Collatz' original drawing

Math and art

Although I seriously doubt if mathematics and art are even close I added http://math-art.net/ to the 'Cool Sites' list. I do agree that math pics can be beautiful. "... Where Mathematics and Art blends into a zen-like state of peace ...".

Venn diagram

Great Britain, The United Kingdom, or is it England: confused? Charlie Eppes ( Numb3rs ) would probably give a mini-lecture on Set Theory before drawing a Venn diagram to explain the difference. We all use math every day.

Collatz conjecture

The 3x + 1 problem or Collatz conjecture is simple in its formulation but until now no proof is known that it is in fact true for any n. A conjecture is generally believed true but a formal proof is still to be found.

The conjecture says that if you repeatedly apply the following rule to a number the end result will always be one (1):
- if even then divide by 2;
- if odd then multiply by 3 and add 1.

Example:
9
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1

Take any number and try it.

What about 27?

Quadratic residu

A number q is called a quadratic residue modulo n if there exists an integer x such that x^2 = q mod n.

For example

a  a^2  a^2 mod 7
1  1    1
2  4    4
3  9    2
4  16   2
5  25   4
6  36   1
7  49   0
8  64   1
9  81   4
etc.

The quadratic residues mod 7 are 1,2 and 4. The set {1,2,4} is a group under multiplication mod 7:

* mod 7   1   2   4
      1   1   2   4
      2   2   4   1
      4   4   1   2

Number theory exercise

Determine the integers n for which there exist integers x and y such that n divides ( x + y - 2 ) and ( 2x - 3y -3 ).

numb3rs

We all use math every day; to predict weather, to tell time, to handle money. Math is more than formulas or equations; it’s logic, it’s rationality, it’s using your mind to solve the biggest mysteries we know.

I like numb3rs. I suppose one of the ideas behind the series is to promote mathematics as one of the cool professions. Well they should, because mathematics is definitely cool.

Bézout's Theorem

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. )

Same topic, different book.

I am studying "Knapp, Basic Algebra", at first read, a well to do self-study 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 fun-studying 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.

Moebius transformations video

( This video has been watched over a million times on YouTube. )
( Update: 10/4-'10 Added Complex Analysis tag. )

Burnside's Lemma

Burnside's Lemma as it can be found on MathWorld:

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.

The Ascent of Man

Jacob Bronowski: A mathematician turned biologist

An IMDB user wrote about The Ascent of Man:

This remarkable series, thirteen fifty-minute 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.