Author: tomcircle

Weekly Homework (hours)

A recent survey in 44 OECD countries reveals for 15-year-old students an average of 5 hours / week of homework.

PISA 2012:

PISA is like the Army IPPT Test on physical fitness. A fit soldier and a weak soldier go to war, whether he can fight with courage under duress to win the battle, has nothing to do with his IPPT scores.

Same for PISA scores…that explains why Americans are poor in PISA but produce many entrepreneurs, Nobel prize / Fields scientists, whereas China, Singapore, Korea, HK have only few.

从来没有状元的老师, 却能培养出状元学生。中国历史的状元, 都是在科举失败的穷秀才教导出来的。看今日的Fields Medalists / Nobel Prize winners, 他们的指导教授多数没拿过这大奖。

法国有两位世界级的数学大师Evariste Galois, Charles Hermite,  相隔15年,同是一位中学数学老师培养。Mr. Richard  是巴黎路易大帝中学(Lycée Louis Le Grand) 的数学老师, 他能创新课材, 加上他本身是”数学迷”, 对学生因材施教, 慧眼视英雄于微时。Galois13 岁前是由妈妈家教(home-schooling), 中一才上学校。Richard看出他有数学天才, 个别教他读当代数学大师的书, 其他科目的老师却视他为功课低劣学生。

结果Galois发现群论 (Group Theory), 开创”抽象代数” (Abstract Algebra)。Galois不被世人接受, 因为他的理论太玄奥 (至今还是数学硕士班的难题), Cauchy, Fourier, Poissons,  Gauss… 这些世界数学泰斗也看不懂! 他的悲剧是法国大革命, 因枪斗而死, 才21岁的生命。可是他对人类的贡献是”Larger than Life”.

Richard 收集Galois的作业, 留给15年后同一班另一学生Charles Hermite, 也是个数学天才。他比Galois的命运好一点, 考进Ecole Polytechnique (X),是排最后一名及格, 好过Galois重考2年都进不了。可是他的脚有问题, X第一年学生是军官训练, 他被X踢出校门。讽刺的是, 很多年后他回校被聘为教授。

Charles Hermite 证明 e是超函数 (Transcendental), 他的学生德国人Lindermann如法炮制, 证明pi也是transcendental. 

Lindemann 从Hermite接过法国数学火种, 回去德国成为一代宗师, 培养了很多大师级的学生 (Félix Klein, Dirichlet, Jacobi, Gauss, …), 20世纪德国数学Gottingen University取代巴黎成为数学王国, 直至二战德国犹太数学家(Émile Noether, Artin, …)逃去美国, 才轮到Princeton University.

一位默默无名的中学/高中数学老师Richard, 百年树人, 改变了世界数学史!

The beauty I see in algebra: Margot Gerritsen at TEDxSTANFORD:

From equations to matrices… to Google search, MRI body scan, …

A friend from China gave us a bag of walnuts plucked from their home-grown walnut tree. I decide to count them by applying math:

A stack of walnuts piled in a pyramid, with base layer 6×6 walnuts, above layers 5×5, 4×4, 3×3, 2x 2, and finally top 1 (1×1).

How many walnuts are there in total ? (Answer: 91)

This is simple math but only taught in A-level (with proof by induction).

Hint: Watch free Khan Academy Math lecture to learn more ….

$latex displaystyle boxed {
sum_{1}^{n} k^2 =frac { n (n+1)(2n+1)} {6}
}$

This is a 400-year-old walnut tree: walnut is called “Wise fruit 聪明果”, it looks like human brain, also has proven nutritious benefits to brain.

Conrad Wolfram provoked the new idea of Computer-Based Math education:

Teach the ‘Why’ of Maths, leave the ‘How’ to the computer.

How: solve quadratic equation, simultaneous equations, differentiation, integration….

He mentioned Singapore is interested in this new approach of teaching Math ? The O & A Level students can now use scientific calculator in Exams.

Stephen Wolfram: Founder & CEO of Mathematica (UK)

Wolfram Alpha: Knowledge-base Computing using public data on the net and private information.

Mathematica: Math tool using Symbolic Functional Language (LISP)

New Kind of Science: Cell Automata

Physics: From Computing World to find new Physical World

Jack Ma of the Alabama.com gives this Arithmetic question to the audience,  only 1% get the answer right !

