The Math Power of Number 4

Despite Chinese & Japanese don’t like ‘4’ for the same sound as \si (death), ‘4’ is the magic number for all ideal compositions:

1) All TCM 药方 has 4 compositions : 君臣佐使
eg. “银翘解毒片” (Chinese Panadol) :
银花 (君药)有毒, 但可以杀flu virus,
连翘(臣药),中和neutralise 银花的毒。
其他的
(佐药 Assistant ) 甘草,..
(使药 Smoothening, sweetening ) 蜂蜜

2)烹饪 Cuisine :
eg. 炒饭
君:冷饭
臣:蛋 x2
佐:mix veggies, 腊肠,烧肉,…
使:香料 (葱花,胡椒粉,garlics,… ), 麻油/鱼露

3) Music
eg. Quartet
君:violin
臣:viola
佐:flute, guitar,…
使:Cello,… Double base

4)Maths
Eg. “4-Color” Problem : the only one Math impossible by mathematicians but proven only by computer.

5) Exemplary Nation Racial Harmony
Eg. Switzerland / Singapore : 4 races & official languages.

6) War against Evils:
Eg. WW2 Allied :
USA, UK, France, China, against evils Japan & German (+Italy).

7) Four seasons :
春夏秋冬

8) Healthy Meal:
Eg.
Fill 1/4 plate with wholegrains.
• Fill 1/4 plate with good sources of protein.
• Fill 1/4 plate with fruit and 1/4 plate of vegetables.

That’s why the 19-year-old Math genius Galois (1832 AD) proved by his invention Abstract Algebra the “Group Theory” that Polynomial equations with radical solution (+ – */,nth root) ONLY possible up to maximum degree 4.
That explains why the “A5-Group” Icosahydron (二十面体) structured SARs-like viruses (eg. COVID-19) WILL NEVER have “radical” drug SOLUTION, only possble killed by our own Antibody pre-trained by Vaccines (eg. mRNA or Sinovac… ).

Also humans impossible to have more than 四代同堂。

The 5th Generation Computer “Prolog Machine” by Japan in 1980s failed miserably.

Anything more than 4G has only “particular case” solution with physical limit beyond which we have to shift paradigm / technology to restart on another track.
Eg 5G by Huawei facing so much difficulty by Western sabotage, only by jumping to new “Quantum Communication Technology” to overcome these physical & artificial human resistance, whether they like it or not by Western countries.

IMO 2020 (6th Question)

2020 IMO (6th problem) only one Chinese 李 student scored full 7 marks.

https://v.ixigua.com/emUQpn1/

This question is similar to Analysis finding epsilon-delta value of N:

(Epsilon-Delta Analysis, aka Advanced Calculus ) definition for “Limit of Series” :
It exists N such that, for all n > N…

[Solution] :

[See also] IMO 2020 (4th Question) : https://m.toutiaoimg.cn/i6881188536880464384/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

3 Math Foundation Skillsets for Engineering / Finance Students

【普通人学数学系列-下集:学到什么程度】

理工/金融科Engineering/Finance 的三个扎实数学基础 :
1) Calculus
2) Linear Algebra
3) Probability / Statistics

Prerequisites for the 3 above in
1) O/A level : Trigonometry, Algebra, Geometry
2) Vector Space (with some basic Algebraic Structures : Group, Ring, Field )
3) O/A level : Permutation/Combinatorics

https://m.ixigua.com/video/6967105499342832166/?app=video_article&timestamp=1622200486&utm_source=native_share&utm_medium=android&utm_campaign=client_share

Leibniz’s original proof : Integration by Parts

莱布尼茨微积分——

Leibniz’s original proof : Integration by Parts (分部积分公式”)

https://m.toutiaocdn.com/i6965860180663976452/?app=news_article&timestamp=1621922350&use_new_style=1&req_id=202105251359090102120610721E3A8184&group_id=6965860180663976452&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

金庸武林江湖与数学江湖

金庸武术的道理 和学数学一样:
1)华山正邪二派 : 气宗 (正) vs 剑宗 (邪)
数学:数学理论 vs 刷题技巧
2)虚竹 :忘掉以前的少林功夫才能学 消遥派
数学: 忘掉A-level 前的思维 (concrete) ,才能学好大学抽象(Abstract ) 数学.

