《追梦赤子心》《中国新歌声》第3期 SING!CHINA

Very good singer and inspirational song. Potential champion of this series.

It’s Time to Redefine Failure BY RICK WARREN

Excellent sermon by Pastor Rick Warren.

Source: http://rickwarren.org/devotional/english%2fit-s-time-to-redefine-failure1?roi=echo7-27381882358-48417402-f0ce079160f08ffc23a2aef23c6680d7&

“Fear of man will prove to be a snare, but whoever trusts in the Lord is kept safe”(Proverbs 29:25 NIV).

Satan’s favorite tool to diminish your faith is the fear of failure. But you cannot serve God and be constantly worried about what other people think. You have to move forward. Proverbs 29:25 says, “Fear of man will prove to be a snare, but whoever trusts in the Lord is kept safe” (NIV).

So how do you get rid of the fear of failure?

One way is to redefine failure. What is failure? Failure is not failing to reach your goal. Failure is not having a goal. Failure is not failing to hit your target. Failure is not having a target. Failure is not falling down. Failure is refusing to get back up. You’re never a failure until you quit. So if you’re attempting something for the glory of God, that’s a good thing. Failure is not trying and not accomplishing anything. Failure is failing to try.

Another way to get rid of the fear of failure is to never compare yourself to anybody else. You’re always going to find somebody who’s doing a better job, and you get discouraged. And, you’re always going to find somebody who’s not doing as good a job as you are, and you become full of pride. Both of them will mess up your life. Discouragement and pride will keep you from serving God’s purpose for your life.

The Bible says in Galatians 6:4, “Each of you must examine your own actions. Then you can be proud of your own accomplishments without comparing yourself to others” (GW)

Did you notice that the Bible says there is a legitimate pride? There’s a good kind of pride and there’s a bad kind of pride. The bad kind of pride is comparing: “I’m better than so and so!” The good kind of pride is, “God, I’m proud of what you’re doing in my family, my business, my life, my walk of faith.” That’s the good kind of pride.

When you get to Heaven, God isn’t going to say, “Why weren’t you more like so and so?” He’s going to say, “Why weren’t you who I made you to be?”

Let go of your fear of failure, because anything you’re attempting for God in faith is a good thing, regardless of the results.

Characterization of Galois Extensions

For a finite extension $E/F$ , each of the following statements is equivalent to the statement that $E/F$ is Galois:

1) $E/F$ is a normal extension and a separable extension.
2) Every irreducible polynomial in $F[x]$ with at least one root in $E$ splits over $E$ and is separable.
3) $E$ is a splitting field of a separable polynomial with coefficients in $F$ .
4) $|\text{Aut}(E/F)|=[E:F]$ , that is, the number of automorphisms equals the degree of the extension.
5) $F$ is the fixed field of $\text{Aut}(E/F)$ .

Fundamental Theorem of Galois Theory

Given a field extension $E/F$ that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.
$H\leftrightarrow E^H$

where $H\leq\text{Gal}(E/F)$ and $E^H$ is the corresponding fixed field (the set of those elements in $E$ which are fixed by every automorphism in $H$ ).
$K\leftrightarrow\text{Aut}(E/K)$

where $K$ is an intermediate field of $E/F$ and $\text{Aut}(E/K)$ is the set of those automorphisms in $\text{Gal}(E/F)$ which fix every element of $K$ .

This correspondence is a one-to-one correspondence if and only if $E/F$ is a Galois extension.

Examples
1) $E\leftrightarrow\{\text{id}_E\}$ , the trivial subgroup of $\text{Gal}(E/F)$ .
2) $F\leftrightarrow\text{Gal}(E/F)$ .

In a PID, every nonzero prime ideal is maximal

Proof:

Let $R$ be a PID. Let $(x)$ be a nonzero prime ideal. Suppose $(x)\subsetneq (y)$ . Then $x=yr$ for some $r\in R$ .

Note that $yr\in(x)$ implies $y\in(x)$ or $r\in(x)$ . Since $(x)\neq(y)$ , we have $r\in(x)$ , so $r=xz$ for some $z\in R$ . Then, $\displaystyle x=yr=yxz$ which implies $1=yz$ , thus $y$ is a unit. Hence $(y)=R$ .

Cook One Fish in Two Ways （一鱼两吃）

Bought this fish from Giant (Vivocity). Seabass, 2 for $8.40. Very delicious!

Chinese Tuition Singapore

Compared to fried fish in western countries, Chinese people always choose to cook fish in different ways, like fish soup, braised fish in brown sauce, or steamed fish.

I bought this sea bass from supermarket. Yet, I found it too big and we didn’t have a pan big enough to cook it. So I decided to divided it into two parts. Fish head for the soup. Fish body and tail for the braised one.

Fish head soup is very popular in China, since many people think this dish is very good for health. Nutritious elements in fish will be contained in the soup.

在中国，鱼头汤是一道很受欢迎的菜。很多人认为鱼头汤对身体很补，鱼里面的营养物质会融合到汤里面去。

It is not difficult to make fish soup.

1. Heat the oil in the pan, and fry two sides of the fish head. When the color turns a little brown, pour in the hot water.

锅内热油，将鱼头两面煎黄，然后加入热水没过鱼头。

2. Add two or three ginger slices, salt and green…

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One-Sided Limit that Does Not Exist

Offhand, it is hard to think of a function that does not have even a one-sided limit. This video shows one!

Convolution Video

This is one of the more illuminating videos on Convolution. I believe after watching it, many will have a better understanding of what is the convolution.

To Reduce Your Fear of Failure, Redefine It

This is a post by Rick Warren, author of “The Purpose Driven Life: What on Earth Am I Here For?“. He is a very good author of Christian books.

(Source: http://rickwarren.org/devotional/english/to-reduce-your-fear-of-failure-redefine-it)

“No matter how often honest people fall, they always get up again.” (Proverbs 24:16a TEV)

Never forget this truth: Failure probably won’t kill you.

We vastly exaggerate the effects of failure. We blow the prospects of failing all out of proportion. Failing is not the end of the world. The fear of failure is far more damaging than failure.

Proverbs 24:16 says, “No matter how often honest people fall, they always get up again” (TEV). Even good guys stumble. They make mistakes, blow it, and stub their toes.

Successful people are not people who never fail. They’re people who get up again and keep going. Successful people just don’t know how to quit.

Ever heard of these famous failures?

George Washington lost two-thirds of all the battles he fought. But he won the Revolutionary War and later became the first U.S. president.
Napoleon graduated 42nd in a class of 43. Then he went out and conquered Europe!
In 21 years Babe Ruth hit 714 home runs, but he struck out 1,330 times. He struck out nearly twice as often as he hit a home run.
The famous novelist John Creasey received 753 rejection slips before he published 564 books.
Rowland Hussey Macy failed seven times at retailing before starting Macy’s department store.

Great people are simply ordinary people who have an extraordinary amount of determination. They just keep on going. They realize they’re never a failure until they quit.

That’s how you reduce your fear of failure. You redefine it.

You don’t fail by not reaching a specific goal. Instead, failure is not having a goal. Failure is refusing to get back up again once you fall. It’s refusing to try.

On the first day of kindergarten, I got in the wrong line and then into the wrong classroom. Can you imagine me going home to my mom and dad and saying, “I’m a failure at education! This school thing just doesn’t work”? Of course not.

You keep going. If at first you don’t succeed, it’s no big deal. You’re never a failure until you give up.

