Maple 2023 - Tasks dosnt work

by: Maple 2023

Just installed Maple 2023 on Macbook Air M1.

/

## A request for the implementation of the inverse...

by: Maple 2023

The inverse problem of a mathematical question is often very interesting.

I'm glad to see that Maple 2023 has added several new graph operations. The GraphTheory[ConormalProduct], GraphTheory[LexicographicProduct], GraphTheory[ModularProduct] and GraphTheory[StrongProduct] commands were introduced in Maple 2023.

In fact, we often encounter their inverse problems in graph theory as well. Fortunately, most of them can find corresponding algorithms, but the implementation of these algorithms is almost nonexistent.

I once asked a question involving the inverse operation of the lexicographic product.

Today, I will introduce the inverse operation of line graph operations. (In fact, I am trying to approach these problems with a unified perspective.)

To obtain the line graph of a graph is easy, but what about the reverse? That is to say, to test whether the graph is a line graph. Moreover, if a graph  g is the line graph of some other graph h, can we find h? (Maybe not unique. **Whitney isomorphism theorem tells us that if the line graphs of two connected graphs are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph K_3 and the claw K_{1,3}, which have isomorphic line graphs but are not themselves isomorphic.)

Wikipedia tells us that there are two approaches, one of which is to check if the graph contains any of the nine forbidden induced subgraphs.

Beineke's forbidden-subgraph characterization:  A graph is a line graph if and only if it does not contain one of these nine graphs as an induced subgraph. This approach can always be implemented, but it may not be very fast. Another approach is to use the linear time algorithms mentioned in the following article.

• Roussopoulos, N. D. (1973), "A max {m,n} algorithm for determining the graph H from its line graph G", Information Processing Letters, 2 (4): 108–112, doi:10.1016/0020-0190(73)90029-X, MR 0424435

Or:

•   Lehot, Philippe G. H. (1974), "An optimal algorithm to detect a line graph and output its root graph", Journal of the ACM, 21 (4): 569–575, doi:10.1145/321850.321853, MR 0347690, S2CID 15036484.

SageMath can do that:

`   root_graph()`

Return the root graph corresponding to the given graph.

`    is_line_graph()`

Check whether a graph is a line graph.

For example, K_{2,2,2,2} is not the line graph of any graph.

```K2222 = graphs.CompleteMultipartiteGraph([2, 2, 2, 2])
C=K2222.is_line_graph(certificate=True)
C.show() # Containing the ninth forbidden induced subgraph.
``` Another Sage example for showing that the complement of the Petersen graph is the line graph of K_5.

```P = graphs.PetersenGraph()
C = P.complement()
sage.graphs.line_graph.root_graph(C, verbose=False)
```

```(Graph on 5 vertices, {0: (0, 1), 1: (2, 3), 2: (0, 4), 3: (1, 3), 4: (2, 4), 5: (3, 4), 6: (1, 4), 7: (1, 2), 8: (0, 2), 9: (0, 3)})
```

Following this line of thought, can Maple gradually implement the inverse operations of some standard graph operations?

Here are some examples:

•   CartesianProduct
•   TensorProduct
•   ConormalProduct
•   LexicographicProduct
•   ModularProduct
•   StrongProduct
•   LineGraph
•  GraphPower

## Maple 2023 is here!

by: Maple 2023 The moment we've all been waiting for has arrived: Maple 2023 is here!

With this release we continue to pursue our mission to provide powerful technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. Bearing this in mind, our team of mathematicians and developers have dedicated the last year to adding new features and enhancements that not only improve the math engine but make that math engine more easily accessible within a user-friendly interface.

And if you ever wonder where our team gets inspiration, you don't need to look further than Maple Primes. Many of the improvements that went into Maple 2023 came as a direct result of feedback from users. I’ll highlight a few of those user-requested features below, and you can learn more about these, and many, many other improvements, in What’s New in Maple 2023.

