MATH NEWS

THE alarm rings. You glance at the clock. The time is 6.30 am. You haven't even got out of bed, and already at least six mathematical equations have influenced your life. The memory chip that stores the time in your clock couldn't have been devised without a key equation in quantum mechanics. Its time was set by a radio signal that we would never have dreamed of inventing were it not for James Clerk Maxwell's four equations of electromagnetism. And the signal itself travels according to what is known as the wave equation.

We are afloat on a hidden ocean of equations. They are at work in transport, the financial system, health and crime prevention and detection, communications, food, water, heating and lighting. Step into the shower and you benefit from equations used to regulate the water supply. Your breakfast cereal comes from crops that were bred with the help of statistical equations. Drive to work and your car's aerodynamic design is in part down to the Navier-Stokes equations that describe how air flows over and around it. Switching on its satnav involves quantum physics again, plus Newton's laws of motion and gravity, which helped launch the geopositioning satellites and set their orbits. It also uses random number generator equations for timing signals, trigonometric equations to compute location, and special and general relativity for precise tracking of the satellites' motion under the Earth's gravity.

Without equations, most of our technology would never have been invented. Of course, important inventions such as fire and the wheel came about without any mathematical knowledge. Yet without equations we would be stuck in a medieval world.

Equations reach far beyond technology too. Without them, we would have no understanding of the physics that governs the tides, waves breaking on the beach, the ever-changing weather, the movements of the planets, the nuclear furnaces of the stars, the spirals of galaxies - the vastness of the universe and our place within it.

There are thousands of important equations. The seven I focus on here - the wave equation, Maxwell's four equations, the Fourier transform and Schrödinger's equation - illustrate how empirical observations have led to equations that we use both in science and in everyday life.

First, the wave equation. We live in a world of waves. Our ears detect waves of compression in the air as sound, and our eyes detect light waves. When an earthquake hits a town, the destruction is caused by seismic waves moving through the Earth.

Mathematicians and scientists could hardly fail to think about waves, but their starting point came from the arts: how does a violin string create sound? The question goes back to the ancient Greek cult of the Pythagoreans, who found that if two strings of the same type and tension have lengths in a simple ratio, such as 2:1 or 3:2, they produce notes that, together, sound unusually harmonious. More complex ratios are discordant and unpleasant to the ear. It was Swiss mathematician Johann Bernoulli who began to make sense of these observations. In 1727 he modelled a violin string as a large number of closely spaced point masses, linked together by springs. He used Newton's laws to write down the system's equations of motion, and solved them. From the solutions, he concluded that the simplest shape for a vibrating string is a sine curve. There are other modes of vibration as well - sine curves in which more than one wave fits into the length of the string, known to musicians as harmonics.

From waves to wireless

Almost 20 years later, Jean Le Rond d'Alembert followed a similar procedure, but he focused on simplifying the equations of motion rather than their solutions. What emerged was an elegant equation describing how the shape of the string changes over time. This is the wave equation, and it states that the acceleration of any small segment of the string is proportional to the tension acting on it. It implies that waves whose frequencies are not in simple ratios produce an unpleasant buzzing noise known as "beats". This is one reason why simple numerical ratios give notes that sound harmonious.

The wave equation can be modified to deal with more complex, messy phenomena, such as earthquakes. Sophisticated versions of the wave equation let seismologists detect what is happening hundreds of miles beneath our feet. They can map the Earth's tectonic plates as one slides beneath another, causing earthquakes and volcanoes. The biggest prize in this area would be a reliable way to predict earthquakes and volcanic eruptions, and many of the methods being explored are underpinned by the wave equation.

But the most influential insight from the wave equation emerged from the study of Maxwell's equations of electromagnetism. In 1820, most people lit their houses using candles and lanterns. If you wanted to send a message, you wrote a letter and put it on a horse-drawn carriage; for urgent messages, you omitted the carriage. Within 100 years, homes and streets had electric lighting, telegraphy meant messages could be transmitted across continents, and people even began to talk to each other by telephone. Radio communication had been demonstrated in laboratories, and one entrepreneur had set up a factory selling "wirelesses" to the public.

This social and technological revolution was triggered by the discoveries of two scientists. In about 1830, Michael Faraday established the basic physics of electromagnetism. Thirty years later, James Clerk Maxwell embarked on a quest to formulate a mathematical basis for Faraday's experiments and theories.

At the time, most physicists working on electricity and magnetism were looking for analogies with gravity, which they viewed as a force acting between bodies at a distance. Faraday had a different idea: to explain the series of experiments he conducted on electricity and magnetism, he postulated that both phenomena are fields which pervade space, change over time and can be detected by the forces they produce. Faraday posed his theories in terms of geometric structures, such as lines of magnetic force.

Maxwell reformulated these ideas by analogy with the mathematics of fluid flow. He reasoned that lines of force were analogous to the paths followed by the molecules of a fluid and that the strength of the electric or magnetic field was analogous to the velocity of the fluid. By 1864 Maxwell had written down four equations for the basic interactions between the electrical and magnetic fields. Two tell us that electricity and magnetism cannot leak away. The other two tell us that when a region of electric field spins in a small circle, it creates a magnetic field, and a spinning region of magnetic field creates an electric field.

But it was what Maxwell did next that is so astonishing. By performing a few simple manipulations on his equations, he succeeded in deriving the wave equation and deduced that light must be an electromagnetic wave. This alone was stupendous news, as no one had imagined such a fundamental link between light, electricity and magnetism. And there was more. Light comes in different colours, corresponding to different wavelengths. The wavelengths we see are restricted by the chemistry of the eye's light-detecting pigments. Maxwell's equations led to a dramatic prediction - that electromagnetic waves of all wavelengths should exist. Some, with much longer wavelengths than we can see, would transform the world: radio waves.

