In this post I have proved the result that \[
\overline{\sin\mathbb{Z}}=\overline{\cos\mathbb{N}}=[-1,1]
\] by purely group theoretical approach. Since I am going to make tutorial notes for the Real Analysis class next semester, I rethink about this problem and further improve the result such that \[
\overline{\sin\mathbb{N}}=[-1,1],
\] based on the findings in that post.
Now by the density we can find intergers $ n_k$ such that $ \sin n_k\to 0$. Let $ p_{k}=|n_k|$, then $ \mathop{\mathrm{sgn}}(n_k) \sin n_k = \sin p_k\to 0$. Pick an $ a\in [-1,1]$, we now show that $a\in \sin \mathbb N$. First of all, there is a sequence of integers $\{h_k\}$ such that $\sin h_k\to a$. Recall the identity that \[\sin x - \sin y = 2\cos \frac{x+y}{2}\sin \frac{x-y}{2},\] we have for each $ k$, \[
\sin (h_k+2p_i)-\sin h_k=2\cos (h_k+p_i)\sin p_i\to 0 \quad i\to \infty,\] meaning that $\displaystyle\lim_{i\to\infty}\sin (h_k+2p_i) = \sin h_k$.
Now we are almost done, for each $n\in \mathbb N$, we can find an $ k_n$ such that $ |\sin h_{k_n} -a|<1/n$. Fix this $ k_n$, by the last limit we obtained, we can find an index $ i_n$ such that $ |\sin (h_{k_n}+2p_{i_n})-\sin h_{k_n}|<1/n$ and $ P_n:= h_{k_n}+2p_{i_n} > 0$. Hence \[|\sin P_n-a|\leq |\sin P_n-\sin h_{k_n}|+|\sin h_{k_n} - a|<2/n,\] we conclude $ a\in \overline{\sin \mathbb N}$.
Consequence: The subsequential limits of $ \{\sin n:n\in \mathbb N\}$ are precisely those in $ [-1,1]$.
Wednesday, August 22, 2012
Friday, July 27, 2012
灰
今天如常地搞 workshop，由一開始的 5 人，減少至今天的 3 人。有一班 (3 個) 原打算今天來的，但因為種種原因今天不想來。沒差，3 個就 3 個，氣氛還算不錯，一氣呵成把第 4 課講完。
希望下一次 workshop 人數不會比今天更低。以下為 ``syllabus"。
希望下一次 workshop 人數不會比今天更低。以下為 ``syllabus"。
Monday, July 23, 2012
Some Example from the net
Drawing midway arrow:
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
Thursday, July 19, 2012
fix the comma in mathptmx
\usepackage{mathptmx}
\mathcode`,"8000
\begingroup
\lccode`\~`\,\lowercase{\endgroup
\def~{\mathpunct{\textrm{,}}}}
\usepackage{eucal}
\begin{document}
\showoutput
,$,$
\end{document}
Monday, June 25, 2012
Tikz 的軟件
無意中發現一套名為 tikzedt 的軟件。
http://www.tikzedt.org/quicktour.html
很方便，不用自己預先設計座標，也把一些搜尋需時的指令放在一個清單上。
為了熟習一下，自製的 vector 版小丑神頭像誔生！
http://ihome.ust.hk/~cclee/document/siuchausang.pdf
http://www.tikzedt.org/quicktour.html
很方便，不用自己預先設計座標，也把一些搜尋需時的指令放在一個清單上。
http://ihome.ust.hk/~cclee/document/siuchausang.pdf
Saturday, June 9, 2012
Record some code
Equations share the same numbering with theorems with ntheorem package
\newtheorem{thm}{Theorem}[section]
\numberwithin{equation}{section}
\makeatletter
\let\c@equation\c@thm
\makeatother
**********
change the font of sections, subsections, chapter to \sf
\usepackage{sectsty}
\allsectionsfont{\sffamily}
\newtheorem{thm}{Theorem}[section]
\numberwithin{equation}{section}
\makeatletter
\let\c@equation\c@thm
\makeatother
**********
change the font of sections, subsections, chapter to \sf
\usepackage{sectsty}
\allsectionsfont{\sffamily}
Wednesday, June 6, 2012
Updated notes in analysis (PW: My ITSC)
I am expanding the notes last year (still working in progress, and should be finished at the beginning of July):
http://ihome.ust.hk/~cclee/document/WorkshopinAnalysis.pdf
Last year I stopped at littlewood's 3 principles (last section in measurable functions). Having the preparation last year, I plan to quickly go through abstract outer measure and measure (skip some of the proofs which can be copied word by word) and go directly into some important extension theorem on "pre-measure" and "pre-sigma algebra" (in my case, semi-ring), they will be used in the construction of product measure.