Class Equation of a Group

The class equation of a group is something that looks difficult at first sight, but is actually very straightforward once you understand it. An amazing equation…

Class Equation of a Group (Proof)

Suppose $G$ is a finite group, $Z(G)$ is the center of $G$ , and $c_1, c_2, \dots, c_r$ are all the conjugacy classes in $G$ comprising the elements outside the center. Let $g_i$ be an element in $c_i$ for each $1\leq i\leq r$ . Then we have: $\displaystyle |G|=|Z(G)|+\sum_{i=1}^r[G:C_G(g_i)].$

Proof:

Let $G$ act on itself by conjugation. The orbits of $G$ partition $G$ . Note that each conjugacy class $c_i$ is actually $\text{Orb}(g_i)$ .

Let $x\in Z(G)$ . Then $gxg^{-1}=xgg^{-1}=x$ for all $g\in G$ . Hence $\text{Orb}(x)$ consists of a single element $x$ itself.

Let $g_i\in c_i$ . Then
$\begin{aligned} \text{Stab}(g_i)&=\{h\in G\mid hg_ih^{-1}=g_i\}\\ &=\{h\in G\mid hg_i=g_ih\}\\ &=C_G(g_i). \end{aligned}$
By Orbit-Stabilizer Theorem, $\displaystyle |\text{Orb}(g_i)|=[G:\text{Stab}(g_i)]=[G:C_G(g_i)].$

Therefore, $\displaystyle |G|=|Z(G)|+\sum_{i=1}^r[G:C_G(g_i)].$

Algebra and Analysis Theorems

The following are two lists of useful algebra and analysis theorems that are covered during university.

Algebra Theorems Mathtuition88

Analysis Theorems Mathtuition88

New Header Images for The Twenty Ten Theme

Just noticed that there are some new header images for the “Twenty Ten Theme” by WordPress.

Some examples include flower, beach, lights, kitchen table. The old images are still there.

Very Motivational: Billionaire Sara Blakely’s Secret of Success and her Favorite Motivational Author

Just read an amazing article about self-made billionaire Sara Blakely.

First amazing story is this.

Early in his own career Sara’s father learned that failure is part of success. That in order to be successful at anything in life, you were going to experience some failures along the way. Sara’s father went to great lengths to instill this simple success principle in the lives and minds of his children.

Once or twice a week at the dinner table the elder Blakely would ask his children what they failed at that week. He would stress that if they had not failed at something it meant that had not tried or attempted something new. This instilled a deep belief in Sara’s mind that failure is not the outcome; the real failure was in not trying.

Being able to see failure as just another stepping stone to success would play a big part in Sara Blakely’s struggles later in life as she began to build her company, SPANX.

This is really interesting. This is one good thing about American culture, which explains why Americans are willing to take risks. How many parents in Asia will ask the same thing? Not many, I would estimate.

Second amazing story is this, the power of motivational books. Most people would think that motivational books are hype, or “BS”, to put it mildly. True, 90% of them may be nonsense, but the top tier ones are good, and possibly life-changing.

Over a relatively short period of time a series of events occurred in young Sara Blakely’s life that would set most young people back in a dramatic way.

Recognizing that his daughter was going through very tough times, the elder Mr. Blakely gave his daughter a set of tapes by Dr. Wayne Dyer titled How to Be a No-Limit Person.

Today, Sara Blakely gives almost all of the credit for her success in life to the success principles she learned as a teenager from that one set of motivational tapes by Dr. Wayne Dyer.

Your Erroneous Zones: Step-by-Step Advice for Escaping the Trap of Negative Thinking and Taking Control of Your Life

This is the top-selling and most popular Wayne Dyer book of all time. Also check out this post on Motivational Books for students.

Sara Blakely on failure:

Mertens’ Theorem

Let $(a_n)$ and $(b_n)$ be real or complex sequences.

If the series $\sum_{n=0}^\infty a_n$ converges to $A$ and $\sum_{n=0}^\infty b_n$ converges to $B$ , and at least one of them converges absolutely, then their Cauchy product converges to $AB$ .

An immediate corollary of Mertens’ Theorem is that if a power series $f(x)=\sum a_kx^k$ has radius of convergence $R_a$ , and another power series $g(x)=\sum b_kx^k$ has radius of convergence $R_b$ , then their Cauchy product converges to $f\cdot g$ and has radius of convergence at least the minimum of $R_a, R_b$ .

Note that a power series converges absolutely within its radius of convergence so Mertens’ Theorem applies.

Tietze Extension Theorem and Pasting Lemma

Tietze Extension Theorem

If $X$ is a normal topological space and $\displaystyle f:A\to\mathbb{R}$ is a continuous map from a closed subset $A\subseteq X$ , then there exists a continuous map $\displaystyle F:X\to\mathbb{R}$ with $F(a)=f(a)$ for all $a$ in $A$ .

Moreover, $F$ may be chosen such that $\sup\{|f(a)|:a\in A\}=\sup\{|F(x)|:x\in X\}$ , i.e., if $f$ is bounded, $F$ may be chosen to be bounded (with the same bound as $f$ ). $F$ is called a continuous extension of $f$ .

Pasting Lemma

Let $X$ , $Y$ be both closed (or both open) subsets of a topological space $A$ such that $A=X\cup Y$ , and let $B$ also be a topological space. If both $f|_X: X\to B$ and $f|_Y: Y\to B$ are continuous, then $f$ is continuous.

Proof:

Let $U$ be a closed subset of $B$ . Then $f^{-1}(U)\cap X$ is closed since it is the preimage of $U$ under the function $f|_X:X\to B$ , which is continuous. Similarly, $f^{-1}(U)\cap Y$ is closed. Then, their union $f^{-1}(U)$ is also closed, being a finite union of closed sets.

Maths Tuition – What are the Benefits?

Maths tuition brings about many benefits that can be seen for the parent, the teacher and especially the student who is struggling with their mathematics subject in school. For starters, it will have a huge impact for the student because their entire future can depend on their academic performance in PSLE, O-level, and A-level examinations – all of which requires the student to take the math subject.

For young kids, academics and performance in school can be everything. Their self-esteem and pride depends on it and it helps to guide them in the right direction. Performing well in all subjects helps us to determine who they will become in the future and what they wish to achieve. However, this can be difficult to do when the poor child is struggling in school, particularly in mathematics. Having the opportunity to participate in private maths tuition can help a student get back on the right track once again.

Maths tuition can be extremely useful for Singaporean parents as well. While we all try our best to help our students be successful, there are simply some areas where we are not knowledgeable enough to help out very much. A lot of parents are simply not equipped enough in maths to be able to guide our children adequately. There are also many changes to the education system such the newer and harder syllabuses that can hinder our ability to help our kids as well.

However, with a private maths tutor, our kids can learn the proper way to craft mathematics answers, draw models, and solve algebraic questions, developing the tools necessary to help them succeed in their mathematics exam. Doing well in mathematics not only helps students regain their confidence and improve their T-scores, it also helps them in the long-run as they develop into working adults, as mental arithmetic ability is useful in many practical situations. A good maths tutor can use their expertise to help guide students back onto the right path so that all of their goals and dreams can become a reality.

Teachers can also benefit from private tuition as well. Since a teacher has many students that they are required to teach at one time, it is difficult for them to have the time necessary to devote to one struggling student. However, when students in the class engage their own private maths tutor from a maths tuition agency, the teacher will not have to focus too much on that one student, hindering any of the other children in the classroom as well.

With the right tuition agency, parents can engage private maths tuition in any location and there are benefits for everyone involved. Also, if the mathematics tutors are specialized; meaning if your student is struggling in math, a private maths tuition teacher can help to bring their grades up and to catch them up with the remainder of the class.

Maths tutors are great for all age groups and can even be beneficial for those in university as well. Choosing to engage private maths tuition for your child is a great decision, and whether you are a student, parent or teacher, a good maths tutor help to make everyone’s lives a little better.

Topological Monster: Alexander horned sphere

Very interesting object indeed. Also see this previous video on How to Unlock Interlocked Fingers Topologically?

