祖暅定理
Author: tomcircle
Math amateur
Modern Math Education : Time for Reform
Modern Math Foundation : Set + Logic (Equivalence Relation).
A level (H2 level) / International Baccalauréat Math should study the big red square box (Number System, Set Theory, Logic).
All STEM university Year 1 & 2 study these 3 Math foundations : Algebra, Analysis & Vector Space.
The Math major university Year 3&4 study more : real /complex analysis, Topology, Galois Theory, graph theory, etc…
新数启蒙 (1-5) – 给16岁以上的中学生
This “Modern Math” Introductory Course is based on the French Baccalaureate “Modern Math” for 16+ years old, simplified & customized for local students in Chinese.
（1）集合： 等价关系 Equivalence Relation
（2）集合与集合的关系：映射 Mapping
(3) 新数 历史： Bourbaki 学派
（4）由外向内看代数结构 : 域Field，环 Ring，群 Group，向量空间 Vector Space
(5) 域 Field : 足球场
Chinese 3AD Arithmetics 东晋. 刘徽 《九章算术》更相减损术
Please explain the Number Theory behind this trick :$latex boxed{frac {a } {b}= frac {frac {a}{b-a}}{frac {b}{b-a}}} &fg=aa0000&s=4$
Example: $latex 246 – 205 = 41$
$latex boxed {frac {205} {246}= frac {frac {205}{41}}{frac {246}{41}}=frac{5}{6}&fg=0000aa&s=3}$
…
Example:
27759 – 10227 = 17532 = 2 x 8766 = 2 x (2 x 4383) = 2 x 2 x (3 x 1461) = 2 x 2 x 3 x (3 x 487 )
$latex boxed {frac {10227} {27759}= frac {frac {10227}{1461}}{frac {27759}{1461}}=frac{7}{19}&fg=00aa00&s=3}$
Explanation:This method is from《九章算术》295AD 刘徽（曹魏/东晋），he invented the “Limit” 割圆法 method with 95-polygons to get the world’s best pi = 3.1416
https://zhidao.baidu.com/question/109475024.html
更相减损术证明：
Bézout’s Theorem :
For a, b CO-PRIME, ie gcd (a, b) = 1
There exist integersxandysuch thatax+by= 1
【数学科普】：函数概念的四重境界 Bourbaki Definition of “Function”
为啥法国数学这么强，平民的算数这么烂 ？
结账的算数：€ 313.5
华人：[给] €323.5 – [找] €10 ＝€313.5
法国人: [给] €320 – [找] ？＝€ 313.5
{? ＝€ 6.5 是这样算的 ：313.5+0.5 [凑整]＝314.0, 然后 314＋6＝320，得[找] 0.5＋6 ＝6.5}
Proof Pi Irrational
“这个人”并不存在，但他却是20世纪最伟大的数学家之 – Bourbaki
Geometry Proof : Sum of Squares
Function Continuous =>? Differentiable
【高等数学】变限积分与变量分离
This young Chinese Math student is quite rigorous :
Given f(x) CONTINOUS function,
Is it wrong to f'(x) right away?
Differential Theorem
Differential Theorem
If f differentiable at point a => f continuous at point a
Converse not true !
‘Differentiability’ stronger than ‘Continuity’
Are all Continuous functions Differentiable ? False!
Counter-example (by Weierstrass):
$Latex f(x)=Sigma{b^n}cos(a^npi x)$ n ∈[0,8], a= odd number, b∈[0,1], ab > 1+3Π/2
f(x) Continuous everywhere (cosine), but non-differentiable everywhere!
Note: Weierstrass Function is the first known fractal. (e.g. Snowflake Koch’s curve).