• The Plot Builder in Maple 2023 now allows you to build interactive plot explorations where parameters are controlled by sliders or dials, and customize them as easily as you can other plots • In Maple 2023, 2-D contour and density plots now feature a color bar to show the values of the gradations. • For those who write a lot of code:  You can now open your .mpl Maple code files directly in Maple’s code editor, where you can  view and edit the file from inside Maple using the editor’s syntax highlighting, command completion, and automatic indenting. • Integration has been improved in many ways. Here’s one of them:  The definite integration method that works via MeijerG convolutions now does a better job of checking conditions on parameters so that they are only applied under proper assumptions. It also tells you the conditions under which the method could have produced an answer, so if your problem does meet those conditions, you can add the appropriate assumptions to get your result.
• Many people have asked that we make it easier for them to create more complex interactive Math Apps and applications that require programming, such as interactive clickable plots, quizzes that provide feedback, examples that provide solution steps. And I’m pleased to announce that we’ve done that in Maple 2023 with the introduction of the Quiz Builder and the Canvas Scripting Gallery.
• The new Quiz Builder comes loaded with sample quizzes and makes it easy to create your own custom quiz questions. Launch the quiz builder next time you want to author interactive quizzes with randomized questions, different response types, hints, feedback, and show the solution. It’s probably one of my favorite features in Maple 2023. • The Scripting Gallery in Maple 2023 provides 44 templates and modifiable examples that make it easier to create more complex Math Apps and interactive applications that require programming. The Maple code used to build each application in the scripting gallery can be easily viewed, copied and modified, so you can customize specific applications or use the code as a starting point for your own work • Finally, here’s one that is bound to make a lot of people happy: You can finally have more than one help page open at the same time! For more information about all the new features and enhancements in Maple 2023, check out the What’s New in Maple 2023.

P.S. In case you weren’t aware - in addition to Maple, the Maplesoft Mathematics Suite includes a variety of other complementary software products, including online and mobile solutions, that help you teach and learn math and math-related courses.  Even avid Maple users may find something of interest!

## Is there an error in dsolve using method=laplace...

Maple 2021

I was looking at symbolically solving a second-order differential equation and it looks like the method=laplace method has a sign error when the coefficients are presented in a certain way.  Below is a picture of some examples with and without method=laplace that should all have the same closed form.  Note that lines (s6) and (s8) have different signs in the exponential than they should have (which is a HUGE problem):

 > >  (1)
 >  (2)
 >  (3)
 >  (4)
 >  (5)
 >  (6)
 >  (7)
 >  (8)
 >  (9)
 >  (10)
 > >  (11)
 >  (12)
 >  (13)
 >  (14)
 > ## Simplifying transfer functions with units and...

by: Maple

Transfer functions are normally not used with units. Involving units when deriving transfer functions can help identify unit inconsistencies and reduce the likelihood of unit conversion errors.

Maple is already a great help in not having to do this manually. However, the final step of simplification still requires manual intervention, as shown in this example.

Given transfer function  (1)

Desired output (derived by hand) where the transfer function is separated in a dimensionless expression and a gain that can be attributed to units with a physical meaning in the context of an application (here displacement per voltage).  (2)  (3)

Units to factor out in the denominator are . Quick check:  (4)  (5)

"Simplification" attempts with the denominator  (6)  (7)

is not effective because all units are wrapped in Unit commands. Example:  (8)

Expand does not expand the argument of Unit commands.

 >  Units:-Unit(kg^2*m^2/s^2/A^2) C1: Expanding Unit command

An expand facility could be a solution that expands a Unit command with combined units to a product of separate Unit commands.

When all units are expanded in a separate Unit command, collect or factor can be used to collect units:  (9)  (10)  (11)

C2: Using the Natural Units Environment

In this environment, no Unit commands are required and the collection of units should work with Maple commands.
However, for the expressions discussed here, this would lead to a naming conflict with the complex variable s of the transfer function and the unit symbol s for seconds. C3: A type declaration or unit assumptions on names

A type declaration as an option of commands like in  (12)

could help Maple in simplification tasks (in its general meaning of making expressions shorter or smaller).
Alternatively, assumptions could provide information of which "unit type" a name is On various occasions (beyond transfer functions) I have looked for such a functionality.

