Thursday 27 January 2011

Carl Friedrich Gauss (30 April 1777 – 23 February 1855)

Carl Friedrich Gauss
Oil painting by G. Biermann
Two hundred years ago, one of the greatest mathematical thinkers ever to have lived was in his prime. In 1807, at the age of 30, Carl Friedrich Gauss was appointed Professor of Astronomy at Göttingen. But this did not inhibit his prolific work in pure mathematics. In 1811, he published a paper on the meaning of integrals with complex limits and examined the dependence of such integrals on the chosen path of integration.

Remarkable as it is, it is not so much this particular piece of work we wish to celebrate in this article. In mathematics Gauss is remembered for so many things, but perhaps most often for the Gaussian or "Normal" statistical distribution.

Gauss was a child prodigy. There is a story, which may or may not be true, about his remarkable mathematical ability at primary school. His teacher, realising Gauss had completed a task well ahead of the rest the class, asked Gauss to calculate the sum of the numbers from 1 to 100. The child thought for a few seconds, then told her the answer: 5050. He had added 1 and 99, 2 and 98, etc, to turn the sum into 49 x 100, after which he added the only unpaired numbers, 50 and 100.

When Gauss was only 14, Charles William Ferdinand, the Duke of Brunswick, who was always on the lookout for bright students, paid for the boy to go to the university in Brunswick.

Subsequently, he studied further in Göttingen. 1796 was Gauss’s most prolific year, in which he documented at least 5 major discoveries and proofs in number theory and geometry. He demonstrated the construction of a 17-sided polygon using only straight-edge and compass, something that had eluded all mathematicians since the ancient Greeks.

Gauss sorely wanted to be accepted by the elite group of Parisian mathematicians. But he presented his ideas in cryptic ways and didn’t believe in showing much working and this ultimately went against him. He was to spend all his life in what is now Germany.

In 1801, there was great excitement in the world of astronomy as an 8th planet was discovered between Mars and Jupiter. It was named Ceres, and today we know it as the largest asteroid. This discovery was considered a great omen for a new dawn of science at the beginning of the 19th century.

Shortly afterwards, however, the astronomers lost sight of the newly discovered object. Gauss announced he knew where to find it, using mathematical techniques now used to analyse data of many different types. To solve the problem of measurements of Ceres, Gauss had invented the Gaussian (or Normal) Distribution, now used widely in statistics, enabling patterns to be seen in seemingly random data.

The breakthrough was his realisation that the measurements, accurate and inaccurate, would, when plotted on a graph, be distributed around the true value in the shape of a bell-shaped curve. With this and other tools, statistical analysis has become a powerful weapon to analyse data, to test hypotheses, to separate statistical fact from fiction.

Gauss had an aversion to teaching and said that a professor “loses his precious time” lecturing students. This led to his taking the non-teaching post as Professor of Astronomy in the University of Göttingen in 1807 and he spent the rest of his career tracking the paths of planets.

The Gaussian distribution crops up everywhere; it is one of the statistician’s tools for understanding the real world, in the fields of chemistry, medicine, engineering, finance and many others.
Although arguably the most famous of Gauss’s mathematical innovations, the Gaussian distribution is probably not the greatest of his many mathematical achievements.

Despite his enormous influence in the fields of statistics and astronomy, the properties of numbers was Gauss’s true mathematical penchant. He once said “Maths is the Queen of the Sciences and Number Theory the Queen of Maths.”
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Wednesday 26 January 2011

Legal Challenge to Gove's Cuts

Michael Gove speaking at the Conservative Part...Image via WikipediaThe coalition government looks set to face yet another problem over education.

After the fiasco with the increases in tuition fees and scrapping of the Education Maintenance Allowance (EMA), the Education Secretary Michael Gove now faces a legal challenge. This revolves around his decision to abolish the Building Schools for the Future (BSF) scheme. The scheme, devised by Labour, was scrapped last July after the coalition government came to power.

Under the scheme, every secondary school would have been given funding for a rebuild. Now, 700 schools will have this money withdrawn. Seven councils (Waltham Forest, Luton Borough Council, Nottingham City Council, Sandwell, Kent County Council and Newham) have brought a legal challenge to the High Court in London over this volte face, saying that stopping the programme in their areas was arbitrary and legally flawed.

