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

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Check out this website for Tyre repairs and replacements!

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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:

$\begin{aligned} \phi(m,n+n')&=\phi(m,n)+\phi(m,n')\\ \phi(m+m',n)&=\phi(m,n)+\phi(m',n)\\ \phi(m\cdot r,n)&=\phi(m,r\cdot n) \end{aligned}$

The first two axioms are essentially bilinearity, while the third is something like associativity.

Something Interesting to do with Paperclips

Study: Kids from affluent families more likely in IP, GEP schools

Source: http://www.straitstimes.com/singapore/education/study-kids-from-affluent-families-more-likely-in-ip-gep-schools

Children from higher socio-economic backgrounds are more likely to attend Integrated Programme (IP) secondary schools and their affiliated primary schools, as well as those that offer the Gifted Education Programme (GEP). – Straits Times

Another related news is Students in IP schools more confident of getting at least a university degree, also published by the Straits Times.

It is like a perpetual virtuous cycle: GEP/IP -> University -> Affluent -> GEP/IP (next generation) -> …, no wonder tuition is so popular in Singapore as no doubt every parent wants their child to get into the virtuous cycle above.

More research needs to be done on how lower-income families can be helped for their children to reach their fullest potential.

“An equation means nothing to me unless it expresses a thought of God. “

Just watched The Man Who Knew Infinity today at Shaw Cinemas @JCube. Very nice and meaningful movie that is different from the typical movie.

One famous quote by Ramanujan is that “An equation means nothing to me unless it expresses a thought of God.”.

Another interesting link online is that Ramanujan did not suffer from tuberculosis, rather he probably had an amoeba infection called amoebiasis.

This parent has hit the nail on the head regarding tuition

Source: http://www.straitstimes.com/forum/letters-on-the-web/tuition-still-effective-for-many

This parent has hit the nail on the head (find exactly the right answer) regarding tuition:

Whether tuition yields results largely depends on pupils’ attitudes towards learning and how motivated they are.

In conclusion, tuition is necessary and can be effective if pupils make full use of it. But parents still need to decide for themselves if tuition is the answer for their children, and not be influenced by societal pressure.

Read the article for the full letter.

Two-dimensional Divergence Theorem is equivalent to Green’s Theorem

The special planar version of Divergence Theorem is actually equivalent to Green’s Theorem, even though on first sight they look quite different.

Getting a degree in Singapore set to become costlier: Study

Source: http://www.straitstimes.com/singapore/education/getting-a-degree-in-singapore-set-to-become-costlier-study

The cost of a university degree in Singapore is set to rise, according to a new study by the Economist Intelligence Unit (EIU).

Released yesterday, the study projected that a four-year degree will cost 70.2 per cent of an individual’s average yearly income in 2030, up from 53.1 per cent last year.

Since 2010, tuition fees at local universities have gone up every year for most undergraduate courses, mainly due to rising operating costs.

2016 PSLE Difficulty — Second Hardest PSLE in history

Watch this interesting video on the PSLE (Primary School Leaving Exam) in Singapore: https://www.facebook.com/cnainsider/videos/1073805505975459/

According to this girl, her teacher says that PSLE 2016 is the 2nd hardest exam in history.

It seems school children these days have longer working hours than even adults. Adults work from 8am-5pm, children have to study from 7am all the way to night time.

One problem from all these cramming may be loss of joy in learning. After all the years of “forced studying”, few if any students have any more joy of learning in their hearts.

The young girl in the video is very optimistic and cheerful despite all the extra classes, keep it up!

How to Excel in DSA

With PSLE getting more and more tough, DSA is more and more important as a backup plan, or even as “Plan A”. A well-planned DSA application could lead to success in entering the secondary school of your choice. Once the PSLE new scoring system is out, DSA is the critical distinguishing factor.

Check out these posts on how to excel in DSA.

WordAds Earnings Rock Bottom

Something is seriously wrong with WordAds earnings last month (April 2016).

I have 3 times the usual number of ad impressions, but earnings have gone down to 1/5 of the usual earnings.

$$…$$ should not be used in LaTeX

Just learnt today that the popular command $$…$$ in LaTeX should not be used as it produces wrong spacing and is an obsolete command.

The correct command should be \[ … \].

Both have the same number of characters and thus are just as easy to type, so it is probably a good idea to switch to \[ … \].

