Professor Wildberger is extremely kind to upload his videos which would be very useful to any Math student studying Topology. Simplices / Simplicial Complexes are usually the first chapter in a Algebraic Topology book.
Check out also Professor Wildberger’s book on Rational Trigonometry, something that is quite novel and a new approach to the subject of Trigonometry. For instance, it can be used for rational parametrisation of a circle.
Wishing Singapore a very happy SG50 birthday this weekend!
Do check out some of the SG50 sales at Qoo10, many of the items are going at half price! Definitely cheaper than buying at retail stores.
WWW.QOO10.SG
WWW.QOO10.SG
Also check out my previous posts on Best Deals at Qoo10!
Tired of “blue screens of death” that are so common in Windows? Don’t want to wait 10 minutes for Windows to “start up”? Switch to Mac OS!
My old computer (ASUS) has lasted me 5 years, but has recently gotten to the point that it slows down to a crawl. Booting up Windows takes up to 15 minutes, and “blue screens of death” occurs extremely frequently. It hasn’t spoiled completely yet, I still use it for printing documents.
I have since switched to MacBook Air, and so far it has been a great experience.
For Math students, some apps that you may want to install are MacTeX. It is the LaTeX Mac version. The initial download is over 2 GB, so it might take a while. I downloaded the installer from the main website, it took around one hour.
Another app is Google Chrome, which works very well on Mac. WordPress.com runs better on Chrome, for instance the LaTeX expressions are rendered better on the Chrome browser versus the Safari browser.
MacBook Air is one of the lightest notebooks around. The downside is that it does not have a few features, for example Ethernet Adapter and Optical Drive. Not to worry, one can purchase add-ons to remedy the problem. (Listed below)
Thunderbolt to Gigabit Ethernet Adapter
Aluminum External USB DVD+RW,-RW Super Drive for Apple–MacBook Air, Pro, iMac, Mini
MacBook takes some time to get used to, hence it is good to play around with it to discover the hidden shortcuts. For instance, Copy on Mac is command-C instead of control-C.
If you have any tips for using the MacBook, do feel free to share it in the comments section below!
Check out the latest new Math book on the Fourth Dimension! The Fourth Dimension is the mysterious dimension which cannot be seen. Check out also our previous post on the Fourth Dimension Explained.
Mathematics popularizer Matt Parker, an Australian based in England, is a self-proclaimed “standup mathematician” perhaps best known for his numerous contributions to the Numberphile YouTube channel. He is also the Public Engagement in Mathematics Fellow at Queen Mary, University of London, and his new book, Things to Make and Do in the Fourth Dimension, is an ambitious and delightful addition to the current age’s plethora of high-quality volumes on recreational mathematics—even if most of the material he covers is focused on 2-D and 3-D. Like the extensive writings of legendaryScientific American columnist Martin Gardner this book seeks to make mathematics come alive for an intelligent and curious audience by engaging the reader in a lively informal style, and with irresistible invocations to roll up one’s sleeve and experiment. Parker also enlivens his chapters with numerous surprises.
Source: http://www.scientificamerican.com/article/how-to-get-to-the-fourth-dimension/?WT.mc_id=SA_WR_20150805
If applying to Harvard, use ‘Father and Mother’ to address your parents, to Stanford, use ‘Dad & Mom’…
By Cédric Villani, Fields Medalist (2010)
– Translated from the French by Malcolm DeBevoise – illustrations by Claude Gondard.” Villani, Cedric
NLB Call Number: English 510.92 VIL
Just to share this site TTMLW (Tips to Make Life Wonderful). Hope it is useful to some of our readers!
Source: Elements of Integration
Students studying Mathematical Analysis, Advanced Calculus, or probability would sooner or later come across the term a.e. or “almost everywhere”.
In layman’s terms, it means that the proposition (in the given context) holds for all cases except for a certain subset which is very small. For instance, if f(x)=0 for all x, and g(x)=0 for all nonzero x, but g(0)=1, the function f and g would be equal almost everywhere.
For formally, a certain proposition holds 
Two functions 
Similarly, this notation can be used in the case of convergence, for example 
The idea of “almost everywhere” is useful in the theory of integration, as there is a famous Theorem called “Lebesgue criterion for Riemann integrability”.
(From Wikipedia)
A function on a compact interval [a, b] is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is known as the Lebesgue’s integrability condition or Lebesgue’s criterion for Riemann integrability or the Riemann—Lebesgue theorem.[4] The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue’s general measure or integral.
Reference book:
Trigonometry in abstract algebra Group Theory… this is a new look of Elementary Math (E. Math) from a higher level (Abstract Algebra : Group Theory) — just as the Tang Poem said “欲穷千里目, 更上一层楼” (To see further distance away, just climb up to higher level).
Modern Algebra: Based on the 1931 influential book “Modern Algebra” written by Van de Waerden (the student of E. Noether). Pioneered by the 20th century german Göttingen school of mathematicians, it deals with Mathematics in an abstract, axiomatic approach of mathematical structures such as Group, Ring, Vector Space, Module and Linear Algebra. It differs from the computational Algebra in 19th century dealing with Matrices and Polynomial equations.
This phase of Modern Algebra emphasises on the algebraization of Number Theory: {N, Z, Q, R, C}
Post-Modern Algebra: The axiomatic, abstract treatment of Algebra is viewed as boring and difficult. There is a renewed interest in explicite computation, reviving the 19th century invariant theory. Also the structural coverage (Group, Ring, Fields, etc) in Modern Algebra is too narrow. There is emphasis on other structures beyond Number Theory, such as Ordered Set, Monoid, Quasigroup, Category, etc.
