Recently here Mapleprimes member @Icz asked about generating all minimal edge cuts of a graph. I gave a brute force algorithm based on testng all bipartitions of the vertices. I then looked into improving the method by using fundamental cutsets to generate the cutsets. A description of the method and the ideas behind it is given here, together with a comparison of the two methods.

[Edit - see below for a newer version.]

Download CutSets.mw

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[1]^2+b[2]^2+b[3]^2+b[4]^2+b[5]^2+b[6]^2+b[7]^2+b[8]^2)^.5, i = 1 .. n+1):

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: 

Download tryHarder.mws

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.

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.
Download Integral_Vector_Calculus_and_Parametrization.mw

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

I have been making animated 3d plots recently; the last time was perhaps three years ago, and I had some problems then.  If I recall correctly, I couldn't make an animated 3d plot that was plotted in non-Cartesian coordinates.

I am very happy to report that this works very smoothly now in Maple 2022, and it's pretty fast, too.  I have a fairly complex function to plot, involving piecewise polynomials on a tensor product grid in the xi and eta variables (actually, I let plot3d pick out the grid; it seems happier to do so) and then plot them on an elliptical base, in coordinates x = d*cosh(xi)*cos(eta) and y=d*sinh(xi)*sin(eta)  (d is just a numerical constant, giving the location of the foci at (d,0) and (-d,0)), for 0 <= xi <= xi[0] (the outer elliptical boundary) and 0 <= eta <= 2Pi.  The straightforward command works, and building a sequence of plots and using plots[display] works.  I put option remember into my procedure w(xi,eta) and because the sample points are consistent for the time-dependent function exp(I*omega*t)*w(xi,eta) the xi-eta grid needs only to be done once and then one can compute (basically) as many frames as one wants in rapid succession.

Works great.  Thanks, folks!

for k to nplots do
    t := evalf(2*Pi*(k - 1)/nplots);
    plts[k] := plot3d([(xi, eta) -> focus*cosh(xi)*cos(eta), (xi, eta) -> focus*sinh(xi)*sin(eta), (xi, eta) -> Re(exp(omega*t*I)*w(xi, eta))], 0 .. xi[0], 0 .. 2*Pi, colour = ((xi, eta) -> Re(exp(omega*t*I)*w(xi, eta))), style = surfacecontour, lightmodel = "none");
end do;
plots[display](seq(plts[k], k = 1 .. nplots), insequence = true);
 

With the winter solstice speeding towards us, we thought we’d create some winter themed documents. Now that they’re here, it’s time to show you all! You’ll see two new puzzle documents in this post, along with three informative documents, so keep reading.

Let’s start with the tromino tree!

First, what’s a tromino? A tromino is a shape made from three equal sized squares, connected to the next along one full edge. In this puzzle, your goal is to take the trominos, and try to fill the Christmas tree shape.

There’s a smaller and larger tree shape, for different difficulties. Try and see how many ways you can fill the trees!

Next, we’ll look at our merry modulo color by numbers.

In this puzzle, your goal is to solve the modulo problems in each square, and then fill in the square with the color that corresponds to the answer. Have fun solving the puzzle and seeing what the image is in the end!

Snowballs are a quintessential part of any winter season, and we’ve got two documents featuring them.

The first document uses a snowball rolling down a hill to illustrate a problem using differential equations. Disclaimer: The model is not intended to be realistic and is simplified for ease of illustration. This document features a unique visualization you shouldn’t miss!

Our second document featuring snowballs talks about finding the area of a 2-dimensional snowman! Using the formula for the area of a circle and a scale factor, the document walks through finding the area in a clear manner, with a cute snowman illustration to match!

The final document in this mini-series looks at Koch snowflakes, a type of fractal. This document walks you through the steps to create an iteration of the Koch snowflake and contains an interactive diagram to check your drawings with!

I hope you’ve enjoyed taking a look at our winter documents! Please let us know if there’s any other documents you’d like to see featured or created.

This command should have been in Physics on day one. Being more familiar with functional differentiation, and Physics:-Fundiff was the first Physics command that ever existed, I postponed writing LagrangeEquations year after year. In general, however, functional differentiation is seen as a more advanced topic. So there is now a new command, Physics:-LagrangeEquations, taking advantage of functional differentiation on background, and distributed for everybody using Maple 2022.2 within the Maplesoft Physics Updates. This is the first version of its help page.

Download LagrangeEquations.mw

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

Welcome back to another Maple Learn blog post! Today we’re going to talk about the gift-wrapping algorithm, used to find the convex hull of a set of points. If you’re not sure what that means yet, don’t worry! We’re going to go through it with four Maple Learn documents; two which are background information on the topic, one that is a visualization for the gift-wrapping algorithm, and another that goes through the steps. Each will be under their own heading, so feel free to skip ahead to your skill level!

