Author Archives: tomcircle

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Math amateur

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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Category Theory in Computing Languages

Originally posted on Math Online Tom Circle:
​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…

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Animation: Linear Algebra 

Originally posted on Math Online Tom Circle:
The Big Picture of Linear Algebra 线性代数 (MIT Open-Courseware by the famous Prof Gilbert Strang) https://m.youtube.com/watch?v=ggWYkes-n6E Abstract Vector Spaces ​向量空间 https://m.youtube.com/watch?v=TgKwz5Ikpc8 Eigenvalues & Eigenvectors (valeurs propres et vecteurs propres) 特征值/特征向量 [ Note: “Eigen-”…

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Cours Diagonalisation 1/2. Cours Maths spé

Originally posted on Math Online Tom Circle:
Maths Spéciales (2nd Year French Engineering University Math ?”Linear Algebra”): The French way of treating Matrices is very general and abstract. Advantage is ?it studies Matrices at a theoretical high-level, disadvantage is it…

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Who cares about topology? (inscribed rectangle problem)

Originally posted on Math Online Tom Circle:
Excellent video for the curious minds! Who cares about Topology such as Torus (aka donut) or Mobius Strip ? They can be used to prove difficult math such as the unsolved problem “Inscribed…

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Homotopy

Originally posted on Math Online Tom Circle:
NJ Wildberger AlgTop24: The fundamental group Homotopy 同伦: When playing skip rope, the 2 ends of the rope are held by 2 persons while a 3rd person jumping over the “swings of rope”…

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Matrix A.X = 0 

Originally posted on Math Online Tom Circle:
1) Row reduction, row-echelon form and reduced row-echelon form https://youtu.be/l69YjkuUym0 2) Rank of Matrix = Rank (A) https://youtu.be/gyc98ynuygI

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Topology Game: 双人脱困 

Originally posted on Math Online Tom Circle:
https://youtu.be/KSlh5cp3roQ

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A Journey from Undergrad Math in China to PhD Math in France

Originally posted on Math Online Tom Circle:
An autobiography of a Chinese PhD student  (France, University  Paris-11) in Number Theory and Algebraic  Geometry. His journey from 武汉大学 4 years undergrad to Beijing, learn mostly from the French-educated Chinese Math professors…

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Counting Homology Using Matrix

Originally posted on Math Online Tom Circle:
https://youtu.be/MntT5juY5Mg Column Space and Null Space of a matrix https://youtu.be/VOXTTgTbj3s Create your own Homology: (Important lecture in Applied Algebraic Topology) https://youtu.be/4BkWL5Rr3Hs http://slideplayer.com/slide/9473277/

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The Euler Characteristic χ

Originally posted on Math Online Tom Circle:
Euler Characteristic?χ?is an invariant in Topology. Interesting to note that a tree with no cycle (loop),?χ =1 https://youtu.be/Q6DLWJX5tbs

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Free Abelian Groups & Free Vector Spaces

Originally posted on Math Online Tom Circle:
https://youtu.be/2HS9ypIe8es Lecture 3: Modular Arithmetic (easy)?https://youtu.be/5YEu-vB2dD8 Lecture 4: Addition and Free Vector Spaces?https://youtu.be/9qL003DG7Og

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Algebraic topology and the brain – The Intrepid Mathematician

Originally posted on Math Online Tom Circle:
https://anthonybonato.com/2016/08/31/algebraic-topology-and-the-brain/

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Homological Algebra 

Originally posted on Math Online Tom Circle:
https://youtu.be/s_22CiNoCQI

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Rigorous Prépa Math Pedagogy

Originally posted on Math Online Tom Circle:
The Classe Prépa Math for Grandes Écoles is uniquely French pedagogy – very rigorous based on solid abstract theories. In this lecture the young French professor demonstrates how to teach students the rigorous…

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Initiation aux Mathématiques Modernes – (TV 1969)

Originally posted on Math Online Tom Circle:
This French Classic (8 series lectures) on “Introduction to Modern Maths” made in 1969 is still valid today for Modern Maths students who need to be “initialized” for abstract, rigorous math, different from…

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Espaces Vectoriels

Originally posted on Math Online Tom Circle:
Cours math sup, math spé, BCPST. The French University (engineering) 1st & 2nd year Prépa Math: “Vector Space” (向量空间), aka Linear Algebra (线性代数), used in Google Search Engine. The French treats the subject…

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Trump’s Speaking Math Formula

Originally posted on Math Online Tom Circle:
The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1) Trump defied most expectation from the world to win the 2017 President of the USA. His…

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Curious Thoughts in Math & Science 

Originally posted on Math Online Tom Circle:
1. Statistical Mechanics: Quantum Mechanics: 2. Ramanujian: Tau Special Function: 3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832…

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Category Theory for Functional Programming

Originally posted on Math Online Tom Circle:
Key Motivations for Category Theory 范畴 : 1. Programming is Math. 2. Object-Oriented is based on Set Theory which has 2 weaknesses: ◇ Set has contradiction: The “Russell’s Paradox”. ◇ Data Immutability for…

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Mathematics: The Next Generation

Originally posted on Math Online Tom Circle:
Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating). This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that…

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What exactly is a Limit ?

Originally posted on Math Online Tom Circle:
such that The above scary ‘epsilon-delta’ definition of “Limit” by the French mathematician Cauchy in 19th century is the standard rigorous definition in today’s Analysis textbooks. It was not taught in my Cambridge…

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Math Foundations

Originally posted on Math Online Tom Circle:
MathFoundations (all videos): http://www.youtube.com/playlist?list=PL5A714C94D40392AB All the Math we learn are taught as such by teachers and professors, but why so? what are the foundations ? These 200 videos answer them ! Good for…

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Circle in Different Representations

Originally posted on Math Online Tom Circle:
Affine Line: Six Representations of a Circle: 1) Euclidean Geometry Unit Circle : 2) Curve: Transcendental Parameterization : Rational Parameterisation : 3) Affine Plane 1-dim sub-Space = Lines thru Origin 4) Polygonal Representation…

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Simplicial Complex

Originally posted on Math Online Tom Circle:
Simplices: 0-dim (Point) 1-dim (Line) 2-dim (Triangle) 3-dim (Tetrahedron) Simplicial Complex: built by various Simplices under some rules. Definitions of Simplex (S) Face Orientation Boundary () Theorem: https://m.youtube.com/watch?v=Uq4dTjHfLpI&from=timeline&isappinstalled=0

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Socratica: Abstract Algebra

Originally posted on Math Online Tom Circle:
Abstract Algebra is scary but not with this pretty lecturer! 1. Group ◇ Definition ◇ Symmetric Group ◇ Cayley Table 2. Linear Groups: ◇ GLn (R) ◇ SLn(R) — Determinant = 1 3. …

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Java Family

Originally posted on Math Online Tom Circle:
Year Ver Code Name Description 1995 1.0 Java Applets 1977 1.1 Java Event, Beans, Internationalization Dec 1978 1.2 Java 2 J2SE, J2EE, J2ME, Java Card 2000 1.3 Java 2 J2SE 1.3 2002 1.4…

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Napoléon Math Problem

Originally posted on Math Online Tom Circle:
The French Emperor Napoleon Bonaparte was himself a Mathematician. He posed a question to his mathematician friends Monge, Laplace, Fourier etc: How to divide a circle equally in 4 sections using only a…

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