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In just a few months, voters across America will head to the polls to decide who will be the next U.S. president. A new study draws on mathematics to break down how humans make decisions like this one.

The researchers, including Zachary Kilpatrick, an applied mathematician at CU Boulder, developed mathematical tools known as models to simulate the deliberation process of groups of people with various biases. They found that decision-makers with strong, initial biases were typically the first ones to make a choice.

“If I want good quality feedback, maybe I should look to people who are a little bit more deliberate in their decision making,” said Kilpatrick, a co-author of the new study and associate professor in the Department of Applied Mathematics. “I know they’ve taken their due diligence in deciding.”

The researchers, led by Samatha Linn of the University of Utah, published their findings August 12 in the journal Physical Review E.

In the team’s models, mathematical decision-makers, or “agents,” gather information from the outside world until, ultimately, they make a choice between two options. That might include getting pizza or Thai food for dinner or coloring in the bubble for one candidate versus the other.

The team discovered that when agents started off with a big bias (say, they really wanted pizza), they also made their decisions really quickly—even if those decisions turned out to run contrary to the available evidence (the Thai restaurant got much better reviews). Those with smaller biases, in contrast, often took so long to deliberate that their initial preconceptions were washed away entirely.

The results are perhaps not surprising, depending on your thoughts about human nature. But they can help to reveal the mathematics behind how the brain works when it needs to make a quick choice in the heat of the moment—and maybe even more complicated decisions like who to vote for.

“It’s like standing on a street corner and deciding in a split second whether you should cross,” he said. “Simulating decision making gets a little harder when it’s something like, ‘Which college should I go to?'”

Pouring water

To understand how the team’s mathematical agents work, it helps to picture buckets. Kilpatrick and his colleagues typically begin their decision-making experiments by feeding their agents information over time, a bit like pouring water into a mop pail. In some cases, that evidence favours one decision (getting pizza for dinner), and in others, the opposite choice (Thai food). When the buckets fill to the brim, they tip over, and the agent makes its decision.

In their experiment, the researchers added a twist to that set up: They filled some of their buckets part way before the simulations began. Those agents, like many humans, were biased.

The team ran millions of simulations including anywhere from 10 to thousands of agents. The researchers were also able to predict the behaviour of the most and least biased agents by hand using pen, paper and some clever approximations.

A pattern began to emerge: The agents that started off with the biggest bias, or were mostly full of water to begin with, were the first to tip over—even when the preponderance of evidence suggested they should have chosen differently. Those agents who began with only small biases, in contrast, seemed to take time to weigh all of the available evidence, then make the best decision available.

“The slowest agent to make a decision tended to make decisions in a way very similar to a completely unbiased agent,” Kilpatrick said. “They pretty much behaved as if they started from scratch.”

Neighbourhood choices

He noted that the study had some limitations. In the team’s experiments, for example, none of the agents knew what the others were doing. Kilpatrick compared it to neighbours staying inside their homes during an election year, not talking about their choices or putting up yard signs. In reality, humans often change their decisions based on the actions of their friends and neighbours.

Kilpatrick hopes to run a similar set of experiences in which the agents can influence each other’s behaviours.

“You might speculate that if you had a large group coupled together, the first agent to make a decision could kick off a cascade of potentially wrong decisions,” he said.

Still, political pollsters may want to take note of the team’s results.

“The study could also be applied to group decision making in human organizations where there’s democratic voting, or even when people give their input in surveys,” Kilpatrick said. “You might want to look at folks carefully if they give fast responses.”

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Credit of the article given to Daniel Strain, University of Colorado at Boulder

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In past years, elections in the U.S. have been marked by stories of long waiting lines at the voting polls. Add other barriers, like long commutes and inadequate transportation, and voting can become inaccessible. But these voting deserts are difficult to quantify.

In a paper, “Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites” in SIAM Review, SFI External Professor Mason Porter (UCLA) and his students applied topological data analysis, which gives a set of mathematical tools that can quantify shape and structure in data, to the problem of quantifying voting deserts in LA County, Chicago, Atlanta, Jacksonville, New York City, and Salt Lake City.

Using a type of topological data analysis called persistent homology, Porter and his co-authors used estimates of average waiting times and commute times to examine where the voting deserts are located.