In 1887, Heinrich Hertz demonstrated radio waves experimentally, but he failed to appreciate their most revolutionary application. If you could impress a signal on such a wave, you could talk to the world. Nikola Tesla, Guglielmo Marconi and others turned the dream into reality, and the whole panoply of modern communications, from radio and television to radar and microwave links for cellphones, followed naturally. And it all stemmed from four equations and a couple of short calculations. Maxwell's equations didn't just change the world. They opened up a new one.

Just as important as what Maxwell's equations do describe is what they don't. Although the equations revealed that light was a wave, physicists soon found that its behaviour was sometimes at odds with this view. Shine light on a metal and it creates electricity, a phenomenon called the photoelectric effect. It made sense only if light behaved like a particle. So was light a wave or a particle? Actually, a bit of both. Matter was made from quantum waves, and a tightly knit bunch of waves acted like a particle.

Dead or alive

In 1927 Erwin Schrödinger wrote down an equation for quantum waves. It fitted experiments beautifully while painting a picture of a very strange world, in which fundamental particles like the electron are not well-defined objects, but probability clouds. An electron's spin is like a coin that can be half heads and half tails until it hits a table. Soon theorists were worrying about all manner of quantum weirdness, such as cats that are simultaneously dead and alive, and parallel universes in which Adolf Hitler won the second world war.

Quantum mechanics isn't confined to such philosophical enigmas. Almost all modern gadgets - computers, cellphones, games consoles, cars, refrigerators, ovens - contain memory chips based on the transistor, whose operation relies on the quantum mechanics of semiconductors. New uses for quantum mechanics arrive almost weekly. Quantum dots - tiny lumps of a semiconductor - can emit light of any colour and are used for biological imaging, where they replace traditional, often toxic, dyes. Engineers and physicists are trying to invent a quantum computer, one which can perform many different calculations in parallel, just like the cat that is both alive and dead.

Lasers are another application of quantum mechanics. We use them to read information from tiny pits or marks on CDs, DVDs and Blu-ray discs. Astronomers use lasers to measure the distance from the Earth to the moon. It might even be possible to launch space vehicles from Earth on the back of a powerful laser beam.

The final chapter in this story comes from an equation that helps us make sense of waves. It starts in 1807, when Joseph Fourier devised an equation for heat flow. He submitted a paper on it to the French Academy of Sciences, but it was rejected. In 1812, the academy made heat the topic of its annual prize. Fourier submitted a longer, revised paper - and won.

The most intriguing aspect of Fourier's prize-winning paper was not the equation, but how he solved it. A typical problem was to find how the temperature along a thin rod changes as time passes, given the initial temperature profile. Fourier could solve this equation with ease if the temperature varied like a sine wave along its length. So he represented a more complicated profile as a combination of sine curves with different wavelengths, solved the equation for each component sine curve, and added these solutions together. Fourier claimed that this method worked for any profile whatsoever, even a one where the temperature suddenly jumps in value. All you had to do was add up an infinite number of contributions from sine curves with more and more wiggles.

Even so, Fourier's new paper was criticised for not being rigorous enough, and once more the French academy refused to publish it. In 1822 Fourier ignored the objections and published his theory as a book. Two years later, he got himself appointed secretary of the academy, thumbed his nose at his critics, and published his original paper in the academy's journal. However, the critics did have a point. Mathematicians were starting to realise that infinite series were dangerous beasts; they didn't always behave like nice, finite sums. Resolving these issues turned out to be distinctly difficult, but the final verdict was that Fourier's idea could be made rigorous by excluding highly irregular profiles. The result is the Fourier transform, an equation that treats a time-varying signal as the sum of a series of component sine curves and calculates their amplitudes and frequencies.

Today the Fourier transform affects our lives in myriad ways. For example, we can use it to analyse the vibrational signal produced by an earthquake and to calculate the frequencies at which the energy imparted by the shaking ground is greatest. A sensible step towards earthquake-proofing a building is to make sure that the building's preferred frequencies are different from the earthquake's.

Other applications include removing noise from old sound recordings, finding the structure of DNA using X-ray images, improving radio reception and preventing unwanted vibrations in cars. Plus there is one that most of us unwittingly take advantage of every time we take a digital photograph.

If you work out how much information is required to represent the colour and brightness of each pixel in a digital image, you will discover that a digital camera seems to cram into its memory card about 10 times as much data as the card can possibly hold. Cameras do this using JPEG data compression, which combines five different compression steps. One of them is a digital version of the Fourier transform, which works with a signal that changes not over time but across the image. The mathematics is virtually identical. The other four steps reduce the data even further, to about one-tenth of the original amount.

These are just seven of the many equations that we encounter every day, not realising they are there. But the impact of equations on history goes much further. A truly revolutionary equation can have a greater impact on human existence than all the kings and queens whose machinations fill our history books.

There is (or may be) one equation, above all, that physicists and cosmologists would dearly love to lay their hands on: a theory of everything that unifies quantum mechanics and relativity. The best known of the many candidates is the theory of superstrings. But for all we know, our equations for the physical world may just be oversimplified models that fail to capture the deep structure of reality. Even if nature obeys universal laws, they might not be expressible as equations.

Some scientists think that it is time we abandoned traditional equations altogether in favour of algorithms - more general recipes for calculating things that involve decision-making. But until that day dawns, if ever, our greatest insights into nature's laws will continue to take the form of equations, and we should learn to understand them and appreciate them. Equations have a track record. They really have changed the world and they will change it again.

The origin of equations

The ancient Babylonians and Greeks knew about equations, though they wrote them using words and pictures. For the past 500 years, mathematicians and scientists have used symbols, the crucial one being the equals sign. Unusually, we know who invented it, and why. It was Robert Recorde, who in 1557 wrote in his treatise The Whetstone of Witte: "To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: bicause noe .2. thynges, can be moare equalle."

