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Free Water Saving Kit (Singapore)

I believe this is not new, but it is the first time I heard of it. Anyway, it is free, so do feel free to request for a sample at the PUB Official Website.

PUB has revamped the packaging of its popular water saving kits featuring its mascot Water Wally. A set of thimbles with three and four holes allows residents to have greater flexibility in regulating their tap’s and showerhead’s flow rates. Water Wally stickers with specific messages are also included in the kit to act as reminders of good water saving habits in homes. ​


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So You Want To Be a Doctor?

This workshop should be useful to students aiming to be a medical doctor. (Note the ticket prices, this event requires ticket for admission.)

Interested students may also check out our previous blog post on BMAT Book Recommendations for NTU Medicine.


Date: 21 July 2018 (Saturday)
Time: 9:30am to 12:30pm (F&B at tea break)
Venue: NUS Shaw Foundation Alumni House, 11 Kent Ridge Drive, Singapore 119244

Calling all IP, JC, polytechnic and university students!

Join us at So You Want To Be a Doctor?, a half-day medical seminar on Saturday, 21 July 2018, from 9:30am to 12:30pm, featuring panel discussions, inspiring keynotes and networking opportunities.

Our seminar provides a unique opportunity for prospective medical students to hear from clinical professors, network with doctors who’ve gone onto diverse careers, and learn about the many job options available in healthcare.

This seminar is targeted at IP, JC, polytechnic and university students, as well as anyone interested in pursuing careers in healthcare, such as biomedical research, allied health, entrepreneurship, technology in medicine, public health and humanitarian projects.

So You Want To Be a Doctor? covers topics such as:
– How will doctors practice in the future?
– Is medicine for me?
– Is clinical research for me?
– The path less wandered: A chat with Dr. Benjamin Seet


09:30 AM: Keynote – “How will doctors practice in the future?”
Professor Wong Tien Yin, Medical Director, Singapore National Eye Centre

09:50 AM: Panel #1 – “Is medicine for me?”
Assistant Professor Isaac Liu Desheng, National University of Singapore
Dr. Tiah Ling, Emergency Medicine Physician, Changi General Hospital
Dr. Kumaran Rasappan, Senior Orthopedic Surgical Resident, Tan Tock Seng Hospital

10:30 AM: Panel #2 – “Is clinical research for me?”
Assistant Professor Swaine Chen, National University of Singapore
Assistant Professor Chester Drum, National University of Singapore

11:10 AM: Tea break

11:40 AM: The path less wandered: A chat with Dr. Benjamin Seet
Dr. Benjamin Seet, Executive Director, Biomedical Research Council, Agency for Science, Technology and Research

12:20 PM: Closing note


Professor Wong Tien Yin (keynote talk)

Prof. Wong Tien Yin is medical director of the Singapore National Eye Centre (SNEC). He is also deputy group CEO (Research & Education) of SingHealth and academic chair of the Ophthalmology and Visual Sciences Academic Clinical Programme at Duke-NUS Medical School. Prof. Wong completed medical school at the National University of Singapore as a 1987 President’s Scholar. He studied on fellowships at the University of Wisconsin-Madison, US, and the University of Sydney, Australia. He obtained his MPH and PhD degrees from the Johns Hopkins University, US.

Dr. Benjamin Seet (keynote talk)

Dr. Benjamin Seet is executive director of the Biomedical Research Council (BMRC) at the Agency for Science, Technology and Research (A*STAR). Prior to joining A*STAR, he served as chief medical officer of the United Nations (UN) Department of Peacekeeping Operations in New York. He oversaw medical support for UN personnel in 16 post-conflict countries. Dr. Seet also served with the Singapore Armed Forces (SAF) for more than 20 years before retiring as chief of the SAF Medical Corps at the rank of Brigadier-General.

Assistant Professor Isaac Liu Desheng (panel #1)

Asst. Prof. Isaac Liu Desheng is a pediatric nephrologist who specializes in dialysis and renal transplantation. In addition to his research on kidney disease, Asst. Prof. Liu is the chief doctor of the Shaw-National Kidney Foundation Children’s Kidney Centre Annual Camp. He is currently an assistant professor (clinician investigator) at the Yong Loo Lin School of Medicine. For his achievements, he was awarded the 2017 Singapore Youth Award, the nation’s highest accolade for youth.

Dr. Tiah Ling (panel #1)

Dr. Tiah Ling is an emergency medical physician consultant in the accident and emergency department at Changi General Hospital. Prior to joining Changi General Hospital, Dr. Tiah participated in the METASHARP project, where she provided technical support, monitoring and evaluation of health facilities in Afghanistan. In 2009, she spent three months in Ghana as a program coordinator in the ‘Systems Improvement at District Hospitals and Regional Training of Emergency Care’ program (sidHARTe). Dr. Tiah received her Masters in Public Health (MPH) degree from the Johns Hopkins University, US.

Dr. Kumaran Rasappan (panel #1)

Dr. Kumaran Rasappan is a senior orthopedic surgical resident with Tan Tock Seng Hospital. In 2012, he became the first Singaporean to scale the summit of Mount Everest for a charitable cause, raising over S$40,000 for needy patients. Since then, Dr. Kumaran has scaled the heights of K2 and Makalu, the second and fifth highest peaks in the world respectively, in an effort to raise funds for the Home Nursing Foundation’s program which aims to address the psycho-social and emotional needs of caregivers. His fundraising effort is titled “No Mountain Too High.”

Assistant Professor Swaine Chen (panel #2)

Asst. Prof. Swaine Chen is an assistant professor of medicine at the National University of Singapore and a senior research scientist at the Genome Institute of Singapore. His research interests include the application of genomics to understanding molecular mechanisms of urinary tract infection, particularly those caused by Escherichia coli. He received a Bachelor’s degree in chemistry and mathematics from Harvard University, US, and an MD-PhD degree from the Stanford University School of Medicine, US.

Assistant Professor Chester Drum (panel #2)

Asst. Prof. Chester Drum is an assistant professor at the National University of Singapore, a consultant cardiologist at the National University Hospital and director of the Clinical Trial Innovation Lab at the Agency for Science, Technology and Research. He received his MD-PhD degree from the University of Chicago, US, and trained at the University of California, San Diego, US, and Harvard Medical School, US. Asst. Prof. Drum has over 20 years of clinical experience in managed care, private, public and academic healthcare settings and holds multiple patents.

About the organizer

Combining savvy communication with technical rigor, Wildtype Media Group is Asia’s leading STEM-focused media company, spanning digital, print, custom publishing and events.

Brands under Wildtype Media Group include the flagship Asian Scientist Magazine and Supercomputing Asia, award-winning titles available in print and online. Through its Asian Scientist Intelligence consulting division, Wildtype Media Group provides a full suite of marketing and communication services to industry and academic clients hoping to engage Asia.