Applying persistent homology to a data set can reveal clusters and holes in that data, and it offers a way to measure how long those holes persist. The combination of waiting times and commute times in the data creates a pattern, with holes filling in as time passes.

The longer the hole takes to fill, the more inaccessible voting is to people in that area. “We are basically playing connect-the-dots in a more sophisticated way, trying to fill in what’s there,” says Porter.

Moving forward, Porter hopes to use this strategy to more accurately determine voting deserts. Finding voting deserts will hopefully be used to make voting more accessible, but it requires better-quality data than what was available to him and his students.

“This is a proof of concept,” Porter said. “We had to make some very severe approximations, in terms of what data we had access to.”

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Credit of the article to be given  Santa Fe Institute

Statisticians from the National University of Singapore (NUS) have introduced a new technique that accurately describes high-dimensional data using lower-dimensional smooth structures. This innovation marks a significant step forward in addressing the challenges of complex nonlinear dimension reduction.

Traditional data analysis methods often rely on Euclidean (linear) dependencies among features. While this approach simplifies data representation, it struggles to capture the underlying complex patterns in high-dimensional data, typically located close to low-dimensional manifolds.

To bridge this gap, manifold-learning techniques have emerged as a promising solution. However, existing methods, such as manifold embedding and denoising, have been limited by a lack of detailed geometric understanding and robust theoretical underpinnings.

The team, led by Associate Professor Zhigang Yao from the Department of Statistics and Data Science, NUS with his Ph.D. student Jiaji Su pioneered a novel method for effectively estimating low-dimensional manifolds hidden within high-dimensional data. This approach not only achieves cutting-edge estimation accuracy and convergence rates but also enhances computational efficiency through the utilization of deep Generative Adversarial Networks (GANs).

This work was conducted in collaboration with Professor Shing-Tung Yau from the Yau Mathematical Sciences Center (YMSC) at Tsinghua University. Part of the work comes from Prof. Yao’s collaboration with Prof. Yau during his sabbatical visit to the Center of Mathematical Sciences and Applications (CMSA) at Harvard University.

Their findings have been published as a methodology paper in the Proceedings of the National Academy of Sciences.

Prof. Yao delivered a 45-minute invited lecture on this research at the recent International Congress of Chinese Mathematicians (ICCM) held in Shanghai, Jan. 2–5, 2024.

Highlighting the significance of the work, Prof. Yao said, “By accurately fitting manifolds, we can reduce data dimensionality while preserving crucial information, including the underlying geometric structure. This represents a major leap in data analysis, enhancing both accuracy and efficiency. By providing a solution that overcomes the limitations of previous methods, our research paves the way for enhanced data analysis and offers valuable insights for diverse applications in the scientific community.”

Looking ahead, Yao’s research team is developing a new framework to process even more complex data, such as single-cell RNA sequence data, while continuing to collaborate with the YMSC team. This ongoing work promises to revolutionize the approach for the reduction and processing of complex datasets, potentially offering new insights into a range of scientific fields.

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

Credit of the article given to National University of Singapore

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10% – a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, WEadvocate for expanding what equity means in mathematics education. WEbelieve policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power – and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom WEinterviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counselor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life – whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, WEcollaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of color and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrollment in advanced courses, WEbelieve true equity will remain elusive. Mathematical power – the ability and confidence to use math to make smart personal and professional decisions – needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, WEbelieve schools, universities and state leaders can create a more expansive and equitable math education system.

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

Credit of the article given to Megan Staples, The Conversation

A simple experiment and mathematical model suggest that when you snack on pistachios, you may need a surprisingly large bowl to accommodate the discarded shells.

Shelling your favourite snack nuts just got a lot easier: physicists have worked out the exact size of bowl to best fit discarded pistachio shells.

Ruben Zakine and Michael Benzaquen at École Polytechnique in Paris often find themselves discussing science in the cafeteria while eating pistachios. Naturally, they began wondering about the mathematics behind storing their snack refuse.

The researchers stuffed 613 pistachios into a cylindrical container to determine “packing density”, or the fraction of space taken up by whole nuts in their shells. Separately, they measured the packing density of the shells alone. In one experiment setup, the researchers poured the shells into a container and let them fall as they may, and in another they shook them into a denser, more efficient configuration.

