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Neural networks have been powering breakthroughs in artificial intelligence, including the large language models that are now being used in a wide range of applications, from finance, to human resources to health care. But these networks remain a black box whose inner workings engineers and scientists struggle to understand.

Now, a team led by data and computer scientists at the University of California San Diego has given neural networks the equivalent of an X-ray to uncover how they actually learn.

The researchers found that a formula used in statistical analysis provides a streamlined mathematical description of how neural networks, such as GPT-2, a precursor to ChatGPT, learn relevant patterns in data, known as features. This formula also explains how neural networks use these relevant patterns to make predictions.

“We are trying to understand neural networks from first principles,” said Daniel Beaglehole, a Ph.D. student in the UC San Diego Department of Computer Science and Engineering and co-first author of the study. “With our formula, one can simply interpret which features the network is using to make predictions.”

The team present their findings in the journal Science.

Why does this matter? AI-powered tools are now pervasive in everyday life. Banks use them to approve loans. Hospitals use them to analyse medical data, such as X-rays and MRIs. Companies use them to screen job applicants. But it’s currently difficult to understand the mechanism neural networks use to make decisions and the biases in the training data that might impact this.

“If you don’t understand how neural networks learn, it’s very hard to establish whether neural networks produce reliable, accurate, and appropriate responses,” said Mikhail Belkin, the paper’s corresponding author and a professor at the UC San Diego Halicioglu Data Science Institute. “This is particularly significant given the rapid recent growth of machine learning and neural net technology.”

The study is part of a larger effort in Belkin’s research group to develop a mathematical theory that explains how neural networks work. “Technology has outpaced theory by a huge amount,” he said. “We need to catch up.”

The team also showed that the statistical formula they used to understand how neural networks learn, known as Average Gradient Outer Product (AGOP), could be applied to improve performance and efficiency in other types of machine learning architectures that do not include neural networks.

“If we understand the underlying mechanisms that drive neural networks, we should be able to build machine learning models that are simpler, more efficient and more interpretable,” Belkin said. “We hope this will help democratize AI.”

The machine learning systems that Belkin envisions would need less computational power, and therefore less power from the grid, to function. These systems also would be less complex and so easier to understand.

Illustrating the new findings with an example

(Artificial) neural networks are computational tools to learn relationships between data characteristics (i.e. identifying specific objects or faces in an image). One example of a task is determining whether in a new image a person is wearing glasses or not. Machine learning approaches this problem by providing the neural network many example (training) images labeled as images of “a person wearing glasses” or “a person not wearing glasses.”

The neural network learns the relationship between images and their labels, and extracts data patterns, or features, that it needs to focus on to make a determination. One of the reasons AI systems are considered a black box is because it is often difficult to describe mathematically what criteria the systems are actually using to make their predictions, including potential biases. The new work provides a simple mathematical explanation for how the systems are learning these features.

Features are relevant patterns in the data. In the example above, there are a wide range of features that the neural networks learns, and then uses, to determine if in fact a person in a photograph is wearing glasses or not.

One feature it would need to pay attention to for this task is the upper part of the face. Other features could be the eye or the nose area where glasses often rest. The network selectively pays attention to the features that it learns are relevant and then discards the other parts of the image, such as the lower part of the face, the hair and so on.

Feature learning is the ability to recognize relevant patterns in data and then use those patterns to make predictions. In the glasses example, the network learns to pay attention to the upper part of the face. In the new Science paper, the researchers identified a statistical formula that describes how the neural networks are learning features.

Alternative neural network architectures: The researchers went on to show that inserting this formula into computing systems that do not rely on neural networks allowed these systems to learn faster and more efficiently.

“How do I ignore what’s not necessary? Humans are good at this,” said Belkin. “Machines are doing the same thing. Large Language Models, for example, are implementing this ‘selective paying attention’ and we haven’t known how they do it. In our Science paper, we present a mechanism explaining at least some of how the neural nets are ‘selectively paying attention.'”

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Credit of the article given to University of California – San Diego

A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and University of Burgundy (uB), have analysed drawings made by children and adults when solving simple problems. The scientists found that, whatever the age of the participant, the most effective calculation strategies were associated with certain drawing typologies.

