Author: tomcircle

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…

Why is the second derivative written as such:

$latex displaystyle
frac{d^2 y}{d x^2}?
$

http://qr.ae/7OFpN8

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).

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…

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.

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

$latex googol = 10^{100}$

Is a (googol + 1) prime?

http://qr.ae/7rppKA

Should 1 be a prime number ?

http://qr.ae/7rpswt

Explain ‘1’ in Abstract Algebra ‘Ideal’ (理想) in ‘Ring'(环) Theory:

http://qr.ae/7royrJ

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

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…

Pandigital numbers

Movie (Maths Problem Scene):

Homeomorphically Irreducible Trees with 10 dots.

Who was the real Good Will Hunting?

5岁儿童背 Pi 600小数位:

There are 158,962,555,217,826,360,000 possibilities of codes in the German Enigma Machine:

Flaw cracked by the genius Mathematician Alan Turing (Father of Artificial Intelligence) :

“A key can never be itself” — this is ‘the straw that breaks the camel’s back’, a critical clue to break the 158,962,555,217,826,360,000 possible codes !

These 3 jobs all require Math: Actuary, Statistician and Mathematician.

Others good to have Math : Software Engineer, Systems Analyst, Data Scientist.

Answer (scroll down)

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费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来现代数学突飞猛进。

FLT 数学长征英雄人物:

1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

集大成:
8. Andrew Wiles (怀尔斯) (证明@1993 -1995)

1.Elliptic Curve (椭圆曲线)
2. Modular Form (模形式)
3. Fermat Last Theorem (费马大定理)

(1) = (2) = (3)

Music and math: The genius of Beethoven – Natalya St. Clair:

Music = Math

Strongly recommended free excellent MIT Math for high school, undergrads/grads and any self-study learners.

Thanks Prof. Gilbert Strang for the unselfish sharing.

http://ocw.mit.edu/faculty/gilbert-strang/

I find extremely pleasure when I discovered his brilliant lecture notes in “Generating Function” – a Discrete Math technique for computing sequencing using function, and the application in complex Combinatorics. Download here:

Example:

丘成桐 (ST Yao 1949~) Fields Medal @1982 [33岁] proved Calabi Conjecture

1. 读私立 培正中学, 高中遇 好数学老师. @香港中文大学, 发觉 Math Beauty, ‘豁然开朗’.
2. Best Math student not necessary Mathematician, only sufficient!
3. 一名数学科学家 都应对 文学,哲学 这类 学科有基本的涉猎. 好的数学 使你体验到庄子讲的
“天地与我并生, 万物与我为一” 的境界
4. 成功 = 要有数学热情.
Strategy:
a. 深入思考
b. 在心中或纸上仔细研究
c. Find clues from book, till get answer.
d. 出题目给自己

“庖丁解牛”数学方法
庄子讲庖丁(butcher)解牛有三个功夫階段：
1st Level: 看见一只全牛 (Whole Cow)

2nd Level: 三年后，不见全牛，只见牛的生理结構(Anatomy) ：骨骼，肌肉，筋腱。

3rd Level: 不以目视而是神视，＂与桑林之舞合拍，与经首之会同律。＂达到了”物我”两忘的境界。Intuition.

数学的方法也如此。

1st Level: Whole Math (Primary school to High School)

2nd Level: Component Structure – (Undergraduate Math)：
Macro- structure (Algebra : Group, Ring, Field, Vector Space… );
Micro-structure (Analysis : Calculus, Topology, etc)

3rd Level: 无处不数 Ubiquitous Math – (Graduate Math)
eg. Fermat’s Last Theorem used all Math theories available today to prove.

Mathematics is roughly divided into 2 categories:

‘Macro’ Math: Algebra

‘Micro’ Math: Analysis (or the outdated name Calculus)

Algebra has been transformed rapidly from 19th century after Galois’s invention of Group Theory, and expanded by David Hilbert and his students E. Noether, Artin, etc in Axiomatic Algebra, takes a very macro view of Mathematical structures in abstract thinking.

Analysis, also after 19th century Cauchy and Wierestrass’s invention of ‘epsilon-delta’ micro view of Calculus, transformed the Newton Calculus into rigourous Math.

The old school of division of Pure and Applied Math is no longer valid. Take for example, the Applied Math used in Google Search Algorithm uses abstract Vector Space of Matrices in Linear Algebra (Pure Math).

GMC – Global Math Challenge

https://www.global-math.com/mobile

Prof CN Yang: Life’s going to get more interesting (and difficult)

結城 浩(1963 -) is a Japanese Math Popular Book Writer for Secondary and High School students. In the “Galois Theory” (Chapter 10) she boldly attempted to explain to them such complicated concepts: Quotient Group, Field Extension, Group Order, Normal Group, Solvable Group …

http://m.ruten.com.tw/goods/show.php?g=21437146332387

http://www.nh.com.tw/nh_bookView.jsp?cat_c=01&stk_c=9789866097010

7+2+5 = ?
(Test the logic pattern)

Ans: (Scroll below — only after you have tried to solve it )
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7+(2)+5= 143547
7×2=14
7×5=35
14 + 35 – (2)=47

No equation, no formula, no calculator, no writing on paper, just do mental calculation.

What is the answer  in the last diagram ?

Scroll down to see the answer, but not until you have tried yourself.
(There are many methods, but one is the fastest and the best !)
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Ans:
Add the first 3 diagrams: 10+20+24 = 54 for 2x (Dog + Cat + Rabbit).

Dog + Cat + Rabbit = 54/2 = 27

http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty

Amazing calculation technique of Indian kids:

It is the advanced level of abacus calculator, where the physical abacus is replaced by a virtual abacus in the mind — see the kids need to move their fingers in the air, as if they touch a real abacus. This is different from the mental calculation sans a real electronic calculator, in which case no fingers playing the pressing buttons of an electronic calculator.

How nice it would be if yesterday were tomorrow, then today would be Friday.

What is ‘today’ here ?

Answer:
A. Wednesday;
B. Thursday;
C. Friday;
D. Sunday

Hints:
either use left brain (by logic) or
right brain (by EQ, after a Sunday evening).

Try before scrolling below for answers:
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Answer (1) : by logic (algebra)
Today = Wednesday

Let today = d
If d-1 = d+1 (yesterday = tomorrow)
Then d = d+2 (today = the day after tomorrow)
and d+2= 5 (today =…

Are crows the ultimate problem solvers?

Scientists have also discovered crows can count up to 3. Weren’t human primitives so too ? In Chinese, English, French, etc, anything upto 3 is equivalent to many.

Crows are ugly black birds, found commonly in the hawker centers here scrounging left over food on the tables or rubbish dumps. They are good children with strong filial piety, bringing food back to feed old parents. Below is a Tang dynasty poem by Bai Juyi白居易 (800AD) praising the crows:

Compare below the Singapore Math versus Algebra Method:

Mrs. Lim 68 (son 36, daughter 32)

Algebra Method: Try yourself !
(It involves solving 2 equations of 2 variables)

Assumptions:
1. Class A (< 1600 cc) Old car (10 years expiring soon) is in good condition:
♢ Less than 200,000 KM
♢ Certified safe by qualified workshop

2. COE last-3-month average price ~$ 60,000 (10 years)
=> $30,000 (5 years)

3. New Car market price of $100,000

4. Parf rebate $7,000 (forfeited if extend car beyond 10 years)

5. Monthly Car Loan (@2.88% for max. bank loan 60% & max 5 years = $60k) = $1,144 (= interest + capital)
5-year car loan = $1,144 x12x5 = $ 68,640
Downpayment Cash = $40,000 (or $20,000 for 5-year Cost of Ownership )

Compare Buy or Extension (Both 5 years)

Description	Buy	Extension
1. Car Price (down payment) for 5 years	$20,000	$…
2.COE (5-Yrs)	$…	$ 30,000
3. Car Loan (5 Yrs) = car price + COE	$ 68,640	$…
4. Parf rebate	$7,000	– $ 7,000
Cost of Ownership…

‘9’ is the biggest number in the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

9 is reserved for the ancient Chinese Emperors whose robes were sewn with 9 dragons, and the palace roof top decorated with 9 dragons too.

http://mp.weixin.qq.com/s?__biz=MzA3MjE0ODAzMw==&mid=204651538&idx=1&sn=33027a35921519fe42c647e1f5a4dadc&scene=1&from=groupmessage&isappinstalled=0#rd

下面有6個考題，就知道你有多聰明。
試一試，古代希臘的智商測試，你敢挑戰嗎？

http://china-dafencun.com/wx/xlcs/?from=timeline&isappinstalled=0

請遵守遊戲規則, 才算准确。

Most of my friends got 135 scores (super genius) – so are you !

This is Singapore PSLE (Primary School Leaving Exams) Maths for 12- year-old kids.

It is not difficult for adults, but I wonder most kids have such analytical skill ?

Answers below (scroll far behind….)
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(1)

(2):
R – B = 1 for every suite of 5 cards
If R-B = 36
36×5= 180 cards

Interesting thing about saturday in 2015
4/4/2015 saturday
6/6/2015 saturday
8/8/2015 saturday
10/10/2015 saturday
12/12/2015 saturday

Calendar of 2015 is same as calendar of 1997

Date & Day even Festivals are same,

Who told history doesn’t repeat ?

So

We are back in 90s

Have a Blessed Happy New Year!

Calendar Math explains:

http://tomcircle.wordpress.com/2013/03/30/calendar-math/

女童记不住乘法表崩溃大哭”三五太難了”:

This 4-year-old Chinese girl cried when her mother ‘tortured’ her to recite Multiplication table by rote learning. She always got stuck at 3 x 5= ?
She complained it is too difficult.

Most parents are teaching the kids Mathematics the wrong way! No wonder they grow up with hate and fear for Maths subject throughout the entire life.

Same in schools and universities, Maths are taught the wrong ways by incompetent Math educators.

Mind, Brain And Education
1. Spaced Repetition
2. Retrieval Practice

Tool: Test
Not to assess what students know, but to reinforce it.
Memory is like a storage tank, a test as a kind of dipstick that measure how much information we’ve put in there.

But that’s not how the brain works.

Every time we pull up a memory, we make it stronger and more lasting, so that testing doesn’t just measure, it changes learning.

Simply reading over materials to be learnt, or even taking notes and making outlines, as many homework assignments require, doesn’t have this effect.

Language learner: 80% retained.
Science: 50% retained.

Self-quizzing (focus less on input of knowledge by passive reading, focus more on output by calling out that same information from brain.)

Cognitive disfluency:
Tough topic, recall better.
Interleaved assignment: mix up different kinds of problems instead of grouping by type.