Author: tomcircle

Category Theory (CT) is like Design Pattern, only difference is CT is a better mathematical pattern which you can prove, also it has no “SIDE-EFFECT” and with strongTyping.

The examples use Haskell to explain the basic category theory : product, sum, isomorphism, fusion, cancellation, functor…

10.1Monads

$latex begin{array}{|l|l|l|}
hline
Analogy & Compose & Identity
hline
Function & : : : : “.” & : : : : Id
hline
Monad & ” >> = ” (bind) & return :: “eta”
hline
end{array}$

Imperative (with side effects eg. state, I/O, exception ) to Pure function by hiding or embellishment in Pure function but return “embellished” result.

Monad = functor T + 2 natural transformations

$latex boxed {text {Monad} = {T , eta , mu} }&fg=aa0000&s=3$

$latex eta :: Id dotto T$
$latex mu :: T^{2} dotto T$
$latex text {Natural Transformation : } dotto $

http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html#functors

In essence, in all kinds of Math, we do 3 things:

1) Find Pattern among objects (numbers, shapes, …),
2) Operate inside the objects (+ – × / …),
3) Swap the object without modifying it (rotate, flip, move around, exchange…).

Category consists of :
1) Find pattern thru Universal Construction in Objects (Set, Group, Ring, Vector Space, anything )
2) Functor which operates on 1).
3) Natural Transformation as in 3).

$latex boxed {text {Natural Transformation}}&fg=000000&s=3$
$latex Updownarrow $

$latex boxed {text {Morphism of Functors}}&fg=aa0000&s=3$

Analogy:

Functors (F, G) :=operation inside a container
$latex boxed { F :: X to F_{X}, : F :: Y to F_{Y}}&fg=0000aa&s=3$

$latex boxed {G :: X to G_{X}, : G :: Y to G_{Y}}&fg=00aa00&s=3$

Natural Transformation ($latex {eta_{X}, eta_{Y}}&fg=ee0000&s=3$) := swap the content ( $latex F_{X} text { with } G_{X}, F_{Y} text { with } G_{Y} $) in the…

https://en.m.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz

See the 1970s FrenchBaccalaureate Math Textbooks:(for UK Cambridge GCE A-level Math students, this is totally new “New Math” to us !)

Curry-Howard-Lambek Isomorphism:

$latex boxed {text {Category Theory = High School Algebra = Logic = Lambda Calculus (IT)}}&fg=aa0000&s=3$

Below the lecturer said every aspect of Math can be folded out from Category Theory, then why not start teaching Category Theory in school.

That was the idea proposed by Alexander Grothendieckto the Bourbakian Mathematicians who rewrote all Math textbooks after WW2, instead of in Set Theory, should switch to Category Theory. His idea was turned down by André Weil.

Facebook rewrote the SPAM rule-based AI engine (“Sigma“) with Haskell functional programming to filter 1 million requests / second.

The Myths about Haskell : Academia, Not for Production ?

Why Facebook choosesHaskell Functional language for Spam rule engine?

Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming:

1. Imperative (or Procedural) – micro-steps or Local 微观世界
2. Declarative (or Functional) – Macro-view or Global 大千世界

In Math:

1. Analysis (Calculus)
2. Algebra (Structures, Category)

In Physics:

1. Newton (Law of Motions), Maxwell (equations)
2. Fermat (*) (Light travels in least time), Feynman (Quantum Physics).

In IT: Neural Network (AI) uses both 1 & 2.

More examples…

In Medicine:

1. Western Medicine: germs/ viruses, anatomy, surgery
2. Traditional Chinese Medicine (中医): Accupunture, Qi, Yin-Yang.

Note (*): Fermat : My alma mater university in Toulouse (France) named after this 17CE amateur mathematician, who worked in day time as a Chief Judge, after works spending time in Math and Physcis. He co-invented Analytic Geometry (with Descartes), Probability (with Pascal), also was the “Father of Number Theory” (The Fermat’s ‘Little’ TheoremandThe Fermat’s ‘Last’ Theorem). He…

[Continued from 1.1 to 2.2]

3.1 MonoidM (m, m)

Same meaning in Category as in Set: Only 1 object, Associative, Identity

Thin / Thick Category:

• “Thin” with only 1 arrow between 2 objects;
• “Thick” with many arrows between 2 objects.

Arrow : relation between 2 objects. We don’t care what an arrow actually is (may be total / partial order relations like = or $latex leq $, or any relation), just treat arrow abstractly.

Note: Category Theory’s “Abstract Nonsense” is like Buddhism “空即色, 色即空” (Form = Emptiness).

Example ofMonoid: String Concatenation: identity = Null string.

Strong Typing: function f calls function g, both types must match.

Weak Typing: no need to match type. eg. Monoid.

Category induces a Hom-Set: (Set of “Arrows”, aka Homomorphism同态, which preserves structure after the “Arrow”)

• C (a, b) : a -> b
• C (a,a) for Monoid…

Top 5:

1. MIT
2. Harvard
3. Stanford
4. Oxford
5. Cambridge

18 University of Tokyo
…
20 Peking University 北京大学
…
22 Ecole Polytechnique (France)
…
26 TsingHua University 清华大学
…
28 Hong Kong University
…
32 Ecole Normale Supérieure (Paris)

https://www.theguardian.com/education/2017/mar/08/qs-world-university-rankings-2017-mathematics

During the 19th century French Revolution, a young French boyEvariste Galoisself-studied Math and invented a totally strange math called “Group Theory“, in his own saying – “A new Mathnot on calculation but on reasoning”. During his short tragic life (21 years) his work was not understood by the world masters like Cauchy, Fourier, Poisson, Gauss, Jacob…

“Group Theory” is the foundation of Modern Math today.

Object-Oriented has 2 weaknesses for Concurrency and Parallel programming :

1. Hidden Mutating States;
2. Data Sharing.

Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>

$latex boxed {text {CT reveals the way how our brain works by analysing, reasoning about structures
!}}&fg=aa0000&s=3$

Our brain works by: 1) Abstraction 2) Composition 3) Identity (to identify)

What is a Category ?
1) Abstraction:

• Objects
• Morphism (Arrow)

2) Composition: Associative
3) Identity

Notes:

• Small Category with “Set” as object.
• Large Category without Set as object.
• Morphism is a Set : “Hom” Set.

Example in Programming:

• Object : Types Set
• Morphism : Function “Sin” converts degree to R: $latex sin frac {pi}{2} = 1$

Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget”…

Haskell is the purest Functional Language which is based on Category Theory.

eBook:

http://learnyouahaskell.com/chapters

http://www.math.niu.edu/%7Ebeachy/abstract_algebra/study_guide/contents.html

The Study Notes on 600 problems and solutions:

http://www.math.niu.edu/~beachy/abstract_algebra/guide/contents.html

Study Tips: http://www.youtube.com/playlist?list=PLi01XoE8jYoi-BCj5m0rRW03vqxaQrH7u

Suitable for Upper Secondary School and Junior College Math Students.

Abstract Algebra is scary because it is abstract, and its Math Profs are mostly fierce – but not with this pretty Math lady…

WHAT IS A FIELD (域) ?

WHAT IS A VECTOR SPACE (向量空间) ?

See all 20+ videos here:

There are 5 great Geometry Masters in history: 欧高黎嘉陈

Euclid (300 BCE, Greece), Gauss (18CE, Germany), Riemann (19CE, Germany), Cartan (20CE, France), Chern (21CE, China).

Jim Simons (Hedge Fund Billionaire, Chern’s PhD Student) quoted Chern had said to him:

“If you do One Thing that is reallygood, that’s all you could really expect in a life time.” 一生作好一件事, 此生无悔矣!

Highlights:

1. Video below @82:00 mins, SS Chern criticised on Hardy’s famous statement: “Great Math is only discovered by young mathematicians before 30.” Chern’s response: “Don’t believe it ! 不要相信它”.

2. Chern’s Conjecture :“21世纪中国将是数学大国。 ” China will be a Math Kingdom in 21st century.

http://www.bilibili.com/mobile/video/av1836134.html

In 15 years, AI driven driverless car will change the transport/work/environment landscape… it is true not futuristic… behind AI is advanced math which teaches computer to learn without a fixed algorithm but by analysing BIG DATA patterns using Algebraic Topology !
世界趨勢，可作參考

矽谷预测AI後的10年大未來

現在因為人工智能(AI)的發展，配合更高速度的積體電路，科技正在加快速度的進展。據悉，在很短的5 -10年後，医療健保、自駕汽車、教育、服務業都將面臨被淘汰的危機。