The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. The exterior is not simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. (Wikipedia)

Lusin’s Theorem and Egorov’s Theorem

Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles.

There are many variations and generalisations, the most basic of which I think are found in Royden’s book.

Lusin’s Theorem:

Informally, “every measurable function is nearly continuous.”

(Royden) Let $f$ be a real-valued measurable function on $E$ . Then for each $\epsilon>0$ , there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F\subseteq E$ for which $\displaystyle f=g\ \text{on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.$

Egorov’s Theorem

Informally, “every convergent sequence of functions is nearly uniformly convergent.”

(Royden) Assume $m(E)<\infty$ . Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise on $E$ to the real-valued function $f$ .

Then for each $\epsilon>0$ , there is a closed set $F\subseteq E$ for which $\displaystyle f_n\to f\ \text{uniformly on}\ F\ \text{and}\ m(E\setminus F)<\epsilon.$

A holomorphic and injective function has nonzero derivative

This post proves that if $f:U\to V$ is a function that is holomorphic (analytic) and injective, then $f'(z)\neq 0$ for all $z$ in $U$ . The condition of having nonzero derivative is equivalent to the condition of conformal (preserves angles). Hence, this result can be stated as “A holomoprhic and injective function is conformal.”

(Proof modified from Stein-Shakarchi Complex Analysis)

We prove by contradiction. Suppose to the contrary $f'(z_0)=0$ for some $z_0\in D$ . Using Taylor series, $\displaystyle f(z)=f(z_0)+f'(z_0)(z-z_0)+\frac{f''(z_0)}{2!}(z-z_0)^2+\dots$

Since $f'(z_0)=0$ , $\displaystyle f(z)-f(z_0)=a(z-z_0)^k+G(z)$ for all $z$ near $z_0$ , with $a\neq 0$ , $k\geq 2$ and $G(z)=(z-z_0)^{k+1}H(z)$ where $H$ is analytic.

For sufficiently small $w\neq 0$ , we write $\displaystyle f(z)-f(z_0)-w=F(z)+G(z),$ where $F(z)=a(z-z_0)^k-w$ .

Since $|G(z)|<|F(z)|$ on a small circle centered at $z_0$ , and $F$ has at least two zeroes inside that circle, Rouche’s theorem implies that $f(z)-f(z_0)-w$ has at least two zeroes there.

Since the zeroes of a non-constant holomorphic function are isolated, $f'(z)\neq 0$ for all $z\neq z_0$ but sufficiently close to $z_0$ .

Let $z_1$ , $z_2$ be the two roots of $f(z)-f(z_0)-w$ . Note that since $w\neq 0$ , $z_1\neq z_0$ , $z_2\neq z_0$ . If $z_1=z_2$ , then $f(z)-f(z_0)-w=(z-z_1)^2h(z)$ for some analytic function $h$ . This means $f'(z_0)=0$ which is a contradiction.

Thus $z_1\neq z_2$ , which implies that $f$ is not injective.

Blueberry pancake （蓝莓松饼）

Chinese Tuition Singapore

Original recipe :

http://m.xiachufang.com/recipe/100460858/

It’s a special way to make pancake, because oven is used instead of pan.
I made some change according to the original recipe above:

Since I don’t have any weighing machine, I am not sure the accurate weight of all the ingredients I used.

Part A:

Flour around 80g

Sugar around 30g

Baking soda 1 tsp

Salt 1/8 tsp

Mix all of them well.

Part B:

Egg 1

Milk around 150g

Yogurt around 25g

Oil ( I used canola oil) around 25g

Vanilla esscence 1 tsp

Mix them well.

Pour Part B into Part A. Stir till no flour can be seen.

Add some blueberries. Stir again.

Preheat oven with 190 degree.

Before put into the oven, spread some blueberries on the surface of batter.

Bake for 45 minutes.

Blueberries will burst and the house is full of fragrance.

You can spread some icing sugar if…

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Why Singapore’s kids are so good at maths

Source: http://www.ft.com/cms/s/0/2e4c61f2-4ec8-11e6-8172-e39ecd3b86fc.html

Sie Yu Chuah smiles when asked how his parents would react to a low test score. “My parents are not that strict but they have high expectations of me,” he says. “I have to do well. Excel at my studies. That’s what they expect from me.” The cheerful, slightly built 13-year-old is a pupil at Admiralty, a government secondary school in the northern suburbs of Singapore that opened in 2002.

To learn more about Singapore Math, check out this comprehensive blog post describing what is Singapore Math.

Munich shooting: Attacker’s Psychology Book

Source: http://edition.cnn.com/2016/07/23/europe/germany-munich-shooting/

(CNN) The teen gunman who killed nine people in a shooting rampage in Munich on Friday was a mentally troubled individual who had extensively researched spree killings and had no apparent links to ISIS, police said.

Condolences to the victims of the Munich shooting.

The next generation of shooting prevention technology lies in psychology, to detect such shooters before they even act. According to Wikipedia, Gun legislation in Germany is considered among the strictest gun control in the world, yet the attacker (Ali Sonboly) managed to get hold of a gun (this fact seemed yet to be explained in the news).

Dr Peter Langman, the world expert on this matter, has written such a book: Why Kids Kill: Inside the Minds of School Shooters. There are some patterns that can be detected, maybe Big Data can help.

Dr Langman's table of correlated traits of school shooters — Dr Langman’s table of correlated traits of school shooters. Once enough traits are gathered, Data Analysis will be useful in finding out potential patterns that emerge.

According to Dr Langman, “The end of the book will not present anything like: A + B + C = School Shooter. The subject is too complicated for that, and there is much that we do not know. Nonetheless, I believe this book will shed light on a phenomenon that, despite massive media coverage, has remained mysterious.”.

Another interesting fact found in Dr Langman’s book is that attackers are not typically loners, unlike what mainstream media usually claims. Dr Langman states that, “A popular sound-bite view of school shooters is that they are loners, a status seen as a contributing factor in their rampages. This is inaccurate. Whereas 9 out of 10 of the shooters we discuss were depressed, only 1 out of 10 was a loner. The others all had friends and acquaintances with whom they engaged in a variety of social activities.”

Overall, review of Dr Langman’s book is highly positive. Definitely useful for teachers in USA to read.

How to cook Beef (pork) Wellington at home

Check out this post by my wife on how to cook Pork / Beef Wellington. All photos taken are original!

Chinese Tuition Singapore

My husband wanted to have beef steak, but we didn’t have any at home. So I cooked Pork Wellington using the recipe of Beef Wellington.

The result was surprisingly good. Pork tested very tender and juicy.

The following is an easy way to cook pork in this way:

1. Season the pork with salt and black pepper.

2. Since the thick steak tests better, I used kitchen twine.

3. Heat some butter in a pan.

4. Put the pork in when the butter is heated. And fry every side of the pork.

5. Frying is done.

6. Stir fry chopped onion, mushroom, and garlic. Add some salt and black pepper powder. ( I don’t have other spices or wine.)

7. Bacon at the bottom, mixture of mushroom and onion in the middle, pork on the top. Do remember to take the kitchen twine off.

8. Roll the pork with bacon…

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Underrated Complex Analysis Theorem: Schwarz Lemma

The Schwarz Lemma is a relatively basic lemma in Complex Analysis, that can be said to be of greater importance that it seems. There is a whole article written on it.

The conditions and results of Schwarz Lemma are rather difficult to memorize offhand, some tips I gathered from the net on how to memorize the Schwarz Lemma are:

Conditions: $f:D\to D$ holomorphic and fixes zero.

Result 1: $|f(z)|\leq|z|$ can be remembered as “Range of f” subset of “Domain”.

$|f'(0)|\leq 1$ can be remembered as some sort of “Contraction Mapping”.

