Month: July 2024

The 14-sided polygon Tile(1,1), on the left, is known as a weakly chiral aperiodic monotile — in other words, if tilings that mix unreflected and reflected tiles are forbidden, then it tiles only aperiodically. However, by modifying its edges, as shown in the centre and right, strictly chiral aperiodic monotiles called “spectres” are created that admit only non-periodic tilings. Credit: University of Waterloo

Recently, an international team of four, that includes Cheriton School of Computer Science professor Dr. Craig Kaplan, discovered a single shape that tiles the plane—an infinite, two-dimensional surface—in a pattern that can never be made to repeat.

The discovery mesmerized mathematicians, tiling enthusiasts and the public alike.

The shape, a 13-sided polygon they called “the hat,” is known to mathematicians as an aperiodic monotile or an “einstein,” the German words that mean “one stone.”

But the team’s most recent discovery has raised the bar once again. They found another shape, related to the first, that meets an even stricter definition. Dubbed the “specter,” the new shape tiles a plane in a pattern that never repeats without the use of mirror images of the shape. For this reason, it has also been called a “vampire einstein”—a shape that tiles aperiodically without requiring its reflection.

“Our first paper solved the einstein problem, but as the shape required reflection to tile aperiodically people raised a legitimate question: Is there a shape that can do what the hat does but without reflection,” Kaplan explains. “It was our good fortune that we found a shape that not only solves this subproblem, but also solved it so soon after the first paper.”

To mathematicians, the hat and its mirror image are a single shape, but in the physical world left-handed and right-handed shapes can behave differently. You can’t, for example, wear a right-handed glove on your left hand.

“If you tiled a large bathroom floor aperiodically with hat-shaped tiles that had been glazed on one side you would need hats and mirror images of hats,” Kaplan says.

But it was not this quibble that motivated the recent discovery.

The discovery of the vampire einstein began with the musings of David Smith, a retired print technician and self-described shape hobbyist from Yorkshire, England, whose curiosity months earlier led to the original einstein discovery.

“Dave emailed us a couple of days after our hat paper went online to say that he had been playing around with a related shape that seemed to be behaving strangely,” Kaplan says. “Yoshiaki Araki, a Japanese mathematician and well-known artist whose work is in the spirit of MC Escher, had posted pictures of Tile(1,1) that got Dave interested in looking at it further.”

Yoshiaki posted an intriguing question on Twitter: “An aperiodic turtle tessellation based on new aperiodic monotile Tile(1, 1.1). In the tiling, it is said that around 12.7% of tiles are reflected. The green one is an instance. One more reflected turtle is hidden in the tiling. Who is the reflected?'”

“Yoshi had turned Tile(1,1) into turtles and it’s a bit hard to see the other reflected turtle in that picture. But it got Dave curious. What if we tile with this shape but without reflections? As he did that, Dave found that he could build tilings progressively outward in a pattern that didn’t stop and didn’t repeat.”

But then this shape came with a different quibble. As Kaplan explains, if you use reflections of Tile(1,1) the pattern does repeat. In other words, it’s periodic. But if Tile(1,1) is modified by replacing its straight edges with curves, it becomes a vampire einstein—a single shape that without reflection tiles the infinite plane in a pattern that can never be made to repeat.

The obvious question for mathematicians and tiling enthusiasts is what’s next?

“We can pose many variations of the problem,” Kaplan says. “The most interesting, for me at least, is whether this can be done in 3D. It would be nice to have a shape that repeats non-periodically in three dimensions. Such constructions are much harder to visualize, but computationally it’s not that much more difficult to prove should we be so lucky as to find a three-dimensional shape—a polyform—that like the hat tiles only aperiodically.”

“Tiling theory as a branch of mathematics is beautiful, tangible, and has a lot of fascinating problems to be solved. There’s no shortage of follow-up work to be done.”

Hatfest, a celebration of the discovery of “the Hat,” will be taking place at the University of Oxford’s Mathematical Institute from July 20 to 21. The event’s first day will feature talks and workshops on tiling aimed at a lay audience, while the second will feature presentations aimed at a broad audience of physicists and mathematicians.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Joe Petrik, University of Waterloo

Researchers at The University of Western Australia’s ARC Industrial Transformation Research Hub for Transforming Energy Infrastructure through Digital Engineering (TIDE) have made a significant mathematical breakthrough that could help transform ocean research and technology.

