# a^a^a^… a

$\boxed{a^{a^{a^ {{...}^{a}}}} =? }$ $\text {.... [A] }$

$\text {Let } a = \sqrt {2}$

Method 0: Wrong !!!

Method 1 : Rigorous!

1) Prove [A] Convergent ?

2) If converge, find Limit ?

https://v.ixigua.com/eUHCEkb/

Method 2: by RMI Homomorphism

# 唐. 王孝通的一元三次方程特例解

Probability * (or＋) between P(A) and P(B) whether A, B are (in-) dependent event.

https://v.ixigua.com/ej7X49B/

# Gauss “高斯绝妙定理”

K1. K2 = K = Constant

［李永乐 数学科普］

https://m.toutiaoimg.cn/a6941285780660683303/?app=news_article&is_hit_share_recommend=0&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) .

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)

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

# Fermat’s Last Theorem for Regular Primes

This FLT (for Regular Primes) is only first part of proof up till Ideal’s Ring Theory by Kumma.

The final complete proof by Andrew Wiles (1994) used more tools: Elliptical curve + Galois Theory.

The complete proof FLT by Andrew Wiles taking him 7 years in solitude, still a short time compared to 350 years before him but failed by the grandmasters Euler/Gauss etc. Today Andrew Wiles is hailed as the greatest 21CE Mathematician, even Fields Medal gave a Special Award to him (even he was > limit age of 40 years old ).

# 《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明：

1. 哲学：希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴， 达芬奇, Golden Ratio

3. 工业革命：Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

https://m.toutiaoimg.cn/group/6919012894134764046/?app=news_article&timestamp=1611057637&group_id=6919012894134764046&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Ideal of Ring, Kernel of Group

Last time 1978 in Maths Supérieures (French Classe Préparatoire ) studying Ideal, never understood the “real” meaning except the definition, until I attended the Harvard online course in 2006, which used the MIT Prof Artin’s textbook 《Algebra》, pioneering in the world by using Linear Algebra (Matrices, etc) as the foundation to study Group, Ring, Vector Space etc.

Like any structure in the nature, it has a core (“kernel”) which encapsulates all the essence of the structure : durian kernel, cell kernel, etc.

With kernel we can partition (分类) the whole structure family, eg. 血型={A, B, O, AB} is a “kernel” which can divide all Blood groups into 4.

In a Group structure (only 1 operation, eg. + or *), the Kernel of Group (G) partitions G into “Quotient Group” , denoted as:

G / Ker f

In Ring structure (2 operations : +), the German Hilbert named it “Ideal” (instead of Kernel), also partitions the whole Ring structure. Eg. IDEAL {Even} partitions whole Integer structure family into Even & Odd. The name “Ideal” bcos it is also found ideal number to the uniqueness factorisation satisfying the 《Fundamental Theorem of Arithmetics》

eg. 6 =3*2 = 2*3 (unique factorization ! )

but not true in uniqueness in Complex number, we have also another factorization !

6 = (1+√-5). (1- √-5)

so the Ideal (I) is found as the gcd of these 4 pairs:
I1=gcd ( 2, 1+√-5)
I2 =gcd (2,1- √-5)
I3=gcd (3,1+√-5)
I4=gcd(3,1- √-5)
such that :
6= I1* I2* I3* I4

https://tomcircle.wordpress.com/2013/04/06/ideals/

Ideal is like a “Black-Hole” which sucks everything outside into it to become inside its “core”. Eg. “Even” × anything outside = “Even”, same to “ZERO” Ideal.

A polynomial P(X)is also a Ring Structure (+*, but not / with zero polynomial) has the ideal.
eg.( X^2+1) if it is a factor of P(X), so we can partition P(X) into
P(X) / (X^2+1) sub-Ring structure.

Note : (X^2+1) factor means P(X) has complex root “i”
(= √-1)

# Free Group and Representation

Abstract Algebra：自由群 ＆表示 Free Group & Representation

https://m.toutiaoimg.cn/group/6661864364057821710/?app=news_article&timestamp=1609069898&group_id=6661864364057821710&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Notes:

1. Answer to What is a free group in abstract algebra? by Cassidy Block https://www.quora.com/What-is-a-free-group-in-abstract-algebra/answer/Cassidy-Block?ch=3&share=83fc0bde&srid=oZzP

2. Free Group (Wikipedia) : https://en.m.wikipedia.org/wiki/Free_group

# Dr. Eugenia Cheng: “How to Bake Pi”

Key Points of the Talk:

1) Math is interesting but is only so after undergraduate school. Before that, Math is taught as computation subject from Elementary to High school.

