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Among the seven problems in mathematics put forward by the Clay Mathematics Institute in 2000 is one that relates in a fundamental way to our understanding of the physical world we live in.

It’s the Navier-Stokes existence and uniqueness problem, based on equations written down in the 19th century.

The solution of this prize problem would have a profound impact on our understanding of the behaviour of fluids which, of course, are ubiquitous in nature. Air and water are the most recognisable fluids; how they move and behave has fascinated scientists and mathematicians since the birth of science.

But what are the so-called Navier-Stokes equations? What do they describe?

The equations

In order to understand the Navier-Stokes equations and their derivation we need considerable mathematical training and also a sound understanding of basic physics.

Without that, we must draw upon some very simple basics and talk in terms of broad generalities – but that should be sufficient to give the reader a sense of how we arrive at these fundamental equations, and the importance of the questions.

From this point, I’ll refer to the Navier-Stokes equations as “the equations”.

The equations governing the motion of a fluid are most simply described as a statement of Newton’s Second Law of Motion as it applies to the movement of a mass of fluid (whether that be air, water or a more exotic fluid). Newton’s second law states that:

Mass x Acceleration = Force acting on a body

For a fluid the “mass” is the mass of the fluid body; the “acceleration” is the acceleration of a particular fluid particle; the “forces acting on the body” are the total forces acting on our fluid.

Without going into full details, it’s possible to state here that Newton’s Second Law produces a system of differential equations relating rates of change of fluid velocity to the forces acting on the fluid. We require one other physical constraint to be applied on our fluid, which can be most simply stated as:

Mass is conserved! – i.e. fluid neither appears nor disappears from our system.

The solution

Having a sense of what the Navier-Stokes equations are allows us to discuss why the Millennium Prize solution is so important. The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equations. The second focuses on whether these solutions are bounded (remain finite).

It’s not possible to give a precise mathematical description of these two components so I’ll try to place the two parts of the problem in a physical context.

1) For a mathematical model, however complicated, to represent the physical world we are trying to understand, the model must first have solutions.

At first glance, this seems a slightly strange statement – why study equations if we are not sure they have solutions? In practice we know many solutions that provide excellent agreement with many physically relevant and important fluid flows.

But these solutions are approximations to the solutions of the full Navier-Stokes equations (the approximation comes about because there is, usually, no simple mathematical formulae available – we must resort to solving the equations on a computer using numerical approximations).

Although we are very confident that our (approximate) solutions are correct, a formal mathematical proof of the existence of solutions is lacking. That provides the first part of the Millennium Prize challenge.

2) The second part asks whether the solutions of the Navier-Stokes equations can become singular (or grow without limit).

Again, a lot of mathematics is required to explain this. But we can examine why this is an important question.

There is an old saying that “nature abhors a vacuum”. This has a modern parallel in the assertion by physicist Stephen Hawking, while referring to black holes, that “nature abhors a naked singularity”. Singularity, in this case, refers to the point at which the gravitational forces – pulling objects towards a black hole – appear (according to our current theories) to become infinite.

In the context of the Navier-Stokes equations, and our belief that they describe the movement of fluids under a wide range of conditions, a singularity would indicate we might have missed some important, as yet unknown, physics. Why? Because mathematics doesn’t deal in infinites.

The history of fluid mechanics is peppered with solutions of simplified versions of the Navier-Stokes equations that yield singular solutions. In such cases, the singular solutions have often hinted at some new physics previously not considered in the simplified models.

Identifying this new physics has allowed researchers to further refine their mathematical models and so improve the agreement between model and reality.

If, as many believe, the Navier-Stokes equations do posses singular solutions then perhaps the next Millennium Prize will go to the person that discovers just what new physics is required to remove the singularity.

Then nature can, as all fluid mechanists already do, come to delight in the equations handed down to us by Claude-Louis Navier and George Gabriel Stokes.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Jim Denier, University of Adelaide

 

How fluids move has fascinated researchers since the birth of science.

Among the seven problems in mathematics put forward by the Clay Mathematics Institute in 2000 is one that relates in a fundamental way to our understanding of the physical world we live in.

It’s the Navier-Stokes existence and uniqueness problem, based on equations written down in the 19th century.

The solution of this prize problem would have a profound impact on our understanding of the behaviour of fluids which, of course, are ubiquitous in nature. Air and water are the most recognisable fluids; how they move and behave has fascinated scientists and mathematicians since the birth of science.

But what are the so-called Navier-Stokes equations? What do they describe?

The equations

In order to understand the Navier-Stokes equations and their derivation we need considerable mathematical training and also a sound understanding of basic physics.

Without that, we must draw upon some very simple basics and talk in terms of broad generalities – but that should be sufficient to give the reader a sense of how we arrive at these fundamental equations, and the importance of the questions.

From this point, I’ll refer to the Navier-Stokes equations as “the equations”.

The equations governing the motion of a fluid are most simply described as a statement of Newton’s Second Law of Motion as it applies to the movement of a mass of fluid (whether that be air, water or a more exotic fluid). Newton’s second law states that:

Mass x Acceleration = Force acting on a body

For a fluid the “mass” is the mass of the fluid body; the “acceleration” is the acceleration of a particular fluid particle; the “forces acting on the body” are the total forces acting on our fluid.

Without going into full details, it’s possible to state here that Newton’s Second Law produces a system of differential equations relating rates of change of fluid velocity to the forces acting on the fluid. We require one other physical constraint to be applied on our fluid, which can be most simply stated as:

Mass is conserved! – i.e. fluid neither appears nor disappears from our system.

The solution

Having a sense of what the Navier-Stokes equations are allows us to discuss why the Millennium Prize solution is so important. The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equations. The second focuses on whether these solutions are bounded (remain finite).

It’s not possible to give a precise mathematical description of these two components so I’ll try to place the two parts of the problem in a physical context.

1) For a mathematical model, however complicated, to represent the physical world we are trying to understand, the model must first have solutions.

At first glance, this seems a slightly strange statement – why study equations if we are not sure they have solutions? In practice we know many solutions that provide excellent agreement with many physically relevant and important fluid flows.

But these solutions are approximations to the solutions of the full Navier-Stokes equations (the approximation comes about because there is, usually, no simple mathematical formulae available – we must resort to solving the equations on a computer using numerical approximations).

Although we are very confident that our (approximate) solutions are correct, a formal mathematical proof of the existence of solutions is lacking. That provides the first part of the Millennium Prize challenge.

2) The second part asks whether the solutions of the Navier-Stokes equations can become singular (or grow without limit).

Again, a lot of mathematics is required to explain this. But we can examine why this is an important question.

There is an old saying that “nature abhors a vacuum”. This has a modern parallel in the assertion by physicist Stephen Hawking, while referring to black holes, that “nature abhors a naked singularity”. Singularity, in this case, refers to the point at which the gravitational forces – pulling objects towards a black hole – appear (according to our current theories) to become infinite.

