Author: IMC

Uncertainty exists – but that’s no excuse for a lack of action.

These are painful times for those hoping to see an international consensus and substantive action on global warming.

In the US, Republican presidential front-runner Mitt Romney said in June 2011: “The world is getting warmer” and “humans have contributed” but in October 2011 he backtracked to: “My view is that we don’t know what’s causing climate change on this planet.”

His Republican challenger Rick Santorum added: “We have learned to be sceptical of ‘scientific’ claims, particularly those at war with our common sense” and Rick Perry, who suspended his campaign to become the Republican presidential candidate last month, stated flatly: “It’s all one contrived phony mess that is falling apart under its own weight.”

Meanwhile, the scientific consensus has moved in the opposite direction. In a study published in October 2011, 97% of climate scientists surveyed agreed global temperatures have risen over the past 100 years. Only 5% disagreed that human activity is a significant cause of global warming.

The study concluded in the following way: “We found disagreement over the future effects of climate change, but not over the existence of anthropogenic global warming.

“Indeed, it is possible that the growing public perception of scientific disagreement over the existence of anthropocentric warming, which was stimulated by press accounts of [the UK’s] ”Climategate“ is actually a misperception of the normal range of disagreements that may persist within a broad scientific consensus.”

More progress has been made in Europe, where the EU has established targets to reduce emissions by 20% (from 1990 levels) by 2020. The UK, which has been beset by similar denial movements, was nonetheless able to establish, as a legally binding target, an 80% reduction by 2050 and is a world leader on abatement.

In Australia, any prospect for consensus was lost when Tony Abbott used opposition to the Labor government’s proposed carbon market to replace Malcolm Turnbull as leader of the Federal Opposition in late 2009.

It used to be possible to hear right-wing politicians in Australia or the USA echo the Democratic congressman Henry Waxman who said last year:

“If my doctor told me I had cancer, I wouldn’t scour the country to find someone to tell me that I don’t need to worry about it.”

But such rationality has largely left the debate in both the US and Oz. In Australia, a reformulated carbon tax policy was enacted in November only after a highly partisan debate.

In Canada, the debate is a tad more balanced. The centre-right Liberal government in British Columbia passed the first carbon tax in North America in 2008, but the governing Federal Conservative party now offers a reliable “anti-Kyoto” partnership with Washington.

Overviews of the evidence for global warming, together with responses to common questions, are available from various sources, including:

• Seven Answers to Climate Contrarian Nonsense, in Scientific American
• Climate change: A Guide for the Perplexed, in New Scientist
• Cooling the Warming Debate: Major New Analysis Confirms That Global Warming Is Real, in Science Daily
• Remind me again: how does climate change work?, on The Conversation

It should be acknowledged in these analyses that all projections are based on mathematical models with a significant level of uncertainty regarding highly complex and only partially understood systems.

As 2011 Australian Nobel-Prize-winner Brian Schmidt explained while addressing a National Forum on Mathematical Education:

“Climate models have uncertainty and the earth has natural variation … which not only varies year to year, but correlates decade to decade and even century to century. It is really hard to design a figure that shows this in a fair way — our brain cannot deal with the correlations easily.

“But we do have mathematical ways of dealing with this problem. The Australian academy reports currently indicate that the models with the effects of CO₂ are with 90% statistical certainty better at explaining the data than those without.

“Most of us who work with uncertainty know that 90% statistical uncertainty cannot be easily shown within a figure — it is too hard to see …”

“ … Since predicting the exact effects of climate change is not yet possible, we have to live with uncertainty and take the consensus view that warming can cover a wide range of possibilities, and that the view might change as we learn more.”

But uncertainty is no excuse for inaction. The proposed counter-measures (e.g. infrastructure renewal and modernisation, large-scale solar and wind power, better soil remediation and water management, not to mention carbon taxation) are affordable and most can be justified on their own merits, while the worst-case scenario — do nothing while the oceans rise and the climate changes wildly — is unthinkable.

Some in the first world protest that any green energy efforts are dwarfed by expanding energy consumption in China and elsewhere. Sure, China’s future energy needs are prodigious, but China also now leads the world in green energy investment.

By blaiming others and focusing the debate on the level of human responsibility for warming and about the accuracy of predictions, the deniers have managed to derail long-term action in favour of short-term economic policies.

Who in the scientific community is promoting the denial of global warming? As it turns out, the leading figures in this movement have ties to conservative research institutes funded mostly by large corporations, and have a history of opposing the scientific consensus on issues such as tobacco and acid rain.

What’s more, those who lead the global warming denial movement – along with creationists, intelligent design writers and the “mathematicians” who flood our email inboxes with claims that pi is rational or other similar nonsense – are operating well outside the established boundaries of peer-reviewed science.

Austrian-born American physicist Fred Singer, arguably the leading figure of the denial movement, has only six peer-reviewed publications in the climate science field, and none since 1997.

After all, when issues such as these are “debated” in any setting other than a peer-reviewed journal or conference, one must ask: “If the author really has a solid argument, why isn’t he or she back in the office furiously writing up this material for submission to a leading journal, thereby assuring worldwide fame and glory, not to mention influence?”

In most cases, those who attempt to grab public attention through other means are themselves aware they are short-circuiting the normal process, and that they do not yet have the sort of solid data and airtight arguments that could withstand the withering scrutiny of scientific peer review.

When they press their views in public to a populace that does not understand how the scientific enterprise operates, they are being disingenuous.

With regards to claims scientists are engaged in a “conspiracy” to hide the “truth” on an issue such as global warming or evolution, one should ask how a secret “conspiracy” could be maintained in a worldwide, multicultural community of hundreds of thousands of competitive researchers.

As Benjamin Franklin wrote in his Poor Richard’s Almanac: “Three can keep a secret, provided two of them are dead.” Or as one of your present authors quipped, tongue-in-cheek, in response to a state legislator who was skeptical of evolution: “You have no idea how humiliating this is to me — there is a secret conspiracy among leading scientists, but no-one deemed me important enough to be included!”