These results, published in the journal Memory & Cognition, open up new perspectives for the teaching of mathematics.

Learning mathematics often involves small problems, linked to concrete everyday situations. For example, pupils have to add up quantities of flour to make a recipe or subtract sums of money to find out what’s left in their wallets after shopping. They are thus led to translate statements into algorithmic procedures to find the solution.

This translation of words into solving strategies involves a stage of mental representation of mathematical information, such as numbers or the arithmetic operation to be performed, and non-mathematical information, such as the context of the problem.

The cardinal or ordinal dimensions of problems

Having a clearer idea of these mental representations would enable a better understanding of the choice of calculation strategies. Scientists from UNIGE, CYU and uB conducted a study with 10-year-old children and adults, asking them to solve simple problems with the instruction to use as few calculation steps as possible.

The participants were then asked to produce a drawing or diagram explaining their problem-solving strategy for each statement. The contexts of some problems called on the cardinal properties of numbers—the quantity of elements in a set—others on their ordinal properties—their position in an ordered list.

The former involved marbles, fishes, or books, for example: “Paul has 8 red marbles. He also has blue marbles. In total, Paul has 11 marbles. Jolene has as many blue marbles as Paul, and some green marbles. She has 2 green marbles less than Paul has red marbles. In total, how many marbles does Jolene have?”

The latter involved lengths or durations, for example: “Sofia traveled for 8 hours. Her trip started during the day. Sofia arrived at 11. Fred leaves at the same time as Sofia. Fred’s trip lasted 2 hours less than Sofia’s. What time was it when Fred arrived?”

Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11–8 = 3; 8–2 = 6; 6 + 3 = 9, but also in a single calculation: 11–2 = 9, using a simple subtraction. However, the mental representations of these problems are very different, and the researchers wanted to determine whether the type of representations could predict the calculation strategy, in 1 or 3 steps, of those who solve them.

‘”Our hypothesis was that cardinal problems—such as the one involving marbles—would inspire cardinal drawings, i.e., diagrams with identical individual elements, such as crosses or circles, or with overlaps of elements in sets or subsets.

“Similarly, we assumed that ordinal problems—such as the one mentioning travel times—would lead to ordinal representations, i.e., diagrams with axes, graduations or intervals—and that these ordinal drawings would reflect participants’ representations and indicate that they would be more successful in identifying the one-step solution strategy,” explains Hippolyte Gros, former post-doctoral fellow at UNIGE’s Faculty of Psychology and Educational Sciences, associate professor at CYU, and first author of the study.

Identifying mental representations through drawings

These hypotheses were validated by analysing the drawings of 52 adults and 59 children. “We have shown that, irrespective of their experience—since the same results were obtained in both children and adults—the use of strategies by the participants depends on their representation of the problem, and that this is influenced by the non-mathematical information contained in the problem statement, as revealed by their drawings,” says Emmanuel Sander, full professor at the UNIGE’s Faculty of Psychology and Educational Sciences.

“Our study also shows that, even after years of experience in solving addition and subtraction, the difference between cardinal and ordinal problems remains very marked. The majority of participants were only able to solve problems of the second type in a single step.”

Improving mathematical learning through drawing analysis

The team also noted that drawings showing ordinal representations were more frequently associated with a one-step solution, even if the problem was cardinal. In other words, drawing with a scale or an axis is linked to the choice of the fastest calculation.

“From a pedagogical point of view, this suggests that the presence of specific features in a student’s drawing may or may not indicate that his or her representation of the problem is the most efficient one for meeting the instructions—in this case, solving with the fewest calculations possible,” observes Jean-Pierre Thibaut, full professor at the uB Laboratory for Research on Learning and Development.

“Thus, when it comes to subtracting individual elements, a representation via an axis—rather than via subsets—is more effective in finding the fastest method. Analysis of students’ drawings in arithmetic can therefore enable targeted intervention to help them translate problems into more optimal representations. One way of doing this is to work on the graphical representation of statements in class, to help students understand the most direct strategies,” concludes Gros.

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Credit of the article given to University of Geneva.

March 14 marks the annual celebration of the mathematical constant pi, aka the Greek letter π. Its infinite number of digits is usually rounded to 3.14, hence the date of Pi Day. For some people, the occasion marks an annual excuse to eat pizza or pie (or both), but to truly honour this wondrously useful number, a serving of mathematics is in order, too. NASA is here to help.