Gove is accused of not consulting properly and failing to give adequate reasons for his decision. His lawyers said in reply that the coalition had inherited the largest ever peacetime deficit and that spending cuts had to be made quickly.

Have you suffered because of the cuts to the BSF programme? Are you teaching or learning in shabby classrooms? Let us know why your school should qualify for a rebuild.

The Government is less than a year old, but is already under considerable pressure, not only in the field of education funding, but in defence, social care, employment, health and just about any area you care to choose. The state of the economy continues to look parlous.

How long can things go on? Will the Coalition Government continue with its controversial cuts? Or will it come up with a Plan B, as shadow chancellor Ed Balls advised yesterday? Will the Conservatives be "out of office for a generation", as Bank of England governor Mervyn King predicted, even before the election.

We live in interesting, if difficult, times.
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Thursday 13 January 2011

What is Randomness?

The BBC Radio 4 programme 'In Our Time' looked at the topic of randomness today. The In Our Time website has a link to the programme on the iPlayer, if you missed it first time round.

What is meant by randomness? Well, a truly random event is not deterministic, i.e. it is not possible to determine the next outcome, based upon the previous outcomes, or on anything else.

In actual fact, random processes are very important in lots of areas of maths, science and life in general, but truly random processes are remarkably difficult to achieve. Why should this be the case? Because in theory, many processes that we consider to be random, such as rolling a dice, are in fact deterministic. You could, theoretically, determine the outcome of the dice roll if you knew its exact position, size, etc.

The ancient Greek philosopher and mathematician Democritus (ca. 460 BC – ca. 370 BC) was a member of the group known as Atomists. This group of ancients were the pioneers of the concept that all matter can be subdivided into its fundamental building blocks, atoms. Democritus decreed there was no such thing as true randomness. He gave the example of two men meeting at a well, both of whom consider their meeting to have been pure chance. What they did not know is that the meeting was probably pre-arranged by their families. This can be considered an analogy for the deterministic dice roll: there are factors determining the outcome, even if we cannot measure or control them precisely.

Epicurus (341 BC – 270 BC), a later Greek mathematician, disagreed. Although he had no idea how small atoms really were, he suggested they swerve randomly in their paths. No matter how well we understand the laws of motion, there will always be randomness introduced by this underlying property of atoms.

Aristotle worked further on probability, but it remained a non-mathematical pursuit. He divided all things into certain, probable and unknowable, for example writing about the outcome of throwing knuckle bones, early dice, as unknowable.

As with many other areas of mathematics, the topic of randomness and probability did not resurface in Europe until the Renaissance. The mathematician and gambler Gerolamo Cardano (24 September 1501 – 21 September 1576) correctly wrote down the probabilities of throwing a six with one dice, a double six with 2 dice, and a triple with three. He was the first person to notice, or at least to record, the fact that you're more likely to throw 7 with 2 dice than any other number. These revelations formed part of his handbook for gamblers. Cardano had suffered terribly because of his penchant for gambling (at times he pawned all his family's belongings, ended up in a poor house, and in fights). This book was his way of telling fellow gamblers how much they should bet and how to stay out of trouble.

In the 17th century, Fermat and Pascal collaborated and developed a more formalised theory of probability and  numbers were assigned to probabilities. Pascal developed the idea of an expected value and famously used a probabilistic argument, Pascal's Wager, to justify his belief in God and his virtuous life.

Today there are sophisticated tests that can be performed on a sequence of numbers to determine whether or not the sequence is truly random, or if it has been determined by formula, human being, or some other means. For example does the number 7 occur one tenth of the time (plus or minus some allowable error)? Is the digit 1 followed by another 1 one tenth of the time?

An increasingly sophisticated series of tests can be fired into action. We have the "poker test", which analyses numbers in groups of 5, to see whether there are two pairs, three of a kind, etc, and compares the frequency of these patterns with those expected in a truly random distribution. The Chi Squared test is another statistician's favourite. Given that a particular pattern that has occurred, it will give a probability, and a confidence level, that it was generated by a random process.

