Category Archives: math

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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Cast Iron Pan Singapore Review

Recently bought a cast iron pan/skillet for home cooking. Cast iron is an ancient technology that has several benefits over the more modern non-stick technology. It is supposed to be cheap (just US$10 in America), but in Singapore it is … Continue reading

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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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Inspirational Scientist: Dan Shechtman

Source: https://www.theguardian.com/science/2013/jan/06/dan-shechtman-nobel-prize-chemistry-interview To stand your ground in the face of relentless criticism from a double Nobel prize-winning scientist takes a lot of guts. For engineer and materials scientist Dan Shechtman, however, years of self-belief in the face of the eminent Linus … Continue reading

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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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Structure Theorem for finitely generated (graded) modules over a PID

If is a PID, then every finitely generated module over is isomorphic to a direct sum of cyclic -modules. That is, there is a unique decreasing sequence of proper ideals such that where , and . Similarly, every graded module … Continue reading

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Persistence Interval

Next, we want to parametrize the isomorphism classes of the -modules by suitable objects. A -interval is an ordered pair with . We may associate a graded -module to a set of -intervals via a bijection . We define for … Continue reading

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The Map of Mathematics (YouTube)

A nicely done video on how the various branches of mathematics fit together. It is amazing that he has managed to list all the major branches on one page. Also see: Beautiful Map of Mathematics.

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Homogenous / Graded Ideal

Let be a graded ring. An ideal is homogenous (also called graded) if for every element , its homogenous components also belong to . An ideal in a graded ring is homogenous if and only if it is a graded … Continue reading

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Donate to help Stray Dogs in Singapore

URL: https://give.asia/movement/run_for_exclusively_mongrels 3 Singaporeans – Dr Gan, A Dentist, Dr Herman, A Doctor, and Mr Ariffin, a Law Undergraduate will be taking on the Borneo Ultra Trail Marathon on Feb 18th 2017 to raise 30k for Exclusively Mongrels Ltd; a welfare … Continue reading

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Water cuts through rock, not because of its strength, but because of its persistence.

Water cuts through rock, not because of its strength, but because of its persistence.

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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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Smooth/Differentiable Manifold

Smooth Manifold A smooth manifold is a pair , where is a topological manifold and is a smooth structure on . Topological Manifold A topological -manifold is a topological space such that: 1)  is Hausdorff: For every distinct pair of … Continue reading

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Persistence module and Graded Module

We show that the persistent homology of a filtered simplicial complex is the standard homology of a particular graded module over a polynomial ring. First we review some definitions. A graded ring is a ring (a direct sum of abelian … Continue reading

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GEP Selection Test Review and Experience

The following is a parent’s review and experience of the GEP Selection Test (2016). Original text (in Chinese) at: http://mp.weixin.qq.com/s/xQpLynFWpZ6QNpI_vlw4cw Interested readers may also want to check out Recommended Books for GEP Selection Test. Translation: One day in September 2016 afternoon, read … Continue reading

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Persistence module and Finite type

A persistence module is a family of -modules , together with homomorphisms . For example, the homology of a persistence complex is a persistence module, where maps a homology class to the one that contains it. A persistence complex (resp.\ … Continue reading

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To Live Your Best Life, Do Mathematics

This article is a very good read. 100% Recommended to anyone interested in math. The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find … Continue reading

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Homotopy for Maps vs Paths

Homotopy (of maps) A homotopy is a family of maps , , such that the associated map given by is continuous. Two maps are called homotopic, denoted , if there exists a homotopy connecting them. Homotopy of paths A homotopy … Continue reading

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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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Universal Property of Quotient Groups (Hungerford)

If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely. This can be used to prove the following proposition: A chain map between chain complexes and induces homomorphisms … Continue reading

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Some Homology Definitions

Chain Complex A sequence of homomorphisms of abelian groups with for each . th Homology Group is the free abelian group with basis the open -simplices of . -chains Elements of , called -chains, can be written as finite formal … Continue reading

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RP^n Projective n-space

Define an equivalence relation on by writing if and only if . The quotient space is called projective -space. (This is one of the ways that we defined the projective plane .) The canonical projection is just . Define , … Continue reading

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Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

Motivation Data is commonly represented as an unordered sequence of points in the Euclidean space . The global `shape’ of the data may provide important information about the underlying phenomena of the data. For data points in , determining the … Continue reading

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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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Equivalence of C^infinity atlases

Equivalence of atlases is an equivalence relation. Each atlas on is equivalent to a unique maximal atlas on . Proof: Reflexive: If is a atlas, then is also a atlas. Symmetry: Let and be two atlases such that is also … Continue reading

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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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Natural Equivalence relating Suspension and Loop Space

Theorem: If , , , , Hausdorff and locally compact, then there is a natural equivalence defined by , where if is a map then is given by . We need the following two propositions in order to prove the … Continue reading

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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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Fundamental Group of S^n is trivial if n>=2

if We need the following lemma: If a space is the union of a collection of path-connected open sets each containing the basepoint and if each intersection is path-connected, then every loop in at is homotopic to a product of … Continue reading

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Tangent Space is Vector Space

Prove that the operation of linear combination, as in Definition 2.2.7, makes into an -dimensional vector space over . The zero vector is the infinitesimal curve represented by the constant . If , then where , defined for all sufficiently … Continue reading

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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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Balance Quote

Never let success go to your head, and never let failure go to your heart.

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Analysis: 97 marks not enough for Higher Chinese cut-off point for Pri 1 pupils

Quite tough to be a primary school kid nowadays, even 97 marks is not enough to be admitted for Higher Chinese classes. From experience, the main underlying reasons behind this scenario could be: Due to intensive tuition starting from preschool, students … Continue reading

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Functors, Homotopy Sets and Groups

Functors Definition: A functor from a category to a category is a function which – For each object , we have an object . – For each , we have a morphism Furthermore, is required to satisfy the two axioms: – For … Continue reading

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H2 Maths Tuition by Ex-RI, NUS 1st Class Honours (Mathematics)

Junior College H2 Maths Tuition About Tutor (Mr Wu): https://mathtuition88.com/singapore-math-tutor/ – Raffles Alumni – NUS 1st Class Honours in Mathematics Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs. Personality: Friendly, patient and … Continue reading

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viXra vs arXiv

viXra (http://vixra.org/) is the cousin of arXiv (http://arxiv.org/) which are electronic archives where researchers can submit their research before being published on a journal. The difference is that viXra allows anyone to submit their article, whereas arXiv requires an academic … Continue reading

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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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Algebraic Topology: Fundamental Group

Homotopy of paths A homotopy of paths in a space is a family , , such that (i) The endpoints and are independent of . (ii) The associated map defined by is continuous. When two paths and are connected in this way … Continue reading

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Multivariable Derivative and Partial Derivatives

If is a derivative of at , then . In particular, if is differentiable at , these partial derivatives exist and the derivative is unique. Proof: Let , then becomes since . By choosing (all zeroes except in th position), … Continue reading

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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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