2) Braid : Bach music, Juggling 3 balls

3) Platonic Icosahedral (20面体) Structure discovered by ancient Greek Plato 2000 years ago, but can’t find a real world Icosahedral object until in Viruses found by Louis Pasteur in 20th century – also now in Sars, Covid19.

4) Group Theory : Battenberg Cake, Bed Mattress Rotate/Flop/ Flipping

5) Mobius Strip & Donnut Cutting.

6) Fermat’s Last Theorem : Andrew Wiles in 1994 proved in 7 years still with a “hole”, but fixed a year later by himself & his student.

7) MacLane Pentagon : Higher-Dimension Categories (PhD Math)

# The Black-Sholes Formula – 诺贝尔经济奖Scholes, Merton与LTCM S

Key Points:

1) From 1950 an unknown PhD Math Thesis by French Dr. Bachelier (a PhD student of the 20CE last Polymath Henri Poincaré, who was not impressed with the ‘gambling’ math, gave only an above-average marks to the thesis) – invention of “Options” Trading to eliminate risk in stock fluctuations.

2) The technique is called “Dynamic Hedging”…

3) …by applying the Japanese mathematician “Ito Math”.

5). Sholes & Merton Company “LTCM” – making tons of money until… 1997 Asian Crisis, bailed out by USA government after loss of billions !

https://m.toutiaoimg.cn/a6906337056042156551/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Octonions & United 4 Forces

This Cambridge Mathematician explains why need so many number system :
N Natural,
Z Integers
R Real,
Complex,
Quartenions (1,i,j,k) ,
Octonions

https://m.toutiaoimg.cn/group/6584683147512775182/?app=news_article&timestamp=1607604783&group_id=6584683147512775182&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

She believes the Octonions is the secret behind united field of the 4 forces :

G gravitational,
E electrical,
M magnetic,
W weak force in particules.

Maxwell United EM,
Einstein United GEM,
She believes her research in Octonions can unite GEM+W.

# Chinese Remainder Theorem 《韩信点兵 》& Ideal

Chinese Remainder Theorem

By 1930, even Bourbaki founder Dieudoné didn’t know “Ideal” when he read 《Algebra》from Noether’s lecture-note compiled by her student Van der Waeden

https://m.toutiaoimg.cn/a6559856147912720909/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Differential Equations Versus Machine Learning

Differential Equations Versus Machine Learning | by Col Jung | Nov, 2020 | Medium

View at Medium.com

# 《Galois Theory》 – Coursera by Ecole Normale Supérieure

Complete Free COURSERA 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)

https://www.coursera.org/learn/theorie-de-galois

# 抽象代数课本教材

＂Abstract Algebra＂抽象代数：法国大一 Math Supérieure 一年就读完 (Set，Group, Linear Algebra / VectorSpace, Ring, Field) ，生吞下肚子，30年后才消化。

https://m.toutiaoimg.cn/a6884573331576586767/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Wolf Prize : Donaldson

The Modern Math is now proved by Physics, unlike the past history in reverse direction.

The first Math Fields Medals awarded to a Quantum Physicists Edward Witten in String Theory, now the 1st Wolf Prize to Donaldson who proved Math with Physics 杨振宁 Yang-Mills Equation (which is still one of the 7 Millenium Math Conjectures worth $1 m Clay Prize, but proven true in Physics Experiment) . Wolf Prize is S$1m for 数学终身奖， unlike \$100K Fields Medal for < 40 years old. Only Harvard Dean 丘成桐 (HK) received both prizes, while his mentor 陈省身 only Wolf (same year as Paul Erdos).

Simon K. Donaldson

# Group Action on Set

Group Action 3 Properties : Orbit, Stabilisers, Fix

# BioMath: SARS-COV-2 in “A5 Simple Group” Icosahedral Structure

Abstract Algebra is about the study of all structures in the univers (Math, Nature, Physics, Chemistry, Bioscience,… ) : “Group Theory” is the tool invented by a 19CE genius Évariste Galois, originally used to solve the 300-year-old problem since 16CE “No Redical Solution for Polynomial Equations with degree 5 & above.

A5 is unbreakable structure called “Simple Group”, just like Atoms, so are fatal SARS & Covid19 hidden in A5 structure.

# 共和国院士风采系列微纪录片《印迹》第六集：

He replaced Cauchy / Wierstrass scary Epsilon-Delta definition of Limit / Continuity by secondary school trigonometry.

https://m.toutiaoimg.cn/a6875217283531309576/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

https://www.amazon.sg/dp/9812704582

Singapore NLB Library (LKC Reference Section, 7th FLOOR, HQ @Victoria Street) has Prof 林群’s book, in which he gives alternate simpler trigonometry definition to Epsilon-Delta Calculus : “Limit, Continuity” – the 2 of the 4 major obstacles to enter Modern Math World (the other 2 are Infinity, Abstract Algebra).