## Archive for May 2009

30 May 2009

I was very excited to see the introduction of Google wave yesterday, and the open source approach it offers.  I thought it would be great to see other embedded support for math (Latex) and music (LilyPond) etc.  Glad to see that Terry Tao is thinking about this.

### Elgar – Nimrod

29 May 2009

Heard a piano arrangement of Elgar’s Nimrod from Enigma Variations on the radio this morning – so beautiful, a great musical idea of just the right length, long enough to stun, but brief enough to not wear out the ears of the listener.

### Hello world!

15 May 2009

After 4 months of blogging only to my local computer, I’ve made the switch-over from blogging on the wordpress.com site.  The export/import transition was painless.  I only had one picture to upload, so that was quick as well.  The largest part of the conversion was reformatting the Latex (math).  Fortunately, there is lots of help WordPress + LaTeX.  Kogler has a posting with a great LaTeX summary.

### Number theory notes

7 May 2009

Back to work on this paper, I’m looking through an intro to number theory to find any helpful ideas.

Random notes

One notational convention I didn’t know was that if $\frac{a}{b}$ is an integer, then we write “b divides a” as $b|a$.  If this is not the case and “b does not divide a” then we write $b \dagger a$.  In the paper I’m working on I use the coprime notation of $a \bot b$.

Fermat’s little theorem – if $p$ is prime then $p | n^p-n$

Wilson’s theorem – if $p$ is prime then $p|[(p-1)! + 1]$

### Prime Obsession

5 May 2009

Read Prime Obsession over the last five days (I got a hold of it just in time for vacation).  It is a really nicely written book on the history and math behind the Riemann hypothesis.  The areas of math involved here are number theory (prime numbers), and complex analysis.  The Euler product (shown on the cover) is the beginning of the bridge between the two fields.

Mathematicians are interested in the number of primes less than a given integer $N$ and call this $\pi(N)$.   While the function $\pi(N)$ can be calculated for any given $N$, there is no closed form (simple) solution for it.  Approximations have been proposed, the first being

$\pi(N) \sim\frac{N}{\log(N)}$

and the second being

$\pi(x) \sim Li(x) = \int \limits_0^x \frac{1}{\log(t)} dt$

and the closest approximation coming in the last chapter.

Meanwhile, also of interest are the convergent infinite series of fractions, given by the Riemann zeta function $\zeta(s)$.  This function is defined as

$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + ...$

The first solution to this I encountered in college for $s=2$ which is amazingly

$1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$

Euler solved this generally for all $N$ even, (and they are all powers of $\pi$ to the $N$).  However, all $N$ odd are unsolved, and it was not until 1978 that $N=3$ was even proved to be irrational!  This summed series can be converted with regular algebra (and some insight using something like the sieve of Eratosthenes) to a product series.  This is the famous Euler product

$\sum \limits_n n^{-s} = \prod \limits_p (1-p^{-s})^{-1}$

where $n$ are all the integers and $p$ stands for all prime numbers.  (The use of $s$ is a convention started by Riemann, and not disturbed by mathematicians over the centuries, it could just as well be $x$).   The Riemann zeta function is continued analytically into the complex plane and gives rise to the Riemann hypothesis,

The Riemann hypothesis – all non-trivial zeros of the zeta function have real part one-half.

(now the zeta function in the complex plane is linked with primes!)

### Everyday Chinese vocab

2 May 2009

Here’s a list of some of the vocab from this year’s trip to Taiwan

### Mathematical physics

1 May 2009

Purchased a copy of Mathematical Physics by Hassani today.  I had previously had it on loan from the library, but it is too good not to have as a reference.   Looking forward to learning from it.