January 2017 – Mathtuition88

Universal Property of Quotient Groups (Hungerford)

If $f:G\to H$ is a homomorphism and $N$ is a normal subgroup of $G$ contained in the kernel of $f$ , then $f$ “factors through” the quotient $G/N$ uniquely.

This can be used to prove the following proposition:
A chain map $f_\bullet$ between chain complexes $(A_\bullet, \partial_{A, \bullet})$ and $(B_\bullet, \partial_{B,\bullet})$ induces homomorphisms between the homology groups of the two complexes.

Proof:
The relation $\partial f=f\partial$ implies that $f$ takes cycles to cycles since $\partial\alpha=0$ implies $\partial(f\alpha)=f(\partial\alpha)=0$ . Also $f$ takes boundaries to boundaries since $f(\partial\beta)=\partial(f\beta)$ . Hence $f_\bullet$ induces a homomorphism $(f_\bullet)_*: H_\bullet (A_\bullet)\to H_\bullet (B_\bullet)$ , by universal property of quotient groups.

For $\beta\in\text{Im} \partial_{A,n+1}$ , we have $\pi_{B,n}f_n(\beta)=\text{Im}\partial_{B,n+1}$ . Therefore $\text{Im}\partial_{A,n+1}\subseteq\ker(\pi_{B,n}\circ f_n)$ .

Some Homology Definitions

Chain Complex
A sequence of homomorphisms of abelian groups $\displaystyle \dots\to C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1}\to\dots\to C_1\xrightarrow{\partial_1}C_0\xrightarrow{\partial_0}0$ with $\partial_n\partial_{n+1}=0$ for each $n$ .

$n$ th Homology Group
$H_n=\text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1}$

$\Delta_n(X)$
$\Delta_n(X)$ is the free abelian group with basis the open $n$ -simplices $e_\alpha^n$ of $X$ .

$n$ -chains
Elements of $\Delta_n(X)$ , called $n$ -chains, can be written as finite formal sums $\sum_\alpha n_\alpha e_\alpha^n$ with coefficients $n_\alpha\in\mathbb{Z}$ .

RP^n Projective n-space

Define an equivalence relation on $S^n\subset\mathbb{R}^{n+1}$ by writing $v\sim w$ if and only if $v=\pm w$ . The quotient space $P^n=S^n/\sim$ is called projective $n$ -space. (This is one of the ways that we defined the projective plane $P^2$ .) The canonical projection $\pi: S^n\to P^n$ is just $\pi(v)=\{\pm v\}$ . Define $U_i\subset P^n$ , $1\leq i\leq n+1$ , by setting $\displaystyle U_i=\{\pi(x^1, \dots, x^{n+1})\mid x^i\neq 0\}.$

Prove
1) $U_i$ is open in $P^n$ .
2) $\{U_1, \dots, U_{n+1}\}$ covers $P^n$ .
3) There is a homeomorphism $\varphi_i: U_i\to\mathbb{R}^n$ .
4) $P^n$ is compact, connected, and Hausdorff, hence is an $n$ -manifold.

Proof:
1) $\pi^{-1}U_i=\{(x^1, \dots, x^{n+1})\mid x^i\neq 0\}$ is open in $S^n$ , so $U_i$ is open in $P^n$ .
2) Let $y=\pi(x^1,\dots, x^{n+1})\in P^n$ . Then since $(x^1,\dots, x^{n+1})\neq(0,\dots,0)$ , so $y\in\bigcup_{i=1}^{n+1}U_i$ . Hence $P^n\subset\bigcup_{i=1}^{n+1}U_i$ .
3) Consider $A=\{(x^1,\dots, x^{n+1})\mid x^i+1\}\cong\mathbb{R}^n$ . Define $\displaystyle \varphi_i(\pi(x^1,\dots, x^{n+1}))=(\frac{x^1}{\|x^i\|},\dots,\frac{x^{i-1}}{\|x^i\|},1,\dots,\frac{x^{n+1}}{\|x^i\|})$ for $x^i>0$ . If $x^i<0$ , then $\varphi_i(\pi(x^1,\dots, x^{n+1}))=\varphi_i(\pi(-x^1,\dots, -x^{n+1}))$ . Then $\varphi_i$ is well-defined.

$\displaystyle \varphi_i^{-1}(x^1,\dots,1,\dots,x^{n+1})=\pi(\frac{x^1}{\|v\|},\dots,\frac{1}{\|v\|},\dots,\frac{x^{n+1}}{\|v\|}),$ where $v=(x^1,\dots, 1,\dots, x^{n+1})$ . Both $\varphi_i$ and $\varphi_i^{-1}$ are continuous, so $\varphi_i: U_i\to A$ is a homeomorphism.
4) Since $S^n$ is compact and connected, so is $P^n=S^n/\sim$ . $P^n$ is a CW-complex with one cell in each dimension, i.e.\ $P^n=\bigcup_{i=0}^n e^n$ . Since CW-complexes are Hausdorff, so is $P^n$ .

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

Motivation
Data is commonly represented as an unordered sequence of points in the Euclidean space $\mathbb{R}^n$ . The global `shape’ of the data may provide important information about the underlying phenomena of the data.