Theorems and theories

Some equations present logical relations between mathematical quantities, and the task of mathematicians is to prove they are valid. Others provide information about an unknown quantity; here the task is to solve the equation and make the unknown known. Equations in pure mathematics are generally of the first kind: they reveal patterns and regularities in mathematics itself. Pythagoras's theorem, an equation expressed in the language of geometry, is an example. Given Euclid's basic geometric assumptions, Pythagoras's theorem is true.

Equations in applied mathematics and mathematical physics are usually of the second kind. They express properties of the universe that could, in principle, have been otherwise. For example, Newton's law of gravity tells us how to calculate the attractive force between two bodies. Solving the resulting equations tells us how planets orbit the sun or how to plot a trajectory for a space probe. But Newton's law isn't a mathematical theorem; the law of gravity might have been different. Indeed, it is different: Einstein's general relativity improves on Newton. And even that theory may not be the last word.

Seven equations that rule your world

It was the holy grail of investors. The Black-Scholes equation, brainchild of economists Fischer Black and Myron Scholes, provided a rational way to price a financial contract when it still had time to run. It was like buying or selling a bet on a horse, halfway through the race. It opened up a new world of ever more complex investments, blossoming into a gigantic global industry. But when the sub-prime mortgage market turned sour, the darling of the financial markets became the Black Hole equation, sucking money out of the universe in an unending stream.

Anyone who has followed the crisis will understand that the real economy of businesses and commodities is being upstaged by complicated financial instruments known as derivatives. These are not money or goods. They are investments in investments, bets about bets. Derivatives created a booming global economy, but they also led to turbulent markets, the credit crunch, the near collapse of the banking system and the economic slump. And it was the Black-Scholes equation that opened up the world of derivatives.

The equation itself wasn't the real problem. It was useful, it was precise, and its limitations were clearly stated. It provided an industry-standard method to assess the likely value of a financial derivative. So derivatives could be traded before they matured. The formula was fine if you used it sensibly and abandoned it when market conditions weren't appropriate. The trouble was its potential for abuse. It allowed derivatives to become commodities that could be traded in their own right. The financial sector called it the Midas Formula and saw it as a recipe for making everything turn to gold. But the markets forgot how the story of King Midas ended.

Black-Scholes underpinned massive economic growth. By 2007, the international financial system was trading derivatives valued at one quadrillion dollars per year. This is 10 times the total worth, adjusted for inflation, of all products made by the world's manufacturing industries over the last century. The downside was the invention of ever-more complex financial instruments whose value and risk were increasingly opaque. So companies hired mathematically talented analysts to develop similar formulas, telling them how much those new instruments were worth and how risky they were. Then, disastrously, they forgot to ask how reliable the answers would be if market conditions changed.

Black and Scholes invented their equation in 1973; Robert Merton supplied extra justification soon after. It applies to the simplest and oldest derivatives: options. There are two main kinds. A put option gives its buyer the right to sell a commodity at a specified time for an agreed price. A call option is similar, but it confers the right to buy instead of sell. The equation provides a systematic way to calculate the value of an option before it matures. Then the option can be sold at any time. The equation was so effective that it won Merton and Scholes the 1997 Nobel prize in economics. (Black had died by then, so he was ineligible.)

If everyone knows the correct value of a derivative and they all agree, how can anyone make money? The formula requires the user to estimate several numerical quantities. But the main way to make money on derivatives is to win your bet – to buy a derivative that can later be sold at a higher price, or matures with a higher value than predicted. The winners get their profit from the losers. In any given year, between 75% and 90% of all options traders lose money. The world's banks lost hundreds of billions when the sub-prime mortgage bubble burst. In the ensuing panic, taxpayers were forced to pick up the bill, but that was politics, not mathematical economics. The Black-Scholes equation relates the recommended price of the option to four other quantities. Three can be measured directly: time, the price of the asset upon which the option is secured and the risk-free interest rate. This is the theoretical interest that could be earned by an investment with zero risk, such as government bonds. The fourth quantity is the volatility of the asset. This is a measure of how erratically its market value changes. The equation assumes that the asset's volatility remains the same for the lifetime of the option, which need not be correct. Volatility can be estimated by statistical analysis of price movements but it can't be measured in a precise, foolproof way, and estimates may not match reality.

The idea behind many financial models goes back to Louis Bachelier in 1900, who suggested that fluctuations of the stock market can be modelled by a random process known as Brownian motion. At each instant, the price of a stock either increases or decreases, and the model assumes fixed probabilities for these events. They may be equally likely, or one may be more probable than the other. It's like someone standing on a street and repeatedly tossing a coin to decide whether to move a small step forwards or backwards, so they zigzag back and forth erratically. Their position corresponds to the price of the stock, moving up or down at random. The most important statistical features of Brownian motion are its mean and its standard deviation. The mean is the short-term average price, which typically drifts in a specific direction, up or down depending on where the market thinks the stock is going. The standard deviation can be thought of as the average amount by which the price differs from the mean, calculated using a standard statistical formula. For stock prices this is called volatility, and it measures how erratically the price fluctuates. On a graph of price against time, volatility corresponds to how jagged the zigzag movements look.

Black-Scholes implements Bachelier's vision. It does not give the value of the option (the price at which it should be sold or bought) directly. It is what mathematicians call a partial differential equation, expressing the rate of change of the price in terms of the rates at which various other quantities are changing. Fortunately, the equation can be solved to provide a specific formula for the value of a put option, with a similar formula for call options.

The early success of Black-Scholes encouraged the financial sector to develop a host of related equations aimed at different financial instruments. Conventional banks could use these equations to justify loans and trades and assess the likely profits, always keeping an eye open for potential trouble. But less conventional businesses weren't so cautious. Soon, the banks followed them into increasingly speculative ventures.

Any mathematical model of reality relies on simplifications and assumptions. The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets. The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different.

