## 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

Mobile: 98348087
Email: chinesetuition88@gmail.com
Website: http://chinesetuition88.com

## 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.

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.

## 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.

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, $\displaystyle I=\bigoplus_{i=0}^\infty (I\cap A_i).$

## Donate to help Stray Dogs in Singapore

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.

## 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.

## 群表示论引言 Introduction to Group Representation

• 群表示论 是什么?
• 有何用 ?

4条 ” + ” 法:

Commutative 交换律, Associative 结合律, Neutral element ” 0″ 零元, Inverse (-) 逆元

2 条 “x ” 法: (exclude ”1″ Unit, WHY ?)

Associative 结合律, Distributive (wrt “+”) 分配律

$latex mathbb {Z} _7 = { bar {0} , bar {1} , bar {2} , bar {3} , bar {4} , bar {5} , bar {6} }$

+ sqrt [3]{frac {q}{2} -sqrt{ { (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}}&fg=0000aa$Quartic Equation: by Cardano’s student Ferrari$latex {a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}&fg=00aa00&s=3$Quintic Equation:$latex {a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}&s=3No radical solution (Unsolvability) was suspected by Ruffini (1799)… View original post 177 more words Posted in math | Leave a comment ## Why call the Algebraic Structure Z a “Ring” ? Posted in math | 1 Comment ## Equivalence of C^infinity atlases Equivalence of $C^\infty$ atlases is an equivalence relation. Each $C^\infty$ atlas on $M$ is equivalent to a unique maximal $C^\infty$ atlas on $M$. Proof: Reflexive: If $A$ is a $C^\infty$ atlas, then $A\cup A=A$ is also a $C^\infty$ atlas. Symmetry: Let $A$ and $B$ be two $C^\infty$ atlases such that $A\cup B$ is also a $C^\infty$ atlas. Then certainly $B\cup A$ is also a $C^\infty$ atlas. Transitivity: Let $A, B, C$ be $C^\infty$ atlases, such that $A\cup B$ and $B\cup C$ are both $C^\infty$ atlases. Notation: \begin{aligned} A&=\{(U_\alpha,\varphi_\alpha)\}\\ B&=\{(V_\beta, \psi_\beta)\}\\ C&=\{(W_\gamma, f_\gamma)\}. \end{aligned} Then $\displaystyle \varphi_\alpha\circ f_\gamma^{-1}=\varphi_\alpha\circ\psi_\beta^{-1}\circ\psi_\beta\circ f_\gamma^{-1}: f_\gamma(U_\alpha\cap W_\gamma)\to\varphi_\alpha(U_\alpha\cap W_\gamma)$ is a diffeomorphism since both $\varphi_\alpha\circ\psi_\beta^{-1}$ and $\psi_\beta\circ f_\gamma^{-1}$ are diffeomorphisms due to $A\cup B$ and $B\cup C$ being $C^\infty$ atlases. Also, $M=\bigcup U_\alpha$, $M=\bigcup W_\gamma$ implies $M=(\bigcup U_\alpha)\cup(\bigcup W_\gamma)$ so $A\cup C$ is also a $C^\infty$ atlas. Let $A$ be a $C^\infty$ atlas on $M$. Define $B$ to be the union of all $C^\infty$ atlases equivalent to $A$. Then $B\sim A$. If $B'\sim A$, then $B'\subseteq B$, so that $B$ is the unique maximal $C^\infty$ atlas equivalent to $A$. Posted in math | Tagged | Leave a comment ## 代 数拓扑 Algebraic Topology 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 Topologyfrom Euler Characteristic (V -E + R = 2) Motivation of Algebraic Topology: Find Invariants[1]of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg. • Vector Space (to + – , × ÷ by multiplier Field scalars); • Ring (to + x), etc. then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants (homology, co-homology, etc). A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调 [2] – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc). Note: [1]… View original post 88 more words Posted in math | Leave a comment ## Natural Equivalence relating Suspension and Loop Space Theorem: If $(X,x_0)$, $(Y,y_0)$, $(Z,z_0)\in\mathscr{PT}$, $X$, $Z$ Hausdorff and $Z$ locally compact, then there is a natural equivalence $\displaystyle A: [Z\wedge X, *; Y,y_0]\to [X, x_0; (Y,y_0)^{(Z,z_0)}, f_0]$ defined by $A[f]=[\hat{f}]$, where if $f:Z\wedge X\to Y$ is a map then $\hat{f}: X\to Y^Z$ is given by $(\hat{f}(x))(z)=f[z,x]$. We need the following two propositions in order to prove the theorem. Proposition \label{prop13} The exponential function $E: Y^{Z\times X}\to (Y^Z)^X$ induces a continuous function $\displaystyle E: (Y,y_0)^{(Z\times X, Z\vee X)}\to ((Y,y_0)^{(Z,z_0)}, f_0)^{(X,x_0)}$ which is a homeomorphism if $Z$ and $X$ are Hausdorff and $Z$ is locally compact\footnote{every point of $Z$ has a compact neighborhood}. Proposition \label{prop8} If $\alpha$ is an equivalence relation on a topological space $X$ and $F:X\times I\to Y$ is a homotopy such that each stage $F_t$ factors through $X/\alpha$, i.e.