Maplesoft Blog

The Maplesoft blog contains posts coming from the heart of Maplesoft. Find out what is coming next in the world of Maple, and get the best tips and tricks from the Maple experts.

This Maplesoft guest blog post is from Prof. Dr. Johannes Blümlein from Deutsches Elektronen-Synchrotron (DESY), one of the world’s leading particle accelerator centres used by thousands of researchers from around the world to advance our knowledge of the microcosm. Prof. Dr. Blümlein is a senior researcher in the Theory Group at DESY, where he and his team make significant use of Maple in their investigations of high energy physics, as do other groups working in Quantum Field Theory. In addition, he has been involved in EU programs that give PhD students opportunities to develop their Maple programming skills to support their own research and even expand Maple’s support for theoretical physics.


 

The use of Maple in solving frontier problems in theoretical high energy physics

For several decades, progress in theoretical high energy physics relies on the use of efficient computer-algebra calculations. This applies both to so-called perturbative calculations, but also to non-perturbative computations in lattice field theory. In the former case, large classes of Feynman diagrams are calculated analytically and are represented in terms of classes of special functions. In early approaches started during the 1960s, packages like Reduce [1] and Schoonship [2] were used. In the late 1980s FORM [3] followed and later on more general packages like Maple and Mathematica became more and more important in the solution of these problems. Various of these problems are related to data amounts in computer-algebra of O(Tbyte) and computation times of several CPU years currently, cf. [4].

Initially one has to deal with huge amounts of integrals. An overwhelming part of them is related by Gauss’ divergence theorem up to a much smaller set of the so-called master integrals (MIs). One performs first the reduction to the MIs which are special multiple integrals. No general analytic integration procedures for these integrals exist. There are, however, several specific function spaces, which span these integrals. These are harmonic polylogarithms, generalized harmonic polylogarithms, root-valued iterated integrals and others. For physics problems having solutions in these function spaces codes were designed to compute the corresponding integrals. For generalized harmonic polylogarithms there is a Maple code HyperInt [5] and other codes [6], which have been applied in the solution of several large problems requiring storage of up to 30 Gbyte and running times of several days. In the systematic calculation of special numbers occurring in quantum field theory such as the so-called β-functions and anomalous dimensions to higher loop order, e.g. 7–loop order in Φ4 theory, the Maple package HyperLogProcedures [7] has been designed. Here the largest problems solved require storage of O(1 Tbyte) and run times of up to 8 months. Both these packages are available in Maple only.

A very central method to evaluate master integrals is the method of ordinary differential equations. In the case of first-order differential operators leading up to root-valued iterative integrals their solution is implemented in Maple in [8] taking advantage of the very efficient differential equation solvers provided by Maple. Furthermore, the Maple methods to deal with generating functions as e.g. gfun, has been most useful here. For non-first order factorizing differential equation systems one first would like to factorize the corresponding differential operators [9]. Here the most efficient algorithms are implemented in Maple only. A rather wide class of solutions is related to 2nd order differential equations with more than three singularities. Also here Maple is the only software package which provides to compute the so-called 2F1 solutions, cf. [10], which play a central role in many massive 3-loop calculations

The Maple-package is intensely used also in other branches of particle physics, such as in the computation of next-to-next-to leading order jet cross sections at the Large Hadron Collider (LHC) with the package NNLOJET and double-parton distribution functions. NNLOJET uses Maple extensively to build the numerical code. There are several routines to first build the driver with automatic links to the matrix elements and subtraction terms, generating all of the partonic subprocesses with the correct factors. To build the antenna subtraction terms, a meta-language has been developed that is read by Maple and converted into calls to numerical routines for the momentum mappings, calls to antenna and to routines with experimental cuts and plotting routines, cf. [11].

In lattice gauge calculations there is a wide use of Maple too. An important example concerns the perturbative predictions in the renormalization of different quantities. Within different European training networks, PhD students out of theoretical high energy physics and mathematics took the opportunity to take internships at Maplesoft for several months to work on parts of the Maple package and to improve their programming skills. In some cases also new software solutions could be obtained. Here Maplesoft acted as industrial partner in these academic networks.