When these assumptions are valid, risk is usually low, because large stock market fluctuations should be extremely rare. But on 19 October 1987, Black Monday, the world's stock markets lost more than 20% of their value within a few hours. An event this extreme is virtually impossible under the model's assumptions. In his bestseller The Black Swan, Nassim Nicholas Taleb, an expert in mathematical finance, calls extreme events of this kind black swans. In ancient times, all known swans were white and "black swan" was widely used in the same way we now refer to a flying pig. But in 1697, the Dutch explorer Willem de Vlamingh found masses of black swans on what became known as the Swan River in Australia. So the phrase now refers to an assumption that appears to be grounded in fact, but might at any moment turn out to be wildly mistaken.

Large fluctuations in the stock market are far more common than Brownian motion predicts. The reason is unrealistic assumptions – ignoring potential black swans. But usually the model performed very well, so as time passed and confidence grew, many bankers and traders forgot the model had limitations. They used the equation as a kind of talisman, a bit of mathematical magic to protect them against criticism if anything went wrong.

Banks, hedge funds, and other speculators were soon trading complicated derivatives such as credit default swaps – likened to insuring your neighbour's house against fire – in eye-watering quantities. They were priced and considered to be assets in their own right. That meant they could be used as security for other purchases. As everything got more complicated, the models used to assess value and risk deviated ever further from reality. Somewhere underneath it all was real property, and the markets assumed that property values would keep rising for ever, making these investments risk-free.

The Black-Scholes equation has its roots in mathematical physics, where quantities are infinitely divisible, time flows continuously and variables change smoothly. Such models may not be appropriate to the world of finance. Traditional mathematical economics doesn't always match reality, either, and when it fails, it fails badly. Physicists, mathematicians and economists are therefore looking for better models.

At the forefront of these efforts is complexity science, a new branch of mathematics that models the market as a collection of individuals interacting according to specified rules. These models reveal the damaging effects of the herd instinct: market traders copy other market traders. Virtually every financial crisis in the last century has been pushed over the edge by the herd instinct. It makes everything go belly-up at the same time. If engineers took that attitude, and one bridge in the world fell down, so would all the others.

By studying ecological systems, it can be shown that instability is common in economic models, mainly because of the poor design of the financial system. The facility to transfer billions at the click of a mouse may allow ever-quicker profits, but it also makes shocks propagate faster.

Was an equation to blame for the financial crash, then? Yes and no. Black-Scholes may have contributed to the crash, but only because it was abused. In any case, the equation was just one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation.

Despite its supposed expertise, the financial sector performs no better than random guesswork. The stock market has spent 20 years going nowhere. The system is too complex to be run on error-strewn hunches and gut feelings, but current mathematical models don't represent reality adequately. The entire system is poorly understood and dangerously unstable. The world economy desperately needs a radical overhaul and that requires more mathematics, not less. It may be rocket science, but magic it's not.

Ian Stewart is emeritus professor of mathematics at the University of Warwick. His new book 17 Equations That Changed the World is published by Profile (£15.99)

The mathematical equation that caused the banks to crash

The brain’s quick interceptions help you navigate the world

When you are about to collide into something and manage to swerve away just in the nick of time, what exactly is happening in your brain? A new study from the Montreal Neurological Institute and Hospital – The Neuro, McGill University shows how the brain processes visual information to figure out when something is moving towards you or when you are about to head into a collision. The study, published in the Proceedings of the National Academy of Sciences of the USA (PNAS), provides vital insight into our sense of vision and a greater understanding of the brain.

Researchers at The Neuro and the University of Maryland have figured out the mathematical calculations that specific neurons employ in order to inform us of our distance from an object and the 3D velocities of moving objects and surfaces relative to ourselves. Highly specialized neurons located in the brain’s visual cortex, in an area known as MST, respond selectively to motion patterns such as expansion, rotation, and deformation. However, the computations underlying such selectivity were unknown until now.

Using mathematical models and sophisticated recording techniques, researchers have discovered how individual MST neurons function. “Area MST is typical of high-level visual cortex, in that information about important aspects of vision can be seen in the firing patterns of single neurons. A classic example is a neuron that only fires when the subject is looking at the image of a particular face.
This type of neuron has to gather information from other neurons that are selective to simpler features, like lines, colors, and textures, and combine these pieces of information in a fairly sophisticated way,” says Dr. Christopher Pack, neuroscientist at The Neuro and senior author. “Similarly, for motion detection, neurons have to combine input from many other neurons earlier in the visual pathway, in order to determine whether something is moving toward you or just drifting past.”
The brain’s visual pathway is made up of building blocks. For example, neurons in the retina respond to very simple stimuli, such as small spots of light. Further along the visual pathway, neurons respond to more complex stimulus such as straight lines, by combining inputs from neurons earlier on.
Neurons further along respond to even more complex stimulus such as combinations of lines (angles), ultimately leading to neurons that can respond to, or recognize, faces and objects for example.

The research team found that a remarkably simple computation lies at the heart of this sophisticated neural selectivity: MST neurons appear to be capable of performing a multiplicative operation on their inputs. These inputs come from neurons one step earlier in the visual pathway, in a well-studied area known as MT. In other words, the inputs of MT neurons are multiplied in order to get the output of MST neurons.
This turns out to be remarkably similar to what has been observed in other brain areas and in other species, suggesting it may reflect a general strategy by which brains process sensory information. “One interesting aspect of the computation is that it appears to be about the same as what other people have found in flies and beetles, suggesting that evolution solved this problem once, at least a few hundred million years ago.”

