## Happy New Year

Happy New Year to all readers of Mathtuition88.com!

Thanks to your support, Mathtuition88.com has reached 885,060 views! We will continue posting Math and Education posts so please continue to visit the website for updates.

## Highly Motivational Math Video (in Chinese)

This is actually one of the best motivational videos on Math I have seen. Unfortunately there is no English translation. It covers how useful Math is, and also some history of Math in ancient China. (It is rarely known, but China discovered negative numbers and calculated pi to high accuracy much earlier than in Western civilization.)

However, (according to the video), Math in ancient China went downhill in the Ming dynasty after it was scrapped from the imperial examination. Seems like removing Math from the examination syllabus is always a bad idea!

Finally, the video ends off with a note not to discourage budding mathematicians. Many budding mathematicians, will face strange looks from well-intentioned friends and society. Will learning math be useful or can it make money? Such thoughts can discourage people from learning mathematics (like the speaker himself).

By the way, at the start of the video, the speaker tells a humorous story of how he used Math to propose to his crush in England. This is related to my earlier post on Valentine’s Day Math on how to draw a heart using math.

## Amos Yee Math Talent

Amos Yee (“famous” for posting controversial videos) actually has good talent and aptitude in Mathematics. His mother is a Math secondary teacher with decades of experience. Many people know him for his infamous videos and his “American English” pronunciation, but few know that he was actually one of the top students in his secondary school in terms of Maths. His English results were good too, and so was his Pure Chemistry. Perhaps even more impressive (and rare), is Amos Yee has a Grade 8 guitar certification (ABRSM merit) (Source: Amos Yee “Happy Teachers’ Day” YouTube video). In March 2011, Yee also won awards for Best Short Film and Best Actor at The New Paper’s First Film Fest (FFF) for his film Jan. He was also an actor in Jack Neo’s movie We Not Naughty.

Unfortunately, Yee seemed to have not used his talents well to benefit society, but instead got himself into a lot of trouble. Who knows, if he turns over a new leaf it could be still possible to have a bright future.

“OK my fellow friends, sorry it’s been so late, I shall announce my O level results.

Apparently I did better than I expected, for all the wrong subjects, so if you truly want to see the innate comedy of my results, you should check out the results which I’d predicted before in my previous post before reading this, and then I think you’ll laugh as much as I did.

E Maths (Dogs truly can get A1 for E Maths)

English: A1 (Well this was surprising, I’d finally gained the coveted A1 for English that I had always hoped for in my Secondary School, and mastered the art of English Comprehension.

A Maths: A2 (So apparently if you leave out 4 entire questions that are 7 marks each, you can still get an A2. So I guess I did really ****ing well for the questions that I did. My mother, being a highly coveted A maths and E maths teacher for 3 decades, threw herself out of the building when she heard her darling son did not attain an A1 for A maths. If you look out of the window, you can still hear the faint cries of ‘****! MY SON IS SUCH A DISAPPOINTMENT’. Well, you threatened to disown me when I became an Atheist, and in the end you didn’t, so I think you’ll do just fine.)

L1R5:11

Raw Aggregate Score: 7(-2 for CCA, -2 for MSP)

Best school I can go to: Nanyang JC”

Source: Amos Yee Facebook, January 14, 2015

Amos also claimed that he “had only studied extensively for the first E maths and A maths papers during the O levels period”. He didn’t really study during the month before the O levels, rather he was “abandoning studying just for that last month, and instead using that month to Complete 4 seasons of Daria, play Spirit Tracks and smash brothers on the DS, create a tuition namecard I now rarely use, and listening to all the best albums of the Beatles”. His Prelim results were also great:

Quote:

“For reference and comparison, this was my mark for prelim 2, the final exam I took in school that isn’t O levels, I think by Prelim 2, the tedium of studying and the uselessness of it was already bearing down on me, and I studied for all the papers, 2 days before and tried to do the best I can with the retained knowledge I had for CA1 (Which I also got an L1R5 of 12 but a slightly higher % of 72 % compared to CA2’s 70%, due largely in part to 93% and 88% for E maths and A maths respectively)
E Maths: A1 (82%)
A maths: A1 (82%)
English: A2 (72%)
Chemistry: A2 (71%)
Chinese: B3 (68%/)
Literature: B3 (65%)
Combined Humans: B3 (65%)
Malay: B4 (64%)
L1R5: 12

Source: Amos Yee Facebook, January 11, 2015

His prediction of his O Level results are also quite accurate:

Quote:

“So here it is, my predicted O level results that are coming out tomorrow:

E maths: A1 (Dogs can get A1 for E maths)

English: A2 (Honestly, I might get an A1 in lieu of the bellcurve,but I never got A1 for English in any full paper before, neither would I feel proud if I did, why would I feel proud about mastering the art of the language of robots.)

Chemistry A2(SPA will help me probably, and though I didn’t study, retention from previous exams was surprisingly good when I did the paper)

A Maths: B3 (Though this was my best subject ever in previous exams(I got 100 plus bonus mark for an A maths paper for God’s sakes), apparently the few weeks I didn’t study was significant enough to make me forget my concepts, to the point that I had skipped an entire 2-3 questions with about 7 marks each, and I think with my inclination to be careless and forget units, it’s going to be a deprovement that will send shock waves)”

Source: Amos Yee Facebook, January 11, 2015

Finally, Amos Yee was also a top student in Secondary 3, 4 (Zhonghua Secondary School) and nominated for a Humanities scholarship. He was consistently getting As for E maths, A maths, Chemistry and English. (Source: Amos Yee WordPress blog, January 25, 2015).

## Math Books for Christmas

Wishing all readers a joyous Christmas ahead! Here are some ideas for a mathematical Christmas gift for your loved ones who are math lovers:

1)

This Christmas-themed Math book is the perfect gift for your child. According to Amazon, it is rated 4.5/5, and one reviewer even remarked that his 7 year old daughter loved reading it:

“I don’t write reviews normally but I was sitting in bed reading it when my 7 year old daughter snuggled up next to me to read it too – she would not let me turn the pages till she finished which was cute even though I had to wait.” (Amazon)

2)

This book is rated very highly on Amazon; it is one of the best sellers in the Math category. It is ideal for homeschoolers, and for Singaporean primary school students who want to learn in advance, during the school holidays. (American Middle School syllabus should be accessible to upper primary Singaporean students) It is written in a very interesting manner as well.

3)

This book is extremely popular in the United States. It is a #1 New York Times bestseller, as well as based on true history. “The phenomenal true story of the black female mathematicians at NASA whose calculations helped fuel some of America’s greatest achievements in space. Soon to be a major motion picture starring Taraji P. Henson, Octavia Spencer, Janelle Monae, Kirsten Dunst, and Kevin Costner.”

## America’s Lost Einsteins

Millions of children from poor families who excel in math and science rarely live up to their potential—and that hurts everyone.

Consider two American children, one rich and one poor, both brilliant. The rich one is much more likely to become an inventor, creating products that help improve America’s quality of life. The poor child probably will not.

