Symmetric Group S5 and Alternating Group A5

An old familiar question comes into my mind: "How to prove A5 is simple?" But I forgot the answer.

I remember there is a proof in Michael Artin's book, by considering some "concrete" rotation of some polyhedron. But I thought that proof is not nice.

Tried to google the proof on web. Found the good website Groupprops. And then found the page for S5 and A5.

Yes, there is a proof by considering the conjugacy classes of A5. And by induction, one can prove that is simple for .

A question: is there an intuitive way to see that in A5, the conjugacy class containing (1 2 3 4 5) and the conjugacy class containing (1 3 5 2 4) are different?

Four Convex Hull Algorithms

Now I am taking COMP573 in HKUST. The title of the course is "Computational Geometry".

We are using the lecture notes written by Prof. David Mount, who are now visiting HKUST. One interesting fact is that the lecturer, Dr. Sunil Arya, is advised by Prof. Mount for PhD degree, while Prof. Mount is teaching an undergraduate algorithm course...

I think it is one of the few best lecture notes I have ever read.

In the lecture notes there are four algorithms to find the convex hull of a set of given points. They are Graham scan, Divide-and-Conquer Methodology, Quick Hull and Jarvis's March.

Another interesting fact I wanna mention here is that these 4 algorithms are analog (strongly or weakly) to insertiont sort, merge sort, quick sort and selection sort respectively.

Read Timothy Chan's algorithm, which combines Graham scan and Jarvis March to form a quicker output sensitive algorithm. The idea is indeed very simple; but this is real intelligence.

Links for Quantum Computing Materials

Dr. Elad Verbin visited HKUST. He told me that he usually learn a topic by some good lecture notes.

He recommended two links to materials of quantum computing and quantum complexity theory:

http://www.scottaaronson.com/democritus/ - Quantum Computing Since Democritus

http://stellar.mit.edu/S/course/6/fa08/6.896/materials.html - Quantum Complexity Theory

A Quotation from "Analytic Combinatorics"

"Analytic Combinatorics" is a book written by Prof. Philippe Flajolet and Prof. Robert Sedgewick. Prof. Flajolet is the post-doc advisor of my MPhil advisor, while Prof. Sedgewick is the PhD advisor of my MPhil advisor.

Here is a quotation in this book:

It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one.

- Jacques Hadamard