Hi again Maple community, and others,

want to share
tiwn_and_cousin_prime_numbers.mw
tiwn_and_cousin_prime_numbers.pdf
(spelling error in file name)

~

just want to share,
some successful code

The lesser of the twin primes are listed
{3,5,11,17,29,41,59,71}
https://oeis.org/A001359
Prime numbers p such that p+2 is also a prime number

also, the lesser of the cousin primes are listed
{3,7,13,19,37,43,67,79,97}
https://oeis.org/A023200
prime numbers p such that p+4 is also a prime number

good fun

also, my webpage has more details
https://mattanderson.fun
okay

regards,
Matt

Would it be possible to include the file path in $File for the proposed header/footer insertions.

Thanks in advance

Peter

I am very pleased to announce the publication of my new book written together with Nic Fillion, "Perturbation Methods Using Backward Error", which uses Maple heavily throughout.

You can find it at

the SIAM bookstore

and I hope that you find it useful and interesting.

You can also find a paper in Maple Transactions, written with Michelle Hatzel, that explains how we generated the image that was chosen for the cover.  Exploring Cover Designs for an Upcoming Book  In the end, the SIAM design people chose a different one than we had thought, but they did pick one of the ones we generated!  

This was fun to do.

Over the past few months, I've created a number of short videos. My intention is to help people use Maple more effectively. I occasionally give workshops introducing Maple and its programming language, and many of the topics come from questions I get from the participants. 

These can be found on our Youtube channel. Here are the ones posted so far.

What is a Workbook?
Creating a Workbook
How to Customize Your Maple Settings with the Options Dialog
What is Maple Transactions?
How to Submit an Article to Maple Transactions
Quotation Marks in Maple
Using Single Quotes to Prevent Evaluation
 

If you find these helpful and have suggestions for future videos, please leave a comment, thanks!

> kernelopts(version);
Maple 2026.0, X86 64 LINUX, Apr 28 2026, Build ID 2011354

Maple 2026.1 

We have just released an update to Maple. Maple 2026.1 includes further enhancements to the new AI Assistant and Plotting Themes, as well as Explore, accessibility, and the math engine. As always, we recommend that all Maple 2026 users install this update. 

In particular, please note that this update includes a fix to the problem where titles do not show up when using the Explore commands title option.  As always, thanks for helping us, and your fellow Maple users, by letting us know! 

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2026.1 download page on web site, where you can also find more details.  

My  post with a valid  question (second time) on the Migration Assistant add-on was deleted by a moderator who thought that it was spam.

I have decided to uninstall Maple Flow altogether and consider Smath Solver or CalcTree instead. It is unfortunate that Maple has such a bad customer service.

my second Question was deleted by a "moderator" !!!!!!!!! what is this censorship????????? (spam i was told)

Download geodesics.mw

it contained only the file geodesics.mw

Jean-Michel 

Hi mapleprimes, and all,

did a little exploration with the Matrix() and ifactor() Maple commands.

prime_factorization_of_one_digit_numbers.mw

have a look
no warnings, and no errors

Regards,
Matt

Hi Maple community, and all,

My intrest in prime numbers continues.

Made a quick example file.

3_tuple_admissible_example.mw

3_tuple_admissible_example.pdf

also, see my webpage for similar content
https://mattanderson.fun
and Norman, in Germany
prime k-tuplets & Primzahlen

Enjoy

Matt

Mathematics often feels precise and deterministic. We solve equations, follow logical steps, and do our best to arrive at exact answers. But sometimes, surprisingly, randomness can also lead us to deep mathematical truths. One of the most famous examples of this idea is a problem from the 18th century known as Buffon’s Needle.

Imagine you have a floor made of long wooden planks placed side by side. The seams between the planks form a set of equally spaced parallel lines across the floor. Now, suppose you take a needle and randomly drop it onto the floor. Sometimes the needle lands entirely on one plank. Other times, it crosses one of the seams between planks, as shown below.

Now here is the curious question posed by the French mathematician Georges-Louis Leclerc, Comte de Buffon in the 1700s:

If we repeatedly drop the needle at random, what is the probability that it crosses one of the lines on the floor?

At first glance, this sounds like a simple probability puzzle. But the answer turns out to involve one of the most famous numbers in mathematics: π.

To keep things simple, assume the distance between the parallel lines on the floor is the same as the length of the needle. We can also imagine that all of our needles are thrown onto the same plank, potentially crossing onto the plank above or below. This configuration is equivalent to throwing the needle onto any plank as long as the planks are equally wide; this modification makes the analysis much simpler.

Every time the needle lands, two things determine whether it crosses a line:

• The distance x from the center of the needle to the nearest line
• The angle θ at which the needle lands with respect to the parallel lines

See a depiction of this below.

To determine the probability of a needle crossing one of these lines, we need to describe what a "random drop" of the needle means mathematically. If the lines are the same length apart as the length of the needle L, then the center of the needle can never be farther than L/2 from the nearest line. Therefore, 0 ≤ x ≤ L/2. Next, we can simplify our domain for θ. The problem is symmetric, so we only need to consider angles between 0 and π/2. Any given half of the needle then has a vertical reach of (L/2)sin(θ).