In the context of the Navier-Stokes equations, and our belief that they describe the movement of fluids under a wide range of conditions, a singularity would indicate we might have missed some important, as yet unknown, physics. Why? Because mathematics doesn’t deal in infinites.

The history of fluid mechanics is peppered with solutions of simplified versions of the Navier-Stokes equations that yield singular solutions. In such cases, the singular solutions have often hinted at some new physics previously not considered in the simplified models.

Identifying this new physics has allowed researchers to further refine their mathematical models and so improve the agreement between model and reality.

If, as many believe, the Navier-Stokes equations do posses singular solutions then perhaps the next Millennium Prize will go to the person that discovers just what new physics is required to remove the singularity.

Then nature can, as all fluid mechanists already do, come to delight in the equations handed down to us by Claude-Louis Navier and George Gabriel Stokes.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jim Denier*

What will be the next number in this sequence?

“At school I was never really good at maths” is an all too common reaction when mathematicians name their profession.

In view of most people’s perceived lack of mathematical talent, it may come as somewhat of a surprise that a recent study carried out at John Hopkins University has shown that six-month-old babies already have a clear sense of numbers. They can count, or at least approximate, the number of happy faces shown on a computer screen.

By the time they start school, at around the age of five, most children are true masters of counting, and many will proudly announce when for the first time they have counted up to 100 or 1000. Children also intuitively understand the regular nature of counting; by adding sufficiently many ones to a starting value of one they know they will eventually reach their own age, that of their parents, grandparents, 2011, and so on.

Counting is child’s play. Photography By Shaeree

From counting to more general addition of whole numbers is only a small step—again within children’s almost-immediate grasp. After all, counting is the art of adding one, and once that is mastered it takes relatively little effort to work out that 3 + 4 = 7. Indeed, the first few times children attempt addition they usually receive help from their fingers or toes, effectively reducing the problem to that of counting:

3 + 4 = (1 + 1 + 1) + (1 + 1 + 1 + 1) = 7.

For most children, the sense of joy and achievement quickly ends when multiplication enters the picture. In theory it too can be understood through counting: 3 x 6 is three lots of six apples, which can be counted on fingers and toes to give 18 apples.

In practice, however, we master it through long hours spent rote-learning multiplication tables—perhaps not among our favourite primary school memories.

But at this point, we ask the reader to consider the possibility—in fact, the certainty—that multiplication is far from boring and uninspiring, but that it is intrinsically linked with some of mathematics’ deepest, most enduring and beautiful mysteries. And while a great many people may claim to be “not very good at maths” they are, in fact, equipped to understand some very difficult mathematical questions.

Primes

Let’s move towards these questions by going back to addition and those dreaded multiplication tables. Just like the earlier example of 7, we know that every whole number can be constructed by adding together sufficiently many ones. Multiplication, on the other hand, is not so well-behaved.

The number 12, for example, can be broken up into smaller pieces, or factors, while the number 11 cannot. More precisely, 12 can be written as the product of two whole numbers in multiple ways: 1 x 12, 2 x 6 and 3 x 4, but 11 can only ever be written as the product 1 x 11. Numbers such as 12 are called composite, while those that refuse to be factored are known as prime numbers or simply primes. For reasons that will soon become clear, 1 is not considered a prime, so that the first five prime numbers are 2, 3, 5, 7 and 11.

Just as the number 1 is the atomic unit of whole-number addition, prime numbers are the atoms of multiplication. According to the Fundamental Theorem of Arithmetic, any whole number greater than 1 can be written as a product of primes in exactly one way. For example: 4 = 2 x 2, 12 = 2 x 2 x 3, 2011 = 2011 and

13079109366950 = 2 x 5 x 5 x 11 x 11 x 11 x 37 x 223 x 23819,

where we always write the factors from smallest to largest. If, rather foolishly, we were to add 1 to the list of prime numbers, this would cause the downfall of the Fundamental Theorem of Arithmetic:

4 = 2 x 2 = 1 x 2 x 2 = 1 x 1 x 2 x 2 = …

In the above examples we have already seen several prime numbers, and a natural question is to ask for the total number of primes. From what we have learnt about addition with its single atom of 1, it is not unreasonable to expect there are only finitely many prime numbers, so that, just maybe, the 2649th prime number, 23819, could be the largest. Euclid of Alexandria, who lived around 300BC and who also gave us Euclidean Geometry, in fact showed that there are infinitely many primes.

Euclid’s reasoning can be captured in just a single sentence: if the list of primes were finite, then by multiplying them together and adding 1 we would get a new number which is not divisible by any prime on our list—a contradiction.

A few years after Euclid, his compatriot Eratosthenes of Cyrene found a clever way, now known as the Sieve of Eratosthenes, to obtain all primes less than a given number.

For instance, to find all primes less than 100, Eratosthenes would write down a list of all numbers from 2 to 99, cross out all multiples of 2 (but not 2 itself), then all multiples of 3 (but not 3 itself), then all multiples of 5, and so on. After only four steps(!) this would reveal to him the 25 primes

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

While this might seem very quick, much more sophisticated methods, combined with very powerful computers, are needed to find really large prime numbers. The current world record, established 2008, is the truly monstrous 2^43112609 – 1, a prime number of approximately 13 million digits.

The quest to tame the primes did not end with the ancient Greeks, and many great mathematicians, such as Pierre de Fermat, Leonhard Euler and Carl Friedrich Gauss studied prime numbers extensively. Despite their best efforts, and those of many mathematicians up to the present day, there are many more questions than answers concerning the primes.

One famous example of an unsolved problem is Goldbach’s Conjecture. In 1742, Christian Goldbach remarked in a letter to Euler that it appeared that every even number greater than 2 could be written as the sum of two primes.

For example, 2012 = 991 + 1021. While computers have confirmed the conjecture holds well beyond the first quintillion (10¹⁸) numbers, there is little hope of a proof of Goldbach’s Conjecture in the foreseeable future.

Another intractable problem is that of breaking very large numbers into their prime factors. If a number is known to be the product of two primes, each about 200 digits long, current supercomputers would take more than the lifetime of the universe to actually find these two prime factors. This time round our inability to do better is in fact a blessing: most secure encryption methods rely heavily on our failure to carry out prime factorisation quickly. The moment someone discovers a fast algorithm to factor large numbers, the world’s financial system will collapse, making the GFC look like child’s play.

To the dismay of many security agencies, mathematicians have also failed to show that fast algorithms are impossible—the possibility of an imminent collapse of world order cannot be entirely ruled out!

Margins of error

For mathematicians, the main prime number challenge is to understand their distribution. Quoting Don Zagier, nobody can predict where the next prime will sprout; they grow like weeds among the whole numbers, seemingly obeying no other law than that of chance. At the same time the prime numbers exhibit stunning regularity: there are laws governing their behaviour, obeyed with almost military precision.

