已經開 sem 一個星期，再加埋前段時間都冇乜再喺個 blog 打野，原因大概都係就我嘅人生，唔係科大就係屋企。所以基本上都冇乜事情可以分享...。我亦大概都會愈黎愈少喺呢個 blog 分享問題，除非對某些問題有另一種見解，從而推出更多有趣嘅 fact。
上個 sem 一科 analysis 都冇，呢個 sem 同上個 sem 最大分別係大部分都係同 analysis 有關。 而且意外自己會走去 take COMP170，我轉性了。
上左兩堂 323，撞到兩個 year 1 sit 323 嘅堂 (自愧當年自己連 topology 有乜 sit 嘅價值都唔知)。為左令到佢地可以有恆心繼續 sit 落去。我決定每星期都贈送一條同 202 進度有關嘅問題。既然教到 differentiation，我就 ``求其" 搵兩條出黎。其實 d 問題大概唔係自己喺 problem book 到抽，就係上 mathxxxxs 搵，我有信心就算自己冇某個 topic，都可以搵到有趣嘅問題黎問 (不過我自己癖好係自己 solve 到先會問人，所以有一兩堂俾唔到問題都唔奇)。
最後，我應該唔會申請人口普查份工。原因係我想嘗試讀完呢個 sem d 野去搵 L 教授做一 d research project。不過我都係要問清楚 d 先，因為我只係隱約記得研究嘅內容同 differential equation 嘅解嘅存在性有關。
Tuesday, February 15, 2011
Thursday, November 4, 2010
Just for fun
Some one ask me to count the number of element in the set $ \{\sigma\in S_5:|\sigma|=2\}$, after doing that I follow the same idea to count $ |\{\sigma\in S_n:|\sigma|=2\}|$, and a step further I count $ |\{\sigma\in S_n:|\sigma|=p\}|$, $ p\leq n$, and I get \[\sum_{k=1}^{[\frac{n}{p}]}\frac{\big((p-1)!\big)^{k}}{(k)!}\prod_{r=0}^{k-1}\binom{n-pr}{p}.\]
The result is summarized in the following document (I hope the result is true):
http://ihome.ust.hk/~cclee/document/sthgen.pdf
To ``verify" the result, let's count $ L(n,3)$ in another way, we count this by considering $ a_n$ defined by
\[a_n=\frac{1}{3}\times \underbrace{\binom{n}{2}}_{\text{choose 2 elements}\atop \text{to form permutation}}\underbrace{\binom{2}{1}}_{\text{choose 1 of the first two}\atop\text{ chosen numbers}}\underbrace{\binom{n-2}{1}}_{\text{choose another 1 to form permutation}\atop\text{with the number in 2C1 }}.\]
The factor $ \frac{1}{3}$ is left there becasue we observe that every length 3 permutation can be written as product of 2 transpositions in exactly 3 ways.
Then clearly the number of ways to form order $ 3$ permutation by multiplying $ k$ disjoint length 3 cycles is given by \[ \frac{\prod_{r=0}^{k-1}a_{n-3r}}{(k)!}=\frac{2^{k}}{(k)!}\prod_{r=0}^{k-1}\binom{n-3r}{3},
\] exactly the same summand appear.
The result is summarized in the following document (I hope the result is true):
http://ihome.ust.hk/~cclee/document/sthgen.pdf
To ``verify" the result, let's count $ L(n,3)$ in another way, we count this by considering $ a_n$ defined by
\[a_n=\frac{1}{3}\times \underbrace{\binom{n}{2}}_{\text{choose 2 elements}\atop \text{to form permutation}}\underbrace{\binom{2}{1}}_{\text{choose 1 of the first two}\atop\text{ chosen numbers}}\underbrace{\binom{n-2}{1}}_{\text{choose another 1 to form permutation}\atop\text{with the number in 2C1 }}.\]
The factor $ \frac{1}{3}$ is left there becasue we observe that every length 3 permutation can be written as product of 2 transpositions in exactly 3 ways.