Research Fellow Dr. Lachlan Astfalck, from UWA’s School of Physics, Mathematics and Computing, and his team developed a new method for spectral density estimation, addressing long-standing biases and paving the way for more accurate oceanographic studies.

The study was published in the journal Biometrika, known for its emphasis on original methodological and theoretical contributions of direct or potential value in applications.

“Understanding the ocean is crucial for numerous fields, including offshore engineering, climate assessment and modeling, renewable technologies, defense and transport,” Dr. Astfalck said.

“Our new method allows researchers and industry professionals to advance ocean technologies with greater confidence and accuracy.”

Spectral density estimation is a mathematical technique used to measure the energy contribution of oscillatory signals, such as waves and currents, by identifying which frequencies carry the most energy.

“Traditionally, Welch’s estimator has been the go-to method for this analysis due to its ease of use and widespread citation, however this method has an inherent risk of bias, which can distort the expected estimates based on the model’s assumption, a problem often overlooked,” Dr. Astfalck said.

The TIDE team developed the debiased Welch estimator, which uses non-parametric statistical learning to remove these biases.

“Our method improves the accuracy and reliability of spectral calculations without requiring specific assumptions about the data’s shape or distribution, which is particularly useful when dealing with complex data that doesn’t follow known analytical patterns, such as internal tides in oceanic shelf regions,” Dr. Astfalck said.

The new method was recently applied in a TIDE research project by Senior Lecturer at UWA’s Oceans Graduate School and TIDE collaborator, Dr. Matt Rayson, to look at complex non-linear ocean processes.

“The ocean is difficult to measure and understand and the work we are doing is all about uncovering some of those mysteries,” Dr. Rayson said.

“The new method means we can better understand ocean processes, climate models, ocean currents and sediment transport, bringing us closer to developing the next generation of numerical ocean models.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to University of Western Australia.

Generations of scientists have compared the universe to a giant, complex game, raising questions about who is doing the playing – and what it would mean to win.

If the universe is a game, then who’s playing it?

The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time for free here.

Is the universe a game? Famed physicist Richard Feynman certainly thought so: “‘The world’ is something like a great chess game being played by the gods, and we are observers of the game.” As we observe, it is our task as scientists to try to work out the rules of the game.

The 17th-century mathematician Gottfried Wilhelm Leibniz also looked on the universe as a game and even funded the foundation of an academy in Berlin dedicated to the study of games: “I strongly approve of the study of games of reason not for their own sake but because they help us to perfect the art of thinking.”

Our species loves playing games, not just as kids but into adulthood. It is believed to have been an important part of evolutionary development – so much so that the cultural theorist Johan Huizinga proposed we should be called Homo ludens, the playing species, rather than Homo sapiens. Some have suggested that once we realised that the universe is controlled by rules, we started developing games as a way to experiment with the consequences of these rules.

Take, for example, one of the very first board games that we created. The Royal Game of Ur dates back to around 2500 BC and was found in the Sumerian city of Ur, part of Mesopotamia. Tetrahedral-shaped dice are used to race five pieces belonging to each player down a shared sequence of 12 squares. One interpretation of the game is that the 12 squares represent the 12 constellations of the zodiac that form a fixed background to the night sky and the five pieces correspond to the five visible planets that the ancients observed moving through the night sky.

But does the universe itself qualify as a game? Defining what actually constitutes a game has been a subject of heated debate. Logician Ludwig Wittgenstein believed that words could not be pinned down by a dictionary definition and only gained their meaning through the way they were used, in a process he called the “language game”. An example of a word that he believed only got its meaning through use rather than definition was “game”. Every time you try to define the word “game”, you wind up including some things that aren’t games and excluding others you meant to include.

Other philosophers have been less defeatist and have tried to identify the qualities that define a game. Everyone, including Wittgenstein, agrees that one common facet of all games is that they are defined by rules. These rules control what you can or can’t do in the game. It is for this reason that as soon as we understood that the universe is controlled by rules that bound its evolution, the idea of the universe as a game took hold.

In his book Man, Play and Games, theorist Roger Caillois proposed five other key traits that define a game: uncertainty, unproductiveness, separateness, imagination and freedom. So how does the universe match up to these other characteristics?

