Month: April 2019

The same freezing which is responsible for transforming liquids into glasses can help to predict some patterns observed in prime numbers, according to a team of scientists from Queen Mary, University of London and Bristol University.

At a low enough temperature, water freezes into ice by arranging its molecules into a very regular pattern called crystal. However many other liquids freeze not into crystals, but in much less regular structures called glasses – window glass being the most familiar example. Physicists have developed theories explaining the freezing phenomena, and built models for understanding the properties of glasses.

Now, a researcher from Queen Mary’s School of Mathematical Sciences, together with his colleagues from Bristol have found that frozen glasses may have something common with prime numbers and the patterns behind them.

Dr Fyodorov explained: “The prime numbers are the elements, or building blocks, of arithmetic. Our work provides evidence for a surprising connection between the primes and freezing in certain complex materials in Physics.”

A prime number is a whole number greater than 1 which can only be divided by 1 or itself. Primes play fundamentally important role in pure mathematics and its applications; and many mathematicians have tried to predict the patterns observed in prime numbers. One theory, called the Riemann Zeta Function is believed to be the most successful in revealing and explaining properties of primes.

The Riemann Zeta Function detects patterns in prime numbers in the same way that you might spot harmonies in music. It can be thought of as a series of peaks and troughs – which may be legitimately called a ‘landscape’ – encoding the properties of primes.

Dr Fyodorov continues: “One of important questions about the Riemann Zeta function relates to determining how large the highest of the peaks in the landscape are. In our paper we have argued that, unexpectedly, answering that question is related to the problem of characterizing the nature of the freezing transition in certain complex materials in Physics, such as glasses.

The team hope that understanding freezing could help mathematicians make progress in attacking some of the grand challenges of number theory.

Dr Fyodorov concludes: “Looking for connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function appeared to be a fruitful and promising approach.”

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Credit of the article given to Queen Mary, University of London

Professor Graeme Pettet is using mathematical equations to better understand how to heal wounds and bone fractures.

Understanding the way our bodies heal is not as easy as 1, 2, 3. But a Queensland University of Technology (QUT) researcher believes mathematics holds the answers to complex biological problems.

Professor Graeme Pettet, a mathematician from QUT’s Institute of Health Biomedical Innovation (IHBI), said maths could be used to better understand the structure of skin and bones and their response to healing techniques, which will eventually lead to better therapeutic innovations.

“Mathematics is the language of any science so if there are spatial or temporal variations of any kind then you can describe it mathematically,” he said.

Professor Pettet is a member of the Tissue Repair and Regeneration group, which in part focuses on growing skin in a laboratory and analysing its growth and repair process.

His team concentrates on the theoretical material, using data to build and solve mathematical equations.

“Skin is very difficult to describe. It’s very messy and very complicated. In fact most of the descriptions that engineers and biologists use are schematic stories (diagrams),” he said.

“Once we understand the structure (of the skin) and how it develops we can begin to analyse how that development impacts upon healing in the skin and maybe also diseases of the skin.”

Professor Pettet said his research would, for the first time, formalise the theories about the way cells interact when healing.

“It is clear that by improving our understanding of how the multiple cellular processes work together in a complex but orchestrated way has great potential in leading to therapeutic innovations,” he said.

Despite skin being our most accessible organ, Professor Pettet said there was a lot more to learn about how and what made it function.

“There’s a lot of structure just in the upper layer of the skin despite the fact it’s only a few cells thick,” he said.

“This can be characterised by how the cells function and change appearance, but what drives them to be that way is not completely known.

“My job is to try and understand and develop ways to describe how the bits and pieces everybody knows about are somehow connected.”

Professor Pettet is also working on applying similar techniques to figure out how to show how small, localised damage at the site of bone fractures can impact on healing.

While the long-term goal of understanding and being able to predict how our bodies heal is still a way off, Professor Pettet’s research is opening up a realm of biological problems where mathematical equations and techniques that have not previously been applied, are providing insights into the biology as well as the maths.

“We’ve discovered essentially new sets of solutions in these contexts and that’s led onto other projects looking at new mathematical tools to describe these new solutions that we’ve never seen before,” he said.

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Credit of the article given to Alita Pashley, Queensland University of Technology

There’s been plenty of commentary recently on the “numeracy crisis” threatening the economies of many developed nations, including Australia.

A 2009 report by the National Academies in the US was not the first to highlight the desperate need to improve mathematical education, particularly at the K-12 level, where so many otherwise talented students either fall behind or lose interest. The report’s summary concluded:

“The new demands of international competition in the 21st century require a workforce that is competent in and comfortable with mathematics.

“There is particular concern about the chronically low mathematics and science performance of economically disadvantaged students and the lack of diversity in the science and technical workforce. Particularly alarming is that such disparities exist in the earliest years of schooling and even before school entry …”

The committee found that, although virtually all young children have the capability to learn and become competent in mathematics, the potential to learn mathematics in the early years of school is not currently realised for most.

This stems from a lack of opportunities to learn mathematics either in early childhood settings or through everyday experiences at home and in communities. And this is especially the case for the economically disadvantaged.

A UK report released last month found that millions of British adults have numerical skills at a level more commonly expected of an 11-year-old. The report also found that young people with poor numeracy skills were twice as likely to drop out of school and twice as likely to be unemployed.

The report’s authors called for a change in society’s attitude to mathematics, so that being bad at maths should no longer be seen as a “badge of honour”.

According to the same UK report, one in five of business members questioned last year said they had to teach remedial mathematics to their employees. As James Fothergill, head of education and skills at the employers’ group CSI, explained:

“It’s really important that [employees] are helped to apply maths skills and concepts in practical situations, such as being able to work out what a 30% discount is without doing it on the till.”

