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…

View original post 86 more words

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%

Udemy Homepage
Udemy Homepage

All Skills Future Courses
All Skills Future Courses

The Complete Innovator Guide to Spark Creative Thinking
The Complete Innovator Guide to Spark Creative Thinking

Career Hacking: Resume/CV, LinkedIn, Interviewing, +More
Career Hacking: Resume/CV, LinkedIn, Interviewing, +More

Win Any Job Interview – TOP Strategies For Job Interviews
Win Any Job Interview – TOP Strategies For Job Interviews

Communication Skills: Become A Superstar Communicator
Communication Skills: Become A Superstar Communicator

The Art of Leadership & Coaching
The Art of Leadership & Coaching

Double Your Social Skills and Instantly Connect With People
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 $G$ .

Amazon CPM Ads VS WordPress WordAds

I have implemented Amazon CPM Ads to my other site http://mathtuition88.blogspot.sg/ which is currently mainly used to host my Javascript Apps like:

Revenue wise, Amazon CPM Ads beats WordAds hands down. Amazon CPM Ads pays at least 3 times the amount WordAds pays, for the same amount of traffic. It can also be integrated with Amazon Associates via the “passback” ad, so that when the CPM Ad is not filled, it shows the Amazon Associates affiliate ad instead.

For WordPress.com, the only form of ad allowed is WordAds, which is still in the development phase.

Engineering matters for Singapore’s future, says PM Lee Hsien Loong

Source: http://www.straitstimes.com/politics/engineering-key-to-singapores-future-as-smart-nation-pm

Excerpt:

Singapore has boosted its water supply by recycling water and increased its physical size by reclaiming land – all feats of engineers.

Indeed, just as engineering helped transform Singapore into a modern state, it will continue to play a key role as the country strives to be a smart nation and overcome its lack of resources, said Prime Minister Lee Hsien Loong yesterday.

…

But it has since become harder to attract top students to study engineering and do engineering jobs, as many opt for the humanities, business and finance, he noted.

Engineering is among the major professions here with the most vacancies in the past few years.

Chinese Tuition

Chinese Tuition for Secondary Level: http://www.chinesetuition88.com

Looking for students (one-to-one tuition) at Secondary Level.

Preferably living in West area: Jurong/Clementi/Boon Lay. Central locations (near MRT) also acceptable.

http://www.chinesetuition88.com

Contact:
Email: chinesetuition88@gmail.com

Cauchy-Riemann Equations

Let $f(x+iy)=u(x,y)+iv(x,y)$ . The Cauchy-Riemann equations are:

$\begin{aligned} u_x&=v_y\\ u_y&=-v_x. \end{aligned}$

Alternative Form (Wirtinger Derivative)

The Cauchy-Riemann equations can be written as a single equation $\displaystyle \frac{\partial f}{\partial\bar z}=0$ where $\displaystyle \frac{\partial}{\partial\bar z}=\frac 12(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y})$ is the Wirtinger derivative with respect to the conjugate variable.

Goursat’s Theorem

Suppose $f=u+iv$ is a complex-valued function which is differentiable as a function $f:\mathbb{R}^2\to\mathbb{R}^2$ . Then $f$ is analytic in an open complex domain $\Omega$ iff it satisfies the Cauchy-Riemann equations in the domain.

dz and dz bar: How to derive the Wirtinger derivatives

Something interesting in Complex Analysis is the Wirtinger derivatives:

$\displaystyle\boxed{\frac{\partial}{\partial z}:=\frac 12(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})}$

$\displaystyle\boxed{\frac{\partial}{\partial \bar z}:=\frac 12(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y})}$

They are often simply defined as such, but one would be curious how to derive them, at least heuristically.

How to derive Wirtinger derivatives

It turns out we can derive them as such. Any complex function $f(z)$ can be viewed as a function $f(x,y)$ by considering $z=x+iy$ . Since $x=\frac 12 (z+\bar z), y=-\frac 12 i(z-\bar z)$ , we can also view $f(x,y)$ as $f(z,\bar z)$ .

Then by the Chain Rule (for multivariable calculus), we have $\displaystyle\frac{\partial}{\partial x}=\frac{\partial z}{\partial x}\frac{\partial}{\partial z}+\frac{\partial\bar z}{\partial x}\frac{\partial}{\partial\bar z}=\frac{\partial}{\partial z}+\frac{\partial}{\partial\bar z}$ .

Similarly, we get $\displaystyle\frac{\partial}{\partial y}=i(\frac{\partial}{\partial z}-\frac{\partial}{\partial\bar z})$ .