The role of uncertainty is interesting. We enter a game because there is a chance either side will win – if we know in advance how the game will end, it loses all its power. That is why ensuring ongoing uncertainty for as long as possible is a key component in game design.

Polymath Pierre-Simon Laplace famously declared that Isaac Newton’s identification of the laws of motion had removed all uncertainty from the game of the universe: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes.”

Solved games suffer the same fate. Connect 4 is a solved game in the sense that we now know an algorithm that will always guarantee the first player a win. With perfect play, there is no uncertainty. That is why games of pure strategy sometimes suffer – if one player is much better than their opponent then there is little uncertainty in the outcome. Donald Trump against Garry Kasparov in a game of chess will not be an interesting game.

The revelations of the 20th century, however, have reintroduced the idea of uncertainty back into the rules of the universe. Quantum physics asserts that the outcome of an experiment is not predetermined by its current state. The pieces in the game might head in multiple different directions according to the collapse of the wave function. Despite what Albert Einstein believed, it appears that God is playing a game with dice.

Even if the game were deterministic, the mathematics of chaos theory also implies that players and observers will not be able to know the present state of the game in complete detail and small differences in the current state can result in very different outcomes.

That a game should be unproductive is an interesting quality. If we play a game for money or to teach us something, Caillois believed that the game had become work: a game is “an occasion of pure waste: waste of time, energy, ingenuity, skill”. Unfortunately, unless you believe in some higher power, all evidence points to the ultimate purposelessness of the universe. The universe is not there for a reason. It just is.

The other three qualities that Caillois outlines perhaps apply less to the universe but describe a game as something distinct from the universe, though running parallel to it. A game is separate – it operates outside normal time and space. A game has its own demarcated space in which it is played within a set time limit. It has its own beginning and its own end. A game is a timeout from our universe. It is an escape to a parallel universe.

The fact that a game should have an end is also interesting. There is the concept of an infinite game that philosopher James P. Carse introduced in his book Finite and Infinite Games. You don’t aim to win an infinite game. Winning terminates the game and therefore makes it finite. Instead, the player of the infinite game is tasked with perpetuating the game – making sure it never finishes. Carse concludes his book with the rather cryptic statement, “There is but one infinite game.” One realises that he is referring to the fact that we are all players in the infinite game that is playing out around us, the infinite game that is the universe. Although current physics does posit a final move: the heat death of the universe means that this universe might have an endgame that we can do nothing to avoid.

Caillois’s quality of imagination refers to the idea that games are make-believe. A game consists of creating a second reality that runs in parallel with real life. It is a fictional universe that the players voluntarily summon up independent of the stern reality of the physical universe we are part of.

Finally, Caillois believes that a game demands freedom. Anyone who is forced to play a game is working rather than playing. A game, therefore, connects with another important aspect of human consciousness: our free will.

This raises a question: if the universe is a game, who is it that is playing and what will it mean to win? Are we just pawns in this game rather than players? Some have speculated that our universe is actually a huge simulation. Someone has programmed the rules, input some starting data and has let the simulation run. This is why John Conway’s Game of Life feels closest to the sort of game that the universe might be. In Conway’s game, pixels on an infinite grid are born, live and die according to their environment and the rules of the game. Conway’s success was in creating a set of rules that gave rise to such interesting complexity.

If the universe is a game, then it feels like we too lucked out to find ourselves part of a game that has the perfect balance of simplicity and complexity, chance and strategy, drama and jeopardy to make it interesting. Even when we discover the rules of the game, it promises to be a fascinating match right up to the moment it reaches its endgame.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Marcus Du Sautoy*

How can you figure out which points lie on a certain curve? And how many possible curves do you count by a given number of points? These are the kinds of questions Pim Spelier of the Mathematical Institute studied during his Ph.D. research. Spelier received his doctorate with distinction on June 12.

What does counting curves mean on an average day? “A lot of sitting and gazing,” Spelier replies. “When I’m asked what exactly do, can’t always answer that easily. Usually give the example about the particle traveling through time.”

All possible curves

Imagine a particle moving through space and you follow the path the particle makes through time. That path is a curve, a geometric object. How many possible paths can the particle follow, if we assume certain properties? For example, a straight line can only pass through two points in one way. But how many paths are possible for the particle if we look at more difficult curves? And how do you study that?

By looking at all possible curves at the same time. For example, all possible directions from a given point form with each other a circle, and that is called a modulspace. And that circle is itself a geometric object.