We will say a needle "crosses" a line precisely when the center lands close enough to a line that one end of the needle can reach across the line. This occurs when x ≤ (L/2)sin(θ).

An important assumption to make is that every pair (x,θ) in the rectangle 0 ≤ x ≤L/2, 0 ≤ θ ≤ π/2 is equally likely. We’re assuming the needle lands with uniform randomness over all vertical positions x and angles θ. This means that the probability of crossing a line is the fraction of this region where the inequalities above hold. That is, 

Probability = (area of favourable region) / (area of total region)

The "rectangle" formed by inequalities has a total area of (L/2) * (π/2) = π*L/4. The needle crosses a line exactly when x ≤ (L/2)sin(θ), so for a fixed angle θ, the allowable x values are 0 ≤ x ≤ (L/2)sin(θ). The favourable area is then:

The probability of a needle crossing a line is therefore:

This result leads to a fascinating idea. If the probability of crossing a line is 2/π, we can rearrange the formula to estimate π itself:

π ≈ 2N / C

where:

• N = the total number of needle drops
• C = the number of times the needle crosses a line

In other words, by performing a simple random experiment and counting how often the needle crosses a line, we can approximate π.

For example, suppose you drop the needle 10,000 times and it crosses a line 6,366 times. Plugging these values into the formula gives

π ≈ (2 × 10,000) / 6,366 ≈ 3.14

With enough trials, the estimate tends to get closer and closer to the true value of π. At the bottom of this post, I attached a Maple worksheet that simulates this phenomenon. Below are results from simulating this result using N = 10, 100 & 1000, respectively. Notice as N increases, our approximation for π tends to become more and more accurate.

Below is a more dynamic simulation from the Maple worksheet to show how the approximation stabilizes as N increases.

What makes Buffon’s Needle so fascinating is the unexpected connection between geometry, probability, and one of mathematics’ most important constants.

π usually appears when dealing with circles (circumference, area, rotation, etc). But in Buffon’s experiment, there are no circles at all. Instead, π emerges from the geometry of all the possible ways a needle can land on a set of parallel lines.

This was one of the earliest examples of what we now call a Monte Carlo method, which is essentially using random experiments to estimate numerical values. Today, similar techniques are used in physics, finance, computer graphics, and machine learning.

One of the best parts of Buffon’s Needle is that you can try it yourself. All you need is:

• A toothpick or needle
• A piece of paper with a sequence of parallel lines, each a distance of the needle's length apart
• A lot of patience

Drop the needle repeatedly (N times), record how many times it crosses a line (C), and compute 2N/C. The more times you repeat the experiment, the closer your estimate will get to π.

After reading about this experiment, I was convinced that mathematics is not only about abstract symbols and formulas. Sometimes, even something as simple as dropping a needle onto the floor can reveal the hidden structure of elements of the universe that we would've otherwise never known were there.

Buffons_Needle_Simulation.mw

I have contacted Maplesoft support with the intend to send them corrupted Maple.ini files (that caused Maple 2026 installation to malfunction) for further analysis.

Before sending I asked whether they were interested. In the email response support replied that they were happy that my problem has been solved. I replied that they apparently did not understand my first mail. Then I got this back.

Hello,

Thank you for clarifying! I apologize for not being more clear in my response.

The issue with the preferences file (Maple.ini) was summarized in the MaplePrimes links that you provided which have been shared with our R&D team. They will be able to investigate the problem further.

Please let us know if you have any questions or concerns.

Best Regards,

XXX(Name removed, Case - 00191471 )
Technical Support Analyst

Apparently I have shared the files already. Not to my knowledge. Without my ini files no one can investigate the case.

For me this answer sounds like an automated AI generated reply. That is not what I expect as a long-time customer and EPM participant (paying full price). Premium products should come with premium support!

In this case I solved the installation problem myself with the help of this forum and wanted to support Maplesoft to make better products. Now I really feel like an idiot. Spending my free time with debugging, offering assistance, talking to a bot(?!?).

Dear upper management and owners:

If you have replaced support staff with bots that do not identify themselves as such, please reconsider what you are doing. Don't squeeze Maple for maximum profit and hide this. Think about your loyal customer base if you have a long-term growth strategy. The value of most companies lies in the people who work for them not in dumb, sloppy working bots. Humans want to deal with competent humans.
And: Do not let AI code Maple. This will lead to sloppy untrustworthy code with definitly more support requests.

Gabriel’s Horn is one of the most famous examples in calculus of how infinity can behave in ways that completely defy our intuition.

The horn-shaped object is created from a very simple curve: y = 1/x for x ≥ 1 (pictured below).

Now imagine rotating this curve around the x-axis. The resulting surface stretches infinitely far to the right while becoming thinner and thinner. Visually, it resembles a long trumpet or horn that continuously narrows to a thickness of zero.

At first glance, nothing about this shape seems particularly mysterious. As x grows larger, the radius 1/x becomes smaller and smaller. It seems reasonable that both the volume contained inside the horn and the area of its surface would remain finite (or at least if the volume was finite, then the surface area would also be finite). After all, the horn gets extremely thin very quickly.

