Author Archives: tomcircle

About tomcircle

Math amateur

The Yoneda Lemma

Originally posted on Math Online Tom Circle:
Representable Functor F of C ( a, -): 4.2?Yoneda Lemma Prove 😕 Yoneda Lemma?: Proof: By “Diagram chasing” below, shows that Left-side: is indeed a (co-variant) Functor. Right-side: Functor “F a“. Note: When…

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Don’t fear the Monad

Originally posted on Math Online Tom Circle:
Brian Beckman:? You can understand Monad without too much Category Theory. Functional Programming = using functions to compose from small functions to very complex software (eg. Nuclear system, driverless car software…). Advantages of…

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Programming and Math

Originally posted on Math Online Tom Circle:
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 strong?Typing. The examples use Haskell to…

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BM Category Theory 10.1: Monads

Originally posted on Math Online Tom Circle:
10.1?Monads 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 http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html#functors

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Category Theory 9: Natural Transformations, BiCategories

Originally posted on Math Online Tom Circle:
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…

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French New Math Lichnerowicz Pedagogy 

Originally posted on Math Online Tom Circle:
https://en.m.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz https://youtu.be/l298jeGgroA See the 1970s French?Baccalaureate Math Textbooks:?(for UK Cambridge GCE A-level ?Math students, this is totally new “New Math” to us !)

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Curry-Howard-Lambek Isomorphism

Originally posted on Math Online Tom Circle:
Curry-Howard-Lambek Isomorphism:? 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…

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Fighting spam with Haskell | Engineering Blog | Facebook Code

Originally posted on Math Online Tom Circle:
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 ?? https://youtu.be/mlTO510zO78 Why Facebook chooses?Haskell…

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Haskell Tutorial in One Video

Originally posted on Math Online Tom Circle:
https://youtu.be/02_H3LjqMr8

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BM Category Theory II 1.1: Declarative vs Imperative Approach

Originally posted on Math Online Tom Circle:
Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming: Imperative (or Procedural) – micro-steps or Local 微观世界 Declarative (or Functional) – Macro-view or Global 大千世界 In Math:?…

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BM Category Theory 3.x Monoid, Kleisli Category (Monad)… Free Monoid 

Originally posted on Math Online Tom Circle:
[Continued from 1.1 to 2.2] 3.1 Monoid?M (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;?…

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QS University Ranking 2017 by Math Subject

Originally posted on Math Online Tom Circle:
Top 5: MIT Harvard? Stanford? Oxford 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…

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Euler’s formula with introductory group theory

Originally posted on Math Online Tom Circle:
During the 19th century French Revolution, a young French boy?Evariste Galois?self-studied Math and invented a totally strange math called “Group Theory“, in his own saying – “A new Math?not on calculation but on…

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Category Theory : Motivation and Philosophy

Originally posted on Math Online Tom Circle:
Object-Oriented ?has 2 weaknesses for Concurrency and Parallel programming 😕 Hidden Mutating States;? Data Sharing. Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra……

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A Category Language : Haskell

Originally posted on Math Online Tom Circle:
Haskell is the purest Functional Language which is based on Category Theory. eBook:? http://learnyouahaskell.com/chapters

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Online Study Guide : Abstract Algebra

Originally posted on Math Online Tom Circle:
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

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How to Take Great Notes – Study Tips – How to be a Great Student – Cornell Notes

Originally posted on Math Online Tom Circle:
Study Tips: http://www.youtube.com/playlist?list=PLi01XoE8jYoi-BCj5m0rRW03vqxaQrH7u

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What is a Field, a Vector Space ? (Abstract Algebra)

Originally posted on Math Online Tom Circle:
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……

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陈省身:数学之美 SS Chern : The Math Beauty

Originally posted on Math Online Tom Circle:
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)…

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转 载: 矩阵的真正含义 The Connotations of Matrix and Its Determinant

Excellent reading for Upper Secondary / High School (JC, IB) Math students.