The mathematical magic can happen because this set of all curves itself has geometrical properties, Spelier says, to which you can apply geometrical tricks. Next, you can make that far more complicated with even more complex curves and spaces. So not counting in three but, for example, in eleven dimensions.

Spelier tries to find patterns that always apply to the curves he studies. His approach? Breaking up complicated spaces into small, easy spaces. You can also break curves into partial curves. That way, the spaces in which you’re counting are easier. But the curves sometimes get complicated properties, because you have to be able to glue them back together.

Spelier says, “The goal is to find enough principles to determine the number of curves exactly.”

In addition to curves, Spelier also counted points on curves. He studied the question: how many solutions does a given mathematical equation have?

These are equations that are a bit more complicated than the a² + b² = c² of the Pythagorean theorem. That equation is about the lengths of the sides of a right triangle. If you replace the squares with higher powers, it is more difficult to investigate solutions. Spelier studied solutions in whole numbers, for example, 3² + 4² = 5².

Meanwhile, there is a method to find those solutions. Professor of Mathematics Bas Edixhoven, who died in 2022, and his Ph.D. student Guido Lido developed an alternative approach to the same problem. But to what extent the two methods match and differ was still unclear. During his Ph.D. research, Spelier developed an algorithm to investigate this.

The first person with an answer

Developing that algorithm is necessary to implement the method. If you want to do it by hand, you get pages and pages of equations. Edixhoven’s method uses algebraic geometry. Through clever geometric tricks, you can calculate exactly the whole number points of a given curve. Spelier proved that the Edixhoven-Lido method is better than the old one.

David Holmes, professor of Pure Mathematics and supervisor of Spelier, praises the proof provided. “When you’re the first person to answer a question that everyone in our community wants an answer to, that’s very impressive. Pim proves that these two methods for finding rational points are similar, an issue that really kept mathematicians busy.”

Doing math together

The best part of his Ph.D.? The meetings with his supervisor. After the first year, it was more collaboration than supervision, both for Spelier and Holmes. Spelier says, “Doing math together is still more fun than doing it alone.”

Spelier starts in September as a postdoc in Utrecht and is apparently not yet done with counting. After counting points and curves, he will soon start counting surfaces.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Leiden University.

Whether you are sharing a cake or a coastline, maths can help make sure everyone is happy with their cut, says Katie Steckles.

One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

When splitting between two parties, you might know a simple rule, proven to be mathematically optimal: I cut, you choose. One person divides the cake (or whatever it is) and the other gets to pick which piece they prefer.

Since the person cutting the cake doesn’t choose which piece they get, they are incentivised to cut the cake fairly. Then when the other person chooses, everyone is satisfied – the cutter would be equally happy with either piece, and the chooser gets their favourite of the two options.

This results in what is called an envy-free allocation – neither participant can claim they would rather have the other person’s share. This also takes care of the problem of non-homogeneous objects: if some parts of the cake are more desirable, the cutter can position their cut so the two pieces are equal in value to them.

What if there are more people? It is more complicated, but still possible, to produce an envy-free allocation with several so-called fair-sharing algorithms.

Let’s say Ali, Blake and Chris are sharing a cake three ways. Ali cuts the cake into three pieces, equal in value to her. Then Blake judges if there are at least two pieces he would be happy with. If Blake says yes, Chris chooses a piece (happily, since he gets free choice); Blake chooses next, pleased to get one of the two pieces he liked, followed by Ali, who would be satisfied with any of the pieces. If Blake doesn’t think Ali’s split was equitable, Chris looks to see if there are two pieces he would take. If yes, Blake picks first, then Chris, then Ali.

If both Blake and Chris reject Ali’s initial chop, then there must be at least one piece they both thought was no good. This piece goes to Ali – who is still happy, because she thought the pieces were all fine – and the remaining two pieces get smooshed back together (that is a mathematical term) to create one piece of cake for Blake and Chris to perform “I cut, you choose” on.

While this seems long-winded, it ensures mathematically optimal sharing – and while it does get even more complicated, it can be extended to larger groups. So whether you are sharing a treat or a divorce settlement, maths can help prevent arguments.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Katie Steckles*

“We wanted to investigate how second pillar pension funds react to financial crises and how to protect them from the crises,” says Kaunas University of Technology (KTU) professor Dr. Audrius Kabašinskas, who, together with his team, discovered a way to achieve this goal. The discovery in question is the development of stress tests for pension funds. Lithuanian researchers were the first in the world to come up with such an adaptation of the stress tests.

