As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

## Sunday, September 30, 2007

### Matrices and the Pacal Triangle

Create a matrix from the first n rows of the Pascal Triangle like
1 0 0 01 1 0 01 2 1 01 3 3 1

Then the inverse of this matrix is...
1 0 0 0-1 1 0 01 -2 1 0-1 3 -3 1

## Thursday, September 20, 2007

### Discrete Mathematics Using Latin Squares

A truly beautiful mathematics book. A book that demands to be read, studied, worked! I will. I love it already. Happiness is easy.

In the past two decades, researchers have discovered a range of uses for Latin squares that go beyond standard mathematics. People working in the fields of science, engineering, statistics, and even computer science all stand to benefit from a working knowledge of Latin squares. Discrete Mathematics Using Latin Squares is the only upper-level college textbook/professional reference that fully engages the subject and its many important applications.

## Sunday, September 16, 2007

### Fibonacci sequence in the Pascal triangle

Where is the Fibonacci sequence in 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

Count as follows.
Start left with a 1.
While on a number:
- Move right
- Move up

Let's go.
First row.
1.
Stop.

Second row.
1
Stop.

Third row.
1, right, up
1
Stop.

Fourth row
1, right, up
2
Stop.

Fifth row
1, right, up
3, right, up
1
Stop.

Sixth row
1, right, up
4, right, up
3
Stop.

Totals were 1, 1, 2, 3, 5, 8.

Fibonacci sequence, obvious.

Now what's the proof?

## Sunday, September 9, 2007

### Discrete Mathematics video

I watched video 11-15-00: Combinations and permutations. A lecture by Shai Simonson. A lecture in the Discrete Mathematics series on the ADUni.org website.

Topics:
Counting principles.
- Multiplication
- Complement
- Counting double
- 'When are things the same'?
Permutation
Combination
n Choose k.
Binomial Theorem
Pascal Triangle
Proofs
- by formula
- by induction
- by combinatorial argument

Doing proofs by combinatorial argument is a powerful technique. The basic identity from 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
...
[n,k]=[n-1,k-1]+[n-1,k]
for example
[6,3]=[5,2]+[5,3]=20=10+10
can easily proven by a combinatorial argument.

Such an argument could be the following.
Let n be the number of employees of a small company with one director. We have to select a k-member team from all employees. The number of teams is [n,k]
Exclude the director from selection and select a k-member team. The number of teams is [n-1,k].
Now select a (k-1)-team, the number of teams is [n-1,k-1] and add the director to it which makes it a k-member team including the director.
Adding the number of k-member teams without a director and k-member teams with a director is the total number of k-member teams or [n-1,k] + [n-1, k-1].

This proves [n,k]=[n-1,k-1]+[n-1,k].

## 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.)