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

A Chinese-English Mathematics Primer

Link: http://maths.anu.edu.au/research/cma-proceedings/chinese-english-mathematics-primer

Interesting book on how to read mathematical texts in Chinese. Even students fluent in Chinese everyday language may have difficulty translating the mathematical terms from Chinese to English (and vice versa), hence this book is a very useful one.

My own interest in the work of Chinese mathematicans arises from their significant contributions to the qualitative theory of ordinary differential equations and, in particular, of plane quadratic systems. However, Chinese mathematics covers a very wide range. This is hardly surprising, since a quarter of the world’s population is Chinese. It is predictable that during the next quarter century the importance of Chinese mathematics will increase greatly.

How to prepare for Changes to PSLE grading

Source: http://www.straitstimes.com/singapore/education/changes-to-psle-grading-what-could-be-in-store

The latest update is that the Primary School Leaving Examination (PSLE) aggregate score is soon going to be scrapped and replaced with simple grade bands such as A, B and C. What effects will there be and how to prepare in advance for it?

Do leave your comments below!

My opinion is that there will be a few crucial changes that parents would have to prepare for as soon as possible:

Chinese (or Mother Tongue) is a top priority. In the past, many students from English-speaking families can get low marks for Chinese, high marks for Math/Science/English and still get a very good PSLE score (e.g. >250). This is unfortunately not possible anymore in the new system. A low Chinese grade will drag down the entire performance. Also read more about the benefits of studying Chinese.

DSA (Direct School Admission) GAT (General Ability Test) and CCA becomes more important. The effect of simple grade bands is that many students will get the perfect score of all As. However, the top schools have limited vacancies and thus will have to use other criteria like DSA and CCA to differentiate students. Check out this previous post on DSA.

The new system benefits all-rounders who are good (but not necessarily excellent) at all subjects, including CCA. All-rounders will manage to get ‘A’s in all subjects. However, the new system is unfortunately not good for those who are excellent in one single subject, but average in others.

This quote from the article is very good:

“The focus should not be on how one performs relative to others, but how well the person himself performs in the exam.”

DR TIMOTHY CHAN, director of SIM Global Education’s academic division, on the use of grade-banding to reflect pupils’ abilities.

Weak Convergence

The sequence $\{x_n\}$ converges weakly to $x$ means that $\lim l(x_n)=l(x)$ for every linear functional $l$ in $X'$ .

The notation for weak convergence is $x_n\rightharpoonup x$ , which is typed as \rightharpoonup in LaTeX.

Topology book for General Audience

Previously I blogged about the book The Shape of Space, here is another book that is also about topology and suitable for a general audience (high school and above)!

Intuitive Topology (Mathematical World, Vol 4)

New Math Recommended Book: Mathematics without Apologies: Portrait of a Problematic Vocation (Science Essentials)

Mathematics without Apologies: Portrait of a Problematic Vocation (Science Essentials)

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers–for the sake of truth, beauty, and practical applications–this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining,Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

Irreducible representations

Let $\rho:G\to \text{GL}(V)$ be a linear representation of $G$ . We say that it is irreducible or simple if $V$ is not 0 and if no vector subspace of $V$ is stable under $G$ , except of course 0 and $V$ . This is equivalent to saying $V$ is not the direct sum of two representations, except for the trivial decomposition $V=0\oplus V$ .

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Hahn-Banach Theorem: Crown Jewel of Functional Analysis

Hahn-Banach Theorem is called the Crown Jewel of Functional Analysis, and has many different versions.

There is a Chinese quote “实变函数学十遍，泛函分析心犯寒”, which means one needs to study real function theory ten times before understanding, and the heart can go cold when studying functional analysis, which shows how deep is this subject.

The following is one version of Hahn-Banach Theorem that I find quite useful:

(Hahn-Banach, Version) If $V$ is a normal vector space with linear subspace $U$ (not necessarily closed) and if $z$ is an element of $V$ not in the closure of $U$ , then there exists a continuous linear map $\psi:V\to K$ with $\psi(x)=0$ for all $x\in U$ , $\psi(z)=1$ , and $\|\psi\|=\text{dist}(z,U)^{-1}$ .

Brief sketch of proof: Define $\phi:U+\text{span}\{z\}\to K$ , $\phi(u+\lambda z)=\lambda$ , and use the Hahn-Banach (Extension version).