That’s the conclusion of a new study by the Equality of Opportunity project, a team of researchers led by the Stanford economist Raj Chetty. Chetty and his team look at who becomes inventors in the United States, a career path that can contribute to vast improvements in Americans’ standard of living. They find that children from families in the the top 1 percent of income distribution are 10 times as likely to have filed for a patent as those from below-median-income families, and that white children are three times as likely to have filed a patent as black children. This means, they say, that there could be millions of “lost Einsteins”—individuals who might have become inventors and changed the course of American life, had they grown up in different neighborhoods. “There are very large gaps in innovation by income, race, and gender,” Chetty told me. “These gaps don’t seem to be about differences in ability to innovate—they seem directly related to environment.”

## From Medical Doctor to Math Professor

Just read about this rather amazing biography: https://today.duke.edu/2017/10/hau-tieng-wu-vital-signs. From a medical doctor, Hau-tieng Wu pursued a Ph.D. in math, and is now a math professor at Duke. Quite an interesting transition, that is quite rare, possibly less than 100 such cases in the world. Most mathematicians know little about medicine, and most medical doctors know little about math. It is rare to have someone know both fields.

Listen to your heartbeat with a stethoscope and you’ll hear a rhythmic lub-dub, lub-dub that repeats roughly 60 to 100 times a minute, 100,000 times a day.

But the normal rhythm of a healthy heart isn’t as steady as you might think, says Hau-tieng Wu, M.D., Ph.D., an associate professor of mathematics and statistical science who joined the Duke University faculty this year.

Rather than beating like a metronome, heart rhythm varies depending on whether you’re asleep or awake, sitting or jogging, calm or driving in rush hour. Breathing rate, brain activity and other physiological signals vary in much the same way, Wu says.

He should know. Before becoming a professor, Wu trained as a medical doctor in Taiwan. In his fifth year of medical school he was doing clinical rotations in the hospital when he was struck by the complex fluctuations in heart rhythm during anesthesia and surgery.

Where some saw noisy patterns — such as the spikes and dips on an electrocardiogram, or ECG — Wu saw hidden information and mathematical problems. “I realized there are so many interesting medical data that aren’t fully analyzed,” Wu said.

When a patient is in the hospital, sensors continuously monitor their heart rate and rhythm, breathing, oxygen saturation, blood pressure, brain activity and other vital signs.

The signals are sent to computers, which analyze and display the results and sound an alarm if anything veers outside normal ranges.

An ECG, for example, translates the heart’s electrical activity into a squiggly line of peaks and valleys whose frequency, size and shape can change from one moment to the next.

Wu is using techniques from differential geometry and harmonic analysis to detect patterns hidden in these oscillating signals and quantify how they change over time.

His methods have been applied to issues in cardiology, obstetrics, anesthesiology, sleep research and intensive care.

## Kids struggling in math? Try this “magic” method from Japan (VIDEO)

URL: Aleteia

It’s an Asian-style mathematics similar to Common Core that’s actually fun to do.

Confession time: I’m terrible at math. I don’t just mean like “struggled with calculus” bad, I mean like “had to watch YouTube videos to relearn long division in order to help my 4th grader with her homework” bad. I don’t know my times tables, except the easy ones. I can’t do fractions or percentages. I count on my fingers.

It’s sad and shameful, and I was determined that my children would not share my fate. So when my oldest daughter was 5, I bought the insanely expensive starter package from Right Start Math and set about teaching her how to do math the right way.

It did not go well, nor did it last long. I found even the very simple activities baffling because I couldn’t grasp the intention. It was like trying to teach my daughter a foreign language I didn’t know.

However, my abject failure to understand it did not diminish my enthusiasm for the Asian method of mathematics. One of the reasons I like Common Core math is because there are lots of similarities. If you’ve never been exposed to the wonder of Asian-style mathematics, allow me to remedy that for you:

Check out the video on the page, it is quite amazing. (Japanese method of multiplying with lines).

URL: Aleteia

## US students aren’t bad at math—they’re just not motivated

It turns out that US students aren’t that bad at math, they just have no motivation to do the PISA test properly. (The PISA test is an external test that has no bearing on their school academic results.)

It’s no secret that young Americans perform poorly on math and science tests, especially compared to their peers in countries like Singapore, Korea and China, where math scores are among the highest in the world. Now, a working paper surfaces a fundamental reason for that weak performance: American students are simply not trying hard enough.

In the latest results of the Programme for International Student Assessment (PISA), US students ranked roughly average among the 75 participating countries. The PISA tests, administered by the OECD every three years, assess 15-year-olds around the world on math, science and reading. Governments and policy makers point to the outcomes when making the case for education reform.

The researchers also ran a simulation, and found that if the 15-year-olds in the US had been given the same cash bonus in 2012 when taking the assessment, America would have ranked 19th in the PISA math test instead of 36th among 65 nations.

## Hokkaido University Photo Gallery and Trip Guide

These photos are from a trip to Hokkaido University in August 2017. Hokkaido University is just walking distance away from Sapporo station, and is worth spending an afternoon there. Admission is free. Even in the hottest summer, Hokkaido has cool weather, ranging from around 17 degrees to 25 degrees.

Some places to visit are the Poplar Avenue, and also the Hokkaido University Museum (check the time, it can close as early as 5pm.) Also, the statue of one of the founders, Dr. William Smith Clark, is also a good place to visit. There is also a monument of the school motto: “Be ambitious!” (少年よ、大志を抱け )

Raven/crows can be spotted all around the campus of Hokkaido University. They are the largest raven I have ever seen, about the size of a small eagle.

The below are some photos from inside the Hokkaido University Museum.

## 8 Facts About Infinity That Will Blow Your Mind

Nice article on infinity. Also little known is the fact that the symbol of infinity was introduced by clergyman and mathematician John Wallis, hundreds of years ago in 1655. Although not well-known, John Wallis was a talented individual as can be deduced from his biography. His works include integral calculus, analytic geometry, and collision of bodies. He was the one who coined the term “momentum”.

Source: ThoughtCo

Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word “lemniscate” comes from the Latin word lemniscus, which means “ribbon,” while the word “infinity” comes from the Latin word infinitas, which means “boundless.”

Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate “countless” in addition to the number. It’s also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.

## Union-closed sets conjecture

The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. (Wikipedia)

It is quite interesting in the sense that the statement is extremely elementary (just basic set notation knowledge is enough to understand it). But it seems that even the experts can’t prove it.

One basic example is: {{1},{2},{1,2}}. The element 2 belongs to 2/3>1/2 of the sets in the family.

## He who loves learning is better than he who knows how to learn (Confucius)

From Baidu Baike:

Translation: He who knows how to learn, is not as good as he who likes learning. He who likes learning, is not as good as he who loves learning. (Confucius)

I guess this applies to mathematics as well. The first step to do well in mathematics is to keep an open mindset and try to get rid of any negative thoughts regarding math. Then, slowly proceed to like and enjoy, and even love math. Only then can one reach his full potential in mathematics.

Like most things, there is a nature and nurture component to this. Some people just naturally love logical things including math. Environment like parents and teachers are very important too, a negative encounter in early childhood can easily give a child a bad impression of learning math.

## It’s mathematically impossible to beat aging, scientists say

According to Math, no one can live forever. So far, the only counterexample that I know of is Turritopsis dohrnii, also known as the “immortal jellyfish”. The article doesn’t seem to address this counterexample though.

Source: Science Daily

Aging is a natural part of life, but that hasn’t stopped people from embarking on efforts to stop the process.

Unfortunately, perhaps, those attempts are futile, according to University of Arizona researchers who have proved that it’s mathematically impossible to halt aging in multicellular organisms like humans.

“Aging is mathematically inevitable — like, seriously inevitable. There’s logically, theoretically, mathematically no way out,” said Joanna Masel, professor of ecology and evolutionary biology and at the UA.

Masel and UA postdoctoral researcher Paul Nelson outline their findings on math and aging in a new study titled “Intercellular Competition and Inevitability of Multicellular Aging,” published in Proceedings of the National Academy of Sciences.

Current understanding of the evolution of aging leaves open the possibility that aging could be stopped if only science could figure out a way to make selection between organisms perfect. One way to do that might be to use competition between cells to eliminate poorly functioning “sluggish” cells linked to aging, while keeping other cells intact.

However, the solution isn’t that simple, Masel and Nelson say.

Two things happen to the body on a cellular level as it ages, Nelson explains. One is that cells slow down and start to lose function, like when your hair cells, for example, stop making pigment. The other thing that happens is that some cells crank up their growth rate, which can cause cancer cells to form. As we get older, we all tend, at some point, to develop cancer cells in the body, even if they’re not causing symptoms, the researchers say.

## Leibniz was a universal genius, but why is Isaac Newton more known? Does it have to do with Newton being British and Leibniz being German?

Answer by Albert Heisenberg, Science Historian M.A. Brown University

Leibniz’s formulation of differential and integral calculus was more refined, elegant, and ‘generalizable’ than Newton’s ‘fluxions.’ Leibniz, the natural genius that he was, became interested in mathematics much later in his life than Newton, and yet was able to generalize Descartes work on analytic geometry into calculus in a way that is so clear that till this day we still use Leibniz’s notation (e.g. dx/dy; his symbolism for time integration/differentiation, etc). Newton’s Principia is a work of incredible genius, but it is riddled with errors and inconsistent notation. As the co-founder of classical physics (along with Galileo) and the culmination of the scientific revolution, his legacy was deeply tied to the spread of natural philosophy as a mathematically rigorous discipline even though Leibniz’s formulation of infinitesimal calculus was superior. Newton achieved greater fame for a few reasons:

## Note on p-divisibility in Bockstein Spectral Sequence

If $u\in H_n(X)$ generates a copy of $\mathbb{Z}$, then $u\notin\ker p^r$.

Write $u=a+b$, where $a\in pH_n(X)$, $b\in\ker p^r$. Note that since $b=u-a\in\ker p^r$, hence $b=u-a$ does not generate a copy of $\mathbb{Z}$. The only way that is possible is when $b=u-a=0$, i.e $u=a$.

## “Actual” GEP Questions 2017 (from Forum)

Since the actual GEP papers are never released, the next best source is from those who have actually taken it and post on forums like Kiasuparents.

Some Maths questions my girl remembers.

“ In a fishing competition, five kids caught 50 fish in total. A is the winner – she got 12 fish. B and C caught the same number of fish and both are at second place. D is at fourth place. E came in last, got only 6 fish. How many fish did B get?“

( my girl couldn’t solve this one. )

“ The red ribbon is twice as long as the blue ribbon. The green ribbon is 2cm shorter than the blue ribbon. A red ribbon and two green ribbon together measure 16cm. How Long is the blue ribbon? “

( she managed to solve this one- but only after spending a lot of time on it. )

Review by mathtuition88: These two questions are not that hard. Can be solved by either model method or algebra.

Some tips from parents: English and GAT is actually harder to prepare than Maths:

Just sharing based on our experience last year. Of the 6 that were selected for GEP eventually from my child’s class, it seems English and GAT were the determining factors. For maths, a lot of kids are already very advanced and well – prepared nowadays. The majority of the balance 14 who went for round 2 found English harder than maths. According to them, English is somewhat like pitched at sec 1 and sec 2 standard, while maths was like up to P6 and Primary Maths Olympiad standard and more manageable. I think it was also more because anything can come out under the sun for English and you can’t really prepare for it. That’s what I heard last year.

For more GEP tips and recommended GEP books, check out: Recommended Books for GEP Selection Test and How to Get Into GEP.

## Why do people get so anxious about math? – Orly Rubinsten

View full lesson: http://ed.ted.com/lessons/why-do-peop… Have you ever sat down to take a math test and immediately felt your heart beat faster and your palms start to sweat? This is called math anxiety, and if it happens to you, you’re not alone: Researchers think about 20 percent of the population suffers from it. So what’s going on? And can it be fixed? Orly Rubinsten explores the current research and suggests ways to increase math performance. Lesson by Orly Rubinsten, animation by Adriatic Animation.

Also view my previous post on Coping with maths anxiety.

## Free Math Games For Everyone

Free Math Games For Everyone

Multiplication of big numbers, complex mathematical problems – there are so many issues that people can face not only at school but also in their everyday lives! People need math everywhere and always! That is why understanding the fundamental principles of math is not less important than being able to read or write! Going shopping or starting your own business – gaining the needed mathematical knowledge and being able to apply it in practice will come in handy for everyone, no matter where they go and what they do!

How to learn this subject? Just like any other science, it requires time and efforts to learn it! However, thanks to the modern technologies and the Internet, everything has become a bit easier today and now, it is enough to find a few useful resources to resolve any academic matters! Some of the most useful resources offer people not only to find the answers for their homework sheets and read the main rules but also to enjoy math games and learn while playing!

It is not a secret that children of a younger age, perceive the information much better if it is presented in a fun and engaging form, for example, while playing. This explains such a high demand for online educational games. However, not only kids can enjoy free maths games, in fact, many adults will also find such activities quite useful and fun!

What are the other benefits? The platforms that offer you to study maths online by means of playing games will help you to master all the possibilities of mathematics easily – you will learn how to add, subtract, divide and multiply. For kids, such activities will be useful for admission to the school. For adults, such activities can help fill in the gaps that they have in their knowledge!

Below you can find a list of top five free sources where you can learn mathematics fast and easily while enjoying an exciting game right from your browser in the online mode!

Math Playground

The website is convenient. There are many different categories, which make it simple to find suitable activities for everyone, while good graphics make the whole process really fun and enjoyable!

Math Game Time

This is one of the best platforms! All the games are organized by grade, but what really makes this site stand out is a wide range of additional opportunities like problem-solving, shapes and geometry, algebra or time and money games!

Cool Math Games

There are many strategy and logic activities. Also, on this platform, you can find some exciting and useful “skill games” that are aimed at developing the basic mathematical skills, and they can come in handy not only for the children but also for the whole family!

Unlike the previous platforms, this website offers a wide variety of fun educational activities on numerous subjects, including spelling, reading, science, etc. All the games are bright and colorful. This creates a pleasant atmosphere and will be especially interesting for younger kids. The highest age for games specified on the site is 12. However, some of them will also be useful for grown-ups!