Stress tests are usually carried out on banks or other financial institutions to allow market regulators to determine and assess their ability to withstand adverse economic conditions.

According to the professor at KTU Faculty of Mathematics and Natural Sciences, this innovative pension fund stress testing approach will benefit both regulators and pension fund managers.

“Making sure your pension fund is resilient to harsh financial market conditions will help you sleep better, save more, and have increased trust in your funds and the pension system itself,” Kabašinskas adds.

Results based on two major crises

First, the study needed to collect data from previous periods. “Two major events that shocked the whole world—COVID-19 and the first year of Russian invasion of Ukraine—just happened to occur during the project. This allowed us to gather a lot of relevant information and data on changes in the performance of pension funds,” says Kabašinskas.

The Hidden Markov Model (HMM), which, according to a professor at KTU Department of Mathematical Modelling, is quite simple in its principle of operation, helped to forecast future market conditions in this study.

The paper is published in the journal Annals of Operations Research.

“The observation of air temperature could be an analogy for it. All year round, without looking at the calendar, we observe the temperature outside and, based on the temperature level, we decide what time of the year it is. Of course, 15 degrees can occur in winter and sometimes it snows in May but these are random events. The state of the next day depends only on today,” he explains vividly.

According to the KTU researcher, this describes the idea of the Hidden Markov Model: by observing the changes in value, one can judge the state of global markets and try to forecast the future.

“In our study, we observed two well-known investment funds from 2019 to 2022. Collected information helped us identify that global markets at any given moment are in one of four states: no shock regime, a state of shock in stock markets, a state of shock in bond markets, and a state of global financial shock—a global crisis,” says Kabašinskas.

Using certain methods, the research team led by a professor Miloš Kopa representing KTU and Charles University in Prague found that these periods were aligned with the global events in question. Once the transition probabilities between the states were identified, it was possible to link the data of pension funds to these periods and simulate the future evolution of the pension funds’ value.

That’s where the innovation of stress testing came in. The purpose of this test is to determine whether a particular pension fund can deliver positive growth in the future when faced with a shock in the financial markets.

“In our study, we applied several scenarios, extending financial crises and modeling the evolution of fund values over the next 5 years,” says a KTU researcher.

This methodology can be applied not only to pension funds but also to other investments.

Example of Lithuanian pension funds

The research and the new stress tests were carried out on Lithuanian pension funds.

Kabašinskas says that the study revealed several interesting things. Firstly, on average, Lithuanian second pillar pension funds can withstand crises that are twice as long.

“However, the results show that some Lithuanian funds struggle to cope with inflation, while others, the most conservative funds for citizens who are likely to retire within next few years or who have already retired, are very slow in recovering after negative shocks,” adds the KTU expert.

This can be explained by regulatory aspects and the related investment strategy, as stock markets recover several times faster than bond markets, and the above-mentioned funds invest more than 90% in bonds and other less risky instruments.

A complementary study has also been carried out to show how pension funds should change their investment strategy to avoid the drastic negative consequences of various financial crises and shocks.

“Funds that invest heavily in stocks and other risky instruments should increase the number of risk-free instruments slightly, up to 10%, before or after the financial crisis hits. Meanwhile, funds investing mainly in bonds should increase the number of stocks in their holdings. In both cases, the end of the crisis should be followed by a slow return to the typical strategy,” advises a mathematician.

Although the survey did not aim to increase people’s confidence in pension funds, the results showed that Lithuania’s second pillar pension funds are resilient to crisis and are worth trust. Historically they have delivered long-term growth, some have even outperformed inflation and price increases.

“Although short-term changes can be drastic, long-term growth is clearly visible,” says KTU professor Dr. Kabašinskas. “Lithuania, by the way, has a better system than many European countries,” he adds.

For more insights like this, visit our website at www.international-maths-challenge.com.

Finding bats is hard. They are small, fast and they primarily fly at night.  But our new research could improve the way conservationists find bat roosts. We’ve developed a new algorithm that significantly reduces the area that needs to be searched, which could save time and cut labour cost.

Of course, you may wonder why we would want to find bats in the first place. But these flying mammals are natural pest controllers and pollinators, and they help disperse seeds. So they are extremely useful in contributing to the health of our environment.