Jack has cash $50, which he uses to buy:
Clothing : $20 (Balance $50-20 = $30)
Shoes: $15 (Balance = $30-15 = $15)
Candy: $9 (Balance = $15- 9 = $6)
Food: $6 (Balance = $6 – 6=$0)

Question:
Add up the Balances = $ 30+15+6 = $51

Where does the extra $1 come from ?

French Concours was influenced by Chinese Imperial Exams (科举ko-gu in ancient Chinese, today in Hokkien dialect) from 7AD till 1910.  The French  Jesuits working in China during the 16th -18th centuries were the culprits to bring them to France, and Napoleon copied it for the newly established Grande École “École Polytechnique” (a.k.a. X).

The “Bachelier” (or Baccalauréat from Latin-Arabic origin) is the Xiucai (秀才), only with this qualification can  a person teach school kids.

With Licencié (ju-ren 举人) a qualification to teach higher education.

Concours was admired in France as meritocratic and fair social system for poor peasants’ children to climb up the upper social strata -” Just study hard to be the top Concours students”! As the old Chinese saying: “十年寒窗无人问, 一举成名天下知” (Unknown poor student in 10 years, overnight fame in whole China once top in Concours).Today,  even in France, the top Concours student in École Polytechnique…

http://www.quora.com/Is-it-worth-it-to-get-good-at-math-olympiads/answer/Milen-Ivanov-4

韓非子是战国法家, 荀子的高徒, 秦始皇宰相李斯的同学。他说”白马非马”, 即白马不是马, 可以用集合論(Set Theory) 证明:

Let 马 = H = {w, b, r, y …}
w : 白马
b : 黑马
r ：红马
y：黄马

Let 白马 = W = {w}

To prove:
H = W
We must prove:
H ⊂ W and H ⊃ W

From definition we know:
$latex w in H supset W $
$latex H nsubseteq W $
$latex implies H neq W $

白马≠马
白马非马
[QED]

其他例子:
木魚非鱼

Definition: $latex text{Sequence } (a_n) $
has limit a

$latex boxed{forall varepsilon >0, exists N, forall n geq N text { such that } |(a_n) -a| < varepsilon}$

$latex Updownarrow $

$latex displaystyle boxed{ lim_{ntoinfty} (a_n) = a }$

What if we reverse the order of the definition like this:

∃ N such that ∀ε > 0, ∀n ≥ N,
$latex |(a_n) -a| < varepsilon$

This means:

$latex boxed {forall n geq N, (a_n) = a }$

Example:

$latex displaystyle (a_n) = frac{3n^{2} + 2n +1}{n^{2}-n-3}$

$latex displaystyletext{Prove: } (a_n) text { convergent? If so, what is the limit ?}$

Proof:
$latex displaystyle (a_n) = 3 + frac{5n +10}{n^{2}-n-3}$

$latex n to infty, (a_n) to 3$

Let’s prove it.

$latex text {Let } varepsilon >0$
$latex text{Choose N such that } forall n geq N, $
$latex displaystyle |(a_n) -3| = Bigr|frac{5n +10}{n^{2}-n-3}Bigr| < varepsilon$

$latex text{Simplify: }…

Cédric Villani (Médaille Fields 2010) “Théorème Vivant”:

“La fameuse ligne directe, quand vous recevez un coup de fil du dieu de la mathématique, et qu’une voix résonne dans votre tête. C’est très rare, il faut l’avouer!”

“The famous direct line, when you receive a ‘telephone call’ from the God of the Mathematic, and that a voice resonates in your head. It is very rare, one has to admit.”

Theoreme Vivant (French Edition)

屈原 QuYuan (343–278 BCE) Symmetry:
http://en.wikipedia.org/wiki/Qu_Yuan

离騷《天问》
1. “九天之际, 安放安属,
隅隈多有, 谁知其数 ?”
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.

If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).

2. “东西南北, 其修孰多,
南北顺橢, 其衍几何。”
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.

几何 = Geometry

3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?

屈原

Mozi

墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton

“墨子” :  “力, 行之所以奋也。”

行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.

F ∝ a
F = m.a

Difference Between Good & Bad Mathematicians

S.S. Chern (陈省身): “The former gives many concrete examples, the latter has only abstract theories.”