Then clearly the number of ways to form order $ 3$ permutation by multiplying $ k$ disjoint length 3 cycles is given by \[ \frac{\prod_{r=0}^{k-1}a_{n-3r}}{(k)!}=\frac{2^{k}}{(k)!}\prod_{r=0}^{k-1}\binom{n-3r}{3},
\] exactly the same summand appear.
Friday, September 17, 2010
305 愈讀愈頭痛……
冇得返轉頭……，我應該要開始將成本嚴教授嘅 topology 啃左佢。(好彩佢個網有齊 homework 答案) 摺拉時數將創我自己新高，得閒記住探我。(睇黎又會好似 year 1 咁淨係顧 204 唔理其他科)
Thursday, September 9, 2010
Some counting
Math 110 students are learning counting, now here is a suitable problem from HKgolden.
Problem. There are 2 guys, A and B, and there is a box containing 9000 balls numbered 1000, 1001, ..., 9999 respectively. Suppose A has chosen a ball with replacement, what is the probability that B chooses a ball with no digit in common with that of A? (Say if A has chosen 1234, then B will choose a number that does not contain any digit in {1, 2, 3, 4})
Sunday, September 5, 2010
開學
期待嘅 fall sem 又開始喇，呢幾日叫做教得幾快 (以第一堂黎計) 嘅 course 可以叫 MATH321 及 MATH305。
(除非特別標明，以下所有 course code 均指 MATH)
星期三. 因為冇野做去 sit 下 217。全埸爆滿...，好多人 sit，原來 A 教授有一種規則，就係你喺佢個 first quiz 到 ``do well" 就可以去 reg 佢個 course (不需 al pure A 或 024 (或其他) 最少 A-)，難怪咁多人 sit 喇。不過聽講 203 一樣好多人 sit，啲學生未免太過份喇，搞到原本班學生冇位……。
今年 203 textbook 係用 rudin，呢屆真係好幸福，第一年就由 metric space 開始講起。有好有壞喇，以嚴教授果份 notes 黎教嘅話我最後學得好少 point-set topology on metric space (大部分相關知識都係李教授嘅 370 到學返黎的)，但相對地我開始接觸 305 嘅野 (204 的 multivariable differentiation 是煉獄的開始，linear algebra 不斷湧現...)。其實我懷疑 204 本身係 introduction to 305 (類似 202 係 introduction to 301)。
呢一日去到中午上 PHYS121，重點都係要知道交功課果個流程，其他都係一堆唔駛理嘅野...。
星期四. 如上一篇 entry 所見...，比較多堂嘅一日。321、305 李教授 (2號) 做左少少比較 deep 嘅 introduction。321 人真係好多...，305 反而好少人 sit (奇怪)。
305 第一堂李教授 (2號) 就恐嚇我地 ``Unlike MATH321, this is not a course for everyone, this course is for advanced student or student with strong background", ``This course is for student who wants to be a professional mathematician in the future" 云云。我諗...，呢啲說話只會令到學生鬥志更加激昂！
星期五. 冇乜特別，number theory...，有見識過出面嘅數學比賽都知道初等數論係非常困難...，論證雖然優美、簡單，但愈簡單嘅野對智力需求愈高 (追求初等證明是好習慣)，希望應付得黎。
(除非特別標明，以下所有 course code 均指 MATH)
星期三. 因為冇野做去 sit 下 217。全埸爆滿...，好多人 sit，原來 A 教授有一種規則，就係你喺佢個 first quiz 到 ``do well" 就可以去 reg 佢個 course (不需 al pure A 或 024 (或其他) 最少 A-)，難怪咁多人 sit 喇。不過聽講 203 一樣好多人 sit，啲學生未免太過份喇，搞到原本班學生冇位……。
