Circles inscribed between curves can be specified by a system of equations relative to the coordinates of the center of the circle and the coordinates of the tangent points. Such a system can have 5 or 6 equations and 6 variables, which are mentioned above.
In the case of 5 equations, we can immediately obtain an infinite set of solutions by selecting the ones we need from it. 
(See the attached text for more details.)
The 1st equation is responsible for the belonging of the point of tangency to one of the curves.
The 2nd equation is responsible for the belonging of the point of tangency to another curve.
In the 3rd equation, the points of tangency on the curves belong to the inscribed circle.
In the 4th and 5th equations, the condition is satisfied that the tangents to the curves are perpendicular to the radii of the circle at the points of contact.
The 6th equation serves either to find a specific inscribed circle or to find an infinite set of solutions. It is selected based on the type of curves and their mutual arrangement.

In this example, we search for a subset of the solution set using the Draghilev method by solving the first five equations of the system: we inscribe circles in two "angles" formed by the intersection of the exponent and the ellipse.
The text of this example, its solution in the form of a picture,"big" option and pictures of similar examples.
INSCRIBED_CIRCLES.mw


 

Addition 09/01/24, 
One curve for the first two equations in coordinates x1,x2 and x3,x4
f1:= x1^2 - 2.5*x1*x2 + 3*x2^2 - 1;
f2:= x3^2 - 2.5*x3*x4 + 3*x4^2 - 1;

This post is inspired by minhthien2016's question.

The problem, denoted 2/N/1, for reasons that will appear clearly further on, is to pack N disks into the unit square in such a way that the sum of their radii is maximum.

I replied this problem using Optimization-NLPSolve for N from 1 (obvious solution) to 16, which motivated a few questions, in particular:

• @Carl Love: "Can we confirm that the maxima are global (NLPSolve tends to return local optima)?" 
  Using NLPSolve indeed does not guarantee that the solution found is the (a?) global maximum. In fact packing problems are generaly tackled by using evolutionnary algorithms, greedy algorithms, or specific heuristic strategies.
  Nevertheless, running NLPSolve a large number of times from different initial points may provide several different optima whose the largest one can be reasonably considered as the global maximum you are looking for.
  Of course this may come to a large price in term of computational time.
   
• @acer: "How tight are [some constraints], at different N? Are they always equality?"
  The fact some inequality constraints type always end to equality constraints (which means that in an optimal packing each disk touches at least one other annd, possibly the boundary of the box) seems hard to prove mathematically, but I gave here a sketch of informal proof.

I found 2/N/1 funny enough to spend some time digging into it and looking to some generalizations I will refer to as D/N/M:  How to pack N D-hypersheres into the unit D-hypercube such that the sum of the M-th power of their radii is maximum?
For the sake of simplicity I will say ball instead of disk/sphere/hypersphere and box instead of square/cube/hypercube.

The first point is that problems D/N/1 do not have a unique solution as soon as N > 1 , indeed any solution can be transformed into another one using symmetries with respect to medians and diagonals of the box. Hereafter I use this convention:

Two solutions s and s' are fundamental solutions if:

1. the ordered lists of radii s and s'  contain are identical but there is no composition of symmetries from s to s',
2. or, the ordered lists of radii s and s'  contain are not identical.
    

It is easy to prove that 2/2/1 and 3/2/1, and likely D/2/1, have an infinity of fundamental solutions: see directory FOCUS__2|2|1_and_3|2|1 in the attached zip file..
At the same time 2/N/2, 3/N/3, and likely D/N/D, have only one fundamental solution (see directory FOCUS__2|N|2 for more details and a simple way to characterize these solutions

 (Indeed the strategy ito find the solution of D/N/D  in placing the biggest possible ball in the largest void D/N-1/D contains. Unfortunately this characterization is not algorithmically constructive in the sense that findind this biggest void is a very complex geometrical and combinatorial problem.
 it requires finding the largest void  in a pack of balls)

Let M_{d, 1}(N)  the maximum value of the sum of balls radii for problem d/N/1.
The first question I asked myself is: How does M_{d, 1}(N) grows with N?
 

(M_{d, 1}(N) is obviously a strictly increasing function of N: indeed the solution of problem d/N/1 contains several voids where a ball of strictly positive radius can be placed, then  M_{d+1, 1}(N) > M_{d, 1}(N) )

The answer seems amazing as intensive numerical computations suggest that
                                      

See D|N|M__Growth_law.mw in the attached sip file.
This formula fits very well the set of points  { [n, S_{d, 1}(n) , n=1..48) } for d=2..6.
I have the feeling that this conjecture might be proven (rejected?) by rigourous mathematical arguments.