Result 2: If $|f(z)|=|z|$ , or $|f'(0)|=1$ , then $f=az$ where $|a|=1$ . Remember it as “ $f$ is a rotation”.

If you have other tips on how to remember or intuitively understand Schwarz Lemma, please let me know by posting in the comments below.

Finally, we proceed to prove the Schwarz Lemma.

Schwarz Lemma

Let $D=\{z:|z|<1\}$ be the open unit disk in the complex plane $\mathbb{C}$ centered at the origin and let $f:D\to D$ be a holomorphic map such that $f(0)=0$ .

Then, $|f(z)|\leq |z|$ for all $z\in D$ and $|f'(0)|\leq 1$ .

Moreover, if $|f(z)|=|z|$ for some non-zero $z$ or $|f'(0)|=1$ , then $f(z)=az$ for some $a\in\mathbb{C}$ with $|a|=1$ (i.e.\ $f$ is a rotation).

Proof

Consider $g(z)=\begin{cases} \dfrac{f(z)}{z} &\text{if }z\neq 0,\\ f'(0) &\text{if }z=0. \end{cases}$
Since $f$ is analytic, $f(z)=0+a_1z+a_2z^2+\dots$ on $D$ , and $f'(0)=a_1$ . Note that $g(z)=a_1+a_2z+\dots$ on $D$ , so $g$ is analytic on $D$ .

Let $D_r=\{z:|z|\leq r\}$ denote the closed disk of radius $r$ centered at the origin. The Maximum Modulus Principle implies that, for $r<1$ , given any $z\in D_r$ , there exists $z_r$ on the boundary of $D_r$ such that $\displaystyle |g(z)|\leq|g(z_r)|=\frac{|f(z_r)|}{|z_r|}\leq\frac{1}{r}.$

As $r\to 1$ we get $|g(z)|\leq 1$ , thus $|f(z)|\leq|z|$ . Thus
$\begin{aligned} |f'(0)|&=|\lim_{z\to 0}\frac{f(z)}{z}|\\ &=\lim_{z\to 0}|\frac{f(z)}{z}|\\ &\leq1. \end{aligned}$
Moreover, if $|f(z)|=|z|$ for some non-zero $z\in D$ or $|f'(0)|=1$ , then $|g(z)|=1$ at some point of $D$ . By the Maximum Modulus Principle, $g(z)\equiv a$ where $|a|=1$ . Therefore, $f(z)=az$ .

Orbit-Stabilizer Theorem (with proof)

Orbit-Stabilizer Theorem

Let $G$ be a group which acts on a finite set $X$ . Then $\displaystyle |\text{Orb}(x)|=[G:\text{Stab}(x)]=\frac{|G|}{|\text{Stab}(x)|}.$

Proof

Define $\phi:G/\text{Stab}(x)\to\text{Orb}(x)$ by $\displaystyle \phi(g\text{Stab}(x))=g\cdot x.$

Well-defined:

Note that $\text{Stab}(x)$ is a subgroup of $G$ . If $g\text{Stab}(x)=h\text{Stab}(x)$ , then $g^{-1}h\in\text{Stab}(x)$ . Thus $g^{-1}hx=x$ , which implies $hx=gx$ , thus $\phi$ is well-defined.

Surjective:

$\phi$ is clearly surjective.

Injective:

If $\phi(g\text{Stab}(x))=\phi(h\text{Stab}(x))$ , then $gx=hx$ . Thus $g^{-1}hx=x$ , so $g^{-1}h\in\text{Stab}(x)$ . Thus $g\text{Stab}(x)=h\text{Stab}(x)$ .

By Lagrange’s Theorem, $\displaystyle \frac{|G|}{|\text{Stab}(x)|}=|G/\text{Stab}(x)|=|\text{Orb}(x)|.$

Field Medallist Prof. Gowers has also written a nice post on the Orbit -Stabilizer Theorem and various proofs.

Udemy + Mathtuition88 Partnership for Singapore Audience

Dear readers of Mathtuition88,

I am pleased to announce that Udemy (famous online course provider) has contacted me to offer readers a special promotion on courses. These are all WDA approved Skills Future Courses and Singaporean citizens can use their Skills Future Credit to reimburse for them.

These courses would be useful for DSA students learning to improve their interview skills, and also parents seeking to improve their career related skills.

Promocode: SINGAPOREMATHS
Discount: 30%

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Win Any Job Interview – TOP Strategies For Job Interviews

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Communication Skills: Become A Superstar Communicator

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The Art of Leadership & Coaching

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Double Your Social Skills and Instantly Connect With People

Rouche’s Theorem

If the complex-valued functions $f$ and $g$ are holomorphic inside and on some closed contour $K$ , with $|g(z)|<|f(z)|$ on $K$ , then $f$ and $f+g$ have the same number of zeroes inside $K$ , where each zero is counted as many times as its multiplicity.

Example

Consider the polynomial $z^5+3z^3+7$ in the disk $|z|<2$ . Let $g(z)=3z^3+7$ , $f(z)=z^5$ , then

$\begin{aligned} |3z^3+7|&<3(8)+7\\ &=31\\ &<32\\ &=|z^5| \end{aligned}$
for every $|z|=2$ .
Then $f+g$ has the same number of zeroes as $f(z)=z^5$ in the disk $|z|<2$ , which is exactly 5 zeroes.

New PSLE System favors “All Rounders” over “Specialists”

The new PSLE system clearly favors “all-rounders” over “specialists”.

Scenario 1: Math-Whiz VS All-Rounder

Imagine a Math/Science-whiz with

Math:100 (AL 1)
Science: 98 (AL 1)
English: 84 (AL 3)
Chinese: 84 (AL 3)

Total marks: 366 (Approx. 275 T-score)
Total AL: 8

With a “all-rounder”:

Math: 90 (AL 1)
Science: 90 (AL 1)
English: 90 (AL 1)
Chinese: 90 (AL 1)

Total marks: 360 (Approx. 270 T-score)
Total AL: 4

The Math/Science whiz (total AL 8) will be getting double the score of the “all-rounder” (total AL 4), effectively eliminating his chance of entering the top schools. The irony is that the total marks of the Math/Science Whiz is a considerable 6 marks more than the “all-rounder”.

Under the old system, both are likely to get around the same T-score (approx. 270+), with the Math/Science whiz having a higher T-score.

Scenario 2: English-Educated Kid VS All Rounder

This scenario is even worse.

Imagine an intelligent English-Educated Kid (with parents who can’t speak Chinese). After a lot of hard work with Chinese enrichment, etc, he manages to pass Chinese, with a score of:

Math:100 (AL 1)
Science: 98 (AL 1)
English: 95 (AL 1)
Chinese: 64 (AL 6)

Total marks: 357 (Approx. 268 T-score)
Total AL: 9

Under the old system, this child is probably one that qualifies to enter any school, including RI/HCI, etc. His T-score will probably be on par with the All-Rounder at around 270, or at most slightly lower. Under the new system, his total AL is almost 10. Really a big difference.

In fact, the O-Levels, A-Levels are also favoring the all-rounders. Only at university (and beyond), do the specialists finally get a chance to shine. That’s why it is common to see top students in universities who were not previously from the top JCs or secondary schools.

My followup post on Kiasuparents:

My concern as Math educator is that students extremely talented in Mathematics/Science but slightly weak in languages will be disadvantaged in the new PSLE system.

To quote from my own blog entry titled “New PSLE System favors “All Rounders” over “Specialists””:

Imagine a Math/Science-whiz with

Math:100 (AL 1)
Science: 98 (AL 1)
English: 84 (AL 3)
Chinese: 84 (AL 3)

Total marks: 366 (Approx. 275 T-score)
Total AL: 8

Previously such a student’s score is more than sufficient to enter the top schools like RI/HCI. But under the new system, his score of 8, chances of entering the top schools are slim.

