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

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

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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?

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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 $\Delta_{i-1}^2$ .

Lebesgue’s Dominated Convergence Theorem (Continuous Version)

This is a basic but very useful corollary of the usual Lebesgue’s Dominated Convergence Theorem.

From what I see, it is basically the Sequential Criterion plus the usual Dominated Convergence Theorem.

From the book: Basic Partial Differential Equations

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.”

Kids of pushy parents ‘face higher risk of depression’: NUS study

Recently there are two articles on “Tiger Moms” and “Kiasu (translated as “overly afraid of losing”) Parents” in Singapore. Interesting to read.

Parents in Singapore are indeed at a dilemma, overly pushing their child will lead to negative consequences (as mentioned in the articles), but not pushing their child may lead to falling behind academically.

This quote sums it up:

A housewife, who wanted to be known only as Mrs Lim, 43, said: “In Singapore, the pressure to do well starts early. Parents have no choice but to set high expectations of their kids’ performance.

“But I will be more mindful of the way I speak to my kids, so that they won’t feel bad about making mistakes in their work.”

The solution, ideally, is for children to be self-motivated rather than being pushed by parents. Check out some motivational educational books here.

The worst consequence of pushing children is that the child may lose the joy of learning.

Source: http://www.straitstimes.com/singapore/kids-of-pushy-parents-face-higher-risk-of-depression

Children with pushy parents are at a much higher risk of developing depression or anxiety symptoms, according to a local university study. The findings, according to researchers, are especially relevant to a society like Singapore’s, in which there is an emphasis on academic excellence.

Said the study’s head, Assistant Professor Ryan Hong: “Parents may set unrealistically high expectations for their children.

“As a result, a sizeable segment of children may become fearful of making mistakes.”

Source: http://www.straitstimes.com/singapore/intrusive-parents-may-lead-children-to-be-overly-self-critical-nus-study

SINGAPORE – Children with intrusive parents may become overly critical of themselves, and such tendencies – at high or increased levels – are reportedly linked to depression or anxiety.

Parents who have high expectations of their children’s academic performance may urge them to achieve good grades or over-react when they make mistakes, but such actions may lead to unintended consequences, a National University of Singapore (NUS) study has found.

The five-year study, conducted by researchers from the department of psychology at NUS, examined how maladaptive perfectionism – commonly known as the “bad” form of perfectionism – develops in primary school children in Singapore.

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.

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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 $\text{Gal}(K/F)$ .

Fried Golden Cicada

Fried Golden Cicada is a delicacy in Shandong, China.

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 $x$ ):

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

Updated LaTex Javascript Converter

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.

Exhaust Repairs Reading

Exhaust repairs Reading From The Car Service Centre

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Check out this website for Tyre repairs and replacements!

Exhaust repairs Reading From The Car Service Centre

This is a sponsored post.

Riesz-Fischer Theorem Proof

Bartle’s proof of the Riesz-Fischer theorem ( $L^p$ spaces are complete) is quite nice. Royden uses a concept called “rapidly Cauchy”, which may complicated things unnecessarily.

Proof from Bartle’s Book Elements of Integration:

What holiday? More kids spending school break at tuition centres

Full article: http://www.tnp.sg/news/singapore-news/more-parents-sending-their-children-study-between-terms

Interesting article with some points to ponder. One quote seems true: “Force-feeding children to learn during their holidays might cause them to develop a resistance to it. This is why some children have low resiliency levels and they eventually don’t want to study.”

Self-motivation is very important for learning, otherwise the child may go for the tuition classes but end up daydreaming during lesson. Check out some motivational books which can motivate your child and improve English at the same time: https://mathtuition88.com/2014/11/16/motivational-books-for-the-student-educational/

Jack, 12, in his own words

We would always go on holidays in June and December, for as long as I can remember.

But in May, when I asked Mummy where we were going in June, I heard the bad news.

She said she was worried and stressed because I had done badly for my term one papers, getting only Bs and Cs.

“We’re not going anywhere,” she said. “You have to study.”

I feel her decision is unfair because I know how to manage my time.

She wants me to get in the school of her choice through Direct School Admission (DSA) – that is why she insisted that I must still continue with my piano and wushu lessons on Sundays.

I used to have tuition only for Mother Tongue and Maths. Now, plus all that, I have to go for intensive study sessions for Mother Tongue, Maths and English.