For data points in $\mathbb{R}^2$ , determining the global structure is not difficult, but for data in higher dimensions, a planar projection can be hard to decipher.
From point cloud data to simplicial complexes
To convert a collection of points $\{x_\alpha\}$ in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance $\epsilon$ ). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$ , the Cech complex (also known as the nerve), $\mathcal{C}_\epsilon$ , is the abstract simplicial complex where a set of $k+1$ vertices spans a $k$ -simplex whenever the $k+1$ corresponding closed $\epsilon/2$ -ball neighborhoods have nonempty intersection.

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$ , the Vietoris-Rips complex, $\mathcal{R}_\epsilon$ , is the abstract simplicial complex where a set $S$ of $k+1$ vertices spans a $k$ -simplex whenever the distance between any pair of points in $S$ is at most $\epsilon$ .

Top left: A fixed set of points. Top right: Closed balls of radius $\epsilon/2$ centered at the points. Bottom left: Cech complex has the homotopy type of the $\epsilon/2$ cover ( $S^1\vee S^1\vee S^1$ ) Bottom right: Vietoris-Rips complex has a different homotopy type ( $S^1\vee S^2$ ). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.

Does Abstract Math belong to Elementary Math ?

Math Online Tom Circle

Yes.

Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):

[X ] Group -> Ring -> Field

It would be better, conceptual wise, to reverse the teaching order as:

Field -> Ring -> Group

or better still as (the author thinks):

Ring -> Field -> Group

Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation to Ring.
Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group…

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In Search for Radical Roots of Polynomial Equations of degree n > 1

Math Online Tom Circle

Take note: Find roots 根 to solve polynomial 多项式方程式equations, but find solution解to solve algebraic equations代数方程式.

Radical : (LatinRadix = root): $latex sqrt [n]{x} $

Quadratic equation (二次方程式) 有 “根式” 解:[最早发现者 : Babylon 和三国时期的吴国数学家赵爽]

$latex {a.x^{2} + b.x + c = 0}&fg=aa0000&s=3$

$latex boxed{x= frac{-b pm sqrt{b^{2}-4ac} }{2a}}&fg=aa0000$

Cubic Equation: 16 CE Italians del Ferro, Tartaglia & Cardano
$latex {a.x^{3} = p.x + q }&fg=0000aa&s=3$

Cardano Formula (1545 《Ars Magna》):
$latex boxed {x = sqrt [3]{frac {q}{2} + sqrt{{ (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}
+ sqrt [3]{frac {q}{2} -sqrt{ { (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}}&fg=0000aa$

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:
$latex {a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}&s=3$

No radical solution (Unsolvability) was suspected by Ruffini (1799)…

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Why call the Algebraic Structure Z a “Ring” ?

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 $A$ .

代数拓扑 Algebraic Topology

Math Online Tom Circle

Excellent Advanced Math Lecture Series (Part 1 to 3) by齊震宇老師

（2012.09.10) Part I:

History: 1900 H. Poincaré invented 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.

AbelianFundamental Group(so as to manipulate by + -);

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]…

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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 $\displaystyle A: [SX, *; Y,y_0]\to [X, x_0; \Omega Y, \omega_0].$

Russian Math Education

Math Online Tom Circle

In the world of Math education there are 3 big schools (门派) — in which the author had the good fortune to study under 3 different Math pedagogies:

“武当派” French (German) -> “少林派” 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),

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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.

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.

群论的哲学 Philosophical Group Theory

Math Online Tom Circle

在一个群体里, 每个会员互动中存在一种”运作” (binary operation)关系, 并遵守以下4个原则:

1) 肥水不流外人田: 任何互动的结果要回归群体。(Closure) = C

2) 互动不分前后次序 (Associative) = A

(a.*b)*c = a*(b*c)

3) 群体有个”中立” 核心 (Neutral / Identity) = N (记号: e)

4) 和而不同: 每个人的意见都容许存在反面的意见 “逆元” (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 )。

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Balance Quote

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

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.

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 $F'\circ (p_\alpha\times 1)=F$ .

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/
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Experience: More than 10 years experience, has taught students from RJC, NJC, ACJC and many other JCs.

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viXra vs arXiv

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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.

Prof ST Yau’s 邱成桐 Talk to Chinese Youth on Math Education

Math Online Tom Circle

Prof ST Yau邱成桐, Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal (at 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…

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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 $\pi_1(X,x_0)=\pi_1(X)$ , $\pi_1(Y,y_0)=\pi_1(Y),\pi_1(X\times Y,(x_0,y_0))=\pi_1(X\times Y)$ .

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 $b_i=\frac{\partial f}{\partial x^i}(p)$ .

Chinese Remainder Theorem

Any short-cut method ? Yes, by L.C.M…

Math Online Tom Circle

How to formulate this problem in CRT ?

Hint: Sunday = 7 , Interval 2 days = mod 2, …

Let d = week days {1, 2, 3, 4, 5, 6, 7} for {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…

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Some Math Connotations Demystified 数学内涵解密

Math Online Tom Circle

This Taiwanese Math Prof is very approachable in clarifying the doubts in an unconventional way different from the arcane textbook definitions. Below are his few key tips to breakthrough the “mystified”concepts :

1. “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 ?

Math Online Tom Circle

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 $

Source: http://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it

Note: Co-homology: (上)同调

Euclid Geometry & Homology:

Isabell Darcy Lecture: cohomology

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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.

Best wishes,
Mathtuition88.com

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

Math Online Tom Circle

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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