Mon
|
Tue
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Wed
|
Thu
|
Fri
| |
09:00 - 09:20
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MATH321 L1
(1504) |
MATH321 L1
(1504) |
MATH311 T1C
(2463) | ||
09:30 - 09:50
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10:00 - 10:20
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10:30 - 10:50
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PHYS121 LA2
(6137) |
MATH305 L1
(1504) |
MATH305 L1
(1504) | ||
11:00 - 11:20
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11:30 - 11:50
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12:00 - 12:20
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MATH315 L1
(4504) | ||||
12:30 - 12:50
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13:00 - 13:20
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LANG208 T12
(5561) |
LANG208 T12
(5561) | |||
13:30 - 13:50
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PHYS121 L1
(4504) |
PHYS121 L1
(4504) | |||
14:00 - 14:20
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PHYS121 T1
(2306) | ||||
14:30 - 14:50
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15:00 - 15:20
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MATH315 T1B
(3584) |
MATH311 L1
(2465) |
MATH311 L1
(2465) | ||
15:30 - 15:50
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16:00 - 16:20
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16:30 - 16:50
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MATH315 L1
(4504) | ||||
17:00 - 17:20
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17:30 - 17:50
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18:00 - 18:20
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MATH321 T1A
4620 |
MATH305 T1A
(4503) | |||
18:30 - 18:50
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Sunday, August 29, 2010
Confirmed enrollment
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 \[\frac{ab}{c(c+a)}+\frac{bc}{a(a+b)}+\frac{ca}{b(b+c)} \geq \frac{a}{c+a}+\frac{b}{a+b}+\frac{c}{b+c}.\]
Friday, August 6, 2010
有關 topology 及 D.G. 的 workshop ...
原來這是一個我不應該去的 workshop =.=...。預備知識多，可見到講者很用心澄清每一個數學用語，但這樣便花了大部分時間 ...。到最後我幾乎還未認識甚麼是 manifold :o。
Wednesday, August 4, 2010
Record of solved math202/3-level problems
首兩條的解題方法在 202 我們多次看到，作為回憶從某論壇 copy 下來。
Problem 1. Let $ f:[a,+\infty)\to \mathbb{R}$ a twice differentiable function on the interval $ (a,+\infty)$ with $ \lim_{n\to\infty}f(x)=l\in\mathbb{R}$ and $ |f''(x)|\leq M,\forall x \in [a,+\infty)$. Prove that $ \displaystyle \lim_{x\to\infty}f'(x)=0$.
Problem 2. Let $ f$ be differentiable on $ (a,+\infty)$. If $ \lim_{x\to\infty}\big(f(x)+xf'(x)\ln x\big)=l$, then show that $ \lim_{x\to\infty}f(x)=l$.
以下這一條取自 2010 Department of Mathematics, Graduate School of Science, Kyoto University subjects of tests : Mathematics I
Problem 3*. Let $ f$ be continuous real valued function on $ [0,1]$ and $ f(0)=0, f(1)=1$. Find \[\lim_{n\to\infty}n\int_0^1f(x)x^{2n}\,dx.\]
基本上 $ f(0)=0, f(1)=1$ 是一多餘的 condition。把它們保留下來作為 unknown 可得到一般結果。我想多餘的資料旨在 confuse 解題的人。把 2 換成 $ k$，結果應該差不多！
以下在 analysis section 裏看到這條，可能與 Lagrange multiplier 有關吧。其實只需中學知識。
Problem 4. Let $ x_1,x_2,\dots,x_n$ be positive real number satisfying $ \displaystyle \sum_{k=1}^n\frac{1}{x_k}=n$. Find the minimum of \[
\sum_{k=1}^n\frac{(x_k)^k}{k}.\]
Problem 5. Find $ \displaystyle\lim_{n\to\infty}\left\{n^{2}\left(\int_{0}^{1}\sqrt[n]{1+x^{n}}\,dx-1\right)\right\} $.
Problem 6. Suppose that $ f:\mathbb{R}-\{0,1\}\to \mathbb{R}$ satisfies the equation
$ \displaystyle f(x)+f\left(\frac{x-1}{x}\right)=1+x$, find $ f(x)$.
Problem 7. Let $ f:[0,1]\to\mathbb{R}$ be differentiable on $ [0,1]$ with $ f(1)=0$, show that
$ \displaystyle \lim_{n\to+\infty}n^{2}\int_{0}^{1}f(x)x^{n}\, dx =-f'(1)$.
都是些 202 基本概念便能夠解決的問題。
Problem 1. Let $ f:[a,+\infty)\to \mathbb{R}$ a twice differentiable function on the interval $ (a,+\infty)$ with $ \lim_{n\to\infty}f(x)=l\in\mathbb{R}$ and $ |f''(x)|\leq M,\forall x \in [a,+\infty)$. Prove that $ \displaystyle \lim_{x\to\infty}f'(x)=0$.