But Mummy does not know that having different teachers makes it more confusing for me.

I cannot focus and they both tell me different things, so I don’t know how to answer the questions best.

I wish we could travel as a family. I’m sad that we are not travelling this time.

Travelling could have been a fun time for me to recharge and then I can focus on the BIG PSLE.

I know Mummy has my welfare and well-being in mind, but it is also hard to pretend that I do not mind.

Just yesterday, I asked if I can have one day free before the holidays are over to go to Universal Studios Singapore (USS) and she said very angrily: “NO.”

She said that I do not know how to prioritise my needs. I feel she is unfair.

She keeps telling me it is “just for a few months”. But June is just one month, and it is the holidays.

Anyway, I hope I can get into DSA so my suffering can be lessened and Mummy will take me on a holiday after PSLE.

She told me I have to set a good example for Mei Mei (Mandarin for sister).

I hope I can.

– See more at: http://www.tnp.sg/news/singapore-news/more-parents-sending-their-children-study-between-terms#sthash.CgqYD8XP.dpuf

Fresh graduates face challenging job search ahead

Source: http://www.todayonline.com/singapore/fresh-graduates-face-challenging-job-search-ahead

SINGAPORE — They’ve sent 20 to 50 job applications but some graduating students are struggling to secure interviews, much less a job offer, amid the Republic’s slowing economy.

With the gross domestic product (GDP) projected to grow 1 to 3 per cent this year — last year’s growth was 2.1 per cent — human resource experts said they have seen a drop of at least 10 per cent in job vacancies open to fresh graduates from last year, with graduates finding it difficult to secure their ideal jobs.

Fatou’s Lemma

Fatou’s Lemma
Let $(f_n)$ be a sequence of nonnegative measurable functions, then $\displaystyle\int\liminf_{n\to\infty}f_n\,d\mu\leq\liminf_{n\to\infty}\int f_n\,d\mu.$

A brilliant graphical way to remember Fatou’s Lemma (taken from the site http://math.stackexchange.com/questions/242920/what-are-some-tricks-to-remember-fatous-lemma).

The first two are ∫f1 and ∫f2 respectively, but even the smaller of these is larger than the area in the third picture, which is ∫inf fn.

Mathematics Enrichment Camp 2016

This camp (series of lectures) is suitable for Pre-University students (e.g. JC / NUS High) who are interested in mathematics!

Saturday, 13th August 2016 8.30am to 2.00pm @ Faculty of Science, NUS Lecture Theatre 25

Tea break & lunch will be provided Prizes to be won!

URL: http://ww1.math.nus.edu.sg/events/MathEnrichmentCamp-2016-brochureregistration.pdf

Finite extension is Algebraic extension (Proof) + “Converse”

These two are useful lemmas in Galois/Field Theory.

Finite extension is Algebraic extension (Proof)
Let $L/K$ be a finite field extension. Then $L/K$ is an algebraic extension.
Proof:
Let $L/K$ be a finite extension, where $[L:K]=n$ . Let $\alpha\in L$ . Consider $\{1,\alpha,\alpha^2,\dots,\alpha^n\}$ which has to be linearly dependent over $K$ since there are $n+1$ elements. Thus, there exists $c_i\in K$ (not all zero) such that $\sum_{i=0}^n c_i\alpha^i=0$ , so $\alpha$ is algebraic over $K$ .

Finitely Generated Algebraic Extension is Finite (Proof)
Let $L/K$ be a finitely generated algebraic extension. Then $L/K$ is a finite extension.

Proof:
Since $L/K$ is finitely generated, $L=K(\alpha_1,\dots,\alpha_n)$ for some $\alpha_1,\dots,\alpha_n\in K$ . Since $L/K$ is algebraic, each $\alpha_i$ is algebraic over $K$ . Denote $L_i:=K(\alpha_1,\dots,\alpha_i)$ for $1\leq i\leq n$ . Then $L_i=L_{i-1}(\alpha_i)$ for each $i$ . Since $\alpha_i$ is algebraic over $K$ , it is also algebraic over $L_{i-1}$ , so there exists a polynomial $g_i$ with coefficients in $L_{i-1}$ such that $g_i(\alpha_i)=0$ . Thus $[L_i:L_{i-1}]\leq\deg g_i<\infty$ . Similarly $[L_1:K]<\infty$ . By Tower Law, $[L:K]=[L_n:L_{n-1}][L_{n-1}:L_{n-2}]\dots[L_1:K]<\infty$ .