“We developed a new motion stimulus with a morphing pattern flow (e.g. dots on a screen that are expansive, swirl around, circle to the right, contract etc) and recorded MST neurons responding to these stimuli,” says Patrick Mineault, Ph.D. candidate at The Neuro and primary author on the study. “We circumvented the issue of increasing complexities of calculations along the various steps of the visual pathway by incorporating known data from neurons just one step earlier in the pathway - area MT, which precedes MST. As we now had measurements of the output of the MST neurons from the study’s recordings, and already knew the input of MT neurons, we could calculate the math linking these two functions – and it turns out to be a multiplicative function.” The mathematical models successfully account for the stimulus selectivity of some of the brain’s complex motion neurons - which are vitally important in helping navigate us through the world.

This work was supported by the Canadian Institutes of Health Research, Le Ministre du Développement économique, de l'Innovation et de l'Exportation du Québec , the National Science Foundation, the Fonds de recherche du Québec – Santé, and the Fonds de recherche du Québec – Nature et technologies.

The Montreal Neurological Institute and Hospital :

The Montreal Neurological Institute and Hospital — The Neuro, is a unique academic medical centre dedicated to neuroscience. Founded in 1934 by the renowned Dr. Wilder Penfield, The Neuro is recognized internationally for integrating research, compassionate patient care and advanced training, all key to advances in science and medicine. The Neuro is a research and teaching institute of McGill University and forms the basis for the Neuroscience Mission of the McGill University Health Centre. Neuro researchers are world leaders in cellular and molecular neuroscience, brain imaging, cognitive neuroscience and the study and treatment of epilepsy, multiple sclerosis and neuromuscular disorders. The Montreal Neurological Institute was named as one of the Seven Centres of Excellence in Budget 2007, which provided the MNI with $15 million in funding to support its research and commercialization activities related to neurological disease and neuroscience.

10 February 2012 by Lisa Grossman

A SUITE of artificial intelligence algorithms may become the ultimate chemistry set. Software can now quickly predict a property of molecules from their theoretical structure. Similar advances should allow chemists to design new molecules on computers instead of by lengthy trial-and-error.

Our physical understanding of the macroscopic world is so good that everything from bridges to aircraft can be designed and tested on a computer. There's no need to make every possible design to figure out which ones work. Microscopic molecules are a different story. "Basically, we are still doing chemistry like Thomas Edison," says Anatole von Lilienfeld of Argonne National Laboratory in Lemont, Illinois.

The chief enemy of computer-aided chemical design is the Schrödinger equation. In theory, this mathematical beast can be solved to give the probability that electrons in an atom or molecule will be in certain positions, giving rise to chemical and physical properties.

But because the equation increases in complexity as more electrons and protons are introduced, exact solutions only exist for the simplest systems: the hydrogen atom, composed of one electron and one proton, and the hydrogen molecule, which has two electrons and two protons.

This complexity rules out the possibility of exactly predicting the properties of large molecules that might be useful for engineering or medicine. "It's out of the question to solve the Schrödinger equation to arbitrary precision for, say, aspirin," says von Lilienfeld.

So he and his colleagues bypassed the fiendish equation entirely and turned instead to a computer-science technique.

Machine learning is already widely used to find patterns in large data sets with complicated underlying rules, including stock market analysis, ecology and Amazon's personalised book recommendations. An algorithm is fed examples (other shoppers who bought the book you're looking at, for instance) and the computer uses them to predict an outcome (other books you might like). "In the same way, we learn from molecules and use them as previous examples to predict properties of new molecules," says von Lilienfeld.

His team focused on a basic property: the energy tied up in all the bonds holding a molecule together, the atomisation energy. The team built a database of 7165 molecules with known atomisation energies and structures. The computer used 1000 of these to identify structural features that could predict the atomisation energies.

When the researchers tested the resulting algorithm on the remaining 6165 molecules, it produced atomisation energies within 1 per cent of the true value. That is comparable to the accuracy of mathematical approximations of the Schrödinger equation, which work but take longer to calculate as molecules get bigger (Physical Review Letters, DOI: 10.1103/PhysRevLett.108.058301). The algorithm found solutions in a millisecond that would take these earlier methods an hour. "Instead of having to wait years to screen lots of new molecules, you might have to wait weeks or a month," says Mark Tuckerman of New York University, who was not involved in the new work.

The algorithm is still mainly a proof of principle. If it can learn to predict something else, such as how well a molecule binds to an enzyme, it could help with designing drugs, fuel cells, batteries or biosensors. "The applications can be as broad as chemistry," von Lilienfeld says.

See graphic: "The not-so-simple Schrödinger equation"

Molecules from scratch without the fiendish physics

Things don't get much slimmer than a sheet of
graphene molecules, just one carbon atom thick
(Image: Pasieka/SPL/Getty)

Astonishing conductivity helped the discoverers of graphene win the Nobel prize in physics in 2010. Now a way to switch off the easy flow of electrons in this wonder form of carbon is bringing superfast graphene computers closer.

A sheet-like molecule just one carbon atom thick, graphene offers much less resistance to the flow of electrons than silicon. It has been hailed for its potential as the basis for computer circuits that operate at unprecedented speed. "It's an extremely promising material," says Konstantin Novoselov, who shared the Nobel prize with his co-discoverer, Andre Geim, both at the University of Manchester, UK.

But the ease of electron flow also creates a problem. To perform calculations, computers need to turn the flow of electricity on and off in their circuits. The gates that open and close to regulate the flow are called transistors. Making graphene-based transistors has proven difficult because it is such a good conductor.

Previous attempts have involved electrons confined to a single layer of graphene, but these still suffer from a leakage of electrons when the transistor is in its "off" state.

Quantum tunnel

Now Novoselov and colleagues have found a way to overcome this leakage problem by sandwiching a layer of molybdenum disulfide between two layers of graphene. The molybdenum acts as an insulator, preventing electrons from flowing in the normal way from one graphene layer to the other. This constitutes an "off" state.

