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.

Professor Writing Math Equations Suspected for being Terrorist

This is quite hilarious.

Sample of what the professor Guido Menzio was scribbling.

Source: http://www.dailymail.co.uk/news/article-3578751/Italian-Ivy-League-economist-pulled-flight-seatmate-suspected-terrorist.html

Italian Ivy League economist pulled off flight and interrogated for ‘mysterious’ scribblings flagged up by another passenger… which turned out to be MATH

Guido Menzio, 40, was on a flight from Philadelphia to Syacuse
The UPenn professor was solving a differential equation during boarding
His seatmate told an American Airlines attendant she was too ill to travel
But after she was escorted off the plane, she revealed her suspicious
Menzio was questioned about his ‘cryptic’ notes before he was allowed to get back on the plane, delayed by two hours
While one traveller thought he looked like a terrorist, another person in an airport mistook him for Sean Lennon and asked for an autograph

Harmonic Series minus terms with “9” Converges!

Something interesting I saw on: http://blogs.scientificamerican.com/roots-of-unity/what-the-prime-number-tweetbot-taught-me-about-infinite-sums/?WT.mc_id=SA_WR_20160504

The harmonic series is the infinite sum 1+1/2+1/3+1/4+…, the sum of the reciprocals of every positive integer. Even though the terms get closer and closer to 0, the series goes to infinity, meaning it eventuallygets bigger than any number you can throw at it. (The question is fun to think about, so I won’t spoil it for you, but Wikipedia has an explanation.)

It seems baffling that if we take away 1/9, 1/19, 1/29, and so on, the series converges instead. In fact, it’s less than 90, which is frankly pitiful compared to the infinitude of the harmonic series.

Functional Analysis List of Theorems

This is a list of miscellaneous theorems from Functional Analysis.

Hahn-Banach: Let $p$ be a real-valued function on a normed linear space $X$ with

(i) Positive homogeneity

(ii) Subadditivity

Let $Y$ denote a linear subspace of $X$ on which $l(y)\leq p(y)$ for all $y\in Y$ . Then $l$ can be extended to all of $X$ : $l(x)\leq p(x)$ for all $x\in X$ .

Geometric Hahn-Banach / Hyperplane Separation Theorem: Let $K$ be a nonempty convex subset, $K=int(K)$ . Let $y$ be a point outside $K$ . Then there exists a linear functional $l$ (depending on $y$ ) such that $l(x)<c$ for all $x\in K$ ; $l(y)=c$ .

Riesz Representation Theorem: Let $l(x)$ be a linear functional on a Hilbert space $H$ that is bounded: $|l(x)|\leq c\|x\|$ . Then $l(x)=\langle x,y\rangle$ for some unique $y\in H$ .

Principle of Uniform Boundedness: A weakly convergent sequence $\{x_n\}$ is uniformly bounded in norm.

Open Mapping Principle/Theorem: Let $X, U$ be Banach spaces. Let $M:X\to U$ be a bounded linear map onto all of $U$ . Then $M$ maps open sets onto open sets.

Closed Graph Theorem: Let $X,U$ be Banach spaces, $M:X\to U$ be a closed linear map. Then $M$ is continuous.

Confucius Quote Chinese Translation (with Poster)

“It doesn’t matter how slow you go as long as you do not stop” – Confucius

The actual quote is “譬如为山，未成一篑，止，吾止也。譬如平地，虽覆一篑，进，吾往也。”

The meaning is “孔子说：“做学问好比积土成山一样，只差一筐土而没有堆成山，停止了，是我自己停止的；好比平整土地一样，即使只倒下一筐土，前进了，也是我自己前进的。” (Source: http://www.bjhrwm.gov.cn/ctwhjy/yryj/t20110805_401107.htm)

In English, it means that studying (gaining wisdom) is like piling soil to make a mountain. If one stops short of one basket of soil to make a mountain, it is a pity to stop. It is also like filling a deep hole with soil, every basket of soil you pour in is progress.

The Lesson of Grace in Teaching

Nice post by Professor Francis Su:

http://mathyawp.blogspot.sg/2013/01/the-lesson-of-grace-in-teaching.html

Excerpt:

And perhaps it will help you frame your own thoughts about teaching. The beginning of that lesson is this:

Your accomplishments are NOT what make you a worthy human being.

It sounds easy for me to say, especially after having some measure of academic ‘success’ and winning this teaching award.

