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

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

《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

Octonions & United 4 Forces

剑桥数学物理学家以八元数打开“万有理论”研究思路

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.

Particle Physics solved by Algebraic Topology with Co-Homology structure

物理学家已经确定了粒子碰撞的代数结构,这将诞生新的物理理论

Keywords: Algebraic Topology 代数拓扑, Co-Homology structure 上同调结构

https://m.toutiaocdn.com/i6886832762934788620/?app=news_article&timestamp=1603662776&use_new_style=1&req_id=20201026055255010014047206182262C3&group_id=6886832762934788620&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

《万物皆数》

[中文书评]

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

Audio book ]: It all adds up – the story of people & Mathematics

https://m.toutiaoimg.cn/i6857428347115766286/?app=news_article_lite&timestamp=1596647454&use_new_style=1&req_id=202008060110540100160281340A44B4BF&group_id=6857428347115766286

化学:金属活动性顺序表

中三化学刘老师教我们背这个金属活动性顺序表, 排前面的元素比排后面的活跃,写化学反应方程式时,非常有用。基本上整个 Physcial Chemistry 就是浓缩在这表里,朗诵上口,终身难忘,50年后还记得!- 化学靠记忆,用中文学化学最方便 。(我在法国精英大学 Classe Preparatoire 念化学,打败全班法国人得第一名,中学用中文读化学的基础, 功不可没。)

铅 (氢)

K Na Ca Mg Al Zn Fe Sn Pb (H)

[嫁 倩 (轻)]

Cu Hg Ag Pt Au

[统 金]

Relativistic Space Time, Fengshui, Isometry

风水 Fengshui (aka Geomancy) is Einstein Relativistic using Compass (东西南北上下 =Space) & Timing (吉凶日/时间 = auspicious date & time).

\Yu (Space) \Zhou (Time) = Space-Time
eg.
宇环 (universal surrounding)
宙时 (instant timing)

https://www.asiatimes.com/2019/08/opinion/why-quantum-physics-needs-asian-philosophy/

Axonometry = Isometry ( dengjiao toushi 等角透视,ie Angle Preserved Projection )

Chinese art is axonometric using isometry geometry, scrolling from right to left along time. Example : this digital animated 12-century painting

清明上河图 Qingming Festival River Scene(北宋)

“Why the world relies on a Chinese ‘perspective’” by Jan Krikke, China, AI and Quantum Physics

https://link.medium.com/z5LaoL0tdZ

New Math applied in Modern Physics

【文小刚:物理学的新革命——凝聚态物理中的近代数学 | 众妙之门】
https://m.toutiaocdn.com/group/6712424299841454595/?app=news_article_lite&timestamp=1562923076&req_id=20190712171755010152027098774DD5D&group_id=6712424299841454595&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

Keywords:

  • Quantum Entanglement (QE) : 量子纠缠
  • Topological Order (Long-range QE) : 拓扑序
  • Topological Insulator (Short-range QE) : 拓扑绝缘体
  • Rigid State : 凝聚态
  • Cohomology : 上同调
  • Tensor : 张量
  • Category Theory : 范畴学
  • Group Theory : 群论
  • Algebraic Topology : 代数拓扑

….

1. Four “Physics” Revolutions aided by Math

Math is applied to explain the Nature (Physics) :

Calculus (Differential Equation) : Newtonian Physics

Riemann Geometry : Einstein General Relativity

Partial Differential Equation: Maxwell Electro-Magnetic Field

Linear Algebra (Matrix) : Quantum Mechanics

Group Theory: Physics / Chemistry Symmetry

Next…

Category Theory : Topological Order of the long-range Quantum Entanglement

Algebraic Topology: Topological Insulator (2016 Nobel Prize Physcis)

2. The 5th Physics Revolution aided by New Math tool : Category Theory

3. Category Theory = “Structural Relationship” Math

4. Algebraic Topology

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http://app.toutiao.com/news_article/?utm_source=link

丘成桐:数理与人文(原创完整版)

【丘成桐:数理与人文(原创完整版)】
https://m.toutiaocdn.com/group/6685925279874941444/?app=news_article_lite&timestamp=1561828635&req_id=20190630011715010152030216506C006&group_id=6685925279874941444&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(想看更多合你口味的内容,马上下载 今日头条)
http://app.toutiao.com/news_article/?utm_source=link