Then, solving the simultaneous equations we get the Wirtinger derivatives.

$\displaystyle i\frac{\partial}{\partial x}+\frac{\partial}{\partial y}=2i\frac{\partial}{\partial z}$ . Thus, $\displaystyle\frac{\partial}{\partial z}=\frac 12(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})$ .

Similarly, we can get that $\displaystyle\boxed{\frac{\partial}{\partial \bar z}:=\frac 12(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y})}$ .

Using Wirtinger derivatives, we can express the Cauchy-Riemann equations in a succinct manner: A function satisfies the Cauchy-Riemann equations iff $\displaystyle\frac{\partial f}{\partial\bar z}=0$ .

Differences in New H2 Math Syllabus (9758)

H2 Math has undergone a revamp to the new syllabus (Subject Code 9758).

So what are the differences between the new syllabus and the old syllabus (Subject Code 9740)?

Main Differences:

Removal of Mathematical Induction
Removal of Recurrence relation
Removal of Loci (Complex Numbers)
Removal of Poisson distribution
Removal of normal approximation to binomial distribution
The new H2 Math Syllabus specifically mentions the importance of the modulus function in inequality.

Most of the changes are actually removal, which means the new syllabus covers less content than the previous syllabus. Many of the topics are being moved to Further Math.

Despite the removal of material, students are advised to study beyond the syllabus and keep an open mind since the concepts will be useful in further studies.

Chinese Tuition 2016 (Secondary)

Looking for Chinese tuition for O levels?

Check out https://chinesetuition88.com/.

Learn tips on how to improve Chinese composition, comprehension, and more!

How to Change Order of Integration

Check out this video by “patrickJMT”. His videos are excellent.

In this case, the reason why we can change the order of integration is Tonelli’s Theorem, since the integrand is non-negative.

Switch to holistic assessment may add pressure on students

Government’s plan to change current methods of assessment to reduce emphasis on academic achievement may be undermined by the fact that Singaporeans will adapt to compete on whatever terms they are given

The winds of change are blowing hard against the Singaporean obsession with examination results that deprives the young of their childhood and propagates despair in society’s pressure-cooker environment.

In April, the Ministry of Education (MOE) announced that the aggregate score for the Primary School Leaving Examination (PSLE) will be scrapped, and replaced with wider scoring bands from 2021. This will be similar to grading at O and A levels.

The current system involves working out a child’s aggregate T-score based on component subject scores – English, Mother Tongue, mathematics and science – weighted against the range of scores within each cohort.

Most of all, I wonder how fair and meritocratic it is for an educational system to systematically reward those who have spent $50,000 pursuing music as a “talent” from age four, when the educational system itself offers students no violins, no violin teachers, and no access to the ABRSM (Associated Board of the Royal Schools of Music) grading certificates schools ask for.

Updated LaTeX to WordPress Converter

WordPress is notorious for not accepting \begin{align} … \end{align} as it is not in math mode.

I have updated the LaTeX to WordPress Converter to change \begin{align} … \end{align} to $l atex\begin{aligned} … \end{aligned}$ which works in WordPress.

Test Input:

Let $h=\chi_{[0,1]}$, the characteristic function of $[0,1]$. We have $\|\chi_{[0,1]}\|_\infty=1$, so $\chi_{[0,1]}\in L^\infty$. Then,
\begin{align*}
(Hh)(x)&=\frac{1}{\pi}\int_0^1\frac{1}{x-t}\ dt\\
&=\frac{1}{\pi}[-\ln|x-t|]_0^1\\
&=\frac{1}{\pi}\ln\frac{|x|}{|x-1|}.
\end{align*}
As $x\to 1$, $(Hh)(x)\to\infty$. Thus, $Hh$ is an unbounded function, so $H$ is not bounded as a map: $L^\infty\to L^\infty$.
\[\frac{a}{b}=c\]

Test Output:

Let $h=\chi_{[0,1]}$ , the characteristic function of $[0,1]$ . We have $\|\chi_{[0,1]}\|_\infty=1$ , so $\chi_{[0,1]}\in L^\infty$ . Then,
$\begin{aligned} (Hh)(x)&=\frac{1}{\pi}\int_0^1\frac{1}{x-t}\ dt\\ &=\frac{1}{\pi}[-\ln|x-t|]_0^1\\ &=\frac{1}{\pi}\ln\frac{|x|}{|x-1|}. \end{aligned}$
As $x\to 1$ , $(Hh)(x)\to\infty$ . Thus, $Hh$ is an unbounded function, so $H$ is not bounded as a map: $L^\infty\to L^\infty$ .
$\displaystyle \frac{a}{b}=c$

Hatcher 2.1.6

Compute the simplicial homology groups of the $\Delta$ -complex obtained from $n+1$ 2-simplices $\Delta_0^2, \dots, \Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge, and for $i>0$ identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$ of $\Delta_i^2$ to a single edge and the edge $[v_0,v_2]$ to the edge $[v_0,v_1]$ of $\Delta_{i-1}^2$ .