Before we can get into the gift-wrapping algorithm we need to define a few terms. Let’s start by defining polygons and simple polygons.

Polygon: A closed shape created by joining a series of line segments.

Simple polygon: A polygon without holes and that does not intersect itself.

So, what are convex and concave polygons? Well, there are three criteria that define a convex polygon. A polygon that is not convex is called concave. The criteria are…

1. Any line segment connecting any two points within the polygon stays within the polygon.
2. Any line intersects a polygon’s boundary at most twice.
3. All interior angles are less than 180 degrees or pi radians.

Because the criteria are equivalent, if any one is missing, the shape is concave. AKA, all three criteria must be present for a shape to be convex. Most “regular shapes”, such as trapezoids, are convex polygons!

A shape that satisfies convex criteria but not the criteria for being a polygon is called a convex set.

As mentioned at the start of this post, the gift-wrapping algorithm is used to find the convex hull of a set of points. Now that we know what convex polygons and convex sets are, we can define the convex hull!

Convex hull: The convex set of a shape or several shapes that fully contains the object and has the smallest possible area.

Why was the convex polygon important? Well, the convex hull of a set of points is always a convex polygon. Some of the points in the set are the vertices of said polygon, and are called extreme points. You can find the convex hull of either concave or convex polygons.

This document amazed me when I tried it for the first time. Here, you can generate a set of points with the “Generate Another” button, and then press the “Visualize” button. The document then calculates the perimeter of the convex hull of the set of points! The set can be further customized below the buttons, by changing the number of points. The other option below it allows you to slow down or speed up the visualization. Pretty cool, huh? It’s like it’s thinking!

Try the document out a few times, or watch the gif below to get a quick idea of it.

This final document walks you through the steps of how to use the gift wrapping algorithm. It is a simple loop of 4 steps, with one set-up step. Unlike the other documents in this post, I won’t be delving too far into the math behind the steps. I want to encourage you to check this one out yourself, as it’s really quite a fun problem to solve once you have some time!

I hope you check out the documents in this post. Please let us know below if there’s any other documents you’d like to see featured!

A failing slinky is another intriguing physics phenome that can be easily reproduced with MapleSim.

The bottom of a vertically suspended slinky does not move when the top is released until the slinky is fully collapsed.

To model this realistically in MapleSim, it is necessary to

• Establish a stretched equilibrium state at the start of the fall
• Avoid penetration of windings when windings collapse (i.e. get into contact)

The equilibrium state is achieved with the snapshot option. Penetration is avoided with the Elasto Gap component. Details can be found in the attached model.

A good overview of “Slinky research” is given here. The paper provides a continuous description of the collapse process (using an inhomogenous wave equation combined with contact modeling!!!) and introduces a finite time for the collapse of all windings. Results for a slinky are presented that collapses after 0.27s. The attached model has sufficient fidelity to collapse at the same time.

Real Slinkies also feature a torsional wave that precedes the compression wave and disturbs an ideal collapse. This can be seen on slowmo footage and advanced computer models. With a torsion spring constant at hand (are there formulas for coil springs?), it could also be modeled with MapleSim.

Falling_slinky.msim

Have you ever heard of the Maurer Rose?

The Maurer Rose was demonstrated in 1987 by Peter Maurer and is created by connecting certain points on a rose curve. This creates petal-like patterns, caused by the oscillation of a sine curve.

So, how are these created? A "rose curve" is created in polar coordinates with the equation sin(nt) for a (positive integer) value of n.  To create the Maurer Rose, straight line segments are drawn connecting points on the curve at incrementing angle values.  The size of this increment (called d in our examples) leads to different patterns of lines across the curve.

This can be done in Maple Learn! One example of the Maurer Rose already exists, complete with a full interactive visualization and a more detailed overview of the Maurer Rose.

Play around with it and look below at some of the different shapes that can be created using this document! The first is created with an n value of 31 and a d value of 65, with blue and red. The second uses an n value of 4 and a d value of 133, and purple and green.

Are there any other concepts you’d like to see represented in Maple Learn’s document gallery? Please let us know in the comments below!

Hello MaplePrimes community,

We just created a Frequently Asked Question article that may address some Primes questions about updates to Physics in Maple 2022.2: 

Why does Maple 2022.2 throw an error executing Physics:-Version(latest)?