Example:
The non-abelian Group S3 (order…
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Mathematics is divided into 2 major branches:
1. Analysis (Continuity, alculus)
2. Algebra (Set, Discrete numbers, Structure)
In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.
Topology specialised in
– ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.
Example:
The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.
This Math technique is very often used by great mathematicians who have the multi-year patience in the laboriously proofs:
Chen Jingrun “1+2” Goldbach Conjecture. (Terrence Tao’s notes)
Zhang Yitang on the “70 million Prime Gap.”
What are the differences of these strings of characters?
1111111111111…
010101………..01…
1001100100001….
The 1st string is all ‘1’
The 2nd string is all ’01’
The 3rd string is complex: random ‘0’, ‘1”
Kolmogrov complexity deals with randomness.
This blogger has a very valid point on Math Self-study Technique : Learning Math ‘Just-in-Time’ using the rich resources in Wikipedia:
http://steve-yegge.blogspot.in/2006/03/math-for-programmers.html
The result is not surprising to China but to USA:
♢Recently China government bans IMO training in schools.
♢Obama was surprised that the USA IMO team consists of predominantly Chinese American students.
IMO Math is like ‘Acrobatics’ to real ‘Kung-fu’, it is not real Math education, but special ‘cute’ techniques to solve tough ‘known’ solution problems. Real Math is long R&D solving problems with UNKNOWN solution (eg. Fermat’s Last Theorem, Riemann Conjecture,…)
2 types of Math: Algorithmic or Deductive (演绎). Chinese long traditional ‘abacus’ mindset, procedural computational Math is Algorithmic, applied to special cases (eg. astronomy, calendar, agriculture, architecture, commerce,…). European Greek’s Euclid deductive, step-by-step axiom-based proofing, is theoretical, generalized in all cases (Geometry, Abstract Algebra,…)
Look at the Fields Medal (aka ‘Nobel Prize’ of Math) super-power – France – which has produced 1/3 of the Fields Medalists, but performing so-so in IMO. In contrast, China has ZERO Fields…
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Click this URL to view more Digital Pianos: http://list.qoo10.sg/sp/7452?jaehu_cust_no=7NCxGWDWWM132kSlYY7/rw==#53572
Is your child beginning in trying out learning piano? Buying an actual upright piano would cost a few thousand dollars, and may not be a good idea, as many children may stop learning after a few months. Practicing piano is an excellent way to improve musical skills and even Math skills, as music has been found to be linked to math. Unfortunately practicing piano requires a lot of patience and perseverance (e.g. 1 hour practice every day), in order to reach a decent level like Grade 8 Piano. Hence, many children understandably cannot sustain the necessary practice required, and stop learning, and the money buying the piano (and space in the house) would be wasted.
Research has found that music helps children learn maths! Listening to music in maths lessons can dramatically improve children’s ability in the subject and help them score up to 40 per cent higher in examinations, a new study has found. Source: http://www.telegraph.co.uk/education/9159802/Music-helps-children-learn-maths.html
Buying a keyboard (non-weighted) is also not a good idea, as classical piano requires dynamics (e.g. forte and piano) which means loud and soft contrasts, which is not achievable using a non-weighted keyboard.
The perfect solution is to purchase a digital piano first, and then upgrade to a upright piano if the child maintains interest after a few years. A very good brand for starters would be Yamaha, a Japanese music brand. An added advantage of digital pianos is the ability to use earphones, which would enable one to practice late at night without disturbing the neighbor.
If the child is interested to pursue MEP (Music Elective Programme), this digital piano would come in handy for musical composition in MIDI, and it would have orchestral sounds like Strings, Brass, Drums, etc. Definitely a very useful instrument to have for serious music students as well as the amateur.
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Click here for even more choices for Digital Pianos: http://list.qoo10.sg/sp/7452?jaehu_cust_no=7NCxGWDWWM132kSlYY7/rw==#53572
Recently, there have been many news on the Straits Times Forum / other newspapers / internet on the phenomenon of tuition in Singapore. There are many mixed opinions on tuition, which are discussed in depth in those articles. I have picked the most interesting articles on the subject of tuition, which would be a familiar topic in Singapore, as 70% or more of Singaporean students have tuition. The links are found near the bottom of the post.
Personally, I think of tutors like a sports coach, like a swimming coach or a badminton coach. Sports coaches help their students to play the sport better. Tuition teachers help their students to perform in the exams better. There are great similarities between their roles. Currently, almost all top athletes would have a coach, it would be unthinkable for an athlete at the international level not to have a coach.
Tuition has also been around since thousands of years ago. Alexander the Great’s father hired Aristotle as a tutor for his son. The Imperial Tutor in ancient China is an extremely prestigious post and is often awarded only to the top scholar in the imperial exams. His job is to tutor the future emperor or other princes / princesses. (See this example of an Imperial Tutor in China). In the past, only the rich and wealthy could afford tutors. However, due to the prosperity in many first world countries like Singapore and South Korea, affording tuition is becoming increasingly possible even for the middle class.
Note: I am currently not giving tuition at the moment, but I have a good recommendation for a very good tuition agency. Interested readers can email me at mathtuition88@gmail.com.
As a former tutor, I don’t really think that tuition (in moderation) can be harmful, like what some of the articles claim. Back to the analogy of sports coaches, it is illogical to suppose that a sport student’s badminton skills can worsen and deteriorate after practice with a qualified coach. That would simply make no sense! Similarly, as long as the tutor is competent and not teaching the wrong thing, it would simply be illogical to say that tuition can harm academic performance. It would be really strange if a student becomes worse at math after more practice.