Plot of Weierstrass Function (Photo credit: Wikipedia)
Note: What it means a curve (function) is :
1. Continuous = not broken curve
2. Differentiable = no pointed ‘V‘ or ‘W‘ shape curve
吴军《数学之美》：数学模型帮您实现认知跨越
20CE 中学数学教育 (陈建功)
20世纪（1950s -1970s）的中国数学教育改革是迎合WW2 战后工业化, 影响我们中学数学 Secondary School Math (13-18 岁) 。
21世纪 (2000s – 2030s) 迎合的是 工业 4.0 (AI, Big Data, IoT, Fintech… )，中学数学改革 是什么呢？
陈建功 (1893-1971) 和 师弟 苏步青是二战前 留学日本派的数学教育家 （微分几何），建立浙江大学和复旦大学数学系 – 与 陈省身 的 留法/德/美 “南开”派，华罗庚的留英/俄 “清华” 派 三派鼎立，融合成特有的 中国数学派 。
这三本数学（中三/中四）书是我们 70s年代 新加坡华文中学的课本 (文言文) ：
- 欧氏几何 (Euclidean Geometry)
- 范氏大代数 (Henry Fine ［*］ Algebra)
- 葛氏平面三角 (Granville Trigonometry ［＃］)
Note ［*］: Henry Fine was the Founder of the Princeton Institute of Advanced Study (IAS) , who hired the first professor – Albert Einstein.
［＃］This 《葛氏平面三角》(Granville Trigonometry) text book was arcane and translated in ‘old’ Chinese (文言文).
Jewish / French Math vs German Math
The French “Bourbaki” School of Abstract Math (based on Set Theory as Foundation of ALL Math) influenced the…
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日本战后数学教育家 远山启
远山启：用数学照亮人性与自由
日本战后的＂新＂数学：把抽象化转成 平易化。出了一群 Fields Medalists, 数学教育家 (这位 远山启，Shimura, ), 科普作家 结城浩 (《Math Girls》Series : Galois Theory, Fermat’s Last Theorem, etc. )…
O-level Math: Exponential
丘成桐：我研究数学的经验
丘成桐：我研究数学的经验
‘路漫漫其修远兮，吾将上下而求索‘ – 屈原 《离骚》(340BC)
丘成桐：我研究数学的经验
‘路漫漫其修远兮，吾将上下而求索‘ – 屈原 《离骚》(340BC)
第一个华人拿 Fields Medal, 而且世界上少数人27岁得奖，ST Yau (岳 – 山=丘) 的成功不是偶然：
- 在香港中文大学遇到美国人年轻数学讲师，引导他进＂数学之大门＂，并推荐他去美国拜他的教授 陈省身 ( 微分几何 的 “Gauss II” 宗师) 读PhD。
- 14岁前父亲给他打下古文学/哲学的基础，”真/善/美” 艺术的直觉 Intuition， 对高深数学理解有大帮助。
- “Read directly from Masters” (Abel 说的) . 读经典原著。
- 多听别人的讲座，多看 journal papers. “抛砖引玉”，“他山之石可以攻玉”。Get inspiration from others.
- 高中/ undergrad 基础功夫要打好，Algebra, Analysis, Geometry, Physics ，都要 “懂” （只有多作习题才知道）。
如何学数学 How to Learn Math
林群院士：
道理 （X 定理）
发明 （X 证明）
林群院士 评学校数学教法颠倒：不教 ＂道理＂(Intuition ) 只教 定理，不教 “发明” 的背景(Discovery background）只教”证明” (Proof) 。
他的数学教法 强调 “all maths go back to the fundamentals.”
He has redefined the abstract “Cauchy ε-δ definition of a limit” in terms of only elementary maths (trigonometry, inequality, gradient).
化学：金属活动性顺序表
中三化学刘老师教我们背这个金属活动性顺序表, 排前面的元素比排后面的活跃，写化学反应方程式时，非常有用。基本上整个 Physcial Chemistry 就是浓缩在这表里，朗诵上口，终身难忘，50年后还记得！- 化学靠记忆，用中文学化学最方便 。(我在法国精英大学 Classe Preparatoire 念化学，打败全班法国人得第一名，中学用中文读化学的基础, 功不可没。)
钾 钠 钙 镁 铝 锌 铁 锡 铅 (氢)
K Na Ca Mg Al Zn Fe Sn Pb (H)
[嫁 那 个 美 女 身 体 细 倩 (轻)]
铜 汞 银 铂 金
Cu Hg Ag Pt Au
[统 供 一 百 金]
数学历史大纲by丘成桐
28 Dec 2019 @中国科学技术大学研究生院
Key Points:
- 中国数学从古到今只注重计算和解题，没有提出自己的问题。 (这是 科举 /Concours / 高考 / IMO Math Olympiad 的心态mindset)。大问题： eg. 法国人 Group Symmetry, Cauchy’s Limit, Poincare Topology， 英国Newton / Leibniz’s Calculus, Russell’s Logic, 德国 Cantor’s Set Theory, Infinity,…
- 清朝数学家花200+年 考古研究古代中国数学 (《九章算数》… ), 这时西方文艺复兴开始搞近代数学 (意大利 虚数 i, Fermat 数论，Descartes 解析几何，Pascal 或然率, Newton Calculus,… ) 。中国数学从此落后西方。
- 要读数学历史，尤其是17/18CE Fermat/Newton 之后的近代数学史， 才有宏观思想，继承他们的精神，从而开拓新的领域 (eg. AI 数学: HoTT = Homotopy Type Theory?)
https://m.toutiaoimg.cn/a6864500678341952013/?app=news_article_lite&is_hit_share_recommend=0 - 他选出 41位 17/18世纪之后至今的伟大数学家, 属于开创新领域的宗师：第一个是 Fermat, 第二个是 Newton…. GALOIS, ABEL,… LIE, KLEIN, HILBERT, Minkowski, Riemann, Euler, Gauss, Cauchy, POINCARÉ, Noether, Weyl, E. Cartan, André Weil, 陈省身，丘成桐, Von Neumann, Turing, … [没有 Bertrand Russell, Ramanujian, Hardy, Terence Tao,… ? ]
- 提到重要的数学：Lie Group, Lie Algebra, Guage Theory, Homology, Co-homology, Homotopy, Riemann Geometry, Zeta Function, etc.
- 他将出书写这些数学家的历史。
- [Q&A 1] : 当明白数学来龙去脉，才知道历史，看未来发展。司马迁写《史记》，司马光 写《资治通鉴》的目的：以古鉴今！
- [Q&A 2] : 数学读得越多，一切数学就变得简单。(这跟学厨艺，钢琴等一样，起初一切都难，但练习煮/弹得多，就越来越容易上手，而且能够创新，举一反三)。40年前在法国大学读抽象代数(Abstract Algebra) 满头雾水，考试心生恐惧。15年前温习并读其他更多的数学书, 顿然开窍，现在爱不释手。
《Galois Theory》 – Cousera by Ecole Normale Supérieure
Complete Free COUSERA Course 《Galois Theory》 (French) from Ecole Normale Supérieure (ENS 巴黎师范) , the university which kicked out Galois as a student in1830, but apologized to him 150 years later.
Today ENS ranked top 3 globally (after Harvard, Princeton IAS) in Math research & education, producing 1/3 of the world’s Fields Medalists by a single university.
For English Text translation (choose Google translate)
IMO Coach : Drill vs Think Style
The USA IMO coach Prof Loh is the son of 2 Singaporean Math Prof (dad) & JC Math teacher (mom).
He uses the Chinese “drill” style + American “think” style to beat China IMO Team last few years.
The Great American Math Teacher 《Stand & Deliver 》
Silver Ratio, Bronze Ratio… A general “Golden Ratio”
These 2 smart Singaporean students invented “Silver Ratio” , “Bronze Ratio” … equally good as the Golden Ratio.
Recap: Golden Ratio
https://plus.maths.org/content/silver-ratio
Without Leap Year 29/2, Winter would become Summer
Music = Math + History
To really appreciate music, you must understand this Math & history :
5 notes to 7 notes to 12 notes.
& fractions: 3/2, 3/4, 2/1 (octave)
Music = Math + History
1. Ancient China 600 BC 管仲 (5 notes 宫商角徵zhǐ羽）
2. Pythagoras 450 BC (7 notes, Chords 和弦)
3. Modern Music: 十二平均律 (明朝 1600 AD 朱载堉- > Bach 1700 AD)
Pythagoras did not believe in Irrational numbers (sqroot 2) , so only 7 integer notes.