Many business leaders also pointed to the fact few of their employees were able to spot “rogue figures” – data that is likely to be in error.

In February this year, speaking at a forum of national educators in Canberra, Australia’s Nobel Prize-winning astronomer Brian Schmidt went so far as to warn that Australia’s resource boom was threatened by a lack of highly-trained engineers, saying:

“Too many kids who are willing and able to excel at maths are taught by teachers without the competency required to teach the subjects they are teaching.”

At the same forum, Australia’s Chief Scientist Ian Chubb said part of the problem was that mathematics and science courses were considered “boring”.

“We need to think about how to deliver the science and mathematics to a generation of students that have many more options available to them,” he said.

The situation is better elsewhere. Finland and Canada, for example, rated an “A” in an international ranking of 17 developed nations in education and skills. Finland has ranked at (or near) the top of the OECD nations in educational performance for more than ten straight years.

Canada’s strength derives in part from the system’s primary focus on K-12 education. On the other hand, Canada faces the challenge of educating and training the three million adults, in a country of under 35m people, who have only “Level 1” literacy. This would seem to show that you do get what you pay for.

Of course, other countries, such as Japan, Taiwan, China, Korea and Singapore are not standing still, with impressive gains in educational performance.

So, what can be done, for the good of everyone? Perhaps all nations can examine the educational programs of highly successful nations such as Finland.

The Finnish educational system eschews standards tests, preferring instead custom tests devised by highly qualified teachers. (Several decades ago, the government required all teachers to have master’s degrees).

Another is Finland’s focus on basic education from age seven until 16, at which point 95% of the population continues in either vocational or academic high schools. According to Pasi Sahlberg, a Finnish educator and author:

“The primary aim of education is to serve as an equalising instrument for society.”

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Credit of the article given to Jonathan Borwein and David H. Bailey

This is a simulation of a network of people playing Prisoner’s Dilemma. Red are cooperators; blue are defectors.

For the past twenty years, there has been a great controversy regarding whether the structure of interactions among individuals (that is, if the existence of a certain contact network or social network) helps to foment cooperation among them in situations in which not cooperating brings benefits without generating the costs of helping. Many theoretical studies have analysed this subject, but the conclusions have been contradictory since the way in which people make decisions is almost always based on a hypothesis of the models with very little basis to justify it.

The study carried out by these university researchers adopts a pioneering perspective on the theoretical study of the emergence of cooperation: rather than postulating that people make decisions according to one procedure or another, it incorporates the results obtained in experiments designed precisely to analyse how people decide whether to cooperate or not. The authors of the study are professors from the Interdisciplinary Complex Systems Group (Grupo Interdisciplinar de Sistemas Complejos – GISC) of the Mathematics Department of Carlos III University of Madrid, José Cuesta and Ángel Sánchez, together with Carlos Gracia and Yamir Moreno, from the Complex Systems and Networks Group (Grupo de Redes y Sistemas Complejos – COSNET Lab) of the Institute for Biocomputation and Physics of Complex Systems (Instituto de Biocomputación y Física de Sistemas Complejos – BIFI) of the University of Zaragoza. Their study was recently published in Scientific Reports, Nature’s new open access magazine.

This work is based on the results of an experiment carried out by the researchers and on information from other previous studies, as well as on the results (as yet unpublished) obtained from their own new experiments. The observations from these studies coincide in indicating that people do not consider what those they interact with gain; rather they think about whether or not they cooperate. In addition, their decisions usually depend on their own mood. That is, the authors noticed that the probability of cooperation occurring was considerably higher if there had been cooperation in the previous interaction. They also observed a certain heterogeneity in behaviour, finding a certain percentage of individuals who cooperated very little, regardless of what those around them did, and other individuals who almost always cooperated, again, no matter what others did.

These researchers have mathematically examined what occurs when groups of people who behave as the experiments say have to decide whether or not to cooperate, and how the existence of cooperation, globally or in the group, depends on the structure of the interactions. Specifically, the study analyses what happens if each person interacts with all of the others, if the people are placed in a square reticule and they interact with their four closest neighbours, or if they are arranged in a network that is more similar to a social network, in which the number of neighbours is highly variable and is dependent on each person. In the first case (each individual interacts with all of the others), the problem can be solved mathematically, so the level of resulting cooperation can be predicted. What the researchers observed is that this depends on the makeup of the population; that is, what proportion of the individuals use the previously described strategy, and what proportion almost always cooperates or almost never does, regardless of what the others do. Afterwards, this prediction can be compared with the results of numeric simulations obtains for the populations placed in each of the two networks, and it can be shown that the result is exactly the same, unlike what had been concluded in previous studies.

The consequences of this prediction are very important, according to the authors of the study, because if they are true, it would rule out the existence of one of the five mechanisms that have been proposed to explain the emergence of cooperation, the so-called “network reciprocity” mechanism. In order to prove the prediction, it will be necessary to carry out a large-scale experiment, something that this group of researchers in currently very involved in. These experiments are very difficult to carry out, given that studying heterogeneous networks in such a way as to obtain significant results, the team must work with hundred of volunteers simultaneously. If, as the team hopes, the experiments confirm what this study predicts, we would be witnessing a paradigm shift in the interpretation of decision-making in cooperative dilemmas: instead of considering what is to be gained, individuals would base their decisions on the cooperation they have received, and this would mean that the way that they interact (the underlying social network) would cease to be important.

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Credit of the article given to Carlos III University of Madrid