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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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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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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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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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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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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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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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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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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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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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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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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 … 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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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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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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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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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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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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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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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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Existence and properties of normal closure

If is an algebraic extension field of , then there exists an extension field of (called the normal closure of over ) such that (i)  is normal over ; (ii) no proper subfield of containing is normal over ; (iii) if is … Continue reading

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Happy New Year to Readers of Mathtuition88.com

Wishing all readers of Mathtuition88.com a happy new year, and may 2017 bring you peace and joy in your life. No matter which stage of life you are in (student/career/parent/retiree), here is my sincere wishes that you will achieve your … Continue reading

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Printable Calendar 2017

2017 Calendar Printable Calendar 2017, with (Singapore) holidays. Generated by http://www.calendarlabs.com/customize/pdf-calendar/monthly-calendar-01.

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A little more perseverance, maybe success is near

Wishing all readers a happy new year ahead. May your dreams and wishes come true! 再努力一下,或许就是成功!

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A finitely generated torsion-free module A over a PID R is free

A finitely generated torsion-free module over a PID is free. Proof (Hungerford 221) If , then is free of rank 0. Now assume . Let be a finite set of nonzero generators of . If , then () if and … Continue reading

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