There’s another way to think about such claims: we have tens-of-thousands of senior scientists in their late-fifties or early-sixties who have seen their retirement savings decimated by the recent stock market plunge. These are scientists who now wonder if the day will ever come when they are financially well-off-enough to do their research without the constant stress and distraction of applying for grants (the majority of which are never funded).

All one of these scientists has to do to garner both worldwide fame and considerable fortune (through book contracts, the lecture circuit and TV deals) is to call a news conference and expose “the truth”. So why isn’t this happening?

The system of peer-reviewed journals and conferences sponsored by major professional societies is the only proper forum for the presentation and debate of new ideas, in any field of science or mathematics.

It has been stunningly successful: errors have been uncovered, fraud has been rooted out and bogus scientific claims (such as the 1903 N-ray claim, the 1989 cold fusion claim, and the more-recent assertion of an autism-vaccination link) have been debunked.

This all occurs with a level of reliability and at a speed that is hard to imagine in other human endeavours. Those who attempt to short-circuit this system are doing potentially irreparable harm to the integrity of the system.

They may enrich themselves or their friends, but they are doing grievous damage to society at large.

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

*Credit for article given to Jonathan Borwein (Jon) and David H. Bailey*

 

These are painful times for those hoping to see an international consensus and substantive action on global warming.

In the US, Republican presidential front-runner Mitt Romney said in June 2011: “The world is getting warmer” and “humans have contributed” but in October 2011 he backtracked to: “My view is that we don’t know what’s causing climate change on this planet.”

His Republican challenger Rick Santorum added: “We have learned to be sceptical of ‘scientific’ claims, particularly those at war with our common sense” and Rick Perry, who suspended his campaign to become the Republican presidential candidate last month, stated flatly: “It’s all one contrived phony mess that is falling apart under its own weight.”

Meanwhile, the scientific consensus has moved in the opposite direction. In a study published in October 2011, 97% of climate scientists surveyed agreed global temperatures have risen over the past 100 years. Only 5% disagreed that human activity is a significant cause of global warming.

The study concluded in the following way: “We found disagreement over the future effects of climate change, but not over the existence of anthropogenic global warming.

“Indeed, it is possible that the growing public perception of scientific disagreement over the existence of anthropocentric warming, which was stimulated by press accounts of [the UK’s] ”Climategate“ is actually a misperception of the normal range of disagreements that may persist within a broad scientific consensus.”

More progress has been made in Europe, where the EU has established targets to reduce emissions by 20% (from 1990 levels) by 2020. The UK, which has been beset by similar denial movements, was nonetheless able to establish, as a legally binding target, an 80% reduction by 2050 and is a world leader on abatement.

In Australia, any prospect for consensus was lost when Tony Abbott used opposition to the Labor government’s proposed carbon market to replace Malcolm Turnbull as leader of the Federal Opposition in late 2009.

It used to be possible to hear right-wing politicians in Australia or the USA echo the Democratic congressman Henry Waxman who said last year:

“If my doctor told me I had cancer, I wouldn’t scour the country to find someone to tell me that I don’t need to worry about it.”

But such rationality has largely left the debate in both the US and Oz. In Australia, a reformulated carbon tax policy was enacted in November only after a highly partisan debate.

In Canada, the debate is a tad more balanced. The centre-right Liberal government in British Columbia passed the first carbon tax in North America in 2008, but the governing Federal Conservative party now offers a reliable “anti-Kyoto” partnership with Washington.

Overviews of the evidence for global warming, together with responses to common questions, are available from various sources, including:

• Seven Answers to Climate Contrarian Nonsense, in Scientific American
• Climate change: A Guide for the Perplexed, in New Scientist
• Cooling the Warming Debate: Major New Analysis Confirms That Global Warming Is Real, in Science Daily
• Remind me again: how does climate change work? on The Conversation

It should be acknowledged in these analyses that all projections are based on mathematical models with a significant level of uncertainty regarding highly complex and only partially understood systems.

As 2011 Australian Nobel-Prize-winner Brian Schmidt explained while addressing a National Forum on Mathematical Education:

“Climate models have uncertainty and the earth has natural variation … which not only varies year to year, but correlates decade to decade and even century to century. It is really hard to design a figure that shows this in a fair way — our brain cannot deal with the correlations easily.

“But we do have mathematical ways of dealing with this problem. The Australian academy reports currently indicate that the models with the effects of CO₂ are with 90% statistical certainty better at explaining the data than those without.

“Most of us who work with uncertainty know that 90% statistical uncertainty cannot be easily shown within a figure — it is too hard to see …”

“ … Since predicting the exact effects of climate change is not yet possible, we have to live with uncertainty and take the consensus view that warming can cover a wide range of possibilities, and that the view might change as we learn more.”

But uncertainty is no excuse for inaction. The proposed counter-measures (e.g. infrastructure renewal and modernisation, large-scale solar and wind power, better soil remediation and water management, not to mention carbon taxation) are affordable and most can be justified on their own merits, while the worst-case scenario — do nothing while the oceans rise and the climate changes wildly — is unthinkable.

Some in the first world protest that any green energy efforts are dwarfed by expanding energy consumption in China and elsewhere. Sure, China’s future energy needs are prodigious, but China also now leads the world in green energy investment.

By blaiming others and focusing the debate on the level of human responsibility for warming and about the accuracy of predictions, the deniers have managed to derail long-term action in favour of short-term economic policies.

Who in the scientific community is promoting the denial of global warming? As it turns out, the leading figures in this movement have ties to conservative research institutes funded mostly by large corporations, and have a history of opposing the scientific consensus on issues such as tobacco and acid rain.

What’s more, those who lead the global warming denial movement – along with creationists, intelligent design writers and the “mathematicians” who flood our email inboxes with claims that pi is rational or other similar nonsense – are operating well outside the established boundaries of peer-reviewed science.