Prelim vs A Level Results

Many students upon entering JC (Junior College) in Singapore are demoralised by their poor grades. Students used to scoring As in secondary schools may now be scoring D, E or even S (subpass), U (ungraded) in JCs.

However, prelim (or promo/block test) results are well known to be traditionally lower than the A level results.

Do speak to your teachers or seniors to get a feel of how to convert your prelim results to the predicted A level results.

A rule of thumb is that if you are in a top tier JC, scoring B’s or C’s in prelims is already commendable.

Quote:

I used to be a Hwachong student. Don’t be alarmed by bad grades during block tests, promo exams or even prelims….the school tends to mark stricter. I never got an A in my other subjects (lit, math, history) except for economics (once), most of the time if you get a B or C it’s pretty commendable. But by A levels many will score As for subjects they used to get Bs or Cs in. I got all As

Another rule of thumb is the percentile is more important than the actual grade for internal school exams. Check your school’s distinction rate. For instance, if your school’s H2 Maths distinction rate is 60%, and you are above 60%, then even if you are getting a D you have a good chance of scoring a distinction in the actual A levels.

A final rule of thumb is that the prelim grades is usually two band grades below the A level grade. E.g. If you score a C in prelims, your predicted A level grade is B or A.

Quote: “It is useful as an indicator about your performance respective to other students, but not about your absolute performance. The top schools deflate your grades by maybe one or two grade bands? Not so sure about the rest.”

The maths formula that proves Brexit will be a disaster

Source: http://www.telegraph.co.uk/news/2016/04/18/the-maths-formula-that-proves-brexit-will-be-a-disaster-accordin/

Leaving the EU would be a disaster: the bods at the Treasury have done the sums. They really have. Today, journalists attending a George Osborne speech were each handed a thumping great 200-page wodge of Treasury bumph, containing the mathematical formulae they’d used. Here’s one from page 159.

1n(Tijt) = α ij + Y t + α₁ 1n(Y it * Y jt ) + α₂ 1n(POP it * POP jt ) + ε ijt

= α ij + Y t + αX ijt + ε ijt

So now you know. Case closed. You can imagine the shockwaves this will send through the country. In pubs across the land they’ll be talking of little else.

“Don’t know about you, Baz, but I’m voting to leave. Get immigration down, take back our country, and stop this lot in Brussels pushing us around.”

“Come off it, Dave. Be realistic. What about 1n(Tijt)?”

“1n(Tijt)?”

“Yeah, 1n(Tijt).”

“What’s 1n(Tijt)?”

“Well, it’s equal to α ij + Y t + α₁ 1n(Y it * Y jt ) + α₂ 1n(POP it * POP jt ) + ε ijt .”

“God, that’s a point. I’d never looked at it like that before.”

“See, it all makes sense when you think about it.”

“Fair enough, got me bang to rights there. And there was me thinking 3x(Tijt) = α it ₁ * Y jt + (X * Y it ) + 2X it ₃ – ε njt .”

At first I thought this was a joke. But apparently it is a real formula.

BV (Bounded Variation) functions

BV functions of one variable

Total variation

The total variation of a real-valued function $f$ , defined on an interval $[a,b]$ , is the quantity $\displaystyle V_a^b(f)=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|$ where the supremum is taken over the set $\mathcal{P}=\{P=\{x_0,\dots,x_{n_P}\}\mid P\ \text{is a partituion of }[a,b]\}$ .

BV function

$f\in BV([a,b])\iff V_a^b(f)<\infty$ .

Jordan decomposition of a function

A real function $f$ is of bounded variation in $[a,b]$ iff it can be written as $f=f_1-f_2$ of two non-decreasing functions on $[a,b]$ .

Free H2 Literature Website

“Unseen” (https://unseenthemagazine.wordpress.com/) is a new website set up by literature teachers and experts specially for H2 students studying literature.

Recently, they are also featured on Straits Times.

It is nice to see people passionate about literature setting up a website to benefit students. The purpose of studying literature (or any other subject) shouldn’t be solely about scoring in exams, but rather gaining knowledge and experiencing the joy of learning.