Facebook page:

Peatix page:

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Cultivating Confidence in Students to Excel Academically

Cultivating Confidence in Students to Excel Academically

Students who struggle academically may lack the self-confidence to continue to learn actively and get their point across during lesson. Educators have a huge influence on their students and hence there is an onus on them to instil confidence in the students and help them to feel confident enough to overcome any obstacles in their path to success despite any initial struggles.

When students possess more confidence in their ability, they will be more willing and ready to study and will put in more efforts to work harder as compared to when they doubt their academic ability due to struggles faced. Hence, self-confidence is extremely important in motivating students to excel academically.

Here are some tips that should be taught to students who struggle with their studies so as to provide them with a proper structure to follow when they are at a loss and also to enable them to regain confidence and guide them to the path of success. The tips can also be used by educators who use them to guide and encourage their students.

  1. Set the right goals

As students, be sure to set goals that are realistic and attainable and this is one way where you will be able to see growth and improvement as compared to setting an unrealistic goal that is too hard to reach. As educators, remember to guide your students in their goal settings and do not impose too high expectations on them, as it will only cause them to falter. Ensure that the goals are what the student wants to attain while also making sure that they are realistic. When learning is goal-oriented, there is more intrinsic motivation to be able to want to attain their goals and hence more confidence in pursuing the academics.

  1. Practice

Practice makes perfect and that is often the case. When studying for quantitative subjects like mathematics, remember that practice is the key to success! But of course you practice when you understand the concepts behind questions and not blindly do questions without any understanding. For more qualitative subjects, one way to practice is to come up with a skeletal / point form answer for the question so as to test your understanding of the subject. For educators, remember to remind your students to practice for topics that they are unfamiliar with in order to familiarise themselves with the chapter.

  1. Use visual aids

If long chunky words don’t interest you and deter you from studying, use visual aids to help you study better! Research has shown that students study better with visual aids such as mind maps and flow charts. So students, sketch those mind maps out to better understand difficult concepts! Similarly when teaching, it is better for educators to use visual aids which capture the attention of their students. Do not include large chunks of works on your notes and presentation slides when teaching, as such chunks deter students from reading and understanding the content.

  1. Seek help

When in doubt, do not hesitate to seek help! It is better to clarify a misunderstanding or something that you do not know at the start. Your teachers and tutors will definitely make themselves available whenever possible. Educators, do not chide your students who seek help from you and instead help them to the best of your abilities when they approach you with questions! Questions are a sign that one is learning.

  1. Do not be afraid to make mistakes

A Japanese proverb goes, “fall down seven times and get up eight”. The meaning behind this is to not be discouraged by any mistakes and failures as it is bound to happen. Remember to always pick yourself after making a mistake and that making mistakes are normal and fine as no one is perfect. Educators should always remind their students of this and be there for students to reach to when students face a failure or a mistake.

  1. Encourage

More for the educators, remember to be a source of encouragement to your students and constantly praise them for whatever progress or good work that has been done! If constructive feedback or criticism needs to be given, it might be better to do it when one on one with the student so that he or she does not feel embarrassed in front of the class. Research about people management has also shown that the “Sandwich Method” works best when one wants to give constructive feedback. How the “Sandwich Method” works is that you begin your feedback with something good as the bread before going into the criticisms and finally ending with encouragement again. These various layers form the “sandwich”.

  1. Enthusiasm

Enthusiasm is extremely important in facilitating interest and confidence in the subject. Try your best to be enthusiastic when learning and it really makes a difference rather than looking at the subject as one that is dreary and boring. The enthusiasm of the educators also matters as an enthusiastic teacher helps students to model after the teacher’s enthusiasm for the subject and hence gain confidence.

  1. Positive learning environment

Last but not least, a positive learning environment is extremely important! Find friends who are willing to help and not those who hate studying and form a study group! Such friends will definitely motivate you to work harder and can also be of aid when help is needed to understand a concept. For educator, a classroom that has a positive environment is one where students feel safe to express themselves and have the desire to learn. These develop confidence in the students.

Hopefully with all these tips, educators will be able to help students to cultivate confidence in order to excel academically and that students themselves will be able to orientate themselves into a structure, which enables them to gain confidence.

“Self-confidence is like a super power, once you believe in yourself, magic starts happening.”

Making Sense Chemistry Tuition instils confidence in students. To find out more about Making Sense Chemistry tuition, click here:

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Maria Agnesi, the Greatest Female Mathematician You’ve Never Heard of

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Boolean Algebra

Math Online Tom Circle

George Boole [2/11/ 1815 – 8/12/ 1864]: 《The Laws of Thought》: symbolic logic representation of thought.

Let x = class of sheep’s

y = white

=> white sheep = xy = yx = sheep white

then Commutativity Law:

$latex boxed {xy = yx}&fg=aa0000&s=3 $

Let x= rivers, y = estuaries河口, z= navigable 通航

then, AssociativityLaw:

$latex boxed {(xy)z= x(yz)}&fg=aa0000&s=3 $

A sheep is a sheep,

$latex boxed {xx = x^{2} = x}&fg=aa0000&s=3 $

Note: x = 0 or 1 fulfills the above equation.

If x = class of men

y = class of women

z = class of adults (either men or women)

$latex boxed {z = x + y}&fg=aa0000&s=3 $

w = European

then Distributive Law:

$latex boxed {w(x+y) = wx + wy}&fg=aa0000&s=3 $

If t = Chinese

then all non-Chinese men = {x – t}

If s = Singaporean,


$latex boxed {s(x – t…

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Zipf’s Law in Linguistic

Math Online Tom Circle


In English, 3 most common words:

  1. the” : occurring 7% of the time;
  2. of” : 3.5% = 7/2
  3. and” : 2.8% ~ 7/3

=> “the” is 2x occurs more often than the 2ndof“, 3x than the 3rdand” …

Zipf’s Law : the frequency of the nth ranked word is proportional to 1/n.


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How Mathematicians Think

Math Online Tom Circle

Hadamard estimated that :

About 90% of mathematicians thinkvisually, 10% think formally.

Usually, they think in steps:

  1. Get the right idea, often think vaguely about structural issues, leading to some kind of strategic vision;
  2. Tactics to implement it;
  3. Rewrite everything in formal terms to present a clean, logical story. (Gauss’s removal of ‘scaffolding’ – middle working steps)

Source: [NLB #510.922]

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Satz & Theorem

Math Online Tom Circle

“A mathematician is a machine for turning coffee into theorems.”

– Alfréd Rényi

It’s a pun in German, where the word Satz means both ‘theorem’ and ‘(coffee) grounds‘ [咖啡渣].

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Quora : How likely is it that a mathematics student can’t solve IMO problems?