信息熵 (Information Entropy) 是怎样炼成的

【信息熵是怎样炼成的 | 纪念信息论之父香农】
https://m.toutiaocdn.com/group/6685498360318657036/?app=news_article_lite&timestamp=1561737552&req_id=201906282359120101520481610092161&group_id=6685498360318657036&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share
(想看更多合你口味的内容,马上下载 今日头条)
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AI with Advanced Math helps in discovering new drugs

https://theconversation.com/i-build-mathematical-programs-that-could-discover-the-drugs-of-the-future-110689?from=timeline

Advanced Mathematical Methods with AI is a powerful tool:

  • Algebraic Topology (Persistent Homology)
  • Differential Geometry
  • Graph Theory

https://sinews.siam.org/Details-Page/mathematical-molecular-bioscience-and-biophysics-1

In Math We Trust: Quantum Computing, AI and Blockchain – The Future of IT

In memory of Prof Zhang Shoucheng 张首晟教授 who passed away on 1 Dec 2018.

Key Points :

  1. Quantum Computing with “Angel Particle” (no anti-particle) : [Analogy] Complex Number (a + i.b) , ‘Anti’ = Conjugate = a – i.b, ‘No anti’ = Real number = a
  2. A. I. Natural Language Algorithm : “Word To Vector” eg. King / Queen (frequently appear together) , etc.
  3. Data Privacy and Big Data Analytics with A. I. : Homomorphic Encryption, ie reveal data but not privacy. (eg. Millionaire Problem)

芯片的基本原理是什么?

芯片 (Chips) 的种类:

  1. CPU 中央处理机
  2. Memory 记忆体储存

材料: 硅 (\gui) Silicon 取自 沙 (sand)

中兴 ZTE 禁令之芯片 为什么这么难做

步骤:

  1. Design 设计 (最难 !)
  2. Manufacture 制作
  3. Test & Packaging 测试&封装

2003 上海交大 微电子学院院长 陈进 的上亿骗局 “汉芯”一号 (其实是 Motorola Chip) 害中国退后十多年 !

Global Foundry: how a CPU is made ?

Joseph Fourier is Still Transforming Science

Key Words: 250 years anniversary

  • Yesterdays: Fourier discovered Heat is a wave , Fourier Series, Fourier Transformation, Signal processing…
  • Today: IT imaging JPEG compression, Wavelets, 3G/4G Telecommunications, Gravitational waves …
  • Friends / bosses: Napoleon, Monge… Egypt Expedition with Napoleon Army.
  • Taught at the newly established Military Engineering University “Ecole Polytechnique”.
  • Scientific Research: Short period but intense.
  • Before Fourier died (he wrapped himself with thick blanket in hot summer), he was reviewing another young Math genius Evariste Galois’s paper on “Group Theory”.

https://news.cnrs.fr/articles/joseph-fourier-is-still-transforming-science

Mind over Matter : Bruce Lipton

Key Points:

1. The Foundation of Science has changed since Newtonian Era, except Biology / Medicine and Psychology which don’t keep up:

Math => Fractal Math

Newton Physics (Matter)=> Quantum Physics (Energy)

(Organic) Chemistry => Electro-Chemistry

Energy = Field = 气 Chi / Qi

3. Environment -> Mind -> Perception -> Genetic Control

4. Consciousness (5%), Subconscious-ness (95%)

Examples:

  • Driving car: inexperienced driver (consciousness), experienced driver (subconscious).
  • Dating (conscious behavior)

Note: The successful Deeplearning AI is using the concept of Perception called “Perceptrons” by analysing the Environmental (BIG DATA) pattern with the help of Mathematics (Calculus : Gradient Descent, Statistics : Bayesian Probability, Algebraic Topology, etc).

英国留学生带你回忆被数学支配的恐惧

这位中国留英数学专业的小伙子有很好的口才: 数学之美, 他的数学遗憾。

华威大学(The University of Warwick),famous for mathematics in UK.

Key Points:

Einstein用 Riemann Geometry 数学救了Newton 物理。

中国数学的没落: 自从明朝科举废除数学考试

3位大师救近代中国数学: 华罗庚, 陈省身, 苏步青

数学光明的未来: AI, Big Data, Cloud Computing

天才的崎岖道路: “扫地僧”张益唐

时代的”指数” exponential快脚步, 不要追风, 要一以贯之 : Prof. Andrew Wiles proved 350-year-old “Fermat’s Last Theorem” (FLT).

