Author: tomcircle

André Weil told the Japanese professor Goro Shimura that Prof GH Hardy talked nonsense, mathematics is not necessary for young men below 35.

Recent math breakthroughs are accomplished by men above 40, because Math needs “Logic as well as Intuition” – both take lengthy research, perseverance and team efforts by other pioneers, as illustrated below:

http://m.scmp.com/lifestyle/technology/article/1256542/zhang-yitang-proof-mathematicians-life-begins-40

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

On April 17, 2013, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Yitang Zhang (Photo: University of New Hampshire)

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang…

Visually cut a cake 1/5 portions of equal size:

1) divide into half:

2) divide 1/5 of the right half:

3) divide half, obtain 1/5 = right of (3)

$latex frac{1}{5}= frac{1}{2} (frac{1}{2}(1- frac{1}{5}))= frac{1}{2} (frac{1}{2} (frac{4}{5}))=frac{1}{2}(frac{2}{5})$

4) By symmetry another 1/5 at (2)=(4)

5) divide left into 3 portions, each 1/5

$latex frac{1}{5}= frac{1}{3}(frac{1}{2}+ frac{1}{2}.frac{1}{5}) = frac{1}{3}.frac{6}{10}$

Verify:2,578,314  x 6,321,737 = 16,299,423,011,418 ?

2,578,314  => (2+5+7+8+3+1+4)) =30 => (3+0) = 3

6,321,737 => (6+3+2+1+7+3+7)=29 => 2+9 = 11 => (1+1) = 2

3 x 2 = 6 

16,299,423,011,418 => (1+6+2+9+9+4+2+3+0+1+1+4+1+8) = 51 => (5+1) = 6

Correct !

The traditional way of cutting cake a la ‘Camembert Cheese’ is wrong.

Terence Tao (1975, Adelaide): Chinese-american, with IQ 230-240, full professor at 24. Fields Medalist.

(French) Groupes de Galois, le cas abélien

Jean Pierre Serre
♢ Youngest Fields Medalist in history at 27.
♢ Wolf Prize in 2000
♢ 1st person to win Abel Prize in 2003.

Listen to Master:

这位石教授的”抽象代数”很棒, 一来是他退休前的最后一课, 二来他总结为何老师教不好, 学生上完课好像听到3个大头”鬼” (群group, 环ring, 域field), 但没实际摸过。

他的第一和第二课很好, 与众不同的花时间讲 “动机”: Motivation – Why study Abstract Algebra ?

抽象代数01: Motivation

https://youtu.be/AGd1TZ-IKr0

抽象代数02: 复数扩域 C
$latex
x^{2} +1=0
$

扩域 (Extended Field)数学思维 = 人解决问题的思维
例: 国内不可行的问题, 跳出国门, 扩大到世界领域, 就找到可行的方法。
马云的Alibaba国内不看好, 跑去美国上市, 让他马上成为中国首富的亿万富翁。

【台湾壹週刊】

速食店员竟然是数学天才

张益唐 (1955 – ) : 北京大学 – 美国数学博士。因为执着数学理论的真理, 得罪美国大学台湾籍论文教授, 毕业后找不到大学教职, 在朋友的 Subway 速食店做会计8年, 潜心业余思考世界数学大难题: Twin Primes Gap, 终于攻破。

他的下一个目标是Riemann Hypothesis, 困扰数学家百年的难题: “素数 (Prime numbers)的分布”都集中在 Zeta function complex plane的 实轴(real = 1/2) 上。大数学家David Hilbert说如果五百年后复活, 第一件事会急着问 “Riemann Hypothesis” 证明了吗?

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…

Simons Foundation

New 2015 Full Documentary #WorldMathsDay” –

Math Mystery Tour (BBC)

Question on @Quora:

In the French Classe Préparatoire 1st year “Mathematiques Supérieures”,  we wanted to test our admired Math Prof whom we think was a “super know-all” mathematician. We asked him the above question. He immediately scolded us in the unique French mathematics rigor:

“L’intégration n’a pas de sense!
Quelle-est la domaine de définition?”

(The integration has no meaning! What is the domain of definition ?)

He was right! Under the British Math education, we lack the rigor of mathematics. We are skillful in applying many tricks to integrate whatever functions, but it is meaningless without specifying the domain (interval) in which the function is defined ! Bear in mind Integration of a function f (curve) is to calculate the Area under the curve f within an interval (or Domain, D). If f is not defined in D, then it is meaningless to integrate f because there won’t be…

99% of my friends get it wrong,  except a 13-year-old boy who can ‘see’ it.

Wrong answer : 25

Answer (below):
Try before you scroll down.

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PSLE is “Primary School Leaving Exams” for 11~12 year-old children sitting at the end of 6-year primary education. The result is used as selection criteria to enter the secondary school of choice.