It is not about the prestige, but rather the resources and enrichment programmes that top schools provides that other schools may not. Some examples include Olympiad training, Laboratory sessions, etc.

For these kind of students, the PSLE score of 100 is not enough to capture their ability in Math/Science, they would score 150/100 if there is such a thing. Hence, their calibre is well above the “All-Rounders” who score 90 for each subject and get 4 points.

Unfortunately, the new PSLE system does not bode well for these students…

ChiefKiasu’s (founder of Kiasuparents) comments:

This is a good analysis. The new system does demand excellence in every subject, which in my opinion will increase stress more than it reduces. And for those who say that it is good because there is no need to count decimal points, consider the fact that Secondary schools will still have COPs. So it is now getting 4 points vs getting above 255 t-scores. Which measure would you consider to be more narrow?

My feeling is that the new system will actually intensify the cookie-cutter education culture and create more average joes than truly outstanding individuals.

The most Striking Theorem in Real Analysis

Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result.

Lebesgue’s Theorem (Monotone functions)

If the function $f$ is monotone on the open interval $(a,b)$ , then it is differentiable almost everywhere on $(a,b)$ .

Absolutely Continuous Functions

Definition

A real-valued function $f$ on a closed, bounded interval $[a,b]$ is said to be absolutely continuous on $[a,b]$ provided for each $\epsilon>0$ , there is a $\delta>0$ such that for every finite disjoint collection $\{(a_k,b_k)\}_{k=1}^n$ of open intervals in $(a,b)$ , if $\displaystyle \sum_{k=1}^n(b_k-a_k)<\delta,$ then $\displaystyle \sum_{k=1}^n|f(b_k)-f(a_k)|<\epsilon.$

Equivalent Conditions

The following conditions on a real-valued function $f$ on a compact interval $[a,b]$ are equivalent:
(i) $f$ is absolutely continuous;

(ii) $f$ has a derivative $f'$ almost everywhere, the derivative is Lebesgue integrable, and $\displaystyle f(x)=f(a)+\int_a^x f'(t)\,dt$ for all $x$ on $[a,b]$ ;

(iii) there exists a Lebesgue integrable function $g$ on $[a,b]$ such that $\displaystyle f(x)=f(a)+\int_a^x g(t)\,dt$ for all $x$ on $[a,b]$ .

Equivalence between (i) and (iii) is known as the Fundamental Theorem of Lebesgue integral calculus.

The Serenity Prayer

This is a wonderful prayer attributed to theologian Reinhold Neibuhr:

The Serenity Prayer

God grant me the serenity

To accept the things I cannot change;

Courage to change the things I can;

And wisdom to know the difference.

Living one day at a time;

Enjoying one moment at a time;

Accepting hardships as the pathway to peace;

Taking, as He did, this sinful world

As it is, not as I would have it;

Trusting that He will make all things right

If I surrender to His Will;

So that I may be reasonably happy in this life

And supremely happy with Him

Forever and ever in the next.

Amen.

PDF File that can be printed (A4 size): the_serenity_prayer

Taken from http://www.lords-prayer-words.com/famous_prayers/god_grant_me_the_serenity.html

Image (JPEG):

Necessary and Sufficient Conditions for Semidirect Product to be Abelian (Proof)

This theorem is pretty basic, but it is useful to construct non-abelian groups. Basically, once you have either group to be non-abelian, or the homomorphism to be trivial, the end result is non-abelian!

Theorem: The semidirect product $N\rtimes_\varphi H$ is abelian iff $N$ , $H$ are both abelian and $\varphi: H\to\text{Aut}(N)$ is trivial.

Proof:
$(\implies)$

Assume $N\rtimes_\varphi H$ is abelian. Then for any $n_1, n_2\in N$ , $h_1, h_2\in H$ , we have
$\begin{aligned} (n_1, h_1)\cdot(n_2,h_2)&=(n_2,h_2)\cdot(n_1, h_1)\\ (n_1\varphi_{h_1}(n_2), h_1h_2)&=(n_2\varphi_{h_2}(n_1), h_2h_1). \end{aligned}$
This implies $h_1h_2=h_2h_1$ , thus $H$ is abelian.

Consider the case $n_1=n_1=n$ . Then for any $n\in N$ , $n\varphi_{h_1}(n)=n\varphi_{h_2}(n)$ . Multiplying by $n^{-1}$ on the left gives $\varphi_{h_1}(n)=\varphi_{h_2}(n)$ for any $h_1, h_2\in H$ . Thus $\varphi_h(n)=\varphi_e(n)=n$ for all $h\in H$ so $\varphi$ is trivial.

Consider the case where $h_1=h_2=e$ . Then we have $n_1n_2=n_2n_1$ , so $N$ has to be abelian.

( $\impliedby$ )

This direction is clear.

There are two kinds of talented students.

Just read this interesting article. Will the new PSLE system reward students of the first kind or second kind? From my experience as student and tutor, Singapore has many talented students of the first kind, but very few talented students of the second kind.

To be a student of the second kind, one needs to “acquire knowledge beyond the school curriculum”, and “read and look at more advanced material”. Check out this page on Math Olympiad books that are suitable for students of the second kind. Parents should encourage, but never force, children to read more of these kinds of books.

What are the Two Kinds of Talented Students

Source: http://www.math.rutgers.edu/~zeilberg/Opinion0.html

There are two kinds of talented students. One kind is that of “obedient students” that do exactly as ordered by their teachers, and do not attempt to acquire knowledge beyond the school curriculum; learning the material is relatively easy for them, and the pressure from the society, their parents, and their teachers, that tells them that study is the only way to acquire a solid socio-economic status is their only motivation. To that group of students also belong less talented students, that have to study much harder, but the “reward” that awaits them in the future, as well as the immediate rewards promised by the parents (“if you will not fail any subject, you would go to an overseas vacation this summer” etc.) prods them to study.

There is yet another kind of talented students, whose natural curiosity lead them, already from a young age, to read and look at more advanced material, in order to satisfy their natural curiosity.

When such a student enters high school (and in fact, already in the higher grades of elementary school) he sees that the material that he has already studied on his own presented in a different way. The learning is induced through severe disciple (all the system of examinations and grades), and the material is taught the same way as in animal training. The fascinating science of Chemistry turns into a boring list of dry formulas, that he has to learn by heart, and the threats and the incentives practiced in school badly offend him. As though out of spite, he does not listen to the commands of his teachers, but instead studies on his own material that is not included in the curriculum. Obviously, even the most talented student can not learn from just sitting in class, (and even during class he often studies other material), and so starts the “tragedy” described in your article.

Inner and Outer Approximation of Lebesgue Measurable Sets

Let $E\subseteq\mathbb{R}$ . Then each of the following four assertions is equivalent to the measurability of $E$ .

(Outer Approximation by Open Sets and $G_\delta$ Sets)

(i) For each $\epsilon>0$ , there is an open set $G$ containing $E$ for which $m^*(G\setminus E)<\epsilon$ .

(ii) There is a $G_\delta$ set $G$ containing $E$ for which $m^*(G\setminus E)=0$ .

(Inner Approximation by Closed Sets and $F_\sigma$ Sets)

(iii) For each $\epsilon>0$ , there is a closed set $F$ contained in $E$ for which $m^*(E\setminus F)<\epsilon$ .

(iv) There is an $F_\sigma$ set $F$ contained in $E$ for which $m^*(E\setminus F)=0$ .

