## 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.