Triangle

1. Not always true!

Sum of 3 internal angles = Π =180 degrees
depends on which geometry:
Euclidean / non-Euclidean / Riemann

2. True always!

Sum of 3 exterior angles = 2 Π = 360 degrees

Kurt Gödel‘s Mathematical Proof of God’s Existence

Axiom 1: (Dichotomy) A property is positive if and only if its negation is negative.

Axiom 2: (Closure) A property is positive if it necessarily contains a positive property.

Theorem 1. A positive is logically consistent (i.e., possibly it has some instance).

Definition. Something is God-like if and only if it possesses all positive properties.

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is (logical, hence) necessary.

Definition. A Property P is the essence of x if and and only if x has P and is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. NE(x): x necessarily exists if it has an essential property.

Axiom 5. Being NE is God-like.

Theorem 3. Necessarily there is…

Pascal Wager:

1. We can choose to believe God exists, or we can choose not to so believe.
2. If we reject God and act accordingly, we risk everlasting agony and torment if He does exist (Type I error in Statistics lingo) but enjoy fleeting earthly delights if He doesn’t exist.
3. If we accept God and act accordingly, we risk little if He doesn’t exist (Type II error) but enjoy endless heavenly bliss if He does exist.
4. It’s in our self-interest to accept God’s existence.
5. Therefore God exists!

Mathematical Proof:
Pascal assumed
Probability of God exists = p
Probability God doesn’t exist = 1-p

I like this analogy:
“Programming without Mathematics is like Sex without Love.” 

Google Search is powerful because of Linear Algebra theory in finding core “EIGENVALUES” in order to manipulate the billion rows X billion columns matrices comprised of PageRanks (another formula invented by 2 Stanford Applied Math Masters degree students who co-founded Google.)

Facebook’s two Harvard undergrads Mark Zuckerberg and roommate Eduardo Savarin (now migrated to Singapore) created the prototype of Campus Facebook to rank Harvard girls with the Elo Formula (applied Normal Distribution Theory, used as standard in Chess and Sport rating). 

Other examples: 
RSA Encryption using Prime number factorization with a public and a private key.

Black-Sholes Formula (won 1997 Nobel Prize in Economics) for Derivatives trading software used by stock traders worldwide. The abuse of this formula was the main culprit of the 2010 Sub-prime global financial crisis.

Number 17th: Black-Sholes is the culprit of 2010 Sub-prime Economic Crisis.

Black-Scholes Equation (1997 Nobel Economics)

Use: Pricing Derivatives (Options): calculate the value of an option before it matures.

1/2 (σS)².∂²V/∂S² + rS.∂V/∂S + (∂V/∂T – rV) =…

Without last 2 terms=> heat equation !

Time T
Price S of the commodity
Price V of the derivative
Risk free interest r (govt bond)
Volatility = σ of the stock = standard deviation

Assumptions: (Arbitrage Pricing Theory)
No transaction costs
No limit on short-selling
Possible to borrow/lend at risk-free rate

Market prices behave like Brownian motion: constant in rate of drift and market volatility

Put option: right to sell at a specific time for an agreed price if you wish.
Call option: right to buy at a specific time for an agreed price if you wish.

One Black-Sholes formula each for Put and Call respectively.

Derivative was invented in 1900 by Mr. Bachelier, a French PhD student of Poincaré, the Mathematics…

It is confusing for students regarding the two forms of the Fermat’s Little Theorem (which is the generalization of the ancient Chinese Remainder Theorem):

General: For any number a

$latex boxed { a^p equiv a mod p, forall a}$

We get,
$latex a^{p} – a equiv 0 mod p$
$latex a. (a^{(p-1)} -1) equiv 0 mod p$
$latex p mid a.(a^{(p-1)} -1)$
If (a, p) co-prime, or g.c.d.(a, p)=1,
then p cannot divide a,
thus
$latex p mid (a^{(p-1)} -1)$
$latex a^{(p-1)} -1 equiv 0 mod p$

Special: g.c.d. (a, p)=1

$latex boxed {a^{(p-1)} equiv 1 mod p, forall a text { co-prime p}}$

Prove : For any positive integers p, k,
$latex (p^k)! text { is divisible by p }$
Proof:
Apply Factorial Formula:
$latex boxed {n!=n.(n-1)! } $

$latex (p^k)! = (p^k). (p^k -1)!
= p.(p^{k-1}).(p^k -1)!
$
hence divisible by p. [QED]

https://mathtuition88.com/2014/11/20/what-is-vixra/

“arXiv” opposite is “viXra”.