Continuing a decade-long tradition, the Education Office at the agency’s Jet Propulsion Laboratory has cooked up a set of illustrated math problems involving real-life NASA science and engineering.

With the NASA Pi Day Challenge, students can use the mathematical constant to:

• determine where the DSOC (Deep Space Optical Communications) technology demonstration aboard NASA’s Psyche spacecraft should aim a laser message containing a cat video so that it can reach Earth (and set a NASA record in the process)
• figure out the change in asteroid Dimorphos’ orbit after NASA intentionally crashed its DART (Double Asteroid Redirection Test) spacecraft into its surface
• measure how much data will be captured by the NISAR (NASA-ISRO Synthetic Aperture Radar) satellite each time it orbits our planet, monitoring Earth’s land and ice surfaces in unprecedented detail
• calculate the distance a small rover must drive to map a portion of the lunar surface as part of NASA’s CADRE (Cooperative Autonomous Distributed Robotic Exploration) technology demonstration that’s headed to the moon

Answers to all four challenge questions will be available on March 15.

The NASA Pi Day Challenge is accompanied by other pi-related resources for educators, K-12 students, and parents, including lessons and teachable moments, downloadable posters, and illustrated web/mobile backgrounds. More than 40 puzzlers from previous challenges are also available.

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Credit of the article given to NASA

Is it possible to deduce the shape of a drum from the sounds it makes? This is the kind of question that Iosif Polterovich, a professor in the Department of Mathematics and Statistics at Université de Montréal, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena involving wave propagation.

Last summer, Polterovich and his international collaborators—Nikolay Filonov, Michael Levitin and David Sher—proved a special case of a famous conjecture in spectral geometry formulated in 1954 by the eminent Hungarian-American mathematician George Pólya.

The conjecture bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.

Pólya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. Until last year, the conjecture was known only for these cases. The disk, despite its apparent simplicity, remained elusive.

“Imagine an infinite floor covered with tiles of the same shape that fit together to fill the space,” Polterovich said. “It can be tiled with squares or triangles, but not with disks. A disk is actually not a good shape for tiling.”

The universality of mathematics

In an article published in the mathematical journal Inventiones Mathematicae, the researchers show that Pólya’s conjecture is true for the disk, a case considered particularly challenging.

Though their result is essentially of theoretical value, their proof method has applications in computational mathematics and numerical computation. The authors are now investigating this avenue.

“While mathematics is a fundamental science, it is similar to sports and the arts in some ways,” Polterovich said.

“Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. And in many cases, beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.”

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Credit of the article to be given Béatrice St-Cyr-Leroux, University of Montreal

When facing a daunting dataset, Principal Component Analysis (PCA), known as PCA, can help distill complexity by finding a few meaningful features that explain the most significant proportion of the data variance.

However, PCA comes with the underlying assumption that all data sources are homogeneous.

The growth in Internet of Things connectivity poses a challenge as the data collected by “clients,” like patients, connected vehicles, sensors, hospitals or cameras, are incredibly heterogeneous. As this increasing array of technologies from smartwatches to manufacturing tools collect monitoring data, a new analytical tool is needed to disentangle heterogeneous data and characterize what is shared and unique across increasingly complex data from multiple sources.

“Identifying meaningful commonalities among these devices poses a significant challenge. Despite extensive research, we found no existing method that can provably extract both interpretable and identifiable shared and unique features from different datasets,” said Raed Al Kontar, an assistant professor of industrial and operations engineering.

To tackle this challenge, the University of Michigan researchers Niaichen Shi and Raed Al Kontar developed a new “personalized PCA,” or PerPCA, method to decouple the shared and unique components from heterogeneous data. The results will be published in the Journal of Machine Learning Research.

“The personalized PCA method leverages low-rank representation learning techniques to accurately identify both shared and unique components with good statistical guarantees,” said Shi, first author of the paper and a doctoral student of industrial and operations engineering.

“As a simple method that can effectively identify shared and unique features, we envision personalized PCA will be helpful in fields including genetics, image signal processing, and even large language models.”