Effective Homotopy Method

The main idea of the effective homotopy method is the following: given some Kan simplicial sets $K_1,\dots,K_n$ , a topological constructor $\Phi$ produces a new simplicial set $K$ . If solutions for the homotopical problems of the spaces $K_1,\dots,K_n$ are known, then one should be able to build a solution for the homotopical problem of $K$ , and this construction would allow us to compute the homotopy groups $\pi_*(K)$ .

Inspirational Story of Sir John Gurdon, Nobel Prize winner

Source: http://www.telegraph.co.uk/news/science/science-news/9594351/Sir-John-Gurdon-Nobel-Prize-winner-was-too-stupid-for-science-at-school.html

A British scientist whose schoolmasters told him he was too stupid to study the subject has been awarded the Nobel Prize in medicine or physiology for his pioneering work on cloning.

At the age of 15, Prof Sir John Gurdon ranked last out of the 250 boys in his Eton year group at biology, and was in the bottom set in every other science subject. Sixty-four years later he has been recognised as one of the finest minds of his generation after being awarded the £750,000 annual prize, which he shares with Japanese stem cell researcher Shinya Yamanaka. Speaking after learning of his award in London on Monday, Sir John revealed that his school report still sits above his desk at the Gurdon Institute in Cambridge, which is named in his honour.

Moral of the story: Teachers may not be always right!

Congrats to Professor Andrew Wiles

http://www.telegraph.co.uk/news/science/science-news/12195189/Oxford-professor-wins-500000-for-solving-300-year-old-mathematical-mystery.html

Oxford professor wins £500,000 for solving 300-year-old mathematical mystery

Sir Andrew Wiles’ proof of Fermat’s Last Theorem has been described as ‘an epochal moment for mathematics’

An Oxford University professor has won a £500,000 prize for solving a three-century-old mathematical mystery that was described as an “epochal moment” for academics.

Sir Andrew Wiles, 62, has been awarded the Abel Prize by the Norwegian Academy of Science and Letters – and almost half a million pounds – for his proof of Fermat’s Last Theorem, which he published in 1994.

Idea for making a map bijective

A technique in algebra to make a homomorphism injective is to “mod out” the kernel.

While, to make a homomorphism surjective, one can restrict the codomain to the image.

This can be illustrated in the first isomorphism theorem (for groups) $G/\ker\phi\cong\text{Im}\ \phi$ .

Regular Representation of G

Let $g$ be the order of $G$ , and let $V$ be a vector space of dimension $g$ , with a basis $(e_t)_{t\in G}$ indexed by the elements $t$ of $G$ . For $s\in G$ , let $\rho_s$ be the linear map of $V$ into $V$ which sends $e_t$ to $e_{st}$ ; this defines a linear representation, which is called the regular representation of $G$ .

Source: Linear Representations of Finite Groups (Graduate Texts in Mathematics) (v. 42)

Happy Pi Day

Wishing all readers a very happy Pi Day!

For US residents, you may want to try out Pizza Hut’s Pi Day Math Contest, which features questions set by famous mathematician John H. Conway.

The prize is 3.14 years of Pizza Hut pizza!

Do your best without comparing yourself to others and without fear or failure

Recently, the A Level results just came out, 93.1% score at least 3 H2 passes, best results since curriculum change in 2006. However, as most students know, 3 H2 passes (low pass e.g. 3 C’s) is far from enough to enter the 3 local universities. For those looking for a rank point calculator, check out my post on how to calculate JC Rank Points.

Just to share some motivational advice for those who may not have done as well for A levels. Source: http://www.kuenselonline.com/do-your-best-without-comparing-yourself-to-others-and-without-fear-or-failure/

In reality, the purpose of education should be to open your mind, gain life skills and help you develop your human qualities. It should not merely be considered as a gateway to a job.

Even if your main reason for studying is motivated by your future career, you still need to first consider what you really want to do before proceeding with your education plans. If, for example, you have a passion for cooking, then it might be better to enter a chef training programme rather than spend two more years in school. On the other hand, if you want to be a teacher then class 12 will be your route to achieve your goal.

Anyway, whatever you do, you should do it to the best of your ability. At the same time, you should have no expectations about the result.

Maybe this example will be helpful: Think of the seeds of a sunflower and a violet. A sunflower seed will produce a large, bright yellow flower, while the seed of a violet will produce a small, dark purple bloom. A violet seed can never produce a sunflower blossom no matter how hard it tries. Likewise, a sunflower seed cannot produce the flower of a violet. Neither flower is better or worse than the other. They are just different. However, it is important that the flowers open fully and are not ashamed whether they are small or large, bright or dark.

In the same way, you might discover that you are great at studies or you might find that you are not so great. Like the seeds, you cannot change your natural inclinations, but like the flowers you have to open fully. This means that you try your very best in every situation.

Practically, to do your best means that whatever assignment you are given, you aim to do it beautifully – not to get high grades, but for your own satisfaction. If you have to compose an essay, for example, write each letter and word clearly and in a way that is easy for others to read. Do the same with a math or science or any other assignment. Make each page of your notebook a work of art.

Most importantly is to have tried your very best. Very nice article!

Tough at home but teen perseveres and scores at A-level exams

Source: http://www.straitstimes.com/singapore/tough-at-home-but-teen-perseveres-and-scores-at-a-level-exams

Very inspirational!

Excerpt:

Richmond Tan does not have a study table in the one-room rental flat that he shares with his father, grandfather and brother.

There were times when he did not even have a roof over his head, after his family was temporarily chased out of the Queenstown flat as a result of spats between his father and the landlord.

Richmond had to sleep in void decks or at Changi Airport.

He said what motivated him to do well in his studies was his role model, Han Xin, one of the heroes of the early Han dynasty in China.

He learnt of Han after reading The Art Of War by Sun Tzu, which inspired him to take China Studies for his A levels.

“Han Xin was so poor he had to beg for food. His circumstances were even worse than mine, but he studied hard and became a capable minister,” he added.

Han Xin was also a mathematician, and one of the earliest to discover the secret of the Chinese Remainder Theorem, a key result in Number Theory. According to legend, he used it to calculate the number of soldiers in his army. See this post for more details: http://chinesetuition88.com/2015/04/25/chinese-remainder-theorem-history-%E9%9F%A9%E4%BF%A1%E7%82%B9%E5%85%B5/

Interview of Michael Atiyah (aged 86!)

Source: https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/

Inspirational interview by Michael Atiyah, winner of both Fields Medal and Abel Prize, currently age 86!

Excerpt from the interview:

Is there one big question that has always guided you?

I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.

People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.

Interesting comment that “A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works.”. Sometimes, intuitive understanding is needed, along with formal proof.

One example in high school mathematics is proving $\displaystyle \sum_{i=1}^n i^2=\frac 16n(n+1)(2n+1)$ . It is possible to prove it by induction without actually understanding how the formula comes about!

Artin-Whaples Theorem

There seems to be another version of Artin-Whaples Theorem, called the Artin-Whaples Approximation theorem.

The theorem stated here is Artin-Whaples Theorem for central simple algebras.

Artin-Whaples Theorem: Let $A$ be a central simple algebra over a field $F$ . Let $a_1,\dots,a_n\in A$ be linearly independent over $F$ and let $b_1,\dots,b_n$ be any elements in $A$ . Then there exists $a_i',a_i''\in A$ for $i=1,\dots,m$ such that the $F$ -linear map $f:A\to A$ defined by $f(x)=\sum_{r=1}^m a_r'xa_r''$ satisfies $f(a_j)=b_j$ for all $j=1,\dots,n$ .

Very nice and useful theorem.

Simple Algebra does not imply Semisimple Algebra

The terminology “semisimple” algebra suggests a generalization of simple algebras, but in fact not all simple algebras are semisimple! (Exercises 1 & 5 in Richard Pierce’s book contain examples)

A simple module is a semisimple module is true though.

Proposition: For a simple algebra A, the following conditions are equivalent:

(i) A is semisimple;

(ii) A is right Artinian;

(iii) A has a minimal right ideal.

Thus to find a algebra that is simple but not semisimple, one can look for an example that is not right Artinian.

Amazon Associates / Affiliates Payment Options (Outside US)

For those using Amazon Associates (or other US based affiliate programs) but are outside USA, there is now an option to transfer the earnings directly to your bank account. Works in Singapore, and most other countries.

The way to do it is through Payoneer. (Direct link: https://share.payoneer.com/nav/FdReUzoZLHa355a95CrNv8uVU528u1dPTojBD2lcYbYvHYFDYT0WxkdmgqlJSsjRs75isVLfZi9fhax47Braxw2)

The other option of receiving cheque from Amazon is bad since the administrative fees are quite expensive (around 20 USD if I remember correctly).

Amazon Associates is quite a good affiliate program. If your blog has just around 100 visits per day, it can already translate to around $20 USD or more per month earnings depending on what you are promoting. This will most likely to cover the cost of hosting your blog. People with extremely popular blogs (see here) have earned more than half a million USD from Amazon Associates.