But twenty years ago, I was a struggling grad student, seeking validation for my mathematical talent but flailing in my research, seeking my identity in my work but discouraged enough to quit. My advisor had even said to me:

“You don’t have what it takes to be a successful mathematician.”

It was my lowest point. Weak and weary, with my identity and my pride stripped away and my PhD nearly out of reach, I realized then that my identity and self-worth could NOT rest on whether I succeeded or failed to get my PhD. So *IF* I were to continue in mathematics, I could not do it for any acclaim that I might receive or for the trappings of what the academic world would call success. I should only do it because math is beautiful, and I feel drawn to it. In my quiet moments, with no one watching, I still found math fun to think about. So I was convinced it was my calling, despite the hurtful thing my advisor had said.

So did I quit? No. I just changed advisors.

This time, I chose differently. Persi Diaconis was an inspiring teacher. More than that, he had shown me a great kindness a couple of years before. The semester I took a class from him, my mother died and I needed an extension on my work. I’ll never forget his response: “I’m really sorry about your mother. Let me take you to coffee.”

I remember thinking: “I’m just some random student and he’s taking me to coffee?” But I really needed that talk. We pondered life and its burdens, and he shared some of his own journey. For me, in a challenging academic environment, with enormous family struggles, to connect with my professor on a deeper level was a great comfort. Yes, Persi was an inspiring teacher, but this simple act of kindness—of authentic humanness—gave me a greater capacity and motivation to learn from him, because we had entered into authentic community with each other, as teacher and student, who were real people to each other.

So when the time came to change advisors, I decided to work with Persi, even though it meant completely starting over in a new area. Only in hindsight did I realize why I had gravitated to him. It’s because he showed me grace.

GRACE: good things you didn’t earn or deserve, but you’re getting them anyway.

By taking me to coffee, he had shown me he valued me as a human being, independent of my academic record. And having my worthiness separated from my performance gave me great freedom! I could truly enjoy learning again. Whether I succeeded or failed would not affect my worthiness as a human being. Because even if I failed, I knew: I am still worth having coffee with!

Knowing my new advisor had grace for me meant that he could give me honest feedback on my dissertation work, even if it was hard to do, without completely destroying my identity. Because, as I was learning, my worthiness does NOT come from my accomplishments. I call this

The Lesson of GRACE:

Your accomplishments are NOT what make you a worthy human being.
You learn this lesson when someone shows you GRACE: good things you didn’t earn or deserve, but you’re getting them anyway.

Weak* convergent sequence uniformly bounded

Theorem 11 (Lax Functional Analysis): A weak* convergent sequence $\{u_n\}$ of points in a Banach space $U=X'$ is uniformly bounded.

We will need a previous Theorem 3: $X$ is a Banach space, $\{l_v\}$ a collection of bounded linear functionals such that at every point $x$ of $X$ , $|l_v(x)|\leq M(x)$ for all $l_v$ . Then there is a constant $c$ such that $|l_v|\leq c$ for all $l_v$ .

Sketch of proof:

Weak* convergence means $\lim u_n(x)=u(x)$ , thus there exists $N$ such that for all $n\geq N$ , we have $|u_n(x)-u(x)|<1$ , which in turns means $|u_n(x)|<1+|u(x)|$ via the triangle inequality. We have managed to bound the terms greater than equals to $N$ .

For those terms less than $N$ , we have $|u_n(x)|\leq\|u_n\|\|x\|$ .

Thus, we may take $M(x)=\max\{\|u_1\|\|x\|,\dots,\|u_{N-1}\|\|x\|,1+|u(x)|\}$ . The crucial thing is that $M(x)$ depends only on $x$ , not $n$ .

Then, use Theorem 3, we can conclude that $\|u_n\|\leq c$ for all $n$ .

Desiderata – An Amazing Poem

Go placidly amid the noise and the haste, and remember what peace there may be in silence. As far as possible, without surrender, be on good terms with all persons.

Speak your truth quietly and clearly; and listen to others, even to the dull and the ignorant; they too have their story.

Avoid loud and aggressive persons; they are vexatious to the spirit. If you compare yourself with others, you may become vain or bitter, for always there will be greater and lesser persons than yourself.

Enjoy your achievements as well as your plans. Keep interested in your own career, however humble; it is a real possession in the changing fortunes of time.