Further increasing its utility, the method can be implemented in a fully federated and distributed manner, meaning that learning can be distributed across different clients, and raw data does not need to be shared; only the shared (and not unique) features are communicated across the clients.

“This can enhance data privacy and save communication and storage costs,” said Al Kontar.

With personalized PCA, different clients can collaboratively build strong statistical models despite the considerable differences in their data. The extracted shared and unique features encode rich information for downstream analytics, including clustering, classification, or anomaly detection.

The researchers demonstrated the method’s capabilities by effectively extracting key topics from 13 different data sets of U.S. presidential debate transcriptions from 1960 to 2020. They were able to discern shared and unique debate topics and keywords.

Personalized PCA leverages linear features that are readily interpretable by practitioners, further enhancing its use in new applications.

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Credit of the article to be given Patricia DeLacey, University of Michigan College of Engineering

With its bright colors, easy-to-learn rules and familiar music, the video game Tetris has endured as a pop culture icon over the last 40 years. Many people, like me, have been playing the game for decades, and it has evolved to adapt to new technologies like game systems, phones and tablets. But until January 2024, nobody had ever been able to beat it.

A teen from Oklahoma holds the Tetris title after he crashed the game on Level 157 and beat the game. Beating it means the player moved the tiles too fast for the game to keep up with the score, causing the game to crash. Artificial intelligence can suggest strategies that allow players to more effectively control the game tiles and slot them into place faster—these strategies helped crown the game’s first winner.

But there’s far more to Tetris than the elusive promise of winning. As a mathematician and mathematics educator, I recognize that the game is based on a fundamental element of geometry, called dynamic spatial reasoning. The player uses these geometric skills to manipulate the game pieces, and playing can both test and improve a player’s dynamic spatial reasoning.

Playing the game

A Russian computer scientist named Alexey Pajitnov invented Tetris in 1984. The game itself is very simple: The Tetris screen is composed of a rectangular game board with dropping geometric figures. These figures are called tetrominoes, made up of four squares connected on their sides in seven different configurations.

The game pieces drop from the top, one at a time, stacking up from the bottom. The player can manipulate each one as it falls by turning or sliding it and then dropping it to the bottom. When a row completely fills up, it disappears and the player earns points.

As the game progresses, the pieces appear at the top more quickly, and the game ends when the stack reaches the top of the board.

Dynamic spatial reasoning

Manipulating the game pieces gives the player an exercise in dynamic spatial reasoning. Spatial reasoning is the ability to visualize geometric figures and how they will move in space. So, dynamic spatial reasoning is the ability to visualize actively moving figures.

The Tetris player must quickly decide where the currently dropping game piece will best fit and then move it there. This movement involves both translation, or moving a shape right and left, and rotation, or twirling the shape in increments of 90 degrees on its axis.

Spatial visualization is partly inherent ability, but partly learned expertise. Some researchers identify spatial skill as necessary for successful problem solving, and it’s often used alongside mathematics skills and verbal skills.

Spatial visualization is a key component of a mathematics discipline called transformational geometry, which is usually first taught in middle school. In a typical transformational geometry exercise, students might be asked to represent a figure by its x and y coordinates on a coordinate graph and then identify the transformations, like translation and rotation, necessary to move it from one position to another while keeping the piece the same shape and size.

Reflection and dilation are the two other basic mathematical transformations, though they’re not used in Tetris. Reflection flips the image across any line while maintaining the same size and shape, and dilation changes the size of the shape, producing a similar figure.

For many students, these exercises are tedious, as they involve plotting many points on graphs to move a figure’s position. But games like Tetris can help students grasp these concepts in a dynamic and engaging way.

Transformational geometry beyond Tetris

While it may seem simple, transformational geometry is the foundation for several advanced topics in mathematics. Architects and engineers both use transformations to draw up blueprints, which represent the real world in scale drawings.

Animators and computer graphic designers use concepts of transformations as well. Animation involves representing a figure’s coordinates in a matrix array and then creating a sequence to change its position, which moves it across the screen. While animators today use computer programs that automatically move figures around, they are all based on translation.

Calculus and differential geometry also use transformation. The concept of optimization involves representing a situation as a function and then finding the maximum or minimum value of that function. Optimization problems often involve graphic representations where the student uses transformations to manipulate one or more of the variables.