1. Uber 是一家軟體公司，它沒有擁用汽車，卻能夠讓你「隨叫隨到」有汽車坐，現在，它已是全球最大的Taxi公司了。
2. Airbnb 也是一家軟體公司，它沒有擁有任何旅館，但它的軟體讓你能夠住進世界各地願出租的房間，現在，它已是全球最大的旅館業了。
3. 今年5月，Google的電腦打敗全球最厲害的南韓圍棋高手，因為它開發出有人工智能(AI)的電腦，使用能夠「自己學習」的軟體，所以它的AI能夠加速度的進步，達到比專家原先預期的、提前10年的成就。
4. 在美國，使用IBM 的Watson電腦軟體，你能夠在幾秒內，就有90%的準確性的法律顧問，比較起只有70% 準確性的人為律師，既便捷又便宜。
所以，你如果還有家人親友在讀大學的法律系，建議他們停學省錢，因為市場已大幅的縮減了，未來的世界，只需要現在10%的專業律師就夠了。
5. Watson 也已經能夠幫病人檢驗癌症，而且比醫生正確4 倍。
6. 臉書也有一套AI的軟體可以比人類更準確的鑒察(辨識)人臉，而且無所不在。
7. 到了2030年，AI的電腦會比世界上任何的專家學者還要聰明。
8. 2017年起，會自動駕駛的汽車就可以在公眾場所使用。

約在2020年，整個汽車工業就會遭遇到全面性的改變，你再不需要擁用汽車。

你可以用手機叫自動駕駛的車，來帶你到你想去的目的地。
9. 未來的世界，你再也不必擁有車，或花時間加油、停車、排隊去考駕照、交保險費，尤其是城市，將會很安靜，走路很安全，因為90%的汽車都不見了，以前的停車場，將會變成公園。
10. 現在，平均每10萬公里就有一次車禍，造成每年全球有約120萬人的死亡。

以後有AI電腦控制的自動駕駛汽車，平均每1000萬公里才有一次車禍，約減少一百萬人死亡。

因為保險費和需要保險的人極少，保險公司會面臨更多的倒閉風潮。
11. 大部份的傳統汽車公司會面臨倒閉。Tesla、 Apple、及 Google 的革命性軟體，將會用在每一部汽車上。

據悉，Volkswagen 和 Audi 的工程師非常擔心Tesla革命性的電池和人工智能軟體技術。
12. 房地產公司會遭遇極大的變化。

因為你可以在車程中工作，距離將不是選住房屋的主要條件之一。市民會選擇住在較遠、但是較空曠且環境優美的鄉村。
13. 電動汽車很安靜，會在2020變成主流。所以城市會很變成安靜，而且空氣乾淨。
14. 太陽能在過去30年也有快速的進展。 去年，全球太陽能的增產超過石油的增產。

預計，到2025年時，太陽能的價格(低廉)會使煤礦業大量的破產。

因為電費非常的便宜，淨化水及海水淡化的費用大減，人類將能解決人口增加的需水問題。
15. 健保：今年醫療設備商會供應如同「星球大戰」電影中的 Tricorder，讓你的手機做眼睛的掃瞄，呼吸氣體及血液的化學檢驗：用54個「生物指標」，就可檢驗出你是否有任何疾病的徵兆。

因為費用低，幾年後，全球人類都可以有世界級的疾病預防服務。
16. 立體列印(3D printing)：預計10 年內，3D列印設備會由近20000美元減到400美元，而速度增加100倍快。

所有的「個人化」設計鞋子，將開始用這種設備生產，其他如大型的機場，其零件也能使用這種設備供應，至於人類太空船，也會使用這種設備。
17. 今年底，你的手機就會有3D掃瞄的功能，你可以測量你的腳送去做「個人化」鞋子。據悉，在中國，他們已經用這種設備製造了一棟6 層樓辦公室，預計到2027年時， 10% 的產品會用3D的列印設備製造。
18. 產業機會：

a. 工作：20年內，70-80% 的工作會消失，即使有很多新的工作機會，但是不足以彌補被智能機械所取代的原有工作。
b. 農業：將有 $100 機械人耕作，不必吃飯、不用住宅、及支付薪水，只要便宜的電池即可。在開發國家的農夫，將變成機械人的經理。溫室建築物可以有少量的水。

到2018年，肉可以從實驗室生產，不必養豬、雞或牛。30%用在畜牧的土地，會變成其他用途的土地。很多初創公司會供給高蛋白質的昆蟲當成食品。
c. 到2020年時 ，你的手機會從你的表情看出，與你說話的人是不是說「假話」？ 是否騙人的？ 政治人物(如總統候選人)若說假話，馬上會被當場揭發。
d. 數位時代的錢，將是Bitcoin ，是在智能電腦中的「數據」。
e. 教育：最便宜的智能手機在非州是$10美元一隻。
f. 到2020年時，全球70%的人類會有自己的手機，所以能夠上網接受世界級的教育，但大部份的老師會被智能電腦取代。所有的「小學生」都要會寫 Code，你如果不會，你就是像住在Amazon森林中的原住民，無法在社會上做什麼。你的國家，你的孩子準備好了嗎？

