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 of row basis elements \hat{e}_i is monotonically decreasing from the top row down. For each fixed column j, \deg e_j is a constant. We have, \deg M_k(i,j)=\deg e_j-\deg \hat{e}_i. Hence, the degree of the elements in each column is monotonically increasing with row. That is, for fixed j, \deg M_k(i,j) is monotonically increasing as i increases.

We may then eliminate non-zero elements below pivots using row operations that do not change the pivot elements or the degrees of the row basis elements. Finally, we place the matrix in (Smith) normal form with row and column swaps.

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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 \mathbb{Z}_2, although the algorithm works for any field.
A filtered simplicial complex with new simplices added at each stage. The integers on the bottom row corresponds to the degrees of the simplices of the filtration as homogenous elements of the persistence module.

The persistence module corresponds to a \mathbb{Z}_2[t]-module by the correspondence in previous Theorem. In this section we use \{e_j\} and \{\hat{e}_i\} to denote homogeneous bases for C_k and C_{k-1} respectively.

We have \partial_1(ab)=-t\cdot a+t\cdot b=t\cdot a+t\cdot b since we are computing over \mathbb{Z}_2. Then the representation matrix for \partial_1 is
\displaystyle M_1=\begin{bmatrix}[c|ccccc]  &ab &bc &cd &ad &ac\\ \hline  d & 0 & 0 & t & t & 0\\  c & 0 & 1 & t & 0 & t^2\\  b & t & t & 0 & 0 & 0\\  a &t &0 &0 &t^2 &t^3  \end{bmatrix}.

In general, any representation M_k of \partial_k has the following basic property: \displaystyle \deg\hat{e}_i+\deg M_k(i,j)=\deg e_j provided M_k(i,j)\neq 0.

We need to represent \partial_k: C_k\to C_{k-1} relative to the standard basis for C_k and a homogenous basis for Z_{k-1}=\ker\partial_{k-1}. We then reduce the matrix according to the reduction algorithm described previously.

We compute the representations inductively in dimension. Since \partial_0\equiv 0, Z_0=C_0 hence the standard basis may be used to represent \partial_1. Now, suppose we have a matrix representation M_k of \partial_k relative to the standard basis \{e_j\} for C_k and a homogeneous basis \{\hat{e}_i\} for Z_{k-1}.

For the inductive step, we need to compute a homogeneous basis for Z_k and represent \partial_{k+1} relative to C_{k+1} and the homogeneous basis for Z_k. We first sort the basis \hat{e}_i in reverse degree order. Next, we make M_k into the column-echelon form \tilde{M}_k by Gaussian elimination on the columns, using elementary column operations. From linear algebra, we know that rank M_k=rank B_{k-1} is the number of pivots in the echelon form. The basis elements corresponding to non-pivot columns form the desired basis for Z_k.

Source: “Computing Persistent Homology” by Zomorodian & Carlsson

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BM Category Theory 10.1: Monads

Math Online Tom Circle

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

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Category Theory 9: Natural Transformations, BiCategories

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 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…

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French New Math Lichnerowicz Pedagogy 

Math Online Tom Circle

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 !)

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Curry-Howard-Lambek Isomorphism

Math Online Tom Circle

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.

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Fighting spam with Haskell | Engineering Blog | Facebook Code

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 ?

Why Facebook choosesHaskell Functional language for Spam rule engine?

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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 but divorced at 30, and there are also those who find love at 40, never to part with them again. Henry Ford was 45 when he designed his revolutionary Model T car. A simple WhatsApp forward message makes so much sense here:

“You are unique, don’t compare yourself to others.

Someone graduated at the age of 22, yet waited 5 years before securing a good job; and there is another who graduated at 27 and secured employment immediately!

Someone became CEO at 25 and died at 50 while another became a CEO at 50 and lived to 90 years.

Everyone works based on their ‘Time Zone’. People can have things worked out only according to their pace.

Work in your “time zone”. Your Colleagues, friends, younger ones might “seem” to go ahead of you. May be some might “seem” behind you. Everyone is in this world running their own race on their own lane in their own time. God has a different plan for everybody. Time is the difference.