Do check out the website!

Galois Group of Polynomial

Separable Polynomial
A polynomial over $F$ is said to be separable if it has no multiple roots (i.e., all its roots are distinct).

Galois Group of Polynomial
Let $f(x)$ be a separable polynomial over $F$ . Let $K$ be the splitting field over $F$ of $f(x)$ . Then the Galois group of $f(x)$ over $F$ is defined to be $\text{Gal}(K/F)$ .

Eisenstein’s Criterion

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ be a polynomial in $\mathbb{Z}[x]$ . If there exists a prime $p$ such that:

(i) $p\mid a_i$ for $i\neq n$ ,
(ii) $p\nmid a_n$ , and
(iii) $p^2\nmid a_0$

then $f$ is irreducible over $\mathbb{Q}$ .

One way to remember Eisenstein’s Criterion is to remember this classic application to show the irreducibility of the cyclotomic polynomials (after substituting $x+1$ for $x$ ):

$\displaystyle\frac{(x+1)^p-1}{x}=x^{p-1}+{p\choose{p-1}}x^{p-2}+\dots+{p\choose 2}x+{p\choose 1}$ .

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I have updated the LaTeX to WordPress Converter to change \[ \] to $l atex\displaystyle and $ respectively.

Note that \[ … \] is preferable over $$ … $$.

Test code:

Input:

If $(X,\Sigma,\mu)$ is a measure space, $f$ is a non-negative measurable extended real-valued function, and $\epsilon>0$, then \[\mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu.\]

Define \[s(x)=\begin{cases}
\epsilon, &\text{if}\ f(x)\geq\epsilon\\
0, &\text{if}\ f(x)<\epsilon.
\end{cases}\]
Then $0\leq s(x)\leq f(x)$. Thus $\int_X f(x)\,d\mu\geq\int_X s(x)\,d\mu=\epsilon\mu(\{x\in X: f(x)\geq\epsilon\})$. Dividing both sides by $\epsilon>0$ gives the result.

Output:

If $(X,\Sigma,\mu)$ is a measure space, $f$ is a non-negative measurable extended real-valued function, and $\epsilon>0$ , then $\displaystyle \mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu.$

Define $\displaystyle s(x)=\begin{cases} \epsilon, &\text{if}\ f(x)\geq\epsilon\\ 0, &\text{if}\ f(x)<\epsilon. \end{cases}$
Then $0\leq s(x)\leq f(x)$ . Thus $\int_X f(x)\,d\mu\geq\int_X s(x)\,d\mu=\epsilon\mu(\{x\in X: f(x)\geq\epsilon\})$ . Dividing both sides by $\epsilon>0$ gives the result.

Markov’s Inequality: No more than 1/5 of the population can have more than 5 times the average income

One way to remember Markov’s Inequality (also called Chebyshev’s Inequality) is to remember this application: No more than 1/5 of the population can have more than 5 times the average income. For instance, if the average income of a certain country is USD $3000 per month, no more than 20% of the citizens can earn more than $15 000!

Brief Explanation

$\mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu$ is Markov’s Inequality, where $\mu$ is the probability measure. Taking $\epsilon=5A$ to be 5 times the average income, the left hand side represents the probability of having more than 5 times the average income. The right hand side is $\frac{1}{5A}\cdot A=\frac 15$ .

Chebyshev’s/Markov’s Inequality (Proof):
If $(X,\Sigma,\mu)$ is a measure space, $f$ is a non-negative measurable extended real-valued function, and $\epsilon>0$ , then $\displaystyle \mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu.$

Proof:
Define $\displaystyle s(x)=\begin{cases} \epsilon, &\text{if}\ f(x)\geq\epsilon\\ 0, &\text{if}\ f(x)<\epsilon. \end{cases}$
Then $0\leq s(x)\leq f(x)$ . Thus $\int_X f(x)\,d\mu\geq\int_X s(x)\,d\mu=\epsilon\mu(\{x\in X: f(x)\geq\epsilon\})$ . Dividing both sides by $\epsilon>0$ gives the result.

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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

Cauchy-Riemann Equations

Alternative Form (Wirtinger Derivative)

Goursat’s Theorem

How to derive Wirtinger derivatives

Main Differences:

Government’s plan to change current methods of assessment to reduce emphasis on academic achievement may be undermined by the fact that Singaporeans will adapt to compete on whatever terms they are given

Test Input:

Test Output:

BV functions of one variable

Total variation

BV function

Jordan decomposition of a function

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

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