今年 203 textbook 係用 rudin，呢屆真係好幸福，第一年就由 metric space 開始講起。有好有壞喇，以嚴教授果份 notes 黎教嘅話我最後學得好少 point-set topology on metric space (大部分相關知識都係李教授嘅 370 到學返黎的)，但相對地我開始接觸 305 嘅野 (204 的 multivariable differentiation 是煉獄的開始，linear algebra 不斷湧現...)。其實我懷疑 204 本身係 introduction to 305 (類似 202 係 introduction to 301)。
呢一日去到中午上 PHYS121，重點都係要知道交功課果個流程，其他都係一堆唔駛理嘅野...。
星期四. 如上一篇 entry 所見...，比較多堂嘅一日。321、305 李教授 (2號) 做左少少比較 deep 嘅 introduction。321 人真係好多...，305 反而好少人 sit (奇怪)。
305 第一堂李教授 (2號) 就恐嚇我地 ``Unlike MATH321, this is not a course for everyone, this course is for advanced student or student with strong background", ``This course is for student who wants to be a professional mathematician in the future" 云云。我諗...，呢啲說話只會令到學生鬥志更加激昂！
星期五. 冇乜特別，number theory...，有見識過出面嘅數學比賽都知道初等數論係非常困難...，論證雖然優美、簡單，但愈簡單嘅野對智力需求愈高 (追求初等證明是好習慣)，希望應付得黎。
Sunday, August 29, 2010
Confirmed enrollment
Mon
|
Tue
|
Wed
|
Thu
|
Fri
| |
09:00 - 09:20
|
MATH321 L1
(1504) |
MATH321 L1
(1504) |
MATH311 T1C
(2463) | ||
09:30 - 09:50
| |||||
10:00 - 10:20
| |||||
10:30 - 10:50
|
PHYS121 LA2
(6137) |
MATH305 L1
(1504) |
MATH305 L1
(1504) | ||
11:00 - 11:20
| |||||
11:30 - 11:50
| |||||
12:00 - 12:20
|
MATH315 L1
(4504) | ||||
12:30 - 12:50
| |||||
13:00 - 13:20
|
LANG208 T12
(5561) |
LANG208 T12
(5561) | |||
13:30 - 13:50
|
PHYS121 L1
(4504) |
PHYS121 L1
(4504) | |||
14:00 - 14:20
|
PHYS121 T1
(2306) | ||||
14:30 - 14:50
| |||||
15:00 - 15:20
|
MATH315 T1B
(3584) |
MATH311 L1
(2465) |
MATH311 L1
(2465) | ||
15:30 - 15:50
| |||||
16:00 - 16:20
| |||||
16:30 - 16:50
|
MATH315 L1
(4504) | ||||
17:00 - 17:20
| |||||
17:30 - 17:50
| |||||
18:00 - 18:20
|
MATH321 T1A
4620 |
MATH305 T1A
(4503) | |||
18:30 - 18:50
|
Saturday, August 14, 2010
Record of some solved inequalities
可嘗試以下問題 (1, 2, 4, 6, 7, 8 都是 AL 知識範圍內)，請不要使用暴力的方法 (暴力通分) 解決問題。
Problem 1. Let $ a,b,c>0$ and $ abc=1$. Prove that \[\frac{a}{a+b+1}+\frac{b}{b+c+1}+\frac{c}{c+a+1}\geq 1.\]
Problem 2. Let $ x,y,z>0$; $ x+y+z=1$ prove that \[\sqrt{\frac{x}{yz}}+\sqrt{\frac{y}{zx}}+\sqrt{\frac{z}{xy}}\ge 2\left(\sqrt{\frac{x}{(x+y)(x+z)}}+\sqrt{\frac{y}{(y+z)(y+x)}}+\sqrt{\frac{z}{(z+x)(z+y)}}\right).\]
以下雖然不難，卻十分漂亮，值得牢記！經驗告訊我 $ ab+bc+ca$ 和 $ a+b+c$ 也是十分常見的因子。
Problem 3. Prove that for any $a,b,c\ge 0$, we always have \[
9(a+b)(b+c)(c+a)\ge 8(a+b+c)(ab+bc+ca)
\] and \[ (a+b+c)(a^{2}+b^{2}+c^{2})+9abc\ge 2(a+b+c)(ab+bc+ca).