Obama retires at 55, Trump resumes at 70. Don’t envy them or mock them, it’s their ‘Time Zone.’ You are in yours!” 

Source: http://www.mensxp.com/special-features/today/34993-if-you-think-you-are-going-nowhere-in-life-take-a-deep-breath-and-read-this.html

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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 \omega on a manifold M is said to be closed if d\omega=0, and exact if \omega=d\tau for some \tau of degree one less.

Corollary:
Since d^2=0, every exact form is closed.

Definition:
Let Z^k(M) be the vector space of all closed k-forms on M.

Let B^k(M) be the vector space of all exact k-forms on M.

Since every exact form is closed, hence B^k(M)\subseteq Z^k(M).

The de Rham cohomology of M in degree k is defined as the quotient vector space \displaystyle H^k(M):=Z^k(M)/B^k(M).

The quotient vector space construction induces an equivalence relation on Z^k(M):

w'\sim w in Z^k(M) iff w'-w\in B^k(M) iff w'=w+d\tau for some exact form d\tau.

The equivalence class of a closed form \omega is called its cohomology class and denoted by [\omega].

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Haskell Tutorial in One Video

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Singular Homology

A singular n-simplex in a space X is a map \sigma: \Delta^n\to X. Let C_n(X) be the free abelian group with basis the set of singular n-simplices in X. Elements of C_n(X), called singular n-chains, are finite formal sums \sum_i n_i\sigma_i for n_i\in\mathbb{Z} and \sigma_i: \Delta^n\to X. A boundary map \partial_n: C_n(X)\to C_{n-1}(X) is defined by \displaystyle \partial_n(\sigma)=\sum_i(-1)^i\sigma|[v_0,\dots,\widehat{v_i},\dots,v_n].

The singular homology group is defined as H_n(X):=\text{Ker}\partial_n/\text{Im}\partial_{n+1}.

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Mapping Cone Theorem

Mapping cone
Let f:(X,x_0)\to (Y,y_0) be a map in \mathscr{PT}. We construct the mapping cone Y\cup_f CX=Y\vee CX/\sim, where [1,x]\in CX is identified with f(x)\in Y for all x\in X.

Proposition:
For any map g: (Y,y_0)\to (Z,z_0) we have g\circ f\simeq z_0 if and only if g has an extension h: (Y\cup_f CX,*)\to(Z,z_0) to Y\cup_f CX.

Proof:
By an earlier proposition (2.32 in \cite{Switzer2002}), g\circ f\simeq z_0 iff g\circ f has an extension \psi: (CX,*)\to (Z,z_0).

(\implies) If g\circ f\simeq z_0, define \tilde{h}: Y\vee CX\to Z by \tilde{h}(y_0,[t,x])=\psi[t,x], \tilde{h}(y,[0,x])=g(y). Note that \tilde{h}(y_0,[0,x])=\psi[0,x]=g(y_0)=z_0. Since \displaystyle \tilde{h}(y_0,[1,x])=\psi[1,x]=gf(x)=\tilde{h}(f(x),[0,x]), \tilde{h} induces a map h: Y\cup_f CX\to Z which satisfies h[y,[0,x]]=\tilde{h}(y,[0,x])=g(y). That is h|_Y=g.

(\impliedby) If g has an extension h: (Y\cup_f CX,*)\to (Z,z_0), then define \psi: CX\to Z by \psi([t,x])=h[y_0,[t,x]]. We have \psi([0,x])=h[y_0,[0,x]]=z_0. Then \displaystyle \psi([1,x])=h[y_0,[1,x]]=h[f(x),[0,x]]=gf(x). That is, \psi|_X=g\circ f.