17CE Chinese 明朝 Prince invented 12 notes music, using a giant 81-row Abacus to compute the12th root of 2 (picture below) ：
https://chinoiseries2014.wordpress.com/2019/02/07/%e5%8d%81%e4%ba%8c%e5%b9%b3%e5%9d%87%e7%8e%87/
Ideal in Ring
Historical Background:
Ideal (理想) was a by-product by mathematicians in the 350-year proof of the 17CE Fermat’s Last Theorem, wherein they found a violation of the existing “Fundemental Law of Arithmetic” (Unique Prime Factorization) . Since it is a Law, there must be an alternative ideal number to satisfy it, hence the birth of the “Ideal”.
Read here: the raison d’être of Ideal : What is an Ideal ?
Note:
Why Integer (Z) is called “Ring” (Dedekind coined it using the German word “Der Ring”) ? because
{1, 2, … , 11, 12 = 0} is clock number “Z/12Z” like a Ring-shaped Clock 🕜
Application:
The ancient “Chinese Remainder Theorem” (aka 韩信点兵 ) since 200 BCE is explained by 19CE Ideal Theory.
[Solve] : “The Problem of 6 Professors”
Whatever inside multiplies outside, still comes back inside.
…
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Where to park your car, according to math
This “Simple Car Parking” is the Decision Making Strategy demonstrated in our daily life, even in the micro-cell biological level in nature.
In the current USA-China Trade War, China Xi is taking the Prudent (中庸) approach of a partial 80% deal, while Trump is too Optimistic (or gung-ho) , insisting to accept either a “Full 100%” deal or None. Math tells you the smartest man’s approach is China Xi “Prudent” strategy.
Opportunity Score:
Meek : 1/5 score = ‘Kiasu’ (怕输 )
Prudent : 3/5 score = (中庸 )
Optimist : 5/5 or 0/5 score = (Gung-Ho)
Differential Equation:
https://m.phys.org/news/2019-09-car-math.html
In 三国演义 Three Kingdoms, the smartest military strategist Zhuge Liang (诸葛亮) was attacking the biggest northern enemy Wei (魏). He chose the Prudent (‘P’) route longer distance but less ambushes, rejected his General 魏延 ‘s Optimistic (‘O ‘) approach of taking the risky “子午谷” mountainous but shortest route.
11 of the Best YouTube Channels for Understanding Mathematics
https://interestingengineering.com/11-of-the-best-youtube-channels-for-understanding-mathematics
Below are some of the best and Free Math channels from Secondary School to University to Public Math Enthusiasts.
1. Khan Academy (School Math, Sponsored by Bill Gates)
2. Numberphile (Math Curiosity)
3. WooTube (School Math)
4. Linear Algebra (Undergrad Math)
5. Rigourous French Math (Classe Prepa, Undergrad Math)
Set Theory Basics (Baccalaureate Math Revision)
Espace Vectorielle (French Abstract Version of Linear Algebra)
Rigorous French Math (Classe Prepa)
From poor in math to Quantum Physics
She is a Cambridge PhD in Quantum Computing.
How to learn Quantum Mechanics on your own :
Data Science: 5 Statistics & Math tools
1. MEAN / MEDIUM
2. PERCENTILE
3. SKEWNESS
4. STANDARD DEVIATION
5. PCA = Pricinpal Component Analysis
1. Probability & Statistics
2. Linear Algebra (Matrix, Eigenvectors, Eigenvalues)
3. Optimisation : eg. Gradient Descent, etc.
4. Discrete Math in Computing
(想看更多合你口味的内容，马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link
林群院士：怎样学好数学？How to learn Math well ?