What About Tutoring Instead of Pills?

Just read this interesting article, which is also related to education. Personally, I am highly skeptical of doctors who prescribe expensive pills / surgery for ailments that require neither. Thankfully there is the internet nowadays so one can research personally instead of believing the doctor who may have his own interests in mind, i.e. maximum profit.

SPIEGEL Interview with Jerome Kagan: ‘What About Tutoring Instead of Pills?’

Harvard psychologist Jerome Kagan is one of the world’s leading experts in child development. In a SPIEGEL interview, he offers a scathing critique of the mental-health establishment and pharmaceutical companies, accusing them of incorrectly classifying millions as mentally ill out of self-interest and greed.

Kagan has been studying developmental psychology at Harvard University for his entire professional career. He has spent decades observing how babies and small children grow, measuring them, testing their reactions and, later, once they’ve learned to speak, questioning them over and over again. For him, the major questions are: How does personality emerge? What traits are we born with, and which ones develop over time? What determines whether someone will be happy or mentally ill over the course of his or her life?

Source: http://www.spiegel.de/international/world/child-psychologist-jerome-kagan-on-overprescibing-drugs-to-children-a-847500.html

Subgroup Isomorphic but Quotient Group Not Isomorphic

The following is a slightly shocking counterexample for beginning students of Group Theory: If $G$ is a group, and $H\cong K$ are normal subgroups of $G$ , it may be possible that $G/H\not\cong G/K$ !

Counter-example: Take $G=\mathbb{Z}/4\times\mathbb{Z}/2$ , $H=\langle (0,1)\rangle$ , $K=\langle (2,0)\rangle$ .

Note that $H\cong K\cong Z/2$ , but $G/H=\{(0,0), (1,0), (2,0), (3,0)\}\cong Z/4$ while $G/K=\{(0,0), (0,1), (1,0), (1,1)\}\cong Z/2\times Z/2$ !

(By $\mathbb{Z}/n$ we mean $\mathbb{Z}/n\mathbb{Z}$ .)

H2 Math JC1 Past Year Practice Papers

Source: https://sellfy.com/p/onQJ/

Need extra practice for JC 1 H2 Maths?

Check out the Highly Condensed H2 Maths Notes, plus the H2 Math JC 1 Past Year Practice Papers (collated out of actual school papers).

Full solution included.

Topics include JC 1 topics like Functions and Graphs, Transformations, Inequalities, Differentiation, and others.

Ideal time to practice this is before JC 1 Promos / Block test, or after JC 1 during the December Holidays for revision.

URL: https://sellfy.com/p/onQJ/

Access WordPress and Gmail in China

When travelling to China, one problem is that one cannot access WordPress, Gmail, Facebook and others.

One solution is to purchase a VPN to bypass the Great Firewall of China. Personally I use PureVPN, and it works. You can use it on both phone / PC / Mac (up to 5 devices).

If you have any other alternative methods, do post in the comments below! I have heard of alternatives like Tor / Psiphon but also comments that they do not work. Also, one should purchase the VPN before entering China, because once you enter China most of the VPN sites / mirrors are blocked by the highly intelligent Great Firewall (which uses machine learning, and also employs more than 50,000 workers to block the internet!)

Click on the links above/below to check out PureVPN!

Schwarz Lemma & Maximum Modulus Principle

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 $|a|=1$ .

Maximum modulus principle

Let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_0$ is a point in $D$ such that $|f(z_0)|\geq|f(z)|$ for all $z$ in a neighborhood of $z_0$ , then the function $f$ is constant on $D$ .

Informally, the modulus $|f|$ cannot exhibit a true local maximum that is properly within the domain of $f$ .

Republic Poly is organising Parents’ Talk

Dear Parents,

Republic Poly is organising Parents’ Talk, happening on 22 June 2016, 7:30pm.

And this time, you will learn the changes to Early Admission Exercise (EAE) and how to go about helping your child to register for it.

Check out RP website for more info.

Cheers!

Balanced Product

For a ring $R$ , a right $R$ -module $M$ , a left $R$ -module $N$ , and an abelian group $G$ , a map $\phi:M\times N\to G$ is said to be $R$ -balanced, if for all $m,m'\in M$ , $n,n'\in N$ , and $r\in R$ the following hold: