This week Staren sent me a puzzle on WeChat. After we discovered a solution for the puzzle, I tried to backtrack the source of it. I found it appeared in the blog post Yet another prisoner puzzle by Oliver Nash. The author seemed to work at a quantitative trading company SIG at that time. Given… Continue reading Communicating Information through Randomness
Crazy Telescoping
In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation. I learnt this technique when I was doing maths olympiad. However until last year I learnt this buzz word ‘telescoping’ since I received my education in China and we call it ‘the method of… Continue reading Crazy Telescoping
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好久没有写日志了。倒也不是因为整个人变得沉默所以变得不想诉说自己了。只是因为发现这个社会的面貌后，不想再说太多了。 但这显然不是我嘛，只是要一点时间恢复而已。 那明天就要二十五岁了，寄语下一个生日前的自己吧。 要多写博客，多拍照，锻炼身体，简单生活。 今年就不搞什么派对聚会了，大家都挺忙的。 还看博客的话，就留个言吧。不想留言，也可以点下面那个小爱心啊。
On Hardy–Littlewood maximal function of singular measure
In this exposition, we explore the behavior of the Hardy–Littlewood maximal function of measures that are singular with respect to Lebesgue measure. We are going to prove that for every positive Borel measure that is singular with respect to Lebesgue measure , for all , where is a torus and is the Hardy–Littilewood maximal function… Continue reading On Hardy–Littlewood maximal function of singular measure
Alternating Fourier Coefficients
Suppose is a periodic function from to with period . Let be its Fourier coefficients, namely for all . Prove for all it is almost surely that function is in where is an infinite sequence of independent and identical random variables indexed by with equals either or with probability . I heard this problem from… Continue reading Alternating Fourier Coefficients
Number of non-isomorphic graphs
This expository essay is to test my understanding of the techniques used in More Bricks – More Walls?, Thirty-three Miniatures by Jiří Matoušek’s. We shall prove the sequence is unimodal, i.e., it is first nondecreasing and then, from some point on, non-increasing, where is the number of non-isomorphic graphs with vertices and edges. In particular,… Continue reading Number of non-isomorphic graphs
Fun with Hex
According to the folklore, The Hex game was invented by the Danish mathematician Piet Hein, who introduced it in 1942 at the Niels Bohr Institute. It was independently re-invented in 1947 by the mathematician John Nash at Princeton University. The rules are really simple. Each player has an allocated color, Red and Blue being conventional.… Continue reading Fun with Hex
The ultimate prisoner puzzle
People like prisoner puzzles. Here is the hardest one I have ever seen. I heard it from Prof. Boris Bukh on the social hour of Maths Department couple of weeks ago, whom heard it on a maths conference. Since recently I am quite into Cryptography, especially setting up communication protocols, I would like to log… Continue reading The ultimate prisoner puzzle
iShattered
Today, probably, is not my day. I dropped my phone on the ground when I was rushing to the first recitation. The screen was totally shattered into pieces. And I had no time mourning on the loss of my first smart phone because I had to arrive in the classroom on time. In the afternoon,… Continue reading iShattered
Looper
昨天去影院看了 Looper，虽然已经上映超过一周，但是影院还是坐得满满的。 影片中间有段很混乱的地方，我感觉大致说的是，在另外一个平行宇宙，男主角一枪干掉被传送回来的未来的自己，他继续活得好好地，在上海找了个媳妇，过上幸福日子之后，又被传送回了那个主人公没有开枪的平行宇宙了。所以无论如何，都是没有开枪的宿命。 想到宿命论的时候，就又觉得特别无力了。