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BM Category Theory II 1.1: Declarative vs Imperative Approach

Math Online Tom Circle

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…

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

Math Online Tom Circle

[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…

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QS University Ranking 2017 by Math Subject

Math Online Tom Circle

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

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

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. 描写父爱——在我的记忆中,爸爸的背是温暖的。
五,开门见山

这是最常见的一种开头。也是最直接的一种方式。

例如:

描写一次难忘的回忆——在我的人生中,有许许多多的人生第一次,令我终身难忘的是第一次游泳。
六,悬念

作文的开头通过描写事情发展中最精彩的部分,即人物的动作或者语言,来引起读者的兴趣。

例如:

“这样的事做不得!”看着背影远去的小明,我从心中发出一声呼唤。当时,我真的应该阻止他的。

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Euler’s formula with introductory group theory

Math Online Tom Circle

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.

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Category Theory : Motivation and Philosophy

Math Online Tom Circle

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”…

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A Category Language : Haskell

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

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

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

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…

WHAT IS A FIELD (域) ?

WHAT IS A VECTOR SPACE (向量空间) ?

See all 20+ videos here:

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陈省身:数学之美 SS Chern : The Math Beauty

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) 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

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

Proposition: \mathbb{C}P^1 is a smooth manifold.

Proof:
Define U_1=\{[z^1, z^2]\mid z^1\neq 0\} and U_2=\{[z^1, z^2]\mid z^2\neq 0\}. Also define g_i: U_i\to\mathbb{C} by g_1([z^1, z^2])=\frac{z^2}{z^1} and g_2([z^1, z^2])=\frac{\overline{z^1}}{\overline{z^2}}.

Let f:\mathbb{C}\to\mathbb{R}^2 be the homeomorphism from \mathbb{C} to \mathbb{R}^2 defined by f(x+iy)=(x,y) and define \phi_i: U_i\to\mathbb{R}^2 by \phi_i=f\circ g_i.

Note that \{U_1, U_2\} is an open cover of \mathbb{C}P^1, and \phi_i are well-defined homeomorphisms (from U_i onto an open set in \mathbb{R}^2). Then \{(U_1,\phi_1), (U_2,\phi_2)\} is an atlas of \mathbb{C}P^1.

The transition function \displaystyle \phi_2\circ\phi_1^{-1}: \phi_1(U_1\cap U_2)\to\mathbb{R}^2,
\begin{aligned}  \phi_2\phi_1^{-1}(x,y)&=\phi_2 g_1^{-1}(x+iy)\\  &=\phi_2([1,x+iy])\\  &=f(\frac{1}{x-iy})\\  &=f(\frac{x+iy}{x^2+y^2})\\  &=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})  \end{aligned}
is differentiable of class C^\infty. Similarly, \phi_1\circ\phi_2^{-1}:\phi_2(U_1\cap U_2)\to\mathbb{R}^2 is of class C^\infty. Hence \mathbb{C}P^1 is a smooth manifold.

Proposition:
S^2 is a smooth manifold.

Proof:
Define V_1=S^2\setminus\{(0,0,1)\} and V_2=S^2\setminus\{(0,0,-1)\}. Then \{V_1, V_2\} is an open cover of S^2.

Define \psi_1: V_1\to\mathbb{R}^2 by \psi_1(x,y,z)=(\frac{x}{1-z},\frac{y}{1-z}) and \psi_2: V_2\to\mathbb{R}^2 by \psi_2(x,y,z)=(\frac{x}{1+z},\frac{y}{1+z}).

We can check that \psi_1^{-1}(x,y)=(\frac{2x}{1+x^2+y^2},\frac{2y}{1+x^2+y^2},\frac{-1+x^2+y^2}{1+x^2+y^2}). Hence \psi_1 is a homeomorphism from V_1 onto an open set in \mathbb{R}^2. Similarly, \psi_2 is a homeomorphism from V_2 onto an open set in \mathbb{R}^2. Thus \{V_1,V_2\} is an atlas for S^2.

The composite \psi_2\circ\psi_1^{-1}: \psi_1(V_1\cap V_2)\to\mathbb{R}^2 is differentiable of class C^\infty since both \psi_2, \psi_1^{-1} are of class C^\infty. Similarly, \psi_1\circ\psi_2^{-1} is of class C^\infty. Thus S^2 is a smooth manifold.

