50 Years of IT Language War

50年编程语言榜单战争史,估计里面的开发语言很多你都不认识

Only C++ stays relevant in Top 10 position for 35 years.

https://m.toutiaoimg.cn/a6880830172367421963/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Feymann Calculus Trick

理查德·费曼非常聪明的求导 differentiate 技巧

Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.

https://m.toutiaocdn.com/i6933218183247626755/?app=news_article&timestamp=1614316375&use_new_style=1&req_id=2021022613125401019410001308000E75&group_id=6933218183247626755&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

林群:先通过案例学道理,后才证明

中科院院士著名数学家林群为大家分享数学怎么学

林群:先通过案例学道理,后才证明

中科院数学院士 林群:"如何学高深 抽象 数学":
先通过案例 (case / motivation) 学道理,后才学证明 (Theorem Proofs) 。

eg. Learn “Group Theory”
Case / Motivation: Symmetry

https://m.toutiaoimg.cn/a6783678419574981120/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Ellipse Proof by Circle : Affine Transformation

Cumbersome Ellipse proof below :

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ

数学中的冷知识:圆幂定理,现在初中课本中都不讲了,但是非常有趣?

圆幂 定理 II: The Affine Transformation FROM Circle to Ellipse. Also applicable to all other 2 Theorems.

https://m.toutiaoimg.cn/a6580683485986423309

Proof : Ellipse Area from Circle :

函数 Function 概念

【数学科普】:

Function = f : R – > R
(with 1& only 1 image)

Mapping 映射 (l’Application) , can be non-value to non-value

仿射变换 Affine Transformation (解析几何 Analytic Geometry 的 Modern Math).

Affine 是 德文 (coined by Euler)

https://m.toutiaoimg.cn/a6791714355483247119/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

丘成桐 的雄心计划: 培养800个数学天才苗子

当年汉朝 霍去病 带 800 骑士 深入漠北驱赶匈奴。

丘成桐 (属牛)的雄心计划: 培养800个数学天才苗子, 每年从中国&世界选16岁起的100个 华人,在清华少年班 8年培训:本科4年+硕2年+博士 2年.

https://m.toutiaoimg.cn/i6928283118424031751/?app=news_article&is_hit_share_recommend=0&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

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 ).

Homology

Part 1: “Homology” without Pre-requisites, except “Function” (he un-rigourously interchanges with “Mapping”, although Function is stricter with 1-and-only-1 Image) .

Part 2: Simplex (单纯形)

Topology History : Euler Characteristic eg. (V – E + R = 2) ,
Poincaré invention

This video uses Algebra of point, line, triangle… to explain a Simplex (plural: Simplices) in R^{n} Space, that is organizing the n-Dimensional “Big Data” data points into Simplices, then (future Part 3, 4…) compute the “holes” (or pattern called Persistent Homology).

Part 3: Boundary

Part 3 justifies why triangles (formed by any 3 data points) called “Simplex” 单纯形(plural: Simplices) are best to fill any Big Data Space.

《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

吴文俊:中学数学教育

吴文俊:数学教育(上)

要点:

1. 文化大革命 / 数学教育革命 灾难

2. 东方数学 : 小学 – 中学/高中数学 (机械化)eg. 四则运算

3. 西方数学 :大学”新”数学 (非机械化)

https://m.toutiaocdn.com/i6917519470810366472/?app=news_article&timestamp=1610825278&use_new_style=1&req_id=202101170327580100080431050833ACBD&group_id=6917519470810366472&wxshare_count=1&tt_from=weixin_moments&utm_source=weixin_moments&utm_medium=toutiao_android&utm_campaign=client_share&share_type=original

吴文俊 批评法国的1970s Bourbaki 数学进入中学/Bac 课本。美国也同时推行 New Math。

问题出在 数学老师的训练不够,太仓促。比如 Singapore Math 1980s 刚推行 也是很"乱",后来新加坡教育部 (MOE) 要求家长们也去学习,才能在家里辅导孩子。

吴文俊:数学教育(下)

他提倡中学数学教育 2 要点:
1) 教少一点 Euclidean Geometry, 用 Analytic Geometry取代, 才可以 computerise (机械化).

2) 中国古代是有 Geometry, 但 用天元(x), 地元(y) 代数(Algebraic) 化。

https://m.toutiaocdn.com/i6917528352391807499/?app=news_article&timestamp=1610902867&use_new_style=1&req_id=20210118010107010204051088185887A1&group_id=6917528352391807499&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

吴杰:拓扑世界的架构师

https://mp.weixin.qq.com/s?__biz=MzA5OTA2MTIzNw==&mid=2656737111&idx=2&sn=e494f6bb6aec2da14ace83220ff103ad&chksm=8b262bcdbc51a2dbcdbc310b96209ba3c47b07478e8db37acecfec050e110896850e5825be07&mpshare=1&srcid=0105hADjmNvluUUjsqfpjNx8&sharer_sharetime=1609899719523&sharer_shareid=82d451518f239e8840f13fc06d759dff&from=timeline&scene=2&subscene=1&clicktime=1609998317&enterid=1609998317&ascene=2&devicetype=android-28&version=27001639&nettype=WIFI&abtest_cookie=AAACAA%3D%3D&lang=en&exportkey=AhuDDklJOctFUKK4k%2Bsfn5o%3D&pass_ticket=gfaC%2F0RZPoeZYEePorezdPv248UbY30mbPoB8ygMtLtw1AlQf1GZnSWDfVaxHpDd&wx_header=1

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

何谓 “Free “ ?

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

3. Free Vector Space

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.

数学竞赛几何理论基础:分角定理、张角定理,推理证明

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

中国古代数学的三个高峰和落没

三高峰时期:

1. 秦汉:张苍&耿寿昌 编 《九章算术》, 3AD 东汉 刘徽 (注解)

2. 4AD 南北朝:祖冲之父子 圆周率π

3. 13-16世纪 元/明朝:珠算盘

落没

只是Applied 应用, 没有 希腊Deductive Theory 理论。

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数学武林的四大门派

数学武林的四大门派比喻:

[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/

英/美/日 数学都是支派,非主流宗派 掌门:

英国 受法国影响 : Newton 说他是站在巨人 Descartes, Fermat 的肩膀. 近代 Bertrand Russell 也是读法国Camile Jordan 的 《Group Theory 》书。

美国 数学 : WW2 德国/东欧 犹太人移民: Noether, Godel, Courant, Artin,…

日本数学 : 唐朝留学 中国 (日本”和算” 鼻祖 关孝和 把中国 “行列式” 传给欧洲Leibniz : Determinant ) ,明治天皇 时期 日本留学欧洲。

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

Baking recipes made by AI | Google Cloud

AI Baking :
“Breakie” = Bread + Cookie
“Cakie” = Cake + Cookie

https://cloud.google.com/blog/topics/developers-practitioners/baking-recipes-made-ai

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

Cake-like “high-class” bread ‘Brioch’ made by my bread machine.
Butter-rich “Brioch”

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”.

4) Noble Prize Economics awarded to “ Black-Sholes Formula”

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