Math Online Tom Circle

How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

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Quora: IMO 1988 Question 3

Math Online Tom Circle

Problem A3

A function f is defined on the positive integers by:

for all positive integers n,

$latex f(1) = 1 $
$latex f(3) = 3$
$latex f(2n) = f(n)$
$latex f(4n + 1) = 2f(2n + 1) – f(n) $
$latex f(4n + 3) = 3f(2n + 1) – 2f(n) $

Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.

What is the explanation of the solution of problem 3 from IMO 1988? by Alon Amit

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Blockchains and Application in Bitcoins

Math Online Tom Circle

Encryption & Decryption: ECC (Elliptic Curve Cryptography):

Sending End: Encryption

1) SHA algorithm generates “Digital Signature” ;

2) Generate random “Private Key”.


3) ECC encrypts the text with “Private Key”;

4) From the Private Key generates a “Public Key”;

5) Send out the “original message” and the “Public Key” with the “encrypted message” from 3);

Receiving End: Decryption

6) ECC with Public Key generates Digital Signature 1 (S1);

7) Use SHA algorithm on the original message generates Digital Signature 2 (S2);

8) If S1 = S2, then accept transaction, otherwise reject.

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Story of Yunhao Fu, 2-time IMO Perfect Scorer

Those who are acquainted with IMO (International Math Olympiad) would know that it is extremely tough to get a gold medal (in fact any medal at all) in the IMO. Fu Yunhao, from China, scored 42/42 perfect score twice in a row. His record can be viewed here. IMO’s difficulty varies from year to year. Fu Yunhao’s two attempts were during “difficult” years, hence he may be one of the best Math Olympians in the history.

However, his life story is quite unique in the sense that after his IMO triumph, things did not go quite smoothly. The below interview (unfortunately only in Chinese) describes it:

Google Translate of interview: English Translation


“The two-time Olympic champion” – although not accurate enough, it also summed up the highest achievements in the first 33 years of Fu Yunsheng’s life: he was the IMO (International Mathematical Olympiad) 2002 and 2003 two consecutive years of full-time gold medalist . In the 30-year history of the Chinese national team, only three players have achieved this result. IMO has the difficulty of being “relatively difficult” and “relatively simple.” Fu Yunyi is the only Chinese player who has competed for two “relatively difficult” matches. The Education Authority of China, Zhu Huawei, commented on Fu Yunxi: “He is a symbol of the Chinese mathematical community.”

To some extent, young people who are qualified to embark on the battlefield of IMO can represent the most outstanding mathematics mind of the generation. Professor Yakovlev, a former communications academy of the Soviet Academy of Sciences and a former chairman of the two-year IMO, made famous assertions: “Now the participating students will become the laborers who hold the golden key of knowledge and wisdom in the world 10 years later. they.”

Most of the IMO contestants will continue to engage in mathematics research in their adulthood. Among the 14 winners of the Fields Prize (recognized as the “Nobel Prize in mathematics”) since 2000, at least 8 of them have IMO’s record. But Fu Yunjun, the much-anticipated Olympic genius who once occupied the high point of IMO, unexpectedly disappeared in academia for the next fifteen years.

Some netizens came up with an old post titled “Fu Yunyi, First Class Cowman”, and asked for advice on the status quo of Fu Yunyi. Like igniting curiosity, those who have heard of their reputation have asked each other in the bottom: where is Fu Yunyi going?

(“Cowman” 牛人 means someone who is really good and excels in doing something.)

(Due to some translational errors, different spelling versions of Fu Yunhao’s name appear. They are all the same person, Fu Yunhao.)

However, the published interview does not reflect the full story. Fu himself posted a follow-up post here: Basically, Fu argues that not following the traditional path of becoming a top mathematician does not mean that it is a failure; on the contrary his chosen path (lecturer at teacher’s college) benefits society by producing more good teachers to teach more students.

Google Translate of Fu’s follow-up post: English Translation

Excerpt: The third reaction: Incomprehension of the values ​​conveyed by the report. According to the author of the article, the point is: Excellent people are engaged in basic work and it is a very shameful thing. Those who have won the imo championship are, if nothing else, their journey must be the sea of ​​higher mathematics instead of teaching a group of “second normal school students” to teach junior high school mathematics knowledge. Two of the normal students who lectured said that the genius had fallen. Although I denied that I was a genius, I still thanked Wu for collecting my example of genius. He is very hardworking and commendable. However, the negative energy brought by his values ​​has forced me to vocalize. I didn’t speak for self-explanatory innocence. I just sang for the positive energy development. First of all, for the Wu students who wrote such values, I expressed my understanding that the author, as a senior student who has not completely taken the ivory tower, has such an idea that it is normal. In his cognitive system, academic research is the top grade, ordinary work is relatively low, and lectures are given to two normal students, and it is even closer to the center of the earth. If a once successful genius did not make achievements in heaven and earth, but lived aliciously, it was failure! In Wu’s report to me, this tendency was obvious during the interview. In an interview on April 1st, I even put forward an example of Mr. Zhang Yitang to mention him. At the same time, I mentioned that many researchers have strong academic abilities, but they always go the wrong way, and they have never been able to conquer their life. Want to overcome the problem, but they are still happy and fulfilling. Those things that I spent 10 days and a half wanting to understand may not be of great value, but how many are some interesting conclusions. Moreover, aside from the results of the research, the process of research itself is a happy thing. Although the author did not ignore this paragraph, I mentioned it slightly at the end of the article, but in his eyes, I would like to give this example in order to balance my talent from the genius champion to the teacher of the “Second Normal School” in front of the interviewer. Heart drop it. However, I do not have such a psychological gap. Young people experienced academic crackdowns, and there were some gaps during the very decadent period, but these have been resolved. For so many years of work and life, I learned that there is a vast world beyond the University’s ivory tower and there are so many things waiting for people to practice. If you have a halo above your head and you are in a high tower, you may be able to refer to the country and spur words, but only if your feet are implemented and everything is done well, you can accumulate less and contribute more to the community. Two of my teammates who were with me in 2003, two of them are still swimming in the sea of ​​mathematics, and the other three are involved in the financial industry, which I mentioned in the interview. Each of them did not regret his choice, nor did I.

If you are interested in Olympiad books, check out Recommended Maths Olympiad Books for Self Learning / Domain Test.