Hint: Without seeing or feeling the weight of the $1 coin, you still can guess the answer. This is the essence of “Singapore Math” — using “Guesstimation“.

Answer (below):
Try before you scroll down.
If wrong answer, please go back to primary school 🙂
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1. Basic:
|y|= 0 or > 0 for all y

2. Limit: $latex displaystylelim_{xto a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex displaystylelim_{xto a}f(x) = L$
$latex iff $
For all ε >0, there exists δ >0 such that
$latex boxed{0<|x-a|<delta}$
$latex implies |f(x)-L|< epsilon$

3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.

Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
$latex |x-a|<delta$

f(x) is continuous at point x = a
$latex iff $
For all ε >0, there exists δ >0 such that
$latex boxed{|x-a|<delta}$
$latex implies |f(x)-f(a)|< epsilon$

For x->0, find limit L of

f(x)= (x³+5x)/x

1) guess L:

f(x)= x(x²+5)/x= x²+5

=> L= 5  when x->0

2) epsilon-delta Proof: find δ in function of ε such that:

|f(x)-5| < ε

|(x²+5)-5| <ε

|x|< √ε

Choose δ=√ε

For all ε, there is δ=√ε such that |x-0|< δ =>|f(x)-5|< ε

If ε=0.5, δ=√0.5=0.25

Why Newton’s Calculus Not Rigorous?

$latex f(x ) = frac {x(x^2+ 5)} { x}$ …[1]

cancel x (≠0)from upper and below => $latex f(x )=x^2 +5 $

$latex mathop {lim }limits_{x to 0} f(x) =x^2 +5= L=5 $ …[2]

In [1]: we assume x ≠ 0, so cancel upper & lower x
But In [2]: assume x=0 to get L=5
[1] (x ≠ 0) contradicts with [2] (x =…)

This is the weakness of Newtonian Calculus, made rigorous later by Cauchy’s ε-δ ‘Analysis’.

Rigorous Analysis epsilon-delta (ε-δ)
Cauchy gave epsilon-delta the rigor to Analysis, Weierstrass ‘arithmatized‘ it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in “Analyse Algébrique” (1821)

2) Cauchy repeatedly used ‘Limit’ in the book Chapter 3 “Résumé des Leçons sur le Calcul infinitésimal” (1823) for ‘derivative’ of f as the limit of

$latex frac{f(x+i)-f(x)}{i}$  when i ->…

3) He introduced ε-δ in Chapter 7 to prove ‘Mean Value Theorem‘: Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f ‘(x)- ε < $latex frac{f(x+i)-f(x)}{i}$ < f'(x)+ ε

4) These ε-δ Cauchy’s proof method became the standard definition of Limit of Function in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it…

1. The electron orbits: first 4 orbits from atom

万门大学抽象代数7:

“定义良好” (Well-defined)

集合: Set (S)
等价关系: Equivalence Relation (~)
商集 : Quotient Set (S/~)
映射 : Mapping (f)

Prove : f is well-defined ?

北京航空航天大学：数学大观 第2讲 数学抽象 无招胜有招”

1. Euclid 5条公理 (Axioms) => 全部 几何 (Geometry)
2. Galois 运算律 (Laws of Operations) => 抽象代数 (Abstract Algebra)

3 Wide Discomforts For Abstract Math Students

1. Group : Coset, Quotient group, morphism…
2. Limit ε-δ: Cauchy
3. Bourbaki Sets: Function f: A-> B is subset of Cartesian Product AxB.

Students should learn from their historical genesis rather than the formal abstract definitions

<a href=”http://http://en.wikipedia.org/wiki/Wu_Wenjun“>Wu Wenjun (吳文俊) on Learning Abstract Math

“…It is more important to understand the ‘Principles’ 原理 behind, à la Physics (eg. Newton’s 3 Laws of Motion), and not blinded by its abstract ‘Axioms’ 公理.”

Prof I.Herstein http://en.wikipedia.org/wiki/Israel_Nathan_Herstein

“… Seeing Abstract Math for the first time, there seems to be a common feeling of being adrift, of not having something solid to hang on to.”

“Do not be discouraged. Stick with it! The best road is to look at examples. Try to understand what a given concept says, most importantly, look at particular, concrete examples of the concept.”

“

Throw 2 dies, the sum of points i, j and probability P (i+j):

2(1/36) 3(1/18) 4(1/12) 5(1/9) 6(5/36) 7(1/6)
8(5/36) 9(1/9) 10(1/12) 11(1/18) 12(1/36)
=> Bell curve symmetric both sides, peak 1/6 (sum 7).

Today Probability is a “money” Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was “A Priori” Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today’s “A Posteriori” Probability.