Without shaking, the shells had about 73 per cent of the original packing density. Shaking decreased this number to 57 per cent. This suggests that, with any pistachio container, an additional half-sized container will hold shell refuse as long you occasionally shake the container while eating.

Zakine and Benzaquen backed up their findings by modelling pistachios as ellipsoids – three-dimensional shapes resembling squashed spheres – and their shells as hollow half-spheres and calculated their packing densities based on mathematical rules. These results confirmed the real-life experiments and suggested that the same ratios would work for other container shapes.

Despite these similarities, the researchers found about a 10 per cent discrepancy between the calculations and the real-life measurements. Zakine says that this is not surprising because pistachios are not perfect ellipsoids and have natural variations in shape. More broadly, it is tricky to calculate how best to pack objects into containers. So far, mathematics researchers have only had luck with doing calculations for spheres, like marbles, and uniform shapes like M&M’s, he says.

Going forward, the researchers want to run more complex calculations on a computer. But for now, they are looking forward to fielding mathematical questions whenever they serve pistachios at dinner parties.

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

*Credit for article given to Karmela Padavic-Callaghan*

Catering for students’ learning needs is something we all aim to do. But it can be challenging. Is it just about differentiation? What is the best way to differentiate? How do we put it into practice? Let’s explore some ideas, strategies and tips.

Differentiation

When you hear the word differentiation, what do you think of? Ability groupings? Open-ended tasks? Educational consultant Jennifer Bowden from the Mathematical Association of Victoria believes differentiation involves teachers considering “a whole range of different pedagogies … and making choices about pedagogical approaches based on the students that they teach”. In a nutshell it comes down to knowing your students and how they learn, so you can cater for their needs.

Find out what students know

Assessment is key to discovering what your students know – and don’t know! You can assess students to find out what knowledge they have, the concepts they understand and the skills they can apply to tasks.

Data from this assessment can then be used as a starting point to plan what you will teach.

Find out how students learn

You can go further than just understanding what your students know. Delve deeper and think about; what are your students’ learning behaviours? What are their attitudes towards learning maths? How do they learn best?

It’s important to note that this Is not about learning styles. It’s about knowing how a student:

• thinks and feels about maths
• becomes engaged in a topic, or problem
• responds to certain scaffolds
• makes connections between concepts
• applies what they have learnt.

When you understand your students on this level you have a greater insight into knowing how to best build their knowledge and skills.

Putting it into practice

Once you know your students well you are better prepared to meet their learning needs, but there are still many aspects to think about. Let’s unpack this further.

Planning for instruction

Maths expert Jennifer Bowden promotes the use of the instructional model known as launch, explore, summarise.

• Launch– begin with a question or a task for students to complete or explore.
• Explore– during this stage the teacher supports students at their different levels. Students can work on the same task, but it can be differentiated to extend or give extra support where needed by scaffolding. You can plan for the learning to be done independently, or in small groups.
• Summarise– upon competition of the lesson or task the students come together to share what they have learnt.

In an excellent podcast on the Maths Hub, Jennifer explains this model in greater detail.

Open-ended tasks

These rich tasks provide differentiation by output. Essentially all students are working on the same, or similar task, and students reach various outcomes, according to their individual knowledge and skill application.

Grouping students

There are times when you can best meet students’ needs through grouping them in certain ways. When doing so, consider the purpose of the groupings, and ensure the groups are flexible.

• You should be clear about the specific purpose of your groupings. What needs are you addressing by grouping students together? Are you extending them? Providing consolidation? Are you supporting them to ‘catch up’ on learning they have missed? Or providing intervention?
• Student groupings should beflexible and change according to their purpose. Sometimes groups are ability based, so students can complete different tasks, at different levels. Sometimes groups have mixed abilities so that students can use their various skills and levels of knowledge to problem solve and use their reasoning skills.

Student agency

Giving students a voice by encouraging them to discuss their learning can help you to understand their individual needs. Ask students about their learning; what they know and want to know, if they are feeling challenged and what helps them to learn. This feedback can help you plan and deliver lessons that cater for all student needs.

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

Credit of the article given to The Mathematics Hub

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

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*

“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.

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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.

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