Exercise caution in your business affairs, for the world is full of trickery. But let this not blind you to what virtue there is; many persons strive for high ideals, and everywhere life is full of heroism.

Be yourself. Especially, do not feign affection. Neither be cynical about love; for in the face of all aridity and disenchantment it is as perennial as the grass.

Take kindly the counsel of the years, gracefully surrendering the things of youth.

Nurture strength of spirit to shield you in sudden misfortune. But do not distress yourself with dark imaginings. Many fears are born of fatigue and loneliness.

Beyond a wholesome discipline, be gentle with yourself. You are a child of the universe no less than the trees and the stars; you have a right to be here.

And whether or not it is clear to you, no doubt the universe is unfolding as it should. Therefore be at peace with God, whatever you conceive Him to be.

And whatever your labors and aspirations, in the noisy confusion of life, keep peace in your soul. With all its sham, drudgery and broken dreams, it is still a beautiful world. Be cheerful. Strive to be happy.

Max Ehrmann, “Desiderata“

Printable version: http://www.stpaulsbaltimore.org/wp-content/uploads/2015/02/desiderata-pamphlet.pdf

Deck Transformations

Consider a covering space $p:\widetilde{X}\to X$ . The isomorphisms $\widetilde{X}\to\widetilde{X}$ are called deck transformations, and they form a group $G(\widetilde{X})$ under composition.

For the covering space $p:\mathbb{R}\to S^1$ projecting a vertical helix onto a circle, the deck transformations are the vertical translations mapping the helix onto itself, so $G(\widetilde{X})\cong\mathbb{Z}$ , where a vertical translate of $n$ “steps” upwards/downwards corresponds to the integer $\pm n$ respectively.

20 Things You Probably Didn’t Know about Singapore Math

Interesting link about Singapore Math: Link here

Some interesting points noted are:

PSLE math is harder than secondary 1,2 math. Totally agree on this.
Majority of math teachers in Singapore are not math majors

Endomorphism ring of Q is a division algebra

We show that $Q$ is not semisimple nor simple, but $\text{End}_\mathbb{Z}(\mathbb{Q})$ is a division algebra.

Consider $A=\mathbb{Z}$ (as a $\mathbb{Z}$ -algebra). Consider $M=\mathbb{Q}$ as a right $\mathbb{Z}$ -module.
Lemma:
$\mathbb{Q}$ is not semisimple nor simple.

Suppose to the contrary $\mathbb{Q}=\bigoplus_{i\in I}N_i$ , where $N_i$ are simple $\mathbb{Z}$ -modules (i.e. $N_i\cong\mathbb{Z}/p_i\mathbb{Z}$ ). Then there exists nonzero $x\in\mathbb{Q}$ such that $x$ has finite order (product of primes). This is impossible in $\mathbb{Q}$ .
Lemma:
$\text{End}_\mathbb{Z}(\mathbb{Q})\cong\mathbb{Q}$ as $\mathbb{Z}$ -algebras.

Define $\Psi:\mathbb{Q}\to\text{End}_\mathbb{Z}(\mathbb{Q})$ where $q\in\mathbb{Q}$ is mapped to $\lambda_q\in\text{End}_\mathbb{Z}(\mathbb{Q})$ , where $\lambda_q(x)=qx$ . Let $k\in\mathbb{Z}$ , $q,q_1,q_2\in\mathbb{Q}$ .

We can check that $\Psi$ is a $\mathbb{Z}$ -algebra homomorphism.

Let $q\in\ker\Psi$ . Then $\Psi(q)=\lambda_q=0$ , $\lambda_q(x)=qx=0$ for all $x\in\mathbb{Q}$ . This implies $q=q\cdot 1=0$ . Hence $\Psi$ is injective.

Let $\phi\in\text{End}_\mathbb{Z}(\mathbb{Q})$ . Let $x=\frac{a}{b}\in\mathbb{Q}$ , where $a,b\in\mathbb{Z}$ . $\phi(x)=a\phi(\frac 1b)=\frac ab\cdot b\phi(\frac 1b)=\frac ab\cdot\phi(1)=\phi(1)\cdot x=\lambda_{\phi(1)}(x)$ . Hence $\Psi$ is surjective.

Thus $\text{End}_\mathbb{Z}(\mathbb{Q})\cong\mathbb{Q}$ is a division algebra, but $\mathbb{Q}$ is not simple.

How Does a Mathematician’s Brain Differ from That of a Mere Mortal?

