Two permutations are conjugate IFF they have the same cycle structure.
So if we calculate the conjugate of
a=(1 2 3)(4 5) and
b=(3 5),
then we know that the conjugate has the same cycle structure as a. Let's find out:
a^b=(3 5)((1 2 3)(4 5))(3 5)
1 2 3 4 5
1 2 5 4 3 : (3 5) applied
2 5 1 3 4 : (1 2 3)(4 5) applied
2 5 4 3 1 : (3 5) applied
In cycles: (1 2 5)(3 4)
There is a smarter way to get at this result. Permute the cycle structure of a with b:
(1 2 3)(4 5)
(1 2 5)(4 3) = (1 2 5)(3 4).
Using this method we can simply find the conjugating permutation of a and b.
For example
a=(1 2 3)(4 5 6) and
b=(1 3 5)(2 4 6).
Since a and b have the same cycle structure there must be a permutation c such that
a=c b c^1.
1 2 3 4 5 6 : cycle a as permutation
1 3 5 2 4 6 : cycle b as permutation
We get from a to b by applying cycle
(2 3 5 4) which must be c.
Let's verify.
1 2 3 4 5 6
1 3 5 2 4 6 : apply c
3 5 1 4 6 2 : apply b
3 4 5 6 1 2: apply c^1 = (2 4 5 3)
In cycles:(1 3 5)(2 4 6) = b.
Monday, November 26, 2007
Sunday, November 25, 2007
Representations and Characters of Groups
I found an electronic version of a book with a high position on my Amazon wish list. With 458 pages including about 60 pages of worked out solutions of all the exercises it is an excellent source for the selfstudy of this fascinating topic. I started on Representation Theory, A First Course by Fulton a while ago but after a few chapters it turned out I wasn't ready for all but parts of the book. My intrest in the subject was firmly set.
In the very beginning of my IT "career" ( since I am still on the first step of the corporate ladder it is clear that I am not the typical corporate type of guy, i adjust enough to be able to survive ) I moved to the position of project leader fast. After a year or two I wanted to write a program but I discovered that the languages I could code in ( Databus, Cobol and RPGII) were dead or not easily available to me. I had to start from scratch. These things don't happen in mathematics, I suppose. I can imagine similar developments in physics though. Imagine string theory to be proved false. If you are a string theory professor you have to come up with something else.
I thought about this when I read "... This set of lecture notes presents a new approach to the representation theory of the symmetric group — more precisely: to the character theory of the symmetric group over a field of characteristic zero ..." in the following book.
Thursday, November 22, 2007
Ken Ward's Math Pages
I read "... to our wives ..." on the first page of "Sequences" by H. Halberstam and K.F. Roth. { Aik. } Ok, printed in 1966. Only fourteen years after Alan Turing was convicted for having a homosexual relationship with a young Manchester man. So maybe "... to our wives ..." means "we are not homosexuals"? Perhaps, not likely. Still, I think it is rather odd to start a mathematics book with "... to our wives ...". Unless it were the wives who were doing the actual mathematics, of course. In that time women were badly discriminated in the academic world. Serious, why write "... to our wives ..." in a mathematics book? Love? Not a very romantic place to put it. A math book! Come on. I think they wrote it out of relief. "... despite your continuous attacks on our concentration by your endless babble talk, loud hoovering and yelling on the phone ..." we managed to finish our book. Something like that.
I added Ken Ward's Math Pages to the Cool Sites box. Although not even close to Eric Weisstein's huge encyclopedic MathWorld I like "Ken's pages" because it has unique content, is the only place on internet where I found a detailed proof of the Vandermonde Convolution theorem, and because I want to create something similar. Different topics of course but of the same ( achievable ) size and quality. Some day this mathematics diary should evolve into a mathematics site with worthwile content.
I added Ken Ward's Math Pages to the Cool Sites box. Although not even close to Eric Weisstein's huge encyclopedic MathWorld I like "Ken's pages" because it has unique content, is the only place on internet where I found a detailed proof of the Vandermonde Convolution theorem, and because I want to create something similar. Different topics of course but of the same ( achievable ) size and quality. Some day this mathematics diary should evolve into a mathematics site with worthwile content.
Tuesday, November 20, 2007
Monday, November 19, 2007
PSTricks
( This morning I asked the account manager what his gut feeling was about the job for which I was interviewed last week. He had an 80% feeling about it. I was more worried because the relationship between the prospect and us was very thin. All we had done was acting upon a request which had been sent to probably every IT company in the country. My position is still 'idle' )
Eventually I want to be able to write mathematics of my own. Besides being able in math, writing math requires some special computer skills. Mathematics is usually not written with a word processor although Word has a nice formula editor nowadays. Writing beautiful mathematics is possible but at the cost of a steep learning curve. The tools to know are MiKTeX and TeXnicCenter. But what about illustrations? For that I discovered an amazing set of tools. PSTricks and LaTeXDraw. PSTricks is a set of commands to include graphics into TeX. LaTeXDraw is a wysyiwig graphics to PSTricks converter. Seeing is believing. This toolset makes it possible to add professional looking graphics to your documents. I think it is worth the steep learning curve. For common graphics a wealth of examples is available anyway.