Austrian-born American physicist Fred Singer, arguably the leading figure of the denial movement, has only six peer-reviewed publications in the climate science field, and none since 1997.

After all, when issues such as these are “debated” in any setting other than a peer-reviewed journal or conference, one must ask: “If the author really has a solid argument, why isn’t he or she back in the office furiously writing up this material for submission to a leading journal, thereby assuring worldwide fame and glory, not to mention influence?”

In most cases, those who attempt to grab public attention through other means are themselves aware they are short-circuiting the normal process, and that they do not yet have the sort of solid data and airtight arguments that could withstand the withering scrutiny of scientific peer review.

When they press their views in public to a populace that does not understand how the scientific enterprise operates, they are being disingenuous.

With regards to claims scientists are engaged in a “conspiracy” to hide the “truth” on an issue such as global warming or evolution, one should ask how a secret “conspiracy” could be maintained in a worldwide, multicultural community of hundreds of thousands of competitive researchers.

As Benjamin Franklin wrote in his Poor Richard’s Almanac: “Three can keep a secret, provided two of them are dead.” Or as one of your present authors quipped, tongue-in-cheek, in response to a state legislator who was skeptical of evolution: “You have no idea how humiliating this is to me — there is a secret conspiracy among leading scientists, but no-one deemed me important enough to be included!”

There’s another way to think about such claims: we have tens-of-thousands of senior scientists in their late-fifties or early-sixties who have seen their retirement savings decimated by the recent stock market plunge. These are scientists who now wonder if the day will ever come when they are financially well-off-enough to do their research without the constant stress and distraction of applying for grants (the majority of which are never funded).

All one of these scientists has to do to garner both worldwide fame and considerable fortune (through book contracts, the lecture circuit and TV deals) is to call a news conference and expose “the truth”. So why isn’t this happening?

The system of peer-reviewed journals and conferences sponsored by major professional societies is the only proper forum for the presentation and debate of new ideas, in any field of science or mathematics.

It has been stunningly successful: errors have been uncovered, fraud has been rooted out and bogus scientific claims (such as the 1903 N-ray claim, the 1989 cold fusion claim, and the more-recent assertion of an autism-vaccination link) have been debunked.

This all occurs with a level of reliability and at a speed that is hard to imagine in other human endeavours. Those who attempt to short-circuit this system are doing potentially irreparable harm to the integrity of the system.

They may enrich themselves or their friends, but they are doing grievous damage to society at large.

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

 

Richard Braatz. Photo: Dominick Reuter

Trial-and-error experimentation underlies many biomedical innovations. This classic method — define a problem, test a proposed solution, learn from failure and try again — is the main route by which scientists discover new biomaterials and drugs today. This approach is also used to design ways of manufacturing these new materials, but the process is immensely time-consuming, producing a successful therapeutic product and its manufacturing process only after years of experiments, at considerable expense.

Richard Braatz, the Edwin R. Gilliland Professor of Chemical Engineering at MIT, applies mathematics to streamline the development of pharmaceuticals. Trained as an applied mathematician, Braatz is developing mathematical models to help scientists quickly and accurately design processes for manufacturing drug compounds with desired characteristics. Through mathematical simulations, Braatz has designed a system that significantly speeds the design of drug-manufacturing processes; he is now looking to apply the same mathematical approach to designing new biomaterials and nanoscale devices.

“Nanotechnology is very heavily experimental,” Braatz says. “There are researchers who do computations to gain insights into the physics or chemistry of nanoscale systems, but do not apply these computations for their design or manufacture. I want to push systematic design methods to the nanoscale, and to other areas where such methods aren’t really developed yet, such as biomaterials.”

From farm to formulas

Braatz’s own academic path was anything but systematic. He spent most of his childhood on an Oregon farm owned by his grandfather. Braatz says he absorbed an engineer’s way of thinking early on from his father, an electrician, by examining his father’s handiwork on the farm and reading his electrical manuals.

Braatz also developed a serious work ethic. From the age of 10, he awoke early every morning — even on school days — to work on the farm. In high school, he picked up a night job at the local newspaper, processing and delivering thousands of newspapers to stores and the post office, sometimes until just before dawn.

After graduating from high school in 1984, Braatz headed to Alaska for the summer. A neighbour had told him that work paid well up north, and Braatz took a job at a fish-processing facility, driving forklifts and hauling 100-pound bags of fishmeal 16 hours a day. He returned each summer for four years, eventually working his way up to plant operator, saving enough money each summer to pay for the next year’s tuition at Oregon State University.

As an undergraduate, Braatz first planned to major in electrical engineering. But finding the introductory coursework unstimulating — given the knowledge he’d absorbed from his father — he cast about for another major.

“There was no Internet back then, so you couldn’t Google; web searches didn’t exist,” Braatz says. “So I went to the library and opened an encyclopedia, and said, ‘OK, what other engineering [is] there?’”

Chemical engineering caught his eye; he had always liked and excelled at chemistry in high school. While pursuing a degree in chemical engineering, Braatz filled the rest of his schedule with courses in mathematics.

After graduation, Braatz went on to the California Institute of Technology, where he earned both a master’s and a PhD in chemical engineering. In addition to his research, Braatz took numerous math and math-heavy courses in electrical engineering, applied mechanics, chemical engineering and chemistry. The combination of real applications and mathematical theory revealed a field of study Braatz had not previously considered: applied mathematics.

“This training was a very good background for learning how to derive mathematical solutions to research problems,” Braatz says.

A systems approach

Soon after receiving his PhD, Braatz accepted an assistant professorship at the University of Illinois at Urbana-Champaign (UIUC). There, as an applied mathematician, he worked with researchers to tackle problems in a variety of fields: computer science, materials science, and electrical, chemical and mechanical engineering.

He spent eight years on a project spurred by a talk he attended at UIUC. In that talk, a representative of Merck described a major challenge in the pharmaceutical industry: controlling the size of crystals in the manufacture of any given drug. (The size and consistency of crystals determine, in part, a drug’s properties and overall efficacy.)