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Math Online Tom Circle

Let N = 2n > 6

哥德巴赫猜想 Conjecture “1+1”: N = p1 + p2 (pj all primes)

陈景润 Chen Theorem “1+2”: N = p1+ p2.p3

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Seven Fields Medalists

Math Online Tom Circle

The 7 Fields Medalists are:

2014 – Maryam Mirzakhani (1977-2017) – 1st lady Fields medalist

2010 – Cédric Villani (1973- )

2006 – Grigori Perelman (1966- ) – 1st declined the award

1998 – Andrew Wiles (1953- ) [silver plaque] – Fermat’s Last Theorem

1990 – Edward Witten (1951- ) – Physicist won Fields medal

1982 – Alain Connes (1947- ) – Quantum Theory

1966 – Alexander Grothendieck (1928-2014) – Hermit mathematician

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Bill Gates Returns to Harvard to Talk : Math55

Math Online Tom Circle

Bill Gates, a top Math student at Harvard entrance exams, recalled his first year Harvard “Math55” Course (Advanced Calculus & Linear Algebra) – the toughest at his time because 4 years of Math coursewares condensed into 1 year (2 semesters) !

Note: Harvard “Math55” is even tougher than the “notorious” French Classe Préparatoire, which is a 3-year Math undergraduate courseware squeezed in 2 years : 1st year (code-name “un-demi” or “1/2”) Mathématiques Supérieures; 2nd year (“trois-demi” or “3/2”) Mathématiques Spéciales.

Math55 Syllabus:
Though Math 55 bore the official title “Honors Advanced Calculus and Linear Algebra”, advanced topics in complex analysis, point set topology, group theory, and/or differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis and abstract linear algebra. In 1970, for example, students studied thedifferential geometryofBanach…

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Abstract “Nonsenses” in Abstract Math make “Sense”

Math Online Tom Circle

After 40 years of learning Abstract Algebra (aka Modern Math yet it is a 200-year-old Math since 19CE Galois invented Group Theory), through the axioms and theorems in math textbooks and lectures, then there is an Eureka “AHA!” revelation when one studies later the “Category Theory” (aka “Abstract Nonsense”) invented only in 1950s by 2 Harvard professors.

A good Abstract Math teacher is best to be a “non-mathematician” , who would be able to use ordinary common-sense concrete examples to explain the abstract concepts: …

Let me explain my points with the 4 Pillars of Abstract Algebra :

$latex boxed {text {(1) Field (2) Ring (3) Group (4) Vector Space}}&fg=aa0000&s=3$

Note: the above “1-2-3 & 4″ sequence is a natural intuitive learning sequence, but the didactical / pedagogical sequence is “3-2-1 & 4″, that explains why most students could not grasp the philosophical essence of Abstract Algebra…

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Book Review: Topology, James R. Munkres

Just updated the book review on Munkres’ Topology book.

Singapore Maths Tuition

Topology (2nd Edition)

This book is the best introductory book on Topology, an upper undergraduate/graduate course taken in university. I have written a short book review on it.


Book Review: Topology
Book’s Author: James R. Munkres
Title: Topology
Prentice Hall, Second Edition, 2000

It is often said that one must not judge a book by its cover. The book with a plain cover, simply titled “Topology”, is truly a rare gem and in a class of its own among Topology books.

One striking aspect of the book is that it is almost entirely self-contained. As stated in the preface, there are no formal subject matter prerequisites for studying most of the book. The author begins with a chapter on Set Theory and Logic which covers necessary concepts like DeMorgan’s laws, Countable and Uncountable Sets, and the Axiom of Choice.

The first part of the book is on General…

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Change traffic light rules to prevent further casualties (Singapore)

This petition is important to save future lives. The traffic rules should be machine controlled, i.e. clearly dictated by traffic lights whether to turn or not. In this way, there will be less error due to human judgement.


Quote: “Two accidents involving fatalities this week have prompted this petition. Roads have been widened significantly in recent years, with new lanes added and as such it is difficult to have field of vision of so many lanes, pedestrians, mobility devices, and fast moving oncoming cars all at once to make an informed and safe judgement.

Requesting LTA to determine if it is still safe to keep to discretionary turning on green given the growth in sizes of junctions (such as the Commonwealth Ave West/Clementi Road and Upper Bukit Timah/Jln Anak Bukit junctions) and to work on improving the safety of all road users.

Thank you”

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The Modular Form

Math Online Tom Circle

Form” : Function with special properties – eg.

  • Space Forms: manifolds with certain shape.
  • Quadratic Forms (of weight 2): $latex x^2+3xy+7z^2 $
  • Cubic Forms (of weight 3): $latex x^3+{x^2}y + y^3 $
  • AutomorphicForms (particular case: ModularForms): auto (self), morphic (shape).

1. Non-Euclidean Geometry

1.1 Hyperbolic Plane : is the Upper-Half in Complex plane H (positive imaginary part) where :

  • Through point p there are 2 lines L1 & L2 (called “geodesic“) parallel to line L.
  • Distance between p & q in H: $latex boxed {int_{L} frac {ds}{y}}&fg=aa0000&s=2$
    where L the “line” segment (the arc of the semicircle or the vertical segment) and $latex ds^2 = dx^2+dy^2$

1.2 Group of Non-Euclidean Motions:
$latex f: H rightarrow H$

  1. Translation: $latex z rightarrow {z + b} quad forall b in mathbb {R}$
  2. Dilation: $latex z rightarrow {az } quad forall a in mathbb {R^{+}}$
  3. Inversion: $latex…

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欧氏几何: 第5公理 : 直线上过一点, 只有 n=1 条平行线


  1. 罗氏: n >1 (空间)
  2. 黎曼: n = 0 (球体)

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The Inventors of the 10 Computer languages

Math Online Tom Circle

  1. Python (Dutch Guido van Rossum, 1956)
  2. Java (Canadian James Gosling 1955)
  3. Javascript (USA Brendan Eich, 1961)
  4. C (USA Dennis Ritchie, 1941 – 2011 )
  5. C++ (Denmark Bjarne Stroustrup, 1950)
  6. Ruby (JAPAN Yukihiro “Matz” Matsumoto, 1965)
  7. Perl (USA Larry Wall, 1954)
  8. Pascal (Switzerland Niklaus Wirth, 1934)
  9. Lisp (USA John McCarthy, 1927 – 2011)
  10. PHP (Denmark Rasmus Lerdorf, 1968)

Below the 3 hotest Functional Programming language influenced by Lisp:

11. Kotlin(Russia Andrey Breslav)

12. Scala (USA Martin Odersky)

13. Haskell (USA)

14. Clojure (USA Rich Hickey)

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Group Theory Game

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School System Video (Do not make a fish climb trees)

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

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11-year old math and chess prodigy in Singapore

Source: Channel News Asia

Aarushi Maheshwari solved the famous “Cheryl’s Birthday Problem” when she was only 9. She is also a chess champion and can play blindfold chess.

Watch the video below to learn more!

Also read our previous post on The Most Accomplished 10-Year-Old (Gifted pupil).

For those who want to learn more about Olympiad Math and International Chess, check out the previous two links. Math and Chess are two of the most intellectually challenging activities that can develop the intelligence of kids.

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Basics of Partial Differential Equations Summary

PDE: Separation of Variables

1) Let u(x,y)=X(x)Y(y).
2) Note that u_x=X'Y, u_y=XY', u_{xx}=X''Y, u_{yy}=XY'', u_{xy}=u_{yx}=X'Y'.
3) Rearrange the equation such that LHS is a function of x only, RHS is a function of y only.
4) Thus, LHS=RHS=some constant k.
5) Solve the two separate ODEs.

Wave Equation
\displaystyle c^2y_{xx}=y_{tt}, where y(t,0)=y(t,\pi)=0, y(0,x)=f(x), y_t(0,x)=0.

Solution of Wave Equation (with Fourier sine coefficients)
\displaystyle y(t,x)=\sum_{n=1}^\infty b_n\sin(nx)\cos(nct) where \displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx.

d’Alembert’s solution of Wave Equation
\displaystyle y(t,x)=\frac{1}{2}[f(x+ct)+f(x-ct)].

Heat Equation
\displaystyle u_t=c^2u_{xx},
u(0,t)=u(L,t)=0, u(x,0)=f(x).

Solution of Heat Equation
\displaystyle u(x,t)=\sum_{n=1}^\infty b_n\sin\left(\frac{n\pi x}{L}\right)\exp\left(-\frac{\pi^2n^2c^2}{L^2}t\right), where \displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx are Fourier sine coefficients of f(x).

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5 Good Habits to Develop for College

As soon as you start your freshman year in the university, you’ll be faced with countless challenges. Getting accepted into college is an accomplishment, but it is only one of the many steps you’ll need to take in order to succeed.

The good news is: there are habits you can develop and continue to practice in preparation for the challenges of college life.

The value of pre-college preps

But first: Before you can develop good habits that will serve you during your college years, it is essential that you establish a strong foundation for success in advance.

The best way to achieve your goals in college is to prepare for college way before you enter the university. While in high school, you can get help from the experts who can help you make smarter decisions about your higher education plans.

By working with college application consultants, you can get expert help every step of the way, including:

  • Assessing your current situation and future academic goals.
  • Establishing an organized system to identify and plan the schools that suit your skills and preferences.
  • Preparing and submitting winning applications on time.
  • Acing your college admissions interview.
  • Finding the best college options, from the institution to the course to study, that matches your goals and capabilities.

Once you find the right college for you, college goals such as developing good habits become easier to achieve.

Habits to achieve your college #goals

It’s important for you to develop good habits early on to have a successful college experience. As an incoming freshman, here are five good habits you should develop.

1. Establish your priorities

When you’re so accustomed to high school life, it may become hard for you to really visualize what college life will be like. The independence needed to cope with different class requirements and strictly following your personal study schedule are some of the things you need to consider.

College is a process of preparing for your future career. You should make it a habit to set priorities, and create and follow a schedule to manage your school workload. Make a list of your priorities from the most urgent projects to those that are due later.

This way, you can tackle them one at a time and reduce stress arising from the pressure of wanting to finish them all at once.

  • Plan ahead. The brain can’t handle things all at once. Overloading your mind with things to remember will only trigger stress. And before you know it, your stress may become unmanageable so you end up making mistakes, and missing assignments and projects.
  • Have a planner. Have a checklist of things you should bring to college. Create or buy an academic planner to help you stay organized. List down due dates for projects, assignments and many others. Make sure to stick to your planner and finish your work before the deadline. While you’re at it, you can also create backup alarms using your smartphone calendar app.
  • Break down big tasks into smaller ones. Writing a 12-page research paper and preparing for a huge exam at the same time can be difficult. Planning ahead means you have time to break down these huge tasks into smaller, manageable chunks so that they seem less daunting.

You can also follow these valuable tips on how to adjust to college life.

2. Participate, get involved

Don’t show up in class just to get your attendance checked. Involve yourself by participating in class discussions.

When you go to college, large lecture halls may appear intimidating to you. Practice the habit of engaging yourself during lectures. This will help you get a better grasp of learning. If there’s a vacant seat in front, occupy it.

Grab the opportunity to sit close to the professor for you to feel more present. You shouldn’t take part just because you want to dominate discussions or to help you get a better grade. Choose to speak out for the sake of engaging yourself.

Foster relationships with professors. A close mentoring relationship with a professor is advantageous as he or she can provide you some guidance as you go about your college education.

3. Challenge yourself

When you get into college, be on the lookout for opportunities. Take some challenging classes to expose yourself to new fields. Be open to new things. The world is too big for you to confine yourself to the things you already know.

Branch out and try everything you can. The more you explore your interests, the earlier you’ll discover your college major. Develop the habit of constantly challenging yourself; it will teach you about the value of hard work and discipline.

4. Build your portfolio

In connection to the third tip, pick activities that’ll advance your knowledge and experience. Make it a habit to save great pieces or project works you’ve done during high school. These will work to your advantage the moment you apply for college.

Your portfolio will serve as a proof of how adept you are in the course you’re applying for. This will also increase your chances of getting in a university that you want.

5. Understand how you learn

Each student has different learning abilities. While some are visual learners, others may be auditory learners. It’s important to consider what type of learner you are.

  • Kinesthetic Learner – gains knowledge through feelings or experience
  • Visual Learner – gains knowledge by seeing
  • Auditory Learner – gains knowledge by hearing

Knowing how you learn will be beneficial to you once you start college. There will be no more spoon-feeding in college so you have to be quicker in learning new things. If you know that you are an auditory learner, grab the chance to sit in front to hear each lecture clearly.

To sum up

Every student is unique, so there really is no one-size-fits-all formula for achieving college success. But by getting expert help in finding the right college and excelling in your chosen institution by building good habits, college can be one of the best years of your life.


Brian Giroux is an experienced college admissions advisor and co-founder of Capital College Consulting. Brian is a Professional Member of Independent Educational Consulting Association (IECA). Brian has worked with students from over 30 countries to help provide guidance through the US admissions process.

Brian’s experience includes 18+ years in education serving multiple roles as educator, athletic director, and college admissions consultant.

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The Most Accomplished 10-Year-Old (Gifted pupil)

Pan Annan is probably the most accomplished 10 Year-old student in Singapore, or perhaps even in the world.

Her list of accomplishments:

  • International rhythmic gymnastics champion
  • Youngest member of the Singapore National Youth Chinese Orchestra (SNYCO), where she plays the Pipa
  • Gifted Education Programme (GEP)
  • Math Olympiad trainee
  • Raffles Girls’ Primary pupil

Any one of the above accomplishments is enough to stand out among 10 year-olds, and she has all of them! The most amazing is how she manages her time. I am familiar with Chinese Orchestra trainings, that alone is enough to account for quite a significant amount of time after school, since there is group practicing, sectional practicing, not to mention practicing alone. Possibly Chinese Orchestra alone adds up to a minimum of 5-10 hours per week.