參考一下；這也是矽谷 VC, Innovators,Entrepreneurs … 談的資料。

辛弃疾的《青玉案·元夕》：“…众里寻他千百度；蓦然回首，那人却在灯火阑珊处。” –表达出了我的一种 (网上)意外相逢的喜悦，又表现出对心中(名师)的追求。

2011 年 北京大学教授丘维声教授被邀给清华大学 物理系(大学一年级) 讲一学期课 : (Advanced Algebra) 高等代数, aka 抽象代数 (Abstract Algebra)。

丘维声（1945年2月－）生于福建省龙岩市[1]，中国数学家、教育家。16岁时以全国高考状元的成绩考入北京大学，1978年3月至今担任北京大学数学科学学院教授，多年坚持讲授数学专业基础课程[2]。截至2013年，共著有包括《高等代数（上册、下册）》、《简明线性代数》两本国家级规划教材在内的40部著述[3]。于1993-97年的一系列文章中逐步解决了n=3pr情形的乘子猜想，并取得了一系列进展[2]。

———————

72岁的丘教授学问渊博, 善于启发, 尤其有别于欧美的”因抽象而抽象”教法, 他独特地提倡用”直觉” (Intuition) – 几何概念, 日常生活例子 (数学本来就是源于生活)- 来吸收高深数学的概念 (见:数学思维法), 谆谆教导, 像古代无私倾囊相授的名师。

全部 151 (小时) 讲课。如果没时间, 建议看第1&第2课 Overview 。

http://www.bilibili.com/mobile/video/av7336544.html?from=groupmessage

第一课: 导言 : n 维 方程组 – 矩阵 (Matrix)-n 维向量空间 (Vector Space) – 线性空间 (Linear Space)

第二课:

上表 (左右对称):

双线性函数 (Bi-linear functions) / 线性映射 (Linear Map)

线性空间 + 度量 norm =>

• Euclidean Space (R) => (正交 orthogonal , 对称 symmetric)变换
• 酉空间 Unitary Space (C)… =>酉变换, Hermite变换

近代代数 (Modern Math since 19CE Galois): 从 研究 结构 (环域群) 开始: Polynomial Ring, Algebraic structures (Ring, Field, Group).

第三课: 简化行阶梯形矩阵 Reduced Row Echelon Matrix

第四课: 例子 (无解)

第五课: 证明 无解/唯一解/无穷解
[几何直觉]: 任何2线 1) 向交(唯一解) ; 2) 平行 (无解) ; 3) 重叠 (无穷解)

n次方程組的解也只有3个情况:

无解:

中国”考研”究生:

考题难, 重视理论基础, 不是技巧。计算量大, 时间(3小时)不够。

国家 “及格” 底线 : 58~ 90分 (总分 : 150 分) – 根据 理工 / 经管系 , 不同重点大学, 底线各异。

http://www.bilibili.com/mobile/video/av2261356.html

[例子] $latex p (x) = a + bx+cx^{2}+dx^{3}$

$latex p(x) – tan x sim x^{3}, text { when } x to 0$

Find a, b, c, d ?

[Solution] :

1. Don’t use l’Hôpital Rule for $latex displaystyle lim frac {f}{g}$

2. Apply Taylor expansion :

$latex tan x = x + frac {1}{3}x^{3} + o (x^{3})$
$latex p (x) – tan x = a + (b -1)x + (c – frac {1}{3})x^{3} + o (x^{3})$

$latex p(x) – tan x sim x^{3}, text { when } x to 0 $

$latex iff boxed {a=0, b=1, c=frac {4}{3}}&fg=aa0000&s=2$

Prelude:

Harvard Lecture:

The Key toopen this secret …

不变子空间: Invariant Sub-space

第一课: Direct Sum 直和 $latex oplus$of Representations

直和 = $latex {oplus}&fg=aa0000&s=3$

第二课: 群表示可约 Reducible Representation

Analogy :
Prime number decomposition
Irreducible Polynomial

外直和 : $latex { dot{ +} }&fg=aa0000&s=3$

$latex boxed { displaystyle phi_{1} dot {+} phi_{2} = tilde {phi_{1}} oplus tilde {phi_{2}}}&fg=aa0000&s=3$

* 第三课: 完全可约表示 Completely Reducible Representation

完全表示是可 完全分解为 不可约表示 的一种表示。

完全可约表示 => 其子表示 也 完全可约。
不可约 一定是完全可约的!
一次表示一定是不可约的!
[Analogy: Polynomial degree 1 (x + 1) is irreducible. ]