Braatz learned that while drug-manufacturing machinery was often monitored by sensors, much of the resulting data went unanalysed. He pored over the sensors’ data, and developed mathematical models to gain an understanding of what the sensors reveal about each aspect of the drug-crystallization process. Over the years, his team devised an integrated series of algorithms that combined efficiently designed experiments with mathematical models to yield a desired crystal size from a given drug solution. They worked the algorithms into a system that automatically adjusts settings at each phase of the manufacturing process to produce an optimal crystal size, based on a “recipe” given by the algorithms.

“Sometimes the recipes are very weird,” Braatz says. “It might be a strange path you have to follow to manufacture the right crystals.”

The automated system, which has since been adopted by Merck and other pharmaceutical companies, provides a big improvement in efficiency, Braatz says, avoiding the time-consuming trial-and-error approach many drug manufacturers had relied on to design a crystallization process for a new drug.

In 2010, Braatz moved to MIT, where he is exploring mathematical applications in nanotechnology and tissue engineering — in particular, models to help design new drug-releasing materials. Such materials have the potential to deliver controlled, continuous therapies, but designing them currently takes years of trial-and-error experiments.

Braatz’s group is designing mathematical models to give researchers instructions, for example, on how to design materials that locally release drugs into a body’s cells at a desired rate. Braatz says approaching such a problem from a systematic perspective could potentially save years of time in the development of a biomedical material of high efficacy.

“Anything is a win if you could reduce those experiments from 10 years to several years,” Braatz says. “We’re talking hundreds of millions, billions of dollars. And the effect on people’s lives, you can’t put a price tag on that.”

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

Credit of the article given to Jennifer Chu, Massachusetts Institute of Technology

There’s far more to the popular maths puzzle than putting numbers in a box.

Last month, a team led by Gary McGuire from University College Dublin in Ireland made an announcement: they had proven you can’t have a solvable Sudoku puzzle with less than 17 numbers already filled in.

Unlike most mathematical announcements, this was quickly picked up by the popular scientific media. Within a few days, the new finding had been announced in Nature and other outlets.

So where did this problem come from and why is its resolution interesting?

As you probably know, the aim of a Sudoku puzzle is to complete a partially-filled nine-by-nine grid of numbers. There are some guidelines: the numbers one to nine must appear exactly once each in every row, column and three-by-three sub-grid.

As with a crossword, a valid Sudoku puzzle must have a unique solution. There’s only one way to go from the initial configuration (with some numbers already filled in) to a completed grid.

Newspapers often grade their puzzles as easy, medium or hard, which will depend on how easy it is at every stage of solving the puzzle to fill in the “next” number. While a puzzle with a huge number of initial clues will usually be easy, it is not necessarily the case that a puzzle with few initial clues is difficult.

Reckon you can complete a 17-clue Sudoku puzzle? (answer below) Gordon Royle

When Sudoku-mania swept the globe in the mid-2000s, many mathematicians, programmers and computer scientists – amateur and professional – started to investigate Sudoku itself. They were less interested in solving individual puzzles, and more focused on asking and answering mathematical and/or computational questions about the entire universe of Sudoku puzzles and solutions.

As a mathematician specialising in the area of combinatorics (which can very loosely be defined as the mathematics of counting configurations and patterns), I was drawn to combinatorial questions about Sudoku.

I was particularly interested in the question of the smallest number of clues possible in a valid puzzle (that is, a puzzle with a unique solution).

In early 2005, I found a handful of 17-clue puzzles on a long-since forgotten Japanese-language website. By slightly altering these initial puzzles, I found a few more, then more, and gradually built up a “library” of 17-clue Sudoku puzzles which I made available online at the time.

Other people started to send me their 17-clue puzzles and I added any new ones to the list until, after a few years, I had collected more than 49,000 different 17-clue Sudoku puzzles.

By this time, new ones were few and far between, and I was convinced we had found almost all of the 17-clue puzzles. I was also convinced there was no 16-clue puzzle. I thought that demonstrating this would either require some new theoretical insight or clever programming combined with massive computational power, or both.

Either way, I thought proving the non-existence of a 16-clue puzzle was likely to be too difficult a challenge.

They key to McGuire’s approach was to tackle the problem indirectly. The total number of completed puzzles (that is, completely filled-in grids) is astronomical – 5,472,730,538 – and trying to test each of these to see if any choice of 16 cells from the completed grid forms a valid puzzle is far too time-consuming.

Instead, McGuire and colleagues used a different, indirect approach.

An “unavoidable set” in a completed Sudoku grid is a subset of the clues whose entries can be rearranged to leave another valid completed Sudoku grid. For a puzzle to be uniquely completable, it must contain at least one entry from every unavoidable set.

If a completed grid contains the ten-clue configuration in the left picture, then any valid Sudoku puzzle must contain at least one of those ten clues. If it did not, then in any completed puzzle, those ten positions could either contain the left-hand configuration or the right-hand configuration and so the solution would not be unique.

Gordon Royle

While finding all the unavoidable sets in a given grid is difficult, it’s only necessary to find enough unavoidable sets to show that no 16 clues can “hit” them all. In the process of resolving this question, McGuire’s team developed new techniques for solving the “hitting set” problem.

It’s a problem that has many other applications – any situation in which a small set of resources must be allocated while still ensuring that all needs are met by at least one of the selected resources (i.e. “hit”) can be modelled as a hitting set problem.

Once the theory and software was in place, it was then a matter of running the programs for each of the 5.5 billion completed grids. As you can imagine, this required substantial computing power.

After 7 million core-CPU hours on a supercomputer (the equivalent of a single computer running for 7 million hours) and a year of actual elapsed time, the result was announced a few weeks ago, on New Year’s Day.

So is it correct?

The results of any huge computation should be evaluated with some caution, if not outright suspicion, especially when the answer is simply “no, doesn’t exist”, because there are many possible sources of error.