The Prime Number Theorem describes the average distribution of the primes; it was first conjectured by both Gauss and Adrien-Marie Legendre, and then rigorously established independently by Jacques Hadamard and Charles Jean de la Vallée Poussin, a hundred years later in 1896.

The Prime Number Theorem states that the number of primes less than an arbitrarily chosen number n is approximately n divided by ln(n), where ln(n) is the natural logarithm of n. The relative error in this approximation becomes arbitrarily small as n becomes larger and larger.

For example, there are 25 primes less than 100, and 100/ln(100) = 21.7…, which is around 13% short. When n is a million we are up to 78498 primes and since 10⁶/ln(10⁶) = 72382.4…, we are only only 8% short.

The Riemann Hypothesis

The Prime Number Theorem does an incredible job describing the distribution of primes, but mathematicians would love to have a better understanding of the relative errors. This leads us to arguably the most famous open problem in mathematics: the Riemann Hypothesis.

Posed by Bernhard Riemann in 1859 in his paper “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given magnitude), the Riemann Hypothesis tells us how to tighten the Prime Number Theorem, giving us a control of the errors, like the 13% or 8% computed above.

The Riemann Hypothesis does not just “do better” than the Prime Number Theorem—it is generally believed to be “as good as it gets”. That is, we, or far-superior extraterrestrial civilisations, will never be able to predict the distribution of the primes any better than the Riemann Hypothesis does. One can compare it to, say, the ultimate 100 metres world record—a record that, once set, is impossible to ever break.

Finding a proof of the Riemann Hypothesis, and thus becoming record holder for all eternity, is the holy grail of pure mathematics. While the motivation for the Riemann Hypothesis is to understand the behaviour of the primes, the atoms of multiplication, its actual formulation requires higher-level mathematics and is beyond the scope of this article.

In 1900, David Hilbert, the most influential mathematician of his time, posed a now famous list of 23 problems that he hoped would shape the future of mathematics in the 20th century. Very few of Hilbert’s problems other than the Riemann Hypothesis remain open.

Inspired by Hilbert, in 2000 the Clay Mathematics Institute announced a list of seven of the most important open problems in mathematics. For the successful solver of any one of these there awaits not only lasting fame, but also one million US dollars in prize money. Needless to say, the Riemann Hypothesis is one of the “Millennium Prize Problems”.

Hilbert himself remarked: “If I were awoken after having slept for a thousand years, my first question would be: has the Riemann Hypothesis been proven?” Judging by the current rate of progress, Hilbert may well have to sleep a little while longer.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Ole Warnaar*

 

Let’s talk numbers for a moment.

The moon is approximately 384,000 kilometres away, and the sun is approximately 150 million kilometres away. The mean distance between Earth and the sun is known as the “astronomical unit” (AU). Neptune, the most distant planet, then, is 30 AU from the sun.

The nearest stars to Earth are 1,000 times more distant, roughly 4.3 light-years away (one light-year being the distance that light travels in 365.25 days – just under 10 trillion kilometres).

The Milky Way galaxy consists of some 300 billion stars in a spiral-shaped disk roughly 100,000 light-years across.

The Andromeda Galaxy, which can be seen with many home telescopes, is 2.54 million light years away. There are hundreds of billions of galaxies in the observable universe.

At present, the most distant observed galaxy is some 13.2 billion light-years away, formed not long after the Big Bang, 13.75 billion years ago (plus or minus 0.011 billion years).

The scope of the universe was illustrated by the astrophysicist Geraint Lewis in a recent Conversation article.

He noted that, if the entire Milky Way galaxy was represented by a small coin one centimetre across, the Andromeda Galaxy would be another small coin 25 centimetres away.

Going by this scale, the observable universe would extend for 5 kilometres in every direction, encompassing some 300 billion galaxies.

But how can scientists possibly calculate these enormous distances with any confidence?

Parallax

One technique is known as parallax. If you cover one eye and note the position of a nearby object, compared with more distant objects, the nearby object “moves” when you view it with the other eye. This is parallax (see below).

The same principle is used in astronomy. As Earth travels around the sun, relatively close stars are observed to move slightly, with respect to other fixed stars that are more distant.

Distance measurements can be made in this way for stars up to about 1,000 light-years away.

Standard candles

For more distant objects such as galaxies, astronomers rely on “standard candles” – bright objects that are known to have a fixed absolute luminosity (brightness).

Since light flux falls off as the square of the distance, by measuring the actual brightness observed on Earth astronomers can calculate the distance.

One type of standard candle, which has been used since the 1920s, is Cepheid variable stars.

Distances determined using this scheme are believed accurate to within about 7% for more nearby galaxies, and 15-20% for the most distant galaxies.

Type Ia supernovas

In recent years scientists have used Type Ia supernovae. These occur in a binary star system when a white dwarf star starts to attract matter from a larger red dwarf star.

As the white dwarf gains more and more matter, it eventually undergoes a runaway nuclear explosion that may briefly outshine an entire galaxy.

Because this process can occur only within a very narrow range of total mass, the absolute luminosity of Type Ia supernovas is very predictable. The uncertainty in these measurements is typically 5%.

In August, worldwide attention was focused on a Type Ia supernova that exploded in the Pinwheel Galaxy (known as M101), a beautiful spiral galaxy located just above the handle of the Big Dipper in the Northern Hemisphere. This is the closest supernova to the earth since the 1987 supernova, which was visible in the Southern Hemisphere.

These and other techniques for astronomical measurements, collectively known as the “cosmic distance ladder”, are described in an excellent Wikipedia article. Such multiple schemes lend an additional measure of reliability to these measurements.

In short, distances to astronomical objects have been measured with a high degree of reliability, using calculations that mostly employ only high-school mathematics.

Thus the overall conclusion of a universe consisting of billions of galaxies, most of them many millions or even billions of light-years away, is now considered beyond reasonable doubt.

Right tools for the job

The kind of distances we’re dealing with above do cause consternation for some since, as we peer millions of light-years into space, we are also peering millions of years into the past.

Some creationists, for instance, have theorised that, in about 4,000 BCE, a Creator placed quadrillions of photons in space en route to Earth, with patterns suggestive of supernova explosions and other events millions of years ago.

Needless to say, most observers reject this notion. Kenneth Miller of Brown University commented, “Their [Creationists’] version of God is one who has filled the universe with so much bogus evidence that the tools of science can give us nothing more than a phony version of reality.”

There are plenty of things in the universe to marvel at, and plenty of tools to help us understand them. That should be enough to keep us engaged for now.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Jonathan Borwein (Jon), University of Newcastle and David H. Bailey, University of California, Davis

This knot has Gauss code O1U2O3U1O2U3. Credit: Graphic by Sam Nelson.