C4: DynamicSystems Package with units

C4.1: The complex variable s could be attributed to the unit 1/s (i.e. Hertz) either by default or as an option. This could enable using units within the dynamic system package which is not possible in Maple 2022. An example what the package provides currently can be found here: The phase plot shows that the package is already implicitly assuming that the unit of s is Hertz. A logical extension would be to have magnitude plots with units (e.g. m/V, as in this example).

C4.2: A dedicated "gain" command that takes units into account and that could potentially simplify the transfer function to an expression like (2) in SI units. In such a way the transfer function is separated into a dimensionless (but frequency depended) term and a gain term with units.
This would make the transfer of transfer functions to MapleSim easy and avoid unit conversion errors.

## One of the basics of the Draghilev method

by: Maple

Does everyone have a good idea of ​​the work of the Draghilev method? For example, in this answer https://www.mapleprimes.com/questions/235407-The-Second-Example-Of-Finding-All-Solutions#answer291268
( https://www.mapleprimes.com/questions/235407-The-Second-Example-Of-Finding-All-Solutions )
there was a very successful attempt by Rouben Rostamian to calculate the line of intersection of surfaces without applying the Draghilev method.
Let now not 3d, but 8d. And how will the solve command work in this case? Imagine that aij ((i=1..7,j=1..8)) are partial derivatives, and xj (,j=1..8) are derivatives, as in the above example. f8 is responsible for the parametrization condition.

```restart;
f1 := a11*x1+a12*x2+a13*x3+a14*x4+a15*x5+a16*x6+a17*x7+a18*x8;
f2 := a21*x1+a22*x2+a23*x3+a24*x4+a25*x5+a26*x6+a27*x7+a28*x8;
f3 := a31*x1+a32*x2+a33*x3+a34*x4+a35*x5+a36*x6+a37*x7+a38*x8;
f4 := a41*x1+a42*x2+a43*x3+a44*x4+a45*x5+a46*x6+a47*x7+a48*x8;
f5 := a51*x1+a52*x2+a53*x3+a54*x4+a55*x5+a56*x6+a57*x7+a58*x8;
f6 := a61*x1+a62*x2+a63*x3+a64*x4+a65*x5+a66*x6+a67*x7+a68*x8;
f7 := a71*x1+a72*x2+a73*x3+a74*x4+a75*x5+a76*x6+a77*x7+a78*x8;
f8 := x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2-1;
allvalues(solve({f1, f2, f3, f4, f5, f6, f7, f8}, {x1, x2, x3, x4, x5, x6, x7, x8}))```

And this is how the Draghilev method works in this case.

```restart; with(LinearAlgebra):
f1 := a11*x1+a12*x2+a13*x3+a14*x4+a15*x5+a16*x6+a17*x7+a18*x8;
f2 := a21*x1+a22*x2+a23*x3+a24*x4+a25*x5+a26*x6+a27*x7+a28*x8;
f3 := a31*x1+a32*x2+a33*x3+a34*x4+a35*x5+a36*x6+a37*x7+a38*x8;
f4 := a41*x1+a42*x2+a43*x3+a44*x4+a45*x5+a46*x6+a47*x7+a48*x8;
f5 := a51*x1+a52*x2+a53*x3+a54*x4+a55*x5+a56*x6+a57*x7+a58*x8;
f6 := a61*x1+a62*x2+a63*x3+a64*x4+a65*x5+a66*x6+a67*x7+a68*x8;
f7 := a71*x1+a72*x2+a73*x3+a74*x4+a75*x5+a76*x6+a77*x7+a78*x8;

n := 7;
x := seq(eval(cat('x', i)), i = 1 .. n+1):
F := [seq(eval(cat('f', i)), i = 1 .. n)]:
A := Matrix(nops(F), nops(F)+1):
for j to nops(F) do for i to nops(F)+1 do A[j, i] := op(1, op(i, op(j, F)))
end do:
end do:

# b[i] and b[n+1] are solutions of a linear homogeneous system and at the
# same time they serve as the right-hand sides of an autonomous ODE.
for i to n do

b[i] := Determinant(DeleteColumn(ColumnOperation(A, [i, n+1]), n+1))
end do:
b[n+1] := -Determinant(DeleteColumn(A, n+1)):
```

Only the original seven linear homogeneous equations with eight variables are needed. We solve them according to Cramer's rule, and in order to have uniqueness when solving the ODE, we use a point on the curve (according to the theory). (This point is sought in any convenient way.)
If we want to get a parameterization, then additionally, directly in dsolve, we can add the following:

```for i to n do
b[i] := simplify(Determinant(DeleteColumn(ColumnOperation(A, [i, n+1]), n+1)))
end do:
b[n+1] := simplify(-Determinant(DeleteColumn(A, n+1)));
deqs := seq(diff(x[i](s), s) = b[i]/(b^2+b^2+b^2+b^2+b^2+b^2+b^2+b^2)^.5, i = 1 .. n+1):```

## Maple should still try harder

by: Maple 2022

In an old post, vv reported a bug in simpl/max, which has been "fixed" in Maple 2018. However, it seem that such repairs are not complete enough.
For example, suppose it is required to find the (squared) distance between the origin and a point on x3 - x + y2 = ⅓ which is closest to the origin. In other words, one needs to minimize x²＋y² among the points on this curve, i.e.,

`extrema(x^2 + y^2, {x^3 + y^2 - x = 1/3}, {x, y}, 's'); # in exact form`

Unfortunately, an identical error message appears again:

 > restart;
 >  (1)
 > >  (2)
 >  (3)
 >  (4)

How about changing the values of parameter ?

```for a from -3 by 3/27 to 3 do
try
extrema(x^2 + y^2, {x^3 + y^2 - x = a}, {x, y});
catch:
print(a);
end;
od;
-1
--
3

-2
--
9

-1
--
9

1
-
9

2
-
9

1
-
3
```

By the way, like extrema, Student[MultivariateCalculus]:-LagrangeMultipliers also executes the Lagrange Multiplier method, but strangely,

`Student[MultivariateCalculus][LagrangeMultipliers](y^2 + x^2, [x^3 + y^2 - x - 1/3], [x, y], output = plot):`

does not cause any errors.

## Integral Vector Calculus and parametrization of...

by: Maple

This is about functionality introduced in Maple 2022, which however is still not well known: Integral Vector Calculus and parametrization using symbolic (algebraic) vector notation. Four new commands were added to the Physics:-Vectors package, implementing the parametrization of curves, surfaces and volumes, as well as the computation of path, surface and volume vector integrals. Those are integrals where the integrand is a scalar or vector function. The computation is done from any description (algebraic, parametric, vectorial) of the region of integration - a path, surface or volume.

There are three kinds of line or path integrals:    NOTE Jan 1: Updated the worksheet linked below; it runs in Maple 2022.

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

## New Maple application center search doesn't work

The search query in the new Maple Application center is broken.  There is no advanced search options and a search for mapleflow or maple flow brings up 0 results.  There should at least be one found, for example the search should have at least brought up The Liquid Volume in a Partially-Filled Horizontal Tank".

## New coordinate-system labels in Physics

by: Maple

Notation is one of the most important things to communicate with others in science. It is remarkable how many people use or do not use a computer algebra package just because of its notation. For those reasons, in the context of the Physics package, strong emphasis is put on using textbook notation as much as possible regarding input and output, including, for that purpose, as people here know, significant developments in Maple typesetting.

Still, for historical reasons, when using the Physics package, the labels used to refer to a coordinate system had been a single Capital Letter, as in X, Y, ...It was not possible to use, e.g. X', or x.

That has changed. Starting with the Maplesoft Physics Updates v.1308, any symbol can be used as a coordinate system label. The lines below demo this change.  1 2 3 4 5 6 7 Last Page 1 of 78