See http://tex.stackexchange.com/questions/503/why-is-preferable-to

The Man Who Knew Infinity Now Showing in Singapore

The amazing movie on Ramanujan’s life is now showing in Singapore. Selected cinemas, for example Shaw, are showing the movie. I am looking forward to watching the movie some time next week. 🙂

Although the movie has mixed reviews, this should be a must-watch for all math fans.

Wonderful Topology Notes for Beginners

Recently found a wonderful topology notes, suitable for beginners at: http://mathcircle.berkeley.edu/archivedocs/2010_2011/lectures/1011lecturespdf/bmc_topology_manifolds.pdf

It starts by pondering the shape of the earth, then generalizes to other surfaces. It also has a nice section on Fundamental Polygons and cutting and gluing, which was what I was looking for at first.

I have backed up a copy on Mathtuition88.com, in case the original site goes down in the future: bmc_topology_manifolds

J-PACT Recommended Books

What is J-PACT / PACT

J-PACT is basically a test that foreigners / overseas Singaporeans must take in order to study in MOE Junior Colleges in Singapore. The benefits of studying in MOE schools (over international schools) is that the school fees are cheaper, and the student gets to integrate better into Singapore’s culture. Another benefit is to study Chinese in schools.

PACT is the primary/secondary school version of J-PACT, for students seeking to enter primary/secondary school (Grade 1-10) in Singapore.

More information: http://www.pact.sg/index.php?option=com_content&view=article&id=59&Itemid=95

J-PACT Books and Past Year Sample Papers

According to this website by experienced expats, the best books to study for the J-PACT / PACT are by the publisher Shinglee.

For J-PACT, you would want to study books from Secondary 1-4. For instance, this is the Secondary 2 Math Textbook by Shinglee (recommended):

New Syllabus Mathematics 2

Shinglee also has Math Workbooks which are more for students to practice on.
New Syllabus Mathematics 2 Workbook

For students who have strong foundation, they can just read the Secondary 3-4 books. For students who are weaker in the subject, it is recommended to work through all the Secondary 1-4 books.

New Syllabus Mathematics 4

For Sample Papers / Past Year Papers for J-PACT, there don’t seem to be any on the web. What students can do is to try out local school papers (Secondary 4) to get an idea of the required level. Do check out this page on Math Resources where there are E Maths/ A Maths notes bundled with exam papers for practice. Students in Singapore at the secondary level usually study double maths (E Maths and A Maths), where E Maths is elementary level maths (e.g. basic geometry and algebra), and A Maths covers more advanced material like calculus and trigonometry.

Abel’s Theorem

Abel’s Theorem is a useful theorem in analysis.

Let $(a_k)$ be any sequence in $\mathbb{R}$ or $\mathbb{C}$ . Let $G_a(z)=\sum_{k=0}^\infty a_k z^k$ . Suppose that the series $\sum_{k=0}^\infty a_k$ converges. Then
$\lim_{z\to 1^-}G_a(z)=\sum_{k=0}^\infty a_k,$ where $z$ is real, or more generally, lies within any Stolz angle, i.e.\ a region of the open unit disk where $|1-z|\leq M(1-|z|)$ for some $M$ .

From Triangle to Mobius Strip

What familiar space is the quotient $\Delta$ -complex of a 2-simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$ , preserving the ordering of vertices?

This is a question from Hatcher. So if we “glue” two edges of the triangle together, preserving order, we get a Mobius Strip!

Try your best. Even if you don’t hit your mark, you’ll be near the mark and that’s all you can hope for in life.

Source: http://www.bdtonline.com/opinion/the-race-of-life-fear-of-failure-can-keep-people/article_b7dfe5c0-1e0c-11e6-ae25-e7a71f04d85b.html

Excerpts from this well written article:

Try your best. Even if you don’t hit your mark, you’ll be near the mark and that’s all you can hope for in life.

I’ve been knocked down in life, but I’m still close to where I want to be. I like writing and I hope to write a book some day. Trying to do something is not as bad as Yoda made it sound to be. “Do or do not there is no try,” Yoda said. Wrong Yoda. Trying is good for a person.

I think life is a bundle of different things. A couple of years ago, an evangelist came to my church and he said the worries we’ve got are first-world problems. Something I needed to hear. He said (most of us) are not worried about eating, where we’ll lay our heads at night or if we’ll be warm. Now, I realize that there are people in our community that are struggling, but for most of us with jobs and a roof over our heads what we’re looking at and what we spend most of our time struggling with are first-world problems. What are first world problems? First-world problems are the worries and cares of this world, in my humble opinion. We get caught up worrying to death about our jobs, whether we’ll do well on an exam or if our performance in the sports arena will land us the next big sign on.

