3 Math Foundation Skillsets for Engineering / Finance Students


理工/金融科Engineering/Finance 的三个扎实数学基础 :
1) Calculus
2) Linear Algebra
3) Probability / Statistics

Prerequisites for the 3 above in
1) O/A level : Trigonometry, Algebra, Geometry
2) Vector Space (with some basic Algebraic Structures : Group, Ring, Field )
3) O/A level : Permutation/Combinatorics


The Black-Sholes Formula – 诺贝尔经济奖Scholes, Merton与LTCM S

Key Points:

1) From 1950 an unknown PhD Math Thesis by French Dr. Bachelier (a PhD student of the 20CE last Polymath Henri Poincaré, who was not impressed with the ‘gambling’ math, gave only an above-average marks to the thesis) – invention of “Options” Trading to eliminate risk in stock fluctuations.

2) The technique is called “Dynamic Hedging”…

3) …by applying the Japanese mathematician “Ito Math”.

4) Noble Prize Economics awarded to “ Black-Sholes Formula”

5). Sholes & Merton Company “LTCM” – making tons of money until… 1997 Asian Crisis, bailed out by USA government after loss of billions !


李彦宏Baidu CEO : Internet 3 Episodes : PC->Mobile ->AI



李彦宏 Baidu CEO Cambridge Speech 剑桥大学演讲
《3 waves of Internet》:
1) PC- based (1997-)

  • Search Webpages
  • 6-month software update cycle

2) Mobile-based (2010 -)

  • “APP” is born
  • Eco-System : eg. Apple Appstore, Google PlayStore
  • O2O (Online to Offline) : Same day Hotel booking/Restaurant /…
  • SW Update everyday few times

3) AI-based (2017 – now)

  • Voice recognition sans keyboard input
  • Image recognition (eg. Customer ePayment :McDonald’s )
  • Natural language Pattern NLP (Salesman Virtual Assistant)

Mathematicians Top List of Hottest Job Titles Yet Again in 2019

In Asian countries, especially Singapore, where there is no proper career guidance from junior colleges / universities, the students and their parents’ herd mentality of following the most trending high-pay career in the past 6 decades, regardless of their personal interest :

  1. Medical doctors (Post WWII, 1950s) – Healthcare of baby boomers
  2. Lawyers (1960s) – Colonial Services
  3. Accountants (1970s) – Trade Business
  4. Engineers (1980s) – Industrialisation
  5. IT (1990s) – Computerisation
  6. BioScience (2000s) – R&D
  7. Applied Math in AI, Big Data
    (Now) – 4th Industrial Revolution.

Advice to parents & students : 行行出状元, 男怕当错行

Follow Your Heart! Every profession is the BEST if you excel in it with a life-long ultmost passion.


AI with Advanced Math helps in discovering new drugs


Advanced Mathematical Methods with AI is a powerful tool:

  • Algebraic Topology (Persistent Homology)
  • Differential Geometry
  • Graph Theory


Maths is a potential life changer — from wages to dating


When you are stuck with too many choices in life, whether buying houses recommended by agents, finding schools for children, hiring staff among the hundreds of CVs…apply ‘e’ the number from nature (logarithm) to make smart and efficient decision under the constraints of time and resources.

Theory of Optimal Stopping” = 1/e ~ 37%


e = 2.7 18281828 45 90 45…

If you do a house search with 20 properties, then by the Theory of Optimal Stopping, pick the first property which is better than the first 7 properties you see (7. 4 = 20 x 37%).

[Note] Ponder over the hidden Philosophy behind – “Brilliant Limit” :

\displaystyle \boxed{ \lim_{n \to \infty}\left({\frac{n!}{n^n}}\right)^{\frac{1} {n}} = \frac{1} {e}}

Blockchains and Application in Bitcoins

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.


Transparent Salary Formula

The 5-year-old Social Media Startup BUFFER has a transparent salary formula, and all the employees’ payroll are listed online :


It is quite ‘humane’ to consider dependents. I once recruited 2 new fresh graduates – one single, the other a father with a baby. By HR policy, they were both paid at fresh graduate starting salary of $ 2,000 (in 1990). There was no consideration of the extra financial burden the father had to support the baby. With appeal to the CEO for special consideration, he was finally given an extra 10%.


James H. Simons,  the mathematician billionaire who cracked Wallstreet

James H. Simons,  the Jewish mathematician who made $14 billion using Math modelling for Hedge Fund.

[Watch from 31:00 mins to 35 mins].  He told the Nobel Physicist Frank Yang (杨振宁) that the Math “Gauge Theory on Fiber Bundles(纤维丛)” which Yang was developing already existed 30 yrs ago in “Differential Geometry” by SS Chern (陈省身) from Berkeley.

“James H. Simons: Mathematics, Common Sense and Good Luck”

[Video 54:00 mins]
After being billionaire, at old age Simons went back to Math in 2004 to take refuge of sadness of the loss of a son.
He beat the German mathematicians in Differential Co-homology (Topology).

5 Guiding Principles of Success:
1) Don’t run with the pack – be original
2) Choose wonderful partner(s) in research, business…
3) Guided by Beauty
4) Don’t give up !
5) Have good luck.

Jim Simons | TED Talks
“A Rare Interview with the Mathematician Who Cracked Wall Street”

What is a Measure? (Measure Theory)

A person can have ‘Measure’ in Internet’s Big Data sense:
A man’s Self = Measure = ∪ {body, psychic, clothes, house, wife, children, ancestors, friends, reputation, job, car, bank-account, websites visited, online buying pattern, hobbies, interests, lifestyle, …}

You become your algorithmic self: identity and identification shifted to an entirely digital (therefore measurable) plane.

Google, Amazon, etc … collect your quantified “Measure”. Not only they amass large amount of data about you, but also use algorithms to make sense of these data.

Your ‘Measure’ is a big business in the age of Big Data, or ‘DT’ (Data Technology) Age — as coined by Jack Ma of Alibaba.com.


Singapore Maths Tuition

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 $latex emptyset$ and measures should be additive over disjoint sets in X.

Definition (from Bartle): A measure is an extended real-valued function $latex mu$ defined on a $latex sigma$-algebra X of subsets of X such that
(i) $latex mu (emptyset)=0$
(ii) $latex mu (E) geq 0$ for all $latex Ein mathbf{X}$
(iii) $latex mu$ is countably additive in the sense that if $latex (E_n)$ is any disjoint sequence ($latex E_n cap E_m =emptyset text{if }nneq m$) of sets in X, then

$latex displaystyle mu(bigcup_{n=1}^infty E_n )=sum_{n=1}^infty mu (E_n)$.

If a measure does not take on $latex +infty$, we say…

View original post 249 more words

Math Will Rock Your World

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 cars, fridges, ovens, washing machines, iWatches, wearables, Google glasses …equipped with wireless Wi-Fi connectivity to Internet…

Top Mathematicians will be the new global elites in the D.T. (Data Technology) Age — as the Alibaba.com Chairman Jack Ma (马云 “数技时代”) predicts.



Mathematician’s Job: High Pay but Lowest Stress


High pay high stress ?

Not really true … among the top 17 high-paying jobs (annual earning above US $100,000) in the USA, Mathematician’s job has the lowest stress below 60 (in the scale from 0 no stress to highest stress at 100).


Other high-pay-high-stress professional jobs (Annual Salary above US$100,000) :
Orthodontists ($196,270 : 67.0)
Computer Hardware Engineer ($106,930 : 67.0)
Astronomers ($110,440 : 62.0)
Political Scientists ($100,900 : 60.8)
Law Teachers ($122,280 : 62.8)
Actuaries ($107,740 : 63.8)
Physicists ($ 117,040 : 61.3)
Optometrists ($111,640 : 70.3)
IT Managers ($132,570 : 64.3)
Geoscientists ($108,420 : 62.5)
Mathematicians ($103,310 : 57.3)


Math Applied in Today’s Society數學在今日社會的應用–丘成桐教授

Prof ST Yau (Fields Medal, Harvard Math Dean)

OUHK – 數學在今日社會的應用–丘成桐教授 (第一部分):



1. Wavelet Data Compression Algorithm:

2. RSA Encryption

OUHK – 數學在今日社會的應用–丘成桐教授 (第二部分):

OUHK – 數學在今日社會的應用–丘成桐教授 (第三部分):

3. Akamai Network Distribution

OUHK – 數學在今日社會的應用–丘成桐教授 (第四部分):

4. Insurance Risks (Actuary)

OUHK – 數學在今日社會的應用–丘成桐教授 (第五部分):

5. GOOGLE Search:

OUHK – 數學在今日社會的應用–丘成桐教授 (第六部分):

6 不急功近利走捷径

7. 做大数学家成功之道:
– 对数学浓厚的兴趣
– 行则的培养: 不肤浅, 不偷功,不炫耀。
– 打好基本功

See also:

丘成桐谈holistic中学教育, 做大学问的态度…

In Conversation with Steve Ballmer at Oxford (4 Mar 2014)

The 3 valuable takeaways (successful and failure experiences) from this 1- hour interview with Steve Ballmer, who co-founded Microsoft with the richest man in the world Bill Gates:

1. Have a great idea before startup.
– The financial,  talent, etc, will come later naturally, attracted by the great idea.
Look at the 3-year-old Whatsapp, a great text-video-picture free SMS idea which attracts 400 million users and $19 billion acquisition from Facebook.

2. Great company does one (or two) trick (or forté 看家本领) well.

– Microsoft has 2.5 tricks: PC operating system (Windows); Microprocessors in data center (Windows Servers); the half trick is X-Box game console. 

– Apple also has 2 tricks: Mac; Mobile computing in iPhone/iPAD/Appstore

– Facebook (Social Network) and Google (Search Engine)  have 1 trick.

– Sony has 2 tricks : Audio & TV, Sony Playstation PS4 Game Console.

– HP has 1 trick: Tester Equipment (now in Agilent).

– IBM has 2 tricks: Enterprise Data Centre Computing; IT Consulting & Solution Services (acquired the trick from Price-Waterhouse).

– Samsung has 1 trick: hardware manufacturer (LCD flat panel, Smartphone…)

– Amazon has 1 trick: Online bookshop

– Alibaba.com has 1 trick: eCommerce

Many great companies degenerate into smallness or extinction, because they try to diversify into too many non-core activities, wasting financial, human ‘energy’ and precious time resources.

Steve Jobs on his final year gave advice to his competitor Google’s Larry Page: “Focus on one thing and do it well.”

Notes :
1). That explains why Microsoft drains the energy to too many fronts (> 2 tricks). The CEO ‘missed 2 BIG boats’ in Mobile Cloud Computing (Smartphone OS, Tablet) and Search engine (Bing) by playing late catch-up.

2). HP did not treasure the acquired tricks from Compaq iPAQ (the world’s first and top Windows Mobile Smartphone in 2006 before iPhone 3), and Digital Equipment’s (DEC) trick on Vax/VMS Networked Clustering Computing (Bill Gates hired the ex-DEC developers to re-create on Intel servers as the Microsoft Windows Server 200x Operating System for Data Centres).

3) Sony wasted energy in Phone (Sony- Ericsson ) and Notebooks (Viao), ignoring the upcoming threat of TV flat-panel technology from a “小さい” (\chisai) i.e. small, insignificant, or 鼻屎 (\pisai) Korean company Samsung.

3. World Centre of Entrepreneuship:
He said the place must satisfy three criteria –
a. Domestic Market,
b. Top Universities,
c. Global Talents flock there to live and work
Steve cited 3 places:
USA (e.g. Silicon Valley) : America is the world’s biggest market, Stanford / MIT / Harvard Universities, global talents immigrated there, English Speaking.
China : potential (barring political and environmental challenges), Beijing/Tsinghua/Fudan/etc top Universities, 1.3 billion people’s domestic market.
UK: Cambridge/Oxford/etc world class top universities, English speaking, European market.

Why not Japan, Singapore, Israel, Australia, France, Germany, India, Canada…? They all fail in 1 or 2 criteria.

The New Language to Watch : Go

Google’s new language Go is invented to replace Java, hence free from Oracle’s threat of Java license in all Android Smartphones.

Go Language is invented by Ken Thompson, the father of Unix and B language, which was transformed into the famous C language by his Bell Lab’s teammate Dennis Ritchie.

Go impresses me of its concurrency best suitable for parallel programming, given today’s most hardware including smartphones are made of multi-core CPU, yet none of today’s languages is designed inherently to take advantage of it, even Java.

If we can say Cobol is for the 1st generation Mainframe computing, C the 2nd generation Unix-based Minicomputer computing, Java the 3rd generation Client-Server computing, then Go will be the 4th generation Mobile-Cloud computing.

Try the Go simulator here:

Note 1: Do not confuse with another Agent-based language “Go! “, differs by “!” sign.

Note 2: Review of Go multicore concurrency feature:

Programming Languages

Programming language, if you are a mathematician, which one you use to test your math formulas, simulate your conjectures, proving the theorems ?

Ranking high on top is Javascript, most popular among the programmer communities, followed by Java, PHP, Python, Ruby, etc.

From Javascript springs off a family of functional programming language Lisp-like cousins: Clojure and Clojurescript, which are compiled into Javascript to run on the ubiquitous Java Virtual Machine.

In the race the new language Go
developed by Google is a dark horse to watch in the next 24 months.

Lisp is one of the oldest language in the 1960s, created by Mathematicians for mathematics (Mathematica was developed in Lisp by Wolfram) and Artificial Intelligence.



Big Data, Brain Storage

Our brain can store up to 3 terabytes of information, equivalent to 3 sets of this 1-terabyte (TB) hard-disk below.


IBM estimates everyday the world generates 2.5 EXABYTES (EB) of BIG DATA, more than 90% of which was created in the last 2 years via mobile devices (smartphones, tablets).

The next decade will be the “Big Data” Age. Companies and countries which can take advantage of such BIG DATA will rule the market and the world, with accurate forecast of consumer purchasing behavior, investment trends, weather prediction, military espionage, terrorist detection, epidemic prevention, etc.

‘Data-mining’ tools in Analytics applying powerful mathematics will be the tomorrow’s ‘Google Search Engine’.

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