My schedule is the following sequence:
Basic Measure Theory
1. Outer measure and Measure
2. Integration
3. Construction of particular measure, go back to R^n
4. ???
5. ???
6. Duality of L^p space on sigma-finite measure space
(possibly need Radon–Nikodym, so 4, 5 may be signed measure and relevant theorems, that's will be big task).
Measure and Topology
7. Brief review of topology material (if necessary)
8. Facts and measure on locally compact Hausdorff space
9. Try to prove the Riesz-Markov theorem
http://en.wikipedia.org/wiki/Riesz_representation_theorem
Hopefully (???) if I have time I plan to apply knowledge in functional analysis and the Riesz-Markov theorem to show Haar measure always exists on compact group, which is unique up to a +ve-constant.
The workshop will begin at the middle of July, and as before, I would ask Kin Li to book me a room. If you are to-be year 2 students and also want to join us, it would be better to digest the material in chapter 2 and 3. We will apply some of the technique implicitly.
http://ihome.ust.hk/~cclee/document/WorkshopinAnalysis.pdf
Last year I stopped at littlewood's 3 principles (last section in measurable functions). Having the preparation last year, I plan to quickly go through abstract outer measure and measure (skip some of the proofs which can be copied word by word) and go directly into some important extension theorem on "pre-measure" and "pre-sigma algebra" (in my case, semi-ring), they will be used in the construction of product measure.
My schedule is the following sequence:
Basic Measure Theory
1. Outer measure and Measure
2. Integration
3. Construction of particular measure, go back to R^n
4. ???
5. ???
6. Duality of L^p space on sigma-finite measure space
(possibly need Radon–Nikodym, so 4, 5 may be signed measure and relevant theorems, that's will be big task).
Measure and Topology
7. Brief review of topology material (if necessary)
8. Facts and measure on locally compact Hausdorff space
9. Try to prove the Riesz-Markov theorem
http://en.wikipedia.org/wiki/Riesz_representation_theorem
Hopefully (???) if I have time I plan to apply knowledge in functional analysis and the Riesz-Markov theorem to show Haar measure always exists on compact group, which is unique up to a +ve-constant.
The workshop will begin at the middle of July, and as before, I would ask Kin Li to book me a room. If you are to-be year 2 students and also want to join us, it would be better to digest the material in chapter 2 and 3. We will apply some of the technique implicitly.