Proof:
( $E$ measurable implies (i)):

Assume $E$ is measurable. Let $\epsilon>0$ . First we consider the case where $m^*(E)<\infty$ . By the definition of outer measure, there is a countable collection of open intervals $\{I_k\}_{k=1}^\infty$ which covers $E$ and satisfies $\displaystyle \sum_{k=1}^\infty l(I_k)<m^*(E)+\epsilon.$

Define $G=\bigcup_{k=1}^\infty I_k$ . Then $G$ is an open set containing $E$ . By definition of the outer measure of $G$ , $\displaystyle m^*(G)\leq\sum_{k=1}^\infty l(I_k)<m^*(E)+\epsilon.$

Since $E$ is measureable and has finite outer measure, by the excision property, $\displaystyle m^*(G\setminus E)=m^*(G)-m^*(E)<\epsilon.$

Now consider the case that $m^*(E)=\infty$ . Since $\mathbb{R}$ is $\sigma$ -finite, $E$ may be expressed as the disjoint union of a countable collection $\{E_k\}_{k=1}^\infty$ of measurable sets, each of which has finite outer measure.

By the finite measure case, for each $k\in\mathbb{N}$ , there is an open set $G_k$ containing $E_k$ for which $m^*(G_k\setminus E_k)<\epsilon/2^k$ . The set $G=\bigcup_{k=1}^\infty G_k$ is open, it contains $E$ and $\displaystyle G\setminus E=(\bigcup_{k=1}^\infty G_k)\setminus E\subseteq\bigcup_{k=1}^\infty (G_k\setminus E_k).$

Therefore
$\begin{aligned} m^*(G\setminus E)&\leq\sum_{k=1}^\infty m^*(G_k\setminus E_k)\\ &<\sum_{k=1}^\infty\epsilon/2^k\\ &=\epsilon. \end{aligned}$
Thus property (i) holds for $E$ .

((i) implies (ii)):

Assume property (i) holds for $E$ . For each $k\in\mathbb{N}$ , choose an open set $O_k$ that contains $E$ such that $m^*(O_k\setminus E)<1/k$ . Define $G=\bigcap_{k=1}^\infty O_k$ . Then $G$ is a $G_\delta$ set that contains $E$ . Note that for each $k$ , $\displaystyle G\setminus E\subseteq O_k\setminus E.$

By monotonicity of outer measure, $\displaystyle m^*(G\setminus E)\leq m^*(O_k\setminus E)<1/k.$

Thus $m^*(G\setminus E)=0$ and hence (ii) holds.

((ii) $\implies E$ is measurable):

Now assume property (ii) holds for $E$ . Since a set of measure zero is measurable, $G\setminus E$ is measurable. $G$ is a $G_\delta$ set and thus measurable. Since measurable sets form a $\sigma$ -algebra, $E=G\cap(G\setminus E)^c$ is measurable.

((i) $\implies$ (iii)):

Assume condition (i) holds. Note that $E^c$ is measurable iff $E$ is measurable. Thus there exists an open set $G\supseteq E^c$ such that $m^*(G\setminus E^c)<\epsilon$ .

Define $F=\mathbb{R}\setminus G$ which is closed. Note that $F\subseteq E$ , and $m^*(E\setminus F)=m^*(G\setminus E^c)<\epsilon$ .

((iii) $\implies$ (i)):

Similar.

((ii) $\iff$ (iv)):

Similar idea. Note that a set is $G_\delta$ iff its complement is $F_\sigma$ .

New PSLE Scoring System, AL1 to AL8 (Singapore)

Source: http://www.straitstimes.com/singapore/education/changes-to-the-psle-scoring-system-7-things-to-note-from-2021

The new, long awaited, PSLE scoring system is now out. Under the new scoring system, T-score is being replaced by Achievement Levels:

AL1: 90 and above
AL2: 85-89
AL3: 80-84
AL4: 75-79
AL5: 65-74
AL6: 45-64
AL7: 20-44
AL8: Below 20

Would this be effective at the target goal of “reducing stress and competition among pupils and parents?”.

Firstly, the good point about this new scoring system is that it is not as fine as the previous T-score system, where every mark matters. Thus, technically there is no difference between a 90 mark and a 100 mark, so there is no need to aim for perfection in a certain sense.

Some concerns of parents are listed out in this article.

Personally, I think that this scoring system is similar to the O Level System Grading of A1, A2, B3, etc. The change in scoring system per se is unlikely to be able to reduce the stress of students, especially those scoring below 90. Those scoring >90 but not close to 100 may breathe a sigh of relief that they don’t have to aim for 100. However, for the students scoring below 90, the stress level remains essentially unchanged.

Note that despite the “wider scoring band” label, the band is not that wide after all. In the higher AL’s the difference from one AL to the next is merely 5 marks, which may be just one problem sum in mathematics. In the previous PSLE it is “every mark counts”. In the new PSLE it is “every question counts”, which is not much of a difference.

For Primary students, the stress comes mostly from the kiasu parents, any superficial change in the scoring system will not have much effect.

Overall, nothing much has changed. It is like changing between Celsius to Fahrenheit, there is no difference in the underlying principle of PSLE, which is to serve as a entry criteria for secondary schools.

The main change, as some parents have noted, is that now all subjects are equally important. It is no longer possible to compensate for one weak subject (e.g. Mother Tongue) by scoring extremely well in other subjects. It can be said that the new system favours “all rounders” or “Jack of all trades” over “specialists” in one or two subjects.

The AL6: 45-64 band looks extremely dangerous to fall in (especially those weak in Mother Tongue) as it does not differentiate between a fail grade (45) and a much higher grade (64). Many English-speaking families should be quite worried now…

Another area of concern is that due to the “wider scoring band”, the importance of DSA (Direct School Admission) has increased tremendously. Due to inevitably many students achieving the perfect score of ‘4’, the top schools (like RGS/NYGH/RI/HCI) may have to resort to DSA/GAT tests to select their students. This will probably increase the stress of students, as other than PSLE, they have to worry about DSA/GAT/CCA and building a portfolio of achievements.

Do post your comments, if any, below!

Is the highest-ranked school the right one for your child? (Singapore)

Sometimes, the highest-ranked school may not be the right school for every child. This issue is also discussed in Malcolm Gladwell’s book David and Goliath: Underdogs, Misfits, and the Art of Battling Giants.

Source: http://www.straitstimes.com/opinion/is-the-highest-ranked-school-the-right-one-for-your-child

Q Is the highest-ranked school the right school for my child?

A As a Singaporean economist working on issues in education, I am often asked by parents to recommend the best school for their children. Invariably, what such parents were really asking me was to identify a highly ranked school that their child had a decent chance of getting into.

But this raises a dilemma – is a highly ranked school really the most suitable school for a child?

Adults may recall school environments as idyllic places, but we forget that classrooms have now become arenas where fierce competition takes place among classmates.

In today’s schools, students take part in academic tournaments where better test results, compared to those of their peers, bring greater opportunities for scholarships and allow access to better schools. Those who do not excel in these tournaments may lose their incentive to compete and ultimately drop out of the academic race altogether.

This is where my work provides some guidelines for parents weighing the pros and cons of being in a more competitive school.

Last year, a fellow researcher, Mr Yoshio Kamijo, and I conducted a two-day experiment where we first tested the maths ability of 132 Secondary 2 students in a school in Shandong, China, through maths pre-tests on the first day.

Afterwards, we categorised the performance of our students into four groups, based on those pre-tests: low maths ability, average maths ability, high maths ability and a mixed group with low-, average- and high-ability students in one class.

We were interested in comparing the performance of students in a mixed class with those in a class with similar-ability students.

Our experiment aimed to see how students in each class performed in another maths test given on the second day, under a competitive environment where winners received rewards and losers were given punishments.

The point of the exercise was to investigate whether being grouped with similar- or dissimilar-ability students mattered to students of different abilities.

Just like a scientific experiment, by controlling for their pre-ability, their performance in our competition captured how such students responded to the knowledge of competing against similar or weaker/stronger opponents.

Our final results were not that surprising. We compared the results of students in the mixed class with similar-ability peers.