Lots of real-world applications use optimization—for example, businesses might want to find out the minimum cost of distributing a product. Another example is figuring out the size of a theoretical box with the largest possible volume.

All of these advanced topics use the same concepts as the simple moves of Tetris.

Tetris is an engaging and entertaining video game, and players with transformational geometry skills might find success playing it. Research has found that manipulating rotations and translations within the game can provide a solid conceptual foundation for advanced mathematics in numerous science fields.

Playing Tetris may lead students to a future aptitude in business analytics, engineering or computer science—and it’s fun. As a mathematics educator, I encourage students and friends to play on.

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Credit of the article to be given Leah McCoy, The Conversation

Intelligent tutoring systems for math problems helped pupils remain or even increase their performance during the pandemic. This is the conclusion of a new study led by the Martin Luther University Halle-Wittenberg (MLU) and Loughborough University in the U.K.

As part of their work, the researchers analysed data from 5 million exercises done by about 2,700 pupilsin Germany over a period of five years. The study found that particularly lower-performing children benefit if they use the software regularly. The paper was published in the journal Computers and Education Open.

Intelligent tutoring systems are digital learning platforms that children can use to complete math problems. “The advantage of those rapid learning aids is that pupils receive immediate feedback after they submit their solution. If a solution is incorrect, the system will provide further information about the pupil’s mistake.

“If certain errors are repeated, the system recognizes a deficit and provides further problem sets that address the issue,” explains Assistant Professor Dr. Markus Spitzer, a psychologist at MLU. Teachers could also use the software to discover possible knowledge gaps in their classes and adapt their lessons accordingly.

For the new study, Spitzer and his colleague Professor Korbinian Moeller from Loughborough University used data from “Bettermarks,” a large commercial provider of such tutoring systems in Germany. The team analysed the performance of pupils before, during and after the first two coronavirus lockdowns.

Their analysis included data from about 2,700 children who solved more than 5 million problems. The data was collected between January 2017 and the end of May 2021. “This longer timeframe allowed us to observe the pupils’ performance trajectories over several years and analyse them in a wider context,” says Spitzer.

The students’ performance was shown to remain constant throughout the period. “The fact that their performance didn’t drop during the lockdowns is a win in and of itself. But our analysis also shows that lower-performing children even managed to narrow the gap between themselves and higher achieving pupils,” Spitzer concludes.

According to the psychologist, intelligent tutoring systems are a useful addition to conventional math lessons. “The use of tutoring systems varies greatly from state to state. However, our study suggests that their use should be expanded across the board,” explains Spitzer. The systems could also help during future school closures, for example in the event of extreme weather conditions, transport strikes or similar events.

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Credit of the article to be given Tom Leonhardt, Martin Luther University Halle-Wittenberg

The tone and tuning of musical instruments has the power to manipulate our appreciation of harmony, new research shows. The findings challenge centuries of Western music theory and encourage greater experimentation with instruments from different cultures.

According to the Ancient Greek philosopher Pythagoras, ‘consonance’—a pleasant-sounding combination of notes—is produced by special relationships between simple numbers such as 3 and 4. More recently, scholars have tried to find psychological explanations, but these ‘integer ratios’ are still credited with making a chord sound beautiful, and deviation from them is thought to make music ‘dissonant,’ unpleasant sounding.

But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong.

Their study, published in Nature Communications, shows that in normal listening contexts, we do not actually prefer chords to be perfectly in these mathematical ratios.

“We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us,” said co-author, Dr. Peter Harrison, from Cambridge’s Faculty of Music and Director of its Center for Music and Science.

The researchers also found that the role played by these mathematical relationships disappears when you consider certain musical instruments that are less familiar to Western musicians, audiences and scholars. These instruments tend to be bells, gongs, types of xylophones and other kinds of pitched percussion instruments. In particular, they studied the ‘bonang,’ an instrument from the Javanese gamelan built from a collection of small gongs.

“When we use instruments like the bonang, Pythagoras’s special numbers go out the window and we encounter entirely new patterns of consonance and dissonance,” Dr. Harrison said.

