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 Call for Authors: Studies in the History of Mathematical Enquiry appeared first on SciTech Connect.

]]>This series publishes across the full range of mathematical enquiry, considered in its thematic, geographical, and historiographical vertices. It consists of germane historical, historiographical, sociological, epistemological, geographical, and biographical treatments pertaining to the development and expansion of mathematics and prominent mathematicians.

The series provides informed studies on mathematicians and their work in historical context, on historiographical topics in the history of mathematics, and on the interrelationships between mathematical ideas, science, and the broader culture. This includes the social status of mathematicians. During the historical development of mathematics there have been large shifts in the way in which mathematical activity has been situated institutionally, what publics it has interacted with and served, how its result have been disseminated, and how the internal structure of the field has been understood. Thus, the series includes research in topics that can be broadly defined as “sociological” or devoted to “historiographical” issues, e.g. books that answer questions concerning the social and economic context, and also the “image” of mathematics received by the large public of non-specialists.

The intended audiences serviced by the series includes mathematicians, scientists in fields related to mathematics, scholars in the history of mathematics and exact sciences, historians of science, undergraduate and graduate students, anyone who seeks to understand the role of mathematics in the human culture.

The series addresses the following key subjects:

- Historical treatments of thematic branches of mathematical inquiry.
- Treatments of the historical development of mathematics in geographical regions.
- The foundations of mathematical inquiry in its epistemology, philosophical, linguistic and biological considerations.
- Interdisciplinary treatments dealing with the intersection between mathematical inquiry and the development of other sciences.
- Sociological aspects of the development of mathematical inquiry.
- Biographical treatments of noted mathematicians.

For prospective authors interested in publishing their work in this series, 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: Umberto Bottazzini

EDITORIAL BOARD: Christian Gilain, Niccolò Guicciardini, Alexander Jones, Han Qi, David E. Rowe, Tilman Sauer

**View the Elsevier Heritage Collection – and a Math Article from 1646!**

Elsevier’s Heritage Collection consists of about 2,000 rare books, of which 1,000 are distinct titles published by the original Elzevier family publishing house between 1580 and 1712. Housed in museum-quality display cases in Elsevier’s headquarters in Amsterdam, the collection is open to researchers, by appointment.

You can also view the books online in the Elsevier Heritage Collection Catalogue which includes a free download of a marked up mathematics text, from 1646, from the journal of Opera Mathematica.

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]]>The post Call for Authors – Pure and Applied Mathematics Series appeared first on SciTech Connect.

]]>*Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas with areas of potential application. All of these activities are crucial to economic growth, national competitiveness, and national security, and this fact should inform both the nature and scale of funding for the mathematical sciences as a whole.*

*The Mathematical* *Sciences in 2025*, National Academies Press (2013)

The mathematical sciences today span an enormous and unprecedented intellectual bandwidth, with research advances encompassing the spectrum from pure theory to myriad applications. With definitions of what constitutes a ‘mathematician’ changing as technological advance dictates, mathematics is today intrinsic to the work of such diverse professionals as computer scientists, systems biologists, materials engineers and economists. Rare is the discipline untouched by mathematics or unencumbered by a debt to it.

Revitalizing the* Pure and Applied Mathematics *series is Elsevier’s response to this unprecedented explosion of literature. Inheriting its title, lineage, and edition base from the highly prestigious and extraordinarily diverse Academic Press series of 142 volumes (discontinued in 2004), series titles are intended to provide a new centerpiece of the books presence in math at Elsevier, as well as an umbrella for non-series projects from which the whole portfolio can leap forward.

Written by leading researchers under the editorship of Dominique Perrin, the series provides superior quality, peerless depth and exhaustive breadth of coverage, advocating solutions to challenging contemporary problems using the symbolic language and concepts of differential equations, calculus, linear algebra, differential geometry, graph theory combinatorics, number theory, the calculus of variations, probability theory, and other branches of mathematics.

The series is intended to provide systematic reviews of many areas of pure math which have application in the real world, particularly in the areas of modelling, computational science, mathematical biology, chemistry, engineering, geosciences, neuroscience, physics and other disciplines at the research and educational levels.

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: DOMINIQUE PERRIN

EDITORIAL BOARD: ALEXANDER BULINSKI, SØREN EILERS, STEPHANE JAFFARD, ANDREW M. ODLYZKO

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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.

To read more from the book, you can visit ScienceDirect, or you can order a print or e-book via the Elsevier Store. Apply discount code STC215 for up to 30% off the list price and free shipping worldwide! Prefer to read it online, access it via ScienceDirect today.

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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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