But in this case, I feel the result is far more likely to be correct than otherwise, and I expect it to be independently-verified before too long. In addition, McGuire’s team built on many different ideas, discussions and computer programs that were thrashed out between interested contributors to various online forums devoted to the mathematics of Sudoku. In this respect, many of the basic components of their work have already been thoroughly tested.

Solution to the 17-clue Sudoku puzzle, above. Gordon Royle

And so back to the question: why is the resolution of this problem interesting? And is it important?

Certainly, knowing that the smallest Sudoku puzzles have 17 clues is not in itself important. But the immense popularity of Sudoku meant that this question was popularised in a way that many similar questions have never been, and so it took on a special role as a “challenge question” testing the limits of human knowledge.

The school students to whom I often give outreach talks have no real concept of the limitations of computers and mathematics. In my past talks, these students were almost always astonished to know that the answer to such a simple question was just not known.

And now, in my future outreach talks, I will describe how online collaboration, theoretical development and significant computational power were combined to solve this problem, and how this process promises to play an increasing role in the future development of mathematics.

 

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

*Credit for article given to Gordon Royle*

 

A team of researchers, led by Daniel Levitin of McGill University, has found after analysing over two thousand pieces of classical music that span four hundred years of history, that virtually all of them follow a one-over-f (1/f) power distribution equation. He and his team have published the results of their work in the Proceedings of the National Academy of Sciences.

One-over-f equations describe the relative frequency of things that happen over time and can be used to describe such naturally occurring events as annual river flooding or the beating of a human heart. They have been used to describe the way pitch is used in music as well, but until now, no one has thought to test the idea that they could be used to describe the rhythm of the music too.

To find out if this is the case, Levitin and his team analysed (by measuring note length line by line) close to 2000 pieces of classical music from a wide group of noted composers. In so doing, they found that virtually every piece studied conformed to the power law. They also found that by adding another variable to the equation, called a beta, which was used to describe just how predictable a given piece was compared to other pieces, they could solve for beta and find a unique number for each composer.

After looking at the results as a whole, they found that works written by some classical composers were far more predictable than others, and that certain genres in general were more predictable than others too. Beethoven was the most predictable of the group studied, while Mozart was the least of the bunch. And symphonies are generally far more predictable than Ragtimes with other types falling somewhere in-between. In solving for beta, the team discovered that they had inadvertently developed a means for calculating a composer’s unique individual rhythm signature. In speaking with the university news group at McGill, Levitin said, “this was one of the most unanticipated and exciting findings of our research.”

Another interesting aspect of the research is that because the patterns are based on the power law, the music the team studied shares the same sorts of patterns as fractals, i.e. elements in the rhythm that occur the second most often happen only half as often, the third, just a third as often and so forth. Thus, it’s not difficult to imagine music in a fractal patterns that are unique to individual composers.

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

Credit of the article given to Bob Yirka , Phys.org

Last month, a team led by Gary McGuire from University College Dublin in Ireland made an announcement: they had proven you can’t have a solvable Sudoku puzzle with less than 17 numbers already filled in.

Unlike most mathematical announcements, this was quickly picked up by the popular scientific media. Within a few days, the new finding had been announced in Nature and other outlets.

So where did this problem come from and why is its resolution interesting.

As you probably know, the aim of a Sudoku puzzle is to complete a partially-filled nine-by-nine grid of numbers. There are some guidelines: the numbers one to nine must appear exactly once each in every row, column and three-by-three sub-grid.

As with a crossword, a valid Sudoku puzzle must have a unique solution. There’s only one way to go from the initial configuration (with some numbers already filled in) to a completed grid.

Newspapers often grade their puzzles as easy, medium or hard, which will depend on how easy it is at every stage of solving the puzzle to fill in the “next” number. While a puzzle with a huge number of initial clues will usually be easy, it is not necessarily the case that a puzzle with few initial clues is difficult.

Reckon you can complete a 17-clue Sudoku puzzle? (answer below) Gordon Royle

When Sudoku-mania swept the globe in the mid-2000s, many mathematicians, programmers and computer scientists – amateur and professional – started to investigate Sudoku itself. They were less interested in solving individual puzzles, and more focused on asking and answering mathematical and/or computational questions about the entire universe of Sudoku puzzles and solutions.

As a mathematician specialising in the area of combinatorics (which can very loosely be defined as the mathematics of counting configurations and patterns), I was drawn to combinatorial questions about Sudoku.

I was particularly interested in the question of the smallest number of clues possible in a valid puzzle (that is, a puzzle with a unique solution).

In early 2005, I found a handful of 17-clue puzzles on a long-since forgotten Japanese-language website. By slightly altering these initial puzzles, I found a few more, then more, and gradually built up a “library” of 17-clue Sudoku puzzles which I made available online at the time.

Other people started to send me their 17-clue puzzles and I added any new ones to the list until, after a few years, I had collected more than 49,000 different 17-clue Sudoku puzzles.

By this time, new ones were few and far between, and I was convinced we had found almost all of the 17-clue puzzles. I was also convinced there was no 16-clue puzzle. I thought that demonstrating this would either require some new theoretical insight or clever programming combined with massive computational power, or both.

Either way, I thought proving the non-existence of a 16-clue puzzle was likely to be too difficult a challenge.

They key to McGuire’s approach was to tackle the problem indirectly. The total number of completed puzzles (that is, completely filled-in grids) is astronomical – 5,472,730,538 – and trying to test each of these to see if any choice of 16 cells from the completed grid forms a valid puzzle is far too time-consuming.

Instead, McGuire and colleagues used a different, indirect approach.

An “unavoidable set” in a completed Sudoku grid is a subset of the clues whose entries can be rearranged to leave another valid completed Sudoku grid. For a puzzle to be uniquely completable, it must contain at least one entry from every unavoidable set.

See the picture below to see what I mean.

If a completed grid contains the ten-clue configuration in the left picture, then any valid Sudoku puzzle must contain at least one of those ten clues. If it did not, then in any completed puzzle, those ten positions could either contain the left-hand configuration or the right-hand configuration and so the solution would not be unique.