\ $x\alpha x'\implies F_t(x)=F_t(x')$, then $F$ induces a homotopy $F':(X/\alpha)\times I\to Y$ such that $F'\circ (p_\alpha\times 1)=F$. Proof of Theorem i) $A$ is surjective: Let $f': (X,x_0)\to ((Y,y_0)^{(Z,z_0)},f_0)$. From Proposition \ref{prop13} we have that $E: (Y,y_0)^{(Z\times X, Z\vee X)}\to ((Y,y_0)^{(Z,z_0)},f_0)^{(X,x_0)}$ is a homeomorphism. Hence the function $\bar{f}: (Z\times X, Z\vee X)\to (Y,y_0)$ defined by $\bar{f}(z,x)=(f'(x))(z)$ is continuous since $(Ef'(x))(z)=f'(z,x)$ and thus $\bar{f}=E^{-1}f'$. By the universal property of the quotient, $\bar{f}$ defines a map $f:(Z\wedge X, *)\to (Y,y_0)$ such that $f[z,x]=\bar{f}(z,x)=(f'(x))(z)$. Thus $\hat{f}=f'$, so that $A[f]=[f']$. ii) $A$ is injective: Suppose $f,g: (Z\wedge X, *)\to (Y,y_0)$ are two maps such that $A[f]=A[g]$, i.e.\ $\hat{f}\simeq\hat{g}$. Let $H': X\times I\to (Y,y_0)^{(Z,z_0)}$ be the homotopy rel $x_0$. By Proposition \ref{prop13} the function $\bar{H}: Z\times X\times I\to Y$ defined by $\bar{H}(z,x,t)=(H'(x,t))(z)$ is continuous. This is because $\bar{H}(z,x,t)=(E\bar{H}(x,t))(z)$ so that $E\bar{H}=H'$, thus $\bar{H}=E^{-1}H'$ where $E$ is a homeomorphism. For each $t\in I$ we have $\bar{H}((Z\vee X)\times\{t\})=y_0$. This is because if $(z,x)\in Z\vee X$, then $z=z_0$ or $x=x_0$. If $z=z_0$, then $(H'(x,t))(z_0)=y_0$. If $x=x_0$, $(H'(x_0,t))(z)=y_0$ as $H'$ is the homotopy rel $x_0$. Then by Proposition \ref{prop8} there is a homotopy $H:(Z\wedge X)\times I\to Y$ rel $*$ such that $H([z,x],t)=\bar{H}(z,x,t)=(H'(x,t))(z)$. Thus $H_0([z,x])=(H_0'(x))(z)=(\hat{f}(x))(z)=f[z,x]$ and similarly $H_1([z,x])=(H_1'(x))(z)=(\hat{g}(x))(z)=g[z,x]$. Thus $[f]=[g]$ via the homotopy $H$. Loop space If $(Y,y_0)\in\mathscr{PT}$, we define the loop space $(\Omega Y, \omega_0)\in\mathscr{PT}$ of $Y$ to be the function space $\displaystyle \Omega Y=(Y,y_0)^{(S^1,s_0)}$ with the constant loop $\omega_0$ ($\omega_0(s)=y_0$ for all $s\in S^1$) as base point. Suspension If $(X,x_0)\in\mathscr{PT}$, we define the suspension $(SX,*)\in\mathscr{PT}$ of $X$ to be the smash product $(S^1\wedge X, *)$ of $X$ with the 1-sphere. Corollary (Natural Equivalence relating $SX$ and $\Omega Y$) If $(X,x_0)$, $(Y,y_0)\in\mathscr{PT}$ and $X$ is Hausdorff, then there is a natural equivalence $\displaystyle A: [SX, *; Y,y_0]\to [X, x_0; \Omega Y, \omega_0].$ Posted in math | Tagged | Leave a comment ## Russian Math Education ​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” Russian (China) -> “华山派” UK (USA). ( ) : derivative of its parent school. eg. China derived from Russian school in 1960s by Hua Luogeng. Note: 武当派 : 内功, 以柔尅刚, 四两拨千斤 <=> “Soft” Math, Abstract, Theoretical, Generalized. 少林派: 拳脚硬功夫 <=> “Hard” Math, algorithmic. 华山派: 剑法轻灵 <=> Applied, Astute, Computer-aided. The 3 schools’ pioneering grand masters (掌门人) since 16th century till 21st century, in between the 19th century (during the French Revolution) Modern Math (近代数学) is the critical milestone, the other (现代数学) is WW2 : – France: Descartes / Fermat / Pascal (17 CE : Analytical Geometry, Number Theory, Probability), Cauchy / Lagrange / Fourier /Galois (19 CE, Modern Math : Analysis, Abstract Algebra), View original post 189 more words Posted in math | Leave a comment ## Fundamental Group of S^n is trivial if n>=2 $\pi_1(S^n)=0$ if $n\geq 2$ We need the following lemma: If a space $X$ is the union of a collection of path-connected open sets $A_\alpha$ each containing the basepoint $x_0\in X$ and if each intersection $A_\alpha\cap A_\beta$ is path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_\alpha$. Proof: Take $A_1$ and $A_2$ to be the complements of two antipodal points in $S^n$. Then $S^n=A_1\cup A_2$ is the union of two open sets $A_1$ and $A_2$, each homeomorphic to $\mathbb{R}^n$ such