References

[1] A.C. Hearn, Applications of Symbol Manipulation in Theoretical Physics, Commun. ACM 14 No. 8, 1971.

[2] M. Veltman, Schoonship (1963), a program for symbolic handling, documentation, 1991, edited by D.N. Williams.

[3] J.A.M. Vermaseren, New features of FORM, math-ph/0010025.

[4] J. Blümlein and C. Schneider, Analytic computing methods for precision calculations in quantum field theory, Int. J. Mod. Phys. A 33 (2018) no.17, 1830015 [arXiv:1809.02889 [hep-ph]].

[5] E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148–166 [arXiv:1403.3385 [hep-th]].

[6] J. Ablinger, J. Blümlein, C .Raab, C. Schneider and F. Wissbrock, Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms, Nucl. Phys. 885 (2014) 409-447 [arXiv:1403.1137 [hep-ph]].

[7] O. Schnetz, φ4 theory at seven loops, Phys. Rev. D 107 (2023) no.3, 036002 [arXiv: 2212.03663 [hep-th]].

[8] J. Ablinger, J. Blümlein, C. G. Raab and C. Schneider, Iterated Binomial Sums and their Associated Iterated Integrals, J. Math. Phys. 55 (2014) 112301 [arXiv:1407.1822 [hep-th]].

[9] M. van Hoeij, Factorization of Differential Operators with Rational Functions Coefficients, Journal of Symbolic Computation, 24 (1997) 537–561.

[10] J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab, C. S. Radu and C. Schneider, Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys. 59 (2018) no.6, 062305 [arXiv:1706.01299 [hep-th]].

[11] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, Precise QCD predictions for the production of a Z boson in association with a hadronic jet, Phys. Rev. Lett. 117 (2016) no.2, 022001 [arXiv:1507.02850 [hep-ph]].

     Happy Easter to all those who celebrate! One common tradition this time of year is decorating Easter eggs. So, we’ve decided to take this opportunity to create some egg-related math content in Maple Learn. This year, a blog post by Tony Finch inspired us to create a walkthrough exploring the four-point egg. The four-point egg is a method to construct an egg-shaped graph using just a compass and a ruler, or in this case, Maple Learn. Here's the final product: 

     The Maple Learn document, found here, walks through the steps. In general, each part of the egg is an arc corresponding to part of a circle centred around one of the points generated in this construction. 

     For instance, starting with the unit circle and the three red points in the image below, the blue circle is centred at the bottom point such that it intersects with the top of the unit circle, at (0,1). The perpendicular lines were constructed using the three red points, such that they intersect at the bottom point and pass through opposite side points, either (-1,0) or (1,0). Then, the base of the egg is constructed by tracing an arc along the bottom of the blue circle, between the perpendicular lines, shown in red below.

 

     Check out the rest of the steps in the Maple Learn Document. Also, be sure to check out other egg-related Maple Learn documents including John May’s Egg Formulas, illustrating other ways to represent egg-shaped curves with mathematics, and Paige Stone’s Easter Egg Art, to design your own Easter egg in Maple Learn. So, if you’ve had your fill of chocolate eggs, consider exploring some egg-related geometry - Happy Easter!  

To celebrate this day of mathematics, I want to share my favourite equation involving Pi, the Bailey–Borwein–Plouffe (BBP) formula:

This is my favourite for a number of reasons. Firstly, Simon Plouffe and the late Peter Borwein (two of the authors that this formula is named after) are Canadian! While I personally have nothing to do with this formula, the fact that fellow Canadians contributed to such an elegant equation is something that I like to brag about.

Secondly, I find it fascinating how Plouffe first discovered this formula using a computer program. It can often be debated whether mathematics is discovered or invented, but there’s no doubt here since Plouffe found this formula by doing an extensive search with the PSLQ integer relation algorithm (interfaced with Maple). This is an example of how, with ingenuity and creativity, one can effectively use algorithms and programs as powerful tools to obtain mathematical results.