Learning Games For Kids

Although from the first glance it seems like this site is created exclusively for children, I am sure that adult users will also find something interesting and useful for them! There is a wide range of choices. All activities are divided by their goals and grades, and there are also addition and random math activities; such divisions help to navigate through the website with ease and find exactly what you were looking for! There are also many other possibilities. The site also features many vocabularies, art, science, health, brain, literature, and some other activities!

There are just a few sources of many! You can look for more opportunities on the Internet. Find out what options you have – test a few games from different platforms to compare their efficiency, and, without a doubt, you should find something suitable for yourself! Also, if you are enjoying playing on the go – there are numerous applications for tablets and smartphones that you can use at any time and from any place, which will be convenient for busy people!

Final Words

Why is math so important in our lives? It is one of the basic sciences that every person should understand. It does not mean that each of you needs to become well-versed on this subject because if you lack certain skills that are needed to cope with your homework, you can always hire a tutor or turn to www.customwriting.com for academic help. However, having the necessary knowledge base is a must! Without it, you will find it difficult to do the most usual things like count the change in the grocery store, and thus, you will feel less confident!

# Best Spectral Sequence Book

So far the most comprehensive book looks like McCleary’s book: A User’s Guide to Spectral Sequences. It is also suitable for those interested in the algebraic viewpoint. W.S. Massey wrote a very positive review to this book.

A User’s Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)

Another book is Rotman’s An Introduction to Homological Algebra (Universitext). This book is from a homological algebra viewpoint. Rotman has a nice easy-going style, that made his books very popular to read.

The classic book may be MacLane’s Homology (Classics in Mathematics). This may be harder to read (though to be honest all books on spectral sequences are hard).

***Update: I found another book that gives a very nice presentation of certain spectral sequences, for instance the Bockstein spectral sequence. The book is Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs) by Joseph Neisendorfer.

## Interesting Facts about Green’s Theorem

Firstly, Green’s Theorem is named after the mathematician George Green (14 July 1793 – 31 May 1841). Something remarkable about George Green is that he is almost entirely self-taught. He only went to school for one year (when he was 8 years old). His father was a baker, and George helped out in the bakery. Later, at the age of 40 he went to Cambridge to get a formal degree, but even before that he had already discovered Green’s Theorem. It is a mystery where did George Green learn his mathematical knowledge from. (During his time there was clearly no such thing as internet.)

It is unclear to historians exactly where Green obtained information on current developments in mathematics, as Nottingham had little in the way of intellectual resources. What is even more mysterious is that Green had used “the Mathematical Analysis,” a form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as LaplaceLacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them.
-Wikipedia

One of the applications of Green’s Theorem that I find interesting is finding the area of the ellipse: https://www.whitman.edu/mathematics/calculus_online/section16.04.html. (Scroll down to Example 16.4.3). I find the proof very neat, you may want to check it out.

## Pierre-Simon de Laplace; French Newton

To increase your interest in mathematics, let me introduce the French mathematician Pierre-Simon de Laplace, also known as the “French Newton” or “Newton of France”. He helped to calculate projectile motion for Napoleon’s artillery. Laplace was also the examiner for Napoleon when he entered military school. Laplace also invented “Laplace transform” and “Laplacian” which will be useful in advanced engineering calculations.

Some quotes:

In September 1785 Laplace subjected Napoleon to a rigorous examination in differential equations and algebra as well as the practical applications of mathematics.
Book on Napoleon

The French Revolution began in 1789. Laplace was fortunately situated for avoiding its dangers, in part because, like Lagrange, his talents were found useful in calculating artillery trajectories. Napoleon esteemed Laplace, and after the Revolution showered him with honors.
https://www.umass.edu/wsp/resources/french/personnnes/laplace.html

Napoleon himself was good at math, he proved a theorem called Napoleon’s Theorem. Napoleon was “close friends with several mathematicians and scientists, including Fourier, Monge, Laplace, Chaptal, Berthollet, and Lagrange.”

Napoleon also made the following quote:

The advancement and perfection of mathematics are intimately connected with the prosperity of the State. — Emperor Napoléon Bonaparte.

Hope the above interesting facts increase your interest in math.

## 5 Skills Students Need To Cope With School Pressures

According to an article published by the American Psychological Association (APA), many teenagers in the USA say they experience stress in patterns comparable to what adults go through. Teenagers also report higher stress levels than adults during the school year.

Tutors from Leaps ‘n Bounds, a learning center in Dubai, also observe that teen stress is not just confined to adolescents in certain countries; it is slowly becoming a widespread issue.

Teen stress can be caused by different factors including the pressure to perform well (or at least to pass) academically and in sports, and to have a great social life. In school, adolescents constantly face tough academic demands and responsibilities and experience social pressure.

Unfortunately, these challenges spill over even after the afternoon school bell rings, which can cause teenagers to feel even more stressed.

## Dealing with Teen Stress

For teenagers to learn how to effectively deal with school pressures, they need to develop and rely on key personal skills. These include:

### 1.    Time management

All teenagers today always seem to be swamped with numerous activities: assignments, studying, extracurricular activities and sports. They need to find time for their friends, too.

Because teenagers need to have enough time to go through and complete these activities, they need to learn how to manage their time properly. Time management is an important skill they need to develop. This skill pertains to their ability to plan and control how they spend the hours in their day to complete their tasks and accomplish their goals.

With proper time management, teens will be able to establish which tasks to prioritize and how to set their goals, and learn how to monitor where their time actually goes. As such, they will be able to avoid the stress of not having enough time on their hands to finish their assignments, complete their projects, meet their friends, and see their maths tutors in Dubai, if they have additional weekly tutorial or learning sessions.

### 2.    Setting realistic goals

Being number one in the class and, at the same time, for example, being the captain of the school football team are goals worth working hard for. However, overachieving teens tend to feel more pressure. When they fail or feel they didn’t perform up to expectations, they may develop low self-esteem and other negative feelings and attitudes.

Teenagers, therefore, are encouraged to lower their goals or set more realistic ones so that they can achieve more. By doing so, teens will also avoid pressure and boost academic success.

### 3.    Positive coping skills

Coping skills are daily strategies and activities everyone uses or relies on to deal with, work through, or process emotions. Examples of positive coping skills include exercising, meditating, talking with friends or other family members, and having healthy hobbies such as reading and gardening.

Teenagers need to develop and practice positive coping skills instead of negative ones so that they will learn how to deal with stress through healthy ways. Positive coping strategies increase long-term resilience and well-being.

Negative coping techniques such as smoking and using drugs, on the other hand, may provide temporary relief from difficult emotions and pressure but lead to substance dependency and abuse.

For teenagers to effectively withstand adversity and deal confidently with daily stress and other challenges, they need to choose and apply positive coping strategies.

### 4.    Self-care

For teens to better cope with pressure, they also need to have strong, healthy bodies. Teenagers, therefore, need to get enough sleep and rest, have a well-balanced diet, and get the right amount of exercise their bodies they need every day.

Adolescents need to take some time to pause from the relentless pace of everyday life and enjoy some creative activities that will help keep them from dwelling on or stressing over school pressures. This, in turn, will help them lower their stress levels.

### 5.    Optimism

Generally, stress is precipitated by stressful thinking. As such, teens can avoid stress and its negative effects by changing the way they think. When they have a positive mental attitude, they will have stronger coping strategies, better health, and a more stable, less stressful emotional life.

Adopting a positive way of thinking also helps teens complete their work and handle all their responsibilities. If they consistently think they won’t finish something or they don’t have enough time on their hands, they will lack the motivation to complete what they already started or even begin their task.

Teenagers only have a few more years before they enter another important phase in their lives: adulthood. But they can still enjoy all the experiences that come with adolescence and, at the same time, cope with all their school work and other activities without all the stress by simply developing the right skills.

AUTHOR BIO

Bushra Manna is one of the founders and Principal of Leaps and Bounds Education Centre – Motorcity. She has 20 years’ experience teaching the British and American curricula internationally at primary level – early middle school level, ages 4-12. Bushra believes in imparting deep learning to a child and not just rote learning, which is why she recommends the Magikats programme at her centre, to promote a genuine understanding with its multisensory, differentiated and interactive approach within a small group setting.

## Best Online Resources to Improve Your Math Skills

Best Online Resources to Improve Your Math Skills

Although Maths is a compulsory subject in all the educational institutions all over the world, many students consider it as a complete waste of time and skip this issue, claiming that they prefer not mathematically oriented disciplines but only social sciences. But will it really help you in your future or it is just something you have to learn because everybody does it? Math, as well as other exact sciences, is essential for the intellectual development of a person from early years, it helps to develop intelligence and better your critical, analytical, deductive and prognostic abilities together with improving your abstract thinking. Very often people understand the importance of logic and algebra when they are already adults and try to make up for lost time, but do not know where to find a qualified help quickly. We think that a man is never too old to study, that is why we have selected the best online sources to improve your knowledge of numerous subjects:

• Online courses

There are plenty of courses available on the web, which can offer math courses online at low prices or even for free. Online learning is a trendy way of getting new knowledge and experience nowadays, but still, there are people, who prefer old and traditional methods and say it is useless and wastefulness. But, if you keep up with the times, you can visit such sites as Academic Earth, Edx, TED, University of the People, Coursera and others and get a unique experience in exploring new things from universities all over the world, just sitting on your sofa and holding a laptop.

• Online libraries

For those, who still hesitate, exist online libraries, where you can download books you need and dive into learning by yourself. Of course, there is nothing better than a smell of a written word, but the theme of books varies depending on the site, and mostly they have a plethora of them available for free, so everybody could find the one he needs. The one drawback of using web libraries is that you have to organize an academic process yourself, thus have good time-management and organizational skills. Nevertheless, if you are a person, who is self-motivated and can move towards the goal on your own, this type of e-learning is exactly what you need. Among the most popular web libraries are The Online Library, Harvard Library, Wiley Library, Open Library and others.

• Online Tutors

There are different types of school teachers, some of them focus on the needs of each student, other – use discipline as a key to successful learning, but the thing that is common for all of them, is strict correspondence to the school program, established in the country. School program doesn’t care whether the student understood the material he had just learned, or he still needs time to master it. This is the time when online tutors come to stage. If you need to learn math or other subjects that you have missed at school or simply improve them, they can assist you as they are 100% students oriented and can present the material according to the students’ abilities and needs.

• Online lessons

This type of lessons is a perfect variant of distance learning for those, who are limited on time and want to master something quickly and get great results. Individual classes can take place in the form of face-to-face conversation by means of various apps or programs such as Skype and other video calls. Among the advantages of this type of e-learning is easy access – you can learn from home and build up your own learning schedule, also there is wide choice of tutors available on the Internet, no matter how far away they are, different educational content – video and audio which you can find for free on the web and, of course, you can choose the subject or skills you want to level up – it could be mathematics assignment writing, organic chemistry, etc.

There are plenty of ways how to study maths online, and it depends on which directions of studying will you choose, the pros and cons of each method based on your goal and preferences. The one thing you should remember is that it is never late to gain new knowledge or skills. As Benjamin Franklin once said, “An investment in knowledge pays the best interest.”

## An ancient Babylonian tablet known as Plimpton 322

Source: NY Times

One of my favorite YouTube Math Professors, Norman Wildberger, has made a historical math discovery: that the ancient Babylonian tablet known as Plimpton 322 is actually a trigonometric table.

“It’s a trigonometric table, which is 3,000 years ahead of its time,” said Daniel F. Mansfield of the University of New South Wales. Dr. Mansfield and his colleague Norman J. Wildberger reported their findings last week in the journal Historia Mathematica.

Check out my other blog posts on Prof. Norman Wildberger:
1) Algebraic Topology Video by Professor N J Wildberger

## Education and the Blockchain – Should We be Teaching Blockchain in Schools?

It goes without saying that tech progress is moving at a rapid pace. Futurists point to Moore’s law – the idea that tech capabilities double every two years – as evidence for tech’s expansion into nearly every facet of our lives.

Teaching Technology

Education has seen its own dramatic tech advances. Kids can learn math from gamified apps while riding in the backseat of the family minivan. Students can hire an online algebra tutor and learn from anywhere via Skype. Aspiring students can virtually attend free Ivy-league classes (Massive Open Online Course, or MOOC) with millions of other learners of all ages and backgrounds. And NASA now collaborates with high school students in inventive hardware and robotics projects.

The most significant advance in computer-based education isn’t AI or virtual-based learning or even big data – it’s the blockchain. Blockchain has its origins in cryptocurrency, i.e. Bitcoin. The blockchain is essentially a way of managing data transactions – and it’s considered a radical disruption of traditional banking.

Plus, its applications in education – both virtual and classroom-based – have the potential to change everything about schools, from instruction to student achievement.

Exposure Versus Creativity

In the US, three-quarters of children have access to a smartphone. But on its own, that’s not necessarily a good thing. Kids who simply learn to operate a phone, just downloading and playing games, become consumers. The future lies with creators.

US Department of Labor statistics tell us that 2020 will bring with it 1.4 million computer specialist job openings. But American universities produce woefully inadequate numbers of graduates in the right fields – enough to fill a mere 29% of the jobs.

So what’s wrong with the picture? Why the big gap? There are many societal reasons we could point to, but one thing seems to stand out. We’re teaching tech literacy the wrong way.

Textbook-style curriculum may have its place, but not in tech ed. When kids are taught to memorize coding sequences and churn out the same answers to the same textbook questions, there’s no creative spark. No outside-the-box thinking.

In the best way, blockchain is wildly unconventional. To advance the world-changing potential of anti-dogmatic thinking, we need to encourage kids’ inventiveness. If the educational focus is on robot-like achievements rather than innovation, where will we find our climate change-tackling problem solvers?

We’ve labeled a generation of kids “tech-savvy” without giving them the tools to move from consumption to creation. It’s a waste of their brain power to hook kids on the addictive side of tech without pulling back the curtains and showing them the remarkable inner workings. Children and teens want to know how things work.

One solution? Teach tech like art. Coding has more in common with drawing than accounting. Yes, there is a necessary foundation in understanding digital languages and principles – but without encouraging creativity, we’re creating a generation of the same brain. Even gamified learning, if done improperly, can be perilously bland.

Tackling the Education Gap

There are few key components of a sound approach to teaching creative thinking around technology.

1. Let it be accessible. Kids will shy away from a big learning curve – learning and doing need an intimate relationship.
2. Remove the achievement roof. Learning platforms and educational approaches which employ standardized tests as the litmus for success – and for what the content can achieve –inhibit creativity. Rather than saying “do this to produce this result,” what if we said, “here are your tools – now, what can you create?” Consider The Lego Movie’s message of the importance of imagination – for future tech innovation, we need makers, not managers.
3. Embrace a shifting curriculum. In other subjects, things might stand as eternal truths; the Magna Carta will always have been signed in 1215. But in technology education, things move at a blistering pace. A particular tool or lesson may become quickly outdated, so the educational format needs flexibility, just like the subject it teaches.

Blockchain is set to change the world. But as we continue to encounter environmental and societal problems, we need amazing minds to solve them. Revolutionizing how we teach technology education might be the answer we didn’t know we needed.

## The scientist nuns: In pursuit of faith and reason

Source: Aleteia

Making a career out of science, just like joining a religious order, requires dedication and discipline. Some tireless souls have managed to do both.

In 1965, Mary Kenneth Keller became the first woman to obtain a PhD in Computer Science. She was also a nun.

Born in Cleveland, Ohio, in 1913, Keller entered the Sisters of Charity of the Blessed Virgin Mary in Dubuque, Iowa, in 1932. Eight years later, she professed her vows, before obtaining B.S. and M.S. degrees in mathematics from DePaul University in Chicago, where she became fascinated by the incipient field of computer science.

As a graduate student, she spent semesters at other schools, including New Hampshire’s Ivy League college Dartmouth, which at that time was not coeducational. For her, however, the school relaxed its policy on gender, and she worked in the computer center, where she contributed to the development of the BASIC programming language that became so instrumental to the early generation of programmers.

## Theorem of the Day

Just to recommend this excellent website: Theoremoftheday where they feature one mathematical theorem each day.

The nice thing is that each theorem is a one-page summary, good for getting acquainted with the theorem, and subsequently you may read it up in more detail.

The website does have a XML feed, though it would be nice if there were a email subscription (with weekly emails).

# The Strange Topology That Is Reshaping Physics

Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing

Charles Kane never thought he would be cavorting with topologists. “I don’t think like a mathematician,” admits Kane, a theoretical physicist who has tended to focus on tangible problems about solid materials. He is not alone. Physicists have typically paid little attention to topology—the mathematical study of shapes and their arrangement in space. But now Kane and other physicists are flocking to the field.

In the past decade, they have found that topology provides unique insight into the physics of materials, such as how some insulators can sneakily conduct electricity along a single-atom layer on their surfaces.

Some of these topological effects were uncovered in the 1980s, but only in the past few years have researchers begun to realize that they could be much more prevalent and bizarre than anyone expected. Topological materials have been “sitting in plain sight, and people didn’t think to look for them”, says Kane, who is at the University of Pennsylvania in Philadelphia.

Now, topological physics is truly exploding: it seems increasingly rare to see a paper on solid-state physics that doesn’t have the word topology in the title. And experimentalists are about to get even busier. A study on page 298 of this week’s Nature unveils an atlas of materials that might host topological effects, giving physicists many more places to go looking for bizarre states of matter such as Weyl fermions or quantum-spin liquids.

## How to do Proof by Cases in LaTeX

If one searches online, one will find many different methods to do “proof by cases” in LaTeX. The most simple and convenient method in my opinion is to use the description environment.

Something like this:

\begin{proof} Proceed by cases.
\begin{description}
\item[Case 1: This.] And so on.
\item[Case 2: That.] And more.
\end{proof}

Source: Reddit

No additional package is needed. One drawback is there is no auto-numbering, but I am sure that is still ok, unless your proof has many many cases.

## (Important Changes) PSLE Math: Arrow -> vs Equal=

For those taking PSLE, please take note of this important update regarding the difference between arrow and equal sign. Forward this to your friends taking PSLE!

Basically, I think MOE is trying to instill students to be mathematically correct. (See update below: Marks will not be deducted in most cases but proper usage is highly encouraged.)

E.g. 100%=40 is wrong as 100%=100/100=1 technically. Similarly, 10 men = 40 hours is wrong as the units do not match (nor make sense).

Trying to enforce “units” instead of “u”, and banning “10 units -> 20” is a bit strict though, in my opinion.

MOE responds

In response to Mothership.sg queries, a Ministry of Education spokesperson clarified that the above information was not provided by the ministry.

The information above was originally sourced from the website of a private tuition centre, whose sources are currently unverified.

While the respective uses of the arrow and equal signs are accurate in the infographic, the MOE spokesperson said full credit will still be awarded to the student even when the signs are used interchangeably, as long as the student demonstrates a full understanding of the question.

Proper use of arrow and equal signs are, nonetheless, encouraged.

## Summary: Shapes, radius functions and persistent homology

This is a summary of a talk by Professor Herbert Edelsbrunner, IST Austria. The PDF slides can be found here: persistent homology slides.

## Biogeometry (2:51 in video)

We can think of proteins as a geometric object by replacing every atom by a sphere (possibly different radii). Protein is viewed as union of balls in $\mathbb{R}^3$.

Decompose into Voronoi domains $V(x)$, and take the nerve (Delaunay complex).

Inclusion-Exclusion Theorem: $\displaystyle Vol(\bigcup B)=\sum_{Q\in D_r(x)}(-1)^{\dim Q}Vol(\bigcap Q).$
Volume of protein $(\bigcup B)$ is alternating sum over all simplices $Q$ in Delaunay complex.

## Nerve Theorem: Union of sets have same homotopy type as nerve (stronger than having isomorphic homology groups).

Wrap (14:04 in video)

Collapses: 01 collapse means 0 dimensional and 1 dimensional simplices disappear (something like deformation retract).

Interval: Simplices that are removed in a collapse (always a skeleton of a cube in appropriate dimension)

Generalised Discrete Morse Function (Forman 1998): Generalised discrete vector field $=$ partition into intervals (for acyclic case only)

Critical simplex: The only simplex in an interval (when a critical simplex is added, the homotopy type changes)

Lower set of critical simplex: all the nodes that lead up to the critical simplex.

Wrap complex is the union of lower sets.

## Persistence (38:00 in video)

Betti numbers in $\mathbb{R}^3$: $\beta_0: \#$ components, $\beta_1:\#$ loops, $\beta_2: \#$ voids.

Incremental Algorithm to compute Betti numbers (40:50 in video). [Deffimado, E., 1995]. Every time a simplex is added, either a Betti number goes up (birth) or goes down (death).

$\alpha$ is born when it is not in image of previous homology group.

Stability of persistence: small change in position of points leads to similar persistence diagram.

Bottleneck distance between two diagrams is length of longest edge in minimizing matching. Theorem: $\displaystyle W_\infty(Dgm(f),Dgm(g))\leq\|f-g\|_\infty.$ [Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.

## Expectation (51:30 in video)

Poisson point process: Like uniform distribution but over entire space. Number of points in region is proportional to size of region. Proportionality constant is density $\rho>0$.

Paper: Expectations in $\mathbb{R}^n$. [E., Nikitenko, Reitones, 2016]

Reduces to question (Three points in circle): Given three points in a circle, what is the probability that the triangle (with the 3 points as vertices) contains the center of the circle? Ans: 1/4 [Wendel 1963].