that $A_1\cap A_2$ is homeomorphic to $S^{n-1}\times\mathbb{R}$. Choose a basepoint $x_0$ in $A_1\cap A_2$. If $n\geq 2$ then $A_1\cap A_2$ is path-connected. By the lemma, every loop in $S^n$ based at $x_0$ is homotopic to a product of loops in $A_1$ or $A_2$. Both $\pi_1(A_1)$ and $\pi_1(A_2)$ are zero since $A_1$ and $A_2$ are homeomorphic to $\mathbb{R}^n$. Hence every loop in $S^n$ is nullhomotopic. Posted in math | Tagged | Leave a comment ## Tangent Space is Vector Space Prove that the operation of linear combination, as in Definition 2.2.7, makes $T_p(U)$ into an $n$-dimensional vector space over $\mathbb{R}$. The zero vector is the infinitesimal curve represented by the constant $p$. If $\langle s\rangle_p\in T_p(U)$, then $-\langle s\rangle_p=\langle s^-\rangle_p$ where $s^-(t)=s(-t)$, defined for all sufficiently small values of $t$. Proof: We verify the axioms of a vector space. Multiplicative axioms: $1\langle s_1\rangle_p=\langle 1s_1+0-(1+0-1)p\rangle_p=\langle s_1\rangle_p$ $(ab)\langle s_1\rangle_p=\langle abs_1-(ab-1)p\rangle_p$ \begin{aligned} a(b\langle s_1\rangle_p)&=a\langle bs_1-(b-1)p\rangle_p\\ &=\langle abs_1-(ab-a)p-(a-1)p\rangle_p\\ &=\langle abs_1-(ab-1)p\rangle_p\\ &=(ab)\langle s_1\rangle_p \end{aligned} Additive Axioms: $\langle s_1\rangle_p+\langle s_2\rangle_p=\langle s_2\rangle_p+\langle s_1\rangle_p=\langle s_1+s_2-p\rangle_p$ \begin{aligned} (\langle s_1\rangle_p+\langle s_2\rangle_p)+\langle s_3\rangle_p&=\langle s_1+s_2-p\rangle_p+\langle s_3\rangle_p\\ &=\langle s_1+s_2-p+s_3-p\rangle_p\\ &=\langle s_1+s_2+s_3-2p\rangle_p \end{aligned} \begin{aligned} \langle s_1\rangle_p+(\langle s_2\rangle_p+\langle s_3\rangle_p)&=\langle s_1\rangle_p+\langle s_2+s_3-p\rangle_p\\ &=\langle s_1+s_2+s_3-2p\rangle_p \end{aligned} $\langle s\rangle_p+\langle s^-\rangle_p=\langle s+s^- -p\rangle_p$ $\frac{d}{dt}f(s(t)+s(-t)-p)|_{t=0}=0=\frac{d}{dt}f(p)|_{t=0}$ Hence $\langle s\rangle_p+\langle s^-\rangle_p=\langle p\rangle_p$. $\langle s_1\rangle_p+\langle p\rangle_p=\langle s_1+p-p\rangle_p=\langle s_1\rangle_p$ Distributive Axioms: \begin{aligned} a(\langle s_1\rangle_p+\langle s_2\rangle_p)&=a\langle s_1+s_2-p\rangle_p\\ &=\langle a(s_1+s_2-p)-(a-1)p\rangle_p \end{aligned} \begin{aligned} a\langle s_1\rangle_p+a\langle s_2\rangle_p&=\langle as_1-(a-1)p\rangle_p+\langle as_2-(a-1)p\rangle_p\\ &=\langle a(s_1+s_2)-2(a-1)p-p\rangle_p\\ &=\langle a(s_1+s_2-p)-(a-1)p\rangle_p\\ &=a(\langle s_1\rangle_p+\langle s_2\rangle_p) \end{aligned} $(a+b)\langle s_1\rangle_p=\langle (a+b)s_1-(a+b-1)p\rangle_p$ $a\langle s_1\rangle_p+b\langle s_1\rangle_p=\langle as_1+bs_1-(a+b-1)p\rangle_p=(a+b)\langle s_1\rangle_p$ Hence $T_p(U)$ is a vector space over $\mathbb{R}$. Since $U\subseteq\mathbb{R}^n$, $T_p(U)$ is $n$-dimensional. Posted in math | Tagged | Leave a comment ## 群论的哲学 Philosophical Group Theory ​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” (Inverse) = I (记号: a 的逆元 =latex a^{-1}$) Agree to disagree = Neutral$latex a*a^{-1} = e $具有这四个性质的群体才是 群体的 “美 : “对称” 如果没有 (3)&(4): 半群 如果没有 (4) 反对者: 么半群 以上是 Group (群 ) 数学的定义: “CAN I” CA = Semi-Group 半群 CAN = Monoid 么半群 群是 19岁Evariste Galois 在法国革命时牢狱中发明的, 解决 300年来 Quintic Equations (5次以上的 方程式) 没有 “有理数” 的 解 (rational roots)。19世纪的 Modern Math (Abstract Algebra) 从此诞生, 群用来解释自然科学(物理, 化学, 生物)里 “对称”现象。Nobel Physicists (1958) 杨振宁/李政道 用群来证明物理 弱力 (Weak Force) 粒子(Particles) 的不对称 (Assymetry )。 View original post Posted in math | Leave a comment ## Balance Quote Never let success go to your head, and never let failure go to your heart. Posted in math | Tagged | Leave a comment ## 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 enter primary 1 already knowing primary 3 syllabus, so everyone is scoring 100/100. So top 25% percentile mark becomes 99/100. • Lack of manpower (Chinese teachers). It is well-known that Singaporeans are not very interested in general in pursuing the career of Mother Tongue teacher (look at the cut-off points of Chinese studies in universities). So only enough manpower for limited number of Higher Chinese classes. • Kiasu principals / HODs who want to “quality-control” those taking Higher Chinese to boost the