Gordon Royle

While finding all the unavoidable sets in a given grid is difficult, it’s only necessary to find enough unavoidable sets to show that no 16 clues can “hit” them all. In the process of resolving this question, McGuire’s team developed new techniques for solving the “hitting set” problem.

It’s a problem that has many other applications – any situation in which a small set of resources must be allocated while still ensuring that all needs are met by at least one of the selected resources (i.e. “hit”) can be modelled as a hitting set problem.

Once the theory and software was in place, it was then a matter of running the programs for each of the 5.5 billion completed grids. As you can imagine, this required substantial computing power.

After 7 million core-CPU hours on a supercomputer (the equivalent of a single computer running for 7 million hours) and a year of actual elapsed time, the result was announced a few weeks ago, on New Year’s Day.

So is it correct?

The results of any huge computation should be evaluated with some caution, if not outright suspicion, especially when the answer is simply “no, doesn’t exist”, because there are many possible sources of error.

But in this case, I feel the result is far more likely to be correct than otherwise, and I expect it to be independently-verified before too long. In addition, McGuire’s team built on many different ideas, discussions and computer programs that were thrashed out between interested contributors to various online forums devoted to the mathematics of Sudoku. In this respect, many of the basic components of their work have already been thoroughly tested.

Solution to the 17-clue Sudoku puzzle, above. Gordon Royle

And so back to the question: why is the resolution of this problem interesting? And is it important?

Certainly, knowing that the smallest Sudoku puzzles have 17 clues is not in itself important. But the immense popularity of Sudoku meant that this question was popularised in a way that many similar questions have never been, and so it took on a special role as a “challenge question” testing the limits of human knowledge.

The school students to whom I often give outreach talks have no real concept of the limitations of computers and mathematics. In my past talks, these students were almost always astonished to know that the answer to such a simple question was just not known.

And now, in my future outreach talks, I will describe how online collaboration, theoretical development and significant computational power were combined to solve this problem, and how this process promises to play an increasing role in the future development of mathematics.

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

Credit of the article given to Gordon Royle, The University of Western Australia

 

Queen Mary, University of London has developed a new educational resource for teachers to help students use amazing magic tricks to learn about maths.

The web resource (www.mathematicalmagic.com), which includes the ‘Manual for Mathematical Magic’ and a series of interactive videos, was led by Queen Mary’s Professor Peter McOwan with the help of the College’s resident stand-up comedian Matt Parker and semi-professional magician and maths teacher Jason Davison.

Professor McOwan said: “It was great fun to be able to work with Matt and Jason on these new videos, showing how maths and magic can fuse together education and entertainment.

“While we explain most of the tricks, we have deliberately included a few where we leave the viewer to figure it out. It’s all just maths, but we wanted to leave some magical mystery in there too!”

Mr Davison said: “Using the fun of magic makes this a really great way to learn some of the fundamentals of maths, the links between maths and magic are strong and a brilliant way to bring excitement into the classroom.”

The educational website builds on a bank of teaching resources led by Professor McOwan, including Illusioneering (www.Illusioneering.org), a website which gives students and teachers the platform to explore science and engineering through a range of magic tricks; and cs4fn (www.cs4fn.org), a web and magazine initiative putting the fun into computer science.

The production of the videos for mathematicalmagic.com was possible due to funding from the UK National Higher Education STEM programme. The Programme supports Higher Education Institutions in the exploration of new approaches to recruiting students and delivering programmes of study within the Science, Technology, Engineering and Mathematics (STEM) disciplines.

Institute of Mathematics and its Applications project manager in HE STEM, Makhan Singh, said: “Once again we see the power of making education fun! Peter McOwan brings alive the mystery of magic whilst showcasing the power of mathematics – sheer brilliance! It’s entertaining, amusing, educational and most definitely relevant in today’s classrooms; well done!”.

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

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

 

We can’t tame the oceans, but modelling can help us better understand them.

Last year will go on record as one of significant natural disasters both in Australia and overseas. Indeed, the flooding of the Brisbane River in January is still making news as the Queensland floods inquiry investigates whether water released from Wivenhoe Dam was responsible. Water modelling is being used to answer the question: could modelling have avoided the problem in the first place?

This natural disaster – as well as the Japanese tsunami in March and the flooding in Bangkok in October – involved the movement of fluids: water, mud or both. And all had a human cost – displaced persons, the spread of disease, disrupted transport, disrupted businesses, broken infrastructure and damaged or destroyed homes. With the planet now housing 7 billion people, the potential for adverse humanitarian effects from natural disasters is greater than ever.

Here in CSIRO’s division of Mathematical and Information Sciences, we’ve been working with various government agencies (in Australia and China) to model the flow of flood waters and the debris they carry. Governments are starting to realise just how powerful computational modelling is for understanding and analysing natural disasters and how to plan for them.

This power is based on two things – the power of computers and the power of the algorithms (computer processing steps) that run on the computers.

In recent years, the huge increase in computer power and speed coupled with advances in algorithm development has allowed mathematical modellers like us to make large strides in our research.

These advances have enabled us to model millions, even billions of water particles, allowing us to more accurately predict the effects of natural and man-made fluid flows, such as tsunamis, dam breaks, floods, mudslides, coastal inundation and storm surges.

So how does it work?

Well, fluids such as sea water can be represented as billions of particles moving around, filling spaces, flowing downwards, interacting with objects and in turn being interacted upon. Or they can be visualised as a mesh of the fluids’ shape.

Let’s consider a tsunami such as the one that struck the Japanese coast in March of last year. When a tsunami first emerges as a result of an earthquake, shallow water modelling techniques give us the most accurate view of the wave’s formation and early movement.

Mesh modelling of water being poured into a glass.

Once the wave is closer to the coast however, techniques known collectively as smoothed particle hydrodynamics (SPH) are better at predicting how the wave interacts with local geography. We’ve created models of a hypothetical tsunami off the northern Californian coastline to test this.