“A Priori” assumes every thing is naturally “like that”: eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

“A Posteriori” by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of “already happened” statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name ‘Probability’ then ):

Addition + Rule: throw a dice, chance to get a “1 and 2” faces:
1/6 +1/6 = 2/6 = 1/3

1. Issac Newton: Hypotheses non fingo (I frame no hypotheses)

2. Gauss: Pauca sed matura (Few but ripe)

3. Descartes: Bene vixit qui bene latuit (he has lived well who has hidden well.)

Galois discovered Quintic Equation has no radical (expressed with +,-,*,/, nth root) solutions, but his new Math “Group Theory” also explains:
$latex x^{5} – 1 = 0 text { has radical solution}$
but
$latex x^{5} -x -1 = 0 text{ has no radical solution}$

Why ?

$latex x^{5} – 1 = 0 text { has 5 solutions: } displaystyle x = e^{frac{ikpi}{5}}$
$latex text{where k } in {0,1,2,3,4}$
which can be expressed in
$latex x= cos frac{kpi}{5} + i.sin frac{kpi}{5} $
hence in {+,-,*,/, √ }
ie
$latex x_0 = e^{frac{i.0pi}{5}}=1$
$latex x_1 = e^{frac{ipi}{5}}$
$latex x_2 = e^{frac{2ipi}{5}}$
$latex x_3 = e^{frac{3ipi}{5}}$
$latex x_4 = e^{frac{4ipi}{5}}$
$latex x_5 = e^{frac{5ipi}{5}}=1=x_0$

=>
$latex text {Permutation of solutions }{x_j} text { forms a Cyclic Group: }
{x_0,x_1,x_2,x_3,x_4} $

Theorem: All Cyclic Groups are Solvable
=>
$latex x^{5} -1 = 0 text { has radical solutions.}$

However,
$latex x^{5} -x -1 =…

2010 Steve Jobs declared Post-PC era has arrived with iPhone/iPad,  little did he know that he had accidentally also brought the Post-TV & Post-Publication (books, Newspapers) on iPhone/iPad platform for YouTube, ebooks.
Today,  you don’t have to sit on sofa at scheduled time to watch TV programmes,  buy/loan/housekeep books, subscribe to political-biased  newspapers.

The advent of Web 2.0 and Internet of Things (IoT) will open up the new era of freedom of “Knowledge Sharing”:
1. Instead of reading 100 books to understand a complex economic/politics/history/science topic, you can go YouTube to attend free seminars by TED, MOOC (Cousera, Khan Academy…), or follow YouTube series by book expert reviewers (罗辑思维, 袁腾飞, 百家论坛, 宋鸿兵货币战争)…
2. You can ask any questions on “Quora”. Anybody with the expertise will volunteer to teach you.
3. You can keep your reading notes in text, video and hyperlink to the vast internet resources (wikipedia, ..) and shared…

We know a Program is a math procedure. 

“A Program without Math is like Sex without Love.”

Do you know in Programming a First-Class Function is a Homomorphism in Abstract Algebra ?

In Functional / Dynamic Programming Language like Lisp, it supports First-Class Function.

Eg.
Map (sqr {1 2 5 4 7})
=> {1 4 25 16 49} 

A First-Class Function like ‘Map’ is a Function call which  accepts  another function ‘sqr’ as argument.

Map means “Apply to All”.
Map applies ‘sqr’ to all members of the list  {1 2 5 4 7}.

In abstract algebra, Map (eg. Linear Map) is a homomorphism !

http://en.m.wikipedia.org/wiki/Map_(higher-order_function)

How to solve this ‘Life’ Algebra ?

The simultatneous inequality equation with 3 unknowns (t, e, m).

It has no solution but we can get the BEST approximation :
Retire after 55 before 60, then you get optimized {e, t, m} — still have good energy (e) with plenty of time (t) and sufficient pension money (m) in CPF & investment saving.

Beyond 60 if continuing to work, the solution of {e, t, m} -> {0, 0, 0}.

I highly recommend this Harvard Online Course “Science & Cooking” for food and Math lover:

http://online-learning.harvard.edu/course/science-and-cooking

Example of the Course :

How much boiled water you need to cook a perfect egg ?

By conservation of heat (energy), the heat (Q) of boiled water is transferred to the egg (assume no loss of heat to the environment: container, air, etc).

Secondary school Physics :

Q = m.C. (T’-T)
m = mass
C=Specific Heat
T’= Final Température
T= Initial Temperature

Chef’s tip: a perfect egg cooked at around 64 C.

If applying to Harvard, use ‘Father and Mother’ to address your parents, to Stanford, use ‘Dad & Mom’…

http://m.fastcompany.com/3049289/most-creative-people/use-these-two-words-on-your-college-essay-to-get-into-harvarda

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

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…

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.”

https://en.m.wikipedia.org/wiki/Sieve_theory

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.

https://en.m.wikipedia.org/wiki/Kolmogorov_complexity