## Brain has 11 dimensions

One of the possible applications of algebraic topology is in studying the brain, which is known to be very complicated.

If you can call understanding the dynamics of a virtual rat brain a real-world problem. In a multimillion-dollar supercomputer in a building on the same campus where Hess has spent 25 years stretching and shrinking geometric objects in her mind, lives one of the most detailed digital reconstructions of brain tissue ever built. Representing 55 distinct types of neurons and 36 million synapses all firing in a space the size of pinhead, the simulation is the brainchild of Henry Markram.

Markram and Hess met through a mutual researcher friend 12 years ago, right around the time Markram was launching Blue Brain—the Swiss institute’s ambitious bid to build a complete, simulated brain, starting with the rat. Over the next decade, as Markram began feeding terabytes of data into an IBM supercomputer and reconstructing a collection of neurons in the sensory cortex, he and Hess continued to meet and discuss how they might use her specialized knowledge to understand what he was creating. “It became clearer and clearer algebraic topology could help you see things you just can’t see with flat mathematics,” says Markram. But Hess didn’t officially join the project until 2015, when it met (and some would say failed) its first big public test.

In October of that year, Markram led an international team of neuroscientists in unveiling the first Blue Brain results: a simulation of 31,000 connected rat neurons that responded with waves of coordinated electricity in response to an artificial stimulus. The long awaited, 36-page paper published in Cell was not greeted as the unequivocal success Markram expected. Instead, it further polarized a research community already divided by the audacity of his prophesizing and the insane amount of money behind the project.

Two years before, the European Union had awarded Markram \$1.3 billion to spend the next decade building a computerized human brain. But not long after, hundreds of EU scientists revolted against that initiative, the Human Brain Project. In the summer of 2015, they penned an open letter questioning the scientific value of the project and threatening to boycott unless it was reformed. Two independent reviews agreed with the critics, and the Human Brain Project downgraded Markram’s involvement. It was into this turbulent atmosphere that Blue Brain announced its modest progress on its bit of simulated rat cortex.

## How to explain this math magic trick?

Quite impressive math magic trick, that even impressed the very strict judge Simon Cowell. I am not sure how he did it, other than possible prearranged volunteers. Another possibility is that the calculator is modified.

London had the volunteers give their best guesses to different questions, including how many No. 1-selling artists Cowell has had on his record label, how many millions of records judge Mel B. sold worldwide with the Spice Girls and what year judge Heidi Klum started modeling.

Meanwhile, London asked host Tyra Banks multiply the three answers together using the calculator on her own phone. He then instructed Banks to close her eyes and add a random eight-digit number to the previous calculation. She revealed that the grand total came out to 73,928,547.

Watch the clip to see why that number left the judges and audience members stunned!

## Yitang Zhang’s Santa Barbara Beach Walk

Professor Yitang Zhang is a famous Math professor who made important progress in number theory (Twin Prime Conjecture). Most strikingly, he made this progress in his fifties, which is kind of rare in the mathematical world.

Source: Quanta Magazine

Yitang Zhang on the beach adjoining the University of California, Santa Barbara, after scratching a function in the sand related to his current work on the Landau-Siegel zeros problem.

As an adolescent during the Cultural Revolution in China, Yitang Zhang wasn’t allowed to attend high school. Later, in his 30s, he worked odd jobs in the United States and sometimes slept in his car. But Zhang always believed he would solve a great math problem someday. Still, despite becoming one of China’s top math students and completing his doctorate at Purdue University in Indiana, for seven years Zhang could not find work as a mathematician. At one point, he worked at a friend’s Subway sandwich restaurant to pay the bills.

“I was not lucky,” Zhang, who is both incredibly reserved and self-confident, told Quanta in a 2015 interview.

At 44, after finally being hired to teach math at the University of New Hampshire, he turned his attention to number theory, a subject he had loved since childhood. He analyzed problems in his head during long walks near his home and the university. In his 50s, well past what many mathematicians consider their prime years (indeed, the Fields Medal is awarded to mathematicians under the age of 40), he began trying to prove the twin primes conjecture, which predicts an infinite number of prime number pairs that have a difference of two, such as 5 and 7, 29 and 31, and 191 and 193. No one had been able to prove this in over 150 years, and top number theorists could not even prove the existence of a bounded prime gap of any finite size.

In 2013, at 58, Zhang published his proof of a bounded prime gap below 70 million in one of the world’s most prestigious journals, the Annals of Mathematics. The paper’s referees wrote that Zhang, who had been unknown to established mathematicians, had proved “a landmark theorem in the distribution of prime numbers.”

## Subtle Error in Wikipedia: Dedekind’s number

On Wikipedia (https://en.wikipedia.org/wiki/Dedekind_number), it is stated that the Dedekind’s number M(n) is the the number of abstract simplicial complexes with n elements.

This is incorrect, at least based on the Wikipedia definition of abstract simplicial complex, which does not allow the empty set as a face.

The correct definition is found in another Wikpedia site: https://en.wikipedia.org/wiki/Abstract_simplicial_complex

The number of abstract simplicial complexes on up to n elements is one less than the nth Dedekind number.

## Math Tricks found in Chess

Just read this very nice article on Quora, on the relationship between Math and Chess: https://www.quora.com/What-math-tricks-are-hidden-in-chess

Also interesting is this YouTube documentary “My Brilliant Brain” featuring Susan Polgar.

Author:

Tutor: Mr Wu (Raffles Alumni, NUS Maths Grad)

Email: mathtuition88@gmail.com

Syllabus: Primary / Secondary Maths Olympiad. Includes Number Theory, Geometry, Combinatorics, Sequences, Series, and more. Flexible curriculum tailored to student’s needs. I can provide material, or teach from any preferred material that the student has.

Target audience: For students with strong interest in Maths. Suitable for those preparing for Olympiad competitions, DSA, GEP, or just learning for personal interest.

Location: West / Central Singapore at student’s home

## Jurong East Maths Tuition

Maths Tuition

Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics

Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).

Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.

Email: mathtuition88@gmail.com

Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)

## Bukit Batok Maths Tuition

Maths Tuition

Tutor (Mr Wu):
– Raffles Alumni
– NUS 1st Class Honours in Mathematics

Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Also has experience teaching Additional Math (O Level, IP).

Personality: Friendly, patient and good at explaining complicated concepts in a simple manner. Provides tips for how to check for careless mistakes, and tackle challenging problems.

Email: mathtuition88@gmail.com

Areas teaching (West / Central Singapore, including Bukit Batok, Dover, Clementi, Jurong)

## Renowned Chinese mathematician Wu Wenjun dies at 98

Wu Wenjun, distinguished mathematician, member of the Chinese Academy of Sciences (CAS), and winner of China’s Supreme Scientific and Technological Award winner, died at the age of 98 on Sunday in Beijing, according to the CAS.
Wu was born in Shanghai on May 12, 1919. In 1940, he graduated from Shanghai Jiao Tong University, and received a PhD from the University of Strasbourg, France in 1947.
In 1951, Wu returned to China and served as a math professor at Peking University. He made great contributions to the field of topology by introducing various principles now recognized internationally.
In the field of mathematics mechanization, Wu suggested a computerized method to prove geometrical theorems, known as Wu’s Method in the international community.
He was elected as a member of the CAS in 1957 and as a member of the Third World Academy of Sciences in 1990.
Wu Wenjun was given China’s Supreme Science and Technology Award by the then President Jiang Zemin in 2000, when this highest scientific and technological prize in China began to be awarded.