For those thinking of signing up Payoneer please sign up using the link above. It is a special “Refer a Friend” program where both you and I will get $25 USD upon you signing up. (* After your friend signs up and receives a total of $100, you both earn a $25 reward.)

Drawing Commutative Diagrams with Tikz-CD

Source: http://www.jmilne.org/not/Mtikz.pdf

The easiest way to draw commutative diagrams is actually with amscd, but this has the drawback of not supporting diagonal arrows.

The next easiest is tikz-CD.

Sample of Commutative Diagram done with tikz-cd.

Advice to a Young Mathematician

The official preview is available here at: http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

Excerpt:

The most important thing that a young mathematician needs to learn is of course mathematics. However, it can also be very valuable to learn from the experiences of other mathematicians. The five contributors to this article were asked to draw on their experiences of mathematical life and research, and to offer advice that they might have liked to receive when they were just setting out on their careers. (The title of this entry is a nod to Sir Peter Medawar’s well-known book, Advice to a Young Scientist.) The resulting contributions were every bit as interesting as we had expected; what was more surprising was that there was remarkably little overlap between the contributions. So here they are, five gems intended for young mathematicians but surely destined to be read and enjoyed by mathematicians of all ages.

The full book can be bought on Amazon:

The Princeton Companion to Mathematics

Direct Sum vs Cartesian Product

Excellent explanation found on Math Stackexchange.

Basically for finite index sets (finite number of factors), the two constructions are the same.

Only when there is an infinite number of factors, the direct sum $\bigoplus_{i\in I}G_i$ is the subgroup of the Cartesian product consisting of all tuples $\{g_i\}$ where there are only finitely many $g_i$ that are nonzero.

Toddlers prepare for their first big interview

Source: BBC

Getting into good schools or universities is tough in many parts of the world, but in Hong Kong the pressure begins earlier. Often parents try to get children into a good kindergarten – and before that, into a good nursery. So there are now classes preparing toddlers for that all-important nursery interview.

Yoyo Chan is preparing for an important interview that could help her succeed in life. She is one-and-a-half years old.

At two she will start nursery, but competition is fierce in Hong Kong, and some of the most prestigious nurseries are selective. Her parents want her to be well-prepared for her first big test in life.

The best nurseries and kindergartens are seen as a gateway into the best primary schools – which in turn, parents believe, pave the way to the best secondary schools and universities.

So the most renowned of them can receive more than 1,000 applications for just a few dozen places. As a result, enterprising tuition companies are now offering interview training for toddlers.

Reliable Tuition Agency Singapore

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There are many excellent tutors from RI, Hwa Chong, etc. at Startutor, teaching various subjects at all levels.

High calibre scholars from NUS/overseas universities like Stanford are also tutoring at Startutor.

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O Level Chinese Exam Date 2016

Mid-Year CL/ML/TL (Written Paper)

Date: Monday, 30 May 2016

Source: http://www.seab.gov.sg/content/examCalendar/2016_ExamCalendar_O-Level.pdf

For Chinese Tuition, check out http://chinesetuition88.com

How to use Kenzo (Algebraic Topology CAS) on Clozure CL

After some trials, finally got Kenzo to run on Mac:

First download it from (https://github.com/gheber/kenzo). Quicklisp needs to be installed and loaded first.

Type:

(ql:quickload :kenzo)

followed by

(in-package “CAT”)

followed by any Kenzo commands.

Screenshot:

How to optimise your brain’s waste disposal system

Source: https://www.theguardian.com/science/neurophilosophy/2015/aug/22/how-to-optimise-your-brains-waste-disposal-system

Summary: Sleeping on the side seems to clear brain’s waste most efficiently.

New research suggests that body posture during sleep may affect the efficiency of the brain’s self-cleaning process

The human brain can be compared to something like a big, bustling city. It has workers, the neurons and glial cells which co-operate with each other to process information; it has offices, the clusters of cells that work together to achieve specific tasks; it has highways, the fibre bundles that transfer information across long distances; and it has centralised hubs, the densely interconnected nodes that integrate information from its distributed networks.

Like any big city, the brain also produces large amounts of waste products, which have to be cleared away so that they do not clog up its delicate moving parts. Until very recently, though, we knew very little about how this happens. The brain’s waste disposal system has now been identified. We now know that it operates while we sleep at night, just like the waste collectors in most big cities, and the latest research suggests that certain sleeping positions might make it more efficient.