A dam break can also be modelled using SPH. The modelling shows how fast the water moves at certain times and in certain places, where water “overtops” hills and how quickly it reaches towns or infrastructure such as power stations.

This can help town planners to build mitigating structures and emergency services to co-ordinate an efficient response. Our models have been validated using historical data from a real dam that broke in California in 1928 – the St. Francis Dam.

Having established that our modelling techniques work better than others, we can apply them to a range of what-if situations.

In collaboration with the Satellite Surveying and Mapping Application Centre in China we tested scenarios such as the hypothetical collapse of the massive Geheyan Dam in China.

We combined our modelling techniques with digital terrain models to get a realistic picture of how such a disaster would unfold and, therefore, what actions could mitigate it.

Our experience in developing and using these techniques over several decades allows us to combine them in unique ways for each situation.

We’ve modelled fluids not just for natural disaster planning but also movie special effects, hot metal production, water sports and even something as everyday as insurance.

Insurance companies have been looking to us for help to understand how natural disasters unfold. They cop a lot of media flak after disasters for not covering people affected. People living in low-lying areas have traditionally had difficulty accessing flood insurance and find themselves unprotected in flood situations.

Insurers are starting to realise that the modelling of geophysical flows can provide a basis for predicting localised risk of damage due to flooding and make flood coverage a viable business proposition. One Australian insurance company has been working with us to quantify risk of inundation in particular areas.

Using data from the 1974 Brisbane floods, the floods of last year and fluid modelling data, an insurance company can reliably assess residents’ exposure to particular risks and thereby determine suitable premiums.

With evidence-based tools such as fluid modelling in their arsenal, decision-makers are better prepared for the future. That may be a future of more frequent natural disasters, a future with a more-densely-populated planet, or, more likely, a combination of both.

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

*Credit for article given to Mahesh Prakash*

Last year will go on record as one of significant natural disasters both in Australia and overseas. Indeed, the flooding of the Brisbane River in January is still making news as the Queensland floods inquiry investigates whether water released from Wivenhoe Dam was responsible. Water modelling is being used to answer the question: could modelling have avoided the problem in the first place?

This natural disaster – as well as the Japanese tsunami in March and the flooding in Bangkok in October – involved the movement of fluids: water, mud or both. And all had a human cost – displaced persons, the spread of disease, disrupted transport, disrupted businesses, broken infrastructure and damaged or destroyed homes. With the planet now housing 7 billion people, the potential for adverse humanitarian effects from natural disasters is greater than ever.

Here in CSIRO’s division of Mathematical and Information Sciences, we’ve been working with various government agencies (in Australia and China) to model the flow of flood waters and the debris they carry. Governments are starting to realise just how powerful computational modelling is for understanding and analysing natural disasters and how to plan for them.

This power is based on two things – the power of computers and the power of the algorithms (computer processing steps) that run on the computers.

In recent years, the huge increase in computer power and speed coupled with advances in algorithm development has allowed mathematical modellers like us to make large strides in our research.

These advances have enabled us to model millions, even billions of water particles, allowing us to more accurately predict the effects of natural and man-made fluid flows, such as tsunamis, dam breaks, floods, mudslides, coastal inundation and storm surges.

So how does it work?

Well, fluids such as sea water can be represented as billions of particles moving around, filling spaces, flowing downwards, interacting with objects and in turn being interacted upon. Or they can be visualised as a mesh of the fluids’ shape.

Let’s consider a tsunami such as the one that struck the Japanese coast in March of last year. When a tsunami first emerges as a result of an earthquake, shallow water modelling techniques give us the most accurate view of the wave’s formation and early movement.

Once the wave is closer to the coast however, techniques known collectively as smoothed particle hydrodynamics (SPH) are better at predicting how the wave interacts with local geography. We’ve created models of a hypothetical tsunami off the northern Californian coastline to test this.

A dam break can also be modelled using SPH. The modelling shows how fast the water moves at certain times and in certain places, where water “overtops” hills and how quickly it reaches towns or infrastructure such as power stations.

This can help town planners to build mitigating structures and emergency services to co-ordinate an efficient response. Our models have been validated using historical data from a real dam that broke in California in 1928 – the St. Francis Dam.

Having established that our modelling techniques work better than others, we can apply them to a range of what-if situations.

In collaboration with the Satellite Surveying and Mapping Application Centre in China we tested scenarios such as the hypothetical collapse of the massive Geheyan Dam in China.

We combined our modelling techniques with digital terrain models to get a realistic picture of how such a disaster would unfold and, therefore, what actions could mitigate it.

Our experience in developing and using these techniques over several decades allows us to combine them in unique ways for each situation.

We’ve modelled fluids not just for natural disaster planning but also movie special effects, hot metal production, water sports and even something as everyday as insurance.

Insurance companies have been looking to us for help to understand how natural disasters unfold. They cop a lot of media flak after disasters for not covering people affected. People living in low-lying areas have traditionally had difficulty accessing flood insurance and find themselves unprotected in flood situations.

Insurers are starting to realise that the modelling of geophysical flows can provide a basis for predicting localised risk of damage due to flooding and make flood coverage a viable business proposition. One Australian insurance company has been working with us to quantify risk of inundation in particular areas.

Using data from the 1974 Brisbane floods, the floods of last year and fluid modelling data, an insurance company can reliably assess residents’ exposure to particular risks and thereby determine suitable premiums.

With evidence-based tools such as fluid modelling in their arsenal, decision-makers are better prepared for the future. That may be a future of more frequent natural disasters, a future with a more-densely-populated planet, or, more likely, a combination of both.

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

Credit of the article given to Mahesh Prakash, CSIRO

 

Our present achievements will look like child’s play in a few years.

What do iPhones, Twitter, Netflix, cleaner cities, safer cars, state-of-the-art environmental management and modern medical diagnostics have in common? They are all made possible by Moore’s Law.