The key to successful life is balance. A role model for children would be Jeremy Lin, the Asian American basketball player. Highly intelligent and an excellent student, he has been admitted and graduated successfully from Harvard. He is also a professional basketball player in the NBA, and at the peak of physical fitness. He is also a humble and devout Christian. He is one guy that all students should take as a role model.
Book on Jeremy Lin:
Hard work and determination DOES pay off! Nine-year-old Filipino pictured studying in the light of a McDonald’s is swamped with donations after the picture goes viral
This boy has inspired and motivated many. Despite having only one pencil, he is studying hard at night by the light of McDonald’s instead of playing. Everyone hopes that he will fulfill his dream of being a policeman and having a good education. He definitely deserves it 100%.
Motivation is very important for studying. Self motivation is key, as the child needs to know the importance of education. In Singapore, almost every child is blessed with good financial resources, and definitely have more than one pencil. Studying environment is also quite good, most children will have a comfy chair and table, not to mention electronic learning devices like iPad or computer. However, all these material things are not as important as motivation. Without the motivation to study, all the fanciful stationery and computers would be no use. It is unfortunate, that in Singapore and other developed countries, sometimes the resources (money, stationary, computer, books, tuition, enrichment) are all present, but the motivation to study is absent!
Motivation can either come from a person, or from books. Countless people have been motivated by motivational books. Do check out these motivational books for the student if you are interested.
The original facebook link is here! The university student, Joyce Gilos Torrefranca, who posted it is also to be commended. Do give it a like! Link: https://www.facebook.com/joyce.torrefranca/posts/1010235928995791
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The exams are coming, but your neighbor decides to do a major renovation? Neighbor has loud music blasting/ very noisy children? An urgent and drastic measure must be taken, as scientific research has shown that loud background noises reduce children’s test scores! Recent research in London (UK) has shown that high levels of environmental noise outside schools reduce children’s scores in standardised nationwide academic tests.
This is hardly surprising, as it is very hard to do any studying in the presence of very loud noises. The source of noises, in Singapore context, may come from the following:
Since Singapore is an urban and densely populated country, it would be quite common to have one of the above situations which causes noise pollution.
Peltor Earmuffs are available on Qoo10 for people who live in Singapore, and also Amazon, for people who live in the USA / worldwide.
Qoo10 (Singapore):
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Amazon (Worldwide):
3M Peltor H10A Optime 105 Earmuff
Whether the situation calls for a earmuff / earplug would be up to a case by case basis. However, if one is desperately looking for a earmuff / earplug (not easy to find in Singapore shops), the above 3M Peltor earmuffs would do a good job. Do choose a earmuff / earplug with at least 20dB NRR (Noise Reduction Rating). Any lower wouldn’t be of any help at all.
I personally bought the 3M Peltor 98 earmuffs as my neighbor is doing a very prolonged renovation which is estimated to last till the end of this year. Drilling and even a crane was involved. It managed to cut down the noise to a more manageable level.
Amazon Peltor Earmuff Review:
Pretty Practical Earmuffs
By Nino Brown on January 20, 2008
Verified PurchaseI got them so that I could better focus on work at home – there can be many distracting noises that obstruct serious thought. While they do not cancel everything out – they drastically reduce noise. If someone directly in front of me starts talking to me, I will hear what they are saying, albeit at a reduced noise level. However, when I went into a connected room, and asked my mom and my sister to have a conversation in the room I was initially in, I couldn’t hear what they were saying.
The really bring me a sense of inner peace and centeredness – I can focus on my work without worrying about becoming distracted with random noises in the house. It allows you to get into your own world. I would advise people, however, to notify their house mates when they are going to use them to tune out – it would be horrible if something happened at the other end of the house and one remained unaware because one could not hear the noise.
The decibel is one excellent application of the logarithm. The formula of the intensity of noise (I) in decibels is:
Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/sound/db.html
Why is the second derivative written as such:
$latex displaystyle
frac{d^2 y}{d x^2}?
$
The Math Theorems & Proofs
If Women propose to Men, then
♡ Women will get the BEST Man, whereas,
♧ Men will get the WORST woman.
See also: The Match Algorithm
James Haris Simon made his billion fortune in Finance business, building Math Model with Statistics and Probability.
He was a Berkerly graduate student of the Chinese mathematician Prof. S.S. Chern (陈省身).
Many people think Math is useless, but here is irrefutable evidence that Math can be useful for many things, including even basketball!
Basketball is a game of tactics, not just brute force, hence other than being physically fit the mental agility of the players and the strategy of the coach is very important. Using the math software described in the above video would definitely help NBA players like Jeremy Lin reach the next highest level and contribute 100% to the team.
This is just for humor!
Jeremy Lin: The Reason for the Linsanity
Jeremy Lin is probably the smartest NBA basketball player in history, graduating from Harvard university! Harvard graduate Jeremy Lin recently became a New York Knicks phenomenon and he’s the NBA’s first American-born player of Taiwanese descent. The book will chronicle Lin’s high school, college and early career in the NBA with particular emphasis on the media explosion surrounding his success as starting point guard with the Knicks. It will explore how Jeremy’s Christian faith, family, education and cultural inheritance have contributed to his success. The book will also include interviews with basketball experts on Jeremy’s future in the NBA, Asian-American thought leaders on the role of race in Jeremy’s rise to stardom, and renowned Christian athletes and pastors on the potent combination of faith and sports.
Ambitious JC students who wish to try their luck for Harvard may wish to take these 5 steps into consideration!
H2 Mathematics has one of the highest distinction rates of all subjects (around 50% each year). This means that around half of all Singaporean A level candidates score an A for H2 Maths!
