# FERMAT & Descartes 的恩怨

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# 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

# IMO Geometry Techniques 几何理论基础：分角定理、张角定理，推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.

https://m.toutiaoimg.cn/a6910816390312100360/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 数学武林的四大门派

［0. 希腊 ＝ 达摩祖师］

1. 法国 ＝少林派
2. 德国 ＝ 武当派（德国祖师 Lindemann 是法国宗师 Charles Hermite 的得意外国徒弟
3. 俄国 ＝ 峨眉派 (俄国祖师奶 Sofya Vasilyevna Kovalevskaya 是 德国大师 Karl Weierstrass 的得意女弟子)
4. 中国 ＝ 古墓派 (‘小龙女’ 的奇特功夫 自成一派：中国古代数学 ＋ 融合 留 法/英/美/日 数学派 )

https://m.sohu.com/a/216801634_404328/

Note:

1. 奇特数学： 东汉 3AD  Chinese Remainder Theorem （韩信点兵 Modular Arithmetics）, Singapore Modelling Math (eg. 鸡兔问题) ，13CE 的 行列式 (Determinant, before “Matrix” was invented in 20th CE by JJ Sylvester )

2. Determinant was invented by the ancient Chinese Algebraists 李冶 / 朱世杰 /秦九韶 in 13th century (金 / 南宋 / 元) in《天元术》.The Japanese “和算” mathematician 关孝和 spread it further to Europe before the German mathematician Leibniz named it the “Determinant” in 18th century. The world, however, had to wait till the 19th century to discover the theory of Matrix 矩阵 by JJ Sylvester (Statistical Math private Tutor of Florence Nightingale, the world’s first nurse) closely linked to the application of Determinant.

# 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)

# HarmonyOS 2.0 Developer Platform

Google is stupid in blocking Huawei from Android, now Huawei Harmony OS 2.0 soon will be a rival to Android not only Mobile Phones but in 8 IoT devices : autopilot car, Smart TV, PC, Earphone, Fridge, etc. By using thru HarmonyOS phone, it is one click away to control all devices at home, without the trouble of AppleOS/Android complicated set-up. This is the “Future Home” made possible by Huawei with 5G.

https://www.theregister.com/2020/12/18/huawei_harmonyos_beta/

There is a reason why Huawei HarmonyOS shy from Python, because Python is not built for mobile phones & 8 IoT devices (TV, autopilot car, earphone, fridge, etc).

Huawei officially reveals Harmony OS, its first party operating system

AI Baking :

Remark:

By the way, the 1789 AD French Revolution Queen Marrie Antoinette’s famous quote of “WHY don’t the hungry farmers go eat Brioch ? “

“Brioch” = 50% Bread + 50% Cake

# 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

# Inequality Problem

For a, b natural numbers , there exists a natural number N:

$\boxed {1- \frac{1} {N} < \frac{a} {b} < 1- \frac{1} {N+1} \implies a+b <= 4N}$

[Proof ] see video for N =2020

Eg. N = 2020
a+b <= 4N =8080

# CRT Chinese Remainder Theorem

x＝ (a+1)+(a+2)+… (a+9) = 9a + 45

x = 9(a +5) +0

x = (b+1)+(b+2)+… (a+10) = 10b + 55

x = 10 (b +5) + 5

x = (c+1)+(c+2)+… (c+11) = 11c + 66

x = 11 (c+6) +0

CRT : (9,10), (9,11), (10,11) are pair-wise coprimes.

x = 0 (mod 9)

x = 5 (mod 10)

x = 0 (mod 11)

Notes: R = Z integer Ring

${X, r_9, r_{10} , r_{11} } \in R$

X = 9*11*5 + 9*10*0 + 10*11*0 (mod 9*10*11)

[/ 10… ]

X = 9*11*5 +0+0 = 495 = 490+5 = 5 (mod 10)

[/9… ] X = 0+0+0= 0 (mod 9)

[/11… ] X = 0+0+0=0 (mod 11)

X = 495 (mod 9*10*11)

Minimum X = 495

Note :

# China Covid19 Test 20 millions in 5 days : Use Math!

Why Chinese hospitals can test Covid19 at the speed of 20 million people in 5 days ?

Use Math !

Pre-requisite conditions :
1) <= 0.1% infected Covid19 patients in local population, then save 90% testing effort. If 9% infected, then only save 50%.

2) 10-in-1 混检 mixed test samples is the optimum algorithm : safety + accurate + efficient.

https://m.toutiaoimg.cn/a6902367747532194317/?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.

# 为啥法国数学这么强，平民的算数这么烂 ？

{?€ 6.5 是这样算的 ：313.5+0.5 [凑整]＝314.0, 然后 314＋6＝320，得[找] 0.566.5}

https://m.toutiaocdn.com/i6896389485832389134/?app=news_article&timestamp=1607277607&use_new_style=1&req_id=202012070200070100140540151610B97E&group_id=6896389485832389134&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# China made 18 q-bit QUANTUM Computer

Intel had made 17-qbit Quantum Computer, recently China just made 18 q-bit but with only 6 photons (vs Intel’s 17).

Application: Decypher a 300-digits password would take 150, 000 years on a traditional computer, but only 1 sec on a Quantum Computer.

18个量子比特纠缠是啥？量子计算机能破解银行密码？李永乐老师开讲

https://m.toutiaoimg.cn/a6575054600494121476/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 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

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# Differential Equations Versus Machine Learning

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

View at Medium.com

# 日本战后数学教育家 远山启

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# 听杨振宁讲物理课

[7.1] 好书推荐

1. E. T. Bell: 《Men of Mathematics” 》 – ［Comments:］ Irish Hamilton & Quartenions – Bell said Halmiton later’s life was a tragedy in trying to make quartenions universally applied in Physics but failed. Prof Yang : Quartenions will be “universally useful” in future.

Note : Quartenions (1,i,j,k) & Special Relativity – Does it ring a bell to you the UNIVERSAL 4 FORCES (Gravity, Electric, Magnetic, Particles Weak force) :

2. E. Segrè (吴健雄的博士导师) & the story of the missing Periodic Table Element ’43’ (Tc) discovered by accident from unwanted trash.

[7.2] 重力势能和弹性势能

https://m.toutiaoimg.cn/group/6608369844179960327/?app=news_article&timestamp=1605036314&group_id=6608369844179960327&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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