But none of these tests are perfect. There are deterministic sequences that look random (pass all the tests) but are not. For example, the digits of the irrational number π look like a random sequence, and pass all the tests for randomness, but of course, it is not. π is a deterministic sequence of numbers - mathematicians can calculate it to as many decimal places as they please, given powerful enough computers.

Another naturally occurring, seemingly random distribution is that of the prime numbers. The Riemann Hypothesis provides a way to calculate the distribution of the primes, but it remains unsolved and nobody knows whether the hypothesis remains valid for very large values. However, like the digits in the irrational number π, the distribution of the primes does pass all the tests of randomness. It remains deterministic, but unpredictable.

Another useful measure of randomness is a statistic called the Kolmogorov Complexity, named after the 20th century Russian mathematician. The Kolmogorov Complexity is the shortest possible description of a sequence of numbers, for example the sequence 01010101.... could be described simply as "Repeat 01". This is a very short description, indicating the sequence is certainly not random.

However, for a truly random sequence, it would be impossible to describe the sequence of digits in any simplified form. The description would be just as long as the sequence itself, which indicates that the sequence would appear to be random.

During the last two centuries, scientists, mathematicians, economists and many others have begun to realise that sequences of random numbers are very important to their work. And so in the 19th century, methods were devised to generate random numbers. Dice, but can be biased. Walter Welden and his wife spent months at their kitchen table rolling a set of 12 dice over 26000 times, but these data were found to be flawed because of biases in the dice, which seems a terrible shame.

The first published collection of random numbers appears in a book of 1927 by Leonard HC Tippet. After that, there were many attempts, many flawed. One of the most successful methods was that used by John von Neumann, who pioneered the middle-square method, in which a 100-digit number is squared, the middle 100 digits are extracted from the result, and squared again, and so on. Very quickly, this process yields a set of digits that pass all the tests of randomness.

In the 1936 US presidential election, all the opinion polls pointed to a close result, with a possible win for the Republican Party's candidate Alf Landon. In the event, the outcome was a landslide to the Democratic Party's Franklin D Roosevelt. The opinion pollsters had chosen bad sampling techniques. In their attempts to be high-tech, they had telephoned people up to ask them about their voting intentions. In the 1930s, it was far more likely for wealthier people - largely Republican voters - to have a telephone, and so the results of the surveys were deeply biased. In surveys, truly randomising the sample population is of prime importance.

Likewise, it is also very important in medical tests. Choosing a biased sample set (e.g. too many women, too many young people, etc.) can make a drug appear more or less likely to work, biasing the experiment, with possibly dangerous consequences.

One thing is certain: humans are not very good at producing random sequences and they are not very good at spotting them either. When tested with two patterns of dots, a human being is particularly bad at deciding which pattern has been generated at random. Likewise, when trying to create a random sequence of numbers, very few people include features such as digits occurring three times in a row, which is a very prominent feature of random sequences.

But is there anything truly random? Going back to the dice we considered at the start, where a knowledge of the precise initial conditions would have allowed us to predict the outcome, surely this is true of any physical process creating a set of numbers.

Well, so far, atomic and quantum physics have come closest to providing us with truly unpredictable events. It is, to date, impossible to determine precisely when a radioactive material will decay. It seems random, but maybe we simply don't understand. At the moment, it remains probably the only way to generate truly random sequences.

Ernie, the UK Government's premium bond number generator, is now on its fourth reincarnation. It must be random, in order to give all the country's premium bond holders an equal chance of a prize. It contains a chip that exploits the thermal noise within itself, i.e. the amount of movement in the electrons. Government statisticians perform tests of the number sequences that this generates, and they do indeed pass the tests for randomness.

Other applications are: the random prime numbers used in internet transactions, encrypting your credit card number. The National Lottery machines use a set of very light balls and currents of air to mix them up, but like the dice, this could, in theory, be predicted.