In the 19th century, Lord Kelvin made the inspired guess that elements are knots in the “ether”. Hydrogen would be one kind of knot, oxygen a different kind of knot—and so forth throughout the periodic table of elements. This idea led Peter Guthrie Tait to prepare meticulous and quite beautiful tables of knots, in an effort to elucidate when two knots are truly different. From the point of view of physics, Kelvin and Tait were on the wrong track: the atomic viewpoint soon made the theory of ether obsolete. But from the mathematical viewpoint, a gold mine had been discovered: The branch of mathematics now known as “knot theory” has been burgeoning ever since.

In his article “The Combinatorial Revolution in Knot Theory”, to appear in the December 2011 issue of the Notices of the AMS, Sam Nelson describes a novel approach to knot theory that has gained currency in the past several years and the mysterious new knot-like objects discovered in the process.

As sailors have long known, many different kinds of knots are possible; in fact, the variety is infinite. A *mathematical* knot can be imagined as a knotted circle: Think of a pretzel, which is a knotted circle of dough, or a rubber band, which is the “un-knot” because it is not knotted. Mathematicians study the patterns, symmetries, and asymmetries in knots and develop methods for distinguishing when two knots are truly different.

Mathematically, one thinks of the string out of which a knot is formed as being a one-dimensional object, and the knot itself lives in three-dimensional space. Drawings of knots, like the ones done by Tait, are projections of the knot onto a two-dimensional plane. In such drawings, it is customary to draw over-and-under crossings of the string as broken and unbroken lines. If three or more strands of the knot are on top of each other at single point, we can move the strands slightly without changing the knot so that every point on the plane sits below at most two strands of the knot. A planar knot diagram is a picture of a knot, drawn in a two-dimensional plane, in which every point of the diagram represents at most two points in the knot. Planar knot diagrams have long been used in mathematics as a way to represent and study knots.

As Nelson reports in his article, mathematicians have devised various ways to represent the information contained in knot diagrams. One example is the Gauss code, which is a sequence of letters and numbers wherein each crossing in the knot is assigned a number and the letter O or U, depending on whether the crossing goes over or under. The Gauss code for a simple knot might look like this: O1U2O3U1O2U3.

In the mid-1990s, mathematicians discovered something strange. There are Gauss codes for which it is impossible to draw planar knot diagrams but which nevertheless behave like knots in certain ways. In particular, those codes, which Nelson calls *nonplanar Gauss codes*, work perfectly well in certain formulas that are used to investigate properties of knots. Nelson writes: “A planar Gauss code always describes a [knot] in three-space; what kind of thing could a nonplanar Gauss code be describing?” As it turns out, there are “virtual knots” that have legitimate Gauss codes but do not correspond to knots in three-dimensional space. These virtual knots can be investigated by applying combinatorial techniques to knot diagrams.

Just as new horizons opened when people dared to consider what would happen if -1 had a square root—and thereby discovered complex numbers, which have since been thoroughly explored by mathematicians and have become ubiquitous in physics and engineering—mathematicians are finding that the equations they used to investigate regular knots give rise to a whole universe of “generalized knots” that have their own peculiar qualities. Although they seem esoteric at first, these generalized knots turn out to have interpretations as familiar objects in mathematics. “Moreover,” Nelson writes, “classical knot theory emerges as a special case of the new generalized knot theory.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to American Mathematical Society

 

My intention with this article is to give an intuitive and non-technical introduction to the field of evolutionary algorithms, particularly with regards to optimisation.

If I get you interested, I think you’re ready to go down the rabbit hole and simulate evolution on your own computer. If not … well, I’m sure we can still be friends.

Survival of the fittest

According to Charles Darwin, the great evolutionary biologist, the human race owes its existence to the phenomenon of survival of the fittest. And being the fittest doesn’t necessarily mean the biggest physical presence.

Once in high school, my lunchbox was targeted by swooping eagles, and I was reduced to a hapless onlooker. The eagle, though smaller in form, was fitter than me because it could take my lunch and fly away – it knew I couldn’t chase it.

As harsh as it sounds, look around you and you will see many examples of the rule of the jungle – the fitter survive while the rest gradually vanish.

The research area, now broadly referred to as Evolutionary Algorithms, simulates this behaviour on a computer to find the fittest solutions to a number of different classes of problems in science, engineering and economics.

The area in which this area is perhaps most widely used is known as “optimisation”.

Optimisation is everywhere

Your high school maths teacher probably told you the shortest way to go from point A to point B was along the straight line joining A and B. Your mum told you that you should always get the right amount of sleep.

And, if you have lived on your own for any length of time, you’ll be familiar with the ever-increasing cost of living versus the constant income – you always strive to minimise the expenditures, while ensuring you are not malnourished.

Whenever you undertake an activity that seeks to minimise or maximise a well-defined quantity such as distance or the vague notion of the right amount of sleep, you are optimising.

Look around you right now and you’ll see optimisation in play – your Coke can is shaped like that for a reason, a water droplet is spherical for a reason, you wash all your dishes together in the dishwasher for a reason.

Each of these strives to save on something: volume of material of the Coke can, and energy and water, respectively, in the above cases.

So we can safely say optimisation is the act of minimising or maximising a quantity. But that definition misses an important detail: there is always a notion of subject to, or satisfying some conditions.

You must get the right amount of sleep, but you also must do your studies and go for your music lessons. Such conditions, which you also have to adhere to, are known as “constraints”. Optimisation with constraints is then collectively termed “constrained optimisation”.

After constraints comes the notion of “multi-objective optimisation”. You’ll usually have more than one thing to worry about (you must keep your supervisor happy with your work and keep yourself happy and also ensure that you are working on your other projects). In many cases these multiple objectives can be in conflict.

Evolutionary algorithms and optimisation

Imagine your local walking group has arranged a weekend trip for its members and one of the activities is a hill climbing exercise. The problem assigned to your group leader is to identify who among you will reach the hill in the shortest time.

There are two approaches he or she could take to complete this task: ask only one of you to climb up the hill at a time and measure the time needed, or ask all of you to run all at once and see who reaches first.

That second method is known as the “population approach” of solving optimisation problems – and that’s how evolutionary algorithms work. The “population” of solutions are evolved over a number of iterations, with only the fittest solutions making it to the next.

This is analogous to the champion girl from your school making to the next round which was contested among champions from other schools in your state, then your country, and finally winning among all the countries.

Or, in our above scenario, finding who in the walking group reaches the hill top fastest, who would then be denoted as the fittest.

In engineering, optimisation needs are faced at almost every step, so it’s not surprising evolutionary algorithms have been successful in that domain.

Design optimisation of scramjets

At the Multi-disciplinary Design Optimisation Group at the University of New South Wales, my colleagues and I are involved in the design optimisation of scramjets, as part of the SCRAMSPACE program. In this, we’re working with colleagues from the University of Queensland.

Our evolutionary algorithms-based optimisation procedures have been successfully used to obtain the optimal configuration of various components of a scramjet.

Some of these have quite technical names, that in themselves would require quite a bit of explanation but, if you want, you can get a feel for the kind of work we do, and its applications for scramjets, by clicking here.