(For year 2010)
H2 Mathematics Distinction Rate: 51.9%
H2 Biology Distinction Rate: 43.7%
H2 Economics Distinction Rate: 33.8%
H1 Mathematics Distinction Rate: 33.1%
H1 Economics Distinction Rate: 33.8%
Literature Distinction Rate: 30.1%
History Distinction Rate: 23.7%
Geography Distinction Rate: 28.3%
Source: http://ajc.edu.sg/pdf/aj_broadcast/newsroom/news_archives/linkaj_may_2011.pdf
Check out the highly summarized and condensed H2 Maths Notes here! (Comes with Free H2 Math Exam Papers.)
Answer: Yes, provided the student does study conscientiously and not lag behind too much. Based on the statistics above, one can easily see that based on probability alone, H2 Maths is the easiest H2 subject to get A. Since more than 50% of students get A for H2 Maths, in a sense it is easier to get A for H2 Maths than flipping a heads on a coin!
H2 Maths is also the easiest to fail! Without sufficient practice and effort to understand the subject material, sub-30 (below 30/100) marks are extremely common for H2 Maths. Last minute cramming will simply not work, and if a student lags too far behind in terms of syllabus, it will take extra effort to just even catch up.
The 50% National Distinction Rate for H2 Maths can be quite misleading to think that every student has 50% chance of getting A for H2 Maths. The truth is that H2 Maths Distinction Rate varies a lot from school to school.
For example, AJC’s H2 Maths Distinction Rate is 62.7%, which is very much higher than the 50% average National Distinction Rate.
Raffles Institution (RI/RJC) Distinction Rate hovers around 70% to 80%!
Victoria JC (VJC)’s H2 Maths Distinction Rate is around 66.6%.
Hwa Chong (HCI) H2 Maths Distinction Rate is around 80% (8 out of 10 students scored an A for H2 Maths in HCI for three consecutive years).
Upon some thinking, one will quickly realize that if so many schools have Distinction Rate significantly above 50%, there has to be many schools with Distinction Rate significantly below 50%, in order for the National Distinction Rate to be around 50%!
The only people who know the exact Distinction Rate for the above mentioned JCs would be the internal staff and students, since the school website will probably not publish the statistics for obvious reasons.
For students who are in schools with super high H2 Maths Distinction Rate, congratulations, your chances of getting A for H2 Maths are very good. However, do not be complacent till the very last day, as the race is not over yet.
For students who are in schools with very low H2 Maths Distinction Rate, the odds are unfortunately stacked against the student. However, do not lose heart, as anything is possible if one puts one’s heart and mind into it.
Good luck!
Check out the highly summarized and condensed H2 Maths Notes here! (Comes with Free H2 Math Exam Papers.)
#1 Recommended BMAT Book
How to Master the BMAT: Unbeatable Preparation for Success in the BioMedical Admissions Test
Thinking of applying to the new Medical School at NTU?
NTU’s application requires the BMAT (Biomedical Admissions Test). Applicants will have to register for the Biomedical Admissions Test (BMAT) and take the BMAT as part of the criteria for entry to the LKCMedicine MBBS programme.
Source: http://www.lkcmedicine.ntu.edu.sg/admissions/Pages/Entry-Requirements-And-Selection-Criteria.aspx
BMAT is useful for applying to Britain’s medical schools too.
NTU only takes in 50-150 students per year, out of Singapore’s entire population! Hence, one can imagine it is definitely not easy to get into NTU medicine.
According to this article by Straits Times, “the final 54 chosen medical students – all Singaporeans – had almost perfect scores in the interviews and also aced their BioMedical Admissions Test (BMAT).”
Getting 4 As is too extremely common in top JCs like RJC or HCI (pick any random guy from the top JCs and he/she is likely to have 4As), hence the distinguishing factor would be your BMAT score.
The BMAT is set by the British, and hence unlike anything students have seen in Singapore. In particular the format and style are different from the standard Singaporean style of testing.
Currently the acceptance rate for NTU medicine is 54/800 (6.75% acceptance rate), which means that getting into NTU medicine is as hard as getting into Ivy League Universities like Harvard / Princeton!!! (Harvard = 5.9% Acceptance Rate, Princeton = 7.4% Acceptance Rate)
Singaporeans are known to be extremely keen when it comes to studying Medicine / Law, and hence competition is definitely going to increase, and a good BMAT book will help you rise above the competition.
#1 Recommended BMAT Book
How to Master the BMAT: Unbeatable Preparation for Success in the BioMedical Admissions Test
#2 Recommended BMAT Book
The Ultimate BMAT Guide – 600 Practice Questions: Fully Worked Solutions, Time Saving Techniques, Score Boosting Strategies, 10 Annotated Essays, 2016 Entry Book (BioMedical Admissions Test)
#3 Recommended BMAT Book
Passing the UKCAT and BMAT: Advice, Guidance and Over 650 Questions for Revision and Practice (Student Guides to University Entrance Series)
#4 Recommended BMAT Book
How to Master the BMAT: Unbeatable Preparation for Success in the BioMedical Admissions Test
#5 Recommended BMAT Book
BMAT Secrets Study Guide: BMAT Exam Review for the BioMedical Admissions Test
Yes, NTU Medicine does have interview, in fact it has Eight Interviews.
Many top scorers (4As, perfect score, perfect portfolio) have unfortunately been weeded out at the interview if they are not adept at interviews or verbally expressing themselves. This is very unfortunate for those who have studied so hard, but yet got eliminated at the interview stage, and have to go to Australia to study Medicine (costs half a million SGD!!!) or even abandon their dream of being a doctor.
