Let a and b be integers with greatest common divisor d. Then there exist integers r and s such that d = ar + bs. Thus, the greatest common divisor of a and b is an integer linear combination of a and b.
( Didn't know that this theorem was called Bézout's Theorem. )
Saturday, December 29, 2007
Same topic, different book.
I am studying "Knapp, Basic Algebra", at first read, a well to do selfstudy book. Chapter I looks like this.
1. Division and Euclidean Algorithms 1
2. Unique Factorization of Integers 4
3. Unique Factorization of Polynomials 9
4. Permutations and Their Signs 15
5. Row Reduction 19
6. Matrix Operations 24
7. Problems 30
A manageable 14 pages on the Euclidean Algorithm for integers and polynomials, but dense. Too dense perhaps. Then I had a look at this other book. "Irving, Integers, Polynomials and Rings", have a look at the table of contents.
1 Introduction: The McNugget Problem
Part I Integers
2 Induction and the Division Theorem
3 The Euclidean Algorithm
4 Congruences
5 Prime Numbers
5.1 Prime Numbers and Generalized Induction
5.2 Uniqueness of Prime Factorizations
5.3 Greatest Common Divisors Revisited
6 Rings
7 Euler’s Theorem
8 Binomial Coefficients
Part II Polynomials
9 Polynomials and Roots
10 Polynomials with Real Coefficients
11 Polynomials with Rational Coefficients
12 Polynomial Rings
13 Quadratic Polynomials
14 Polynomial Congruence Rings
Part III All Together Now
15 Euclidean Rings
16 The Ring of Gaussian Integers
17 Finite Fields
I'll guess I stop for a while on my route and do a bit of funstudying in Irving. If I don't study it now I probably never will. It is a beautiful book. I was almost forgotten that I had it.
Wednesday, December 19, 2007
Moebius transformations video
( This video has been watched over a million times on YouTube. )
( Update: 10/4'10 Added Complex Analysis tag. )
Wednesday, December 12, 2007
Burnside's Lemma
Burnside's Lemma as it can be found on MathWorld:
Today I deepened my understanding of Burnside's Lemma considerably.
Let J be a finite group and the image R(J) be a representation which is a homomorphism of J into a permutation group S(X), where S(X) is the group of all permutations of a set X. Define the orbits of R(J) as the equivalence classes under x~y ,which is true if there is some permutation p in R(J) such that p(x)=y. Define the fixed points of p as the elements x of X for which p(x)=x. Then the arithmetic mean number of fixed points of permutations in R(J) is equal to the number of orbits of R(J).
Today I deepened my understanding of Burnside's Lemma considerably.
Saturday, December 1, 2007
The Ascent of Man
Jacob Bronowski: A mathematician turned biologist
An IMDB user wrote about The Ascent of Man:
This remarkable series, thirteen fiftyminute episodes, is one of television's highest achievements. Jacob Bronowski takes the viewer literally around the world, to discuss Mankind's greatest accomplishments and lowest depths. One outstanding quality of this extraordinary series is that Bronowski speaks to the viewer directly, in a very personal fashion, through the lense of the camera.
The book, that derives from the episodes themselves, is a virtual transcript of Bronowski's remarks. These are not "lectures', but rather discussions presenting his "personal view." The episodes are sprinkled with delightful and moving anecdotes of people Bronowski knew and worked with, such as Leo Szilard (who first thought of the nuclear "chain reaction") and John von Neumann (the "Father of Electrionic Computing").
Anyone interested in the history of science  and of thought in general  will be astonished, delighted and deeply moved by "The Ascent of Man." The production value is of the highest order throughout.
Highest recommendation.
ASCENT OF MAN LINKS
 Museum of Broadcasting
 IMDB
 Digitally Remastered series on DVD
Many years ago when I was a school dropout without any qualification whatsoever I thought there was nothing interesting to learn. This series is part of what brought me back on track. I thought about it many times. I wanted to see it again just to check if I still like it, if it still makes an impression. I couldn't find it for many years because I was forgotten the series name, even Bronowski's name. But I wasn't forgotten Bronowski's face, voice and the way he lectures. I found the series by accident. It's on internet if you know where to look. I have episodes 1 to 8 and 9 to 13 soon. Sofar I have seen episode 1. Yes. It is great television. Well worth the time.
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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.)