We can also compute the transition function explicitly:
\displaystyle  \psi_2\psi_1^{-1}(x,y)=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}).
Note that \psi_2\psi_1^{-1}=\phi_2\phi_1^{-1}.

Define h:\mathbb{C}P^1\to S^2 by h(\phi_1^{-1}(x,y))=\psi_1^{-1}(x,y) and h(\phi_2^{-1}(x,y))=\psi_2^{-1}(x,y).

We see that h is well-defined since if \phi_1^{-1}(x,y)=\phi_2^{-1}(u,v) then \displaystyle (u,v)=\phi_2\phi_1^{-1}(x,y)=\psi_2\psi_1^{-1}(x,y) so that \psi_2^{-1}(u,v)=\psi_1^{-1}(x,y).

Similarly, we have a well-defined inverse h^{-1}: S^2\to\mathbb{C}P^1 defined by h^{-1}(\psi_1^{-1}(x,y))=\phi_1^{-1}(x,y) and h^{-1}(\psi_2^{-1}(x,y))=\phi_2^{-1}(x,y).

We check that (from our previous workings)
\begin{aligned}  \psi_1 h\phi_1^{-1}(x,y)&=(x,y)\\  \psi_2 h\phi_1^{-1}(x,y)&=\psi_2\psi_1^{-1}(x,y)\\  \psi_1 h\phi_2^{-1}(x,y)&=\psi_1\psi_2^{-1}(x,y)\\  \psi_2 h\phi_2^{-1}(x,y)&=(x,y)  \end{aligned}
are of class C^\infty. So h is a smooth map. Similarly, h^{-1} is smooth. Hence h is a diffeomorphism.

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Tangent space (Derivation definition)

Let M be a smooth manifold, and let p\in M. A linear map v: C^\infty(M)\to\mathbb{R} is called a derivation at p if it satisfies \displaystyle v(fg)=f(p)vg+g(p)vf\qquad\text{for all}\ f,g\in C^\infty(M).
The tangent space to M at p, denoted by T_pM, is defined as the set of all derivations of C^\infty(M) at p.

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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’ slots will start to fill up as the year progresses, so by mid year May/June it is going to be very hard to find a good tutor.

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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

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 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。

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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
Email: chinesetuition88@gmail.com

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Homology Group of some Common Spaces

Homology of Circle
\displaystyle H_n(S^1)=\begin{cases}\mathbb{Z}&\text{for}\ n=0,1\\  0&\text{for}\ n\geq 2.  \end{cases}

Homology of Torus
\displaystyle H_n(T)=\begin{cases}\mathbb{Z}\oplus\mathbb{Z}&\text{for}\ n=1\\  \mathbb{Z}&\text{for}\ n=0, 2\\  0&\text{for}\ n\geq 3.  \end{cases}

Homology of Real Projective Plane
\displaystyle H_n(\mathbb{R}P^2)=\begin{cases}  \mathbb{Z}&\text{for}\ n=0\\  \mathbb{Z}_2&\text{for}\ n=1\\  0&\text{for}\ n\geq 2.  \end{cases}

Homology of Klein Bottle
\displaystyle H_n(K)=\begin{cases}  \mathbb{Z}&\text{for}\ n=0\\  \mathbb{Z}_2\oplus\mathbb{Z}&\text{for}\ n=1\\  0&\text{for}\ n\geq 2.  \end{cases}

Also see How to calculate Homology Groups (Klein Bottle).

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北大 高等代数 Beijing University Advanced Algebra

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 (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)… =>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
:

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中国数学考研 Graduate Math Exams

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

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

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 $

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

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

Math Online Tom Circle

Prelude:


Harvard Lecture:

The Key toopen this secret

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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 K and we wish to find its kth homology H_k. In each dimension the homology of complex K^i becomes a vector space over a field, described fully by its rank \beta_k^i. (Over a field F, H_k is a F-module which is a vector space.)

We need to choose compatible bases across the filtration (compatible bases for H_k^i and H_k^{i+p}) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module \mathscr{M} corresponding to K, which is a direct sum of these vector spaces (\alpha(\mathscr{M})=\bigoplus M^i). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.