There are, at the risk of sounding over-zealous, no limits to the application of evolutionary algorithms.

Has this whetted your appetite? Have you learnt something new today?

If so, I’m glad. May the force be with you!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Amit Saha*

We know the universe is vast, but how do we measure the distances between things? Dave Scrimshaw.

Let’s talk numbers for a moment.

The moon is approximately 384,000 kilometres away, and the sun is approximately 150 million kilometres away. The mean distance between Earth and the sun is known as the “astronomical unit” (AU). Neptune, the most distant planet, then, is 30 AU from the sun.

The nearest stars to Earth are 1,000 times more distant, roughly 4.3 light-years away (one light-year being the distance that light travels in 365.25 days – just under 10 trillion kilometres).

The Milky Way galaxy consists of some 300 billion stars in a spiral-shaped disk roughly 100,000 light-years across.

The Andromeda Galaxy, which can be seen with many home telescopes, is 2.54 million light years away. There are hundreds of billions of galaxies in the observable universe.

At present, the most distant observed galaxy is some 13.2 billion light-years away, formed not long after the Big Bang, 13.75 billion years ago (plus or minus 0.011 billion years).

The scope of the universe was illustrated by the astrophysicist Geraint Lewis in a recent Conversation article.

He noted that, if the entire Milky Way galaxy was represented by a small coin one centimetre across, the Andromeda Galaxy would be another small coin 25 centimetres away.

Going by this scale, the observable universe would extend for 5 kilometres in every direction, encompassing some 300 billion galaxies.

But how can scientists possibly calculate these enormous distances with any confidence?

Parallax

One technique is known as parallax. If you cover one eye and note the position of a nearby object, compared with more distant objects, the nearby object “moves” when you view it with the other eye. This is parallax (see below).

Booyabazooka

The same principle is used in astronomy. As Earth travels around the sun, relatively close stars are observed to move slightly, with respect to other fixed stars that are more distant.

Distance measurements can be made in this way for stars up to about 1,000 light-years away.

Standard candles

For more distant objects such as galaxies, astronomers rely on “standard candles” – bright objects that are known to have a fixed absolute luminosity (brightness).

Since light flux falls off as the square of the distance, by measuring the actual brightness observed on Earth astronomers can calculate the distance.

One type of standard candle, which has been used since the 1920s, is Cepheid variable stars.

Distances determined using this scheme are believed accurate to within about 7% for more nearby galaxies, and 15-20% for the most distant galaxies.

Type Ia supernovas

In recent years scientists have used Type Ia supernovae. These occur in a binary star system when a white dwarf star starts to attract matter from a larger red dwarf star.

As the white dwarf gains more and more matter, it eventually undergoes a runaway nuclear explosion that may briefly outshine an entire galaxy.

Because this process can occur only within a very narrow range of total mass, the absolute luminosity of Type Ia supernovas is very predictable. The uncertainty in these measurements is typically 5%.

In August, worldwide attention was focused on a Type Ia supernova that exploded in the Pinwheel Galaxy (known as M101), a beautiful spiral galaxy located just above the handle of the Big Dipper in the Northern Hemisphere. This is the closest supernova to the earth since the 1987 supernova, which was visible in the Southern Hemisphere.

These and other techniques for astronomical measurements, collectively known as the “cosmic distance ladder”, are described in an excellent Wikipedia article. Such multiple schemes lend an additional measure of reliability to these measurements.

In short, distances to astronomical objects have been measured with a high degree of reliability, using calculations that mostly employ only high-school mathematics.

Thus the overall conclusion of a universe consisting of billions of galaxies, most of them many millions or even billions of light-years away, is now considered beyond reasonable doubt.

Right tools for the job

The kind of distances we’re dealing with above do cause consternation for some since, as we peer millions of light-years into space, we are also peering millions of years into the past.

Some creationists, for instance, have theorised that, in about 4,000 BCE, a Creator placed quadrillions of photons in space en route to Earth, with patterns suggestive of supernova explosions and other events millions of years ago.

Needless to say, most observers reject this notion. Kenneth Miller of Brown University commented, “Their [Creationists’] version of God is one who has filled the universe with so much bogus evidence that the tools of science can give us nothing more than a phony version of reality.”

There are plenty of things in the universe to marvel at, and plenty of tools to help us understand them. That should be enough to keep us engaged for now.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jonathan Borwein (Jon)*

Australia faces many big challenges – in the economy, health, energy, water, climate change, infrastructure, sustainable agriculture and the preservation of our precious biodiversity.

To meet these, we need creative scientists and engineers drawn from many disciplines, and a technologically skilled workforce.

The many world-changing advances and achievements of Australian research and development (R&D) are encouraging. Indeed, the Australian Academy of Science, of which I’m president, believes our country’s scientific potential has never been greater.

But our ability to improve this performance in the future, or even maintain it, is not assured.

Four things threaten our ongoing R&D performance and, as a consequence, our economic security and prosperity, and I’ll address each of these in turn.

1) The level of investment in R&D

Over the past decade, successive Australian governments have recognised the need to properly invest in research and innovation.

Total investment by the current government has increased by almost 43% and is projected to amount to $9.4 billion dollars over the current financial year. This is very commendable.

It’s heartening to see Australia’s business sector is also increasing its investment – although admittedly this boost is coming off a low base compared to many other OECD nations. (Australia ranks 14th for business expenditure on R&D as a percentage of GDP).

But to remain competitive internationally we need even greater investment.

Australia spends around 2.2% of its GDP (around AU$900 per person per year) on research and development.

Iceland, the next best-ranked country, devotes 2.6% cent of GDP. Top of the list is Israel, with 4.6%, followed by Finland and Sweden, each of which spend 3.6%.

We have around 92,000 full-time equivalent researchers which, again, is only middle order. According to the OECD, in 2008 the proportion of R&D personnel in our total labour force puts Australia 16th, well short of Canada, which ranks ninth.

China has more than 1.6 million people working on research and development, a number that’s increasing rapidly. (China is ranked 33rd, with 2.5 R&D personnel per thousand in the workforce, from a total population of 1.3 billion)

Worryingly, Australia sits well within the bottom half of OECD countries (ranked 20th of 30) when it comes to the number of university graduates emerging with a science or engineering degree per capita.

These are sobering statistics.

The Australian Academy of Science therefore calls on the government to create a Sovereign Fund for Science, to secure the future prosperity of the nation.

The goal should be to increase Australia’s research and development expenditure to at least 3% of GDP by 2020.

2) International collaboration

By its very nature, science is a collaborative enterprise. It transcends generations, individual scientific disciplines and, increasingly, national boundaries. To paraphrase Sir Isaac Newton, we see further by standing on the shoulders of giants.

Australia produces only 2% of the world’s knowledge. To gain access to the other 98%, we must ensure our scientists are well-connected internationally.

Getting involved with major international projects at inception allows Australia to stay abreast of new scientific developments, to have a say in their direction, to take the knowledge further, and to apply it.