[纠正]: Andrew Wiles 超过40岁, 没赶上Fields Medal, 只得个”奖励”。他看到椭圆(Ellipse)气球, 得到突破 FLT “工具”的灵感 – “Elliptic Curve” 。

数学是什么: 爱情, 艺术, 音乐, 科技

Symmetry, Algebra and the “Monster”

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.

https://www.quantamagazine.org/symmetry-algebra-and-the-monster-20170817/

Summary:

  • Symmetry of a Square
  • Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
  • What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
  • Monster Group = God ?
  • String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry),是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

Is There a Multi-dimensional Mathematical World Hidden in the Brain’s Computation?

Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) &  Computers.

Homology = compute the number of “holes” in multi-dim space. 

Neurons formed in the brain can be modeled in Math (Topology) by Simplex 单纯 (plural : Simplices), billions of them interconnected into a complex – “Simplicial Complex” (单纯复体)。

https://singularityhub.com/2017/06/21/is-there-a-multidimensional-mathematical-world-hidden-in-the-brains-computation/?from=timeline#.WUvPNsvmjqD

Curious Thoughts in Math & Science 

1. Statistical Mechanics: e^ {- Ht}

Quantum Mechanics: e^{iHt}

2. Ramanujian:

1 +2 + 3 + …+ n = –  1/12

Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

Tau Special Function:

\boxed {\displaystyle \sum_{n=1}^{\infty}\tau (n) x^{n} = x \{(1-x)(1-x^{2})(1-x^{3})... \}^{24}}

3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843).

AND\boxed {x.y} 

NOT\boxed {1-x} 

XOR\boxed {x+y-2x.y} 

Extra constraints ”  \boxed {x^{2}=x} 

 4. Solomon Golomb, Sol: “Linear Feedback Shift Register” (LFSR)  — shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

Maximal Length = The shift register of size n will repeat every 2^{n}-1 steps (exclude all ‘0’ sequence).

Which arrangement of “Taps” will produce the maximal length ? 

Solomon applied Pure Math : represent the above sequence of  registers algebraically  by: 

\boxed {x^{7}+x^{6}+ 1} 

in reducible modulo 2 (prime in polynomial, ie can’t be factored).

=> the sequence is Maximal length

LFSR Applications: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance,  etc.

Arrow’s Theorem

The Axioms for a fair voting system (eg. Political Election) :

1. Non-dictatorship:
◇ Outcome decided not by one ‘dictator’, but more than one person.

2. Unanimity
◇ If everyone votes that A is better than B, then A will be ranked higher than B in the final result.

3. Independence of irrelevant alternatives
◇ The ranking of A relative to B should not be affected by someone changing their mind about C.

Arrow’s Theorem says that if there are more than 2 people (parties) to vote for, then there is no fair voting system.

Note:
Most democratic voting systems violate the 3rd axiom (independence of irrelevant alternatives).

The Future of World Economy: Technologies

An excellent article on the Future of world economy.  A view from Stanford University….

3 Keywords: Clean Energy, Robotics,  3D-Printing

Governments, businesses, and economists have all been caught off guard by the geopolitical shifts that happened with the crash of oil prices and the slowdown of China’s economy. Most believe that the price of oil will recover and that China will continue its rise. They are mistaken. Instead of worrying about the rise of China, we need to fear its fall; and while oil prices may oscillate over the next four or five years, the fossil-fuel industry is headed the way of the dinosaur. The global balance of power will shift as a result.

LED light bulbs, improved heating and cooling systems, and software systems in automobiles have gradually been increasing fuel efficiency over the past decades. But the big shock to the energy industry came with fracking(水力压裂), a new set of techniques and technologies for extracting more hydrocarbons from the ground. Though there are concerns about environmental damage, these increased the outputs of oil and gas, caused the usurpation of old-line coal-fired power plants, and dramatically reduced America’s dependence on foreign oil.