“The shape of some percussion instruments means that when you hit them, and they resonate, their frequency components don’t respect those traditional mathematical relationships. That’s when we find interesting things happening.”

“Western research has focused so much on familiar orchestral instruments, but other musical cultures use instruments that, because of their shape and physics, are what we would call ‘inharmonic.'”

The researchers created an online laboratory in which over 4,000 people from the US and South Korea participated in 23 behavioural experiments. Participants were played chords and invited to give each a numeric pleasantness rating or to use a slider to adjust particular notes in a chord to make it sound more pleasant. The experiments produced over 235,000 human judgments.

The experiments explored musical chords from different perspectives. Some zoomed in on particular musical intervals and asked participants to judge whether they preferred them perfectly tuned, slightly sharp or slightly flat.

The researchers were surprised to find a significant preference for slight imperfection, or ‘inharmonicity.’ Other experiments explored harmony perception with Western and non-Western musical instruments, including the bonang.

Instinctive appreciation of new kinds of harmony

The researchers found that the bonang’s consonances mapped neatly onto the particular musical scale used in the Indonesian culture from which it comes. These consonances cannot be replicated on a Western piano, for instance, because they would fall between the cracks of the scale traditionally used.

“Our findings challenge the traditional idea that harmony can only be one way, that chords have to reflect these mathematical relationships. We show that there are many more kinds of harmony out there, and that there are good reasons why other cultures developed them,” Dr. Harrison said.

Importantly, the study suggests that its participants—not trained musicians and unfamiliar with Javanese music—were able to appreciate the new consonances of the bonang’s tones instinctively.

“Music creation is all about exploring the creative possibilities of a given set of qualities, for example, finding out what kinds of melodies can you play on a flute, or what kinds of sounds can you make with your mouth,” Harrison said.

“Our findings suggest that if you use different instruments, you can unlock a whole new harmonic language that people intuitively appreciate, they don’t need to study it to appreciate it. A lot of experimental music in the last 100 years of Western classical music has been quite hard for listeners because it involves highly abstract structures that are hard to enjoy. In contrast, psychological findings like ours can help stimulate new music that listeners intuitively enjoy.”

Exciting opportunities for musicians and producers

Dr. Harrison hopes that the research will encourage musicians to try out unfamiliar instruments and see if they offer new harmonies and open up new creative possibilities.

“Quite a lot of pop music now tries to marry Western harmony with local melodies from the Middle East, India, and other parts of the world. That can be more or less successful, but one problem is that notes can sound dissonant if you play them with Western instruments.”

“Musicians and producers might be able to make that marriage work better if they took account of our findings and considered changing the ‘timbre,’ the tone quality, by using specially chosen real or synthesized instruments. Then they really might get the best of both worlds: harmony and local scale systems.”

Harrison and his collaborators are exploring different kinds of instruments and follow-up studies to test a broader range of cultures. In particular, they would like to gain insights from musicians who use ‘inharmonic’ instruments to understand whether they have internalized different concepts of harmony to the Western participants in this study.

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Credit of the article to be given University of Cambridge

Studies by applied mathematicians at the University of Surrey are helping to identify ways of reducing how much liquids slosh around inside tanks.

Baffles slow down the movement of fluid by diverting its flow. The research found that two or three porous baffles dividing a tank calms sloshing better than a single separator, but the returns diminish as more baffles are added. The paper is published in the Journal of Engineering Mathematics.

The findings and improved understanding into how external movement impacts the way liquids slosh could help mathematicians and engineers design better tankers to transport liquids on land or at sea.

The findings could also be used in tuned liquid dampers, which reduce the sway of skyscrapers in earthquakes and high winds.

Dr. Matthew Turner, a mathematician at the University of Surrey and expert in fluid dynamics who conducted the research using mathematical modeling, said, “Sloshing liquids can impact safety and efficiency. For example, if a tanker transporting liquids via road stopped suddenly, extreme movement of liquid inside the tanker could move the vehicle forwards, and unstable fuel loads in a space rocket could be catastrophic. Porous baffles inserted within a tank can help stabilize loads and reduce sloshing. Our research helps clarify how many it’s worth using.”

Jane Nicholson, EPSRC’s director of research base, said, “This fundamental research demonstrates the potential impact of math research, as a result of our mathematical sciences small grants investment. It is motivated by real-world applications to ensure the safer and more efficient transportation of liquids and will bring new solutions in a wide range of sectors.”