阅读理解中常见典故解说—”千里马”和”伯乐” Common Literary Quotation In Reading Comprehension

Chinese Tuition Singapore

千里马指日行千里，善跑的骏马。

伯乐本名孙阳，是古代春秋时期秦穆公时人。他擅长相马，对马很有研究。

唐代文学家韩愈写过一篇文章《马说》。文章中写道”世有伯乐，然后有千里马。千里马常有，而伯乐不常有。”意思是：世界上有了伯乐才会有千里马。千里马经常会有，但是伯乐却很少。”

现在的”千里马”通常用来比喻人才，而”伯乐”则是用来比喻发现，推荐，培养和使用人才的人。

“千里马”，即人才有很多，但是能够赏识人才的”伯乐”却很少。如果人才不被发现和重用，那就和普通人没有区别。只有当伯乐发掘了他们，他们的价值才会被体现出来，才会发挥一个人才的价值。所以先有伯乐，才有千里马。

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Install Common Lisp on Mac

I am intending to install Common Lisp on Mac. Installing Lisp is quite troublesome, for normal users who are used to installing everything in one click.

Clozure CL (App for Mac) seems to be the easiest option. However, I have been faced with Clozure CL App error: “Unknown Error” message, followed by “We Could not complete your request”. This error appears when I click on install the app.

The next method I am trying is http://www.jonathanfischer.net/modern-common-lisp-on-osx/. Currently at the step of installing Aquamacs, everything seems to be going smoothly so far.

Update: The above website’s method is not as easy as it looks. http://www.yellosoft.us/installing-common-lisp may be another easier option.

Some interesting Lisp books:

ANSI Common LISP

If x^2 is in F, x not in F, then x is a Pure Quaternion

Proposition: If $x^2\in Z(A)=F$ and $x\notin F$ , then $x$ is a pure quaternion.

Proof: Let $x=c+z$ , with $c\in F$ and $z\in A_+$ ( $z$ is a pure quaternion).

Then $x^2=c^2+2cz+z^2=c^2-\nu(z)+2cz$ . The key observation is that if $z=c_1i+c_2j+c_3k$ , then $z^2=-\nu(z)=ac_1^2+bc_2^2-abc_3^2$ .

Since $x\in F$ , this means that $2cz=0$ , i.e. $c=0$ , so $x$ is a pure quaternion.

Representing map of x

Proposition:
(Representing map of $x$ ). Let $X$ be a simplicial set and let $x\in X_n$ . There exists a unique simplicial map $f_x:\Delta[n]\to X$ such that $f_x(\sigma_n)=x$ .
Proof:
Let $(i_0,i_1,\dots,i_k)\in\Delta[n]$ . $(i_0,i_1,\dots,i_k)$ can be written as iterated compositions of faces and degeneracies of $\sigma_n$ , i.e. $(i_0,i_1,\dots,i_k)=g\sigma_n$ where $g$ is an iterated composition of faces $(d_j)$ and degeneracies ( $s_j$ ). Then
$\begin{aligned} f_x(i_0,i_1,\dots,i_k)&=f_x(g\sigma_n)\\ &=gf_x(\sigma_n)\\ &=gx \end{aligned}$
This defines a unique simplicial map $f_x$ such that $f_x(\sigma_n)=x$ .

My philosophy for a happy life | Sam Berns | TEDxMidAtlantic

GEP / DSA Pattern Recognition Book

Visual Discrimination, Grades 2 – 8

Some readers of my blog has bought this book on Amazon. Upon closer inspection, I realised this is actually an excellent book for preparing for pattern recognition (visual discrimination) which is tested in GEP / DSA under the logic portion.

The above is a sample of what the book teaches. If you are familiar with the GEP / DSA test, this is exactly what the logic part of the test is about. Not just the GEP / DSA tests use this, basically any IQ test worldwide will test something like this.

Getting familiar with such tests will obviously be an advantage. No harm giving it some practice rather than seeing such tests the first time in the GEP / DSA test.

Pattern recognition is a skill that can be learnt, and one can argue that many intellectual activities like chess / math are just advanced forms of pattern recognition!

Further Maths Versus H2 Maths

Source: http://www.seab.gov.sg/content/syllabus/alevel/2017Syllabus/9649_2017.pdf

From the syllabus, it looks like Further Maths has some overlap with H2 Maths. This is a good thing, since students can reinforce their knowledge by learning it twice. It is also like killing two birds with one stone!