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矽谷预测AI後的10年大未來

Originally posted on Math Online Tom Circle:
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…

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北大 高等代数 Beijing University Advanced Algebra

Originally posted on Math Online Tom Circle:
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…

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中国数学考研 Graduate Math Exams

Originally posted on Math Online Tom Circle:
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] Find a, b, c, d ? [Solution] :?…

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Barry Mazur  – Harvard Lecture on Primes and the Riemann Hypothesis for High School Students

Originally posted on Math Online Tom Circle:
Prelude: https://youtu.be/sD0NjbwqlYw Harvard Lecture:? https://youtu.be/way0jAWpjZA&start=480 The Key to?open this ?secret … https://youtu.be/VTveQ1ndH1c

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Part 4 群的线性表示的结构

Originally posted on Math Online Tom Circle:
不变子空间: Invariant Sub-space 第一课: ?Direct Sum 直和 of Representations 直和 = ? 第二课: 群表示可约 Reducible Representation Analogy : Prime number decomposition Irreducible Polynomial? 外直和 : * 第三课: 完全可约表示 Completely Reducible Representation 完全表示是可 完全分解为…

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Part 3 (b) 群的线性表示和例

Originally posted on Math Online Tom Circle:
第七课 Group Action?群作用 … 第11课:??Cyclic Group (循环群) Representation , Dihedron 二面体 ???3 阶 Cyclic Group (循环群) Representation

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苏联老师Arnold 如何 教中小学 抽象”群”

Originally posted on Math Online Tom Circle:
Download eBook (PDF) here:?

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Part 3 群的线性表示和例

Originally posted on Math Online Tom Circle:
[?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…

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Part 2:   群表示论的基本概念和Abel群的表示

Originally posted on Math Online Tom Circle:
[?引言 : Part 1 温习] 第一课:?映射(f) 集合A,B https://youtu.be/3xps19FOiDA (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…

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Pure Mathematicians versus Applied Mathematicians

Originally posted on Math Online Tom Circle:
“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…

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数学是什么 ? What is Mathematics? 

Originally posted on Math Online Tom Circle:
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 抽象 (Abstraction) : 概念, 建立 模型 (Model)…

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2016 Nobel-Prize Winning Physics Explained Through Pastry 

Originally posted on Math Online Tom Circle:
https://m.youtube.com/watch?feature=youtu.be&v=zO8esJuQIMs 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…

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群表示论引言 Introduction to Group Representation

Originally posted on Math Online Tom Circle:
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!…

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NOUVEAU : découvrez l’appli mobile d’Optimal Sup Spé !

Originally posted on Math Online Tom Circle:
https://youtu.be/zx2cflESEjk

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【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 Geometry ) 交换代数  (Commutative Algebra ) 微分流形 (Differential Manifold )

Originally posted on Math Online Tom Circle:
​【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)

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Morphism Summary Chart

Originally posted on Math Online Tom Circle:
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…

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Isomorphism = Congruence, Homomorphism = Similar

Originally posted on Math Online Tom Circle:
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)

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Homomorphism History

Originally posted on Math Online Tom Circle:
1830 Group Homomorphism (1831 Galois) 1870 Field Homomorphism (1870 Camile Jordan Group Isomorphism) (1870 Dedekind: Automorphism Groups of Field) 1920 Ring Homomorphism (1927 Noether)

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Quora: Galois Field Automorphism for 15/16 year-old kids

Originally posted on Math Online Tom Circle:
3 common Fields: with 4 operations : {+ – × ÷} Automorphism = “self” ?isomorphism (Analogy: ?look into mirror of yourself, ?image is you <=> Automorphism of yourself). The trivial Field Automorphism of…

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Does Abstract Math belong to Elementary Math ? 

Originally posted on Math Online Tom Circle:
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,…

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In Search for Radical Roots of Polynomial Equations of degree n > 1

Originally posted on Math Online Tom Circle:
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 : (Latin Radix = root): 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]…

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Why call the Algebraic Structure Z a “Ring” ?

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代 数拓扑 Algebraic Topology

Originally posted on Math Online Tom Circle:
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 Topology?from Euler Characteristic (V -E + R = 2) Motivation of Algebraic Topology: Find Invariants?[1]of…

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Russian Math Education

Originally posted on Math Online Tom Circle:
​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”…

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群论的哲学 Philosophical Group Theory

Originally posted on Math Online Tom Circle:
​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”…

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Prof ST Yau’s 邱成桐 Talk to Chinese Youth on Math Education 

Originally posted on Math Online Tom Circle:
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…

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Chinese Remainder Theorem

Originally posted on Math Online Tom Circle:
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…

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Some Math Connotations Demystified 数学内涵解密

Originally posted on Math Online Tom Circle:
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.…

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Homology: Why Boundary of Boundary = 0 ?

Originally posted on Math Online Tom Circle:
This equation puzzles most people. WHY ? It is analogous to the Vector Algebra: Let the boundary of {A, B} = ? 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…

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Cours Raisonnements (Logics) , Ensembles ( Sets), Applications (Mappings)

Originally posted on Math Online Tom Circle:
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 /…

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