For searchability, the specific error in question is

Error, (in Physics:-Version) unable to determine the Physics Updates version, could you please report the problem to support [at] maplesoft [dot] com

• Maplesoft will work to improve package updating in future versions of Maple.
• In Maple 2022.2, the workaround is to install and/or update the Maplesoft Physics Updates using the MapleCloud toolbar.

An example of uniform motion along a generalized coordinate using the Draghilev method. (This post was inspired by school example in one of the forums.)
The equations used in the program are very simple and, I think, do not require any special comments. DM is a procedure that implements the Draghilev method with "partial parameterization".
DM_V.mw

When K = 1, parameterization is carried out by changing the angle of rotation of the wheel. That is, uniform rolling is carried out.

For K = 4, the coordinate corresponding to the position of the slider is parametrized.

When K = 6, the slider moves with acceleration, according to a given equation. Hence, we have carried out the parameterization with respect to “time”.

With the help of such techniques, we can obtain the calculation of the kinematics of both lever mechanisms and various types of manipulators.

Have you heard of Maple Scripting before? Do you want to extend your Maple Learn documents with your Maple knowledge? Scripting is the process of using Maple to create Maple Learn documents. If you’re already used to Maple, this may be a piece of cake for you, but we wanted to start from the basics for anyone who wants to extend their Maple Learn and Maple knowledge. This process can be used for many different types of documents, from quizzes to intensive 3D visualizations.

So, let’s get started! All Maple Learn document scripting needs the DocumentTools:-Canvas package. The canvas, as you know, is that white space in a Maple Learn document. Therefore, this package is the core content of a scripted document! Always put:

with(DocumentTools:-Canvas):

At the top of your code, or put

uses DocumentTools:-Canvas:

At the start of your procedures.

Now that we’ve told Maple to use the DocumentTools:-Canvas, we need to create a canvas.

Canvases are created as variables, using the command NewCanvas. Inside NewCanvas, you will add a square-bracket list of all the content you want to see inside. For now, just know that you can add text cells with Text(“YOUR TEXT”) and a math cell with Math(YOUR MATH). On the next line, make sure to put either ShareCanvas(YOUR CANVAS VARIABLE) or ShowCanvas(YOUR CANVAS VARIABLE).  ShareCanvas creates a Maple Learn sharelink, while ShowCanvas shows the canvas directly in Maple. Note that ShowCanvas does not have every Maple Learn feature, but makes quick work of fast error checking.

canvas := NewCanvas([Text(“My first canvas”), Math(3*x+2*y)]):

ShareCanvas(canvas);

There are two more things I want to show you in this post: How to make a group have multiple cells (instead of just the one), and how to position your items on the canvas. Let’s start with group making.

To create a group with multiple cells, use the Group() command within the NewCanvas command, and separate the cells with commas, in a list. You don’t need to specify Text() or Math() when using Group().

canvas := NewCanvas([Group([“This is the first cell…”, “The second….”, “and the third.”])]):

At the end of any command/canvas element, within the brackets, you can define position=[x,y] to specify where on the canvas the object should go. You can adjust the precision pixel by pixel until you are happy with the layout.

When we put all these together, we get code that looks like this:

with(DocumentTools:-Canvas):

canvas := NewCanvas([

Group(["This is the first cell…", "The second…", "and the third."], position=[200,200]),

Math(3*x+2, position=[100,100]),

Text("This is text!", position=[400,400])]):

ShareCanvas(canvas);

And in the end, your scripted document looks like this.

We hope this helps you get started with Maple Scripting. There will be another post on even more of what we can do with Maple Scripting, and how we can make these documents even more interactive. Let us know if there’s anything specific you want to see in that post!

We've just released Maple Flow 2022.2. The update enhances the user experience in many areas, including user interaction, performance, and the interface.

Performance is a signficant focus.

• Maple Flow prioritizes the evaluation of the math you see on screen, giving you faster calculation updates for the part of the worksheet you’re working on, with more math being evaluated as you scroll down.
• We also have more users developing larger documents. Adding white space to large documents, and interacting with sections is now more response and snappier.

In response to many user requests for faster interaction, a new optional evaluation method lets you simply hit equals to evaluate math and display results.

We've also refreshed the in-product Application Gallery with a new look and many new applications (this includes a library of section properties).

You can also optionally restrict printing to the left-most column of pages, allowing you to have off-screen supporting calculations not displayed in the final report.

You'll find a complete list of enhhacements here, and you can download the update here.

There is something wrong with the search in mapleprimes. 

I noted one users comment last week about using the search to find something being very difficult to find.  The results don't often match the search.  Only just this morning I was looking for something and the results were less than satisfying often pointing to a list of more results which didn't seem to help. 

This has been a problem for a long time, is this going to be looked at or fixed soon?