3) 少林寺僧好高鹜远:还没学精本派“一阳指”,就想去换 印度鸠摩罗的功夫。
数学: 先打好 大学基础数学 (Epsilon-Delta, Abstract Algebraic Structures) ,才去学其他的高深东西 (Category, Differential Geometry, Algebraic Topology,… )

https://v.ixigua.com/e5Y62ye/


金庸的武林江湖 宗师 vs 数学江湖

东邪(孤僻冷漠) :黄药师 vs 德国. Gauss

西毒 (妒忌,狠毒) :欧阳锋 vs 法国. Cauchy (迫害 年轻天才 Abel, Galois )

南僧(避世隐士): 大理国王 一灯大师 vs 法国.Fermat

北丐 (流浪天涯):洪七公 vs 匈牙利. Paul Erdos

French Baccalaureate Math Paper 2 (Analysis)

【「给我进来做题啦!」法国高考数学题难度如何?居然只有四道题?】

Baccalaureate Scientifique Math 2019 (Analyse).

Yet to see the Baccalaureate (S) Paper 1 “Algèbre” which is tougher on Abstract Algebra (Group, Ring, Field, Vector Space / Linear Algebra,…)

https://m.ixigua.com/video/6867459333400035851/?app=video_article&timestamp=1621316174&utm_source=native_share&utm_medium=android&utm_campaign=client_share

吴文俊 Wu WenJun

吴文俊,陈省身在 WW2昆明西南联大的弟子, 留法Strasbourg University, 兼任 博士生的tutor, 教导 法国未来的 大师Grothendieck (两人都是新数Bourbaki 学派 最后一批会员) 。

吴文俊在1975 文革后才研究 中国古代数学,从中得灵感,发明 电脑 机器化 AI 证明axiom-based 几何定理。得 1st batch Run- Run Shaw (东方Nobel Prize) $1 m Prize.

数学家清心寡慾不爱斗争,一心专注 “数学之美” ,其他生活繁琐的事(文革 批斗) 都看得开。所以多长寿 (Newton, 陈省身, Hadamard, 杨振宁, 也都90+) ,吴文俊也高寿活到98岁。

https://v.ixigua.com/efrQhx8/

Princeton 《Calculus Lifesaver》

This is a free video lecture series of the Princeton Math Textbook 《Calculus Lifesaver》(link Amazon.com) for university non-math majors.

He explains the “funny” but smart Calculus basics : First & 2nd Fundamental Theorems of Calculus (by Newton’s co-inventor Leibniz), some how never taught in the 1970s GCE A level (now also ? ) bcos not Newton’s Anglo-saxxon invention. Neither the French textbooks teach so (bcos Leibniz was German ?). They only appear in pure American/German Calculus books.

“Funny” bcos by “1st Theorem” you could ‘d/dx’ any intergral to get the anti-derivative . Richard Feymann self-learnt this trick in High school to solve complex integrations.

Also why the “Coset” (+ C) is explained by the 2nd Theorem in this video.

Coset & Index 陪集,指数

https://v.ixigua.com/eU7pf3V/

群论的原本是Galois 发明,没有现在这么抽象的Axiom-based, 是100年后 Noether 在 WW2 归纳的抽象化。Galois 只发明 左/右陪集 Left /Right Coset (l’ensemble à gauche / à droite),当 左陪集=右陪集,就是Normal subgroup (正规子群,l’invariant), 破解300年的数学难题:5次方程以上(Quintic equations & above ) 无根式 (radical root) 解, 从而诞生Group Theory, 开辟抽象代数Abstract Algebra / New Math.

这位中国年轻老师 教得很棒, 证明严谨rigorous, 很好的方法:eg. Test “well-defined”, necessary & sufficient condition (Set Prove Technique : ⊂ left inclusion, ⊃ right inclusion, then equal =), Bijection 双射 (Surjective 满射 trivial + Injective proof).