Monday, June 4, 2012
Workhop in Analysis
正準備第二次 workshop 的 notes。上一個暑假邊教邊製作 notes，進度緩慢之餘，到後期沒有動力由頭 develop 那些已經用慣的 theorem (所以沒有教 integration)。有了上次經驗，今次 workshop 在 abstract outer measure, measure 還有 measurable function 的 proof 不用重覆再教 (照抄可也)，所以 focus 可放在 extension of set function 和 integration。因為所認識的同學都有學 topology，我打算一起學 locally compact hausdorff space 和在這類 space 上的 measure theory。一年前很``粗略"地讀過一次，故大概知道這些 theory ``發生甚麼事"。正打算以 rudin 的 approach 學學看...。
Tuesday, May 22, 2012
本日摘要
朝早 9 點， in UST IA 前 check MATH4221(320) 卷，份卷考得唔理想。咁 check 卷時不忘吹水，有其他同學問我差咩科未考，我話仲有科 N展。個同學問返我邊科 N展 黎，Avery 答：「Civl 黎架嘛。」我：「...？！」追問下佢話有``好奇心"所以 search 我 facebook =.=。
去到 10 點，開始去 in IA，十點半先正式到我 in。今日嘅衣着我以為 dress code smart casual 就得，咁我就 casual 少少，最後俾 Priscilla 話係最 casual 果個。而我嘅前一位 interviewee 係着西裝...。interview 過程 ... 超快結束。
**********
我啊姨教我啊媽玩 facebook，拿拿臨將 facebook 所有野變哂做 friend 先睇得。唔係驚有野俾佢睇到，而家我 d 野佢睇完都唔知係咩黎，廢事佢知 d 唔知 d 又喺到問長問短。
去到 10 點，開始去 in IA，十點半先正式到我 in。今日嘅衣着我以為 dress code smart casual 就得，咁我就 casual 少少，最後俾 Priscilla 話係最 casual 果個。而我嘅前一位 interviewee 係着西裝...。interview 過程 ... 超快結束。
**********
我啊姨教我啊媽玩 facebook，拿拿臨將 facebook 所有野變哂做 friend 先睇得。唔係驚有野俾佢睇到，而家我 d 野佢睇完都唔知係咩黎，廢事佢知 d 唔知 d 又喺到問長問短。
Friday, May 11, 2012
UG 生涯完結
今日係我 UST year 3 最後一日上學日。本來呢個 sem 應該輕輕鬆鬆，不過嫌自己太少野做，所以最後都 take 多少少 course。咁最後一日就 expect 應該專心準備考試就得，但噚日 MATH5112 突如奇來咁整左份 take-home exam #2 ...！而某個 special topic course 又有一份功課係 due 25th of May，自己本身仲有一篇 essay 要寫。又要準備考試，又要做功課。忙死！攞黎！
Sunday, May 6, 2012
Sunday, March 25, 2012
完了
幾天前科大 reject 了我 mphil 的申請，而我亦沒有報讀其他的院校，半年前在想 334 應該會需要更多的人手負責教學，應該沒問題。但我期後聽某些教授說因為 "productivity" 之類的資源 ``計算" 方法，PGS (Postgraduate Studentship) 被削減了，收生也不比上年多。
完了...，沒有 offer，沒有申請任何工作，也沒有考任何語文試，半年後的我會是怎樣呀......。難得辛辛苦苦二年學了不少知識 (第一年比較渾渾噩噩)，好像都白讀了。可能半年後的我不會再接觸數學........。
完了...，沒有 offer，沒有申請任何工作，也沒有考任何語文試，半年後的我會是怎樣呀......。難得辛辛苦苦二年學了不少知識 (第一年比較渾渾噩噩)，好像都白讀了。可能半年後的我不會再接觸數學........。
Monday, March 12, 2012
哈哈...
科 number theory 走向變成 algebra course 的路線。上一堂講到對於 finite field $ F$，若 $ f\in F[X]$ 係 irreducible，$ F[X]/(f)$ 同樣係 finite field。因為以前冇乜點諗過呢類 example，開頭諗都係諗返有 division algorithm 云云，同覺得幾神奇，咁就證到 $ (f)$ 係 maximal。但其實我己經學過喺 PID 裏面所有 prime ideal 都係 maximal ...，而 $ (f)$ prime $ \iff$ $ f$ prime (i.e., irreducible in UFD) ...。學完又係唔記得！
Friday, March 2, 2012
Vocabulary
Cavalier (adj) 不經思考 (e.g. we shouldn't be completely cavalier in doing something)
In the sequel = In the following
haphazard (adj) 無計劃地 (it is not easy to haphazardly guess a functor that ...)
trump (verb) 勝過 e.g.:
You should always follow the style guide given to you by a particular publisher or company. That trumps everything else.
ironclad (adj) 不可打破、不可變動
I know there’re no ironclad rules regarding this subject.
dread to do something 不敢去想做某事 (因太可怕)
have yet to do something 仍未做某事
stubborn (adj.) 頑固
compromise (n.) 妥協方案
intricate (adj.) 難以理解
miraculously (adj.) 神奇地
gratuitous (adj.) 無必要的
e.g. The proof is gratuitously complicated.
honestly (adv.) 老實講
In the sequel = In the following
haphazard (adj) 無計劃地 (it is not easy to haphazardly guess a functor that ...)