International collaborations also attract scientists from overseas to spend time in Australia, bringing us new skills and knowledge. Importantly, many return and become part of our scientific workforce.

Work arising from such collaborations often attracts great attention and gets cited more frequently. Take the recently announced kangaroo genome sequence, which garnered international media attention.

This work was done by a consortium of more than 100 researchers from Australia, the US, the UK, Germany and Japan, headed by my friend and Academy colleague Professor Marilyn Renfree. The “kangaroo” was in fact the Tammar wallaby.

Its genome is yielding many unexpected insights that may have significance for humans as well as for wallabies – for example the genes that make antibiotics in the mother’s milk to protect the tiny newborns from harmful bacteria.

There are many such examples.

We hope to bring international astronomers to Australia by winning the bid to build a giant collection of radio telescopes in the Western Australian desert. Known as the Square Kilometre Array, or SKA, this international project – which could go to either South Africa or Australia – will give astronomers huge insights into the formation and evolution of the first stars and galaxies after the Big Bang.

Barriers that have impeded the use of Australian research grants for international collaborations are being dismantled.

Today many grants and fellowships provided by the Australian Research Council, National Health and Medical Research Council and CSIRO support projects that include international partners.

Many of these linkages were initially catalysed by the federal government’s International Science Linkages (or ISL) program.

With funding of about $10 million per year, the ISL program has supported bilateral and multilateral relations with many other countries.

Regrettably, the ten-year program ended in June this year as funding was not renewed in the 2011-2012 Budget.

Put simply, it would be a grave blow if our ability to compete on the international stage were to be diminished.

I strongly urge the Federal Government to fund in its next Budget a new program to provide strategic support for Australia’s International Science Linkages.

3) Science capability in the workforce

We are a lucky nation: we have access to immense mineral wealth. But resources are finite. Even the minerals sector acknowledges that we cannot ride the current boom indefinitely.

Further, the Minerals Council of Australia warns skills shortages and structural weaknesses in the Australian economy have been masked by the boom.

And so, when the end of the mining boom comes, where will Australia be?

There is broad consensus among minds more economically astute than mine that our future prosperity will depend upon:

• a skilled workforce
• innovation
• entrepreneurship
• high productivity
• the creation of the kind of knowledge-intensive goods and services that can only result from robust research and development.

Certain skills are already in short supply in Australia.

In fact, the No More Excuses report issued by the Industry Skills Council earlier this year points to an alarming deficit in even basic skills.

According to that report, “millions of Australians have insufficient language, literacy and numeracy skills to benefit fully from training or to participate effectively at work”.

A recent project looking at the maths skills of bricklaying apprentices at a regional TAFE showed:

• 75% could not do basic arithmetic.
• 80% could not calculate the area of a rectangle, or the pay owed for working four-and-a-half hours.

Such figures are particularly worrying at a time when the demand for higher-level skills is increasing.

It’s essential we act now to ease the bottleneck and put in place measures that will create the technologically competent workforce we need for the future.

We can, and should, be “the clever country”. But this will only happen if we place appropriate emphasis on properly educating our young people.

4) Science and maths education

Without a robust and inspiring science and maths education system, it’s impossible to create an internationally competitive workforce.

Myriad jobs – apart from the obvious research, engineering and technology careers – require a basic understanding of science and maths.

And, as a parent, a mentor of young scientists and a passionate advocate for quality education, I know that all children are natural born scientists.

“Why?”, “How?”, and “What happens if …?” are questions asked frequently by young children, whose natural spirit of inquiry is crucial to understanding the big, exciting world around them.

We need to harness this natural curiosity and nurture it with inspiring education.

Australian public expenditure on education as a percentage of GDP is just 4.2% – significantly below the OECD average of 5.4%.

A decade ago, a review of Australian science education, revealed many students were disappointed with their high school science.

Today, this disenchantment continues, as evidenced by the declining number of students choosing to study science in senior secondary school. Consider the following:

• In 1991, more than a third of Year 12 students chose to study biology. That now sits at less than a quarter.
• 23% of Year 12 students studied chemistry ten years ago, compared with 18% now.
• In the same period, physics has fallen from 21% to 14%.

While Australian students have been losing interest in science, their international peers have been taking it up with great enthusiasm.

The OECD Program for International Student Assessment (PISA) examines the scientific literacy of teenagers in 57 different countries.

In 2000, the only nations that performed better than Australia were Korea and Japan. In 2009 – the most recent figures available – Australia ranked behind Shanghai, Finland, Hong Kong, Singapore, Japan and Korea.

What happened? The Assessment indicated that the performance of other countries has improved while Australia’s has remained stationary.

Maths

Australia’s early secondary mathematical literacy scores have significantly declined over the last decade. Our Year 4 and Year 8 students ranked 14th internationally in the most recent Trends International Mathematics and Science Study, conducted in 2007.

The decline in Australia’s mathematical literacy is of grave concern because mathematics is an enabling science, without which it’s not possible to make use of other sciences – either in the lab or in the workforce.

A recent survey conducted by Science and Technology Australia and the Academy of Science showed Australians clearly value science – 80% of respondents acknowledged science education is absolutely essential or very important to the national economy.

But it also revealed some alarming holes in the basic science understanding of the average Australian.

• Three in ten believe humans were around at the time of dinosaurs.
• More than a fifth of our university graduates think that it takes just one day for the Earth to travel around the sun.
• Almost a third of Australians do not think evolution is currently occurring.
• About a quarter say human activity is not influencing the evolution of other species: a worrying statistic given the impact that human activity is having on the environment.

In other words, many of us do not understand even the most basic science.

How can we halt this slide in science and maths in our schools and attain an internationally enviable position?

Thankfully, our government is already investing significantly in school infrastructure and in rolling out a national high-speed internet network.

Last December, education ministers approved the content for new national curricula in English, history, maths and science. In coming months, they’ll be asked to sign off on the standards for these curricula. This is an important initiative, and the Academy of Science applauds it.

But we also need investment in teachers, and in inspiring curriculum programs.

This is a responsibility for both the Commonwealth and the States, who must work together rather than reverting to the blame game.

Inspired (and inspiring) teachers will be the most important agents for improving educational outcomes.

We must place a much higher societal value on teachers and do everything we can to recruit some of our brightest and best into teaching.

We must support these educators with the best tools and resources available and provide them with stimulating opportunities for ongoing training.

I agree with the prime minister that we live in a crucial time for science in Australia and around the world.

In fact, I could not agree more.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Suzanne Cory, WEHI (Walter and Eliza Hall Institute of Medical Research)

Are we getting closer to solving one of life’s greatest mysteries?

During a lunch in the summer of 1950, physicists Enrico Fermi, Edward Teller and Herbert York were chatting about a recent New Yorker cartoon depicting aliens abducting trash cans in flying saucers. Suddenly, Fermi blurted out, “Where is everybody?”