Specifically, each \mathcal{P}-interval (i,j) describes a basis element for the homology vector spaces starting at time i until time j-1. This element is a k-cycle e that is completed at time i, forming a new homology class. It also remains non-bounding until time j, at which time it joins the boundary group B_k^j.

A natural question is to ask when e+B_k^l is a basis element for the persistent groups H_k^{l,p}. Recall the equation \displaystyle H_k^{i,p}=Z_k^i/(B_k^{i+p}\cap Z_k^i). Since e\notin B_k^l for all l<j, hence e\notin B_k^{l+p} for l+p<j. The three inequalities \displaystyle l+p<j,\ l\geq i,\ p\geq 0 define a triangular region in the index-persistence plane, as shown in Figure below.

triangle

The triangular region gives us the values for which the k-cycle e is a basis element for H_k^{l,p}. This is known as the k-triangle Lemma:

Let \mathcal{T} be the set of triangles defined by \mathcal{P}-intervals for the k-dimensional persistence module. The rank \beta_k^{l,p} of H_k^{l,p} is the number of triangles in \mathcal{T} containing the point (l,p).

Hence, computing persistent homology over a field is equivalent to finding the corresponding set of \mathcal{P}-intervals.

Source: “Computing Persistent Homology” by Zomorodian and Carlsson

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Part 4 群的线性表示的结构

Math Online Tom Circle

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…

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Part 3 (b) 群的线性表示和例

Math Online Tom Circle

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

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

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苏联老师Arnold 如何 教中小学 抽象”群”

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Part 3 群的线性表示和例

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 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…

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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$15 in America Lodge L8SK3 Cast Iron Skillet, Pre-Seasoned, 10.25-inch), but in Singapore it is quite expensive probably due to import fees.

I bought the mid-range USA brand Lodge 10.25-inch skillet (around $60 SGD). It can be found in Qoo10:

Where to buy Cast Iron Pan/Pot/Skillet Singapore

Lodge Pre-Seasoned 8-inch Cast Iron Skillet: http://www.qoo10.sg/su/412118339/Q100000595

Lodge Pre-Seasoned Cast Iron Skillet 10.25-inch: http://www.qoo10.sg/su/412118386/Q100000595

Lodge Cast Iron Skillet (Midrange, affordable)

The high-end brands include Le Creuset, Staub. These are very expensive (at least $100 SGD).

Le Creuset Skillet (High-end, Super Expensive)

Benefits of Cast Iron Cookware vs Non-stick:

  • Non-stick Teflon, even with extreme care, tends to flake off and end up in food. Also it is released as fumes during cooking. It has dubious, unknown effects on humans, but is scientifically proven to be toxic to birds and rats.
  • Adds iron supplementation to cooked food. Iron is essential for human health to make hemoglobin in blood.
  • Technically lasts forever, as it is very durable. Save cost in the long run, as you don’t have to keep replacing the pan.

Also, other benefits include:

  • Works with induction cookers. (Iron is magnetic.)
  • It is moderately non-stick, almost as non-stick as Teflon. If anything sticks, just boil with water. Also, the more you use and season it, the more non-stick it becomes.

Downsides include: Heavy weight, needs seasoning (wipe dry and coat with oil) after cooking otherwise it can rust.

The third popular alternative, Aluminum pans, are definitely not good as it may be linked to Alzheimer’s and dementia.

Check out my wife’s food blog making Cheese Zucchini Patties using the new cast iron skillet.

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Part 2:   群表示论的基本概念和Abel群的表示

Math Online Tom Circle

[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

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Pure Mathematicians versus Applied Mathematicians

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 is time to learn more mathematics and find better tools”

Distinguished differential geometer EugenioCalabi

Ref:

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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 Pauling‘s criticisms led him to the ultimate accolade: his own Nobel prize.