Eventually I want to be able to write mathematics of my own. Besides being able in math, writing math requires some special computer skills. Mathematics is usually not written with a word processor although Word has a nice formula editor nowadays. Writing beautiful mathematics is possible but at the cost of a steep learning curve. The tools to know are MiKTeX and TeXnicCenter. But what about illustrations? For that I discovered an amazing set of tools. PSTricks and LaTeXDraw. PSTricks is a set of commands to include graphics into TeX. LaTeXDraw is a wysyiwig graphics to PSTricks converter. Seeing is believing. This toolset makes it possible to add professional looking graphics to your documents. I think it is worth the steep learning curve. For common graphics a wealth of examples is available anyway.
Friday, November 16, 2007
Exercise ( Aigner ) 1.21
The exercise above is from the book .
I worked on this exercise for a great deal of the week.
I had considerably more time to do math this week. In a sense I even made some discoveries. At least for myself. I am going to try to document them in 'paperformat' during the weekend. If all goes well, I''ll publish them there.
I had time to go after new books as well. Too many topics fascinate me. I want to select a topic in math to which I could eventually contribute. That means that I must be good at it, i.e. find it easily accessible. Becoming 'current' is the first step, I suppose, being able to read the journals and understanding who is who in the world of mathematics. At the moment I am going in the direction of computational group theory, representation theory and combinatorics. Still much too broad, I know.
As far as computational tools are concerned up until now I have worked with GAP and Mathematica. Since a year or so GAP can be used as part of SAGE, a more or less complete math workbench, comparable with Mathematica. The philosophy of the SAGE team is 'best of breed', if I understood them correct. Where Mathematica has to develop all their tools with their own payrolled staff, SAGE draws the best of breed from the Open Source Community, i.e. GAP for Group Theory in SAGE. I don't see Mathematica overtaking the lead from GAP real soon and the same is true for other specialized areas like for example Commutative Algebra. But as a productivity tool Mathematica beats SAGE by far.
Another identity in Pascal's Triangle
Sunday, November 11, 2007
Michael Pogorsky's proof of Fermat's last theorem
Every now and then I ask the guru's at PlanetMath for help. I did so again today and while I was there I visited the Cafe section in the forum. There was a discussion with a rather long thread about Fermat's last theorem. Because Andrew Wiles proved the theorem with late 20th century mathematics people are still trying to prove the theorem using mathematics from Fermat's days. Every now and again a proof comes up. Usually professional mathematicians don't even bother to look. The mostly unknown authors are called crackpots in mathematical circles. How do these so called 'proofs' look like? Like this for example. It's a proof of Fermat's last theorem by Michael Pogorsky.  Read an analysis of the proof by a PM guru here.
Tuesday, November 6, 2007
Property of the Pascal Triangle
Another interesting property of the Pascal Triangle.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
Since Choose(n,k) = (k+1)/(nk)*Choose(n,k+1) we get ( for example ) for row 7:
1/7 * 7 = 1
2/6 * 21 = 7
3/5 * 35 = 21
4/4 * 35 = 35
5/3 * 21 = 35
6/2 * 7 = 21
7/1 * 1 = 7
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
Since Choose(n,k) = (k+1)/(nk)*Choose(n,k+1) we get ( for example ) for row 7:
1/7 * 7 = 1
2/6 * 21 = 7
3/5 * 35 = 21
4/4 * 35 = 35
5/3 * 21 = 35
6/2 * 7 = 21
7/1 * 1 = 7
Thursday, November 1, 2007
Wolfram 2,3 Turing Machine
News update from Mathematica:
I am still reading the biography of Alan Turing. I am reading chapter 6 now. World war II has ended and Alan is writing a proposal to get a project for building a general purpose computer ( the first one ever ) funded. He wrote his paper on the Universal Turing Machine long before the war started.
We're excited to announce that the $25,000 Wolfram 2,3 Turing Machine Research Prize has been won.
Alex Smith, a 20yearold undergraduate in Birmingham, UK, has given a 40page proof that Wolfram's 2,3 Turing machine is indeed universal.
This result ends a halfcentury quest to find the simplest possible universal Turing machine.
It also provides strong further evidence for Wolfram's Principle of Computational Equivalence.
The official prize ceremony is planned for November at Bletchley Park, UK, site of Alan Turing's wartime work.
For more information about the prize and the solution, see: http://www.wolframprize.org
Stephen Wolfram has posted his personal reaction to the prize at:
http://blog.wolfram.com/2007/10/the_prize_is_won_the_simplest.html
I am still reading the biography of Alan Turing. I am reading chapter 6 now. World war II has ended and Alan is writing a proposal to get a project for building a general purpose computer ( the first one ever ) funded. He wrote his paper on the Universal Turing Machine long before the war started.
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(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.)