Calculus allows us to test that intuition.

To compute the volume of the horn, we use the disk method. Each slice perpendicular to the x-axis forms a circular disk of radius r = 1/x, each with an area of π*r² = π*(1/x²).

The total volume is the sum of an infinite number of these disc areas with thickness dx. As an integral,

V = π ∫₁^∞ (1/x²) dx.

This is a simple integral that converges to a value of 1. We could use the power or rule or our favourite computing software (I used Maple below).

Hence, V = π ∫₁^∞ 1/x² dx = π*1 = π. This means the horn contains only π cubic units of space, even though it extends infinitely far. 

Now let’s compute the surface area of the horn. For a surface of revolution, the surface area is

A = 2π ∫₁^∞ y √(1 + (y′)²) dx.

Since y = 1/x, we have y′ = −1/x². Substituting into the formula gives

A = 2π ∫₁^∞ (1/x) √(1 + 1/x⁴) dx.

Software like Maple can easily handle this integral. It tells us the integral diverges to infinity.

However, this is difficult to solve analytically. To understand what happens to this integral, notice that for large x, the square root term is very close to 1, since 1/x⁴ can be approximated as 0 as x grows large. This means the integrand behaves roughly like 1/x (it's actually slightly larger than 1/x). But

∫₁^∞ 1/x dx diverges, and ∫₁^∞ (1/x) √(1 + 1/x⁴) dx > ∫₁^∞ 1/x dx, so ∫₁^∞ (1/x) √(1 + 1/x⁴) dx must also diverge. As a result, the surface area of Gabriel’s Horn is infinite.

This leads to the famous, surprising conclusion:

• The horn has finite volume.
• The horn has infinite surface area.

In other words, it could be filled with a finite amount of paint, but it would require an infinite amount of paint to coat its inside surface.

Of course, real paint has thickness, so the paradox disappears in the physical world. Eventually, the horn would become thinner than the paint layer itself. But mathematically, the result is perfectly consistent.

The key idea lies in how quickly the function 1/x shrinks. The cross-sectional area of the disks scales like (1/x)² = 1/x², and the integral of 1/x² converges.

But the circumference of each slice scales like 1/x, and the integral of 1/x diverges.

So as the horn extends outward, the added volume decreases quickly enough to sum to a finite value, while the added surface area decreases too slowly and accumulates forever.

Gabriel’s Horn beautifully illustrates one of the central themes of calculus: infinite processes can produce results that feel deeply counterintuitive.

Volume and surface area seem closely related, but can behave in completely different ways when infinite limits are involved. A shape can stretch endlessly yet still contain a finite amount of space.

This strange object reminds me that mathematics isn’t just about calculating numbers, but is also about exploring the strange and fascinating consequences of simple ideas pushed to their limits.

I was wondering whether a water molecule exhibits the intermediate axis theorem (IAT) effect.

I was also curious if a tool like MapleSim could be used for simulations, as it is designed with technical applications in mind. (It would have been handy to enter parameters in Angstroms and atomic mass, but MapleSim only provides length in mm and mass in grams for small objects.)  

The atoms of the H2O molecule form an obtuse triangle. An obtuse triangle exhibits the effect when rotated about its intermediate axis (horizontal axis in the image below) provided it consists of equal point masses (more details here) which is not the case for oxygen and hydrogen. 

Building and running a model is quick since geometry and masses are available online. Two molecules are modeled in the attached file with parameters from different sources. The left model in the image below represents the atoms as three discrete point masses without individual rotational inertia. The system's total inertia is derived entirely from the spatial distribution of these masses. On the right, the entire molecule is modeled as a single lumped mass located at the center of mass (CoM), with the molecule’s full rotational inertia tensor applied at that point.

IAT_H2O.msim

Both models are rotating about the assumed intermediate axis at 3 THz (3 trillion rotations per second), which is about two orders of magnitude less than near-infrared light (just beyond the visible spectrum). To generate the IAT effect an orthogonal tiny initial “kick” of 0.001 Hz was added.

Visualizing was not so easy since the models are in the picometer range. It was difficult to locate the molecules by zooming with the mouse wheel (the button “Fit scene” on the 3-D Playback window did not work). Running an animation to visualize the IAT effect was not possible because the display speed could “only” be slowed down by a factor of 2^31 (see A in the image below). However, the slider C to move back and forth in time worked. The time (B) was always rounded to zero due to the very short simulation time span of 1E-10 seconds (i.e. 10 picoseconds).

Export of an animation movie worked better. Over the displayed timespan of 10 ps the molecules flip 2 times back and forth. This corresponds to a flip frequency of 200 GHz.

Time is still not displayed in the movie but that is pretty much all what did not work. Overall, a good performance. Also the numerical results did not show signs of loss of fidelity despite the atomic scale  of the objects and the very short time span.

In reality, the rotation of a dipole results in the emission of electromagnetic radiation, which dissipates energy and decelerates the molecule. How can this damping effect be modeled in MapleSim? Someone has an idea?