The atoms in a solid material are arranged in an orderly fashion and that order is usually periodic and will have a particular rotational symmetry. A square arrangement, for example, has four-fold rotational symmetry – turn the atoms through 90 degrees and it will look the same. Do this four times and you get back to its start point. Three-fold symmetry means an arrangement can be turned through 120 degrees and it will look the same. There is also one-fold symmetry (turn through 360 degrees), two-fold (turn through 180 degrees) and six-fold symmetry (turn through 60 degrees). Five-fold symmetry is not allowed in periodic crystals and nothing beyond six, purely for geometric reasons.

Shechtman’s results were so out of the ordinary that, even after he had checked his findings several times, it took two years for his work to get published in a peer-reviewed journal. Once it appeared, he says, “all hell broke loose”.

Many scientists thought that Shechtman had not been careful enough in his experiments and that he had simply made a mistake. “The bad reaction was the head of my laboratory, who came to my office one day and, smiling sheepishly, put a book on x-ray diffraction on my desk and said, ‘Danny, please read this book and you will understand that what you are saying cannot be.’ And I told him, you know, I don’t need to read this book, I teach at the Technion, and I know this book, and I’m telling you my material is not in the book.

“He came back a couple of days later and said to me, ‘Danny, you are a disgrace to my group. I cannot be with you in the same group.’ So I left the group and found another group that adopted a scientific orphan.”

He says that the experience was not as traumatic as it sounded. Scientists around the world had quickly replicated Shechtman’s discovery and, in 1992, the International Union of Crystallography accepted that quasi-periodic materials must exist and altered its definition of what a crystal is from “a substance in which the constituent atoms, molecules or ions are packed in a regularly ordered, repeating three-dimensional pattern” to the broader “any solid having an essentially discrete diffraction diagram”.

That should have been the end of the story were it not for Linus Pauling, a two-time Nobel laureate, once for chemistry and a second time for peace. Shechtman explains that at a science conference in front of an audience of hundreds Pauling claimed, “Danny Shechtman is talking nonsense, there are no quasi-crystals, just quasi-scientists.”

Pauling told everyone who would listen that Shechtman had made a mistake. He proposed his own explanations for the observed five-fold symmetry and stuck to his guns, despite repeated rebuttals. “Everything he did was wrong and wrong and wrong and wrong; eventually, he couldn’t publish his papers and they were rejected before they were published,” says Shechtman. “But he was very insistent, was very sure of himself when he spoke; he was a flamboyant speaker.”

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数学是什么 ? What is Mathematics? 

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
$latex downarrow$
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)

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2016 Nobel-Prize Winning Physics Explained Through Pastry 

Math Online Tom Circle

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).

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

If R is a PID, then every finitely generated module M over R is isomorphic to a direct sum of cyclic R-modules. That is, there is a unique decreasing sequence of proper ideals (d_1)\supseteq(d_2)\supseteq\dots\supseteq(d_m) such that \displaystyle M\cong R^\beta\oplus\left(\bigoplus_{i=1}^m R/(d_i)\right) where d_i\in R, and \beta\in\mathbb{Z}.

Similarly, every graded module M over a graded PID R decomposes uniquely into the form \displaystyle M\cong\left(\bigoplus_{i=1}^n\Sigma^{\alpha_i}R\right)\oplus\left(\bigoplus_{j=1}^m\Sigma^{\gamma_j}R/(d_j)\right) where d_j\in R are homogenous elements such that (d_1)\supseteq(d_2)\supseteq\dots\supseteq(d_m), \alpha_i, \gamma_j\in\mathbb{Z}, and \Sigma^\alpha denotes an \alpha-shift upward in grading.

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Secondary Level Chinese Tuition

Looking for O Level / IP / JC Chinese Tuition?

Ms Gao specializes in teaching secondary level chinese (CL/HCL) tuition in Singapore. Ms Gao has taught students from various schools, including RI (Raffles Institution IP Programme).

Teaches West / Central Area: E.g. Clementi, Jurong East, Bukit Timah, Dover, Bishan, Marymount

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Website: http://chinesetuition88.com

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Persistence Interval

Next, we want to parametrize the isomorphism classes of the F[t]-modules by suitable objects.

A \mathcal{P}-interval is an ordered pair (i,j) with 0\leq i<j\in\mathbb{Z}^\infty=\mathbb{Z}\cup\{+\infty\}.