And finally (most importantly), with some clever rearranging, it can be used to compute arbitrary digits of Pi!

Digit 2024 is 8
Digit 31415 is 5
Digit 123456 is 4
Digit 314159 is also 4
Digit 355556 is… F?

That last digit might look strange… and that’s because they’re all in hexadecimal (base-16, where A-F represent 10-15). As it turns out, this type of formula only exists for Pi in bases that are powers of 2. Nevertheless, with the help of a Maple script and an implementation of the BBP formula by Carl Love, you can check out this Learn document to calculate some arbitrary digits of Pi in base-16 and learn a little bit about how it works.

After further developments, this formula led to project PiHex, a combined effort to calculate as many digits of Pi in binary as possible; it turns out that the quadrillionth bit of Pi is zero! This also led to a class of BBP-type formulas that can calculate the digits of other constants like (log2)*(π^2) and (log2)^5.

Part of what makes this formula so interesting is human curiosity: it’s fun to know these random digits. Another part is what makes mathematics so beautiful: you never know what discoveries this might lead to in the future. Now if you’ll excuse me, I have a slice of lemon meringue pie with my name on it 😋

 

References
BBP Formula (Wikipedia)
A Compendium of BBP-Type Formulas
The BBP Algorithm for Pi

     On International Women’s Day we celebrate the achievements of women around the world. One inspiring story of women in STEM is that of Sophie Germain (1776-1831), a French mathematician and physicist who made groundbreaking strides in elasticity theory and number theory. She learned mathematics from reading books in her father’s library, and while she was not permitted to study at the École Polytechnique, due to prejudice against her gender, she was able to obtain lecture notes and decided to submit work under the name Monsieur LeBlanc. Some prominent mathematicians at the time, including Joseph-Louis Lagrange and Carl Friedrich Gauss, with whom Germain wrote, recognized her intellect and were supportive of her work in mathematics. 

     Sophie Germain is remembered as a brilliant and determined trailblazer in mathematics. She was the first woman to win a prize from the Paris Academy of Sciences for her contributions in elasticity theory and was among the first to make significant contributions toward proving Fermat’s Last Theorem. Among her many accomplishments, one special case of Fermat’s Last Theorem she was able to prove is when the exponent takes the form of what is now known as a Sophie Germain prime: a prime, p, such that 2p+1 is also a prime. The associated prime, 2p+1, is called a safe prime. 

     To mark International Women’s Day, I’ve created a document exploring the Ulam spiral highlighting Sophie Germain primes and safe primes, as an adaptation of Lazar Paroski’s Ulam spiral document. The image below displays part of the Ulam spiral with Sophie Germain primes highlighted in red, safe primes highlighted in blue, primes that are both a Sophie Germain prime and safe prime highlighted in purple, and primes that are neither in grey. 

  

     The document also contains small explorations of these types of prime numbers. For instance, one interesting property of safe primes is that they must either be 5, 7 or take the form 12k-1 for some k≥1. This can be shown from the fact a safe prime q must equal 2p+1 for some prime, p (a Sophie Germain prime), by definition. Then, since q and p are prime, for q > 7 we can determine through contradiction that q ≡ 3 (mod 4) and q ≡ 5 (mod 6), to conclude q ≡ 11 (mod 12) ≡ -1 (mod 12). And so, q = 12k-1 for some k≥1. The Maple Learn document can be found here along with its Maple script. The document also includes a group where you can test whether some positive integer of your choice, n, is a Sophie Germain prime or a safe prime. Alternatively, given n, a button press will display the next Sophie Germain prime greater than n, using Maple’s NextSafePrime command in the number theory package.  

     In mathematics, there is no shortage of interesting rabbit holes to dive into; many of which are the result of past and present women in mathematics, like Sophie Germain, who have persevered despite their hardships. 

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Attention Maple enthusiasts! It gives me great pleasure to announce Maple 2024! Maple 2024 brings together a collection of new features and enhancements carefully designed to enrich your mathematical explorations. Maple 2024 is the result of a lot of hard work by a lot of people, and there is far more in it than I can cover here. But I’d like to share with you some of my favorite features in this release.

