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

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

Source: http://www.takepart.com/photos/ten-surprising-facts-finlands-education-system-americans-should-not-ignore/finland-knows-whats-best

Just to summarize the 10 Surprising Facts:

1. School Starts Later
2. More Recess
3. No Testing (and little homework)
4. Extra Teachers for Struggling Students
5. More Languages
6. Instruction Guidelines, Not Prescriptions
7. Less Teaching
8. Professional Teacher
9. Teachers Stick with Students
10. 43 Percent Attend Vocational School

Read more at: http://www.takepart.com/photos/ten-surprising-facts-finlands-education-system-americans-should-not-ignore/more-languages

Featured book:
The Temperament God Gave Your Kids: Motivate, Discipline, and Love Your Children

This book is highly recommended for those who are interested in the 4 Temperaments and related research (Introvert, Extrovert, Sanguine, Choleric, Melancholic, Phlegmatic, etc.)

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]

Sounds bizarre, but it is true! Something “uniquely Singapore” as 92 marks is usually the top echelon for most other countries.

Students need not be disappointed, as 88 to 92 marks is already a very respectable score. Most important is to understand the subject well, and try to improve.

Source: http://news.asiaone.com/news/education/tuition-no-enough

Despite scoring between 88 and 92 marks for his subjects, he was placed in the third quartile of his cohort.

“Which means he was not among the top 50 per cent,” says his mum, Mrs Jaclyn Chew, 41.

Her son, who attends an all-boys school in north-eastern Singapore, was crestfallen after he asked his parents how he fared.

“I had to tell him the truth about where he stood compared to his peers,” Mrs Chew says.

“It’s just bizarre that with his grades and the tuition, he’s still in the lower half of the grade spectrum.”

– See more…

“It is more blessed to give than to receive.” – Acts 20:35

Source: http://www.bbshare.sg/

Join the Boys’ Brigade in spreading the Christmas cheer to needy families in Singapore.

To most of us, items like rice and biscuits are ordinary and common, but to the needy and elderly in Singapore, they are necessities and much needed items.

There are 3 ways to help:

1. Volunteer to Deliver Food Hampers
2. Donate Essential Items
3. Online Donation

Wishing all our readers a Merry Christmas!

Featured Book:
Bright Minds, Poor Grades: Understanding and Motivating your Underachieving Child

For any parent who has ever been told, “your child isn’t performing up to his or her potential,” this book has the answer. Renowned clinical psychologist Michael Whitley, Ph.D. offers a proven ten-step program to motivate underachieving children. This easy-to follow book identifies the six types of underachievers from the procrastinator to the hidden perfectionist to the con artist, and it…

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:

Recently I was giving a talk at a school in London and had cause to think about sums of kth powers, because I wanted to show people the difference between unenlightening proofs and enlightening ones. (My brief was to try to give them some idea of what proofs are and why they are worth bothering about.) On the train down, I realized that there is a very simple general argument that can be used to work out formulae for sums of kth powers. After a brief look online, I haven’t found precisely this method, though I’ve found plenty of methods in the vicinity of it. It’s bound to be in the literature somewhere, so perhaps somebody can point me towards a reference. But even so, it’s worth a short post.

I’ll start with the case $latex k=1$. I want to phrase a familiar argument in a way that will make…

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