Moore’s Law stems from a seminal 1965 article by Intel founder Gordon Moore. He wrote:

“The complexity for minimum component costs has increased at a rate of roughly a factor of two per year … Certainly over the short term this rate can be expected to continue, if not to increase. Over the longer term, the rate of increase is a bit more uncertain, although there is no reason to believe it will not remain nearly constant for at least ten years. That means, by 1975, the number of components per integrated circuit for minimum cost will be 65,000.”

Moore noted that in 1965 engineering advances were enabling a doubling in semiconductor density every 12 months, but this rate was later modified to roughly 18 months. Informally, we may think of this as doubling computer performance.

In any event, Moore’s Law has now continued unabated for 45 years, defying several confident predictions it would soon come to a halt, and represents a sustained exponential rate of progress that is without peer in the history of human technology. Here is a graph of Moore’s Law, shown with the transistor count of various computer processors:

Where we’re at with Moore’s Law

At the present time, researchers are struggling to keep Moore’s Law on track. Processor clock rates have stalled, as chip designers have struggled to control energy costs and heat dissipation, but the industry’s response has been straightforward — simply increase the number of processor “cores” on a single chip, together with associated cache memory, so that aggregate performance continues to track or exceed Moore’s Law projections.

The capacity of leading-edge DRAM main memory chips continues to advance apace with Moore’s Law. The current state of the art in computer memory devices is a 3D design, which will be jointly produced by IBM and Micron Technology, according to a December 2011 announcement by IBM representatives.

As things stand, the best bet for the future of Moore’s Law are nanotubes — submicroscopic tubes of carbon atoms that have remarkable properties.

According to a recent New York Times article, Stanford researchers have created prototype electronic devices by first growing billions of carbon nanotubes on a quartz surface, then coating them with an extremely fine layer of gold atoms. They then used a piece of tape (literally!) to pick the gold atoms up and transfer them to a silicon wafer. The researchers believe that commercial devices could be made with these components as early as 2017.

Moore’s Law in science and maths

So what does this mean for researchers in science and mathematics?

Plenty, as it turns out. A scientific laboratory typically uses hundreds of high-precision devices that rely crucially on electronic designs, and with each step of Moore’s Law, these devices become ever cheaper and more powerful. One prominent case is DNA sequencers. When scientists first completed sequencing a human genome in 2001, at a cost of several hundred million US dollars, observers were jubilant at the advances in equipment that had made this possible.

Now, only ten years later, researchers expect to reduce this cost to only US$1,000 within two years and genome sequencing may well become a standard part of medical practice. This astounding improvement is even faster than Moore’s Law!

Applied mathematicians have benefited from Moore’s Law in the form of scientific supercomputers, which typically employ hundreds of thousands of state-of-the-art components. These systems are used for tasks such as climate modelling, product design and biological structure calculations.

Today, the world’s most powerful system is a Japanese supercomputer that recently ran the industry-standard Linpack benchmark test at more than ten “petaflops,” or, in other words, 10 quadrillion floating-point operations per second.

Below is a graph of the Linpack performance of the world’s leading-edge systems over the time period 1993-2011, courtesy of the website Top 500. Note that over this 18-year period, the performance of the world’s number one system has advanced more than five orders of magnitude. The current number one system is more powerful than the sum of the world’s top 500 supercomputers just four years ago.

 

Linpack performance over time.

Pure mathematicians have been a relative latecomer to the world of high-performance computing. The present authors well remember the era, just a decade or two ago, when the prevailing opinion in the community was that “real mathematicians don’t compute.”

But thanks to a new generation of mathematical software tools, not to mention the ingenuity of thousands of young, computer-savvy mathematicians worldwide, remarkable progress has been achieved in this arena as well (see our 2011 AMS Notices article on exploratory experimentation in mathematics).

In 1963 Daniel Shanks, who had calculated pi to 100,000 digits, declared that computing one billion digits would be “forever impossible.” Yet this level was reached in 1989. In 1989, famous British physicist Roger Penrose, in the first edition of his best-selling book The Emperor’s New Mind, declared that humankind would likely never know whether a string of ten consecutive sevens occurs in the decimal expansion of pi. Yet this was found just eight years later, in 1997.

Computers are certainly being used for more than just computing and analysing digits of pi. In 2003, the American mathematician Thomas Hales completed a computer-based proof of Kepler’s conjecture, namely the long-hypothesised fact that the simple way the grocer stacks oranges is in fact the optimal packing for equal-diameter spheres. Many other examples could be cited.

Future prospects

So what does the future hold? Assuming that Moore’s Law continues unabated at approximately the same rate as the present, and that obstacles in areas such as power management and system software can be overcome, we will see, by the year 2021, large-scale supercomputers that are 1,000 times more powerful and capacious than today’s state-of-the-art systems — “exaflops” computers (see NAS Report). Applied mathematicians eagerly await these systems for calculations, such as advanced climate models, that cannot be done on today’s systems.

Pure mathematicians will use these systems as well to intuit patterns, compute integrals, search the space of mathematical identities, and solve intricate symbolic equations. If, as one of us discussed in a recent Conversation article, such facilities can be combined with machine intelligence, such as a variation of the hardware and software that enabled an IBM system to defeat the top human contestants in the North American TV game show Jeopardy!, we may see a qualitative advance in mathematical discovery and even theory formation.

It is not a big leap to imagine that within the next ten years tailored and massively more powerful versions of Siri (Apple’s new iPhone assistant) will be an integral part of mathematics, not to mention medicine, law and just about every other part of human life.

Some observers, such as those in the Singularity movement, are even more expansive, predicting a time just a few decades hence when technology will advance so fast that at the present time we cannot possibly conceive or predict the outcome.

Your present authors do not subscribe to such optimistic projections, but even if more conservative predictions are realised, it is clear that the digital future looks very bright indeed. We will likely look back at the present day with the same technological disdain with which we currently view the 1960s.

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

*Credit for article given to Jonathan Borwein (Jon) and David H. Bailey*