## 2009年12月28日 星期一

### Polynomials and topology

Recently my friend is working on a problem in topology, and out of his work, it appears that the there is a special pattern in the coefficients of the following polynomial, when m,n are relatively prime:

$\frac{(x^{mn}-1)(x-1)}{(x^m-1)(x^n-1)} (x^{4n}-x^{2n}+1)$

It appears that if one expands this polynomial out, collects terms and arranges them in decreasing powers of x, then the non-zero coefficients are all either 1 and -1, and they appear to alternate as the power decreases. (e.g. when m=4, n=3, the polynomial is

$x^{18}-x^{17}+x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x+1$

It is not known whether this pattern really exists. But I thought this is cute and may be of interest to some of you. Does any of you have any idea about how to prove/disprove it?

(The case of interest in topology is when m > 3n, but it looks like this pattern persists as long as m,n are relatively prime.)

## 2009年12月24日 星期四

### Elementary number theory

Someone say that 167588402882520529579353108764873470755823697 is the smallest positive integer k such that all digit of 1989k are the same.

Do you agree?

## 2009年12月19日 星期六

=====================================================

Probability from a gambler's viewpoint --- A taste of Martingale Theory

Suppose you keep flipping a fair coin until 10 heads occur consecutively. How many times of flipping do you need on average?

We will solve this and other related problems as an application of the Optional Stopping Theorem. The basic notions and properties of discrete martingales will be introduced in an informal manner, with emphasis on the intuitive ideas.

Prerequisite: Basic probability in secondary school level.

## 2009年12月11日 星期五

### Two Analysis Problems

Recently I heard two problems in analysis, both I think are interesting, and they do not require too deep knowledge in analysis, which is the kind of questions I like most. Share here.

1) Suppose $\sum_{j=1}^\infty |a_j|$ converges. Also, for each positive integer k, it is known that $\sum_{j=1}^\infty a_{jk} = 0$ (just to avoid confusion, allow me clarify here that jk means "j times k"). Prove that $a_i=0$ for all positive integers i.

2) S contains all elements $x\in[0,1]$ such that for any $\epsilon>0$, there exists a rational number $\frac{p}{q}$ (where p,q are positive integers) satisfying $\left| x - \frac{p}{q}\right| < \frac{\epsilon}{q^3}$. Prove that S is uncountable.

## 2009年11月26日 星期四

### Log Is Everywhere

1) $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\Theta(\ln n)$

2) 一種不穩定的物質進行radioactive decay。若它在k秒內質量由1變成$\frac{1}{d}$，則該物質的half-year為$\frac{k}{\log_2 d}$

3) 絕大部分在現實使用將數字排序（sorting）的算法（algorithm），其運算時間為$\Theta(n\log n)$

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## 2009年11月20日 星期五

### 電子簽名…(上)

(i) 在數碼世界中，到底有冇沒有電子郵票呢？

(ii) 一張証書經掃瞄器傳入電腦後的檔案是電子証書嗎？

A : "Could you put your electronic signature in the document discussed earlier?"
B : "As I don't have any electronic signature, I signed it on the hard copy which your helper is keeping."
A : "What I meant was if you could scan your signature and merge it into the document."
B : (His mind: Ok....WXF...) "Done, the signed document is attached"

(1) 你能理解B先生 心裏浮現 WXF 的理由嗎？
(2) 到底B先生的簽署是否一個電子簽名呢？
(3) B先生最後送出的文件的簽名有法律效力嗎？

ami~wkc

p.s.

## 2009年11月13日 星期五

### Bertrand's Postulate

Bertrand's Postulate says that for any integer n > 1 there is a prime number p such that n < p < 2n. Here's a link to a beautiful and elementary proof of this fact: http://mathforum.org/library/drmath/view/51527.html.

## 2009年11月12日 星期四

### A beautiful solution

Suppose you want to prove that the sum of 1/(m^2 + n^2) diverges as m,n ranges over all positive integers. What do you do?

Here's a very beautiful solution, from one of my students in an undergraduate complex analysis class:

Every prime of the form 4k+1 is expressible as the sum of two squares. Hence the previous sum is bounded below by the sum of 1/p, where p ranges over all primes that are congruent to 1 mod 4. The latter sum diverges. Q.E.D.

## 2009年11月4日 星期三

### Daily Life Application of Number Theory

A message from UC Berkeley (exact situation unknown):

Warning: Due to a known bug, the default Linux document viewer evince prints N*N copies of a PDF file when N copies requested. As a workaround, use Adobe Reader acroread for printing multiple copies of PDF documents, or use the fact that every natural number is a sum of at most four squares.

## 2009年10月23日 星期五

### 數學講座

（如果無法載入，可複製網址 http://hkage.org.hk/b5/new/Students/ma/0910003/poster.pdf）

## 2009年10月14日 星期三

### AM-GM 不等式的幾個證明（六）

$e^x - e^a \geq e^a (x-a)$

$x_k = e^{\log x_k} \geq e^a + e^a (\log x_k - a), \qquad k = 1,\dots,n$

$\frac{x_1+\dots+x_n}{n} \geq e^a + e^a \left(\frac{\log x_1 + \dots + \log x_n}{n} - a\right)$ ------(1)

$a = \frac{\log x_1 + \dots + \log x_n}{n}$，則

$\frac{x_1+\dots+x_n}{n} \geq e^{\frac{\log x_1 + \dots + \log x_n}{n}} = (x_1 \dots x_n)^{\frac{1}{n}}$

$\lambda_1 x_1 + \dots + \lambda_n x_n \geq x_1^{\lambda_1} \dots x_n^{\lambda_n}$

## 2009年10月11日 星期日

### AM-GM 不等式的幾個證明（五）

\begin{align*} &\ a_1b_n+a_2b_{n-1}+\cdots+a_nb_1\\ \le&\ a_1b_{f(1)}+a_2b_{f(2)}+\cdots+a_nb_{f(n)}\\ \le&\ a_1b_1+a_2b_2+\cdots+a_nb_n \end{align*}

$\frac{y_1}{y_2}+\frac{y_2}{y_3}+\cdots+\frac{y_n}{y_1}\ge n$

（不失一般性設 $y_1\le y_2\le\cdots\le y_n$，並設 $a_i=y_i$$b_{n-i}=\frac1{y_i}$，再使用「亂序和 ≧ 逆序和」即可）。

$y_1=\frac1{x_1},\quad y_2=\frac1{x_1x_2},\quad\ldots,\quad y_n=\frac1{x_1x_2\cdots x_n}=1$

$x_1+x_2+\cdots+x_n=\frac{y_n}{y_1}+\frac{y_1}{y_2}+\cdots+\frac{y_{n-1}}{y_n}\ge n$

## 2009年10月6日 星期二

### Gil Kalai's Seminar

Prof. Kalai講了三個猜想，其中一個比較易懂和有趣，可以在這裏講講。

$f:\{-1,1\}^n\rightarrow\{-1,1\}$。我們可以進行一個"Fourier Transform"，即寫成$f(x)=\sum_{S\subset\{1,2,\cdots,n\}} \hat{f}(S) W_S$，當中$W_S=\Pi_{i\in S} x_i$（這與一般Fourier Series中的$e^{i n x}$的角色相同。）假設$x=(x_1,x_2,\cdots,x_n)$，若每一個$x_i$均獨立地有機會率 t 改變值（即由-1變成1或由1變成-1），若$f(x)=f(y)$的機會很高，我們就可說 f 是noise stable。