distinction rate of the cohort (a common but unethical tactic to improve the cohort’s performance in national exams is to force those who are not doing well to drop the subject) • Lastly, it is not known if 97 is the overall mark, or just one of the marks in the continual assessment. It is possible to score 97 in one test, but the average can be much lower. This is quite a serious issue as Chinese is no longer a minor/unimportant subject, like in the past it was. In fact, under the new PSLE scoring system, Chinese is one of the major game-changing core components, a severe Achilles’ heel for those in English-speaking families. Getting proficient in Chinese from an early age is a must for the new PSLE system, so no doubt many parents are anxious about Higher Chinese. http://www.straitstimes.com/singapore/education/how-can-97-marks-be-not-good-enough Parents of some children in a well-known primary school have complained about the selection process for Higher Chinese. St Hilda’s Primary pupils are routed into Higher Chinese classes in Primary 2 based on continual assessment test results in Primary 1. What upset the parents was that pupils who scored as high as 97 marks in Chinese last year were told that they had failed to make the cut for Higher Chinese. Posted in math | Tagged | Leave a comment ## Functors, Homotopy Sets and Groups Functors Definition: A functor $F$ from a category $\mathscr{C}$ to a category $\mathscr{D}$ is a function which – For each object $X\in\mathscr{C}$, we have an object $F(X)\in\mathscr{D}$. – For each $f\in\hom_\mathscr{C}(X,Y)$, we have a morphism $\displaystyle F(f)\in\hom_\mathscr{D}(F(X),F(Y)).$ Furthermore, $F$ is required to satisfy the two axioms: – For each object $X\in\mathscr{C}$, we have $F(1_X)=1_{F(X)}$. That is, $F$ maps the identity morphism on $X$ to the identity morphism on $F(X)$. – For $f\in\hom_{\mathscr{C}}(X,Y)$, $g\in\hom_\mathscr{C}(Y,Z)$ we have $\displaystyle F(g\circ f)=F(g)\circ F(f)\in\hom_\mathscr{D}(F(X),F(Z)).$ That is, functors must preserve composition of morphisms. Definition: A cofunctor (also called contravariant functor) $F^*$ from a category $\mathscr{C}$ to a category $\mathscr{D}$ is a function which – For each object $X\in\mathscr{C}$, we have an object $F^*(X)\in\mathscr{D}$. – For each $f\in\hom_\mathscr{C}(X,Y)$ we have a morphism $\displaystyle F^*(f)\in\hom_\mathscr{D}(F^*(Y),F^*(X))$ satisfying the two axioms: – For each object $X\in\mathscr{C}$ we have $F^*(1_X)=1_{F^*(X)}$. That is, $F^*$ preserves identity morphisms. – For each $f\in\hom_\mathscr{C}(X,Y)$ and $g\in\hom_\mathscr{C}(Y,Z)$ we have $\displaystyle F^*(g\circ f)=F^*(f)\circ F^*(g)\in\hom_\mathscr{D}(F^*(Z),F^*(X)).$ Note that cofunctors reverse the direction of composition. Example Given a fixed pointed space $(K,k_0)\in\mathscr{PT}$, we define a functor $\displaystyle F_K:\mathscr{PT}\to\mathscr{PS}$ as follows: for each $(X,x_0)\in\mathscr{PT}$ we assign $F_K(X,x_0)=[K,k_0; X,x_0]\in\mathscr{PS}$. Given $f: (X,x_0)\to (Y,y_0)$ in $\hom((X,x_0),(Y,y_0))$ we define $F_K(f)\in\hom([K,k_0; X,x_0],[K,k_0;Y,y_0])$ by $\displaystyle F_k(f)[g]=[f\circ g]\in[K,k_0; Y,y_0]$ for every $[g]\in [K,k_0; X,x_0]$. We can check the two axioms: – $F_k(1_X)[g]=[1_X\circ g]=[g]$ for every $[g]\in[K,k_0; X, x_0]$. – For $f\in\hom((X,x_0),(Y,y_0))$, $h\in\hom((Y,y_0),(Z,z_0))$ we have $\displaystyle F_K(h\circ f)[g]=[h\circ f\circ g]=F_K(h)\circ F_K(f)[g]\in[K,k_0; Z,z_0]$ for every $[g]\in[K,k_0; X,x_0]$. Similarly, we can define a cofunctor $F_K^*$ by taking $F_K^*(X,x_0)=[X,x_0; K,k_0]$ and for $f:(X,x_0)\to (Y,y_0)$ in $\hom((X,x_0),(Y,y_0))$ we define $\displaystyle F_K(f)[g]=[g\circ f]\in[X,x_0; K,k_0]$ for every $[g]\in[Y,y_0; K,k_0]$. Note that if $f\simeq f'$ rel $x_0$, then $F_K(f)=F_K(f')$ and similarly $F_K^*(f)=F_K^*(f')$. Therefore $F_K$ (resp.\ $F_K^*$) can also be regarded as defining a functor (resp.