trump (verb) 勝過 e.g.:
You should always follow the style guide given to you by a particular publisher or company. That trumps everything else.
ironclad (adj) 不可打破、不可變動
I know there’re no ironclad rules regarding this subject.
dread to do something 不敢去想做某事 (因太可怕)
have yet to do something 仍未做某事
stubborn (adj.) 頑固
compromise (n.) 妥協方案
intricate (adj.) 難以理解
miraculously (adj.) 神奇地
gratuitous (adj.) 無必要的
e.g. The proof is gratuitously complicated.
honestly (adv.) 老實講
Sunday, February 19, 2012
Record some code
Use xeCJK and mtpro2 at the same time:
\documentclass[cm-default,no-math]{article}
\usepackage[BoldFont,SlantFont]{xeCJK}
\setCJKmainfont[Mapping=tex-text]{PMingLiU}
\setmainfont[Mapping=tex-text]{Times New Roman}
\usepackage[scaled=0.92]{helvet}
\usepackage[lite,subscriptcorrection,nofontinfo,zswash]{mtpro2}
\straightbraces
\documentclass[cm-default,no-math]{article}
\usepackage[BoldFont,SlantFont]{xeCJK}
\setCJKmainfont[Mapping=tex-text]{PMingLiU}
\setmainfont[Mapping=tex-text]{Times New Roman}
\usepackage[scaled=0.92]{helvet}
\usepackage[lite,subscriptcorrection,nofontinfo,zswash]{mtpro2}
\straightbraces
Tuesday, February 14, 2012
Material of presentation for SVD
Prepared a draft for my presentation:
http://ihome.ust.hk/~cclee/document/PRESENTSVD.pdf
The proof in the original text is too short and seemingly incomplete to me, so I modified the proof and filled in many details. Let me first describe terminology used in this notes, and let me for the moment describe everything in $ \mathbb R$ instead of $ \mathbb C$. Let $ \mathbb{R}^{m\times n}$ denote the collection of $ m\times n$ dimensional real matrices. For $ x\in \mathbb R^k$, write $ x = (x^1,x^2,\dots,x^k)$ (i.e., $ x^k$ is the $ k$th coordinate of $ x$), define $ \|x\|_2=\sqrt{\sum_{i=1}^k |x^i|^2}$.
We also define the $ (n-1)$-sphere to be $ S^{n-1}:=\{x\in \mathbb R^n:(x^1)^2+\cdots+(x^n)^2=1\}$. For $ A\in \mathbb R^{m\times n}$ we define the operator norm
\[ \|A\|_2= \sup_{x\in \mathbb R^n, \|x\|_2=1}\|Ax\|_2.\]
We assume, and what we are trying to prove in my draft, that $ AS^{n-1}=\{Ax:x\in S^{n-1}\}$ is a ``hyperellipse" (or just an ellipse when $ n \leq 3$). Suppose also that $ m\ge n$ and $ A$ is of full rank, then it is legitimate to assume there are unit vectors $ u_1,\dots,u_n$ which point in the direction of semi-axises of the hyperellipse.
We can describe a general construction for $ u_i$'s: first find a vector in $ AS^{n-1}$ that has largest length, call it $ a\in \mathbb R^{m}$, then find another vector $ b$ in $ AS^{n-1} \cap (\mathbb R a)^\perp$ that has largest length, then find another such vector in $ AS^{n-1} \cap (\mathbb R a)^\perp\cap (\mathbb R b)^\perp$ and continue the process until we find $ n$ such vectors. This is indeed how we find semi-axises of an ellipse in $ \mathbb R^3$ and $ \mathbb R^2$.