註: (*) 深奥课, 可以越过直接跳到结果。(证明 待以后 复习)。

集合证明: 交(和)⊇和(交)

如果 也是⊆ , 则 交(和) =和(交)
Ref 2 《高代》 Pg 250 命题 1

$latex boxed {U cap (U_{1} oplus W) supseteq (U cap U_{1} ) oplus (U cap W)}&fg=aa0000&s=3$
Also,
$latex U cap (U_{1} oplus W) subseteq (U cap U_{1} ) oplus (U cap W)$
Then,
$latex boxed {U…

第七课 Group Action群作用

$latex x_{i} in Omega = big{0.x_{1},0.x_{2},…, 0.x_{i-1},1.x_{i}, 0.x_{i+1}, ….0.x_{n} big}$

…

第11课:Cyclic Group (循环群) Representation , Dihedron 二面体

$latex begin{pmatrix}
0 & 0 & 1
1 & 0 &…
0 & 1 &…
end{pmatrix} = P (a) $3 阶 Cyclic Group (循环群) Representation

$latex boxed{ Bigr|D_{n} Bigr| = 2n }&fg=aa0000&s=3$

Download eBook (PDF) here:

[Part 1 引言 : 温习]
[Part 2 群的基础概念 : 温习]

北大: 丘维声

Part 1 & 2 : 本科班 (Undergraduate) 数学 温习

Part 3 开始: 研究班 (Graduate) 数学

第一课 群表示 Group Representation

Φ: Group homomorphism 群同态
V: Linear Space 线性空间 (K 域上 Over Field K) => 表示空间

有限 V => deg (Φ) : 次数 / 维数

无限 V => 无限维

$latex boxed {text {Group Representation : }(phi, V)}&fg=aa0000&s=3$

群表示: 通过研究 1)Φ 同态 2) 像 = 线性空间3)Φ核 = Normal Subgroup => 了解 群

KerΦ = {e} =>Φinjective =>ΦFaithful 忠实表示

KerΦ = G =>Φ平凡表示 (全部G 都映射到 零, 平凡)

若 平方表示Φ 是一次的 ( 即V 是 1 维) => 主表示 (或 单位表示)

$latex boxed {GL(V) cong GL_{n} (K)}&fg=aa0000&s=2$ 可逆矩阵

$latex boxed { Phi : G to GL_{n} (K)}&fg=aa0000&s=2 $ G…

[引言 : Part 1 温习]

第一课:映射(f) 集合A,B

$latex f: A to B$
$latex f: a mapsto b , a in A, b in B$
$latex f(A) = { f(a) | a in A } subseteq B$ (f的值域， Im f)

A : 象域 domain:
B : 陪域 co-domain: 唯一
满射 Surjective， 单射 Injective ， 双射 Bijective

第二课: 线性空间， 线性变化， 同态

Projection 投影 $latex P_{U} implies $ 线性变化

$latex V = U oplus W$ , W non-unique

$latex V = U oplus U^{perp}$

北大 丘维声的 “群论” List of All Videos:http://www.youtube.com/playlist?list=PLwzFfIxhEkcxvU7-c8rPBbPLHUeacPIpa

“A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools”

— Distinguished differential geometer EugenioCalabi

Ref:

北京大学：丘维声教授

第1讲 数学的思维方式

3000 年前 希腊，巴比伦，中国，印度， 10世纪阿拉伯， 16世纪欧洲文艺复兴 数学 – 经典数学

1830 年 数学的革命 – 近代数学: 法国天才少年 伽瓦罗 (Evariste Galois 1811 – 1832)

观察 (Observe): 客观现象
$latex downarrow$
抽象 (Abstraction) : 概念, 建立 模型 (Model)
$latex downarrow$
探索 (Explore): 自觉 (Intuition), 解剖 , 类比(Analogy), 归纳 (Induction), 联想, 推理 (Deduction) 等…
$latex downarrow$
猜测 (Conjecture) : eg. Riemann Conjecture (unsolved)
$latex downarrow$
论证 (Prove): 只能用公理 (Axioms)(以知的共识), 定义 (概念), 已经证明的定理 (Theorems), 进行逻辑推理并计算.
$latex downarrow$
揭示 (Reveal): 事物的内在规律 (井然有序)

2016 Nobel Prize Physics is Mathematics (Topology) applied in SuperConductor and SuperFluid to explain the Phase Transitions and Phase matters.

Phase matters: Solid, Liquid, Gas

Phase Transition: Solid -> Liquid -> Gas

Superconductor below Tc (critical temperature) : zero resistance.