Fundamental solutions raise several open problems:

• Are D/2/1 problems the only one with more than one fundamental solutions?
  
  The truth is that I have not been capable to find any other example (which does not mean they do not exist).
  A quite strange thing is the behaviour of NLPSolve: as all the solutions of D/2/1 are equally likely, the histogram of the solutions provided by a large number of NLPSolve runs from different initial points is far from being uniform.
  For more detail refer ro directory FOCUS__2|2|1_and_3|2|1 in the attached zip file. 
  I do not understand where this bias comes from: is it due to the implementation of SQP in NLPSolve, or to SQP itself?
   
• For some couples (D, N) the solution of D/N/1 is made of balls of same radius.
  For N from 1 to 48 this is (numerically) the case for 2/1/1 and2/2/1, but the three dimensional case is reacher as it contains  3/1/1, 3/2/1,  3/3/1,  3/4/1 and 3/14/1 (this latter being quite surprising).
  Is there only a finite number of values N such that D/N/1 is made of balls with identical radii?
  If it is so, is this number increasing with D?
  It is worth noting that those values of N mean that the solution of problems D/N/1 are identical to those of a more classic packing problem: "What is the largest radius N identical balls packed in a unit bow may have?".
  For an exhaustive survey of this latter problem see Packomania.
   
• A related question is "How does the number of different radii evolves as N increases dor given values of D and M?
  Displays of 2D and 3D packings show that the set of radii has significantly less elements than N... at least for values of N not too large. So might we expect that solution of, let us say, 2/100/1 can contain 100 balls of 10 different radii, or it is more reasonable to expect it contains 100 balls of 100 different radii?
   
• At the opposite numerical investigations of  2/N/1 and  3/N/1 suggest that the number of different radii a fundamental solution contains increases with N (more a trend than a continuous growth).
  So, is it true that very large values of N correspond to solutions where the number of different radii is also very large?
  Or could it be that the growth of the number of different radii I observed is simply the consequence of partially converged results?
   
• Numerical investigations show that for a given dimension d and a given number of balls n,  solutions of d/n/1 and d/n/M (1 < M < d) problems are rather often the same. Is this a rule with a few exceptions or a false impression due to the fact that I did not pushed the simulations to values of n large enough to draw a more solid picture)?

It is likely that some of the questions above could be adressed by using a more powerful machine than mine.


All the codes and results are gathered in  a zip file you can download from OneDrive Google  (link at the end of this post, 262 Mb, 570 Mb when unzipped, 1119 files).
Install this zip file in the directory of your choice and unzip it to get a directory named PACKING. 
Within it:

• README.mw contains a description of the different codes and directories
• Repository.rtf must contain a string repesenting the absolute path of directory PACKING. 

Follow this link OneDrive Google

As AI becomes increasingly relevant in the tech world, Maplesoft has taken steps to integrate AI into our products. We recently launched two new features: Ask AI in Maple Learn and Word Problem Solver in Maple Calculator. 

Ask AI - Maple Learn

As a Math Content Creator at Maplesoft, sometimes I find myself in a creative rut. What documents would be engaging for students? How can I address certain math topics in a fun and interactive way?

I've had the pleasure of creating several collections during my time, including Extreme Value Theorem, Intermediate Value Theorem, and Polynomial Long Division. Nonetheless, each collection took a lot of storyboarding and creativity before I even began drafting them, and I've missed out on creating so many more collections because of this long idea generation process. Having a tool in my back pocket to reignite those creative juices would make it so much easier and faster to create new and exciting Maple Learn documents. 

Luckily, our new Ask AI feature in Maple Learn can help with that! 

Whenever you enter text into a Maple Learn document, a new Context Panel operation called "Ask AI" will pop up. Simply click that button to receive an AI response related to your prompt.

One of my favourite uses of Ask AI is to pick a random subject or phrase and see what the AI responds with. The Ask AI feature is designed to respond with a mathematics-centric answer so it will twist even the least mathematical of concepts into a math problem! The prompt "tacos" resulted in some formulas about sharing tacos with friends, and a prompt of "celebrity gossip" introduced statistical functions to compute the number of celebrity mentions per day. 

I also found that completing part of a tongue twister will result in some funny AI responses!

Here are a couple of my favorites below:

"She sells sea shells..."

Ask AI completes this tongue twister, then offers some formulas to compute the profit of selling S shells!

"How much wood..."