【中国科学院林群院士告诉你，怎样才能学好数学？】复制这条信息€80cKc€ZjlaWx€后打开👉今日头条极速版👈
林群院士很谦虚，自称很笨, 看到200页的数学教科书头也会晕：
他用学习Calculus例子，批评＂教科书不教真理，只教证明 (Proof)＂ :
…把简单道理讲得很复杂:
…其实一个日常生活的案例(火车 Case), 就可以解释清楚。
Note: 林群院士是第一个把复杂吓人的Cauchy Epsilon-Delta Analysis (aka Advanced Calculus) 用中学数学简单化。
Video: 大自然的悟性发现数学:
… 文学 《The War & The Peace》： 托尔斯泰用Calculus 证明Napoleon 必败。
＂不是你听不懂数学，是数学老师教得不好。＂
https://baike.baidu.com/wikisecond/videoflow?secondId=332699&lemmaId=11364&from=lemma&feedInlemma=1
Singlish Math : Toa-Cah-Soh大脚嫂
In Chinese school Sec 3 we had to memorize by hard these 3 Trigonometric formula:
正弦 sin, 馀弦cos, 正切 tan,
对边比斜边＝ sin,
邻边比斜边＝ cos,
对边比邻边＝ tan,
…
凶数学老师叫一个个学生轮流从座位上站起来背诵，不会背就罚站。
她说学高数”三角” (Trigonometry) 第一步就是死记这个公式。我最反感无趣味的学习，违背”数学精神” (Mathematical Spirit) 。尤其讨厌 “三角恒等式” (Trigonometric Identities Proof) 证明，用死记的公式纠缠地 (messy) 变来变去，＂剪不断, 理还乱＂，考试差点不及格 (60%)。
- Example: (below a meaningless proof)
80年代 英校数学老师发明一个记忆妙招 ：Singlish TOA CAH SOH 大脚嫂👍
Who says Singlish is useless? Our Sec 3 Math students are Top PISA (#1) in the World !
Another smart Singlish Math Chant:
Primary School Math
Given: 3 circles of radius 2
Find: Shaded Area?
Hint:
Sum of 3 angles in the triangle
= π = 1/2 (2π) ＝1/2 circle
=> its Area = 1/2(π. 2^2) = 2π = Sum of 3 arcs’ area
The World’s Most Beautiful Equation : Euler’s Identity
Prof Loh (CMU) is the 2019 USA IMO coach whose team (5 ABC, 1 White) co-won the IMO Championship with China.
…
Math High-Pay Career in Data Science
https://towardsdatascience.com/data-science-jobs-with-their-salaries-171acd3bf9be
Gone are the days where Math graduates were destined to low-pay teachers in the 60s to 90s.
Now Math career is the hottest high pay job in the 2 key engines of the “4th Industrial Revolution” , ie Mind Automation by Machine Learning and its ‘Mother’ Artificial Intelligence.
The 3 top jobs (by order of pay) :
1. Data Scientist (US $120K) :
Skills : Math (Stats, Linear Algebra, Calculus, Probability, & potentially Algebraic Topology / Homological Algebra) + A. I.
Mathematician PhD preferred.
2. Data Architect / Engineer (US $100K)
Skill: IT, especially in Big Data technologies
3. Data Analyst / (US $65K)
Skills : IT, Business
A Programmer’s Regret: Neglecting Math at University – Adenoid Adventures
Advanced Programming needs Advanced Math: eg.
Video Game Animation: Verlet Integration
AI: Stats, Probability, Calculus, Linear Algebra
Search Engine : PageRank: Linear Algebra
Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad
Program “Proof” : Propositions as Types, HoTT
https://awalterschulze.github.io/blog/post/neglecting-math-at-university/
Abstraction: Monoid, Category
Category
IMO 2019
【打破美国35年纪录的天才，与中国奥数、信息学双料冠军的巅峰对决】
https://m.toutiaocdn.com/group/6714952524924715531/?app=news_article_lite×tamp=1563521442&req_id=20190719153042010152029169696E814&group_id=6714952524924715531&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
USA Team (2016 & 2018 World Champion) :
China IMO Team :
…
…
【2019国际数学奥赛结果公布 中美两国并列团体冠军】
https://m.toutiaocdn.com/group/6715594110440309261/?app=news_article_lite×tamp=1563682066&req_id=20190721120745010025067136905FD11&group_id=6715594110440309261&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
Two observations:
1. Why almost all Chinese in top China, USA, Canada, NZ, Singapore teams?
Either Chinese chase the wrong Math (IMO) education, or the western Math Power countries (UK, France, Germany…) are RIGHT to ignore IMO education?