He reasoned: “Since there are likely many other technological civilisations in the Milky Way galaxy, and since in a few tens of thousands of years at most they could have explored or even colonised many distant planets, why don’t we see any evidence of even a single extraterrestrial civilisation?”

This has come to be known as Fermi’s Paradox.

Clearly the question of whether other civilisations exist is one of the most important questions of modern science. Any discovery of a distant civilisation – say by analysis of microwave data – would rank as among the most far-reaching of all scientific discoveries.

Drake equation

At a 1960 conference regarding extraterrestrial intelligence, Frank Drake (1930 —) sketched out what is now the Drake equation, estimating the number of civilisations in the Milky Way with which we could potentially communicate:

where

N = number of civilisations in our galaxy that can communicate.

R* = average rate of star formation per year in galaxy.

f_p = fraction of those stars that have planets.

n_e = average number of planets that can support life, per star that has planets.

f_l = fraction of the above that eventually develop life.

f_i = fraction of the above that eventually develop intelligent life.

f_c = fraction of civilisations that develop technology that signals existence into space.

L = length of time such civilisations release detectable signals into space.

The result? Drake estimated ten such civilisations were out there somewhere in the Milky Way.

This analysis, led to the Search for Extraterrestrial Intelligence (SETI) project, looking for radio transmissions in a region of the electromagnetic spectrum thought best suited for interstellar communication.

But after 50 years of searching, using increasingly powerful equipment, nothing has been found.

So where is everybody?

Proposed solutions to Fermi’s paradox

Numerous scientists have examined Fermi’s paradox and proposed solutions. The following is a list of some of the proposed solutions, and common rejoinders:

• Such civilisations are here, or are observing us, but are under orders not to disclose their existence.

Common rejoinder: This explanation (known as the “zookeeper’s theory”) is preferred by some scientists including, for instance, the late Carl Sagan. But it falls prey to the fact that it would take just one member of an extraterrestrial society to break the pact of silence – and this would seem inevitable.

• Such civilisations have been here and planted seeds of life, or perhaps left messages in DNA.

Common rejoinder: The notion that life began on Earth from bacterial spores or the like that originated elsewhere, known as the “panspermia theory”, only pushes the origin of life problem to some other star system – scientists see no evidence in DNA sequences of anything artificial.

• Such civilisations exist, but are too far away.

Common rejoinder: A sufficiently advanced civilisation could send probes to distant stars, which could scout out suitable planets, land and construct copies of themselves, using the latest software beamed from home.

So the entire Milky Way galaxy could be explored within, at most, a few million years.

• Such civilisations exist, but have lost interest in interstellar engagement.

Common rejoinder: As with the zookeeper theory, this would require each civilisation to forever lack interest in communication and transportation – and someone would most likely break the pact of silence.

• Such civilisations are calling, but we don’t recognise the signal.

Common rejoinder: This explanation doesn’t apply to signals sent with the direct purpose of communicating to nascent technological societies. Again, it is hard to see how a galactic society could enforce a global ban.

• Civilisations invariably self-destruct.

Common rejoinder: This contingency is already figured into the Drake equation (the L term, above). In any event, we have survived at least 100 years of technological adolescence, and have managed (until now) not to destroy ourselves in a nuclear or biological apocalypse.

Relatively soon we will colonise the moon and Mars, and our long-term survival will no longer rely on Earth.

• Earth is a unique planet in fostering long-lived ecosystems resulting in intelligent life.

Common rejoinder: Perhaps, but the latest studies, in particular the detections of extrasolar planets point in the opposite direction. Environments like ours appear quite common.

• We are alone in the Milky Way galaxy. Some scientists further conclude we are alone in the entire observable universe.

Common rejoinder: This conclusion flies in the face of the “principle of mediocrity,” namely the presumption, popular since the time of Copernicus, that there’s nothing special about human society or environment.

Numerous other proposed solutions and rejoinders are given in by Stephen Webb in his 2002 book, If the Universe Is Teeming with Aliens … Where is Everybody?.

Two of Drake’s key terms – fp (the fraction of stars that have planets) and ne (the average number of planets that can support life, per star that has planets) are subject to measurement.

Scientists once thought stable planetary systems and Earth-like planets were a rarity. But recent evidence suggests otherwise.

Thanks to Kepler and other projects, these two terms have been found to have reasonable values, although not quite as optimistic as Drake and his colleagues first estimated.

With every new research finding in the area of extrasolar planets and possible extraterrestrial living organisms, the mystery of Fermi’s paradox deepens.

“Where is everybody?” is a question that now carries even greater resonance.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jonathan Borwein (Jon)*

Australia faces many big challenges – in the economy, health, energy, water, climate change, infrastructure, sustainable agriculture and the preservation of our precious biodiversity.

To meet these, we need creative scientists and engineers drawn from many disciplines, and a technologically-skilled workforce.

The many world-changing advances and achievements of Australian research and development (R&D) are encouraging. Indeed, the Australian Academy of Science, of which I’m president, believes our country’s scientific potential has never been greater.

But our ability to improve this performance in the future, or even maintain it, is not assured.

Four things threaten our ongoing R&D performance and, as a consequence, our economic security and prosperity, and I’ll address each of these in turn.

1) The level of investment in R&D

Over the past decade, successive Australian governments have recognised the need to properly invest in research and innovation.

Total investment by the current government has increased by almost 43%, and is projected to amount to $9.4 billion dollars over the current financial year. This is very commendable.

It’s heartening to see Australia’s business sector is also increasing its investment – although admittedly this boost is coming off a low base compared to many other OECD nations. (Australia ranks 14th for business expenditure on R&D as a percentage of GDP).

But to remain competitive internationally we need even greater investment.

Australia spends around 2.2% of its GDP (around AU$900 per person per year) on research and development.

Iceland, the next best-ranked country, devotes 2.6% cent of GDP. Top of the list is Israel, with 4.6%, followed by Finland and Sweden, each of which spend 3.6%.

We have around 92,000 full-time equivalent researchers which, again, is only middle order. According to the OECD, in 2008 the proportion of R&D personnel in our total labour force puts Australia 16th, well short of Canada, which ranks ninth.

China has more than 1.6 million people working on research and development, a number that’s increasing rapidly. (China is ranked 33rd, with 2.5 R&D personnel per thousand in the workforce, from a total population of 1.3 billion)

Worryingly, Australia sits well within the bottom half of OECD countries (ranked 20th of 30) when it comes to the number of university graduates emerging with a science or engineering degree per capita.

These are sobering statistics.

The Australian Academy of Science therefore calls on the government to create a Sovereign Fund for Science, to secure the future prosperity of the nation.

The goal should be to increase Australia’s research and development expenditure to at least 3% of GDP by 2020.

2) International collaboration

By its very nature, science is a collaborative enterprise. It transcends generations, individual scientific disciplines and, increasingly, national boundaries. To paraphrase Sir Isaac Newton, we see further by standing on the shoulders of giants.