After relating that this tongue twister is not a mathematical problem, Ask AI then builds a simple formula for computing how much wood a woodchuck would (hypothetically) chuck.

There are many more applications of this feature, and I hope you all enjoy exploring them as you create documents on Maple Learn. If you're having trouble inputting text into your documents, or looking for a quick introduction to Maple Learn, check out the Walkthrough Tutorial. Beginner Tutorial (slide 8) addresses adding text to your document. Check out this blog post if you aren't sure how to access the Walkthrough Tutorial. 

Word Problem Solver - Maple Calculator

Maple Calculator now offers support for word problems by leveraging AI. Simply take a picture of your word problem and Maple Calculator will provide a solution generated by AI.

Here is a quick example:

I wrote on paper, “Alice and Bob have 17 apples total. Alice has double the number of apples as Bob plus two. How many apples does Bob have?”. Then I took a picture of this in Maple Calculator, and it gave me a breakdown of the problem using linear equations. See screenshots of my Maple Calculator below.

AI can be an amazing tool, but it can also make mistakes. We ensure that all our tools that incorporate AI clearly indicate its use, so that our users can know when AI is used and choose whether to use it. We're committed to remaining transparent about AI as our journey continues and we are always open to feedback. 

For our community of educators, a valuable exercise for students might be to show examples where AI makes mistakes and encourage students to find and explain the errors.

As an example, here is an algebra problem answered by Ask AI in Maple Learn – but it made a mistake! See if your students can spot where it went wrong and explain what should happen instead.

Building these skills will translate into good critical thinking skills that will benefit students inside and outside the classroom. For example, these exercises aim to help students identify their own mistakes in math and critically evaluate online sources. We would love to hear feedback about these exercises if you try them.

We hope these features will come in handy next time you use Maple Learn and Maple Calculator! 

Download Attractors.mw

 Introduction
Maple Coding Expert is a GPT-based AI tool designed to assist with various mathematical tasks using Maple software. It offers step-by-step guidance and detailed explanations for a range of functions, making it a valuable resource for students, educators, and professionals.

 Core Features and Functions

1.Graph Creation:

   - Function Plotting: Users can plot a wide range of mathematical functions. For instance, to plot the function y = x², the user would input the command `plot(x^2, x = -10..10);` in Maple. The expert helps in setting up the plotting parameters to visualize the function effectively.
   - Advanced Graphing: Beyond simple functions, the expert can guide users through plotting more complex functions and customizing plots with labels, legends, and different styles.

2. Equation Definition and Manipulation:

   - Defining Equations: The tool assists in defining equations for various calculus operations. For example, to differentiate a function, the command might be `diff(f(x), x);`. This helps in accurately modeling the equations necessary for solving real-world problems.
   - Solving Integrals: For integral calculus, users can get assistance in setting up both definite and indefinite integrals. Commands like `int(f(x), x);` are used to perform integration in Maple.

3. Calculus Problem Solving:
   - Differentiation and Integration: The expert provides guidance on solving derivatives and integrals, which are fundamental operations in calculus. It supports both symbolic and numerical methods, allowing users to choose the best approach for their problem.
   - Differential Equations: Users can solve ordinary and partial differential equations using commands like `dsolve({equations}, {variables});`. The expert offers advice on choosing solution methods and interpreting results.

I recently tried using the Maple Coding Expert for solving some calculus problems. It worked well overall and provided detailed solutions, though sometimes it approached the problems in a more complicated way than expected. Despite this, the accuracy and depth of the explanations were impressive and very helpful for understanding the underlying concepts.

Maple Coding Expert stands out as a comprehensive tool for anyone involved with Maple software for mathematical computing. It enhances learning, supports professional tasks, and aids in solving complex mathematical problems with ease.

For more information, you can explore the Maple Coding Expert on [GPTs Hunter](https://www.gptshunter.com/gpt-store/MzExMzI2MzYyMzJlNTAxMjM2) and [YesChat.ai](https://www.yeschat.ai).

I’m thrilled to introduce the updated Q&A Cards Creator! Michael Barnett had created the original Flash Cards Creator, inspired by the quiz creators in the Maple Learn gallery. I added some of the features (mentioned later in this post) that will help you use this tool to make more comprehensive quizzes. Students can use the creator to quiz themselves before a test, and instructors can integrate more practice quizzes into their lesson plans. One feature I particularly love is the ability to link full solutions to the back of each card, allowing users to understand the answers in depth (as shown in this document). Additionally, you can link a general solutions package (as seen here) if individual solutions aren’t necessary for each question. Below is an example of what the Q&A cards can look like from the users point of view.
This creator is a great example of the Maple Learn documents you can create through scripting in Maple. With a single script, you can create an infinite amount of content and quizzes. If you are interested in Maple scripting, here is a link to the Q&A cards script. If this script looks intimidating, feel free to check out this blog post on the basics of Maple scripting!