Problem 2. Let $ f$ be differentiable on $ (a,+\infty)$. If $ \lim_{x\to\infty}\big(f(x)+xf'(x)\ln x\big)=l$, then show that $ \lim_{x\to\infty}f(x)=l$.
以下這一條取自 2010 Department of Mathematics, Graduate School of Science, Kyoto University subjects of tests : Mathematics I
Problem 3*. Let $ f$ be continuous real valued function on $ [0,1]$ and $ f(0)=0, f(1)=1$. Find \[\lim_{n\to\infty}n\int_0^1f(x)x^{2n}\,dx.\]
基本上 $ f(0)=0, f(1)=1$ 是一多餘的 condition。把它們保留下來作為 unknown 可得到一般結果。我想多餘的資料旨在 confuse 解題的人。把 2 換成 $ k$，結果應該差不多！
以下在 analysis section 裏看到這條，可能與 Lagrange multiplier 有關吧。其實只需中學知識。
Problem 4. Let $ x_1,x_2,\dots,x_n$ be positive real number satisfying $ \displaystyle \sum_{k=1}^n\frac{1}{x_k}=n$. Find the minimum of \[
\sum_{k=1}^n\frac{(x_k)^k}{k}.\]
Problem 5. Find $ \displaystyle\lim_{n\to\infty}\left\{n^{2}\left(\int_{0}^{1}\sqrt[n]{1+x^{n}}\,dx-1\right)\right\} $.
Problem 6. Suppose that $ f:\mathbb{R}-\{0,1\}\to \mathbb{R}$ satisfies the equation
$ \displaystyle f(x)+f\left(\frac{x-1}{x}\right)=1+x$, find $ f(x)$.
Problem 7. Let $ f:[0,1]\to\mathbb{R}$ be differentiable on $ [0,1]$ with $ f(1)=0$, show that
$ \displaystyle \lim_{n\to+\infty}n^{2}\int_{0}^{1}f(x)x^{n}\, dx =-f'(1)$.
都是些 202 基本概念便能夠解決的問題。
!!!
Workshop on Differential Topology & Differential Geometry
The workshop would be based on the lecture notes of MATH323 by Prof. Yan Min, MATH305 by Prof. Li Wei-Ping and MATH321 by Prof. Li Wei-Ping & Prof. Hu Jishan. Some review on Linear Algebra will be covered at the beginning.
The workshop would contain some exercises and discussions, the details are as follows:
Please contact Hoi (hoien14@gmail.com) for the soft copy of notes above.
The workshop would be based on the lecture notes of MATH323 by Prof. Yan Min, MATH305 by Prof. Li Wei-Ping and MATH321 by Prof. Li Wei-Ping & Prof. Hu Jishan. Some review on Linear Algebra will be covered at the beginning.
The workshop would contain some exercises and discussions, the details are as follows:
- Date: Aug 5, 2010 (Thu) & Aug 6, 2010 (Fri)
- Time: 1:00pm-4:00pm
- Venue: Room 3209D (near Lifts 19 & 21, via Room 3209B)
Please contact Hoi (hoien14@gmail.com) for the soft copy of notes above.
Monday, August 2, 2010
今天的星期日生活，有關動漫。
我愛日本漫畫﹕全球化浪潮下 再閱讀日本動漫
【明報專訊】過去幾十年，日本動漫對香港讀者有什麼影響？這個問題，很「通識」，難答，但要急切處理。
通識，是因為多年來，日本動漫在香港累積過百萬讀者，經典動漫如《小甜甜》、《機動戰士》、《美少女戰士》及《鋼之鍊金術師》等亦製造了幾代人的集體回憶；換句話說，日本動漫已紮根港人生活，問動漫對香港讀者的影響，其實自然不過，很「通識」。難答——主流論述動輒把動漫的色情及暴力成分，看成年輕人越軌行為的主因（有媒體就把近日的家庭慘劇和《多啦A夢》掛鹇），而近年亦常把動漫迷等同電車男。這些答案，稍有批判力的人都知道扭曲事實；然而沒有更適切的角度，要另外提供有說服力的答案又很困難。急切——如此「通識」的問題還未有答案，其實要急切處理。
【明報專訊】過去幾十年，日本動漫對香港讀者有什麼影響？這個問題，很「通識」，難答，但要急切處理。
通識，是因為多年來，日本動漫在香港累積過百萬讀者，經典動漫如《小甜甜》、《機動戰士》、《美少女戰士》及《鋼之鍊金術師》等亦製造了幾代人的集體回憶；換句話說，日本動漫已紮根港人生活，問動漫對香港讀者的影響，其實自然不過，很「通識」。難答——主流論述動輒把動漫的色情及暴力成分，看成年輕人越軌行為的主因（有媒體就把近日的家庭慘劇和《多啦A夢》掛鹇），而近年亦常把動漫迷等同電車男。這些答案，稍有批判力的人都知道扭曲事實；然而沒有更適切的角度，要另外提供有說服力的答案又很困難。急切——如此「通識」的問題還未有答案，其實要急切處理。
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