\ cofunctor) $\mathscr{PT}'\to\mathscr{PS}$. Homotopy Sets and Groups Theorem: If $(X,x_0)$, $(Y,y_0)$, $(Z,z_0)\in\mathscr{PT}$, $X$, $Z$ Hausdorff and $Z$ locally compact, then there is a natural equivalence $\displaystyle A: [Z\wedge X, *; Y,y_0]\to [X, x_0; (Y,y_0)^{(Z,z_0)}, f_0]$ defined by $A[f]=[\hat{f}]$, where if $f:Z\wedge X\to Y$ is a map then $\hat{f}: X\to Y^Z$ is given by $(\hat{f}(x))(z)=f[z,x]$. We need the following two propositions in order to prove the theorem. Proposition 1: The exponential function $E: Y^{Z\times X}\to (Y^Z)^X$ induces a continuous function $\displaystyle E: (Y,y_0)^{(Z\times X, Z\vee X)}\to ((Y,y_0)^{(Z,z_0)}, f_0)^{(X,x_0)}$ which is a homeomorphism if $Z$ and $X$ are Hausdorff and $Z$ is locally compact\footnote{every point of $Z$ has a compact neighborhood}. Proposition 2: If $\alpha$ is an equivalence relation on a topological space $X$ and $F:X\times I\to Y$ is a homotopy such that each stage $F_t$ factors through $X/\alpha$, i.e.\ $x\alpha x'\implies F_t(x)=F_t(x')$, then $F$ induces a homotopy $F':(X/\alpha)\times I\to Y$ such that $F'\circ (p_\alpha\times 1)=F$. Posted in math | Tagged | 2 Comments ## 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 good at explaining complicated concepts in a simple manner SMS: 98348087 Email: mathtuition88@gmail.com Areas teaching (West / Central Singapore): • Clementi • Jurong East • Buona Vista • West Coast • Dover • Central Areas like Bishan/Toa Payoh/Marymount (near MRT) Posted in math | Tagged , | Leave a comment ## 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 affiliation to recommend before submitting. There are pros and cons to viXra, the pros being freedom of submission open to everyone on the world. The cons is that, naturally, there may be more crackpots who submit nonsense. There are, however, some serious papers on viXra. After submitting, the viXra admin will send an email something like this: Thank you for your submission request to viXra.org Your submission has now been uploaded and is available at Please check and let us know is there are any errors. If everything is OK please do not reply to this mail. If you do not see it on some of the listing pages this may be due to caching in your web browser, please clear the cache and reload If you need to replace the document you should use the replacement form and enter the viXra number If you need to make any changes on the abstract page without changing the PDf document (e.g. authors, title, comment, abstract), use the change web form. Links to the forms are provided at http://vixra.org/submit please note that viXra.org does not do cross-listings to other subject categories The search feature on viXra is dependent on Google and will update with changes when they reindex. This may take a few days. The details of the submission were as follows: Thank you for using our service at viXra.org Regards, the viXra admin Featured book: An Introduction to the Theory of Numbers An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today’s students through the key milestones and developments in number theory. Posted in math | Tagged , | Leave a comment ## Prof ST Yau’s 邱成桐 Talk to Chinese Youth on Math Education 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 27, proving Calabi Conjecture), Crafoord Prize(1994),Wolf Prize(2010). Key Takeaways: 1. On Math Education: ◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc. ◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after it do the problems. ◇ Read Math topics even though you do not understand in first round, re-read few more times, then few days / months / years / decades later you will digest them. (做学问的程序). ◇ Do not consult students in WHAT to teach, because they don’t know what to learn. ◇ Love of Math beauty is the “pull-factor” for motivating students’ interest in Math. ◇ Parental… View original post 106 more words Posted in math | Leave a comment ## Algebraic Topology: Fundamental Group Homotopy of paths A homotopy of paths in a space $X$ is a family $f_t: I\to X$, $0\leq t\leq 1$, such that (i) The endpoints $f_t(0)=x_0$ and $f_t(1)=x_1$ are independent of $t$. (ii) The associated map $F:I\times I\to X$ defined by $F(s,t)=f_t(s)$ is continuous. When two paths $f_0$ and $f_1$ are connected in this way by a homotopy $f_t$, they are said to be homotopic. The notation for this is $f_0\simeq f_1$. Example: Linear Homotopies Any two paths $f_0$ and $f_1$ in $\mathbb{R}^n$ having the same endpoints $x_0$ and $x_1$ are homotopic via the homotopy $\displaystyle f_t(s)=(1-t)f_0(s)+tf_1(s).