The former “arXiv” is administered by Cornell University for Math paper publishing online. The traditional math journals would take 2 years to review and publish. 

The Russian Mathematician G. Perelman was fed up of the long and bureaucratic review process, sent his proof of the 100-year-old unsolved “Poincaré Conjecture” to arXiv site. Later it was recognized to be correct but Perelman refused to accept the Fields Medal and $1 million Clay Prize. 

See http://tomcircle.wordpress.com/2013/03/31/grigory-perelman-arxiv/

The new site “viXra” is open to  anybody in the world while “arXiv” is still restricted to academics. 

This young Singapore mathematician William Wu proved his new found Math Theorem on “viXra” site:

Prove that: if p is prime, for any number k,
$latex boxed {(p – 1)^{p^k} equiv -1 mod {p^k}}$

[By using the Binomial Theorem and Legendre’s Theorem.]

Example: p = 3, k=2, 3^2=9

2^9 = 512…

This “Butterfly Lovers Violin Concerto” (梁山伯与祝英台) composed 50 years ago by 2 Chinese music students, now played so lovely by a Japanese lady violinist. 

Only in the kingdom of Music (the other one is Mathematics) where human political hatred does not exist between countries due to past wars: Japan and China, Germany and the Allied Nations, … Just only yesterday China President Xi and Japan PM Abe both showed awkward “poker face” hand-shake at the APEC Beijing meeting; contrast to the 20th century’s greatest mathematician David Hilbert from Nazi Germany was welcome  in America to chair  the inauguration of the International Conference of Math.

If more students love Math and Music, the world of tomorrow will be more peaceful.

Watch 諏訪內晶子 -《梁祝小提琴協奏曲》   Butterfly Lovers Violin Concerto:

Notes:
1. The legendary love story is the Chinese version of “Romeo & Juliette”:

http://en.m.wikipedia.org/wiki/Butterfly_Lovers’_Violin_Concerto

2. On [13:27mins] the only single…

我现正在上2010全国华乐比赛-大师班。
大师音乐评分标准：
1. 大法（基本功）
2. 小法（特色）
3. 无法（风格）

数学大师Math Masters 也是如此，
1. 大法（基本功 Classical Math）
2. 小法（特色 Math Olympiad techniques）
3. 无法（风格 Abstract Math -> French “Math Composition“）

Answer: (don’t scroll down until you have tried)

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Answer : 2889 = 5

Draw ONE line to divide the following diagram into 2 triangles:

Anything inside x outside still comes back inside

=> Zero x Anything = Zero

=> Even x Anything = Even

Mathematically,

1. nZ is an Ideal, represented by (n)

Eg. Even subring (2Z) x anything big Ring Z = 2Z = Even

2. (football) Field F is ‘sooo BIG’ that

(inside = outside)

=> Field has NO Ideal (except trivial 0 and F)

Why was Ideal invented ? because of ‘failure” of UNIQUE Primes Factorization” for this case (example):

6 = 2 x 3
but also
$latex 6=(1+sqrt{-5})(1-sqrt{-5})$
=> two factorizations !
=> violates the Fundamental Law of Arithmetic which says UNIQUE Prime Factorization

Unique Prime factors exist called Ideal Primes: $latex mbox{gcd = 2} , mbox{ 3}$, $latex (1+sqrt{-5})$, $latex (1-sqrt{-5}) $

Greatest Common Divisor (gcd or H.C.F.):
For n,m in Z
gcd (a,b)= ma+nb
Example: gcd(6,8) = (-1).6+(1).8=2
(m=-1, n=-1)

Dedekind’s Ideals (Ij):
6 =2×3= u.v =I1.I2.I3.I4…

【一分鐘學物理】在雨中應該走還是跑比較好？【木瓜貓翻譯】:

I remembered it was a Pre-U One (JC-1) Physics Quiz:

It is common sense to run as fast as possible – but here is the Physics explanation.

Test p is prime:

$latex (x-1)^p – (x^p -1) =
$
all coefficients are divisible by p

孤岛天才 超强记忆:

http://finance.yahoo.com/news/harvard-tops-us-news-world-150403455.html

These 20 are purely anglophone universities — USA (16), UK(3), Canada (1).