A quantum mechanical effect means a small number of electrons can "tunnel" through the molybdenum. This normally happens very rarely but applying a voltage across the barrier boosts the energy of the electrons, making tunnelling much more probable – a sizable current starts to flow. This is the "on" state. By varying the voltage, the researchers could turn the flow on and off, making the device a transistor.

The graphene sandwich reduces leakage by a factor of 10 compared with previous graphene-based transistors. The team suggests reducing leakage further by increasing the thickness of the insulating layer. "It really opens a new dimension in our research," says Novoselov.

Journal reference: Science, DOI: 10.1126/science.1218461

Slow graphene down, speed computers up

Julie Clutterbuck, a research fellow at the Australian National University, has joined
a global protest against Elsevier over increased prices for individual journals.
Picture: Penny Bradfield Source: The Australian

DON'T expect to see Julie Clutterbuck's name again in the Journal of Differential Equations. Or in any other journal owned by Elsevier, the Amsterdam-based behemoth of scholarly publishing.

Clutterbuck, a mathematician at the Australian National University, has joined a global protest against Elsevier.

At last count, more than 4300 academics had put their names to a website, The Cost of Knowledge, which accuses the publisher of jacking up prices for individual journals so that libraries have to buy bundles of journals including titles they do not want.

Elsevier is also charged with lending its weight to US measures that would frustrate open access to the fruits of taxpayer-funded research.

Won't publish, won't referee, won't do editorial work -- that is the anti-Elsevier pledge. "It's kind of weird for us -- we're both the market for these journals but we're also creating the raw product," says Clutterbuck.

It was a Cambridge mathematician, Timothy Gowers, who inspired thecostofknowledge.com. In a blog post with the title "Elsevier: my part in its downfall", Gowers says he believes Elsevier's practices are doomed -- "the internet will see to that."

Not necessarily, says the Economist magazine: "Academic journals generally get their articles for nothing and may pay little to editors and peer reviewers. They sell to the very universities that provide that cheap labour."

The result: in 2010 Elsevier made pound stg. 724 million ($1.06 billion), representing an operating-profit margin of 36 per cent.

"Academics are heroic complainers and not always well disposed to profit-maximising businesses," The Economist says.

It is true that for some boycotters the grievance is nothing less than the commodification of knowledge. For others, it's more specific to Elsevier.

The first complaint of Peter Drummond, a physicist at Swinburne, is Scopus, the Elsevier database used for the Excellence in Research for Australia audit.

"I've just found it incredibly full of bugs," he says. Scopus says it has fixed a problem of missing citations in physics.

At ANU, researcher Michael Young says Elsevier asked him to update and expand his entry on the anthropologist Malinowski in the International Encyclopedia of Social and Behavioural Sciences. "It would involve a week's work -- for the standard fee of $US100 ($92.60). I have declined the invitation," he says on the boycott site.

Says Harald Boersma, Elsevier's senior manager (corporate relations): "We respect the freedom of authors to make their own decisions."

But he protests that investment in digitised content has in fact reduced the cost of journals per article.

Andrew Wells, from the Council of Australian University Librarians, is puzzled by the singling out of Elsevier.

"It's not the only commercial scholarly publisher making large profits," he said.

"The reality of it is that they [Elseiver, Springer, Taylor & Francis and Wiley-Blackwell] are commercial organisations, so bottom line and shareholder return are very important to them."

"They've got tremendous market power because of their size, it's not as though we can go to other people for this stuff."

On open access, Mr Wells said there was a range of views among university librarians but he was a sceptic of mandatory open access.

One reason for the Elsevier boycott is the publisher's support for the US Research Works Act, which would stop public funding agencies mandating open access without prior approval from commercial publishers.

Mr Wells believed there was poor compliance with such mandates and in any case, commercial publishers did allow open access in the sense that researchers could deposit an author's version of their articles in university archives.

He doubted the Elsevier boycott would overturn the knowledge dissemination model.

"I don't think that this is going to be the death of commercial publishing, especially when it is so deeply engaged with the academy and how universities are run," he said.

"[The publishers] are our suppliers and the material they generate is essential to teaching and learning."

He thought the wiser tactic was to keep talking to the publishers and try to get better deals, for example, more flexibility in the bundling of journals.

Academics boycott publisher Elsevier

In today's global village, national coffers are more interconnected than ever before. And as the current economic crisis has proven, a downturn in one country can travel in a wave across the globe, like a financial tsunami. Now, researchers from Tel Aviv University, in collaboration with the Kiel Institute of World Economy in Germany, have developed a market "seismograph" — a new methodology that measures the interconnections between stock markets across the globe. It has the potential to serve as an early warning system and provide measures to manage and mitigate the spread of financial crisis.

The method sheds new light on the structure of the global financial village, says Ph.D. student Dror Kenett, working with Prof. Eshel Ben-Jacob of TAU's School of Physics and Astronomy and Matthias Raddent and Prof. Thomas Lux from the Kiel Institute. Recently published in the journal PLOS ONE, the research investigates connections among individual major world markets by analyzing the concurrent behavior of the stock market as a global whole.

Their approach studies individual economies in the context of the global financial village, exploring the flow of information between financial markets, says Kenett. "It has become both vital and critical to understand the relationships and dependencies among the world's markets," he explains, suggesting that each country could use this method as a tool for analyzing the extent of its connection to particular foreign markets and identifying where they are at risk — prompting protective financial measures.

Financial ties that bind

There's nothing new in analyzing the correlations between stocks in an individual market, using parameters such as market index and volatility to determine whether prices of stocks will rise or fall in tandem. But with this project, the researchers have introduced the concept of the "meta-correlation," in which they measure the average correlation of countries' stock markets against one another. The result is a precise understanding of how changes in one market impact another. At worst, these connections can lead to a fast spread of financial crisis.