$ Simply-connected A space is called simply-connected if it is path-connected and has trivial fundamental group. A space $X$ is simply-connected iff there is a unique homotopy class of paths connecting any two parts in $X$. Path-connectedness is the existence of paths connecting every pair of points, so we need to be concerned only with the uniqueness of connecting paths. ($\implies$) Suppose $\pi_1(X)=0$. If $f$ and $g$ are two paths from $x_0$ to $x_1$, then $f\simeq f\cdot \bar{g}\cdot g\simeq g$ since the loops $\bar{g}\cdot g$ and $f\cdot\bar{g}$ are each homotopic to constant loops, due to $\pi_1(X,x_0)=0$. ($\impliedby$) Conversely, if there is only one homotopy class of paths connecting a basepoint $x_0$ to itself, then all loops at $x_0$ are homotopic to the constant loop and $\pi_1(X,x_0)=0$. $\pi_1(X\times Y)$ is isomorphic to $\pi_1(X)\times \pi_1(Y)$ if $X$ and $Y$ are path-connected. A basic property of the product topology is that a map $f:Z\to X\times Y$ is continuous iff the maps $g:Z\to X$ and $h:Z\to Y$ defined by $f(z)=(g(z),h(z))$ are both continuous. Hence a loop $f$ in $X\times Y$ based at $(x_0,y_0)$ is equivalent to a pair of loops $g$ in $X$ and $h$ in $Y$ based at $x_0$ and $y_0$ respectively. Similarly, a homotopy $f_t$ of a loop in $X\times Y$ is equivalent to a pair of homotopies $g_t$ and $h_t$ of the corresponding loops in $X$ and $Y$. Thus we obtain a bijection $\pi_1(X\times Y, (x_0,y_0))\approx \pi_1(X,x_0)\times \pi_1(Y,y_0)$, $[f]\mapsto([g],[h])$. This is clearly a group homomorphism, and hence an isomorphism. Note: The condition that $X$ and $Y$ are path-connected implies that $\pi_1(X,x_0)=\pi_1(X)$, $\pi_1(Y,y_0)=\pi_1(Y),\pi_1(X\times Y,(x_0,y_0))=\pi_1(X\times Y)$. Posted in math | Tagged | Leave a comment ## Multivariable Derivative and Partial Derivatives If $L_p(x)=c+\sum_{i=1}^n b_ix^i$ is a derivative of $f$ at $p$, then $\displaystyle b_i=\frac{\partial f}{\partial x^i}(p),$ $1\leq i\leq n$. In particular, if $f$ is differentiable at $p$, these partial derivatives exist and the derivative $L_p$ is unique. Proof: Let $h=x-p$, then $\lim_{x\to p}\frac{f(x)-L_p(x)}{\|x-p\|}=0$ becomes $\displaystyle \lim_{h\to 0}\frac{f(p+h)-f(p)}{\|h\|}-\lim_{h\to 0}\frac{L_p(h)-c}{\|h\|}=0$ since $L_p(p+h)=L_p(p)+L_p(h)-c=f(p)+L_p(h)-c$. By choosing $h=(0,\dots,h^i,\dots, 0)$ (all zeroes except in $i$th position), then as $h\to 0$, $\displaystyle \lim_{h\to 0}\frac{f(p+h)-f(p)}{\|h\|}=\frac{\partial f}{\partial x^i}(p)$ and $\displaystyle \lim_{h\to 0}\frac{L_p(h)-c}{\|h\|}=\lim_{h^i\to 0}\frac{b_ih^i}{h^i}=b_i.$ So $b_i=\frac{\partial f}{\partial x^i}(p)$. Posted in math | Tagged | Leave a comment ## Chinese Remainder Theorem Any short-cut method ? Yes, by L.C.M… 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 {Monday (Prof M), tuesday (Prof t), Wednesday (Prof W), Thursday (Prof T), Friday (Prof F), saturday (Prof s), Sunday (Prof S)} d : 1 2 3 4 5 6 [7] 1 2 3 4 5 6 [7] 1 2 M: m 0 m 0 m 0 [m] ==> fell on 1st sunday t: - t 0 0 t 0 [0 ] t 0 0 t 0 0 [t ] ==> fell on 2nd sunday W: – - w 0 0 0 [w] 0 0 0 w 0 0 [0] ==> fell on 1st sunday T: - - - T T T [T] ==> fell on 1st sunday (TRIVIAL CASE!) F: - - - - f 0… View original post 626 more words Posted in math | Leave a comment ## Some Math Connotations Demystified 数学内涵解密 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. “Dual Space“(对偶空间) : it is the evaluation of a “Vector Space”. Example: A student studies few subjects {Math, Physics, English, Chemistry…}, these subjects form a “Subject Vector Space” (V), if we associate the subjects with weightages (加权) , say, Math 4, Physics 3, English 2, Chemistry 1, the “Weightage Dual Space” of V will be W= {4, 3, 2, 1}. 2. Vector: beyond the meaning of a physical vector with direction and value, it extends to any “object” which can be manipulated (抵消) by the 4 operations “+, – , x, / ” in a FieldF = {R or Z2 …}. Eg.$latex alpha_{1}.v_{1} + alpha_{2}.v_{2} + alpha_{3}.v_{3}, forall alpha_{j} in…