Source: http://www.scientificamerican.com/article/how-does-a-mathematician-s-brain-differ-from-that-of-a-mere-mortal/?WT.mc_id=SA_WR_20160420

Interesting article!

The main question I am curious is, how do the differences in brain structure come about? Is it cause or effect, i.e. does difference in brain lead to becoming a mathematician, or does working on mathematics lead to a change in brain structure?

Also read: How I Learned the Art of Math [Excerpt]

Covering map is an open map

We prove a lemma that the covering map $p:\tilde{X}\to X$ is an open map.

Let $U$ be open in $\tilde{X}$ . Let $y\in p(U)$ , then $y$ has an evenly covered open neighborhood $V$ , such that $p^{-1}(V)=\coprod A_i$ , where the $A_i$ are disjoint open sets in $\tilde{X}$ , and $p|_{A_i}:A_i\to V$ is a homeomorphism. $A_i\cap U$ is open in $\tilde{X}$ , and open in $A_i$ , so $p(A_i\cap U)$ is open in $U$ , thus open in $X$ .

There exists $x\in U$ such that $y=p(x)$ . Thus $x\in p^{-1}(y)\subseteq p^{-1}(V)$ so $x\in A_i$ for some $i$ . Thus $x\in A_i\cap U$ and thus $y\in p(A_i\cap U)\subseteq p(U)$ . This shows $y$ is an interior point of $p(U)$ . Hence $p(U)$ is open, thus $p$ is an open map.

WordPress.com Slow

Recently, experienced unusual slowness in the WordPress.com Login website. Other websites load fast, except WordPress.com. It can take up to 10 seconds to load the Dashboard, which is unusual for WordPress.com.

Quite strange. Not sure if the WordPress.com App for Mac will be faster, haven’t tried that out yet.

Coping with maths anxiety

Source: http://www.straitstimes.com/singapore/coping-with-maths-anxiety

This is an article on the Straits Times on children who experience difficulty learning mathematics.

The highlight of the article are the words of Dr Mighton, who is an expert on math learning and has a PhD in Mathematics from the University of Toronto.

This is a highly recommended book that he wrote:

The Myth of Ability: Nurturing Mathematical Talent in Every Child

The following is from the Straits Times (see link above):

I knew we were in trouble when my son looked uncomprehendingly at me, then nodded slowly.

I had been trying for several futile minutes to explain, in growing decibels, the solution to a maths problem sum. Finally, I snapped in frustration: “So do you get it or not?”

He obviously did not, but was scared of admitting it lest it fuelled my irritation.

…

The most reassuring words come from Dr John Mighton, a former maths tutor in Toronto who went on to develop Jump (Junior Undiscovered Math Prodigies) Math as a charity in 2001. Its website offers free teaching guides and lesson plans for educators and parents.

Everyone, he says, can learn maths at a very high level, to the point where they can do university-level maths courses.

His Jump Math curriculum, based on breaking things down into minute steps to slowly build confidence, bears this out. It has yielded impressive results in some Canadian and British schools, which adopted the programme for students who struggled the most with maths.

Dr Mighton, who is also a playwright and author, designed Jump Math based on his own experience. He nearly failed his first-year calculus course, but trained himself to break down complicated tasks and practise them until he got the hang of things. He went on to do a PhD in mathematics at the University of Toronto.

The Strange Location of Your Second Brain

Check out this interesting video on the human body’s second brain — the intestines.

Children who are studying may want to drink more Yakult and yoghurt to have a healthy intestine, which impacts the brain.

Changes to PSLE: Less stress for students but don’t dumb down education system

Source: http://www.straitstimes.com/singapore/education/changes-to-psle-less-stress-for-students-but-dont-dumb-down-education-system

Latest Straits Times article on the PSLE.

The Ministry of Education’s move is laudable. In effect, though, kiasu parents will still find a way to put the screws on their children. Mark my words. No system is perfect. But the problem of stress lies largely with parents who cannot accept that their children are anything less than the best.

Topology Puzzle

Assume you are a superman who is very elastic, after making linked rings with your index fingers and thumbs, could you move your hands apart without separating the joined fingertips?

In other words, is it possible to go from (a) to (b) without “breaking” the figure above?

Figure taken from Intuitive Topology (Mathematical World, Vol 4).

The answer is yes!

This is an animation of the solution: https://vk.com/video-9666747_142799479