Fortunately, Medicine Interview is something that you can prepare for. Do be prepared for an answer to the question “Why do you want to study Medicine?”. The interviewers are looking for compassionate doctors, not money-minded individuals who want to fatten their bank account.
The interview would be a bit difficult for quiet / introverted people, which is a pity, since introverts can be very good doctors too. Interviews tend to favor extroverts, or those who are adept at self-promotion (do be humble though). Hence, if you are more of an introvert, you would need to work doubly hard to prepare for the interview.
#1 Medicine Interview Book
Medicine Interview questions and answers with full explanations: The comprehensive guide to the medicine interview for 2013-2014 applicants
#2 Medicine Interview Book
Why Medicine?: And 500 Other Questions for the Medical School and Residency Interviews
#3 Medicine Interview Book
The Medical School Interview: Secrets and a System for Success
#4 Medicine Interview Book
The Medical School Interview: Winning Strategies from Admissions Faculty
#5 Medicine Interview Book
The Medical School Interview: From preparation to thank you notes: Empowering advice to help you succeed
Good luck and all the best!
Look through this video to discover the 4 types of Klein Bottles!
Math Geek: From Klein Bottles to Chaos Theory, a Guide to the Nerdiest Math Facts, Theorems, and Equations
If you are fascinated by Klein Bottles, you can check out the website mentioned in the video: http://www.kleinbottle.com/
The website sells Klein Bottles under the name Acme Klein Bottles, made by Cliff Stohl.
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Just to share a recommendation for BESTA English Chinese Electronic Dictionary Translator, sold online at Qoo10.
BESTA
ITS AN ENCYCLOPAEDIA, OXFORD DICTIONARY,
SPELLCHECKER & THESAURUS IN ONE!
BESTA CD-580+ English-Chinese Electronic Dictionary have Back Lighting
(Note:2 AAA batteries are not included)
(Voice: Mandarin/English)
Chinese and English Operation Menu
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In the last two posts, I discussed the idea of status. First, I talked about why status matters, then I talked about how teachers can see it in the classroom.
Sometimes, after I have explained how status plays out in the classroom, somebody will push back by saying, “Yeah, but status is going to happen. Some kids are just smarter than others.”
I am not naive: I do not believe that everybody is the same or has the same abilities. I do not even think this would be desirable. However, I do think that too many kids have gifts that are not recognized or valued in school — especially in mathematics class.
Let me elaborate. In schools, the most valued kind of mathematical competence is typically quick and accurate calculation. There is nothing wrong with being a fast and accurate calculator: a facility with numbers and algorithms no…
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I have added more Math Olympiad books suitable for students training for GEP Math / DSA Math.
These are books actually bought by a viewer of my website through my Amazon affiliate link.
Just to share, and hope it is helpful!
The first book is written by Professor Derek Holton. Prof Holton writes a nice column for a Math magazine, which gives out books as prizes to correct solutions. I tried some of the problems here: Maths Olympiad Magazine Problems.
If you are searching for GEP Math Olympiad Books to prepare for the GEP Selection Test, you may search for Math Olympiad Books for Elementary School. Note that Math Olympiad Books for IMO (International Mathematics Olympiad) are too difficult even for a gifted 9 year old kid!
A suitable book would be The Original Collection of Math Contest Problems: Elementary and Middle School Math Contest problems
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These top 10 tough Maths: also the top 10 potential to win Fields Medal or Nobel Prize, or another billionaire business like Google, RSA Encription, …
Weil’s Rosetta stone (or Conjecture):
Number Theory (1) | Curves over Finite Fields (2) | Riemann Surfaces (3)
Weil wanted to link up these 3 distinct Maths, as in the Langands Program.
Langrands’ original idea on the Left Column (1) Number Theory & the Middle Column (2):
1. He related :
representations of the Galois groups of number fields (objects studied in number theory)
to:
automorphic functions (objects in harmonic analysis).
2. The middle column (2):
Galois group relevant to curves over finite fields.
Also there exists a branch of harmonic analysis for automorphic functions.
3. How to translate column (3) Riemann Surfaces ?
We have to find geometric analogues of the Galois groups and automorphic functions in the theory of Riemann surfaces.
Next we have to find suitable analogues of the automorphic functions ?
It was a mystery until 1980 solved by the Russian Vladimir Drinfeld (Fields medalist for…
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Langlands’ Program:
Langland wrote to André Weil in 1967:
Analysis & Algebra linked up by L-function, which converts algebraic data from Galois theory into Analytic functions in complex numbers.
He goes beyong Modular Form to the Automorphic Forms (complex functions whose symmetries are described by larger matrices).
Key concepts on Algebra marry up with those from Analysis.
Just to share with my Singaporean readers on 5 ways to save money while shopping!
This is a very good bargain, Qoo10 is selling the Universal Studios Singapore Ticket at half the retail price. Normal price is $74. If you haven’t visited USS yet, this June holidays is a good time.
WWW.QOO10.SG
Everyone knows that the official Apple cable is very expensive, overpriced in fact. Most people just need a basic iPhone cable that can charge and connect to computer. I personally bought this cable, at $3.70 it is extreme value for money. So far so good, it charges and transfer data well. I don’t think you can find another place with such low price for a cable.
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At $99, it is one of the cheapest electronic items out there (as compared to iPhone, iPad, etc.) Also, Kindle is ideal as a gift to students as it is very education oriented, and has less games than the iPad (a upside for children).
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This notebook (from HP) would be a good choice if you need a basic laptop for school work. There are other brands (Asus, Lenovo) at similar prices at Qoo10. Very few physical shops have computers at this price.