 

AI Formula Assistant

The AI Formula Assistant in Maple 2024 is undoubtedly the feature that excites me the most, especially considering how often I’m asked the question: 'When will Maple include AI features?' This assistant serves as your new mathematical companion and will change the way you look up and enter formulas and equations. Driven by advanced AI technology, it presents a range of relevant options based on your search query. Alongside suggestions, you'll also receive detailed explanations for each formula and its parameters so you can select the one you need, and then you can insert the formula into your document at a click of a button, as a proper Maple expression.

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NaturalLanguage Package

The Formula Assistant is built on top of the new NaturalLanguage package, which integrates powerful language models like GPT-4 and ChatGPT from OpenAI into Maple. With this feature, you can leverage large language models to process natural language within Maple. Ask the AI to explain concepts, provide additional details, find specific Maple commands, and more. Of course, since this is a Maple package, you can also use it as a basis to build your own AI-powered applications inside Maple. We’re really looking forward to seeing what you will do with it!

 

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Argument Completion
We’ve had a lot of requests from people who wanted Maple’s command completion features to do even more, and I’m happy to say that Maple 2024 delivers. The new argument completion feature in Maple 2024 is poised to significantly enhance your experience with commands. For many users, including myself, not being aware of all the options a command takes is a challenge, often leading me to refer to help pages for clarification. With argument completion, that's no longer a concern. Just enter the command with the help of the existing command completion feature, then automatic argument completion takes over to guide you through the rest. Give it a try with the 'plot' command!

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Check My Work

A personal favorite feature that's gotten even better in Maple 2024 is Check My Work. As someone who has tutored students, I vividly recall their stress before exams, often receiving emails and text messages from them seeking last-minute help. At the time, I found myself wishing the students had a way to check their work themselves, so we would all be less stressed! So I was super excited when we added the first Check My Work feature a couple of years ago, and am very happy that it gets better ever year. In Maple 2024, we’ve expanded its capabilities to support problems involving factoring, simplification, and limits.

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Scrollable Matrices:
This feature will definitely resonate with many of the engineers in the Maple Primes community. If you're someone who works with worksheets containing large matrices, you've likely wished that you could scroll the matrices inside your document instead of having to launch a separate matrix browser. With Maple 2024, your wish has come true.

 

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Color Bars

And finally, for those of you who appreciated the addition of color bars in Maple 2023 but wanted to see them extended to more 2D and 3D plots, you'll be delighted to know that this is exactly what we’ve done. We’ve also added new customization options, providing you with greater control over appearance.

 

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This is just a partial glimpse of what's new in Maple 2024. For a comprehensive overview, visit What’s New in Maple 2024.

The Lunar New Year is approaching and 2024 is the Year of the Dragon! This inspired me to create a visualization approximating the dragon curve in Maple Learn, using Maple. 

The dragon curve, first described by physicist John Heighway, is a fractal that can be constructed by starting with a single edge and then continually performing the following iteration process:  

Starting at one endpoint of the curve, traverse the curve and build right triangles on alternating sides of each edge on the curve. Then, remove all the original edges to obtain the next iteration. 

visual of dragon curve iteration procedure 

This process continues indefinitely, so while we can’t draw the fractal perfectly, we can approximate it using a Lindenmayer system. In fact, Maple can do all the heavy lifting with the tools found in the Fractals package, which includes the LSystem subpackage to build your own Lindenmayer systems. The subpackage also contains different examples of fractals, including the dragon curve. Check out the Maple help pages here: 

Overview of the Fractals Package  

Overview of the Fractals:-LSystem Subpackage 

Using this subpackage, I created a Maple script (link) to generate a Maple Learn document (link) to visualize the earlier iterations of the approximated dragon curve. Here’s what iteration 11 looks like: 

eleventh iteration of dragon curve approximation  

You can also add copies of the dragon curve, displayed at different initial angles, to visualize how they can fit together. Here are four copies of the 13th iteration: