material 準備完畢

今個 sem 準備左兩堆 material，到今個星期終於完成哂～。可以話下面係第 2 代嘅 notes。

Record some Problems

May be used in class of analysis:

Problem. Suppose $f'(x)<0<f''(x)$ for $x<a$; and $f'(x)>0>f''(x)$ for $x>a$. Prove that $f$ is not differentiable at $a$.  

Problem. If $f:\R\to \C^\times$ is a group homomorphism and is continuous, prove that $f(x)=e^{cx}$ for some $c\in \C$.

Problem. Let $C[a,b]$ be the ring of continuous functions on the finite closed interval $[a,b]\subseteq \R$.
(a) For $c\in [a,b]$, prove that $I_c$ given by \[I_c=\{f\in C[a,b]:f(c)=0\}\] is a maximal ideal of $c[a,b]$.
(b) Prove that every maximal ideal of $C[a,b]$ is $I_c$ for a unique $c\in [a,b]$.

Problem. Define a sequence by $a_1 = 1$, $a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$, if it exists.

Problem. Denote $A = \{ x \in \mathbb{R} : \lim_{n\to\infty} \{ 2^n x\} = 0 \}$. Prove that $m(A) = 0$. Is $A$ countable or uncountable?

Notation: $\{x\}$ denotes the fractional part of $x$, given by $\{x\} = x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the unique integer for which $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

Problem. Find $\dis  \prod_{k=0}^\infty(e^{a\cdot 2^{-k}}+e^{-a \cdot 2^{-k}}-1)$.

Problem. Find $\dis \lim_{n\to\infty}\left(\prod_{k=1}^{n}\cos{\frac{\pi k}{2n}}\right)^{\frac{1}{n}}$.

Problem.  Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable ($C^1$) function such that $\dis f(x) = f(\frac{x}{2})+ \frac{x}{2} f'(x)$.

Show that $f(x)$ is a linear function, i.e. $\exists (a,b) \in \mathbb{R}^2$, $f(x)= ax+b$.

Problem. Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that \[\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0.\] Then prove that \[\lim_{N\to\infty}\int_0^\pi\frac{\sin(N+\frac12)x}xdx=\pi/2.\] Finally, prove that \[\int_0^\infty\frac{\sin x}xdx=\pi/2.\]

Problem. Let $a>0$ and $\{x_n\}$ a bounded sequence of real numbers. If $\{x_n\}$ doesn't converge and $\lim_{n \to \infty} (x_{n+1}+ax_n)=0$, prove that $a=1$.
Remark. We can generalize the problem by merely assuming $\lim_{n \to \infty} (x_{n+1}+ax_n)$ exists.

Problem. Let $f: \mathbb{R} \to (0,\infty)$ be a continuous differentiable function such that $f'(x)=f(f(x))$ for all $x\in \mathbb{R}$. Show that there is no such function. 

Problem. Suppose $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that, $|f(x)-f(y)| \ge \frac{1}{2}|x-y|$ for all $x,y \in \mathbb{R}$. Is $f$ necessarily one to one and onto?

Problem. Let $f:[0,1]\to \R$ be continuous and differentiable on $(0,1)$ such that $f(0)>0,f(\frac{1}{2})<0$ and $f(1)>1$. Show that there is a $\xi\in (0,1)$ such that $f'(\xi)=f(\xi)$. 

一些歷史遺物

在 photobucket 那尋回九張作品，暫時年份不明 (可以肯定每一張都在某論壇發表過，但懶得找了 ...)。

The application of Lagrange multiplier's method with multiple constraints to linear algebra

In this post we record an application of a standard result in multivariable calculus (whose rigorous proof is still beyond the standard multivariable calculus course) to proving the following standard fact in linear algebra:

Theorem (Spectral). Let $A$ be an $n\times n$ real symmetric matrix, then $A$ is orthogonally diagonalizable. 