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Many households (including myself, until recently) are still using Cat5 / Cat5e old cables which would impact your internet speed. No point subscribing to the best internet plan, and have it all slowed down by an outdated Cat5 cable. This Cat6 Ethernet Cable may be what you need to boost your internet connection.
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In layman’s terms, “measures” are functions that are intended to represent ideas of length, area, mass, etc. The inputs for the measure functions would be sets, and the output would be a real value, possibly including infinity.
It would be desirable to attach the value 0 to the empty set 
Definition (from Bartle): A measure is an extended real-valued function 
(i)
(ii) 
(iii) 
If a measure does not take on 
(a) Let X be any nonempty set and let X be the 
Let 
The most famous measure is definitely the Lebesgue measure. If X=R, and X=B, the Borel algebra, then (shown in Bartle’s Chapter 9) there exists a unique measure 
Today’s world is Big Data, with explosive unstructured data from Internet, Mobile phones, tablets, soon the IoT (Internet of Things), ie devices such as car, fridge, oven, washing machines…equipped with wireless Wi-Fi connectivity to Internet…
Top Mathematicians will be the global elites of the D.T. (Data Technology) Age — as the Alibaba.com Chairman Jack Ma predicts.
http://www.bloomberg.com/bw/stories/2006-01-22/math-will-rock-your-world
Should 1 be a prime number ?
Explain ‘1’ in Abstract Algebra ‘Ideal’ (理想) in ‘Ring'(环) Theory:
CN Yang (杨振宁) Yang-Mills Equation in Quantum Physics and ‘Fibre Bundle’ in Differential Geometryby SS Chern (陈省身) are equivalent, however both men worked independently for 30 years!
1957, 35岁第一个华人诺贝尔奖。
2004, 82岁娶28岁。
2015, 93岁牵39岁的夫妻。
翁帆(28)从敬仰伟人而爱杨振宁(82)。
三大科学贡献:
1. 宇恒不对称 : nuclei weakforce (弱力)不对称
2. Yang-Baxter Law (Proved by Quantum Group Mathematics)
3. Yang-Mills Conjecture (Clay Prize US $1 million, 7 Millennium Unsolved Math Problems)
如果3)被证明, 可能杨振宁得第二个诺贝尔奖。
父亲杨武之是”庚子赔款”留美的中国第一位数学博士(Chicago University), 把现代代数(Modern Algebra)介绍进中国。儿子杨振宁15岁就教儿子群论(Group Theory) — 后来证明”宇恒不对称” — 用的就是他美国恩师 Dickson的名著。
1962 two American economists David Gale & Lloyd Shipley designed The “Stable Marriage Problem” aka “The Match“.
Note: ‘Stable’ means nobody would be unhappy or breakup after the match.
Applications:
1. Matching couples
2. Matching hospitals & doctor graduates
3. Match schools to students
4. Match HDB house to families
5. …
Scenario: An island with 4 men (m1, m2, m3, m4) and 4 women (w1, w2, w3, w4). You are to match 4 couples of opposite sex.
Each man would propose to a woman. However both men and women could list down their preferences with ranking, the higher ranked person would be given the choice.
Suppose the women preferences are (Table 1):
| Choices | 1st | 2nd | 3rd | 4th |
| w1 | m1 | m2 | m3 | m4 |
| w2 | m2 | m4 | m1 | m3 |
| w3 | m3 | m4 | m1 | m2 |
| w4 | m4 | m3 | m2 | m1 |
Suppose the men preferences are (Table 2):
| Choices |
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For this post we will explain what is a homotopy of paths.
Source: Topology (2nd Economy Edition)
The book above is a nice introductory book on Topology, which includes a section of introductory Algebraic Topology.
Definition: If f and f’ are continuous maps of the space X into the space Y, we say that f is homotopic to f’ if there is a continuous F: X x I -> Y such that
F(x, 0)=f(x) and F(x,1) = f'(x)
for each x. The map F is called a homotopy between f and f’. If f is homotopic to f’, we write 
If f and f’ are two paths in X, there is a stronger relation, called path homotopy, which requires that the end points of the path remain fixed during the deformation. We write 
Next, we will prove that the relations 
If f is a path, we shall denote its path-homotopy equivalence class by [f].
Proof: We shall verify the properties of an equivalence relation, namely reflexivity, symmetry and transitivity.
Reflexivity:
Given f, it is rather easy to see that 
F(x,0)=f(x) and F(x,1)=f(x) is clearly satisfied.
If f is a path, then F is certainly a path homotopy, since f and f itself has the same initial point and final point.
Symmetry:
Next we shall show that given 
G(x,0) = F(x, 1)=f’ (x)
G(x,1) = F(x, 0) = f(x)
Furthermore, if F is a path homotopy, so is G.
G(0,t)=F(0, 1-t) =
G(1,t)=F(1,1-t) =
Transitivity:
Next, suppose that 
First, we need to check if the map G is well defined at t=1/2. When t=1/2, we have F(x,2t) = F(x,1)=f'(x) = F'(x,2t-1).
Because G is continuous on the two closed subsets X x [0, 1/2] and X x [1/2, 1] of XxI, it is continuous on all of X x I, by the pasting lemma.
Thus, we may see that G is the required homotopy between f and f”.
G(x,0)=F(x,0) = f(x)
G(x,1) = F’ (x, 1) = f”(x)
We can also check that if F and F’ are path homotopies, so is G.
G(0,t) = F(0, 2t) =
G(1, t)=F'(1, 2t-1) =
http://en.m.wikipedia.org/wiki/Augustin-Louis_Cauchy
We mentioned Augustin Louis Cauchy in the tragic stories of Galois and Abel. Had Cauchy been more generous and kind enough to submit the two young mathematicians’ papers to the French Academy of Sciences, their fates would have been different and they would not have died so young.
