Using Mathematica ( what else? ):
Pascal[n_]:=MatrixExp[Table[If[i == (j+1),j+1,0],{i,0,n},{j,0,n}]]
Usage: Pascal[n]//MatrixForm
Sunday, October 28, 2007
Pascal's Triangle as a Matrix Exponential
For me, this is an amazing result. When I read about it in "Accessing BernoulliNumbers by MatrixOperations, Gottfried Helms 3'2006 Version 2.3" I immediately started Mathematica and tried it myself. The picture above is the result. It's what I call some deep mathematics, although the formula for calculating the Matrix Exponential is easily understood.
Friday, October 26, 2007
Dangerous Knowledge
Dangerous Knowledge is a BBC Four documentary about the lives and work of Georg Cantor, Kurt Godel and Alan Turing. ( Find it on Google Video. ) Cantor, Godel and Turing worked on paradoxical stuff. The paradox about the documentary is that it is about mathematics without showing any of it. I suppose it is a nice documentary if you are familiar with Cantor, Godel and Turing. The documentary in that respect becomes a " complementary ".
Sunday, October 21, 2007
Thursday, October 18, 2007
Generating Functions
Generating functions are one of the most surprising, useful, and clever inventions in discrete mathematics.
Generating functions transform problems about sequences into problems about functions.
For example:
* sequence {1, 4, 9, 16, 25, ... }
* closed form a(n) = n^2
* generating function: F(x)=x(1+x)/(1x)^3.
Interested? An introduction to generating functions (pdf document) can be found [ here ].
Monday, October 15, 2007
Difference sequences
Rather early in our math education we learn about functions and derivative functions.
Something similar can be defined for sequences, in that case the 'derative' is called the difference sequence.
Here also we see that Pascal's Triangle has a crucial meaning.
f(x+h)f(x)
f'(x) = lim 
h> 0 h
For example:
f(x) = x^n
f'(x) = n*x^(n1)
Something similar can be defined for sequences, in that case the 'derative' is called the difference sequence.
a_n = {1, 16, 81, 256, ... }
f(n) = n^4
f'(n) = 1 + 4n + 6n^2 + 4n^3
f''(n) = 14 + 24n + 12n^2
f(3)(n) = 36 + 24n
f(4)(n) = 24
For an arbitrary sequence f:
f(m)(n) = Sum[(1)^(k)*Binomial[m,k]*f[nk+m],{k,0,m}]
Here also we see that Pascal's Triangle has a crucial meaning.
Thursday, October 4, 2007
Binomial Theorem
(x + y)^n = Sum(n,k) Choose(n,k) * x^(nk) * y^k
So what? I can now prove it. ( By induction. )
So what? I can now prove it. ( By induction. )
Tuesday, October 2, 2007
Literary Mathematics
"... A generating function is a clothesline on which we hang up a sequence of numbers for display. ..." ( by Herbert S. Wilf in Generatingfunctionology )
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Mathematics: is it the fabric of MEST?
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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.)