Superfluid below Tc : zero viscosity.

Reason explained by Mathematics : Topological invariance increased step-wise.

Eg. Disk (0 hole), Circle (1 hole), Donut (2 holes), Coffee Cup (2 holes)… XYZ (n holes). [n increased by steps from 0, 1, 2, 3… ]

We say donut and coffee cup are homeomorphic (同胚) because they have the same topological invariant 拓扑不变量(2 holes).

北京大学数学系 丘维声 教授

引言: 基本数学强化班 — 深入浅出介绍

• 群表示论 是什么?
• 有何用 ?

第一课:环Ring
丘教授 不愧是大师, 也和一些良师一样, 认同 “数”的(代数)结构先从“环” (Ring)开始教起, 再域, 后群 : 美国/法国/英国 都从 “群”(Group)开始, 然后 “环”, “域” (Field) , 是错误的教法, 好比先穿鞋后穿袜, 本末倒置!

精彩的”环” (Ring) 引出 6 条 axioms 公理:

4条 ” + ” 法:

Commutative 交换律, Associative 结合律, Neutral element ” 0″ 零元, Inverse (-) 逆元

2 条 “x ” 法: (exclude ”1″ Unit, WHY ?)

Associative 结合律, Distributive (wrt “+”) 分配律

如果:

环 + 交换 = 交换环 (Commutative Ring)

环 + 单位 ‘1’ =单位环 (Unit Ring)

第二课: 域 Field

星期: 子集的划分 Partitions

$latex mathbb {Z} _7 =
{ bar {0} , bar {1} , bar {2} , bar {3} , bar {4} , bar {5} , bar {6} } $

模m剩余类 : Mod m
$latex mathbb {Z} _ m =
{ bar {0} , bar {1} , bar…

​【区别：代数拓扑 (Algebraic Topology) 微分拓扑 (Differential Topology ) 微分几何 ( Differential Geometry ) 代数几何 (Algebraic Grometry ) 交换代数 (Commutative Algebra ) 微分流形 (Differential Manifold ) ？】月如歌：并不能理解什么叫做楼主所说的配对。我简要谈下我对于上述所列名词的理解。… http://www.zhihu.com/question/23848852/answer/26771912 （分享自知乎网）

The more common morphisms are:

1. Homomorphism (Similarity between 2 different structures) 同态
Analogy: Similar triangles of 2 different triangles.

2. Isomorphism (Sameness between 2 different structures) 同构
Analogy: Congruence of 2 different triangles

Example: 2 objects are identical up to an isomorphism.

3. Endomorphism (Similar structure of self) = {Self + Homomorphism} 自同态
Analogy: A triangle and its image in a magnifying glass.

4. Automorphism (Sameness structure of self) = {Self + Isomorphism} 自同构
Analogy: A triangle and its image in a mirror; or
A triangle and its rotated (clock-wise or anti-clock-wise), or reflected (flip-over) self.

5. Monomorphism 单同态 = Injective + Homomorphism

6. Epimorphism 满同态 = Surjective + Homomorphism

New Math <=> Old Math

1. Isomorphism of Groups (or any structures)
<=> Congruence Triangles
(Faithful Representation)

2. Homomorphism of Groups (or any structures)
<=> Similar Triangles
(unFaithful Representation)

1830 Group Homomorphism
(1831 Galois)

1870 Field Homomorphism
(1870 Camile Jordan Group Isomorphism)
(1870 Dedekind: Automorphism Groups of Field)

1920 Ring Homomorphism
(1927 Noether)

3 common Fields: $latex mathbb{R, Q, C}$ with 4 operations : {+ – × ÷}

Automorphism = “self” isomorphism (Analogy: look into mirror of yourself, image is you <=> Automorphism of yourself).

The trivial Field Automorphism of : $latex mathbb{R, Q}$ is none other than Identity Automorphism (mirror image of itself).

Best example for Field Automorphism : : $latex mathbb{C}$ and its conjugate. (a+ib) conjugate with (a-ib)

Field automorphisms using terms a 15/16/ year oldwould understand? by David Joyce

What interesting results are there regardingautomorphisms of fields? by Henning Breede

Yes.

Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):

[X ] Group -> Ring -> Field

It would be better, conceptual wise, to reverse the teaching order as:

Field -> Ring -> Group

or better still as (the author thinks):

Ring -> Field -> Group

• Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
• Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation to Ring.
• Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group…

Take note: Find roots 根 to solve polynomial 多项式方程式equations, but find solution解to solve algebraic equations代数方程式.

