Category Archives: math

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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SO(3) diffeomorphic to RP^3

} Proof: We consider as the group of all rotations about the origin of under the operation of composition. Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. We … Continue reading

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SU(2) diffeomorphic to S^3 (3-sphere)

(diffeomorphic) Proof: We have that Since , we may view as Consider the map It is clear that is well-defined since if , then . If , it is clear that . So is injective. It is also clear that … Continue reading

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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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Echelon Form Lemma (Column Echelon vs Smith Normal Form)

The pivots in column-echelon form are the same as the diagonal elements in (Smith) normal form. Moreover, the degree of the basis elements on pivot rows is the same in both forms. Proof: Due to the initial sort, the degree … Continue reading

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Persistent Homology Algorithm

Algorithm for Fields In this section we describe an algorithm for computing persistent homology over a field. We use the small filtration as an example and compute over , although the algorithm works for any field. A filtered simplicial complex … Continue reading

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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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Everyone’s Unique Timezone (Motivational)

Relax. Take a deep breath. Don’t compare yourself with others. The world is full of all kinds of people – those who get successful early in life and those who do later. There are those who get married at 25 … Continue reading

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De Rham Cohomology

De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined. Definition: A differential form on a manifold is said to be closed if , and exact … Continue reading

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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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Singular Homology

A singular -simplex in a space is a map . Let be the free abelian group with basis the set of singular -simplices in . Elements of , called singular -chains, are finite formal sums for and . A boundary … Continue reading

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Mapping Cone Theorem

Mapping cone Let be a map in . We construct the mapping cone , where is identified with for all . Proposition: For any map we have if and only if has an extension to . Proof: By an earlier … Continue reading

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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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记叙文开头的几种方式 How to Write The Start of Narrative Composition

Originally posted on Chinese Tuition Singapore:
记叙文是以记人,叙事,写景,状物(描绘事物)为主,主要内容是人物的经历和事物的发展变化。 记叙文有五种主要表达方式:叙述,描写,议论,抒情,说明。而记叙文的开头主要有以下几种形式: 一,叙述 把人物的经历和事物的发展变化过程表现出来。用简单的话说,就是,这件事怎么发生的,过程是什么,结果怎么样。当然,如果用这种方式开头,就不需要把整个事情的过程交代清楚,一般只要把事情的起因表述清楚即可。过程和结果可以在正文中体现。 比如: 1. 描写母爱——上幼儿园的时候,妈妈给我买了一把可爱的小花伞。伞的大小对我来说刚刚好,因为正好能遮住我小小的身体。妈妈说,自从我有了小花伞,就特别喜欢下雨天。只要一下雨,我就会把小花伞找出来,拉着妈妈往外跑。妈妈就撑起一把大伞来遮住我的小伞,陪着我在雨里玩。 2. 描写一次难忘的经历——清晨,大街上异常忙碌,人来人往,像一条畅流的小溪。忽然,两辆自行车撞在一起,像一块石头横挡在小溪中间,小溪变得流动缓慢,渐渐停止了。 3. 描写一次闯祸——在故事发生时,他还是个七八岁的孩子,他常常做些让大人们意想不到的恶作剧。但是,因为他还只是个孩子,所以大人们除了偶尔斥责几句之外,都不把他做的那些调皮捣蛋的事放在心上。就这样,他的胆子越来越大,闯的祸也越来越大。 作文开头交代了事情的起因,下面就可以直接写事情的经过。 二, 描写 主要是对人物的外貌,动作,心理,事物的形态,样貌等具体的刻画。通常对人物的这种描写会从侧面反映出人物的性格特点。 例如: 1. 描写邻居——我有一位小邻居,她的名字叫小红,今年九岁。她远远的小脑袋上扎着两条小辫子,有着一双水灵灵的大眼睛。她的耳朵粉红小巧,像贝壳一样。红嘟嘟的小嘴整天叽叽喳喳不知疲倦。 2. 描写亲人——我弟弟很可爱,他那圆圆的小脸蛋上嵌着一双水灵灵的大眼睛。嘴唇薄薄的,一笑小嘴一咧,眼睛一眯,还生出一堆小酒窝,非常可爱。要是谁惹他生气了,他就会瞪大眼睛,撅起小嘴。 如果作文中需要写关于某个人的事情,那在作文的一开始就告诉读者这个人的性格特点,将会为作文的正文做好铺垫。 三, 抒情 通过文中要描写的人或事来表达自己内心的情感。 例如: 描写母爱——如果说我有向全世界的人宣布一件事情的权力的的话,我一定会说,我要感谢那个赋予我生命,教会我勇敢,关爱我成长的,我心中最漂亮的女人-妈妈。 如果作文题目是关于“后悔”,“感激”,“难过”等对一个人或一件事的心情,在作文开头就表现出这种情感是一个很好的选择。 四, 回忆 通常用于写时间比较久的事情,比如,童年,几年前,几个月前,等发生的事情。 例如: 1. 描写童年——在偌大的世界上,人人都有一个栖息之地—家庭。有的家庭富丽堂皇,有的家庭美满甜蜜。对无忧无虑的小孩子来说,这是一块充满慈爱和乐趣的生命之地。然而,我是个不幸的孤儿,从小失去了父母,跟姐姐住在外婆家。回忆起自己在外婆家度过的那几年,我的泪水就像断了线的珠子。 2.…

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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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CP1 and S^2 are smooth manifolds and diffeomorphic (proof)

Proposition: is a smooth manifold. Proof: Define and . Also define by and . Let be the homeomorphism from to defined by and define by . Note that is an open cover of , and are well-defined homeomorphisms (from onto … Continue reading

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Tangent space (Derivation definition)

Let be a smooth manifold, and let . A linear map is called a derivation at if it satisfies The tangent space to at , denoted by , is defined as the set of all derivations of at .

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Looking for Home Tutors?

If you are looking for home tutors (any subject, e.g. Mathematics, Chinese, English, Science, etc.) contact us at: SMS/Whatsapp: 98348087 Email: mathtuition88@gmail.com We are able to recommend you highly qualified tutors, free of charge, no obligations. Note that usually tutors’ … Continue reading

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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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Secondary Chinese Tuition (IP / O Level)

Ms Gao specializes in tutoring Secondary Level Chinese. Can teach composition, comprehension, etc, according to student’s weaknesses. Has taught students from RI (IP Programme), MGS, and more. Familiar with IP and O Level (HCL/CL) Chinese syllabus. Website: https://chinesetuition88.com/ Contact: 98348087 … Continue reading

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Homology Group of some Common Spaces

Homology of Circle Homology of Torus Homology of Real Projective Plane Homology of Klein Bottle Also see How to calculate Homology Groups (Klein Bottle).

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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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Summary of Persistent Homology

We summarize the work so far and relate it to previous results. Our input is a filtered complex and we wish to find its th homology . In each dimension the homology of complex becomes a vector space over a … Continue reading

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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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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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