If you are interested in creating your own Q&A quiz, you can go to this document to get started. If you get stuck at any point creating your card set, check out the instructions included in the document for clarification. We hope you enjoy creating some quizzes with this document!

Download predator_prey2.mw

Dear Maple Community,

It has been a year since the passing of Stefan Vorkoetter, who started contributing to the Maple project in the 80s and was a long term member of our development team. 

Here are a few recently published articles about Stefan, that I'd like to share with you:

https://mapletransactions.org/index.php/maple/article/view/18269
https://mapletransactions.org/index.php/maple/article/view/18681

we shall not forget

We are pleased to announce that the registration for the Maple Conference 2024 is now open.

Like the last few years, this year’s conference will be a free virtual event. Please visit the conference page for more information on how to register.

This year we are offering a number of new sessions, including more product training options and an Audience Choice session.
You can find an overview of the program on the Sessions page. Those who register before September 10th, 2024 will have a chance to vote for the topics they want to learn more about during the Audience Choice session.

We hope to see you there!

Maple Learn has so much to offer, but it can be tricky to know where to start! Even for those experienced with Maple Learn, sometimes, we miss an update with new features or fall out of practice with older ones. Luckily, we have the perfect solution for you–and it shows up right when you open your first document.

Introducing our brand-new Walkthrough Tutorial!

The tutorial covers all the main features of Maple Learn: from assigning functions, to using the Plot commands and Context Panel operations, all the way to creating your own visualizations with the Geometry commands. Stuck? Hints are provided throughout, or just click "Next" and the step will be completed automatically. 

If you're just starting out with Maple Learn, try the Beginner tutorial and work up to Advanced. This will introduce you to a holistic view of Maple Learn's capabilities along with some Maple Learn terminology. If you have some experience, starting with the Beginner tutorial is still a great option, but you may wish to begin with the Intermediate and Advanced tutorials. The Intermediate and Advanced sections cover how to use newer features of Maple Learn and you might discover something you haven't seen before!

How do I access the tutorial?

The tutorial will automatically launch when you open a new document. Head to https://learn.maplesoft.com and click "Open new document".

If the tutorial doesn't open automatically, it may have been disabled. You can manually open it by clicking the "Help" button in the top right, then clicking "Walkthrough Tutorial". 

There you have it! I had been using Maple Learn for the past few months and only recently discovered these two incredible features:

Silencing Groups (Intermediate - 6/7)

Live Sessions (Advanced - 6/6)

I found these features thanks to the Walkthrough Tutorial and my experience on Maple Learn hasn't been the same since! The Walkthrough Tutorial is a great introduction for new users, and a quick refresher for experts, but isn't the end of exploring Maple Learn's capabilities. See our How to Use Maple Learn (maplesoft.com) collection and our Getting Started with Maple Learn (youtube.com) video for more. You can also challenge The Treasure of Maple Learn (maplesoft.com) – a collection of documents designed to gamify exploring Maple Learn's features. Check out our blog post on The Case of the Mysterious Treasure - MaplePrimes to learn more about this collection. 

Hope you enjoy our new tutorial and let us know what you think!

This is a reminder that presentation applications for the Maple Conference are due July 17, 2024.

The conference is a a free virtual event and will be held on October 24 and 25, 2024.

We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn. You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page.

I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or an attendee!

Kaska Kowalska
Contributed Program Co-Chair

The Proceedings of the Maple Conference 2023 is now out, at

mapletransactions.org

The presentations these are based on (and more) can be found at https://www.maplesoft.com/mapleconference/2023/full-program.aspx#schedule .

There are several math research papers using Maple, an application paper by an undergraduate student, an engineering application paper, and an interesting geometry teaching paper.

Please have a look, and don't forget to register for the Maple Conference 2024.

A simple visual way to show that the parametric equation of a circle is a helix in our three-dimensional space.
Parametric equation of a circle f1 and f2.
The helix is ​​defined by the intersection of two mutually perpendicular cylindrical surfaces f1 and f2.