Cauchy was excellent in language. He was the 2nd most prolific writer (of Math papers) after Euler in history. When he was a math prodigy, his neighbor — the great French mathematician and scientist Pierre-Simon Laplace — advised Cauchy’s father to focus the boy on language before touching mathematics. (Teachers / Parents take note of the importance of language in Math education.)
Cauchy’s language education made him very rigorous in micro-details. This was the man who developed the most rigorous epsilon-delta Advanced Calculus (called Analysis) after Newton / Lebniz had invented the non-rigorous Calculus (why?).
Rigorous epsilon-delta…
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Big Data: A Revolution That Will Transform How We Live, Work, and Think
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. (Wikpedia)
Big data is a broad term for data sets so large or complex that traditional data processing applications are inadequate. Challenges include analysis, capture, data curation, search, sharing, storage, transfer, visualization, and information privacy. The term often refers simply to the use of predictive analytics or other certain advanced methods to extract value from data, and seldom to a particular size of data set. Accuracy in big data may lead to more confident decision making. And better decisions can mean greater operational efficiency, cost reductions and reduced risk. (Wikipedia)
Big Data is said to be the next biggest scientific advance since the internet. Algebraic Topology is one branch of Mathematics that is directly related to Big Data.
Topological data analysis (TDA) is a new area of study aimed at having applications in areas such as data mining and computer vision. The main problems are:
The human brain can easily extract global structure from representations in a strictly lower dimension, e.g. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images, e.g. dot-matrix printers and televisions communicate images via arrays of discrete points.
The main method used by topological data analysis is:
Source: Wikipedia
Very interesting!
In the previous chapters, Bartle showed that that if f is in M(X,X), then the functions 
The case of the measurability of the product fg when f, g belong to M(X,X) is a little bit more tricky. If 
Let 
Proof:
Each 
All of the above sets are in X.
Thus, we may use an earlier Lemma 2.6 to show that the product 
We also have 
Finally, (fg)(x)=f(x)g(x)=
This is a very powerful result of Lebesgue integration, since we can see that the theory includes extended real-valued functions, and prepares us to integrate functions that can reach infinite values!
This is a continuation of the study of the book The Elements of Integration and Lebesgue Measure
Example:
If X is the set R of real numbers, and X is the Borel algebra B, then any monotone function is Borel measurable.
Proof:
Suppose that f is monotone increasing, i.e. 
Then, 
Both cases can indeed occur. For example, if f(x)=x, then the set will be of the form 
Lemma: An extended real-valued function f is measurable if and only if the sets 
This lemma is often useful when dealing with extended real-valued functions.
Proof: If f is in M(X,X), it is proven earlier in the book by Bartle that A and B belong to X. Let 
If 
Hence,
Conversely, if 
Very sad news…. Rest in peace, Professor John Nash.
Source: https://sg.news.yahoo.com/beautiful-mind-mathematician-john-nash-killed-us-police-143603056.html
Nobel Prize-winning US mathematician John Nash, who inspired the film “A Beautiful Mind,” was killed with his wife in a New Jersey car crash.
Nash, 86, and his 82-year-old wife Alicia were riding in a taxi on Saturday when the accident took place, State Police Sergeant Gregory Williams told AFP.
“The taxi passengers were ejected,” Williams said, adding that they were both killed.
The Princeton University and Massachusetts Institute of Technology (MIT) mathematician is best known for his contribution to game theory — the study of decision-making — which won him the Nobel economics prize in 1994.
His life story formed the basis of the Oscar-winning 2001 film “A Beautiful Mind” in which actor Russell Crowe played the genius, who struggled with mental illness.
“Stunned… my heart goes out to John & Alicia & family. An amazing partnership. Beautiful minds, beautiful hearts,” Crowe said on Twitter.
A Beautiful Mind
Synopsis: “HOW COULD YOU, A MATHEMATICIAN, BELIEVE THAT EXTRATERRESTRIALS WERE SENDING YOU MESSAGES?” the visitor from Harvard asked the West Virginian with the movie-star looks and Olympian manner. “Because the ideas I had about supernatural beings came to me the same way my mathematical ideas did,” came the answer. “So I took them seriously.”
Thus begins the true story of John Nash, the mathematical genius who was a legend by age thirty when he slipped into madness, and who—thanks to the selflessness of a beautiful woman and the loyalty of the mathematics community—emerged after decades of ghostlike existence to win a Nobel Prize for triggering the game theory revolution. The inspiration for an Academy Award–winning movie, Sylvia Nasar’s now-classic biography is a drama about the mystery of the human mind, triumph over adversity, and the healing power of love.
I have added a new addition to the Recommended Books for Undergraduate Math, which is one of my most popular posts!
The new book is The Elements of Integration and Lebesgue Measure
I am personally reading this book as well, as I didn’t manage to study it in university, but it is a key component for graduate level analysis. Students interested in advanced Probability (see this post on Coursera Probability course) would be needing Lebesgue theory too!
Do you wish there is a method to improve your grades? How do you improve your grades after failing a Common Test for Secondary School or JC?
The Four Quadrant Method is an ideal method for students (especially higher level students like O Level or A Level students) to plan their study schedule and revision time table.
Many students do ok in primary school, but start to falter and fail in secondary school or JC. This may be due to many factors, some of which can be remedied using effective time management.