We can’t get lost in this or we’ll miss life’s precious moments. The time with friends, the time with grandparents are little moments we don’t think about as we pass through our ordinary days. I urge you to spend time with those who are closest to you. You may spend a lifetime regretting not spending time with those you love because you became caught up in the rat race of life.

King Solomon said all is vanity and chasing after the wind. I’d say that aside from saving souls, he’s right. We want more. We get more. We want more again. Most of us are middle class, struggling with life’s issues. A bad boss, an angry customer, a student who’s on the edge of giving up. These things are all shaping us. Making us better. Gearing us up for the next round. Life’s a race, the bible says. Let us run with endurance the race placed before us.

Is your heart in your race? Are you in the wrong race? Do you just want to give up and throw in the towel. Maybe you’re way behind the race and you feel like you’ll never get to the end. Everyone is running the race of life. Encourage those who you’re running with. Help pick up those who’ve stumbled during the race. A lot of times it’s not what place you finish in the race, but how you ran the race. Integrity, helping others, putting others before yourself — those are the keys to truly winning the race at hand.

Half a Million views

Mathtuition88.com has hit half a million views! 505,224 views to be exact.

Thanks for all the support by the readers and subscribers!

F&N Alive Yoghurt Review

F&N Alive Yoghurt is one of the very few yoghurts in Singapore that have the following bacterial strains:

Bifidobacterium BB-12
Lactobacillus acidophilus LA-5

Most yoghurts only have the standard Lactobacillus bulgarius and Streptococcus thermophilus. So this is a plus point for F&N Alive Yoghurt.

Unfortunately, the texture of F&N Alive Yoghurt is too watery, it is more like a yoghurt drink instead of yoghurt. Also, it is too sweet.

H2 Topical Package with Highly Condensed Summary

All products are listed on the main page: https://mathtuition88.com/math-notes-worksheets-sale/

The new product is the H2 Topical Package with Highly Condensed Summary.

Highly Condensed H2 Maths Notes, bundled with hundreds of Topical questions (arranged according to topic) for H2 Mathematics.

Topical questions have full solution.

Will be useful for students preparing for H2 Maths (A Levels / Prelims / Common Tests).

The list of topics ranges from Binomial, AP/GP, all the way to Statistics and Complex Numbers. Basically all topics in H2 Maths. The filelist can be seen in the screenshots above.

Suitable for both old and new syllabus (9740/9758).

Basically this package would be recommended for students in JC1 / Mid JC2 who are not taught all the topics yet, hence are unable to attempt full prelim papers.

For JC2 students already in the final months of JC, they can purchase the other package featuring prelim papers instead.

Outer Measure Zero implies Measurable

One quick way is to use Caratheodory’s Criterion:

Let $\lambda^*$ denote the Lebesgue outer measure on $\mathbb{R}^n$ , and let $E\subseteq\mathbb{R}^n$ . Then $E$ is Lebesgue measurable if and only if $\lambda^*(A)=\lambda^*(A\cap E)+\lambda^*(A\cap E^c)$ for every $A\subseteq\mathbb{R}^n$ .

Suppose $E$ is a set with outer measure zero, and $A$ be any subset of $\mathbb{R}^n$ .

Then $\lambda^*(A\cap E)+\lambda^*(A\cap E^c)\leq\lambda^*(E)+\lambda^*(A)=\lambda^*(A)$ by the monotonicity of outer measure.

The other direction $\lambda^*(A)\leq\lambda^*(A\cap E)+\lambda^*(A\cap E^c)$ follows by countable subadditivity of outer measure.

VCH OPEN HOUSE 2016

Wed, 25 May 2016, 8am – 7pm
Victoria Concert Hall

Join us at the Victoria Concert Hall for a day of musical fun! From educational workshops, to backstage and historical tours of the building, and not forgetting indoor and outdoor performances – there’s definitely something for everyone.
FREE PERFORMANCES

Time	Event	Venue
10.30AM – 11.00AM	Clarinet Duet	Atrium
11.30AM – 12.30PM	More than Music*	Concert Hall
12.30PM – 1.30PM	Bloco Singapura	Empress Lawn
1.30PM – 2.00PM	Flute Duet	Atrium
2.30PM – 3.30PM	Nadi Singapura	Empress Lawn
4.00PM – 4.30PM	Guzheng & Percussion	Atrium
5.00PM – 6.00PM	Singapore National Youth Sinfonia*	Concert Hall

*Seating in the concert hall is limited and on a first-come-first serve basis.