The report is too biased. I am sure there are some non-anglophone universities in Europe, Australia and Asia which are equally good, if not better, than some of those in this list.

Mathematics and sex | Clio Cresswell TEDxSydney:

She had 2/20 in French Math, but loves the “Mathematics & Sex”:

Watch “Mathematics and sex | Clio Cresswell TEDxSydney” on YouTube :

Flatland: 2D World

I came across this review at Amazon in 2007 on how to study Advanced Math on your own. Wonderful advice !

Give yourself 10-15 years, with passion, interest, dedicated commitment, disciplined, you could self-study Math to be a next Fermat, or Hua Luogen – both learned Math by themselves through self-learning from books.

http://www.amazon.com/gp/richpub/syltguides/fullview/R1GE1P236K3YSV/ref=cm_syt_dtpa_f_1_rdssss1/102-4263436-5550568?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=sylt-center&pf_rd_r=00JK8KDA3S1T2JRNBCVC&pf_rd_t=201&pf_rd_p=253457301&pf_rd_i=0821839675

http://www.onlineconversion.com/faq_11.htm

A French math teacher’s insight in math classes in Germany (15 years old = Secondary 3) and France (16/17 years old in Sec 4 & Pre-U / JC 1):

The French Math is more theoretical while the German Math (like English Math) is applied. So the result is Germany produces excellent precision engineers with Applied Math, while France produces 1/3 of the World’s Fields Medalists in theoretical Pure Math.

Many English GCE A-level top Math students from Singapore studying in French Universitues face the same dilemma: while their French Math professors think they are “weak” in Math (i.e. French abstract Pure Math), yet they beat the French classmates in Applied Math.

06.08.06.大家_王元·数学的魅力:

华罗庚的学生, 陈景润的大师兄。

“数学很美,也很神秘。”

他说55岁是数学家的顶峰期, 之后不能再有突破。这”道理”被2013张益唐58岁的数学突破所推翻。

两百万分钟 “中美教育差异”全记录:

解密神算超人

Excellent free course for non mathematicians.

This Philosophical Math course has started half way but past videos are still hosted on the site.

The course is taught by prof Wang of Shanghai Jiaotong Technology University上海交通大学 (SJT), the Alma Mata of former China President Jiang (江泽民), Prime minister Chu (朱镕基), and Prof Qian XueSheng (钱学森) “The Father of Chinese Space and Missile” (China exchanged his country home return with USA FBI for 4 American generals from Korean War prisoners of War) who sent Chinese Taikongnauts (太空人) to space. 

SJT was formed initially as the ‘Classe Préparatoire’ (Bachelor degree, post-High school Prep-college) for graduate engineering to MIT,  while Qing Hua 清华 University was a prep-college for graduate Science/ Math to Harvard, Chicago, Cornell,  etc.

Go to Lesson 3: He explains from a game of Go what is “Space” in maths: Geometrical n-dimensional Space, Linear space, vector space.. why study functional space (in  which…

High pay high stress ?

Not really true … among the top 17 high-paying jobs (yearly 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).

Source:
http://www.businessinsider.sg/high-paying-low-stress-jobs-2014-7/9/#.VAYQrYEZ7qA

ICM2014 VideoSeries PL9: Manjul Bhargava:

(Part 1 of video – difficult part ): His PhD supervisor Benedict Gross (Harvard Math Dean) gave the laudation speech.

(Part 2 of video: Excellent and Understandable presentation) by the Fields medal 2014  receipient: the Canadian/American Indian Prof Manjul Bargarva.

When teaching Quadratic equation in Algebra class using the conventional math pedagogy, this is what you get …

Teaching Math without word, especially for autistic and dyslexic kids, using pictures, video games… look how easy is to explain difficult concepts – even for adults – why (-2)x (-3) = + 6 ?

http://www.transum.org/software/SW/YouTube/Video.asp?Movie=7odhYT8yzUM

This teacher draws a perfect circle free-hand on the blackboard:

http://www.transum.org/software/SW/YouTube/Video.asp?Movie=eAhfZUZiwSE

Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
1. Arithmetic
2. Geometry
3. Trigonometry
4. Algebra
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.

Euler influenced not only in Russia schools, but in schools worldwide today.

Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)

MF87: Logical weakness in modern pure mathematics:

See also Declining Rigor in Modern Pure Math.