Also, the workload from RGPS GEP is very demanding. Her schedule and timetable can only be achieved with 100% efficiency and focus. (She even does her homework in the car to maximize efficiency and save time.)

Parents may want to read my previous blog post on Book by Truly Gifted Kid (GEP Book), where a similarly prodigious child genius Moshe Kai Cavalin outlines his secret, with input from his mom on parenting. Also, as you can see, the standard for GEP students nowadays is very high, you may read Recommended Books for GEP Selection Test and How to Get Into GEP for some tips on how to do some foundational preparation for the GEP.

Sincerely all the best to Pan Annan for achieving her dreams of being a gymnastic champion.

Read more at: Channel News Asia

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Population Differential Equations and Laplace Transform

Malthus Model
\displaystyle \frac{dN}{dt}=BN-DN=kN

N: Total population

B: Birth-rate per capita

D: Death-rate per capita


Solution to D.E.:
\displaystyle \boxed{N(t)=\widehat{N}e^{kt}},

where \widehat{N}=N(0).

Logistic Equation
\begin{aligned}  D&=sN\\  \frac{dN}{dt}&=BN-sN^2\\  \widehat{N}&=N(0)\\  N_\infty&=B/s  \end{aligned}

Logistic Case 1: Increasing population (\widehat{N}<N_\infty)
\begin{aligned}  N(t)&=\frac{B}{s+(\frac{B}{\widehat{N}}-s)e^{-Bt}}\\  &=\frac{N_\infty}{1+(\frac{N_\infty}{\widehat{N}}-1)e^{-Bt}}  \end{aligned}

The second expression can be derived from the first: divide by s in both the numerator and denominator.

Logistic Case 2: Decreasing population (\widehat{N}>N_\infty)
\begin{aligned}  N(t)&=\frac{B}{s-(s-\frac{B}{\widehat{N}})e^{-Bt}}\\  &=\frac{N_\infty}{1-(1-\frac{N_\infty}{\widehat{N}})e^{-Bt}}  \end{aligned}

Logistic Case 3: Constant population (\widehat{N}=N_\infty)
\displaystyle N(t)=N_\infty

Basic Harvesting Model: \displaystyle \boxed{\frac{dN}{dt}=(B-sN)N-E}.

E: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: \boxed{\dfrac{B^2}{4s}}.

\displaystyle \boxed{\beta_1,\beta_2=\frac{B\mp\sqrt{B^2-4Es}}{2s}}.

\beta_1: Unstable equilibrium population

\beta_2: Stable equilibrium population

Extinction Time: \displaystyle \boxed{T=\int_{\widehat{N}}^0\frac{dN}{N(B-sN)-E}}.

Laplace transform of f
\displaystyle F(s)=L(f)=\int_0^\infty e^{-st}f(t)\,dt

Tip: Use this equation when the questions contains the words “show from the definition”.

Inverse transform of F(s)
\displaystyle f(t)=L^{-1}(F(s))

\begin{aligned}  L(af(t)+bg(t))&=aL(f)+bL(g)\\  L^{-1}(aF(s)+bG(s))&=aL^{-1}(F)+bL^{-1}(g)  \end{aligned}

List of common Laplace Transforms

\begin{aligned}  L(e^{at})&=\frac{1}{s-a}\\  L(1)&=\frac{1}{s}\\  L(\cos wt)&=\frac{s}{s^2+w^2}\\  L(\sin wt)&=\frac{w}{s^2+w^2}\\  L(t^n)&=\frac{n!}{s^{n+1}}\\  L(f')&=sL(f)-f(0)\\  L(f'')&=s^2L(f)-sf(0)-f'(0)\\  L(f^{(n)})&=s^nL(f)-s^{n-1}f(0)\\  &\quad -s^{n-2}f'(0)-\dots-f^{(n-1)}(0)\\  L\left(\int_0^t f(\tau)\,d\tau\right)&=\frac{1}{s}L(f)  \end{aligned}

If L(f)=F(s), s>a, then \displaystyle \boxed{L(e^{ct}f(t))=F(s-c)},

Tip: Use this when doing Laplace Transform of a function with an exponential factor e^{ct}. Note that the reverse direction can sometimes be used as well: \displaystyle L^{-1}[F(s-c)]=e^{ct}f(t).

If L(f(t))=F(s), then \displaystyle \boxed{L(f(t-a)u(t-a))=e^{-as}F(s)}.

Tip: Frequently, we use the reverse direction \displaystyle L^{-1}[e^{-as}F(s)]=f(t-a)u(t-a).

Delta function
\delta(t): infinitely tall and narrow spike at t=0.

\delta(t-a): infinitely tall and narrow spike at t=a.


Two properties of delta function
\begin{aligned}  \int_0^\infty\delta(t-a)\,dt&=1\\  \int_0^\infty \delta(t-a)g(t)\,dt&=g(a)  \end{aligned}
for a\geq 0.

Tip: Use delta function when the keywords “suddenly”, “burst”, etc. appear.

Unit step function
\displaystyle u(t-a)=\begin{cases}  0, &t<a\\  1, &t>a.  \end{cases}

For 0<a<b, \displaystyle u(t-a)-u(t-b)=\begin{cases}  0, &t<a\\  1, &a<t<b\\  0, &t>b.  \end{cases}

Tip: Use unit step function for questions that require a force to “switch on / switch off” at certain times.

\displaystyle \boxed{L(u(t-a))=\frac{e^{-as}}{s}}

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Free Money from PayLah (Quick! Before it is fully redeemed)

Last call to redeem the offer by PayLah! $5 just for downloading the app is pretty worth it.

Singapore Maths Tuition

Hey! You can get S$5 when you register for PayLah! with my Referral Code WILDOF087 (last 3 digits are numbers zero-eight-seven) by 31 December 2018. Download PayLah! from now and enter the above Code during sign-up before this offer is fully redeemed. T&Cs apply.–

PayLah! can be downloaded from Apple Store or Android store (Singapore). Getting your $5 should be instantaneous. Be sure to redeem it today since there is a cap of 40000 people who can redeem. (The offer is still available now, but who knows next week it may be fully redeemed.)

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NTUC Fairprice Receipt Number

For those participating in lucky draws, often it requires you to SMS the NTUC Fairprice Receipt Number. However, the latest version of NTUC receipts, upon closer inspection contains many numbers but none that are explicitly specified as “receipt number”.

In the old NTUC receipt, the receipt number is clearly stated.

What is NTUC Receipt Number?

It is extremely confusing to find the NTUC Receipt Number (out of the many numbers in the receipt). After some “research”, I come to the conclusion is that the receipt number is the same as the transaction number, labelled by Tr!

How I came to this conclusion: Look at the image above. The receipt number is 65539, the transaction number is Tr: 65539. It matches, right? Hence, it is logical that the NTUC Fairprice receipt number is precisely the Tr (transaction) number.