In other words, there is an orthonormal  basis $\{v_1,\dots,v_n\}$ such that each of $v_i$'s is an eigenvector. Througout the proof the following will be assumed:

Theorem (Lagrange's Multiplier). Let $n\ge m$, suppose that:

• $G=(g_1,\dots,g_m):\R^n\to \R^m$ is $C^1$ on $\R^n$ and $G'(x_0)$ has full rank.
•  $f$ is a real-valued function defined near $x_0$ and is differentiable at $x_0$.
• $f$ has a  local extreme at $x_0$ under the constraint $G(x_0)=0$.

 Then there are $\lambda_1,\lambda_2,\dots,\lambda_m\in \R$ such that \begin{equation}\label{equality of lagrange}\nabla f(x_0)=\lambda_1\nabla g_1(x_0)+\lambda_2\nabla g_2(x_0)+\cdots +\lambda_m \nabla g_m(x_0).\end{equation}

A detailed proof of this multiplier method has been recorded in my lengthened version of Math3033 tutorial note 3.

Proof of Spectral Theorem. The maximum of \[
\psi (x) = x^TAx
\] w.r.t. the constraint $x^Tx=1$ can be attained since $S^{n-1}$ is compact. Now it is easy to show that \[\psi'(x)=2x^TA\quad \text{and}\quad \nabla(x^Tx)(x)=2x^T,
\] therefore the existence of the constrained extreme implies that there is $\lambda\in \R$ and a vector $v\in \R^n$ such that

1. $v^Tv=1$.
2. $\nabla \psi (v)=\lambda\nabla (x^Tx-1)(v)\iff 2v^TA=2\lambda v^T\iff Av=\lambda v$.

Therefore we have obtained our first eigenvalue and eigenvector.

Now in view of the statement of Spectral Theorem let's suppose that there are already $\lambda_1,\dots,\lambda_k\in \R$ and $v_1,\dots,v_k$ orthonormal such that $Av_i=\lambda_iv_i$ for each $i=1,\dots,k$. We try to find $\mu\in \R$ and $u\in S^{n-1}$ such that $Au=\mu u$ and $v_i\cdot u=0$ for all $1\leq i\leq k$. After that our proof is completed by induction. 

Following the same idea in the first paragraph of the proof we know that the constrained extreme problem with the following data can be attained: \[ x^TAx = \text{max},\quad x^Tx=1,\quad v_i^Tx=0,\forall i=1,\dots,k.
\] Therefore there are $\mu_1,\dots,\mu_k,\mu\in \R$ and a $u\in\R^n$ satisfying the constraints such that \[
\nabla  (x^TAx)(u)=  \mu \nabla(x^Tx-1)(u)+\sum_{i=1}^k \mu_i\nabla(v_i^Tx)(u),
\] on simplification we get \[
2u^TA=2 \mu u^T+\sum_{i=1}^k \mu_i v_i^T.
\] From our hypothesis in the last paragraph, by applying $v_i$'s on both sides above we have for $i=1,2,\dots,k$, \[
0=0+\mu_i \|v_i\|^2,
\] therefore $\mu_1=\mu_2=\cdots=\mu_k=0$, and we have \[
2u^TA=2\mu u^T\iff Au=\mu u.
\] Recall that $u$ satisfies the constraints $v_i\cdot u=0$ for $i=1,\dots,k$ and $u\cdot u=1$.$\qed$

重新開張

Wordpress 說起

大概從中五開始用 blog (xanga 時期)，那時亦開始間中打一些數學 post，到中七時為方便輸入數式而轉往 wordpress。在我認知裏，方便打 math 的 blog 只有 blogger 及 wordpress。而我一開始是試用 wordpress 的，因而沒有嘗試 blogger。