We found that those in the mixed class had different reactions towards their competitors depending on their ability level: the low-ability students were discouraged and performed poorly, the middle-ability students were more motivated and did better in a mixed class than in a class with similar-ability students, while for the high-ability students, no real difference was seen.

The difference in performance was significant.

小学生厌恶数学写诗:数学是死亡之源

Source: http://news.sina.com.cn/s/2015-02-08/031931495241.shtml

武汉的董女士前天在家帮女儿清理书包，从书包里搜出一张纸，上面赫然写着一首诗：数学是死亡之源，它像入地狱般痛苦。它让孩子想破脑汁，它让家长急得转圈。它让校园死气沉沉，它使生命慢慢离去。生命从数学中走去，一代代死得超快。那是生命的敌人，生命从数学中走去。珍惜宝贵的生命吧，一代代死得超快。数学是死亡之源。

读完这首诗，董女士惊呆了：“没想到她厌恶数学到了这般田地。”

董女士的女儿晶晶，今年10岁，读小学五年级，从进小学开始就特别不喜欢数学，尤其讨厌应用题，只要碰到追及问题和工程问题，晶晶那绝对是“一个脑袋两个大”。这次期末考试晶晶的数学考了70分，全班倒数第七。

在董女士的逼问下，晶晶终于交代，这首诗是和班上另外两个女生一起创作的。她们三个都对数学不感冒，联合创作了此诗，抒发忧伤。

网友们看到这首诗后，也表达了不同的观点。网友“@飞不动的咋咋鸟”说，“很有才的小学生，这搞不好以后是余秀华第二啊！”网友“@左边追寻”则说，“想用自己的血泪史告诉这位妹妹，学好数学很重要！”据《武汉晚报》

（原标题：小学生厌恶数学写诗:数学是死亡之源）

The Dangerous Math of Chinese Island Disputes

Source: http://www.wsj.com/articles/SB10001424052970203922804578082371509569896

China’s standoff with Japan over the rocky Senkaku (Diaoyu) Islands has entered its second month. The current confrontation, however, is more dangerous than is commonly believed. China’s past behavior in other territorial disputes demonstrates why the Senkaku standoff is primed to explode.

Since 1949, China has been involved in 23 territorial disputes with its neighbors on land and at sea. Seventeen of them have been settled, usually through compromise agreements. Nevertheless, China has used force, often more than once, in six of these disputes. And it’s these cases that most closely parallel the Senkaku impasse.

To start, China has usually only used force in territorial disputes with its most militarily capable neighbors. These include wars or major clashes with India, Russia and Vietnam (several times), as well as crises involving Taiwan. These states have had the greatest ability to check China’s territorial ambitions. In disputes with weaker states, such as Mongolia or Nepal, Beijing has eschewed force because it could negotiate from a position of strength. Japan is now China’s most powerful maritime neighbor, with a modern navy and a large coast guard.

Nick Vujicic was top in class for Mathematics!

Time: 1.16:05

“Every time I was frustrated in something I could not do, or when people bullied and teased me, I came back to the knowledge that I was top in my class in Mathematics. And that was a confidence booster.” – Nick Vujicic

Nick Vujicic’s Academic Advice to a Singaporean Mom

Time: 1.09:32

A Singaporean mother asks Nick Vujicic, “What would you say to a kid who has totally lost all interest in a subject?”.

Can see Nick is slightly caught off-guard by this very typical Singaporean question that is probably rarely asked in the US. His answer is slightly off-topic (talks about drugs, etc, which is not that relevant in Singaporean context) but nonetheless very inspirational!

Watch the video to hear about Nick Vujicic’s answer!

Nick Vujicic is one of the most inspirational motivational speakers ever. He also runs a nonprofit charity organization that help kids in poverty around the world. Do check out some of his books.

Life Without Limits: Inspiration for a Ridiculously Good Life

Excision Property in Measure Theory

Excision property of measurable sets (Proof)

If $A$ is a measurable set of finite outer measure that is contained in $B$ , then $\displaystyle m^*(B\setminus A)=m^*(B)-m^*(A).$

Proof:

By the measurability of $A$ ,
$\begin{aligned} m^*(B)&=m^*(B\cap A)+m^*(B\cap A^c)\\ &=m^*(A)+m^*(B\setminus A). \end{aligned}$
Since $m^*(A)<\infty$ , we have the result.

How to remember the Divergence Theorem

The Divergence Theorem:
$\displaystyle \int_U\nabla\cdot\mathbf{F}\,dV_n=\oint_{\partial U}\mathbf{F}\cdot\mathbf{n}\,dS_{n-1}$

is a rather formidable looking formula that is not so easy to memorise.

One trick is to remember it is to remember the simpler-looking General Stoke’s Theorem.

One can use the general Stoke’s Theorem ( $\int_{\Omega}d\omega=\int_{\partial\Omega}\omega$ ) to equate the $n$ -dimensional volume integral of the divergence of a vector field $\mathbf{F}$ over a region $U$ to the $(n-1)$ -dimensional surface integral of $\mathbf{F}$ over the boundary of $U$ .

On Semidirect Products

Outer Semidirect Product

Given any two groups $N$ and $H$ and a group homomorphism $\phi:H\to\text{Aut}(N)$ , we can construct a new group $N\rtimes_\phi H$ , called the (outer) semidirect product of $N$ and $H$ with respect to $\phi$ , defined as follows.
(i) The underlying set is the Cartesian product $N\times H$ .
(ii) The operation, $\bullet$ , is determined by the homomorphism $\phi$ :

$\bullet: (N\rtimes_\phi H)\times (N\rtimes_\phi H)\to N\rtimes_\phi H$

$(n_1,h_1)\cdot(n_2,h_2)=(n_1\phi_{h_1}(n_2),h_1h_2)$

for $n_1,n_2\in N$ and $h_1,h_2\in H$ .

This defines a group in which the identity element is $(e_N, e_H)$ and the inverse of the element $(n,h)$ is $(\phi_{h^{-1}}(n^{-1}), h^{-1})$ .
Pairs $(n, e_H)$ form a normal subgroup isomorphic to $N$ , while pairs $(e_N, h)$ form a subgroup isomorphic to $H$ .

Inner Semidirect Product (Definition)

Given a group $G$ with identity element $e$ , a subgroup $H$ , and a normal subgroup $N\lhd G$ ; then the following statements are equivalent:

(i) $G$ is the product of subgroups, $G=NH$ , where the subgroups have trivial intersection, $N\cap H=\{e\}$ .
(ii) For every $g\in G$ , there are unique $n\in N$ and $h\in H$ , such that $g=nh$ .

If these statements hold, we define $G$ to be the semidirect product of $N$ and $H$ , written $G=N\rtimes H$ .

Inner Semidirect Product Implies Outer Semidirect Product

Suppose we have a group $G$ with $N\lhd G$ , $H\leq G$ and every element $g\in G$ can be written uniquely as $g=nh$ where $n\in N$ , $h\in H$ .

Define $\phi: H\to\text{Aut}(N)$ as the homomorphism given by $\phi(h)=\phi_h$ , where $\phi_h(n)=hnh^{-1}$ for all $n\in N$ , $h\in H$ .

Then $G$ is isomorphic to the semidirect product $N\rtimes_{\phi}H$ , and applying the isomorphism to the product, $nh$ , gives the tuple, $(n,h)$ . In $G$ , we have
$\displaystyle (n_1h_1)(n_2h_2)=n_1h_1n_2(h_1^{-1}h_1)h_2=(n_1\phi_{h_1}(n_2))(h_1h_2)=(n_1,h_1)\cdot(n_2,h_2)$
which shows that the above map is indeed an isomorphism.

Sylow Theorems

Let $G$ be a finite group.