Similarly, I believe that this logic applies to Giant, Sheng Siong, Cold Storage receipt numbers too, it should be the transaction number, in the event that the receipt number is not clearly stated in the receipt. Same for NTUC Finest, NTUC Extra and all other types of NTUC Fairprice receipts.

I hope this helps those who are desperately searching for the receipt number!

PayLah Promotion

Free Money from PayLah (Quick! Before it is fully redeemed) If you haven’t signed up for PayLah yet, here is your chance to get $5 free (limited time only).

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100 Changi Rewards points for you

Sign up here!

Changi points are quite useful, can be used to redeem the following:

  • Attraction Passes
  • Exclusive Items
  • Exclusive Events
  • Travel and Airline Miles
  • Changi Rewards e-Vouchers
  • 1-for 1 Deals
  • Hotel Stays
  • and other Changi Rewards’ Partners Deals

URL to sign up and get 100 points instantly: Changi Points

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Hans Rosling’s Best TED Talk & Book “Factfulness”

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Why Prime Numbers Still A Mistery

Math Online Tom Circle

Prime Numbers since Euclid 2300 years ago remains a mystery today…

Riemann’s Hypothesis is the Top Conjecture Unsolved in 21st Century.

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Free Money from PayLah (Quick! Before it is fully redeemed)

Hey! You can get S$5 when you register for PayLah! with my Referral Code WILDOF087 (last 3 digits are numbers zero-eight-seven) by 31 December 2018. Download PayLah! from now and enter the above Code during sign-up before this offer is fully redeemed. T&Cs apply.–

PayLah! can be downloaded from Apple Store or Android store (Singapore). Getting your $5 should be instantaneous. Be sure to redeem it today since there is a cap of 40000 people who can redeem. (The offer is still available now, but who knows next week it may be fully redeemed.)

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110 Nanyang Girls’ High students fall sick during boarding school programme

Quite worrying. Based on my personal experience and online research, food in Singapore cannot be left in room temperature for more than 4 hours, due to our hot and humid temperature. To be safe, food needs to be put in the refrigerator as soon as possible if not consumed immediately.

Hope all affected students get well soon.

Source: 110 Nanyang Girls’ High students fall sick during boarding school programme

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Second Order Linear ODE Summary

Second Order Linear D.E. Summary

Homogenous D.E.

Solve the Characteristic Equation: \lambda^2+a\lambda+b=0.
Case 1) Two real roots \lambda_1,\lambda_2: \implies \boxed{y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}}

Case 2) Real double root \lambda: \implies \boxed{y=c_1e^{\lambda x}+c_2xe^{\lambda x}}

Case 3) Complex Conjugate root \lambda_1,\lambda_2=-\frac{a}{2}\pm iw, where w=\sqrt{b-\frac{a^2}{4}}: \implies \boxed{y=e^{-\frac{a}{2}x}(c_1\cos wx+c_2\sin wx)}

Non-homogenous D.E.
General solution of non-homogenous D.E.: \displaystyle y=y_h+y_p, where y_h is the general solution of the homogenous equation, and y_p is the particular solution (with no arbitrary constants).

Method of Undetermined Coefficients (Guess and try method)

Only works if r(x) is polynomial, exponential, sine or cosine (or sum/product of these).

Polynomial: Try y=Polynomial (e.g. y=Ax^2+Bx+C or y=Bx+C.)

Exponential (e^{kx}): Try y=ue^{kx}, where u is a function of x.

Trigonometric (\sin kx or \cos kx): Convert to complex differential equation by replacing y with z, replace \sin kx/\cos kx by e^{ikx}.

Try z=ue^{ikx}, where u is a function of x. After solving for z, take real/imaginary part of z for cosine/sine respectively.

Method of variation of parameters

[Step 1)] Solve the homogenous D.E. y''+p(x)y'+q(x)y=0.

Get solution of the form y_h=c_1y_1+c_2y_2.

[Step 2)]
Let \displaystyle u=-\int\frac{y_2r}{W}\,dx and \displaystyle v=\int\frac{y_1r}{W}\,dx where W is the Wronskian \displaystyle W=y_1y_2'-y_1'y_2.

Particular solution: y_p=uy_1+vy_2.

General solution: y=y_h+y_p.

Forced Oscillations
Let F_0 be the amplitude of the driving (external) force. If F_0=0, by Newton’s Second Law, m\ddot{x}=-kx, hence \displaystyle \boxed{\ddot{x}=-\omega^2 x},
where \omega=\sqrt{k/m}. The value \omega is called the natural frequency.

If F_0\neq 0, then \displaystyle \boxed{m\ddot{x}+kx=F_0\cos\alpha t},
where \alpha is the driving (external) frequency.

At resonance (when \alpha=\omega), \displaystyle \boxed{x=\frac{F_0t}{2m\omega}\sin(\omega t)}.

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Joseph Fourier is Still Transforming Science

Math Online Tom Circle

Key Words: 250 years anniversary

  • Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
  • Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
  • Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
  • Taught at the newly established Military Engineering University “Ecole Polytechnique”.
  • Scientific Research: Short period but intense.
  • Before Fourier died (wrapped himself in thick carpet in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

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10 Things You Don’t Know About Albert Einstein

Source: Thoughtco

Some of the 10 facts are:

  • He Loved to Sail
  • Einstein’s Brain
  • Einstein and the Violin
  • Presidency of Israel

For those interested in the theory of relativity (special and general), which is the main core of Einstein’s research, do check out this book Relativity: The Special and the General Theory, 100th Anniversary Edition. Highly rated on Amazon, it is one of the few books that tells the story of Einstein and also includes real scientific content.

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Drawing Chemistry Diagrams in LaTeX

The web application “mol2chemfig” is very amazing. (

Especially the search function, where one can just type the name of the chemical and out comes the chemfig code where it can be pasted to LaTeX. The mol2chemfig package needs to be imported in LaTeX before the code can be compiled. Alternatively, one can just save the PDF image (vector graphics) generated by the web app.

Below is a screenshot of what is happening:

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Linear First Order ODE, Bernoulli Equations and Applications

Linear First Order ODE
DE of the form: y'+P(x)y=Q(x).

Integrating factor: R(x)=e^{\int P(x)\,dx}.
\begin{aligned}  R'&=RP\\  Ry'+RPy&=RQ\\  (Ry)'&=RQ\\  Ry&=\int RQ\,dx  \end{aligned}
\displaystyle \boxed{y=\frac{\int RQ\,dx}{R}}
(Remember to have a constant C when integrating the numerator \int RQ\,dx.)

Integration by parts
\displaystyle  \boxed{\int uv'\,dx=uv-\int u'v\,dx}

Acronym: LIATE (Log, Inverse Trig., Algebraic, Trig., Exponential), where L is the best choice for u. (This is only a rough guideline.)