Finally, the Met Office uses sets of random numbers for its ensemble forecasts. Sometimes it is difficult to predict the weather because of the well-known "chaos theory" - that the final state of the atmosphere is highly dependent on the precise initial conditions. It is impossible to measure the initial conditions to anything like the precision required, so atmospheric scientists feed their computer models various different scenarios, with the initial conditions varying slightly in each. This results in a set of different forecasts and a weather presenter who talks in percentage chances, rather than in certainties.
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Tuesday 4 January 2011

Erwin Schrödinger 12 August 1887 – 4 January 1961

Erwin Schrödinger, 1933

Fifty years ago today, one of the twentieth century's finest mathematicians died. Erwin Schrödinger was one of those rare mathematicians responsible for a concept that many non-mathematically minded people will know of and refer to: Schrödinger's Cat. He is also famed for a number of important contributions to physics, especially the Schrödinger Equation, for which he received the Nobel Prize in Physics in 1933.

In 1887 Schrödinger was born in Vienna, Austria to Rudolf Schrödinger, a botanist and Georgine Emilia Brenda, the daughter of Alexander Bauer, a Professor of Chemistry. His family home was prosperous and well-educated and he grew up in an academic environment. Living in Austria, but having a half-English mother, meant that Schrödinger grew up bilingually - he learned both English and German.

During the 1920s, Schrödinger made a name for himself at various European universities. While at the University of Zürich, in January 1926, Schrödinger published the paper "Quantisierung als Eigenwertproblem" or "Quantization as an Eigenvalue Problem" on wave mechanics and what is now known as the Schrödinger Equation. In this paper he gave a "derivation" of his famous wave equation, and showed that it predicted correct energy levels for hydrogen atoms. This paper has been universally celebrated as one of the most important achievements of the twentieth century. Along with three follow-up papers, it was the greatest achievements of Schrödinger's academic career. The collection of papers created a revolution in the new field of quantum mechanics, and shook the worlds of physics and chemistry. For the first time, the quantum structure of the atom was understood and mathematically described.

Although the Schrödinger Equation is probably his finest achievement, his most well-known legacy must be his cat. So, what is Schrödinger's Cat and why is the idea so well-known? This paradox presents a cat that might be alive or dead, depending on a random event. The cat, along with a flask containing a poison and a radioactive source, are all contained inside a sealed box. If an internal Geiger counter detects radiation, the flask is shattered, releasing the poison that kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is both alive and dead, simultaneously. Yet, when we look in the box, of course, we see the cat either alive or dead, not both alive and dead. So it is only when observed, by an external observer, that the cat ceases to be in both states at once. In the course of developing this experiment, he coined the term Verschränkung (entanglement).

The idea of an object being in two states at the same time (especially dead and alive) was so new and counter-intuitive to science that many scientists and mathematicians rebuked the concept. It stems from the remarkable discovery that some properties of sub-atomic particles cannot be measured simultaneously, for example position and momentum: only probabilities can be assigned. But if such concepts send your head into a spin, do not worry. Richard Feynman, one of the recent greats at popularising science, said "I think I can safely say that nobody understands quantum mechanics" and "Nobody knows how it can be like that." Even Einstein, during the 1950s, made an unsuccessful attempt to refute the accepted interpretation of quantum mechanics.

Such revolutionary mathematical thinking, spawning ideas that truly shake up the world of science and mathematics, are rare, and this is one sign of a truly remarkable mind. The field of quantum mechanics is probably still in its infancy. Some think that the coming century is when this strange academic pursuit will find real applications: will there be super-fast quantum computers? What about quantum cryptography? The basic idea behind this possible revolution is that the very attempt to decipher a quantum code disturbs that code, bringing new levels of security to internet transactions.

Because of these things, Schrödinger takes his place in the highest echelons of the great mathematical minds of the 20th century, but there was so much more to his life and work. His private life, for example, was intriguing and controversial. His tenure at Oxford University was cut short because, at that time, he was living with two women.

Like his friend Einstein (and like many other notable mathematicians), Schrödinger was a deeply political man. He took a strong stance against the anti-semitism going on in Germany during the 1930s and eventually emigrated.

There is a huge Schrödinger crater, on the far side of the Moon named after him.

Schrödinger died of tuberculosis in Vienna on 4 January 1961, at the age of 73. He left a widow, Anny (Annemarie Bertel), and was buried in Alpbach, Austria.