Theorem 1

For every prime factor $p$ with multiplicity $n$ of the order of $G$ , there exists a Sylow $p$ -subgroup of $G$ , of order $p^n$ .

Theorem 2

All Sylow $p$ -subgroups of $G$ are conjugate to each other, i.e.\ if $H$ and $K$ are Sylow $p$ -subgroups of $G$ , then there exists an element $g\in G$ with $g^{-1}Hg=K$ .

Theorem 3

Let $p$ be a prime such that $|G|=p^nm$ , where $p\nmid m$ . Let $n_p$ be the number of Sylow $p$ -subgroups of $G$ . Then:
1) $n_p\mid m$ , which is the index of the Sylow $p$ -subgroup in $G$ .
2) $n_p\equiv 1\pmod p$ .

Theorem 3b (Proof)

We have $n_p=[G:N_G(P)]$ , where $P$ is any Sylow $p$ -subgroup of $G$ and $N_G$ denotes the normalizer.

Proof

Let $P$ be a Sylow $p$ -subgroup of $G$ and let $G$ act on $\text{Syl}_p(G)$ by conjugation. We have $|\text{Orb}(P)|=n_p$ , $\text{Stab}(P)=\{g\in G:gPg^{-1}=P\}=N_G(P)$ .

By the Orbit-Stabilizer Theorem, $|\text{Orb}(P)|=[G:\text{Stab}(P)]$ , thus $n_p=[G:N_G(P)]$ .

Orbit-Stabilizer Theorem

Let $G$ be a group which acts on a finite set $X$ . Then $\displaystyle |\text{Orb}(x)|=[G:\text{Stab}(x)]=\frac{|G|}{|\text{Stab}(x)|}.$

Flyers surface in Batam warning of bomb attacks targeting Singaporeans

This is indeed scary news. Just went to Batam not long ago.

On a side note, the flyers seem to be counter-intuitive, why on earth would the terrorists send a warning letter stating where they are going to bomb? More likely a scare tactic or a hoax, but better be safe than sorry.

Update: Outdated MFA advisory falsely linked to letters warning of attacks on Batam

Source: http://news.asiaone.com/news/singapore/flyers-surface-batam-warning-bomb-attacks-targeting-singaporeans?utm_campaign=Echobox&utm_medium=Social&utm_source=Twitter&link_time=1467873063#xtor=CS1-2

Flyers and letters warning that bomb attacks will be carried out at a number of locations in Batam and Bintan in the Riau archipelago have surfaced, an Indonesian news website has reported.

According to Batam Today, the flyers claim that attacks would occur at the Batam Center Ferry Terminal and Nagoya on the island of Batam, as well as the Bintan Telani Ferry Terminal and Tanjung Pinang on the island of Bintan.

– See more at: http://news.asiaone.com/news/singapore/flyers-surface-batam-warning-bomb-attacks-targeting-singaporeans?utm_campaign=Echobox&utm_medium=Social&utm_source=Twitter&link_time=1467873063#xtor=CS1-2

MIT Students Won $8 Million in the Massachusetts Lottery

Old but interesting news. Those students used math to successfully invest in lottery tickets!

Source: http://newsfeed.time.com/2012/08/07/how-mit-students-scammed-the-massachusetts-lottery-for-8-million/

Several years ago, while doing research for a school project, a group of MIT students realized that, for a few days every three months or so, the most reliably lucrative lottery game in the country was Massachusetts’ Cash WinFall, because of a quirk in the way a jackpot was broken down into smaller prizes if there was no big winner. The math whizzes quickly discovered that buying about $100,000 in Cash WinFall tickets on those days would virtually guarantee success. Buying $600,000 worth of tickets would bring a 15%–20% return on investment, according to the New York Daily News.

Why Differentiability in Higher Dimensions is defined as it is?

Source: http://www.math.caltech.edu/~dinakar/08-Ma1cAnalytical-Notes-chap.2.pdf

The above paragraph describes nicely the intuitive meaning of the idea behind the definition of differentiability in higher dimensions! It is a very neat idea.

Critique on the Modern Axiomatic Approach of Mathematics

This video is a rare critique of the axiomatic approach of modern mathematics. Worth viewing, to gain an alternative viewpoint. Very interesting and well-argued!

Quote by Professor Wildberger (PhD Yale):

I believe it was by having a closer look and think about Euclid. What he is doing is so very different from what modern mathematics is up to, that it naturally leads one to suspicions. One can tell that Euclid really honestly meant to start a logical development. The current axiomatics are exactly the opposite–they were tacked on at the end of a long development when all else failed, and clearly are just a backward attempt to keep up a framework for which other more direct supports proved impossible. When in doubt, resort to wishful thinking.

Prime Minister Lee Hsien Loong’s Message to Youths

PM to youth: Go for your dreams, don’t be afraid to make mistakes

Source: http://www.straitstimes.com/politics/pm-to-youth-go-for-your-dreams-dont-be-afraid-to-make-mistakes

Prime Minister Lee Hsien Loong marked Youth Day by posting a jump shot on Facebook yesterday accompanied with a message to the “young and young at heart” to experiment and try things out, saying they should not be afraid of making mistakes.

Borel measurability

A function $f$ is said to be Borel measurable provided its domain $E$ is a Borel set and for each $c$ , the set $\displaystyle \{x\in E\mid f(x)>c\}=f^{-1}(c,\infty)$ is a Borel set.

Borel set

A Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement.

FTFGAG: Fundamental Theorem of Finitely Generated Abelian Groups

Fundamental Theorem of Finitely Generated Abelian Groups

Primary decomposition

Every finitely generated abelian group $G$ is isomorphic to a group of the form $\displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{q_1}\oplus\dots\oplus\mathbb{Z}_{q_t}$ where $n\geq 0$ and $q_1,\dots,q_t$ are powers of (not necessarily distinct) prime numbers. The values of $n, q_1, \dots, q_t$ are (up to rearrangement) uniquely determined by $G$ .

Invariant factor decomposition

We can also write $G$ as a direct sum of the form $\displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{k_1}\oplus\dots\oplus\mathbb{Z}_{k_u},$ where $k_1\mid k_2\mid k_3\mid\dots\mid k_u$ . Again the rank $n$ and the invariant factors $k_1,\dots,k_u$ are uniquely determined by $G$ .

Characterization of Galois Extensions

Fundamental Theorem of Galois Theory

In a PID, every nonzero prime ideal is maximal

Proof:

Class Equation of a Group (Proof)

Proof:

Mertens’ Theorem

Tietze Extension Theorem

Pasting Lemma

Proof:

Lusin’s Theorem:

Egorov’s Theorem

Schwarz Lemma

Proof

Orbit-Stabilizer Theorem

Proof

Rouche’s Theorem

Example

Scenario 1: Math-Whiz VS All-Rounder

Scenario 2: English-Educated Kid VS All Rounder

Lebesgue’s Theorem (Monotone functions)

Absolutely Continuous Functions

Definition

Equivalent Conditions

What are the Two Kinds of Talented Students

(Outer Approximation by Open Sets and Sets)

(Inner Approximation by Closed Sets and Sets)

Excision property of measurable sets (Proof)

Outer Semidirect Product

Inner Semidirect Product (Definition)

Inner Semidirect Product Implies Outer Semidirect Product

Sylow Theorems

Theorem 1

Theorem 2

Theorem 3

Theorem 3b (Proof)

Proof

Orbit-Stabilizer Theorem

PM to youth: Go for your dreams, don’t be afraid to make mistakes

Borel measurability

Borel set

Fundamental Theorem of Finitely Generated Abelian Groups

Primary decomposition

Invariant factor decomposition

(Outer Approximation by Open Sets and $G_\delta$ Sets)

(Inner Approximation by Closed Sets and $F_\sigma$ Sets)