Despite their importance though, bat habitats are threatened by human activities such as increased lighting, noise and land use. To ensure that we can study and enhance the health of our bat population, we need to locate their roosts. But finding bat roosts is a bit like finding a needle in a haystack.

Our previous work measured and modelled the motion of greater horseshoe bats in flight. Having such a model means we can predict where bats will be, depending on their roost position. But the position of the roost is something we often don’t know.

Our new research combines our previous mathematical model of bat motion with data gathered from acoustic recorders known as “bat detectors”. These bat detectors are placed around the environment and left there for several nights.

Seeing with sound

Bats use echolocation, which allows them to “see with sound” when they’re flying. If these ultrasonic calls are made within ten to 15 metres of a bat detector, the device is triggered to make a recording, providing an accurate record of where and when a bat was present.

The sound recordings also provide clues about the identity of the species. Greater horseshoe bats make a very distinctive “warbling” call at almost exactly 82kHz in frequency, so we can easily tell whether the species is present or not.

Assuming that a bat detector’s batteries last for a few nights, its memory card is not full, and the units are not stolen or vandalised, then we can use the bat call data to generate a map that shows the proportion of bat calls at each detector location.

Our model can also be used to predict the proportion of bat calls based on a given roost location. So, we split the environment up into a grid and simulate bats flying from each grid square. The grid square, or squares, whose simulations best reproduce the bat detector data will then be the most likely locations of the roost.

This simple algorithm can then be applied to whole terrains, meaning that we can create a map of likely roost locations. Cutting out the regions that are least likely to contain the roost can mean we shrink the search space to less than 1% of the initially surveyed area. Simplifying the process of finding bat roosts allows more of an ecologist’s time to be spent on conservation projects, rather than laborious searching.

In 2022, we developed an app that uses publicly available data to predict bat flight lines. At the moment the app can help ecologists, developers or local authority planners, know how the environment is used by bats. However, it needs a roost location to be specified first, and this information is not always known. Our new research removes this barrier, making the app easier to use.

Our work offers a way of identifying likely roost locations. These estimates can then be verified either by directly observing particular features, or by capturing bats at a nearby location and following them back home, using radiotracking.

Over the past two decades, bat detectors have gone from simple hand-held machines to high-performance devices that can collect data for days at a time. Yet they are usually deployed only to identify bat species. We have shown they can be used to identify the areas most likely to contain bat roosts, uncovering critical information about these most secretive of animals.

We hope that this will provide further tools for ecologists to optimise the initial microphone detector locations, thereby providing a holistic way of detecting bat roosts.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Thomas Woolley and Fiona Mathews, The Conversation

By rapidly analysing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviours.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper, published in the Proceedings of the National Academy of Sciences, they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

“Computer algorithms are increasingly dominant in scientific research, a practice now broadly called ‘AI for Science,'” Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

“However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so.”

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

“On a philosophical level, our work explores the interplay between algorithms and mathematicians,” Elimelech and Kaminer explained. “Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants.”

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

“The ‘conservative matrix field’ is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad,” Elimelech and Kaminer explained. “In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

“Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications.”

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

“The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space),” Elimelech and Kaminer said. “Finding non-trivial matrix fields that are also conservative is challenging.”

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

“Each sequence represents a path hidden in the conservative matrix field,” Elimelech and Kaminer explained. “From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC, an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible.”

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a “virtual lab.” Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

“Such hypotheses, or conjectures, are what drives mathematical research forward,” Elimelech and Kaminer said. “The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing.”

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project’s early days and relied on contributions by a large network of volunteers.

“Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research,” Elimelech and Kaminer said. “We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website. This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper.”

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

“Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant,” Elimelech and Kaminer said.

“Another example is the Riemann zeta function, central in number theory, with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians.”

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

“Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise,” Elimelech and Kaminer added. “Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics,” inspired by the long history and impact of user facilities for experimental physics.

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Credit of the article given to Ingrid Fadelli , Phys.org

Direct reciprocity facilitates cooperation in repeated social interactions. Traditional models suggest that individuals learn to adopt conditionally cooperative strategies if they have multiple encounters with their partner. However, most existing models make rather strong assumptions about how individuals decide to keep or change their strategies. They assume individuals make these decisions based on a strategy’s average performance. This in turn suggests that individuals would remember their exact payoffs against everyone else.