To develop their method, the researchers looked at data from six major world markets — the U.S., the U.K., Germany, Japan, China, and India — from the beginning of 2000 to the end of 2010. Choosing the leading stocks in each market, the team then mapped the correlations between the groups of stocks from each country over the 11-year period. With the exception of China, which tended to operate independently, the researchers discovered an interesting pattern of interdependencies between these markets. Some markets, such as the U.K. and U.S., were closely connected, as predicted. But there were also surprising findings, such the fact that Japan fluctuates in its financial alignment between western and eastern countries.

Predicting economic disaster

According to the researchers, this method of understanding market connections could help each country predict when a financial crisis is imminent, allowing it to set up policies that will protect their own markets from becoming dangerously intertwined with struggling markets. "In the current era, when the global financial village is highly prone to systematic collapses, our approach can provide a sensitive 'financial seismograph' to detect early signs of global crisis," Prof. Ben-Jacob says.

There are different safety mechanisms that each country can implement, continues Kenett, citing Greece's financial problems and their impact on the European market as a whole. "Germany is so invested in Greece that they don't have an option other than to bail Greece out," he says, noting that if it had been able to see the extent of their dangerous connection with Greece, Germany could have opted to reduce its investments earlier.

Having attracted the interest of governmental financial ministries, the project will now be extended to include even more markets. "With such high frequency data, you can have almost real-time or short-time predictions on how economic information flows throughout the world," Kenett notes.

To read the article, see:
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0031144 For more business and management news from Tel Aviv University, click here Keep up with the latest AFTAU news on Twitter: http://www.twitter.com/AFTAUnews

Storm Warning: Financial Tsunami Heading This Way

The findings by a team of demographers at the social science research organization NORC at the University of Chicago contradict a long-held belief that mortality rates level off above age 80. They also explain why the number of Americans age 100 and above is less than half of what the Census Bureau predicted as recently as six years ago.

The research is based on a new way of accurately measuring mortality among people 80 years of age and older, an issue that has proven remarkably elusive in the past. Instead of using self-reports of age to gauge mortality rates, the new study compared previous projections against real, observed death rates. The work will be significant in arriving at more accurate cost projections for programs such as Social Security and Medicare, which are based in part on mortality rates.

The research, done by Leonid A. Gavrilov and Natalia S. Gavrilova and published in the current edition of the North American Actuarial Journal, is based on highly accurate information about the date of birth and the date of death of more than nine million Americans born between 1875 and 1895. The data is publicly available in the Social Security Administration Death Master File. “It is a remarkable resource that allowed us to build what is called an extinct birth cohort that corrects or explains a number of misunderstandings about the mortality rate of our oldest citizens,” said Leonid Gavrilov.

A stark example of the problem of estimating the number of people over 100 came recently when the U.S. Census Bureau revised sharply downward the number of living centenarians. Six years ago, the bureau predicted that by 2010 there would be 114,000 people age 100 or older. The actual number turned out to be 53,364. The projection was wrong by a factor of two.

The newly published paper, titled “Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File,” explains the discrepancy and is likely to make a difference in the way mortality projections for the very old are done in the future.

The key finding is straightforward—the rise in death rates that occurs as people grow older is the same for the oldest Americans as for those who are younger. Previous studies had supposed that the mortality rate flattens out above age 80. But the new NORC research reveals that the expected mortality deceleration does not take place.

Anne Zissu, chair of the Department of Business NYC College of Technology/CUNY, said the research provides “an essential tool” for developing models on seniors’ financial assets.

Zissu said the research “will alter our financial approach to this valuation of mortality/longevity risk. Demographers and financiers need to work on this issue together, and their models must adapt to each other.”

The mortality rate for people between the ages of 30 and 80 follows what is called the Gompertz Law, named for its founder, Benjamin Gompertz, who observed in 1825 that a person’s risk of death in a given year doubles every eight years of age. It is a phenomenon that holds up across nations and over time and is an important part of the foundation of actuarial science.

For approximately 70 years, demographers have believed that above age 80 the Gompertz Law did not hold and that mortality rates flattened out. The work done by the Gavrilovs, a husband-and-wife team, reveals that the Gompertz Law holds at least through age 106, and probably higher, but the researchers say mortality data for those older than 106 is unreliable.

The Gavrilovs say the extinct birth cohort of people born between 1875 and 1895, which they built using the Social Security Administration Death Master File, reveals beyond question that the mortality rate of people in that cohort aligns with the Gompertz Law.

“It amazes me that the Gompertz model fits so well nearly 200 years after he proposed it,” said Tom Edwalds, Assistant Vice President of Mortality Research for the Munich American Reassurance Company.

Prior estimates of the number of centenarians in the United States were made in less direct ways that were subject to error. They depended, for example, on people self-reporting their age in the U.S. Census, which is less reliable than having actual birth and death data.

Gavrilov and Gavrilova work at the Center on the Economics and Demography of Aging, one of the Academic Research Centers of NORC. The study is supported by the National Institute on Aging.

The Chicago Actuarial Association has invited the authors to present their new findings at the CAA annual meeting in Chicago on March 13.

Odds of living a very long life lower than formerly predicted

Professor Ngo Bao Chau

Famous Vietnamese mathematician Prof. Ngo Bao Chau has received the French Legion of Honor medal from president Nicolas Sarkozy.

The French Legion of Honor, created by Napoleon Bonaparte in 1802, is the highest decoration in France. It was awarded to Chau in Paris late last month, he told local media recently.

Chau, who teaches at the University of Chicago, returned to Vietnam in mid-January to chair a math conference and prepare for the opening of Vietnam Institute of Advanced Math.

In 2010, he won the Fields Medal, the mathematics version of the Nobel Prize. Since the prize was founded by the International Mathematical Union in 1936, only three Asian mathematicians have won it, all of whom were from Japan. Up to four medals are awarded every four years.