## How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)

Set $A_{n,p}^1=H_n(X_p)$ and $E_{n,p}^1=H_n(X_p,X_{p-1})$. The diagram then has the following form:

When we pass to the derived couple, each group $A_{n,p}^1$ is replaced by a subgroup $A_{n,p}^2=\text{Im}\,(i_1: A_{n,p-1}^1\to A_{n,p}^1)$. The differentials $d_1=j_1k_1$ go two units to the right, and we replace the term $E_{n,p}^1$ by the term $E_{n,p}^2=\text{Ker}\, d_1/\text{Im}\,d_1$, where the $d_1$‘s refer to the $d_1$‘s leaving and entering $E_{n,p}^1$ respectively.

The maps $j_2$ now go diagonally upward because of the formula $j_2(i_1a)=[j_1a]$. The maps $i_2$ and $k_2$ still go vertically and horizontally, $i_2$ being a restriction of $i_1$ and $k_2$ being induced by $k_1$.

## Relative Homology Groups

Given a space $X$ and a subspace $A\subset X$, define $C_n(X,A):=C_n(X)/C_n(A)$. Since the boundary map $\partial: C_n(X)\to C_{n-1}(X)$ takes $C_n(A)$ to $C_{n-1}(A)$, it induces a quotient boundary map $\partial: C_n(X,A)\to C_{n-1}(X,A)$.

We have a chain complex $\displaystyle \dots\to C_{n+1}(X,A)\xrightarrow{\partial_{n+1}}C_n(X,A)\xrightarrow{\partial_n}C_{n-1}(X,A)\to\dots$ where $\partial^2=0$ holds. The relative homology groups $H_n(X,A)$ are the homology groups $\text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1}$ of this chain complex.

Relative cycles
Elements of $H_n(X,A)$ are represented by relative cycles: $n$– chains $\alpha\in C_n(X)$ such that $\partial\alpha\in C_{n-1}(A)$.

Relative boundary
A relative cycle $\alpha$ is trivial in $H_n(X,A)$ iff it is a relative boundary: $\alpha=\partial\beta+\gamma$ for some $\beta\in C_{n+1}(X)$ and $\gamma\in C_n(A)$.

Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
\begin{aligned} \dots\to H_n(A)\xrightarrow{i_*}H_n(X)\xrightarrow{j_*}H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)&\xrightarrow{i_*}H_{n-1}(X)\to\dots\\ &\dots\to H_0(X,A)\to 0. \end{aligned}

The boundary map $\partial:H_n(X,A)\to H_{n-1}(A)$ is as follows: If a class $[\alpha]\in H_n(X,A)$ is represented by a relative cycle $\alpha$, then $\partial[\alpha]$ is the class of the cycle $\partial\alpha$ in $H_{n-1}(A)$.

## Exact sequence (Quotient space)

Exact sequence (Quotient space)
If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence
\begin{aligned} \dots\to\widetilde{H}_n(A)\xrightarrow{i_*}\widetilde{H}_n(X)\xrightarrow{j_*}\widetilde{H}_n(X/A)\xrightarrow{\partial}\widetilde{H}_{n-1}(A)&\xrightarrow{i_*}\widetilde{H}_{n-1}(X)\to\dots\\ &\dots\to\widetilde{H}_0(X/A)\to 0 \end{aligned}
where $i$ is the inclusion $A\to X$ and $j$ is the quotient map $X\to X/A$.

Reduced homology of spheres (Proof)
$\widetilde{H}_n(S^n)\cong\mathbb{Z}$ and $\widetilde{H}_i(S^n)=0$ for $i\neq n$.

For $n>0$ take $(X,A)=(D^n,S^{n-1})$ so that $X/A=S^n$. The terms $\widetilde{H}_i(D^n)$ in the long exact sequence are zero since $D^n$ is contractible.

Exactness of the sequence then implies that the maps $\widetilde{H}_i(S^n)\xrightarrow{\partial}\widetilde{H}_{i-1}(S^{n-1})$ are isomorphisms for $i>0$ and that $\widetilde{H}_0(S^n)=0$. Starting with $\widetilde{H}_0(S^0)=\mathbb{Z}$, $\widetilde{H}_i(S^0)=0$ for $i\neq 0$, the result follows by induction on $n$.

## Reduced Homology

Define the reduced homology groups $\widetilde{H}_n(X)$ to be the homology groups of the augmented chain complex $\displaystyle \dots\to C_2(X)\xrightarrow{\partial_2}C_1(X)\xrightarrow{\partial_1}C_0(X)\xrightarrow{\epsilon}\mathbb{Z}\to 0$ where $\epsilon(\sum_i n_i\sigma_i)=\sum_in_i$. We require $X$ to be nonempty, to avoid having a nontrivial homology group in dimension -1.

Relation between $H_n$ and $\widetilde{H}_n$
Since $\epsilon\partial_1=0$, $\epsilon$ vanishes on $\text{Im}\,\partial_1$ and hence induces a map $\tilde{\epsilon}:H_0(X)\to\mathbb{Z}$ with $\ker\tilde{\epsilon}=\ker\epsilon/\text{Im}\,\partial_1=\widetilde{H}_0(X)$. So $H_0(X)\cong\widetilde{H}_0(X)\oplus\mathbb{Z}$. Clearly, $H_n(X)\cong\widetilde{H}_n(X)$ for $n>0$.

## Mayer-Vietoris Sequence applied to Spheres

Mayer-Vietoris Sequence
For a pair of subspaces $A,B\subset X$ such that $X=\text{int}(A)\cup\text{int}(B)$, the exact MV sequence has the form
\begin{aligned} \dots&\to H_n(A\cap B)\xrightarrow{\Phi}H_n(A)\oplus H_n(B)\xrightarrow{\Psi}H_n(X)\xrightarrow{\partial}H_{n-1}(A\cap B)\\ &\to\dots\to H_0(X)\to 0. \end{aligned}

Example: $S^n$
Let $X=S^n$ with $A$ and $B$ the northern and southern hemispheres, so that $A\cap B=S^{n-1}$. Then in the reduced Mayer-Vietoris sequence the terms $\tilde{H}_i(A)\oplus\tilde{H}_i(B)$ are zero. So from the reduced Mayer-Vietoris sequence $\displaystyle \dots\to\tilde{H}_i(A)\oplus\tilde{H}_i(B)\to\tilde{H}_i(X)\to\tilde{H}_{i-1}(A\cap B)\to\tilde{H}_{i-1}(A)\oplus\tilde{H}_{i-1}(B)\to\dots$ we get the exact sequence $\displaystyle 0\to\tilde{H}_i(S^n)\to\tilde{H}_{i-1}(S^{n-1})\to 0.$
We obtain isomorphisms $\tilde{H}_i(S^n)\cong\tilde{H}_{i-1}(S^{n-1})$.