Prime Gap : Oxford James Maynard

Oxford post-doc proved the Prime Gap (= 600) six months after 张益唐 had proved with70 million gap (which was further improved by Terence Tao’s PolyMath Project team to ~260) .

Till today nobody has proved the gap = 2 (Twin Prime).

https://www.quantamagazine.org/james-maynard-solves-the-hardest-easy-math-problems-20200701/

The Mathematical Revolution That Was Bred on a Sheep Farm | OpenMind

https://www.bbvaopenmind.com/en/science/mathematics/the-mathematical-revolution-that-was-bred-on-a-sheep-farm/

In 17CE a similar plague like COVID19 forced a young man Newton to be quarantined in a sheep farm for 2 years, he changed the universe with Differential & Integration Calculus, to “move” the planets.

Will COVID19 give birth to another “Newton” in some “kampong” at one corner in USA or China or Europe ?

https://www.bbvaopenmind.com/en/science/mathematics/the-mathematical-revolution-that-was-bred-on-a-sheep-farm/

Chinese 3AD Arithmetics 东晋. 刘徽 《九章算术》更相减损术

Please explain the Number Theory behind this trick :\boxed{\frac {a } {b}= \frac {\frac {a}{b-a}}{\frac {b}{b-a}}}

Example: 246 - 205 = 41

\boxed {\frac {205} {246}= \frac {\frac {205}{41}}{\frac {246}{41}}=\frac{5}{6}}

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 )

\boxed {\frac {10227} {27759}= \frac {\frac {10227}{1461}}{\frac {27759}{1461}}=\frac{7}{19}}

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 integers x and y such that ax + by = 1

Conway Surreal Numbers

John Conway (82) passed away last month due to COVID19.

He made Math fun by playing Math games.

Surreal Number = 超现实数

Conway 是"大师级"数学家,他的这个 Toronto lecture “Sur-Real Numbers” 值得 花 1hr 细听, 茅塞顿开:
1) 数学是"玩"出来的 : 他玩”Go” 围棋,发现Surreal number (a Field structure, its addition operation forms a “Additive Group”).

2) Cantor 的逸事: 发明 Set Theory, Infinity,Unionize Mathematicians by 1st ICM (International Conference of Math)。法国打算 boycott 战争敌人德国,Poincare 带30法国数学家半场突然出席 – 数学无国籍 !

3) French “Positif” Integers include ‘0’ , better than English’s “Non-negative” Integers.

4) 他的 Q&A 更精彩… 可以"看"出大师的思路。

Cédric Vilani (Chinese) Interview

French Fields Medalist Cédric Vilani 中文interview: 他苦思证明数学/物理定理,在第 1001 th 夜 @4am,”好像上帝给他打电话 – Un coup de fil du Dieu” , 突然开窍…

他从政加入Macron的小党 “En March”,当MP, 今年竞选 Paris Mayor.

2017 年 他引入 “Singapore Math” 进法国小学。

https://m.toutiaocdn.com/i6827717478164988419/?app=news_article_lite&timestamp=1589724887&req_id=2020051722144701001404813006329A50&group_id=6827717478164988419

NLB Library has 13 copies of Cedric Vilani’s book for public loan.

书法”九宫格” 的”均” =黄金分割

书法是”字如人品” 。

大自然的美表现在 “黄金比率” (Golden Ratio) = , 暗藏在 唐初 欧阳询 发明的楷书”九宫格” : 一个比率 “均” (即:均称,对称 Symmetry) 。

1, 1, 2, 3, 5, 8,… (Fibonacci Series)

黄金分割= 0.618 ~ 1:2 ¦ 2:3 ¦ 3:5 ¦ 5:8

Note : Golden Ratio 黄金比率 =1.618 (长:短)

欧阳询 楷书 《九成宫》

晋/隋/唐的行书, 草书是在汉朝的楷书 (aka 真书) 的”“基础上”速度化” : “” (角度), (笔画突) ,尚保存”均” 的美。

东晋. 王羲之 《兰亭序》行书

唐.张旭 草书

毛泽东 草书沁园春· 雪

https://m.toutiaocdn.com/group/6821832651129750030/?app=news_article_lite&timestamp=1589048685&req_id=202005100224450100140481400C466DDA&group_id=6821832651129750030

Terence Tao Math Tests at 7 & 8 years old

https://m.toutiaocdn.com/group/6824291273520710151/?app=news_article_lite&timestamp=1588923284&req_id=2020050815344401001407019805217EB6&group_id=6824291273520710151

数学老师陶母的儿子教育

Terence at 8 reading these 22 Math books: 《Matrices & Vectors》, 《Logic》, 《Calculus》…

Math Test : [T = Terence Tao at 7 Years old]

On Distribution Law :

Terence Tao showed that a good Math “Abstract Thinking” Mind can AVOID the need of using concrete examples.

That explains why MOST good students excel in High-school A-level Math but average in university Abstract Math.

Math Olympiad is harmful

王文湛 教授 (80) is a Math Prof in 清华,yet he can’t solve his 10-year-old grandson (Primary 4) Math Olympiad questions in Combinatorics.
The harms to kids math education:
1. Teach too early the higher math, only “acrobatic technique” , not genuine math education.
2. Waste parents’ money in unnecessary tuition for Olympiad math.
3. Pressure on kids.

"新数"中重要的3个抽象概念

华罗庚精辟地宏观解释:

初等数学 (Elementary Math) : 高中 / 初级学院 JC (A-level / Baccalaureate) 毕业前的12年数学。

高等数学 (Superior Math, “Math Sup” in French) : 大学本科 (undergraduate) 的数学。

[高深数学 (Advanced Math) = Graduate PhD Math]

李善兰 (晚清 19CE):继续翻译 晚明 Ricci Mateo / 徐光启 《几何原理》(《The Elements》13 books)没翻译的最后4本。

李善兰引进中国数学 "Function" (法国Descartes “Relationship” 概念 / 德国 Leibniz’s 引进 / 瑞士 Euler 正式化 y = f(x) ), Felix Klein 带入中学数学) 。他翻译成 "函数", 为什么?因为他用信函举例子: Function 好比 一封信 (f) ,写信人集合 Set {A} ,收信人集合 Set {B} :

f(A) – > B

每封信只有一个收信人B.

每个B人 可以收到 0 1 封信。

不可能 同一封信 发给 > 1 个 B人。

Key Points (Read below link) :

高等数学的三个"挡路虎"(Road Blocks) ,排除他们三个障碍, 才能登堂入室”高等数学” ,游刃自如。

  1. 函数 Function,
  2. 连续 Continuity ,
  3. 极限 Limit

https://m.zjurl.cn/answer/6809263684867604750/?app=news_article&app_id=35

Amazing math bridge extended beyond Fermat’s Last theorem

D <=> A

D : Diaphantine equations
eg.
x^{n} + y^{n} = z^{n}

A = Automorphic Form (eg. Elliptic Curves used in highest Cryptography “ECC” )

Fermat’s Last Theorem: after failures of 350 yrs, Andrew Wiles in 1994 proved easier from reversed direction A to D.

Lycée Pierre de Fermat (Maths Supérieures & Spéciales @Toulouse, France) was my Alma mater。

https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/