The post Did Artists Lead the Way in Mathematics? appeared first on SciTech Connect.

]]>Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny.

From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other.

Does art sometimes spur mathematical discovery? There’s no simple answer to this question, but in some instances it seems very likely.

Consider Islamic ornament, such as that found in the Alhambra in Granada, Spain.

In the 14th and 15th centuries, the Alhambra served as the palace and harem of the Berber monarchs. For many visitors, it’s a setting as close to paradise as anything on earth: a series of open courtyards with fountains, surrounded by arcades that provide shelter and shade. The ceilings are molded in elaborate geometric patterns that resemble stalactites. The crowning glory is the ornament in colorful tile on the surrounding walls, which dazzles the eye in a hypnotic way that’s strangely blissful. In a fashion akin to music, the patterns lift the onlooker into an almost out-of-body state, a sort of heavenly rapture.

It’s a triumph of art – and of mathematical reasoning. The ornament explores a branch of mathematics known as tiling, which seeks to fill a space completely with regular geometric patterns. Math shows that a flat surface can be regularly covered by symmetric shapes with three, four and six sides, but not with shapes of five sides.

It’s also possible to combine different shapes, using triangular, square and hexagonal tiles to fill a space completely. The Alhambra revels in elaborate combinations of this sort, which are hard to see as stable rather than in motion. They seem to spin before our eyes. They trigger our brain into action and, as we look, we arrange and rearrange their patterns in different configurations.

An emotional experience? Very much so. But what’s fascinating about such Islamic tilings is that the work of anonymous artists and craftsmen also displays a near-perfect mastery of mathematical logic. Mathematicians have identified 17 types of symmetry: bilateral symmetry, rotational symmetry and so forth. At least 16 appear in the tilework of the Alhambra, almost as if they were textbook diagrams.

The patterns are not merely beautiful, but mathematically rigorous as well. They explore the fundamental characteristics of symmetry in a surprisingly complete way. Mathematicians, however, did not come up with their analysis of the principles of symmetry until several centuries after the tiles of the Alhambra had been set in place.

Stunning as they are, the decorations of the Alhambra may have been surpassed by a masterpiece in Persia. There, in 1453, anonymous craftsmen at the Darbi-I Imam shrine in Isfahan discovered quasicrystalline patterns. These patterns have complex and mysterious mathematical properties that were not analyzed by mathematicians until the discovery of Penrose tilings in the 1970s.

Such patterns fill a space completely with regular shapes, but in a configuration which never repeats itself – indeed, is infinitely nonrepeated – although the mathematical constant known as the Golden Section occurs over and over again.

Daniel Schectman won the 2001 Nobel Prize for the discovery of quasicrystals, which obey this law of organization. This breakthrough forced scientists to reconsider their conception of the very nature of matter.

In 2005, Harvard physicist Peter James Lu showed that it’s possible to generate such quasicrystalline patterns relatively easily using girih tiles. Girih tiles combine several pure geometric shapes into five patterns: a regular decagon, an irregular hexagon, a bow tie, a rhombus and a regular pentagon.

Whatever the method, it’s clear that the quasicrystalline patterns at Darbi-I Imam were created by craftsmen without advanced training in mathematics. It took several more centuries for mathematicians to analyze and articulate what they were doing. In other words, intuition preceded full understanding.

Geometric perspective made it possible to portray the visible world with a new verisimilitude and accuracy, creating an artistic revolution in the Italian Renaissance. One could argue that perspective also led to a major reexamination of the fundamental laws of mathematics.

According to Euclidian mathematics, two parallel lines will remain parallel into infinity and never meet. In the world of Renaissance perspective, however, parallel lines eventually do meet in the far distance at the so-called “vanishing point.” In other words, Renaissance perspective present a geometry which follows regular mathematical laws, but is non-Euclidian.

When mathematicians first devised non-Euclidian mathematics in the early 19th century, they imagined a world in which parallel lines meet at infinity. The geometry they explored was, in many ways, similar to that of Renaissance perspective.

Non-Euclidian mathematics has since moved on to explore space which has 12 or 13 dimensions, far outside the world of Renaissance perspective. But it’s worth asking whether Renaissance art may have made easier to make that initial leap.

An interesting modern case of art that broke traditional boundaries – and that has suggestive parallels with recent developments in mathematics – is that of the paintings of Jackson Pollock.

To those who first encountered them, the paintings of Pollock seemed chaotic and senseless. With time, however, we’ve come to see that they have elements of order, though not a traditional sort. Their shapes are simultaneously predictable and unpredictable, in a fashion similar to the pattern of dripping water from a faucet. There’s no way to predict the exact effect of the next drip. But, if we chart the pattern of drips, we find that they fall within a zone that has a clear shape and boundaries.

Such unpredictability was once out of bounds for mathematicians. But, in recent years, it has become one of the hottest areas of mathematical exploration. For example, chaos theory explores patterns that are not predictable but fall within a definable range of possibilities, while fractal analysis studies shapes that are similar but not identical.

Pollock himself had no particular interest in mathematics, and little known talent in that arena. His fascination with these forms was intuitive and subjective.

Intriguingly, mathematicians have not been able to accurately describe what Pollock was doing in his paintings. For example, there have been attempts to use fractal analysis to create a numerical “signature” of his style, but so far the method has not worked – we can’t mathematically distinguish Pollock’s autograph work from bad imitations. Even the notion that Pollock employed fractal thoughts is probably incorrect.

Nonetheless, Pollock’s simultaneously chaotic and orderly patterns have suggested a fruitful direction for mathematics. At some point, it may well be possible to describe what Pollock was doing with mathematical tools, and artists will have to move on and mark out a new frontier to explore.

*The author of this article is Henry Adams, Ruth Coulter Heede Professor of Art History, Case Western Reserve University. Read the original article here.*

If you found this article interesting, you may also like a new book *called **Understanding Emotions in Mathematical Thinking and Learning**, *a multidisciplinary approach to the role of emotions in numerical cognition, mathematics education, learning sciences, and affective sciences.

We are pleased to offer you a look at the book by providing you with a chapter from the book called *“An Overview of the Growth and Trends of Current Research on Emotions and Mathematics” *that gives an overview of the current study of the relationship between emotions and mathematics below:

If you would like to read additional chapters from the book, visit ScienceDirect. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

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]]>The post With New Technology, Mathematicians Turn Numbers into Art appeared first on SciTech Connect.

]]>Once upon a time, mathematicians imagined their job was to discover new mathematics and then let others explain it.

Today, digital tools like 3-D printing, animation and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.

Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.

**Chapter Download: Basic Finance**

The atmosphere was electric. Talks ran the gamut, from wildly creative thinkers who apply mathematics in the world of design to examples of pure mathematical results discovered through computer experimentation and visualization. It shed light on how powerful visualization has become for studying and sharing mathematics.

Visualization plays a growing role in mathematical research. According to John Sullivan at the Technical University of Berlin, mathematical thinking styles can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.

Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles.

Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity.

The “bubble” version of the configuration, shown below, is minimal, in the sense that it is the shortest possible shape where the tubes around the rings do not overlap. It’s as if you were to blow a soap bubble around each of the rings in the configuration. Techniques for proving that configurations like this are optimal often involve concepts of flow: If a given configuration is not the best, there are often ways to tell it to move in a direction that will make it better. This topic has great potential for visualization.

At the workshop, Sullivan dazzled us with a video of the three bands flowing into their optimal position. This animation allowed the researchers to see their ideas in action. It would never be considered as a substitute for a proof, but if an animation showed the wrong thing happening, people would realize that they must have made an error in their mathematics.

Visualization tools have helped mathematicians share their work in creative and surprising ways – even to rethink what the job of a mathematician might entail.

Take mathematician Fabienne Serrière, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.

Edmund Harriss of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”

Mathematical algorithms power the products made by Nervous System, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology.

Their first 3-D printed dress consists of thousands of interlocking pieces designed to fit a particular model. In order to print the dress, the designers folded up their virtual version, using protein-folding algorithms. A selective laser sintering process fused together parts of a block of powder to make the dress, then let all the unwanted powder fall away to reveal its shape.

Meanwhile, a delightful collection called Geometry Games can help everyone, from elementary school students to professional mathematicians, explore the concept of space. The project was founded by mathematician Jeff Weeks, one of the rock stars of the mathematical world. The iOS version of his “Torus Games” teaches children about multiply-connected spaces through interactive animation. According to Weeks, the app is verging on one million downloads.

My own work, described in my book “Creating Symmetry: The Artful Mathematics of Wallpaper Patterns,” starts with a visualization technique called the domain coloring algorithm.

I developed this algorithm in the 1990s to visualize mathematical ideas that have one dimension too many to see in 3-D space. The algorithm offers a way to use color to visualize something seemingly impossible to visualize in one diagram: a complex-valued function in the plane. This is a formula that takes one complex number (an expression of the form *a*+_b_i, which has two coordinates) and returns another. Seeing both the 2-D input and the 2-D output is one dimension more than ordinary eyes can see, hence the need for my algorithm. Now, I use it to create patterns and mathematical art.

My main pattern-making strategy relies on a branch of mathematics called Fourier theory, which involves the superposition of waves. Many people are familiar with the idea that the sound of a violin string can be broken down into its fundamental frequencies. My “wallpaper functions” break down plane patterns in just the same way.

My book starts with a lesson in making symmetric curves. Taking the same idea into a new dimension, I figured out how to weave polyhedral solids – think cube, dodecahedron, and so on – from symmetric bands made from these waves. I staged three of these new shapes, using Photoshop’s 3-D ray-tracing capacity, in the “Platonic Regatta” shown below. The three windsails display the symmetries of Platonic solids: the icosahedron/dodecahedron, cube/octahedron and tetrahedron.

About an hour after I spoke at the workshop, mathematician Mikael Vejdemo-Johansson had posted a Twitter bot to animate a new set of curves every day!

5-fold symmetry, seed: 701035, complex Fourier coefficients pic.twitter.com/HsLUZuVhIw

— Symmetric Curves (@symmetric_curve) March 19, 2017

Mathematics in the 21st century has entered a new phase. Whether you want to crack an unsolved problem, teach known results to students, design unique apparel or just make beautiful art, new tools for visualization can help you do it better.

*The author of this article is Frank A. Farris, Associate Professor of Mathematics, Santa Clara University. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may also like a new book out called *The Joy of Finite Mathematics: The Language and Art of Math**. *This book teaches step-by-step procedures, and clearly defined formulae, to help readers learn to apply math to subjects ranging from reason (logic) to finance (personal budget), making this interactive and engaging book appropriate for non-science, undergraduate students in the liberal arts, social sciences, finance, economics, and other humanities areas.

We are pleased to offer you a look at the book by providing you with a chapter from the book called *“Basic Finance” *that covers basic financing including sinking funds and amortization, various savings situations, and comparison shopping (credit versus cash, leasing versus purchasing, and renting versus owning).

There is also a section on personal finance (how to create a monthly budget), insurance (what every homeowner should know), and your credit report.

If you would like to read additional chapters from the book, visit ScienceDirect here. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

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]]>The post Universities Should Require Science, Engineering and Commerce Students to Know Their Maths appeared first on SciTech Connect.

]]>In 2013, a meeting of academics specialising in teaching first year undergraduate mathematics (known as the FYiMaths network) identified that the broad removal of mathematics prerequisites for many undergraduate degrees had created the biggest challenge they faced in teaching.

Many individuals had made attempts to pass this message up the management line at their universities. But at that time, staff believed that reintroducing prerequisites would never happen.

However, earlier this year The University of Sydney announced it would do exactly that, by requiring students studying science, engineering, commerce and IT to have completed at least intermediate level mathematics in high school.

The Australian Academy of Science’s Decadal Plan for the Mathematical Sciences, launched in Canberra yesterday, continues this push. One of its key recommendations is the reinstatement of mathematics prerequisites for science, engineering and commerce degrees.

But will it improve the level of maths education? Will it bolster mathematics skills in those studying science, engineering and commerce?

**Chapter Download: Equations and Functions**

A prerequisite study for entry to a degree is considered to be essential background knowledge that students need in order to be successful in that degree. A student cannot be selected into the degree if they do not have the stated prerequisite or an equivalent to it.

Over the past two decades, most universities have moved away from mathematics prerequisites, replacing them with assumed knowledge statements. This means that students can be selected without verifying that they have in fact completed this background study.

So what’s wrong with that?

In most cases, the assumed knowledge statements are unclear and often difficult to find, so students may not be aware of the assumed requirements. The removal of mathematics prerequisites also grossly underplays the level of mathematical facility required for these courses and trivialises the learning and skill development required to acquire it.

It places the burden on students to decide what should or should not be known in order to succeed in a course, and to assume the risk of those decisions, even though they are in no position to know what the risks are.

As a consequence, large numbers of students have been enrolling in mathematics-dependent courses without the assumed knowledge.

Over the last decade or more, numbers of students studying intermediate and advanced level mathematics in school has been in steady decline. Students have been free to make subject choices based on maximising their ATAR score rather than choosing the subjects that will best prepare them for their chosen career.

Since intermediate and advanced mathematics subjects are seen as hard and deemed not necessary for entry, students have been allowed – in some cases even encouraged – to opt out.

On the other side of the enrolment gate, consequences for students include being required to undertake bridging courses (some at extra cost) and having limited pathways through their degrees. Students do not generally know this at the end of Year 10 when they decide on which subjects they will choose for their Year 12.

Neither do they know that these choices may impact on their ability to succeed in their tertiary studies. Failure and attrition rates are generally high in first-year STEM subjects. And lack of the requisite background in mathematics plays a significant part in this.

Students who enter university without the assumed knowledge in mathematics also generally have lower success rates than students who have the assumed knowledge from school, even after they have completed bridging courses. In consumer terms, this buyer beware approach is not working.

So, where does that leave us?

Universities have a responsibility to determine what minimum background knowledge students require to be successful in a course. Once that determination is made, they should be required to ensure that the students they accept have that required knowledge.

Reintroducing appropriate mathematics prerequisites should increase participation in intermediate and advanced level mathematics at school. It has to.

We want students to take full advantage of the excellent education that is available to them through our secondary school system rather than trying to play catchup for years later.

Engaging students in the study of mathematics at school needs to be addressed on many levels. Certainly, making strong statements about prerequisites is one piece of the puzzle, but not the only one.

The Decadal Plan also calls for an urgent increase in the provision of professional development for teachers, especially those teaching mathematics out-of-field. It is essential that we support our teachers at all levels of education, so that we can give students the best possible education in mathematics that we must.

*The authors of this article are Deborah King, Associate Professor in Mathematics, University of Melbourne, and John Rice, Honorary Professor, University of Sydney. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

If you found this article interesting, you may also like a new book out called *Engineering Mathematics with Examples and Applications* that provides a compact and concise primer in the field, starting with the foundations, and then gradually developing to the advanced level of mathematics that is necessary for all engineering disciplines.

If you would like to read additional chapters from the book, visit ScienceDirect here. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** and save up to 30% off the list price and free global shipping.

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]]>The post Uncertainties in GPS Positioning: A Mathematical Discourse appeared first on SciTech Connect.

]]>In 2014, I read in *Mathematics Today* about a competition that was taking place called the Mathematical Competitive Game 2014-2015. It concerned GPS positioning and one had to try to estimate the uncertainty in the position of a GPS receiver from actual data. In due course, I submitted an entry called “Uncertainties in GPS Positioning.”

The intended readership for the entry was individuals interested in GPS, such as university students. The reader will benefit from being able to: understand how a GPS receiver calculates its position; understand why the calculated position is only an approximation to the true position; gain some appreciation of the factors which contribute to the difficulties in calculating an approximation of the true position; gain some appreciation of the mathematical steps that are employed in order to reduce errors in the approximation.

I was fortunate enough to achieve some success by winning the Joint First prize, Individual Category.

As a result of my research into the topic of GPS positioning, I submitted a book proposal to Elsevier called *Uncertainties in GPS Positioning: A Mathematical Discourse**, *it was accepted, and the book published in January, 2017.

Today, we all know what a GPS receiver is: it communicates with a satellite system and lets you know where you are on a map. A receiver receives signals from several orbiting satellites and processes them. The receiver has a built-in map. Uncertainties in GPS Positioning: A Mathematical Discourse describes the calculations performed by a GPS receiver and describes the problem associated with making sure that the estimated location is in close agreement with the actual location.

** Uncertainties in GPS Positioning: A Mathematical Discourse** provides a brief introduction to positioning and navigation systems, followed by the main topics that cover an introduction to GPS, basic GPS principles, signals from satellites to receiver for GPS, GPS modernization, signals from satellites to receiver for other satellite navigation systems, the solution of an idealized problem, and sources of inaccuracy. An example positioning problem with estimated inaccuracies is presented in detail, including a step-by-step mathematical solution. For each topic, background information is provided to aid the reader comprehend the subject matter. The future of satellite navigation systems is also discussed.

Dr. Alan Oxley is a tutor in the Faculty of Engineering, Design, and Information & Communications Technology (EDICT) at Bahrain Polytechnic, Kingdom of Bahrain. He and his former postgraduate students have published a number of researcher papers. In 2014-2015 the Mathematical Competitive Game took place with the topic ‘Uncertainties in GPS Positioning.’ Dr. Oxley’s entry received First Prize ex-aequo. His research interests are wide-ranging in both mathematics and computer science.

We are pleased to offer a free chapter of his book, “*Introduction to GPS.”*

If you would like to view more chapters, you can access the book on ScienceDirect. If you prefer a print or e-copy, visit the Elsevier Store. Apply discount code **STC317** for up to 30% off the list price and free global shipping.

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]]>The post Call for Authors – ‘Spatial Econometrics and Spatial Statistics’ Book Series appeared first on SciTech Connect.

]]>**Series Editor: Giuseppe Arbia**

**Editorial Advisory Board: Luc Anselin, Badi Baltagi, Raymond Florax, Dan Griffith, James LeSage, Daniel McMillen, Ingmar Prucha**

*Spatial Econometrics and Spatial Statistics* consolidates the dispersed methodological contributions of the growing field, unifying a body of empirical content relevant to graduate and PhD students conducting spatial research or studying spatial problems.

Primarily focusing on high-level original research publishing in monographs and edited handbooks, the series address themes from within classical spatial econometrics and spatial statistical approaches as well as novel research developed in response to changing technologies. Volumes cover classical methods and techniques for the analysis of regression models using data observed both within discrete portions of space and on individuals. They also describe models and theoretical instruments used to analyse various spatially-relevant economic effects such as externalities, spillovers, interactions, peer effects, copycatting, network effects, spatial concentration and many others.

Methods developed within the series transcend the paradigm of conventional regression models, and include cluster analysis and other multivariate methods, probability auto-models, simultaneous testing issues, error propagation, finite sample bias correction and bootstrap, dynamic spatial panels, endogeneous spatial weight matrices, cross-sectional dependence and factor models, social networks, point pattern analysis and Bayesian estimation methods. Computational approaches for econometric spatial analysis are covered extensively and at volume length, including both individual programming languages as well as critical comparison of them.

Adding the rigor and expertise of spatial econometricians and spatial statisticians to spatially-relevant problems from both econometrics and related disciplines, audiences will stretch across regional economics, transportation, criminology, public finance, industrial organization, political sciences, psychology, agricultural economics, health economics, demography, epidemiology, managerial economics, urban planning, education, land use, social sciences, economic development, innovation diffusion, environmental studies, history, labor, resources and energy economics, food security, real estate, and marketing.

**Key features of Series Volumes: **

- Focuses on original research and systematic reviews, in monograph-length contributions and edited handbooks.
- Concentrates on the most empirically relevant spatial econometric theory but limiting lengthy proofs and obscure theory.
- Emphasizes ‘how-to’ methods and empirical applications, often in specific disciplinary contexts and problem sets.
- Evaluates, compares and contrasts the use of spatially-relevant, mathematically-intensive computer programs and datasets.
- Enhances the research outputs of non-specialized but spatially-interested researchers by linking to the rigor and expertise of spatial econometricians and spatial statisticians.

For prospective authors interested in publishing their work in this series, please contact Senior Acquisitions Editor Graham Nisbet (G.Nisbet@elsevier.com).

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]]>The post How Math and Driverless Cars Could Spell the End of Traffic Jams appeared first on SciTech Connect.

]]>Being stuck in miles of halted traffic is not a relaxing way to start or finish a summer holiday. And as we crawl along the road, our views blocked by by slow-moving roofboxes and caravans, many of us will fantasise about a future free of traffic jams.

As a mathematician and motorist, I view traffic as a complex system, consisting of many interacting agents including cars, lorries, cyclists and pedestrians. Sometimes these agents interact in a free-flowing way and at other (infuriating) times they simply grind to a halt. All scenarios can be examined – and hopefully improved – using mathematical modelling, a way of describing the world in the language of maths.

Mathematical models tell us for instance that if drivers kept within the variable speed limits sometimes displayed on a motorway, traffic would flow consistently at, say, 50mph. Instead we tend to drive more aggressively, accelerating as soon as the opportunity arises – and being forced to brake moments later. The result is greater fuel consumption and a longer overall journey time. Cooperative driving seems to go against human nature when we get behind the wheel. But could this change if our roads were taken over by driverless cars?

Incorporating driverless cars into mathematical traffic models will prove key to improving traffic flow and assessing the various conditions in which traffic reaches a traffic jam threshold, or “jamming density”. The chances of reaching this point are affected by changes such as road layout, traffic volume and traffic light systems. And crucially, they are affected by whoever is in control of the vehicles.

In mathematical analysis, dense traffic can be treated as a flow and modelled using differential equations which describe the movement of fluids. Queuing models consider individual vehicles on a network of roads and the expected time they spend both in motion and waiting at junctions.

Another type of model consists of a grid in which cars’ positions are updated, according to certain rules, from one grid cell to the next. These rules can be based on their current velocity, acceleration and deceleration due to other vehicles and random events. This random deceleration is included to account for situations caused by something other than other vehicles – a pedestrian crossing the road for example, or a driver distracted by a passenger.

Adaptations to such models can take into account factors such as traffic light synchronisation or road closures, and they will need to be adapted further to take into account the movement of driverless cars.

In theory, autonomous cars will typically drive within the speed limits, have faster reaction times allowing them to drive closer together and will behave less randomly than humans, who tend to overreact in certain situations. On a tactical level, choosing the optimum route, accounting for obstacles and traffic density, driverless cars will behave in a more rational way, as they can communicate with other cars and quickly change route or driving behaviour.

So driverless cars may well make the mathematician’s job easier. Randomness is often introduced into models in order to incorporate unpredictable human behaviour. A system of driverless cars should be simpler to model than the equivalent human-driven traffic because there is less uncertainty. We could predict exactly how individual vehicles respond to events.

In a world with only driverless cars on the roads, computers would have full control of traffic. But for the time being, to avoid traffic jams we need to understand how autonomous and human-driven vehicles will interact together.

Of course, even with the best modelling, cooperative behaviour from driverless cars is not guaranteed. Different manufacturers might compete to come up with the best traffic-controlling software to ensure their cars get from A to B faster than their rivals. And, like the behaviour of individual human drivers, this could negatively affect everyone’s journey time.

But even supposing we managed to implement rules that optimised traffic flow for everyone, we could still get to the point where there are simply too many cars on the road, and jamming density is reached. Yet there is still potential for self-driving cars to help in this scenario.

Some car makers expect that eventually we will stop viewing cars as possessions and instead simply treat them as a transport service. Again, by applying mathematical techniques and modelling, we could optimise how this shared autonomous vehicle service could operate most efficiently, reducing the overall number of cars on the road. So while driverless cars alone might not rid us of traffic jams completely by themselves, an injection of mathematics into future policy could help navigate a smoother journey ahead.

*The author of this article is Lorna Wilson, Commercial Research Associate, University of Bath. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

*On-Road Intelligent Vehicles*, provides a comprehensive account of the technology of autonomous vehicles, with a special focus on the navigation and planning aspects, including information on the use of different sensors to perceive the environment, the problem of motion planning, and the macroscopic concepts related to Intelligent Transportation Systems. Essential reading for Postgraduate students, researchers and practitioners, working in the areas of Intelligent Vehicles, Intelligent Transportation Systems, Autonomous Vehicles, Robot Motion Planning, Special Topics in Robotics, Cooperative Systems, Planning and Navigation.

*Intelligent Vehicular Networks and Communications* examines cognitive radio, big data, and the cloud in vehicular communications and the current and future evolution of today’s transportation system. Chapters examine how intelligent transportation systems make more efficient transportation in urban environments and outline next generation vehicular networks technology. Vehicular and Wireless Network researchers, instructors, students, designers, and engineers will find the book particularly useful.

These books are also available in print on the Elsevier Store at 30% off the list price and with free global shipping. Apply discount code **STC215** at checkout.

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]]>The post What Makes a Mathematical Genius? appeared first on SciTech Connect.

]]>The film The Man Who Knew Infinity tells the gripping story of Srinivasa Ramanujan, an exceptionally talented, self-taught Indian mathematician. While in India, he was able to develop his own ideas on summing geometric and arithmetic series without any formal training. Eventually, his raw talent was recognised and he got a post at the University of Cambridge. There, he worked with Professor G.H. Hardy until his untimely death at the age of 32 in 1920.

Despite his short life, Ramanujan made substantial contributions to number theory, elliptic functions, infinite series and continued fractions. The story seems to suggest that mathematical ability is something at least partly innate. But what does the evidence say?

There are many different theories about what mathematical ability is. One is that it is closely tied to the capacity for understanding and building language. Just over a decade ago, a study examined members of an Amazonian tribe whose counting system comprised words only for “one”, “two” and “many”. The researchers found that the tribe were exceptionally poor at performing numerical thinking with quantities greater than three. They argued this suggests language is a prerequisite for mathematical ability.

But does that mean that a mathematical genius should be better at language than the average person? There is some evidence for this. In 2007, researchers scanned the brains of 25 adult students while they were solving multiplication problems. The study found that individuals with higher mathematical competence appeared to rely more strongly on language-mediated processes, associated with brain circuits in the parietal lobe.

However, recent findings have challenged this. One study looked at the brain scans of participants, including professional mathematicians, while they evaluated mathematical and non-mathematical statements. They found that instead of the left hemisphere regions of the brain typically involved during language processing and verbal semantics, high level mathematical reasoning was linked with activation of a bilateral network of brain circuits associated with processing numbers and space.

In fact, the brain activation in professional mathematicians in particular showed minimal use of language areas. The researchers argue their results support previous studies that have found that knowledge of numbers and space during early childhood can predict mathematical achievement.

For example, a recent study of 77 eight- to 10-year-old children demonstrates that visuo-spatial skills (the capacity to identify visual and spatial relationships among objects) have an important role in mathematical achievement. As part of the study, they took part in a “number line estimation task”, in which they had to position a series of numbers at appropriate places on a line where only the start and end numbers of a scale (such as 0 and 10) were given.

The study also looked at the children’s overall mathematical ability, visuospatial skills and visuomotor integration (for example, copying increasingly complex images using pencil and paper). It found that children’s scores on visuospatial skill and visuomotor integration strongly predicted how well they would do on number line estimation and mathematics.

An alternative definition of mathematical ability is that it represents the capacity to recognise and exploit hidden structures in data. This may account for an observed overlap between mathematical and musical ability. Similarly, it could also explain why training in chess can benefit children’s ability to solve mathematical problems. Albert Einstein famously claimed that images, feelings and musical structures formed the basis of his reasoning rather than logical symbols or mathematical equations.

However, the extent to which mathematical ability relies on innate or environmental factors remains controversial. A recent large scale twin and genome-wide analysis of 12-year-old children found that genetics could explain around half of the observed correlation between mathematical and reading ability. Although this is quite substantial, it still means that the learning environment has an important role to play.

So what does all this tell us about geniuses like Ramanujan? If mathematical ability does stem from a core non-linguistic capacity to reason with spatial and numerical representation, this can help explain how a prodigious talent could blossom in the absence of training. While language might still play a role, the nature of the numerical representations being manipulated could be crucial.

The fact that genetics seems to be involved also helps shed light on the case – Ramanujan could have simply inherited the ability. Nevertheless, we should not forget the important contribution of environment and education. While Ramanujan’s raw talent was sufficient to attract attention to his remarkable ability, it was the later provision of more formal mathematical training in India and England that allowed him to reach his full potential.

*The author of this article is David Pearson, Reader of Cognitive Psychology, Anglia Ruskin University. This article was originally published in The Conversation under a Creative Commons Attribution No Derivatives license. Read the original article here.*

A new book by the power team of Daniel Berch, David Geary, and Kathleen Mann Koepke, is a review of how contemporary brain imaging techniques and genetic methods can inform our understanding of mathematical cognitive development and ways to improve it. *Development of Mathematical Cognition* is available in print or as an e-book, and is also available online via ScienceDirect.

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]]>The post Call for authors – ‘Mathematics in Science and Engineering’ book series appeared first on SciTech Connect.

]]>**It is an exciting time to do applied mathematics.**

From the mathematical physics and engineering underpinning the Large Hadron Collider, quantum computing, the search for flight MH370, to the route-finding systems powering self-driving cars, modern science and technology is filled with innovations propelled by advances in modeling, nonlinear mathematics and high-performance computing.

Contemporary applied and applicable fields of mathematics stretches across fields as diverse as: renewable energy (biomass, solar and wind power, smart grids), earth and atmospheric sciences (climate studies, global warming, oceanography, turbulence), fossil fuel (geology, oil and gas exploration, fracking, reaction-diffusion combustion dynamics), aerospace technology (drones, new generation hypersonic airplane and spacecraft designs), conflict resolution (control and game theory for quantitative social sciences), homeland security (terror cell detection, tomography), biomedicine (pharmaceutical drug and protein designs, war on cancer, mathematical biology, computational chemistry), material science (graphene, nano-colloids), manufacturing (robotics, 3D lithography), big data (analysis, probability and statistics), and economics and finance (mathematical finance models, stochastic equations in investment and banking, chaos).

In response to this tremendous new interest in applied and applicable mathematics, Elsevier is pleased to re-launch the famous Academic Press ‘red series’, *Mathematics in Science and Engineering [MSE]*, which originally developed under the leadership of Richard Bellman (1960-1984) and William F. Ames (1991-2006).

Under new Series Editor Goong Chen, this collection of books will aim to provide researchers, graduate students, and higher-division undergraduates in mathematics, science, engineering and technology domains with quality books straightforwardly focusing on the inter- and multidisciplinarity, nonlinearity, complexity, and large-scale scientific computing which now characterize much of the application environment.

The Series Editor and Editorial Board are here pleased to announce a call for authors for the prestigious and highly visible series. The Editors are particularly interested in considering works which address:

**Inter- and multi-disciplinarity:**problem solving for modern complex problems;**Nonlinearity:**providing methodology to analyze and control nonlinearities, bifurcations and chaos**Modeling:**quantifying and approximating the physical and engineering systems under study**Computation:**providing concrete data for validation, visualization, optimization and design, and analysis, estimation and prediction on big data**Contemporaneity:**giving timely treatments on “hot” or trendy events or topics- Applied mathematics disciplinary topics are also welcomed

For prospective authors interested in publishing their work in MSE, please download a __New Book Proposal Form__ and submit, together with copies of your CVs, and a sample chapter of 15-25 pages (if available), to Graham Nisbet (G.Nisbet@elsevier.com).

SERIES EDITOR: Goong Chen

EDITORIAL BOARD: Helene Frankowska, Jordi Boronat Medico, Vicentiu Radulescu, Ulrich Stadtmuller, Stephen Wiggins, Pengfei Yao

SENIOR ACQUISITIONS EDITOR: Graham Nisbet (G.Nisbet@elsevier.com)

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]]>The post How Many Ways Can You Arrange 128 Tennis Balls? Researchers Solve An Apparently Impossible Problem appeared first on SciTech Connect.

]]>A bewildering physics problem has apparently been solved by researchers, in a study which provides a mathematical basis for understanding issues ranging from predicting the formation of deserts, to making artificial intelligence more efficient.

In research carried out at the University of Cambridge, a team developed a computer program that can answer this mind-bending puzzle: Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

The answer, it turns out, is something like 10^{250} (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

Being able to calculate configurational entropy would, in theory, eventually enable us to answer a host of seemingly impossible problems – such as predicting the movement of avalanches, or anticipating how the shifting sand dunes in a desert will reshape themselves over time.

These questions belong to a field called granular physics, which deals with the behaviour of materials such as snow, soil or sand. Different versions of the same problem, however, exist in numerous other fields, such as string theory, cosmology, machine learning, and various branches of mathematics. The research shows how questions across all of those disciplines might one day be addressed.

Stefano Martiniani, a Gates Scholar at St John’s College, University of Cambridge, who carried out the study with colleagues in the Department of Chemistry, explained: “The problem is completely general. Granular materials themselves are the second most processed kind of material in the world after water and even the shape of the surface of the Earth is defined by how they behave.”

“Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that.”

At the heart of these problems is the idea of entropy – a term which describes how disordered the particles in a system are. In physics, a “system” refers to any collection of particles that we want to study, so for example it could mean all the water in a lake, or all the water molecules in a single ice cube.

When a system changes, for example because of a shift in temperature, the arrangement of these particles also changes. For example, if an ice cube is heated until it becomes a pool of water, its molecules become more disordered. Therefore, the ice cube, which has a tighter structure, is said to have lower entropy than the more disordered pool of water.

At a molecular level, where everything is constantly vibrating, it is often possible to observe and measure this quite clearly. In fact, many molecular processes involve a spontaneous increase in entropy until they reach a steady equilibrium.

The brute force way of doing this would be to keep changing the system and recording the configurations. Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store them, because there isn’t enough matter in the universe. – Stefano Martiniani

In granular physics, however, which tends to involve materials large enough to be seen with the naked eye, change does not happen in the same way. A sand dune in the desert will not spontaneously change the arrangement of its particles (the grains of sand). It needs an external factor, like the wind, for this to happen.

This means that while we can predict what will happen in many molecular processes, we cannot easily make equivalent predictions about how systems will behave in granular physics. Doing so would require us to be able to measure changes in the structural disorder of all of the particles in a system – its configurational entropy.

To do that, however, scientists need to know how many different ways a system can be structured in the first place. The calculations involved in this are so complicated that they have been dismissed as hopeless for any system involving more than about 20 particles. Yet the Cambridge study defied this by carrying out exactly this type of calculation for a system, modelled on a computer, in which the particles were 128 soft spheres, like tennis balls.

“The brute force way of doing this would be to keep changing the system and recording the configurations,” Martiniani said. “Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store the configurations, because there isn’t enough matter in the universe with which to do it.”

Instead, the researchers created a solution which involved taking a small sample of all possible configurations and working out the probability of them occurring, or the number of arrangements that would lead to those particular configurations appearing.

Based on these samples, it was possible to extrapolate not only in how many ways the entire system could therefore be arranged, but also how ordered one state was compared with the next – in other words, its overall configurational entropy.

Martiniani added that the team’s problem-solving technique could be used to address all sorts of problems in physics and maths. He himself is, for example, currently carrying out research into machine learning, where one of the problems is knowing how many different ways a system can be wired to process information efficiently.

“Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one,” he said. “By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find.”

The paper, Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings, is published in the journal, Physical Review E.

The text in this work is licensed under a Creative Commons Attribution 4.0 International License and first appeared on The University of Cambridge website. Click here for the original article.

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]]>The post How Much Math Do You Need to Win Your March Madness Pool? appeared first on SciTech Connect.

]]>Deciding which teams to pick in your NCAA basketball pool? Then you’re faced with a classic decision problem – and here, science can help.

On one hand, you want to pick good teams, the “favorites,” because those teams seem more likely to win. On the other hand, you want to pick some weaker teams, the “underdogs,” so your bracket will stand out from the rest and win the pool. These two opposing forces make for an interesting math problem, because somewhere in the middle is an optimal solution.

In my heart, I always know which teams will win, or at least which teams I want to root for. As an academic, though, I’d rather squeeze all the fun out of it by overanalyzing the situation. Let’s do that here!

To find the best way to build our own brackets, we need to first build a mathematical model for simulating the tournament.

Suppose we model the tournament by replacing basketball games with coin flips, except with coins that don’t land evenly heads or tails but rather are weighted to reflect each game’s actual odds. For example, when Baylor plays Yale on Thursday, instead of playing the game, we just flip a coin that gives the higher-seeded Baylor a greater chance of winning. We’d need to flip one of these coins for every first-round game, every potential second-round game, and for each possible matchup in the tournament. Each coin must be weighted in a way that models the actual game, so its probabilities must be determined by the specific matchup.

Where should we get these probabilities? The NCAA provides you with a handy little number next to each team, the team’s seed. For the first few rounds, each game has a favorite, and that choice was made by people with a tremendous amount of basketball knowledge. You could look back over history and observe that when a #5 seed plays a #12 seed, the #5 seed wins 65 percent of the time.

But there are plenty of other methods: Las Vegas betting odds give a point spread for each game, and based on those teams’ scoring averages, you can convert the point spread into a probability of winning. Computer rating systems abound, and you can convert these ratings into probabilities by considering the ratings difference between two teams – a method known as the Bradley-Terry model. Some more sophisticated systems can even produce a probability custom fit to the two teams in the game.

So, pick your favorite method. Even then, things aren’t as simple as they seem. The most likely outcome of the tournament is not necessarily that all favorites win. Look at this example:

Imagine a four-team tournament with teams A, B, C and D as shown. Assume that A always beats B, and C beats D with probability 0.6. Finally, A always beats D, but has only 0.5 probability of beating C. The only possible outcomes are: A wins over C (probability 0.3), C wins over A (probability 0.3) and A wins over D (probability 0.4). The most likely outcome contains the upset D beats C.

Further complicating the situation, the rules of your office or friends’ pool probably mean that picking correctly in later-round games earns more points than early-round picks. How do you pick a bracket that gets you those crucial late-round points?

In one of the first analytic papers on this subject, Kaplan and Garstka gave an algorithm for deciding which picks are expected to score the highest. Their method builds a list of 64 brackets backwards, round-by-round, starting each one with a different team as the winner. For example, Duke’s bracket starts with just Duke, and adds one round at a time, doubling in size but always keeping Duke as the winner. In the end, the algorithm selects the best from each of the 64 team-specific brackets.

This doesn’t sound like something a human would do, and in fact it is best implemented by a computer. The brackets produced tend to be “chalk” – in which higher-ranked teams are most likely to win – but do not always select the higher seed. And Kaplan and Garstka did observe that their algorithm did better than just automatically picking the high seeds.

To this point our model is ignoring an important fact: the goal of picking your bracket is not to achieve a high score, but to win a pool against other people. And people behave irrationally.

In a psychological experiment, McCrea and Hirt found evidence that pool participants pursue “probability matching”: if a collection of games (say, the 5-12 matchups) has historically produced an upset one-third of the time, people will attempt to predict upsets in about one-third of those games in their brackets. In fact, people do no better than random chance at making such predictions, and so hurt their overall chances in the pool.

On the other hand, when choosing the tournament winner, people flock to the favorites. Every year, ESPN Tournament Challenge publishes data on its 11 million entries. Last year, 48 percent of their players had selected prohibitive favorite Kentucky as champion. Picking the correct champion is important, but if everyone else has the same opinion then you need to pick a bunch of other games well, too.

This brings us back to what makes this problem interesting: you need to pick teams that win, but not the same teams as everyone else – so you come out on top in your pool.

To improve your odds in your pool, you need to model the other players you’re up against. Each year, large, free, Internet pools publish data on player behavior, and they publish it before your brackets are due on Thursday morning.

Let’s assume people make their picks the same way we modeled the games, by flipping biased coins for each game in the bracket. The national Internet pools give exactly the data you need to properly bias the coins. Nobody I know actually picks their bracket this way, but it turns out that real (human-picked) brackets and randomized brackets have nearly the same score distribution.

In my own research, we used this model to calculate optimal picks. The brackets produced tend to be very conservative in the first two rounds, include one or two surprises in the Final Four, and a strong but not heavily favored champion. They never, ever, pick an upset in a 5-12 game. According to the computers, these picks increase the chances of winning a big Internet pool by a factor of 100 to 1,000.

This sounds great. It is great! But there’s a catch: the NCAA basketball tournament happens only once a year. And your probability of winning is very low indeed – even with a boost from math and computer analytics. It will likely take thousands of years before the strategy pays off.

And that’s the beautiful thing about scientific studies of the NCAA tournament. Serious modeling and data analysis quail before the absurdity of predicting such a notoriously unpredictable event. After a decade of study, the only things we really know are that the tournament is madness and that your friend whose picks are based on mascots will probably win your pool.

Don’t want to just pick your favorite team and depend on luck? Here’s a chapter from the book *The Joy of Finite Mathematics* called *“Game Theory” *to help increase your odds:

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