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.

## Archive for May 2009

### Google Wave

30 May 2009### Elgar – Nimrod

29 May 2009Heard 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.

(to hear what this sounds like, click here)

### Hello world!

15 May 2009After 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 2009Back 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 is an integer, then we write “b divides a” as . If this is not the case and “b does not divide a” then we write . In the paper I’m working on I use the coprime notation of .

**Fermat’s little theorem **– if is prime then

**Wilson’s theorem **– if is prime then

### Prime Obsession

5 May 2009Read 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 and call this . While the function can be calculated for any given , there is no closed form (simple) solution for it. Approximations have been proposed, the first being

and the second being

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 . This function is defined as

The first solution to this I encountered in college for which is amazingly

Euler solved this generally for all even, (and they are all powers of to the ). However, all odd are unsolved, and it was not until 1978 that 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**

where are all the integers and stands for all prime numbers. (The use of is a convention started by Riemann, and not disturbed by mathematicians over the centuries, it could just as well be ). 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 2009Here’s a list of some of the vocab from this year’s trip to Taiwan

新流行感冒

流行

餛飩

甜菜

櫃臺

猴子

熊

獅子

螃蟹

打噴嚏

不會掉

一隻蚊子

拉鏈

漱口

買一送一

心安平安

### Mathematical physics

1 May 2009Purchased 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.