2009年1月29日 星期四

Finite simple group of order 2

I'm sure many of you have seen this before. This is a really mathematical song, by a group of students from Northwestern University who call themselves "The Klein Four Group". Pay attention to the lyrics; it's amazing how many mathematical terminologies they could fit in a simple song of finite length!

Finite Simple Group (of Order Two)
The Klein Four Group

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

2009年1月22日 星期四






某國家的憲法規定總統必須由普選產生,選舉方法則由政府制定。支持度只有 1% 的現任總統任期快將屆滿,正設法尋求在 1% 的支持度下通過普選成功連任。


總統宣佈,來屆總統選舉將採用「層遞」的選舉模式,1250 萬名國民先分成 5 人一「小小小組」,每個小小小組先互選一名代表,然後每 5 名小小小組代表組成一小小組並互選代表,之後每 5 名小小組代表組成一小組並互選代表,如此類推,經過 8 輪選舉後選出 32 人組成選舉委員會,再投票選出總統。分組名單由政府決定。

政府卻在分組時作了巧妙的安排。在第一輪選舉中,總統的 125000 名支持者被分派到 41666 組,每組 3 人(餘下 2 人已不重要),結果在這 41666 個小小小組中,選出的代表都是總統的支持者。換句話說,經過首輪選舉後,總統的支持者在 2500000 名代表中佔 41666 人。之後的各輪選舉中,政府都以同樣方法分組,於是總統的「支持度」越來越高:

全體選民:   125000 / 12500000(1.00%)
第 1 輪選舉後: 41666 / 2500000(1.67%)
第 2 輪選舉後: 13888 / 500000(2.78%)
第 3 輪選舉後: 4629 / 100000(4.63%)
第 4 輪選舉後: 1543 / 20000(7.72%)
第 5 輪選舉後: 514 / 4000(12.85%)
第 6 輪選舉後: 171 / 800(21.38%)
第 7 輪選舉後: 57 / 160(35.63%)
第 8 輪選舉後: 19 / 32(59.38%)

最終,總統的支持者在 32 人中佔了 19 人,總統自然得以成功連任。

2009年1月14日 星期三


在紐約開完conference後,去了Princeton探望其中一位MathDB前會長Polam。在Princeton數學系原來是可以找到人一起打PES 2009的。再次聲明,千其唔好以為讀MPhil讀PhD的人唔打機呀。

之後去Boston,住的地方是MIT。在Boston的Museum of Science逛時,赫然發現有一個展館叫"Mathematica",顧名思義,是關於數學(更正確來說是關於那個同名的數學軟件)的。貼幾張圖吧。




2009年1月6日 星期二

Attending Conference in New York City

This is long time from my last entry. Recently I am busy with my trip to USA, training MO in my mother school and preparing my presentation in the conference I am attending.

My first station is New York City. This is my second time to visit here. The conference I am attending is called Symposium on Discrete Algorithms. It is a Computer Science conference, but I would say many of the stuff are really Mathematics, and a few of them are related to Physics and Biology.

Okay, let me bring back to topic to Mathematics. In a talk, the speaker gave the following cute problem: In how many permutations of size n will two elements i and j in the same cycle? There is a one-line solution.

BTW, as some of you may know, I am the teaching assistant of the course Game Theory in HKUST in the recent two years. I have just found in the conference that a game "Atomic Game Flow" is an excellent example of applications of Game Theory to Computer Science. And its nature is so simple that undergraduate students should have no difficulty to understand it (certainly, as in most branches of Mathematics, if you go deeper, it can be very difficult :p).