他的习题答案:
|Z : nZ|Index is easy ?
意思:
把 Z 用coset 分类 (partitioned) 为
0Z,1Z, 2Z,… (n-1)Z 个陪集,
Total = n个

|Z : nZ|= n

Note: Coset in Calculus ( Indefinite Integration) :

Integral Solution “+ C” 就是 Right Coset = solution的右陪集:意思所有 “+C” 都是solution Set 的 右陪伴。

2021人口普查的契机:大学普及改革先废高考

2021人口普查数据 :14.1亿,新生婴儿减少,男多于女,老人(>60岁)佔20%…

教育需要改革来应对新的人口趋势:
(1)缩短基本 小学-中学教育 :废除高考,省下2年准备高考的时间,12年减2年。

(2)高中到 大学本科 :直通车(看高中学校成绩)。

(3)名牌大学(清/北/211/918 等):不设本科undergrads,直接 研究生 Masters/PhD , 像法国Grandes Ecoles 制度 (2nd Cycle Tertiary education, 本科设在普通 universities / 工程科预备班 Classes Préparatoires).

(4)所有 普通大学universities 只 教 本科undergrad 学生为主。

这样普及全国人口教育素质=
10年 (强迫 中/小学) +2 or 4年 (专校 or 大学), 即20岁 / 22 岁 专科/大学本科毕业, 解决“人才荒”的问题。

笔者认为

– 大学改革不可操之过急,要按步就班, 先试点 一些top高中附属大学的 “直通车”(例如:人民大学附属中学), 这样就省下2年高考准备时间。慢慢推行到全国各省内大学和结盟高中。十年完成直通车,百年树人才。

– 名牌大学集中精力/资金在培养 硕士/博士科研人才,把本科教育的资源分散给各省普通大学,直接提升后者的教育水平。这样也可留住中国人才,防止外流去美国读博深造而不回国。

https://m.toutiaocdn.com/i6960914813585392158/?app=news_article&timestamp=1620752197&use_new_style=1&req_id=20210512005637010212205224381BC609&group_id=6960914813585392158&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

FTA Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic | by Maths and Musings | Cantor’s Paradise

https://www.cantorsparadise.com/the-fundamental-theorem-of-arithmetic-37470aa1a7a0

Note:

The Fundamental Theorem of Arithmetic (FTA):

2300 years ago Euclid 《Elements》proved prime factorisation :
If p | x. y then p | x or p |y

19CE Gauss proved FTA. Uniqueness of prime factorisation.

French 《Bezout Theorem》 is the useful tool: if p, q co-prime, then there exist n, m integers such that
np + mq = 1
eg. Prove 5, 7 co-prime,
(3) *5+ (-2) *7 = 15-14=1

Strange, why Bezout Theorem not taught in A level ?

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.

https://v.ixigua.com/e6mRFnP/

(1)集合: 等价关系 Equivalence Relation

https://v.ixigua.com/e6mDFhw/

(2)集合与集合的关系:映射 Mapping

https://v.ixigua.com/e6mfC4C/

(3) 新数 历史: Bourbaki 学派

https://v.ixigua.com/e6mjedR/

(4)由外向内看代数结构 : 域Field,环 Ring,群 Group,向量空间 Vector Space

https://v.ixigua.com/e6mj5sd/

(5) 域 Field : 足球场

https://v.ixigua.com/e6mkau7/

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…

https://m.toutiaocdn.com/i6743479520755450380/?app=news_article&timestamp=1615286356&use_new_style=1&req_id=202103091839150102120751990F03C964&group_id=6743479520755450380&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

mRNA剪接Splicing 原理 – 施一公

3 Step Splicing : first & last steps are linear, 2nd step 3D syructure (non-linear) .

施一公 returned from USA, established the private 西湖大学 (funded by Alibaba, Baidu, Huawei etc), a first pure research university modeled on Caltech. His team first decoded the COVID19 genetic code sequence just 1 month after the Wuhan outbreak & announced the result FREE to the world.

The Hungarian lady scientist Katalin Karikó jumped on his newly published DNA code to invent the mRNA PfizerBioNTech vaccines.

https://m.toutiaoimg.cn/a6936774993665589790/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

数学与物理的关系

(小学生与诺奖得主杨振宁的对话)

Math was a tool for Physics (Newtonian Physics)

Math & Physics were independently developed ( Physics Gauge Field = Math Fibre Bundle )

Physics is a tool for discovering Math (Quantum Physics : String Theory)

杨振宁 Yang-Mills Conjecture is one of the unsolved Millenium Math Problems.

https://m.toutiaoimg.cn/a6936190545714741798/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share