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Wednesday, April 29, 2009

My favourite posts

Ordered partitions

Example. The partitions of 5 are:
4 1
3 2
3 1 1
2 2 1
2 1 1 1
1 1 1 1 1

The -ordered- partitions of 5 are:
4 1
1 4
3 2
2 3
3 1 1
1 3 1
1 1 3
2 2 1
2 1 2
1 2 2
2 1 1 1
1 2 1 1
1 1 2 1
1 1 1 2
1 1 1 1 1

1 ( of 1 ) = C(4,0)
4 ( of 2 ) = C(4,1)
6 ( of 3 ) = C(4,2)
4 ( of 4 ) = C(4,3)
1 ( of 5 ) = C(4,4)

The number of ordered partitions of n = 2^(n-1). In the case of 5: 2^4=16.

Try 3,4 or 6 as an ( interesting ) exercise.

Saturday, April 25, 2009

Results MS221 - TMA01

MS221 - TMA01 consisted of six questions.
1. Solving a second order recurrence equation.
2. Proving a recurrence identity.
3. About a conjecture regarding Fibonacci numbers.
4. Given was a quadratic equation. One had to show it was elliptic. Many more questions about the conic.
5. Several questions which required knowledge of the trig identities.
6. About an affine transformation consisting of a translation, rotation and scaling.
My score is 97/100.

Sofar so good, both on MST121 and MS221. What I must do now is to remain focused and continue to work as I did. Never change a method that provides the results you want. I implemented a few changes though. More about that another time.

Monday, April 20, 2009

Math podcast from BBC Radio 4

More or Less: Behind the Stats

More or Less makes sense of the numbers that surround us and that rule our world. With wit, freshness and simplicity Tim Harford shows the genuine significance of statistics, productivity, performance indicators, measurements and quantification of every kind. We all use numbers in so many ways to argue about, understand, help make sense of the world around us. More or Less hopes to make that task easier, more entertaining, more surprising. This half-hour programme is broadcast at 1330 on Friday afternoons and repeated at 2000 on Sundays on Radio 4.

Tuesday, April 14, 2009

TeXnicCenter or Texclipse as a LaTeX IDE ??

Currently I have both IDE's installed, both do the job of compiling my in TeX written TMAs to .pdf documents. The TMA sets I sent in were close to 20 pages and contained several TeX and external images.

I haven't decided yet, here are some preliminary thoughts.

TeXnicCenter is modelled after Visual Studio ( I suppose ) while Texclipse is a plugin for the Eclipse platform.

Both applications are free.

No major new functionalities seem to be planned for Texclipse. The last bugfix-only release was in mid 2008. It looks as though Texclipse will be abondoned soon. The platform Eclipse is updated regularly.

For TeXnicCenter work on release 2.0 has been started.

TeXnicCenter is closed source, while Texclipse is open source Java.

Sofar about comparing the two. More on functionality after I have completed my next TMA. I´ll prepare the TMA with Texclipse.

More later.

Monday, April 13, 2009

Mad mathematicians

It is a beautiful Easter. What does this mean? Close to a million tourists in our already overpopulated small country. It's not that I don't welcome tourists, on the contrary. But I don't want to meet them today. I have been out early this morning, getting drinks and other groceries and the roads were already crowded. Almost as busy as on an ordinary working day. I want to do something different today though. It is Easter for me too. So I have been searching for documentaries on math. ( I can't give you exact links because Google might ban my Blogger account but seek and you will find. ) Yesterday I found the documentary about the Black Scholes option pricing formula. Today I found the following documentary.

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt G?del and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

The film also talks to the latest in the line of thinkers who have continued to pursue the question of whether there are things that mathematics and the human mind cannot know. They include Greg Chaitin, mathematician at the IBM TJ Watson Research Center, New York, and Roger Penrose.

Dangerous Knowledge tackles some of the profound questions about the true nature of reality that mathematical thinkers are still trying to answer today.

Part 1: God’s messenger
The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God’s messenger and was eventually driven insane trying to prove his theories of infinity.

Ludwig Boltzmann’s struggle to prove the existence of atoms and probability eventually drove him to suicide.

Part 2: The Enigma
Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death.

Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and homosexual, died trying to prove that some things are fundamentally unprovable.