Australia produces only 2% of the world’s knowledge. To gain access to the other 98%, we must ensure our scientists are well-connected internationally.

Getting involved with major international projects at inception allows Australia to stay abreast of new scientific developments, to have a say in their direction, to take the knowledge further, and to apply it.

International collaborations also attract scientists from overseas to spend time in Australia, bringing us new skills and knowledge. Importantly, many return and become part of our scientific workforce.

Work arising from such collaborations often attracts great attention and gets cited more frequently. Take the recently announced kangaroo genome sequence, which garnered international media attention.

This work was done by a consortium of more than 100 researchers from Australia, the US, the UK, Germany and Japan, headed by my friend and Academy colleague Professor Marilyn Renfree. The “kangaroo” was in fact the Tammar wallaby.

Its genome is yielding many unexpected insights that may have significance for humans as well as for wallabies – for example the genes that make antibiotics in the mother’s milk to protect the tiny newborns from harmful bacteria.

There are many such examples.

We hope to bring international astronomers to Australia by winning the bid to build a giant collection of radio telescopes in the Western Australian desert. Known as the Square Kilometre Array, or SKA, this international project – which could go to either South Africa or Australia – will give astronomers huge insights into the formation and evolution of the first stars and galaxies after the Big Bang.

Barriers that have impeded the use of Australian research grants for international collaborations are being dismantled.

Today many grants and fellowships provided by the Australian Research Council, National Health and Medical Research Council and CSIRO support projects that include international partners.

Many of these linkages were initially catalysed by the federal government’s International Science Linkages (or ISL) program.

With funding of about $10 million per year, the ISL program has supported bilateral and multilateral relations with many other countries.

Regrettably, the ten-year program ended in June this year as funding was not renewed in the 2011-2012 Budget.

Put simply, it would be a grave blow if our ability to compete on the international stage were to be diminished.

I strongly urge the Federal Government to fund in its next Budget a new program to provide strategic support for Australia’s International Science Linkages.

3) Science capability in the workforce

We are a lucky nation: we have access to immense mineral wealth. But resources are finite. Even the minerals sector acknowledges that we cannot ride the current boom indefinitely.

Further, the Minerals Council of Australia warns skills shortages and structural weaknesses in the Australian economy have been masked by the boom.

And so, when the end of the mining boom comes, where will Australia be?

There is broad consensus among minds more economically astute than mine that our future prosperity will depend upon:

• a skilled workforce
• innovation
• entrepreneurship
• high productivity
• the creation of the kind of knowledge-intensive goods and services that can only result from robust research and development.

Certain skills are already in short supply in Australia.

In fact, the No More Excuses report issued by the Industry Skills Council earlier this year points to an alarming deficit in even basic skills.

According to that report, “millions of Australians have insufficient language, literacy and numeracy skills to benefit fully from training or to participate effectively at work”.

A recent project looking at the maths skills of bricklaying apprentices at a regional TAFE showed:

• 75% could not do basic arithmetic.
• 80% could not calculate the area of a rectangle, or the pay owed for working four-and-a-half hours.

Such figures are particularly worrying at a time when the demand for higher-level skills is increasing.

It’s essential we act now to ease the bottleneck and put in place measures that will create the technologically-competent workforce we need for the future.

We can, and should, be “the clever country”. But this will only happen if we place appropriate emphasis on properly educating our young people.

4) Science and maths education

Without a robust and inspiring science and maths education system, it’s impossible to create an internationally-competitive workforce.

Myriad jobs – apart from the obvious research, engineering and technology careers – require a basic understanding of science and maths.

And, as a parent, a mentor of young scientists and a passionate advocate for quality education, I know that all children are natural born scientists.

“Why?”, “How?”, and “What happens if …?” are questions asked frequently by young children, whose natural spirit of inquiry is crucial to understanding the big exciting world around them.

We need to harness this natural curiosity and nurture it with inspiring education.

Australian public expenditure on education as a percentage of GDP is just 4.2% – significantly below the OECD average of 5.4%.

A decade ago, a review of Australian science education, revealed many students were disappointed with their high school science.

Today, this disenchantment continues, as evidenced by the declining number of students choosing to study science in senior secondary school. Consider the following:

• In 1991, more than a third of Year 12 students chose to study biology. That now sits at less than a quarter.
• 23% of Year 12 students studied chemistry ten years ago, compared with 18% now.
• In the same period, physics has fallen from 21% to 14%.

While Australian students have been losing interest in science, their international peers have been taking it up with great enthusiasm.

The OECD Program for International Student Assessment (PISA) examines the scientific literacy of teenagers in 57 different countries.

In 2000, the only nations that performed better than Australia were Korea and Japan. In 2009 – the most recent figures available – Australia ranked behind Shanghai, Finland, Hong Kong, Singapore, Japan and Korea.

What happened? The Assessment indicated that the performance of other countries has improved while Australia’s has remained stationary.

Maths

Australia’s early secondary mathematical literacy scores have significantly declined over the last decade. Our Year 4 and Year 8 students ranked 14th internationally in the most recent Trends International Mathematics and Science Study, conducted in 2007.

The decline in Australia’s mathematical literacy is of grave concern because mathematics is an enabling science, without which it’s not possible to make use of other sciences – either in the lab or in the workforce.

A recent survey conducted by Science and Technology Australia and the Academy of Science showed Australians clearly value science – 80% of respondents acknowledged science education is absolutely essential or very important to the national economy.

But it also revealed some alarming holes in the basic science understanding of the average Australian.

• Three in ten believe humans were around at the time of dinosaurs.
• More than a fifth of our university graduates think that it takes just one day for the Earth to travel around the sun.
• Almost a third of Australians do not think evolution is currently occurring.
• About a quarter say human activity is not influencing the evolution of other species: a worrying statistic given the impact that human activity is having on the environment.

In other words, many of us do not understand even the most basic science.

How can we halt this slide in science and maths in our schools and attain an internationally enviable position?

Thankfully, our government is already investing significantly in school infrastructure and in rolling out a national high-speed internet network.

Last December, education ministers approved the content for new national curricula in English, history, maths and science. In coming months, they’ll be asked to sign off on the standards for these curricula. This is an important initiative and the Academy of Science applauds it.

But we also need investment in teachers, and in inspiring curriculum programs.

This is a responsibility for both the Commonwealth and the States, who must work together rather than reverting to the blame game.

Inspired (and inspiring) teachers will be the most important agents for improving educational outcomes.

We must place a much higher societal value on teachers and do everything we can to recruit some of our brightest and best into teaching.

We must support these educators with the best tools and resources available and provide them with stimulating opportunities for ongoing training.

I agree with Prime Minister Julia Gillard that science is one of the fundamental platforms upon which our conception of a modern advanced society is based.

I agree with the prime minister that we live in a crucial time for science in Australia and around the world.

In fact, I could not agree more.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Suzanne Coryter *