\]
Problem 4. When $ a+b+c=3$, $ a,b,c\ge 0$, prove that \[\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab}\ge 3.\]
Problem 5. Let $ a,b,c>0$, show that \[a^{2}+b^{2}+c^{2}+2abc+1\ge 2(ab+bc+ac).\]
Problem 6. Let $ x,y,z>0$, prove that \[\frac{xy}{x^{2}+y^{2}+2z^{2}}+\frac{yz}{y^{2}+z^{2}+2x^{2}}+\frac{zx}{z^{2}+x^{2}+2y^{2}}\leq\frac{3}{4}.\]
Problem 7. Let $ a,b,c$ be positive real numbers such that $ abc=1$. Prove that \[\frac{1}{a+b^{2}+c^{3}}+\frac{1}{b+c^{2}+a^{3}}+\frac{1}{c+a^{2}+b^{3}}\leq 1.\]
Problem 8. $ a,b,c$ are real positive numbers, prove that \[\frac{ab}{c(c+a)}+\frac{bc}{a(a+b)}+\frac{ca}{b(b+c)} \geq \frac{a}{c+a}+\frac{b}{a+b}+\frac{c}{b+c}.\]
Problem 1. Let $ a,b,c>0$ and $ abc=1$. Prove that \[\frac{a}{a+b+1}+\frac{b}{b+c+1}+\frac{c}{c+a+1}\geq 1.\]
Problem 2. Let $ x,y,z>0$; $ x+y+z=1$ prove that \[\sqrt{\frac{x}{yz}}+\sqrt{\frac{y}{zx}}+\sqrt{\frac{z}{xy}}\ge 2\left(\sqrt{\frac{x}{(x+y)(x+z)}}+\sqrt{\frac{y}{(y+z)(y+x)}}+\sqrt{\frac{z}{(z+x)(z+y)}}\right).\]
以下雖然不難，卻十分漂亮，值得牢記！經驗告訊我 $ ab+bc+ca$ 和 $ a+b+c$ 也是十分常見的因子。
Problem 3. Prove that for any $a,b,c\ge 0$, we always have \[
9(a+b)(b+c)(c+a)\ge 8(a+b+c)(ab+bc+ca)
\] and \[ (a+b+c)(a^{2}+b^{2}+c^{2})+9abc\ge 2(a+b+c)(ab+bc+ca).
\]
Problem 4. When $ a+b+c=3$, $ a,b,c\ge 0$, prove that \[\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab}\ge 3.\]
Problem 5. Let $ a,b,c>0$, show that \[a^{2}+b^{2}+c^{2}+2abc+1\ge 2(ab+bc+ac).\]
Problem 6. Let $ x,y,z>0$, prove that \[\frac{xy}{x^{2}+y^{2}+2z^{2}}+\frac{yz}{y^{2}+z^{2}+2x^{2}}+\frac{zx}{z^{2}+x^{2}+2y^{2}}\leq\frac{3}{4}.\]
Problem 7. Let $ a,b,c$ be positive real numbers such that $ abc=1$. Prove that \[\frac{1}{a+b^{2}+c^{3}}+\frac{1}{b+c^{2}+a^{3}}+\frac{1}{c+a^{2}+b^{3}}\leq 1.\]
Problem 8. $ a,b,c$ are real positive numbers, prove that \[\frac{ab}{c(c+a)}+\frac{bc}{a(a+b)}+\frac{ca}{b(b+c)} \geq \frac{a}{c+a}+\frac{b}{a+b}+\frac{c}{b+c}.\]
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