Bernoulli Equations
DE of the form: y'+p(x)y=q(x)y^n.


Set \boxed{y^{1-n}=z}.

Then (1-n)y^{-n}y'=z'. The given DE becomes
\displaystyle  \boxed{z'+(1-n)p(x)z=(1-n)q(x)}.

Fundamental Theorem of Calculus (FTC)
Part 1: \displaystyle \frac{d}{dx}\int_a^x f(t)\,dt=f(x)

Part 2: \displaystyle \int_a^b F'(t)\,dt=F(b)-F(a)

Hyperbolic Functions
\begin{aligned}  \sinh x&=\frac{e^x-e^{-x}}{2}\\  \cosh x&=\frac{e^x+e^{-x}}{2}\\  \cosh^2 x-\sinh^2 x&=1\\  \end{aligned}
\begin{aligned}  \frac{d}{dx}\sinh x&=\cosh x\\  \frac{d}{dx}\cosh x&=\sinh x\\  \frac{d}{dx}\sinh^{-1}x&=\frac{1}{\sqrt{x^2+1}}\\  \frac{d}{dx}\cosh^{-1}x&=\frac{1}{\sqrt{x^2-1}}  \end{aligned}
\displaystyle \int \tanh(ax)\,dx=\frac{1}{a}\ln(\cosh(ax))+C.

Uranium-Thorium Dating
Starting Equations:
\displaystyle \begin{cases}  \frac{dU}{dt}=-k_U U\implies U=U_0e^{-k_Ut}\\  \frac{dT}{dt}=k_UU-k_TT.  \end{cases}

\frac{dT}{dt}+k_T T=k_U U_0e^{-k_Ut}

R=e^{\int k_T}=e^{k_T t}.

\displaystyle \boxed{T(t)=\frac{k_U}{k_T-k_U}U_0(e^{-k_Ut}-e^{-k_Tt})}

\displaystyle \boxed{\frac{T}{U}=\frac{k_U}{k_T-k_U}[1-e^{(k_U-k_T)t}]}

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Louis-Le-Grand, un lycée d’élite 法国(巴黎)精英学校: 路易大帝中学

Math Online Tom Circle

Lycée Louis-Le-Grand is the best high school (lycée) for Math in France – if not in the world – it produced many world-class mathematicians, among them “the Father of Modern Math” in 19th century the genius Evariste Galois. (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Victor Hugo, 3 French Presidents, etc.

Its Baccalaureat (A-level) result 100% passed with 75% scoring distinctions. Each year 1/4 of Ecole Polytechnique (France Top Engineering Grande Ecole) students come from here.

More surprisingly, the “Seconde” (Secondary 4) students learn Chinese Math since 6ème (PSLE Primary 6).

Note: Louis Le Grand (= Louis 14th). He sent the Jesuits (天主教的一支: 耶稣会传教士) as the “French King’s Mathematicians”(eg. Bouvet 白晋) to the 16-year-old Chinese Emperor (康熙) KangXi’s Court.

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How to draw polygons with angles (LaTeX Tikz)

Just learnt some amazing angle-drawing techniques using LaTeX Tikz.

Although cumbersome to code, the benefit of using LaTeX to draw diagrams is that it matches the font of your document, and is easier to customize and edit. Also, it is “vector graphics” in the sense that even if you zoom in, it is in perfect resolution.

Sample code shown below

Important: In order to use Tikz to draw angles you need to load the following packages:


Sample Tikz code for drawing the above figure (polygon angles)

\coordinate (v0) at (-2.2,1.2);
\coordinate (v1) at (0,0);
\coordinate (v2) at (2.2,1.8);
\coordinate (v3) at (0,3);

(v1) circle (2pt) node[align=left, below] {$v_1$}
-- (v2) circle (2pt) node[align=center, below, right] {$v_2$}
-- (v3) circle (2pt) node[align=left, above] {$v_3$}
-- (v0) circle (2pt) node[align=left, above, left] {$v_0$}
-- (v1);
\pic [draw, -, "$\alpha_0$", angle eccentricity=1.5] {angle = v1--v0--v3};
\pic [draw, -, "$\alpha_1$", angle eccentricity=1.5] {angle = v2--v1--v0};
\pic [draw, -, "$\alpha_2$", angle eccentricity=1.5] {angle = v3--v2--v1};
\pic [draw, -, "$\alpha_3$", angle eccentricity=1.5] {angle = v0--v3--v2};

Clearly, the code can be modified for any polygon be it triangle, pentagon, etc. Also, if you are curious how to post source code in, it is by surrounding your source code with [ code ] [/ code]. It also supports a “lang=” option, see more at

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Bukit Batok / Hillview Maths Tuition

Please contact us for one-to-one Maths Tuition at Bukit Batok / Hillview area.

Nearby areas such as Clementi / Jurong East are also suitable.

Patient and dedicated Maths Tutor.

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Senior Wrangler: Singapore Prime Minister

Math Online Tom Circle

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek…

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Topological Data Analysis

Math Online Tom Circle

Three Key Ideas:

  1. Cloud of Points
  2. Filter function: f (x,y,z) –> x
  3. Cluster: Overlapping bins
  4. Draw network:
  • Vertices: clusters
  • Edges: connecting lines between clusters.

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Crash course in Category Theory

Math Online Tom Circle

Key Point:

Haskell & any FP compiler don’t check the Category Theory proof if your codes (eg. fmap) follow Functor’s Laws (eg. Preserve structure, identity) or Monad’s Laws !

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God’s Book of Perfect Proof

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Canadian mathematician Langlands wins Abel Prize

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Michael Bishop’s Science

Math Online Tom Circle

Michael Bishop (Nobel Medicine 1989)
School Science subjects should be taught in this sequence (unless in single ‘Combined Science’):
1st Physics
2nd Chemistry
3rd Biology

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Mind over Matter : Bruce Lipton

Math Online Tom Circle

Key Points:

1. The Foundation of Science has changed since Newtonian Era, except Biology / Medicine and Psychology which don’t keep up:

Math => Fractal Math

Newton Physics (Matter)=> Quantum Physics (Energy)

(Organic) Chemistry => Electro-Chemistry

Energy = Field = 气 Chi / Qi

3. Environment -> Mind -> Perception -> Genetic Control

4. Consciousness (5%), Subconscious-ness (95%)


  • Driving car: inexperienced driver (consciousness), experienced driver (subconscious).
  • Dating (conscious behavior)

Note: The successful DeeplearningAI is using the concept of Perception called “Perceptrons” by analysing the Environmental (BIG DATA) pattern with the help of Mathematics (Calculus : Gradient Descent, Statistics : Bayesian Probability, Algebraic Topology, etc).

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