2. Singapore team skewed heavily in 2 schools: RI (5) + HCI (1). Also 0 girl.
【解读中美数学奥赛并列第一，敬请期待对话美国队总教练 | 袁岚峰】
https://m.toutiaocdn.com/group/6716112894641046024/?app=news_article_lite×tamp=1563733425&req_id=20190722022345010152042134236B8C3&group_id=6716112894641046024&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
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10部关于数学的顶级纪录片，带你进入奇幻之门
10 Math Videos:
- The Joy of Math
- The Joy of Stats
- The Joy of Logic
- The Code
- A Brilliant Madness
- The Great Math Mystery
- How Long is a piece of string
- Hunting the hidden dimension
- Dimension: A walk thru mathematics
- The secret life of chaos
(想看更多合你口味的内容，马上下载 今日头条)
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What to read to fall in love with Math
Find the last 2 in Singapore NLB libraries:
“The Joy of X” , “How to bake Pi”
New Math applied in Modern Physics
【文小刚：物理学的新革命——凝聚态物理中的近代数学 | 众妙之门】
https://m.toutiaocdn.com/group/6712424299841454595/?app=news_article_lite×tamp=1562923076&req_id=20190712171755010152027098774DD5D&group_id=6712424299841454595&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
Keywords:
- Quantum Entanglement (QE) : 量子纠缠
- Topological Order (Long-range QE) : 拓扑序
- Topological Insulator (Short-range QE) : 拓扑绝缘体
- Rigid State : 凝聚态
- Cohomology : 上同调
- Tensor : 张量
- Category Theory : 范畴学
- Group Theory : 群论
- Algebraic Topology : 代数拓扑
….
1. Four “Physics” Revolutions aided by Math
Math is applied to explain the Nature (Physics) :
Calculus (Differential Equation) : Newtonian Physics
Riemann Geometry : Einstein General Relativity
Partial Differential Equation: Maxwell Electro-Magnetic Field
Linear Algebra (Matrix) : Quantum Mechanics
Group Theory: Physics / Chemistry Symmetry
Next…
Category Theory : Topological Order of the long-range Quantum Entanglement
Algebraic Topology: Topological Insulator (2016 Nobel Prize Physcis)
2. The 5th Physics Revolution aided by New Math tool : Category Theory
3. Category Theory = “Structural Relationship” Math
4. Algebraic Topology
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5G Polar Codes Dr. Erdal Arikan
Dr. Erdal Arikan (1958 -) of Bilkent University, Turkey, recieved his PhD in MIT at the age of 27.
Huawei Award: The Father of Polar Codes (极化码)
Industry watchers believe that the selection of polar coding as the channel coding technique for control channels for 5G communications system may have put Huawei at the forefront in the 5G race(27 Gbps in 2016).
1 Gbps = 1000 Mbps
…
5G Basic Explained:
Part 1: The Math behind 5G Polar Codes
..
Part 2:
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Notations:
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$latex displaystyle F = begin {bmatrix}
1 &…
1 & 1
end{bmatrix}$
$latex A =
begin{pmatrix}
a_{11} & a_{12}& ldots & a_{1n}
a_{21} & a_{22} & ldots & a_{2n}
vdots & vdots & ddots & vdots
a_{n1}& a_{n2} &ldots & a_{nn}
end{pmatrix}
$$latex B =
begin{pmatrix}
b_{11} & b_{12}& ldots & b_{1n}
b_{21} & b_{22} & ldots &…
View original post 70 more words
“Poisson Distribution Intuition (and derivation)”
“Poisson Distribution Intuition (and derivation)” by Aerin Kim 🙏
https://link.medium.com/bho611jxSX
Poisson was a 19CE French mathematician, or Statistician, who invented the Poisson Distribution.
Note: “Applied” Math vs “Pure” (aka Modern / Abstract) Math
Poisson, being an Applied Math expert in the French Science Academy, did not understand the “abstract” Math paper submitted to him by the 19-year-old Evariste Galois in “Group Theory” for the “Insolvabilityof Quintic Polynomial Equations“. Poisson rejected the paper commenting it was too vague, destroying the final hope of Galois to make his invention the “Modern Math” known to the world. The dejected poor Galois died at 21 after a duel during the chaotic era of the French Revolution.