Next Dr. Turner wants to investigate whether actively varying how porous the baffles are could offer further benefits, “A mechanism which controls the rate of flow through the baffle could help us optimize designs. It could also be helpful when designing wave energy converters.”

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Credit of the article to be given UK Research and Innovation

Human coexistence depends on cooperation. Individuals have different motivations and reasons to collaborate, resulting in social dilemmas, such as the well-known prisoner’s dilemma. Scientists from the Chatterjee group at the Institute of Science and Technology Austria (ISTA) now present a new mathematical principle that helps to understand the cooperation of individuals with different characteristics. The results, published in PNAS, can be applied to economics or behavioural studies.

A group of neighbours shares a driveway. Following a heavy snowstorm, the entire driveway is covered in snow, requiring clearance for daily activities. The neighbours have to collaborate. If they all put on their down jackets, grab their snow shovels, and start digging, the road will be free in a very short amount of time. If only one or a few of them take the initiative, the task becomes more time-consuming and labor-intensive. Assuming nobody does it, the driveway will stay covered in snow. How can the neighbours overcome this dilemma and cooperate in their shared interests?

Scientists in the Chatterjee group at the Institute of Science and Technology Austria (ISTA) deal with cooperative questions like that on a regular basis. They use game theory to lay the mathematical foundation for decision-making in such social dilemmas.

The group’s latest publication delves into the interactions between different types of individuals in a public goods game. Their new model, published in PNAS, explores how resources should be allocated for the best overall well-being and how cooperation can be maintained.

The game of public goods

For decades, the public goods game has been a proven method to model social dilemmas. In this setting, participants decide how much of their own resources they wish to contribute for the benefit of the entire group. Most existing studies considered homogeneous individuals, assuming that they do not differ in their motivations and other characteristics.

“In the real world, that’s not always the case,” says Krishnendu Chatterjee. To account for this, Valentin Hübner, a Ph.D. student, Christian Hilbe, and Maria Kleshina, both former members of the Chatterjee group, started modeling settings with diverse individuals.

A recent analysis of social dilemmas among unequals, published in 2019, marked the foundation for their work, which now presents a more general model, even allowing multi-player interaction.

“The public good in our game can be anything, such as environmental protection or combating climate change, to which everybody can contribute,” Hübner explains. The players have different levels of skills. In public goods games, skills typically refer to productivity.

“It’s the ability to contribute to a particular task,” Hübner continues. Resources, technically called endowment or wealth, on the other hand, refer to the actual things that participants contribute to the common good.

In the snowy driveway scenario, the neighbours vary significantly in their available resources and in their abilities to use them. Solving the problem requires them to cooperate. But what role does their inequality play in such a dilemma?

The two sides of inequality

Hübner’s new model provides answers to this question. Intuitively, it proposes that for diverse individuals to sustain cooperation, a more equal distribution of resources is necessary. Surprisingly, more equality does not lead to maximum general welfare. To reach this, the resources should be allocated to more skilled individuals, resulting in a slightly uneven distribution.

“Efficiency benefits from unequal endowment, while robustness always benefits from equal endowment,” says Hübner. Put simply, for accomplishing a task, resources should be distributed almost evenly. Yet, if efficiency is the goal, resources should be in the hands of those more willing to participate—but only to a certain extent.

What is more important—cooperation efficiency or stability? The scientists’ further simulations of learning processes suggest that individuals balance the trade-off between these two things. Whether this is also the case in the real world remains to be seen. Numerous interpersonal nuances also contribute to these dynamics, including aspects like reciprocity, morality, and ethical issues, among others.

Hübner’s model solely focuses on cooperation from a mathematical standpoint. Yet, due to its generality, it can be applied to any social dilemma with diverse individuals, like climate change, for instance. Testing the model in the real world and applying it to society are very interesting experimental directions.

“I’m quite sure that there will be behavioural experiments benefiting from our work in the future,” says Chatterjee. The study could potentially also be interesting for economics, where the new model’s principles can help to better inform economic systems and policy recommendations.

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Credit of the article to be given Institute of Science and Technology Austria