 

restart; with(plots): 
R := 1.;
f1 := x1-R*cos(x3); 
f2 := x2-R*sin(x3); 
PT := implicitplot3d([f1, f2], x1 = -6 .. 6, x2 = -6 .. 6, x3 = -2 .. 12, numpoints = 10000, style = surface, color = [blue, green], transparency = .5):
IT := intersectplot(f1, f2, x1 = -1 .. 1, x2 = -1 .. 1, x3 = -2 .. 12, thickness = 3, axes = normal, grid = [10, 10, 30]): 
display(PT, view = [-6 .. 6, -6 .. 6, -2 .. 12]);
display(PT, IT, view = [-6 .. 6, -6 .. 6, -2 .. 12]); 
display(IT, view = [-R .. R, -R .. R, -2 .. 12], scaling = constrained)

We have just released an update to Maple. Maple 2024.1 includes improvements to the math engine, PDF export, the Physics package, command completion, and more. As always, we recommend that all Maple 2024 users install this update. In particular, please note that this update includes fixes to ODESteps and simplifying integrals, as reported on Maple Primes. Thanks for helping us, and other users, by letting us know!

At the same time, we have also released an update to MapleSim. MapleSim 2024.1.1 includes improvements to FMU import/export, plotting, co-simulation, and more, as well as enhancements to the Web Handling Library.

These updates are available through Tools>Check for Updates in Maple or MapleSim, and are also available from the Download Product Updates section of our web site, where you can find more details.

This post summarizes links for those who have not studied numerical integration methods from scratch and are interested in simulation settings in MapleSim (like me).

The MapleSim help pages simulation settings and advanced simulation settings give first guidance for the trained user but do not provide explanations or links for the terms used in the description of the settings (as for example: stiffness, constraint stabilization, constraint projection, events and event iteration,...).

It can easily be overlooked that Maple help pages provide further information for most of the terms. Under the assumption that MapleSim uses the same terminology as Maple, I recommend to first have a look at Maple help topics before consulting the web or other resources. Since searching and retrieving can be time consuming, I made a list of helpful links.

There are still some open points. I would be happy for more links and help in filling these gaps.

How Maple simulates

?MapleSimUserGuide,Chapter04:
section 4.1 How MapleSim Simulates a Model

?tasks,generatingCode

Solvers

An overview of solvers: ?dsolve,numeric

Differential Algebraic Equation introduction: ?MaplePortal,DAE

Overview of numeric differential-algebraic equation solvers (index reduction, constraint drift, projection):
 ?examples/numeric_DAE and ?dsolve,numeric,DAE_extension

Stiffness and stiff solvers

Stiffness and stiff IVPs: ?dsolve,Stiffness

Events

?dsolve,numeric,Events

Time events and state events

Event handling:

?MapleSimUserGuide,Chapter04:
section 4.1 How MapleSim Simulates a Model

Event iteration:

?MapleSimUserGuide,Chapter05:
section 5.5 Selecting the Code Generation Options

Iteration, hysteresis, Intermediate steps: ?tasks,generatingCode

Hysteresis:

Hysteresis in value or also in time?

Do variable solvers adapt the value of event hysteresis during runtime?

Baumgarte constraint stabilization, unconstrained dynamics, constrained dynamics

?MapleSim,Multibody,Dynamic_Exports
(in combination with ?MapleSim,Multibody,Kinematic_Exports)

?examples/numeric_DAE

?tasks,generatingCode

?MapleSimUserGuide,Chapter05:
section 5.5 Selecting the Code Generation Options

Error control

              ?dsolve,numeric,Error_Control

              Absolute error: ?dsolve,numeric,IVP

              Relative error: (relative to what?)

Index1 error control and Index1 Tollerance: see solvers

Scaling

scalemethod (this does not seem to exist in Maple)

Examples (Multibody)

Events

                            Catapult
                            (from MapleSim>Help>Examples>Physical Domains>Multibody)
                            contact events

                          Catapult_-_Events.msim

                            Throwing a ball
                            (from MapleSim>Help>Examples>Physical Domains>Multibody)

                            conditional events (with boolean logic)

                          Throwing_a_Ball_-_Events.msim

              Solvers

              Conservation of energy of a pendulum depends on solvers.
                           Euler increases energy, implict Euler dissipates energy.

             Pendulum_for_solver_comparision.msim

Constraint dirft/projection

              2-d rigid slider crank

               (from MapleSim>Help>Examples>Physical Domains>Multibody)

              projection off leads to assembly desintegration after 2000 s simulation

             2D_Rigid_Slider_Crank_-_constraint_projection.msim

                         A stiff solver improves constraint drift, but only delays desintegration

                         Baumgarte constraint stabilization prevents simulation error but shows dislocated rigid body frames