丘成桐：数理与人文（原创完整版）
(想看更多合你口味的内容，马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link
China Gaokao 2019 Interesting Math Question
Gaokao = Baccalaureate = GCE A Level
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Let Ф = 0.618
… Golden Ratio Ф = 0.618
defined as:
“Head to Throat” vs “Throat to Navel” = a/b = Ф ,
also
“Head to Navel” vs “Navel to bottom of Legs” =(a+b) / c= Ф
Knowing :
Head to Neck = 26 cm &
Legs = 105 cm
Find the height (H) ?
Note: The tricky part is to know “the neck is below the throat” & “the legs start some distance below the navel”, hence 2 inequalities :
a <…
View original post 62 more words
Viral Chinese 2019 Gaokao Question
Gaokao 高考 ＝ Baccalaureate = Cambridge GCE A-Level
Hint:
f(x+1) = 2f(x)
tells the height = f(x) double when x moves to x +1
=> periodicity is 1
Like Sine or Cosine curves have periodicity of 2π.
sin (2π + A) = sin (A)
Ref: More complicated curve drawing by French Method.
数学大师的数学启蒙书
Abel / Gauss 自修数学经验：
Read Directly from the Masters – 自接读大师的书 （而非读其他人注解的书）
下面这2本数学经典 ＂葵花宝典＂(有中译版) 是WW1＆2 前 世界第一流 宗师 (Noether’s ‘Abstract Algebra’ , Hardy’s ‘Number Theory’ ) 的杰作，值得阅读，收藏。
1. Modern Algebra 史上第一本抽象代数的书: by Van der Waerden (Noether的学生: Artin, Waerden)
2. Number Theory 史上第一本数论启蒙书：
(想看更多合你口味的内容，马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link
Simple groups, Lie groups, and the search for symmetry
Math History | NJ Wildberger
Summary:
- Normal subgroup
- Quotient Group
- Finite Simple Group: Z5, D5,…
- Tesselation (regular)
- Platonic solids
… Two families of Group :
- Symmetric Groups: Sn
- Linear Group: GL(n) = M, det M ≠0
…
An = Alternating Group
[even permutation, det = +1]
n>=5, An is a Simple (non-commutative) Group
Lie Groups :
SO(2)
SO(3)
Simple Lie Groups : Matrix Group M
- Special Linear Group: SL(n), det M = 1
- Orthogonal Group: O(n), $latex M. M^T = I$
- Symplectic Group : Sp(2n)
…
Alibaba “Damo” Global Math Competition
https://m.damo.alibaba.com/alibaba-global-mathematics-competition/
Jack Ma is a kongfu fan, his Alibaba offices all named under the Kongfu heroes, eg. Damo Academy.
“Damo” 达摩 (Bodhidharma) was an indian Buddhist monk who came to China in 5th Century, established ShaoLin Monastery, also invented Chinese Kongfu.
Problem 1
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Hint: Answers in the above link.
RSA Encryption – Number Theory
https://dev.to/renegadecoder94/understanding-the-number-theory-behind-rsa-encryption-1pdo
Background knowledge:
1. Group
2. Cardinality : Euler Totient Function
Note:
A linguistic remark: The Latin word “totiens” (or “toties”) means “that/so many times”.
Homotopy Type Theory: Vladimir Voevodsky
A self-study Math genius Vladimir Voevodsky, Fields Medalist, Russian.
Voevodsky borrowed idea from Computer Science (“Interface” =hide implementation details) to Math, and vice-versa, from Math to Computer Science :
Homotopy Type Theory (HoTT)
Result: Computer to prove math theorems.
Computer Science Idea:
- Hiding Implementation Detail
- Black Box “Interface”
- Type (类) : Boolean, Integer, String, … (stronger structure than ＂Set＂ 集合)
Math Idea (Abstract Geometry : Algebraic Geometry) :
- Homotopy (同伦)
- eg. Cup ~ Donut, both are homotopic or same “Type” (in Computer)
- Isomorphism is Equality “=”
- Homotopy Type Theory (HoTT):
- Univalent Theorem: both objects are considered “=” if they are isomorphic (prove if 1-to-1 correspondence 对应)