Radical : (LatinRadix = root): $latex sqrt [n]{x} $

Quadratic equation (二次方程式) 有 “根式” 解:[最早发现者 : Babylon 和 三国时期的吴国 数学家 赵爽]

$latex {a.x^{2} + b.x + c = 0}&fg=aa0000&s=3$

$latex boxed{x= frac{-b pm sqrt{b^{2}-4ac} }{2a}}&fg=aa0000$

Cubic Equation: 16 CE Italians del Ferro, Tartaglia & Cardano
$latex {a.x^{3} = p.x + q }&fg=0000aa&s=3$

Cardano Formula (1545 《Ars Magna》):
$latex boxed {x = sqrt [3]{frac {q}{2} + sqrt{{ (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}
+ sqrt [3]{frac {q}{2} -sqrt{ { (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}}&fg=0000aa$

Quartic Equation: by Cardano’s student Ferrari
$latex {a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}&fg=00aa00&s=3$

Quintic Equation:
$latex {a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}&s=3$

No radical solution (Unsolvability) was suspected by Ruffini (1799)…

Excellent Advanced Math Lecture Series (Part 1 to 3) by齊震宇老師

（2012.09.10) Part I:

History: 1900 H. Poincaré invented Topologyfrom Euler Characteristic (V -E + R = 2)

Motivation of Algebraic Topology: Find Invariants[1]of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.

• AbelianFundamental Group(so as to manipulate by + -);

• Vector Space (to + – , × ÷ by multiplier Field scalars);

• Ring (to + x), etc.

then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants (homology, co-homology, etc).

A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调 [2] – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
Note: [1]…

​In the world of Math education there are 3 big schools (门派) — in which the author had the good fortune to study under 3 different Math pedagogies:

“武当派” French (German) -> “少林派” Russian (China) -> “华山派” UK (USA).

( ) : derivative of its parent school. eg. China derived from Russian school in 1960s by Hua Luogeng.

Note:
武当派 : 内功, 以柔尅刚, 四两拨千斤 <=> “Soft” Math, Abstract, Theoretical, Generalized.

少林派: 拳脚硬功夫 <=> “Hard” Math, algorithmic.

华山派: 剑法轻灵 <=> Applied, Astute, Computer-aided.

The 3 schools’ pioneering grand masters (掌门人) since 16th century till 21st century, in between the 19th century (during the French Revolution) Modern Math (近代数学) is the critical milestone, the other (现代数学) is WW2 : –

France: Descartes / Fermat / Pascal (17 CE : Analytical Geometry, Number Theory, Probability), Cauchy / Lagrange / Fourier /Galois (19 CE, Modern Math : Analysis, Abstract Algebra),

​在一个群体里, 每个会员互动中存在一种”运作” (binary operation)关系, 并遵守以下4个原则:

1) 肥水不流外人田: 任何互动的结果要回归 群体。(Closure) = C

2) 互动不分前后次序 (Associative) = A

(a.*b)*c = a*(b*c)

3) 群体有个”中立” 核心 (Neutral / Identity) = N (记号: e)

4) 和而不同: 每个人的意见都容许存在反面的意见 “逆元” (Inverse) = I (记号: a 的逆元 = $latex a^{-1}$)

Agree to disagree = Neutral

$latex a*a^{-1} = e $

具有这四个性质的群体才是

群体的 “美 : “对称”

如果没有 (3)&(4): 半群

如果没有 (4) 反对者: 么半群
以上是 Group (群 ) 数学的定义: “CAN I”

CA = Semi-Group 半群

CAN = Monoid 么半群

群是 19岁Evariste Galois 在法国革命时牢狱中发明的, 解决 300年来 Quintic Equations (5次以上的 方程式) 没有 “有理数” 的 解 (rational roots)。19世纪的 Modern Math (Abstract Algebra) 从此诞生, 群用来解释自然科学(物理, 化学, 生物)里 “对称”现象。Nobel Physicists (1958) 杨振宁/李政道 用群来证明物理 弱力 (Weak Force) 粒子(Particles) 的不对称 (Assymetry )。

Prof ST Yau邱成桐, Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal (at 27, proving Calabi Conjecture), Crafoord Prize(1994),Wolf Prize(2010).

Key Takeaways:

1. On Math Education:
◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc.
◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after it do the problems.
◇ Read Math topics even though you do not understand in first round, re-read few more times, then few days / months / years / decades later you will digest them. (做学问的程序).
◇ Do not consult students in WHAT to teach, because they don’t know what to learn.
◇ Love of Math beauty is the “pull-factor” for motivating students’ interest in Math.
◇ Parental…

How to formulate this problem in CRT ?