Now in the way above, the length of semi-axis in the direction of $ u_i$, $ \sigma_i$, is in nonincreasing order, i.e., $ \sigma_1\ge \sigma_2\ge\cdots\ge \sigma_n>0$. Take the unit vector $ v_i \in A^{-1}(\sigma_i u_i)$, one gets \[A\underbrace{\left[\begin{array}{c|c|c}v_1&\cdots&v_n\end{array}\right]}_{:=V}=\left[\begin{array}{c|c|c}u_1&\cdots&u_n\end{array}\right]\begin{bmatrix}\sigma_1&& \\ &\ddots & \\ & & \sigma_n\end{bmatrix}.\]
Let $ u_{n+1},\dots, u_m$ be orthonormal basis in $ (\mathrm{span}_{\mathbb R}\{u_1,\dots,u_n\})^\perp$, then RHS of the above equation becomes
\[\underbrace{\left[\begin{array}{c|c|c}u_1&\cdots&u_m\end{array}\right]}_{:= U}\underbrace{\left[\begin{array}{ccc}\sigma_1&& \\ &\ddots & \\ & & \sigma_n \\ \hline &\mathcal O& \end{array}\right]}_{:=\Sigma},\]
where $ \mathcal O$ denotes a matrix with only 0 entries. One can rewrite the above as: $ AV = U\Sigma$. It will be proved in my draft that indeed $ \{v_i\}$ is orthonormal, hence we can conclude both $ V,U$ are unitary, and we arrive to the expression $ A=U\Sigma V^*$. Owing to this decomposition those $ u_i$'s are called left-singular vector, $ v_i$'s are called right-singular vector and ``diagonal" elements (i.e., if $ \Sigma=(d_{ij})$, diagonal elements are $ d_{ii}$'s) are called singular values. These basically are all the motivation of the general result in my draft.
http://ihome.ust.hk/~cclee/document/PRESENTSVD.pdf
The proof in the original text is too short and seemingly incomplete to me, so I modified the proof and filled in many details. Let me first describe terminology used in this notes, and let me for the moment describe everything in $ \mathbb R$ instead of $ \mathbb C$. Let $ \mathbb{R}^{m\times n}$ denote the collection of $ m\times n$ dimensional real matrices. For $ x\in \mathbb R^k$, write $ x = (x^1,x^2,\dots,x^k)$ (i.e., $ x^k$ is the $ k$th coordinate of $ x$), define $ \|x\|_2=\sqrt{\sum_{i=1}^k |x^i|^2}$.
We also define the $ (n-1)$-sphere to be $ S^{n-1}:=\{x\in \mathbb R^n:(x^1)^2+\cdots+(x^n)^2=1\}$. For $ A\in \mathbb R^{m\times n}$ we define the operator norm
\[ \|A\|_2= \sup_{x\in \mathbb R^n, \|x\|_2=1}\|Ax\|_2.\]
We assume, and what we are trying to prove in my draft, that $ AS^{n-1}=\{Ax:x\in S^{n-1}\}$ is a ``hyperellipse" (or just an ellipse when $ n \leq 3$). Suppose also that $ m\ge n$ and $ A$ is of full rank, then it is legitimate to assume there are unit vectors $ u_1,\dots,u_n$ which point in the direction of semi-axises of the hyperellipse.
We can describe a general construction for $ u_i$'s: first find a vector in $ AS^{n-1}$ that has largest length, call it $ a\in \mathbb R^{m}$, then find another vector $ b$ in $ AS^{n-1} \cap (\mathbb R a)^\perp$ that has largest length, then find another such vector in $ AS^{n-1} \cap (\mathbb R a)^\perp\cap (\mathbb R b)^\perp$ and continue the process until we find $ n$ such vectors. This is indeed how we find semi-axises of an ellipse in $ \mathbb R^3$ and $ \mathbb R^2$.