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## Homology: Why Boundary of Boundary = 0 ?

This equation puzzles most people. WHY ?
$latex boxed {{delta}^2 = 0 { ?}}&fg=aa0000&s=3$

It is analogous to the Vector Algebra:
Let the boundary of {A, B} =
$latex delta (A,B) = overrightarrow{AB }$

$latex overrightarrow{AB } + overrightarrow {BA} =overrightarrow{AB } – overrightarrow {AB} = vec 0$

Note: Co-homology: (上)同调

Euclid Geometry & Homology:

Isabell Darcy Lecture: cohomology

View original post

## Existence and properties of normal closure

If $E$ is an algebraic extension field of $K$, then there exists an extension field $F$ of $E$ (called the normal closure of $E$ over $K$) such that
(i) $F$ is normal over $K$;
(ii) no proper subfield of $F$ containing $E$ is normal over $K$;
(iii) if $E$ is separable over $K$, then $F$ is Galois over $K$;
(iv) $[F:K]$ is finite if and only if $[E:K]$ is finite.

The field $F$ is uniquely determined up to an $E$-isomorphism.

Proof:
(i) Let $X=\{u_i\mid i\in I\}$ be a basis of $E$ over $K$ and let $f_i\in K[x]$ be the minimal polynomial of $u_i$. If $F$ is a splitting field of $S=\{f_i\mid i\in I\}$ over $E$, then $F=E(Y)$, where $Y\supseteq X$ is the set of roots of the $f_i$. Then $F=K(X)(Y)=K(Y)$ so $F$ is also a splitting field of $S$ over $K$, hence $F$ is normal over $K$ as it is the splitting field of a family of polynomials in $K[x]$.