大約二年前我發覺在 wordpress.com 打數學文章異常辛苦，原因是每次打數式，如 $1+1=2$，都必須輸入

$\$\text{latex }1+1=2 \$$。

那種繁複程度磨滅了我打數學文章的意欲。直至上年見識到孟國武教授用 blogger 上 lecture，我才知道原來 blogger 可以用一套名為 mathjax 的 javascript typeset 漂亮的數式。這在 wordpress.com (直至現在) 是辦不到的，因它不讓你在 header 內放入任何 javascript。現在 blogger 打數學不用像從前 wordpress 時在每句數式上加上 ``latex'' 。

有了一年以 blogger 為 tutorial 的 blog 的經驗，我開始熟習在 blogger 輸入數式的「方法」。「方法」所指的是 spacing 的控制，因在 blogger 以平常用 latex 的方式輸入數式會容易造成過多的換行。

幸運地在網上很容易找到由 wordpress $\to$ exported xml $\to$ blogger xml $\to$ import to blogger 的方法，所以最近耗了半天將 wordpress 移植到 blogger，當中出現了很多錯誤 =.=。

現在在 UST 教 tutorial，某程度上正在 represent 數學系，所以不宜讓學生知道太多我過去的「祕密」，尤其是我以前所畫的畫對我的形像 (？) 有一定的影響，所以決定 private 這 blog 及只讓熟人看。

BLOG 定向

本想在這 blog 上記錄一些發洩性言論，但這些易得罪他人的評論少放上網為妙了 (雖然這 blog 是 private 的)，因難以判斷哪些身邊的人是女王的線人。所以這 blog 的主旨傾向自己的無謂「日常」及討論數學問題。一如以往，我以"題"為本，不會無謂地將一套 theory 重新在 blog 上打一次。

Record some Problems

In this post I want to record 3 problems:

The following is a problem in GRE math subject test:

Problem 1. Let $\F_p$ be a finite field of order $p$, show that the number of noninvertible $2\times 2$ matrices over $\F_p$ is \[p^4-p^3-p^2+p.\]

Can we generalize to $n\times n$ matrices?

The following is a problem found somewhere by someone in Taiwan's forum, actually I also find this in mainland textbook on real functions:

Problem 2. Let $E,E_1,E_2\subseteq \R$ be such that $E=E_1\cup E_2$. If $E$ is measurable such that \[
m(E)=m^*(E_1)+m^*(E_2)<\infty,
\] show that $E_1$ and $E_2$ are measurable.
HINT: Approximate $E_1,E_2$ from outside by measurable set.

The following is a problem in the midterm exam of Math4061 (fall 2013), which I want to record a simple proof here.

Problem 3. Let $X,Y$ be metric spaces. Suppose that $f:X\to Y$ is a function such that

• $f$ is 1-1.
• $K$ compact in $X$ $\implies$ $f(k)$ compact in $Y$.

Prove that $f$ is continuous.

Dr Li's original solution proves by contradiction, I want to give a direct proof using the sequential criterion of continuity for metric spaces.

Solution to Problem 3. Suppose that $x_n\to x$ in $X$, let's prove that $f(x_n)\to f(x)$ in $Y$. First we observe that \[
L:=\{x_n:n\ge 1\}\cup \{x\}
\] is compact. Second we define $g:=f|_{L}:L\to f(L)$ and show that $g^{-1}:f(L)\to L$ is continuous. Indeed, let $K\subseteq L$ be any closed set, since $L$ is compact, so is $K$, and thus \[
f(K)=g(K)=(g^{-1})^{-1}(K)
\] is a compact set, hence closed set, in $Y$ by hypothesis. Therefore any closed set in $L$ is pulled back to a closed set in $f(L)$, $g^{-1}$ is continuous. Since $f(L)$ is compact and $L$ is Hausdorff (as it is a metric space), a standard training in topology enables us to show $g^{-1}$ is a homeomorphism, so $g$ is continuous.

Finally we check that $f(x_n)\to f(x)$. Since $x_n\to x$ in $X$, we have $x_n\to x$ in $L$, so $g(x_n)\to g(x)$ in $f(L)$, but then $f(x_n)=g(x_n)\to g(x)=f(x)$ in $Y$, so we are done.$\qed$