Hint: Sunday = 7 , Interval 2 days = mod 2, …

Let d = week days {1, 2, 3, 4, 5, 6, 7} for {Monday (Prof M), tuesday (Prof t), Wednesday (Prof W), Thursday (Prof T), Friday (Prof F), saturday (Prof s), Sunday (Prof S)}

d : 1 2 3 4 5 6 [7] 1 2 3 4 5 6 [7] 1 2
M: m 0 m 0 m 0 [m] ==> fell on 1st sunday
t: - t 0 0 t 0 [0 ] t 0 0 t 0 0 [t ] ==> fell on 2nd sunday
W: – - w 0 0 0 [w] 0 0 0 w 0 0 [0] ==> fell on 1st sunday
T: - - - T T T [T] ==> fell on 1st sunday (TRIVIAL CASE!)
F: - - - - f 0…

This Taiwanese Math Prof is very approachable in clarifying the doubts in an unconventional way different from the arcane textbook definitions. Below are his few key tips to breakthrough the “mystified”concepts :

1. “Dual Space“(对偶空间) : it is the evaluation of a “Vector Space”.

Example: A student studies few subjects {Math, Physics, English, Chemistry…}, these subjects form a “Subject Vector Space” (V), if we associate the subjects with weightages (加权) , say, Math 4, Physics 3, English 2, Chemistry 1, the “Weightage Dual Space” of V will be W= {4, 3, 2, 1}.

2. Vector: beyond the meaning of a physical vector with direction and value, it extends to any “object” which can be manipulated (抵消) by the 4 operations “+, – , x, / ” in a FieldF = {R or Z2 …}.

Eg. $latex alpha_{1}.v_{1} + alpha_{2}.v_{2} + alpha_{3}.v_{3}, forall alpha_{j} in…

This equation puzzles most people. WHY ?
$latex boxed {{delta}^2 = 0 { ?}}&fg=aa0000&s=3 $

It is analogous to the Vector Algebra:
Let the boundary of {A, B} =
$latex delta (A,B) = overrightarrow{AB }$

$latex overrightarrow{AB } + overrightarrow {BA} =overrightarrow{AB } – overrightarrow {AB} = vec 0 $

Source: http://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it

Note: Co-homology: (上)同调

Euclid Geometry & Homology:

Isabell Darcy Lecture: cohomology

This is an excellent quick revision of the French Baccalaureat Math during the first month of French university. (Unfortunately common A-level Math syllabus lacks such rigourous Math foundation.)

Most non-rigourous high-school students / teachers abuse the use of :

“=> ” , “<=>” .

Prove by “Reductio par Absudum” 反证法 (by Contradiction) is a clever mathematical logic :

$latex boxed {(A => B) <=> (non B => non A)} &s=3$

Famous Examples: 1) Prove $latex sqrt 2 $ is irrational ; 2) There are infinite prime numbers (both by Greek mathematician Euclid 3,000 years ago)

The young teacher showed the techniques of proving Mapping:

Surjective (On-to) – best understood in Chinese 满射 (Full Mapping)

Injective (1-to-1) 单射

Bijective (On-to & 1-to-1) 双射

He used an analogy of (the Set of) red Indians shooting (the Set of bisons 野牛):

All bisons are shot by arrows from1 or more Indians. (Surjective…

​Is there any connection between category theory and the way computer languages work?by Thorsten Altenkirch

Yes, lots.

Just one example: a function with 2 inputs from A and B and results from C would have the type A x B -> C but in functional languages like Haskell we are using A -> (B -> C), i.e. a function that returns a function. This “currying” is exactly a the categorical definition of a cartesian closed category as one where Hom(AxB,C) is isomorphic to Hom(A,B -> C) and in this false you can replace Hom(X,Y) with X -> Y.

It is well known that effects in functional programming can be modelled by monads which is a concept from category theory. Nowadays a weaker structure called applicative functors has become very popular – needless to say also a concept from Category Theory.

Not all languages are functional (yet) but…

Abstract Vector Spaces ​向量空间

Eigenvalues & Eigenvectors (valeurs propres et vecteurs propres) 特征值/特征向量

[ Note: “Eigen-” is German for Characteristic 特征.]

The Essence of Determinant (*): (行列式)

(*) 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.

[NOTE] 金庸 武侠小说 《神雕侠女》里 元朝初年的 黄蓉 破解 大理国王妃 瑛姑 苦思不解的 “行列式”, 大概是求 eigenvalues & eigenvectors ? 🙂