Now in the way above, the length of semi-axis in the direction of $ u_i$, $ \sigma_i$, is in nonincreasing order, i.e., $ \sigma_1\ge \sigma_2\ge\cdots\ge \sigma_n>0$. Take the unit vector $ v_i \in A^{-1}(\sigma_i u_i)$, one gets \[A\underbrace{\left[\begin{array}{c|c|c}v_1&\cdots&v_n\end{array}\right]}_{:=V}=\left[\begin{array}{c|c|c}u_1&\cdots&u_n\end{array}\right]\begin{bmatrix}\sigma_1&& \\ &\ddots & \\ & & \sigma_n\end{bmatrix}.\]
Let $ u_{n+1},\dots, u_m$ be orthonormal basis in $ (\mathrm{span}_{\mathbb R}\{u_1,\dots,u_n\})^\perp$, then RHS of the above equation becomes
\[\underbrace{\left[\begin{array}{c|c|c}u_1&\cdots&u_m\end{array}\right]}_{:= U}\underbrace{\left[\begin{array}{ccc}\sigma_1&& \\ &\ddots & \\ & & \sigma_n \\ \hline &\mathcal O& \end{array}\right]}_{:=\Sigma},\]
where $ \mathcal O$ denotes a matrix with only 0 entries. One can rewrite the above as: $ AV = U\Sigma$. It will be proved in my draft that indeed $ \{v_i\}$ is orthonormal, hence we can conclude both $ V,U$ are unitary, and we arrive to the expression $ A=U\Sigma V^*$. Owing to this decomposition those $ u_i$'s are called left-singular vector, $ v_i$'s are called right-singular vector and ``diagonal" elements (i.e., if $ \Sigma=(d_{ij})$, diagonal elements are $ d_{ii}$'s) are called singular values. These basically are all the motivation of the general result in my draft.
Saturday, February 11, 2012
嗯...
最近發覺和不會主動接觸任何 pure math 的 (UG) applied math 學生相處是一件非常困難的事。我認為十分 concrete 和自然的 example 他們也會感動反感，abstract 一點 (實際上也不抽像...) 他們會感到不實際，沒有意義。難道 applied math 只須學習 compute，compute 和 compute 嗎？費解。
我做 presentation，多說一點，多舉一些例子，只務求聽眾能學得深入。抽像的東西不一定是難，我反而覺得 concrete 的東西很多時更難。學會抽像的部分能加深對 concrete case 的了解。讀 algebra 時就是很多 concrete case 更需要思考呀...。
我做 presentation，多說一點，多舉一些例子，只務求聽眾能學得深入。抽像的東西不一定是難，我反而覺得 concrete 的東西很多時更難。學會抽像的部分能加深對 concrete case 的了解。讀 algebra 時就是很多 concrete case 更需要思考呀...。
Saturday, February 4, 2012
Confirmed Enrollment
應該唔會有大變動：
CIVL 1160 - Civil Engg & Society
HUMA 1710 - Art of Thinking in HK Context
MATH 3426 - Sampling
MATH 4221 - Euclid & Non-Euclid Geom
MATH 4822A - Intro to Modern Num Theory
MATH 4981D - COMP Linear Algebra
MATH 5112 - Advanced Algebra II
除某二個 4 credit math course 外其他都係 3 credit。因為今個 sem 有幾科 math course 都係冇 tutorial，所以讀 7 科但時間表都唔係太密。
HUMA 1710 - Art of Thinking in HK Context
MATH 3426 - Sampling
MATH 4221 - Euclid & Non-Euclid Geom
MATH 4822A - Intro to Modern Num Theory
MATH 4981D - COMP Linear Algebra
MATH 5112 - Advanced Algebra II
除某二個 4 credit math course 外其他都係 3 credit。因為今個 sem 有幾科 math course 都係冇 tutorial，所以讀 7 科但時間表都唔係太密。
Monday, January 16, 2012
開始自修 Algebra
因為感覺到自己的 algebra 底十分十分弱，所以這個 winter break 的時間都用來磨練一下自己的 algebra。遇到過的 proof 也盡可能自己做一次。有一些重點想記下 (也為了方便回顧)，所以就以 bullet point 形式打了份 notes。
http://ihome.ust.hk/~cclee/document/notes_on_algebra.pdf (notes 不斷更新，鏈結地址不變)
不是所有 proof 都會記下。太長 (e.g. 所有 field 都可以嵌入一個 algebraically closed field) 的不會記下。若證明只需數行又或是雖然非常長但證明技巧有機會再用到的都會記下 (有時用來引證自己記下的重點)。
http://ihome.ust.hk/~cclee/document/notes_on_algebra.pdf (notes 不斷更新，鏈結地址不變)
不是所有 proof 都會記下。太長 (e.g. 所有 field 都可以嵌入一個 algebraically closed field) 的不會記下。若證明只需數行又或是雖然非常長但證明技巧有機會再用到的都會記下 (有時用來引證自己記下的重點)。
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