The life of Turing was both fascinating and dramatic. Turing foresaw the profession of computer programmer long before computers were invented. He also won WW2 practically by himself by cracking the Enigma code. This super-hero status was not enough to forgive him the fact that he was homosexual. Instead he got convicted because he was a practising homosexual. He didn't have to go to jail if he took a libido eliminating injection every month. I can imagine that such an intelligent person couldn't live with so much brutality. He died of eating a poisoned apple. He wanted to spare his mother the misery of having a son who killed himself.

Sunday, April 12, 2009

Mathematics and the economic crisis

As far as I can understand the current economic problems have their root cause in derivatives like Credit Default Swaps. And that is all math. There is a BBC program about a company which failed in the nineties but the story seems relevant even today. The founders of LCTM thought to have found a math formula with which you could always win at the stock market.

This is the extraordinary story of a beautiful mathematical formula that changed the world, the financial markets, and indeed capitalism itself. It could do the unthinkable - it took the risk out of playing the money-markets. To its inventors it brought the Nobel Prize for economics. To those who used it, it brought great wealth. But this glittering tale would end in tragedy.

The Black Scholes formula was invented 25 years ago, by three young mathematicians. They had been trying to solve a problem that had plagued economists for centuries - how to counter the randomness of market forces and the irrationality of human behaviour that made the markets dangerously turbulent. Whilst pondering this dilemma, they made a remarkable discovery.

The search for a way to price option contracts began in earnest when the thesis of an unknown student named Louis Bachelier was unearthed in the 1950s. Working at the beginning of this century, Bachelier had set out to do something no-one had ever done before - using a series of equations he created the first complete mathematical model of the markets. He had realised that stock prices moved at random and that it was impossible to make exact predictions about them, but Bachelier said he had also found a solution - through the pricing of a financial contract called an option.

The risk in the stock market is that if you buy a stock today the price can drop in the future and you could lose money but if you pay for an option contract this gives you the right to wait and buy the stock if it reaches some agreed price in the future, but there’s no obligation. If the stock fails to reach that price you can opt out and you would lose only the cost of the option. In theory options are a perfect way to get rid of risk, but there was a problem. How much would someone pay for such absolute peace of mind?

Bachelier believed that if someone could discover a formula that would allow option contracts to be widely used, they would be able to tame the markets completely, but he died before he could find it. By the end of the 60s, academics were no nearer to pricing options than they’d ever been. But all this was about to change when Myron Scholes and his colleague Fischer Black set out to tackle the problem of options…

At its simplest level, the Black Scholes formula could be used to hedge against losing any bet, by working out how to place another bet in the opposite direction. That way, you couldn’t lose. The formula had the almost magical ability to allow you to make a fortune with the minimum of risk. But there was one problem. In the time it took to make the calculation, the fast moving markets had moved on and the calculation would effectively be out-of-date.

However, unbeknown to them, the problem had already been solved by a financial genius called Bob Merton. Using an idea taken from rocket science, the value of an option could now be constantly recalculated and the risk eliminated continually.

Myron Scholes and Bob Merton joined forces with the greatest dealers on Wall Street, and started a legendary company - Long Term Capital Management (LTCM). Relying on mathematics, the company traded and borrowed on a scale never seen before. But the mathematical model was based on normal market behaviour and unforeseen events were about to send the markets wild. The calculations in LTCM’s models became hopelessly out of kilter, and when the company collapsed last year, it nearly brought down the entire global economy.

Saturday, April 11, 2009

MIT differential equations lectures

Watched lecture 3 from the series.

About solving linear DE's. I suppose it is all stuff which is covered in MST209 at the OU. It is nevertheless very handy to have such a course available online.

Tuesday, April 7, 2009

MS221 - TMA01

Shipped to tutor. MS221 Book A is about the Fibonacci sequence and other 2nd order recurrences, the family of conic sections and trigonometry ( again ).

Sunday, April 5, 2009

Mathematics on video

The number of complete mathematics courses on video keeps growing. Academic Earth has a listing of the best of what is available.

The more free courses there will be 'out there' on the web the more students will enroll some paid university course. There will always be a demand for teachers and course authors no matter how much free videos there will be available. Education should not be seen as a simple business anyway although I am sure many universities are managed as though they were businesses. But now I am talking economics and that is 'off-topic' for this blog.

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Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)