(iii) If $E$ is separable over $K$, then each $f_i$ is separable. Therefore $F$ is Galois over $K$ as it is a splitting field over $K$ of a set of separable polynomials in $K[x]$.

(iv) If $[E:K]$ is finite, then so is $X$ and hence $S$. Say $S=\{f_1,\dots,f_n\}$. Then $F=E(Y)$, where $Y$ is the set of roots of the $f_i$. Then $F$ is finitely generated and algebraic, thus a finite extension. So $[F:K]$ is finite.

(ii) A subfield $F_0$ of $F$ that contains $E$ necessarily contains the root $u_i$ of $f_i\in S$ for every $i$. If $F_0$ is normal over $K$ (so that each $f_i$ splits in $F_0$ by definition), then $F\subset F_0$ (since $F$ is the splitting field) and hence $F=F_0$.

Finally let $F_1$ be another extension field of $E$ with properties (i) and (ii). Since $F_1$ is normal over $K$ and contains each $u_i$, $F_1$ must contain a splitting field $F_2$ of $S$ over $K$ with $E\subset F_2$. $F_2$ is normal over $K$ (splitting field over $K$ of family of polynomials in $K[x]$), hence $F_2=F_1$ by (ii).

Therefore both $F$ and $F_1$ are splitting fields of $S$ over $K$ and hence of $S$ over $E$: If $F=K(Y)$ (where $Y$ is set of roots of $f_i$) then $F\subseteq E(Y)$ since $E(Y)$ contains $K$ and $Y$. Since $Y\supseteq X$, so $K(Y)$ contains $E=K(X)$ and $Y$, hence $F=E(Y)$. Hence the identity map on $E$ extends to an $E$-isomorphism $F\cong F_1$.

## 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 goals in 2017, and more importantly be happy in the process.

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## Cours Raisonnements (Logics) , Ensembles ( Sets), Applications (Mappings)

This is an excellent quick revision of the French Baccalaureat Math during the first month of French university. (Unfortunately common A-level Math syllabus lacks such rigourous Math foundation.)

Most non-rigourous high-school students / teachers abuse the use of :

“=> ” , “<=>” .

Prove by “Reductio par Absudum” 反证法 (by Contradiction) is a clever mathematical logic :

$latex boxed {(A => B) <=> (non B => non A)} &s=3$

Famous Examples: 1) Prove $latex sqrt 2$ is irrational ; 2) There are infinite prime numbers (both by Greek mathematician Euclid 3,000 years ago)

The young teacher showed the techniques of proving Mapping:

Surjective (On-to) – best understood in Chinese 满射 (Full Mapping)

Injective (1-to-1) 单射

Bijective (On-to & 1-to-1) 双射

He used an analogy of (the Set of) red Indians shooting (the Set of bisons 野牛):

All bisons are shot by arrows from1 or more Indians. (Surjective…

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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.

## A little more perseverance, maybe success is near

So close yet so far, to the heap of diamonds…

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

A finitely generated torsion-free module $A$ over a PID $R$ is free.
Proof
(Hungerford 221)

If $A=0$, then $A$ is free of rank 0. Now assume $A\neq 0$. Let $X$ be a finite set of nonzero generators of $A$. If $x\in X$, then $rx=0$ ($r\in R$) if and only if $r=0$ since $A$ is torsion-free.

Consequently, there is a nonempty subset $S=\{x_1,\dots,x_k\}$ of $X$ that is maximal with respect to the property: $\displaystyle r_1x_1+\dots+r_kx_k=0\ (r_i\in R) \implies r_i=0\ \text{for all}\ i.$

The submodule $F$ generated by $S$ is clearly a free $R$-module with basis $S$. If $y\in X-S$, then by maximality there exist $r_y,r_1,\dots,r_k\in R$, not all zero, such that $r_yy+r_1x_1+\dots+r_kx_k=0$. Then $r_yy=-\sum_{i=1}^kr_ix_i\in F$. Furthermore $r_y\neq 0$ since otherwise $r_i=0$ for every $i$.

Since $X$ is finite, there exists a nonzero $r\in R$ (namely $r=\prod_{y\in X-S}r_y$) such that $rX=\{rx\mid x\in X\}$ is contained in $F$:

If $y_i\in X-S$, then $ry=r_{y_1}\dots r_{y_n}y_i\in F$ since $r_{y_i}y_i\in F$. If $x\in S$, then clearly $rx\in F$ since $F$ is generated by $S$.

Therefore, $rA=\{ra\mid a\in A\}\subset F$. The map $f:A\to A$ given by $a\mapsto ra$ is an $R$-module homomorphism with image $rA$. Since $A$ is torsion-free $\ker f=0$, hence $A\cong rA\subset F$. Since a submodule of a free module over a PID is free, this proves $A$ is free.