Some new things in Further Math:

Conic sections in Polar Form
Arc length of curves
Integrating factor (Differential Equations)
Second order recurrence relations
Matrices and Linear Spaces (This in my opinion is one of the biggest difference between Further Maths and H2 Maths)
Numerical methods
Geometric Distribution
Uniform and exponential distribution
Non-parametric tests

Further Math is a good subject to take for those intending to enter physical science or engineering in university, since it will cover many of the topics learnt in the first year of university.

Overall, there are quite a lot of advanced topics crammed into Further Math, so I doubt the teachers have time to teach everything in depth in two years. Matrices and Linear Spaces (Linear Algebra) alone will take half a year to teach it properly. Most likely it will be a touch-and-go event and students will get to see the tip of the iceberg. Students wishing to learn more can read up some undergraduate math books listed here.

Minimal Simplicial Sets

Let $X$ be a space. We can have a fibrant simplicial set, namely the singular simplicial set $S_*(X)$ , where $S_n(x)$ is the set of all continuous maps from the $n$ -simplex to $X$ . However $S_*(X)$ seems too large as there are uncountably many elements in each $S_n(X)$ . On the other hand, we need the fibrant assumption to have simplicial homotopy groups. This means that the simplicial model for a given space cannot be too small. We wish to have a smallest fibrant simplicial set which will be the idea behind minimal simplicial sets.

Let $X$ be a fibrant simplicial set. For $x,y\in X_n$ we say that $x\simeq y$ if the representing maps $f_x$ and $f_y$ are homotopic relative to $\partial\Delta[n]$ .

A fibrant simplicial set is said to be minimal if it has the property that $x\simeq y$ implies $x=y$ .

Let $X$ be a fibrant simplicial set. $X$ is minimal iff for any $0\leq k\leq n+1$ , $v,w\in X_{n+1}$ such that $d_iv=d_iw$ for all $i\neq k$ implies $d_kv=d_kw$ .

In other words, it means that a fibrant simplicial set is minimal iff for any two elements with all faces but one the same, then the missed face must be the same.

Higher Homotopy Groups

We can generalise the idea to higher homotopy groups as follows.

Let $X$ be a pointed fibrant simplicial set. The fundamental group $\pi_n(X)$ is the quotient set of the spherical elements in $X_n$ subject to the relation generated by $x\sim x'$ if there exists $w\in X_{n+1}$ such that $d_0w=x$ , $d_1w=x'$ and $d_jw=*$ for $j>1$ .

The product structure in $\pi_n(X)$ is given by: $[x]+[x']=[d_1w]$ , where $w\in X_{n+1}$ such that $d_0w=x'$ , $d_2w=x$ and $d_jw=*$ for $j>2$ . Furthermore, the map $|\cdot|:\pi_n(X)\to\pi_n(|X|)$ preserves the product structure.

Simplicial objects and higher categories, part I

Very nice!

ErdosNinth

Hi all! Today I’ll introduce simplicial sets, and talk about how they relate to higher categories. Simplicial sets are basically combinatorial models for topological things; in fact, a particular kind of simplicial set (a Kan complex) is essentially equivalent to a topological space! So more general kinds of simplicial sets should model something more general than a topological space. Exploring this will be the topic of this post. The reader should note that these are directly from the (rather terse) notes I took while preparing for the Intel International Science and Engineering Fair last year (so this blog post is currently also serving as the notes for preparing for the Intel Science Talent Institute), so there may be small errors.

We will begin by talking about some combinatorial constructions. Let $latex [n]&s=1$ denote the set $latex {0,…,n}&s=1$ equipped with the linear ordering. Define $latex mathbf{Delta}&s=1$ to be the category whose objects are the sets $latex…

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Is it safe to say that anyone with a doctorate in math was probably a math prodigy when he/she was growing up?

My second grade teacher was convinced that I had a learning disability. Now I am in my third year at Yale, working on a PhD degree in mathematics (I’m into analytic number theory, if you are curious).

“Prodigy” describes neither me nor anyone that I know. I think that it is a word that is far too overused, abused, and misused. My experience tells me that (with possibly a few, singular exceptions that we don’t yet understand well enough to properly gauge) people don’t become experts in something by being innately good at it, but by putting in the 10,000 hours necessary to make the subject an inherent part of their make-up.

I have told this to students before—there is no shame in not studying higher math because you think your time would be better used elsewhere. But for the love of God, don’t quit just because you feel like you aren’t good enough.