Plus Blog

April 18, 2008
Friday, April 18, 2008

Edward Lorenz, American mathematician and meteorologist, died in his Cambridge Massachusetts home on April 16 aged 90. Lorenz was the "father of chaos theory" and discovered the Lorenz attractor that often occurs in chaotic systems.

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posted by westius @ 4:27 PM

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April 15, 2008
Tuesday, April 15, 2008

The Enigma machine was once considered unbreakable, and the cracking of the "unbreakable code" by the allies changed the course of World War 2. Plus talks to Nadia Baker from the Enigma Project about the history of codes and code-breaking, why the Enigma machine was considered unbreakable, the mathematics behind codes, and how it was finally cracked. The Enigma Project travels all over the United Kingdom and abroad, visiting over 100 schools and organisations, reaching over 12,000 people of all ages every year.

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posted by Plus @ 4:13 PM

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April 11, 2008
Friday, April 11, 2008

Sixty-three year-old Avraham Trakhtman has solved one of the current generation's toughest mathematical problems — the 38 year-old road colouring problem. The solution will shortly be published in the Israel Journal of Mathematics.

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posted by Plus @ 3:11 PM

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April 10, 2008
Thursday, April 10, 2008

I would do anything for maths, but I won't do that

One of the latest group emails that I have received concerns maths and music. Good to see that mathematics has embarked on viral marketing. See if you can pick the songs referenced in the pictures below.

posted by westius @ 5:02 PM

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At 8:33 PM, Blogger Tenon_Saw said...

I thought this was great! Thanks for posting it.

 
April 10, 2008
Thursday, April 10, 2008

Attacks on the foundations of calculus

George Berkeley

Born: 12 March 1685 in Kilkenny, County Kilkenny, Ireland

Died: 14 Jan 1753 in Oxford, England

George Berkeley was an Irish bishop and philosopher who is best known for his attacks on the logical foundations of the calculus as developed by Newton.

George Berkeley.

Berkeley started at Trinity College, Dublin at the tender age of 14 and graduated with a masters when only 18 years old. His attacks on the foundations of calculus were first aired in 1734 when he published The analyst: A discourse addressed to an infidel mathematician. The infidel mathematician is believed to have been either Edmond Halley or Isaac Newton. He argued that although the calculus led to true results, its foundations were no more secure than those that underpin religion. He stated that the calculus involved a logical fallacy and described derivatives thus:

"And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?"

In modern language, this could be read as:

"What are these 'instantaneous' rates of change? The ratios of vanishing increments? And what are these 'vanishing' increments? They are neither finite quantities nor 'infinitesimal' quantities, nor yet nothing. May we not call them the ghosts of departed quantities?"

His interesting theory as to why the calculus actually worked was that it was the result of two compensating errors.

As a consequence of the controversy surround Berkeley's publication, the foundations of calculus were rewritten in a much more formal and rigorous manner using limits. It was not until 1966, with the publication of Abraham Robinson's book Non-standard Analysis, that the concept of the infinitesimal was made rigorous. This gave an alternative way of overcoming the difficulties that Berkeley found in Newton's approach.

Berkeley's influence in mathematics is reflected by the fact that the University of California, Berkeley, and the city of Berkeley that grew up around it, are named after him. Berkeley died of a heart attack on 14 January 1753, sitting with his family listening to his wife reading. He died so peacefully that the event went unnoticed, his family thinking he had fallen asleep. He left instructions that he was not to be buried for at least five days and was buried at Christ Church, Oxford on 20 January.

For more information, see the MacTutor biography of Berkeley.

posted by westius @ 5:00 PM

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April 1, 2008
Tuesday, April 01, 2008

Climate change and composing computers

Last night, Plus attended an excellent talk at the Edinburgh International Science Festival by Professor Chris Rapley, who was awarded the Edinburgh Medal. The Edinburgh Medal is awarded each year to those within science whose professional achievements are judged to have made a significant contribution to the understanding and well-being of humanity.