The 40-year-old mathematician received his doctoral degree from Université Paris-Sud in 1997. He returned to Vietnam upon the government's request to help implement the national 2010-2020 math development program, after accepting a professorship at the University of Chicago.

Recent Vietnamese winners of the French honor include Nguyen Thi That Peel in 2010, a Vietnamese-French citizen who arranged for a lot of medicines and medical equipment to be sent to Vietnam; and had many French doctors visit the country for charity healthcare projects.

Another winner, in 2011, was Nguyen Thi Xuan Phuong, a former war reporter and documentary director of Vietnam Television who wrote many reports and translated different documents to introduce Vietnam to French readers for more than 30 years.

Vietnamese math genius receives top French honor

Professor Robert McKibbin

Robert McKibbin, a Professor of Applied Mathematics based at the Institute of Information and Mathematical Sciences at Albany, received the 2012 ANZIAM (Australian and New Zealand Industrial and Applied Mathematics group) Medal for his lifelong work in applied and industrial mathematics.

He is known as one of the pre-eminent applied mathematicians in New Zealand, with a particular focus on geophysical and industrial applications, from modelling hydrothermal eruptions in areas such as Rotorua and the distribution of volcanic dust from eruptions, to fluid motion and pollution transport in groundwater aquifers, ground subsidence and aluminium and iron smelting.

Professor McKibbin says he had always been good at maths at school, but never realised until he reached university how diverse its applications and uses could be in a wide range of industrial, agricultural and other scientific areas.

“In mathematical modelling, we take an interdisciplinary approach. You need to understand the physics, chemistry or biology of a phenomenon as well as having the mathematical tools to address whatever the problem is,” he says.

Creating conceptual models for invisible or unpredictable phenomena – like volcanic dust particles and underground hydrothermal activity – is both challenging and fascinating, he says. “You are dealing with ‘what if’ scenarios, like 'what if Mount Taranaki blew its top?' What might the impact be, and how would the surrounding population and landscape be affected?”

Professor McKibbin, who was recognised at the awards ceremony for his contribution to research and enhancing the profile of applied and industrial maths through teaching and mentoring, including supervising more than 20 PhD and masters students, says budding high school mathematicians need to be made aware of the exciting job prospects available. "Mathematicians are a fairly rare breed, and are highly sought-after by a range of industries for their logical thinking and conceptual skills that are needed in problem solving.”

Professor McKibbin received the medal on February 1 at a presentation at the group’s conference in Warrnambool, Victoria, Australia. It has been awarded biennially since 1995, making him the ninth person, and only the second New Zealand-based mathematician to receive it. The other was Professor Graeme Wake, also from the Albany-based institute, who received it in 2006.

Industrial-strength award for Massey mathematician

Shelley Zacks

Many of Shelley Zacks’ most influential findings have arisen out of a need to address real-world problems — how to track inventory, aiding the military in “seeing” an enemy in blind situations, training IBM engineers in basic statistical quality control and helping doctors treat cancer.

“Some mathematicians like to work on abstract things — they don’t care whether it is applicable or not,” Zacks says. “Usually what I like is to get a good applied problem and try to develop the mathematics around it. A model should not be built around some mathematical analysis just for the sake of doing mathematics alone.”

This interest in addressing real-world problems started right after he earned his bachelor’s degree in 1955 from Hebrew University of Jerusalem and began helping scientists at the Research Council of Israel understand their data using statistical methods.

“I loved those opportunities to work directly with scientists,” he says. “They were not working on routine problems and, hence, any conventional analysis was rarely appropriate.” So, to move the experiments forward, he had to be innovative, developing his own theories and methods.

Zacks kept his eye on the applicable during several appointments throughout the 1960s and ’70s. In 1976, because of his work on survival probabilities for particle crossing, a field having absorption points, a scientist at the White Sands Missile Range called to ask if he could apply his research to crossing minefields. Zacks said yes, and the collaboration lasted 10 years.

The research expanded as he worked with the Office of Naval Research and the U.S. Army Research Office to understand blind spots in other theaters as well. A big challenge facing submarines is icebergs obscuring much of the North Sea. Armies have a tough time fighting in forests because there are so many random variables that hide the target — the number of trees, tree width, enemy movement, ally movement, heaviness of foliage, etc. Put simply, Zacks helped the military figure out the probabilities of how much of the enemy could be seen and how much couldn’t.

While conducting that research, Zacks landed at Binghamton University in 1980, right when the mathematics academy was debating how mathematics could, or should, address the needs of private industry, which was clamoring for practical applications. Zacks revealed his position by establishing Binghamton University’s Center for Statistics to train IBM engineers in basic statistical quality control, reliability and design of experiments.

He worked with a colleague in the Thomas J. Watson School of Engineering and Applied Science to create simulations that ran on IBM’s first generation of PCs, which were used during seminars for the engineers. Later, other companies joined, including General Electric, Universal Instruments and Dupont, and Zacks partnered with them on several research projects.

At about the same time, he started working with a biostatistician at the Medical Center at Charleston, S.C., to develop cancer treatment experiments that would yield the best-quality data possible. Later, the design became more and more sophisticated as he worked with researchers at Philadelphia’s Fox Chase Cancer Center.

“We developed the methodology that tells them how to do the research — how many patients to take, how to randomize the patients, how long to do the trial,” Zacks says. “Basically, we are building the framework for the experiments. The question is, when they start getting results from the patients and see how they react, how do you incorporate these results into the theoretical framework so that it will tell them how to continue their research?”

Zacks, who was named a distinguished professor by the State University of New York last year, doesn’t maintain the editing load he used to (at one point he was the editor or on the editorial board of five scholarly journals simultaneously), but he’s still teaching and researching. Most recently, he has looked at distributions of stopping times, which has a wide range of applications, from telecommunications and computer servers to coffeehouse staffing and hospital wait times.

Mathematician seeks practical solutions