According to this model, which comes from the book First Things First
Quadrant 1) Important and Urgent (crises, deadline-driven projects)
Quadrant 2) Important, Not Urgent (preparation, prevention, planning, relationships)
Quadrant 3) Urgent, Not Important (interruptions, many pressing matters)
Quadrant 4) Not Urgent, Not Important (trivia, time wasters)
The key to doing well in school and exams is actually Quadrant 2! It is highly related to human psychology. Most people would think Quadrant 1 is more important, but actually Quadrant 2 is the most important type of activity for students.
Quadrant 1 activities (in the Singapore context) are activities like assignment due next day, test next day, exam the next day, and so on. They are important and also urgent. The thing is, these things are usually done by most people since there is a time pressure factor to it. Most students will actually do and complete Quadrant 1 activities. However, as you would know by now, just doing the homework the teacher assigns is not enough to do well for the test / exam under the Singapore syllabus. Firstly, the work that the teacher assigns may be basic material, while in Singapore, the school tests and exams all contain advanced and challenging material.
Quadrant 2 activities are long-ranged planning and strategies, like preparing for a test that is 3 months later, preparing for the Promo Exam that is half a year later. Since these activities are not urgent, most people skip them altogether. However, it is highly important to do Quadrant 2 activities everyday. Stephen R. Covey is a genius for discovering that Quadrant 2 is the secret to time management. Students should set aside some time everyday to do long-ranged preparation, e.g. preparing for a test that is a few months into the future.
Quadrant 3 activities are things that are urgent but not important. Examples are checking Email, checking Whatsapp for class group notifications. Yes, checking email and Whatsapp is compulsory nowadays, but it is not considered an important activity in the grand scheme of things. One should set a minimum amount of them for these activities. CCA may also be classified under this category. This Quadrant is highly deceptive, and a huge time sink, but in the end the activities in Quadrant 3 rank very low in importance.
Quadrant 4 activities are things that are not urgent and not important. Examples are checking Facebook, playing computer games, and so on. These activities should be kept to a bare minimum, and only during scheduled breaks for destressing.
The Four Quadrant technique can be coupled with the Pomodoro Technique which is another good technique for time management.
Hope it helps! This method is for parents to teach their child about Time Management, provided their child is motivated and wishes to improve. For children that are not motivated to study / not interested in learning, parents should check out these Motivational books to motivate students instead.
Startutor is Singapore’s most popular online agency, providing tutors to your home. There are no extra costs for making a request. The tutors’ certificates are carefully checked by Startutor.
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There are many excellent tutors from RI, Hwa Chong, etc. at Startutor, teaching various subjects at all levels.
High calibre scholars from NUS/overseas universities are also tutoring at Startutor.
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现在的家长都很注重孩子的学习，但是有时候孩子不专注，或者对学习没兴趣怎么办？
俗话说：你可以把马牵到水边，但你无法强迫它饮水（意指有的事情必需本人自愿，强迫无济于事）；老牛不喝水，不能强按头。
可见，强逼孩子读书是没有用的，反而会造成孩子厌倦学习。最重要是培养孩子读书的兴趣，这样往往事半功倍，孩子的学习成绩突飞猛进。
在此，让我介绍一些帮忙孩子读书厉害的书：
现在很多孩子都有多动症，就算没有多动症也很难静下来读书，这是21世纪普遍的问题。因为现在太多引诱，比如电脑，手机，电视。毋庸置疑，专注的孩子更聪明，学习也肯定比较好。《我的第一本专注力训练书 》为《看到找不到》系列之精彩合集。精选本系列中最经典和最受欢迎的形象页面，按综合难度由易到难编排，增加了专注能量级的划分和“目标锁定”等小细节， 提升孩子寻找之后的成就感，逐步提升专注力、记忆力、观察力三大能力。《我的第一本专注力训练书 》足足128页，让孩子一次玩得过瘾、找得开心。
2） 学会提问(原书第10版)
学会提问是一门很大的学问。批判性思维领域“圣经”之作！权威大师30年畅销不衰的经典！史上最有内涵的思维训练书！亚马逊思维科学领域no.1！俞敏洪高度推荐 美国大学生人手一本！打开心智，提早具备未来创新人才的核心竞争力！
3) 棚车少年(套装共8册)(中英双语)（当孩子遇到挫折，这本书能让他们笑着面对人生）
天下没有100%顺利的事，孩子总有一天会遇到挫折。遇到挫折该怎么办，怎么面对？读了这本书能让孩子笑着面对人生，不如买给孩子看看。这本书是中英双语，还能帮助孩子练习语文。《棚车少年》：亨利、杰西、维莉、班尼四兄妹从小就是孤儿，他们知道自己有一个爷爷在绿野镇，但是他们不喜欢他。为了躲避爷爷，孩子们在一个破旧的棚车里 安了家，开始相依为命的生活。他们积极向上，阳光开朗，不惧生活的挫折。不仅如此，他们还相互帮助，一起寻找生活的乐趣，在树林里安家、收留小狗望望、探 宝、自由赛……最后爷爷找到了他们，原来爷爷很年轻、慈祥、很爱他们。最后他们跟爷爷一起回家，过上幸福快乐的日子。
![G(x,t) = \begin{cases} F(x,2t) &\text{for }t\in [0,\frac{1}{2}],\\ F'(x, 2t-1) &\text{for } t\in [\frac{1}{2}, 1].\end{cases}](https://s0.wp.com/latex.php?latex=G%28x%2Ct%29+%3D+%5Cbegin%7Bcases%7D+F%28x%2C2t%29+%26%5Ctext%7Bfor+%7Dt%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+F%27%28x%2C+2t-1%29+%26%5Ctext%7Bfor+%7D+t%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D.%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
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