Profesor Rapley's talk was entitled "Great while it lasted; now what?" referring to how humans have exploited carbon-based energy sources for the last century, but now must confront the fact that we are exceeding the safe limits of such use. Professor Rapley is a systems scientist and has studied the Earth mathematically as a complex system through roles as Director of the British Antarctic Survey, Executive Director of the International Geosphere-Biosphere Programme, and Professor of Remote Sensing Science and Associate Director of University College London's Mullard Space Science Laboratory.

Professor Rapley is now Director of the Science Museum in London, and considers that the best way to study the Earth is to bring together "many -ologists — geologists, biologists, sociologists etc." As a complex system, the Earth has many inbuilt checks and balances — the oceans and the biosphere can act as carbon sinks, reducing the amount of carbon dioxide in the atmosphere. However, Rapley thinks that the Earth will also have a "tipping point" whereby the ocean and biosphere can no longer take-up carbon dioxide. Using recent ice melts in the Antarctic and Greenland as examples, he highlighted the incredibly difficult task in modelling the Earth as a complex system. Describing the Earth as "the most complex system in the Universe", he said that the Earth system "has many surprises and can't be modelled too much." He added, soberly, that whilst the greenhouse effect was understood in the 1800s by mathematician Joseph Fourier , the Earth now:

"...has no user's manual, and no spares... . Many senior economists act like they don't understand this."

He likened our current situation to that of the next day after a great party. Burning immense amounts of carbon has given humans a much higher quality of life than before the use of carbon fuels, however the carbon dioxide is the "hangover." He attacked sceptics of climate change by saying that the weight of scientific opinion supports the assertion that climate change is real and man-made, highlighting that 90% of climate change is anthropogenic.

Another interesting, and perhaps unfortunate, point (well, for me anyway) was that meat-eaters contribute far more than vegetarians to climate change. This is because much more energy goes into meat production than vegetarian foods.

Professor Rapley's talk was preceded by an oration by Dr James Lovelock. Dr Lovelock is a systems modeller and is well-known for proposing the Gaia hypothesis — that living and non-living parts of the Earth are a complex interacting system that can be thought of as a single organism, and that all living things have a regulatory effect on the Earth's environment that promotes life overall. Lovelock and Rapley have recently proposed a radical idea to "Help the Earth heal herself" in a letter to Nature Magazine. The proposal is to use huge pipes to pump lower ocean water, rich in nutrients, to the surface, stimulating the growth of algae and plankton in the upper ocean. The surface layer of the ocean is increasingly starved of nutrients such as phosphates and nitrates. These nutrients are needed by the algae to grow, but as the surface warms, there is less mixing with the nutrient-rich water layers beneath. The mixing of ocean layers would encourage algal blooms — the algae take up more carbon dioxide, which is eventually locked up in the tiny shells of plankton when they die and fall to the bottom of the ocean. This proposal essentially assumes that it's too late for carbon emission controls and other alternatives to have any effect on climate change — the damage is already done.

To read more on the mathematics of climate change, read the Plus article Maths and climate change: the melting Arctic.

Another interesting session that Plus attended was on a computer that had been taught using hidden markov models to harmonise melodies in the style of Johann Sebastian Bach. A hidden Markov model is a statistical model in which the system being modelled is assumed to be a Markov process with unknown parameters, and the hidden parameters need to be determined from the observable parameters. Hidden Markov models are known for their application in pattern recognition such as speech, handwriting, musical score following and bioinformatics.

You can read more on Machine Learning at the Edinburgh Festival on yesterday's blog, and you can read more about Markov Processes in the Plus article Formulaic football. And to read more on using hidden Markov Models to analyse Bach, see the group's publication Harmonising chorales by probabilistic inference.

This afternoon, I eagerly await the screening of Pi in the maths film festival. For more information and a comprehensive program, see the Festival website. The event runs until April 6.

posted by westius @ 10:12 AM

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