We may associate a graded F[t]-module to a set \mathcal{S} of \mathcal{P}-intervals via a bijection Q. We define \displaystyle Q(i,j)=\Sigma^i F[t]/(t^{j-i}) for a \mathcal{P}-interval (i,j). When j=+\infty, we have Q(i,+\infty)=\Sigma^iF[t].

For a set of \mathcal{P}-intervals \mathcal{S}=\{(i_1,j_1),(i_2,j_2),\dots,(i_n,j_n)\}, we define \displaystyle Q(\mathcal{S})=\bigoplus_{k=1}^n Q(i_k, j_k).

We may now restate the correspondence as follows.

The correspondence \mathcal{S}\to Q(\mathcal{S}) defines a bijection between the finite sets of \mathcal{P}-intervals and the finitely generated graded modules over the graded ring F[t].

Hence, the isomorphism classes of persistence modules of finite type over F are in bijective correspondence with the finite sets of \mathcal{P}-intervals.

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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 A=\bigoplus_{i=0}^\infty A_i be a graded ring. An ideal I\subset A is homogenous (also called graded) if for every element x\in I, its homogenous components also belong to I.

An ideal in a graded ring is homogenous if and only if it is a graded submodule. The intersections of a homogenous ideal I with the A_i are called the homogenous parts of I. A homogenous ideal I is the direct sum of its homogenous parts, that is, \displaystyle I=\bigoplus_{i=0}^\infty (I\cap A_i).

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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 group set up for Mongrels in Singapore. (https://www.facebook.com/exclusivelymongrels/)

Do support them in their cause, if you can. And share this story so as to spread the word (maintenance and upkeep of the dogs can be a huge cost). Mongrels are actually highly intelligent, and can be more healthy and robust as compared to pedigrees, which may have hereditary diseases. For example, the popular Golden Retriever breed is prone to hip dysplasia.

A story told by Dr Gan summarizes everything — The state and welfare of stray dogs in Singapore, supposedly a first-world country, is actually worse than jungle dogs in Borneo. The Orang Asli, primitive junglers in Sabah, apparently treat dogs better than the average layperson in Singapore:

When Dr Gan, an EM member, was running through the trails of Sabah in Oct 2016, he stumbled upon a stray dog.

Being an avid dog lover and the proud father to three rescued Mongrels, he had to stop in his tracks. He fed the dog and it even ran alongside him for a mile or two. Further along the route, he encountered more stray dogs too.

All of the stray dogs he encountered seemed well-fed and were very approachable. They all displayed no aggression, despite being in the middle of a jungle. To Dr Gan, this was a tell-tale sign that the Orang Asli, who lived in villages in these jungles, took care of the dogs by feeding them. The fact that these Orang Aslis were living in harmony with these strays was indeed very commendable in his eyes.

These thoughts stuck with him throughout the run, and on the journey home too.

He couldn’t help but compare the Orang Asli’s hospitality to how a Singaporean layperson would react upon encountering a stray dog. More often than not, even in the absence of aggressive behaviour, a Singaporean who sees a stray dog would view it as no more than a pest and would either chase it away or even, call the authorities. As it so often is when the latter option is exercised, the authorities would have a hard time rehoming the dog and EM has to step in to ‘bail’ the dog out before the authorities euthanize it.

It is strange, he remarked, how the Orang Asli from the jungle can treat these strays with reverence while many Singaporeans would report a stray to the authorities without the slightest hesitation.

“Would the situation end up the same way if, instead of a stray mongrel, there was a stray pedigree dog?”

Armed with the notion that more needs to be done not just for these dogs but also to empower and educate the general public in Singapore about the plight of these strays and what can be done to help them, he then called on his two running buddies to undertake this journey with him.

It was going to be a journey that united his two passions – running and dogs; a journey back to the jungles where he first encountered the strays; back to where he first witnessed the hospitality of the Orang Asli; back to where where the spark was first ignited. He, and his Team, hope to bash through the jungles of Borneo, all in the hopes of blazing a new trail for Mongrels back home, in Singapore.

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