The next shock will come from clean energy. Solar and wind are now advancing on exponential curves. Every two years, for example, solar installation rates are doubling, and photovoltaic-module costs are falling by about 20 percent. Even without the subsidies that governments are phasing out, present costs of solar installations will, by 2022, halve, reducing returns on investments in homes, nationwide, to less than four years. By 2030, solar power will be able to provide 100 percent of today’s energy needs; by 2035, it will seem almost free — just as cell-phone calls are today.

This seems hard to believe, given that solar production provides less than one percent of the Earth’s energy needs today. But this is how exponential technologies advance. They double in performance every year or two and their prices fall. Given that California already generates more than 5 percent of its electricity from utility-scale solar, it is not hard to fathom what the impact of another few doublings would be: the imminent extinction of the fossil-fuel industry. Exponential technologies are deceptive because they move very slowly at first, but one percent becomes two percent, which becomes four, eight, and sixteen; you get the idea. As futurist Ray Kurzweil says, when an exponential technology is at one percent, you are halfway to 100 percent, and that is where solar and wind energies are now.

Anyone tracking the exponential growth of fracking and the gradual advances that were being made in conservation and fuel efficiency should have been able to predict, years ago, that by 2015, the price of oil would drop dramatically. It wasn’t surprising that relatively small changes in supply and demand caused massive disruptions to global oil prices; that is how markets work. They cause commodities futures and stock prices to fall dramatically when slowdowns occur. This is what is happening to China’s markets also. The growth of China’s largest industry, manufacturing, has stalled, causing ripple effects throughout China’s economy.

For decades, manufacturing was flooding into China from the U.S. and Europe and fueling its growth. And then a combination of rising labor and shipping costs and automation began to change the economics of China manufacturing. Now, robots are about to tip the balance further.

Foxconn (Apple Contract Manufacturer for iPhones) had announced in August 2011 that it would replace one million workers with robots. This didn’t occur, because the robots then couldn’t work alongside human workers to do sophisticated circuit board assembly. But a newer generation of robots such as ABB’s Yumi and Rethink Robotics’ Sawyer can do that. They are dextrous enough to thread a needle and cost as much as a car does.

China is aware of the advances in robotics and plans to take the lead in replacing humans with robots. Guangdong province is constructing the world’s first “zero-labor factor,” with 1,000 robots which do the jobs of 2,000 humans. It sees this as a solution to increasing labor costs.

The problem for China is that its robots are no more productive than their counterparts in the West are. They all work 24×7 without complaining or joining labor unions. They cost the same and consume the same amount of energy. Given the long shipping times and high transportation costs it no longer makes sense to send raw materials across the oceans to China to have them assembled into finished goods and shipped to the West. Manufacturing can once again become a local industry.

It will take many years for Western companies to learn the intricacies of robotic manufacturing, build automated factories, train workers, and deal with the logistical challenges of supply chains being in China. But these are surmountable problems. What is now a trickle of manufacturing returning to the West will, within five to seven years, become a flood.

After this, another technology revolution will begin: digital manufacturing.

In conventional manufacturing, parts are produced by humans using power-driven machine tools, such as saws, lathes, milling machines, and drill presses, to physically remove material to obtain the shape desired. In digital manufacturing, parts are produced by melting successive layers of materials based on 3D models — adding materials rather than subtracting them. The “3D printers” that produce these use powdered metal, droplets of plastic, and other materials — much like the toner cartridges that go into laser printers. 3D printers can already create physical mechanical devices, medical implants, jewellery, and even clothing. But these are slow, messy, and cumbersome — much like the first generations of inkjet printers were. This will change.

In the early 2020s we will have elegant low-priced printers for our homes that can print toys and household goods. Businesses will use 3D printers to do small-scale production of previously labor-intensive crafts and goods. Late in the next decade, we will be 3D-printing buildings and electronics. These will eventually be as fast as today’s laser printers are. And don’t be surprised if by 2030, the industrial robots go on strike, waving placards saying “stop the 3D printers: they are taking our jobs away.”

The geopolitical implications of these changes are exciting and worrisome. America will reinvent itself just as does every 30-40 years; it is, after all, leading the technology boom. Oil producers such as Venezuela will go bankrupt; the Middle East will become a cauldron of instability. Countries that have invested in educating their populations, built strong consumer economies, and have democratic institutions that can deal with social change will benefit — because their people will have had their basic needs met and can figure out how to take advantage of the advances in technology.