In a recent study, researchers from the Max Planck Institute for Evolutionary Biology, the School of Data Science and Society, and the Department of Mathematics at the University of North Carolina at Chapel Hill examine the effects of realistic memory constraints. They find that cooperation can evolve even with minimal memory capacities. The research is published in the journal Proceedings of the Royal Society B: Biological Sciences.

Direct reciprocity is based on repeated interactions between two individuals. This concept, often described as “you scratch my back, I’ll scratch yours,” has proven to be a pivotal mechanism in maintaining cooperation within groups or societies.

While models of direct reciprocity have deepened our understanding of cooperation, they frequently make strong assumptions about individuals’ memory and decision-making processes. For example, when strategies are updated through social learning, it is commonly assumed that individuals compare their average payoffs.

This would require them to compute (or remember) their payoffs against everyone else in the population. To understand how more realistic constraints influence direct reciprocity, the current study considers the evolution of conditional behaviours when individuals learn based on more recent experiences.

Two extreme scenarios

This study first compares the classical modeling approach with another extreme approach. In the classical approach, individuals update their strategies based on their expected payoffs, considering every single interaction with each member of the population (perfect memory). Conversely, the opposite extreme is considering only the very last interaction (limited memory).

Comparing these two scenarios shows that individuals with limited payoff memory tend to adopt less generous strategies. They are less forgiving when someone defects against them. Yet, moderate levels of cooperation can still evolve.

Intermediate cases

The study also considers intermediate cases, where individuals consider their last two or three or four recent experiences. The results show that cooperation rates quickly approach the levels observed under perfect payoff memory.

Overall, this study contributes to a wider literature that explores which kinds of cognitive capacities are required for reciprocal altruism to be feasible. While more memory is always favourable, reciprocal cooperation can already be sustained if individuals have a record of two or three past outcomes.

This work’s results have been derived entirely within a theoretical model. The authors feel that such studies are crucial for making model-informed deductions about reciprocity in natural systems.

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Credit of the article given to Michael Hesse, Max Planck Society

The purpose of education is to ensure that students acquire the skills necessary for succeeding in a world that is constantly changing. Self-assessment, or teaching students how to examine and evaluate their own learning and cognitive processes, has proven to be an effective method, and this competence is partly based on metacognitive knowledge.

A new study conducted at the University of Eastern Finland shows that metacognitive knowledge, i.e., awareness of one’s cognitive processes, is also a key factor in the learning of mathematics. The work is published in the journal Cogent Education.

The study explored thinking skills and possible grade-level differences in children attending comprehensive school in Finland. The researchers investigated 6th, 7th and 9th graders’ metacognitive knowledge in the context of mathematics.

“The study showed that ninth graders excelled at explaining their use of learning strategies, while 7th graders demonstrated proficiency in understanding when and why certain strategies should be used. No other differences between grade levels were observed, which highlights the need for continuous support throughout the learning path,” says Susanna Toikka of the University of Eastern Finland, the first author of the article.

The findings emphasize the need to incorporate elements that support metacognitive knowledge into mathematics learning materials, as well as into teachers’ pedagogical practices.

Self-assessment and understanding of one’s own learning help to face new challenges

Metacognitive knowledge helps students not only to learn mathematics, but also more broadly in self-assessment and lifelong learning. Students who can assess their own learning and understanding are better equipped to face new challenges and adapt to changing environments. Such skills are crucial for lifelong learning, as they enable continuous development and learning throughout life.

“Metacognitive knowledge is a key factor in learning mathematics and problem-solving, but its significance also extends to self-assessment and lifelong learning,” says Toikka.

In schools, metacognitive knowledge can be effectively developed as part of education. Based on earlier studies, Toikka and colleagues have developed a combination of frameworks for metacognitive knowledge, which helps to identify students’ needs for development regarding metacognitive knowledge by offering an alternative perspective to that of traditional developmental psychology.

“This also supports teachers in promoting students’ metacognitive knowledge. Teachers can use the combination of frameworks to design and implement targeted interventions that support students’ skills in lifelong learning.”

According to Toikka, the combination of frameworks enhances understanding of metacognitive knowledge and helps to identify areas where individual support is needed: “This type of understanding is crucial for the development of metacognitive knowledge among diverse learners.”

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Credit of the article given to University of Eastern Finland