# 如何平分 Pizza?

https://vt.tiktok.com/ZSdQmPupt/?k=1

# 韦神瞪眼法

Sufficient to prove the yellow circle term is ≥ 0

(Hint: use determinant of quadratic b^2-4ac)

https://v.ixigua.com/FxU5sQP/

# 数学知识一定都是有用的吗？

【【数学漫谈】数学知识一定都是有用的吗？】

1 Applied Math: eg Calculus, Euclidean Geometry

2.Pure Math aka Theoretical :
eg.
2a – Group Theory (Symmetry in Physics/Chemistry)
2b – Ramanujian 4000+ mysterious formula (Blackhole)

1. Repair Math Theories:
eg. 3a – Cauchy “epsilon-delta” repairs Calculus,
3b – Non-Euclidean Geometry repairs Geometry

# Music Geometry

I long suspected “Music” is “Math in Sound form”, this Geometry interpretation confirms the music chords “diminished”, “augmented” etc.

# Intégration by Complex Analysis

He used Complex Analysis, what about by conventional method still gets the same answer ?

# Excel Data from Web

Select Data > Get & Transform > From Web. Press CTRL+V to paste the URL into the text box, and then select OK. In the Navigator pane, under Display Options, select the Results table. Power Query will preview it for you in the Table View pane on the right.

https://vt.tiktok.com/ZSdxLJRqx/?k=1

# Tensor

“Tensor” (张量) was invented for the General Relativity, since the then-existing math had no such algebraic structure.

# Russian Multiplication

Cute Russian primary school Multiplication method

# Math Maturity

This american prof explained well what “Math Maturity” is using the first 2 years of univereity maths (analysis: epsilon-delta, linear algebra etc).

In 知乎you can read some China 省高考状元 “吐槽”，after entering 北大清华，discovered their “Maths ” no longer top as in High School but became “poor Math” students, eg one IMO 2 times Gold Medalist 符xx failed 北大 数学系 after first year. That is BCOS University Math is theoretical pure /abstract, proof-oriented. Those who do well are either math talent or have this “math maturity”.

18:30 mins [Q&A]
Math Maturity ≠ Math Talent
At every stage from Primary to Sec to Uni to PhD…different stage of math maturity : new understanding of the same topic.

eg. After studying all the Group, Ring, Field , Category structures, you have at the final stage (many years later) a new understanding of kernel (of Group, of Endommphism )/ Ideal (Ring, Polynomial)…are just the “invariant ” core which characterise the whole structure, just like a durian 核 kernel reveals what D24, 猫山王， 黑刺…. These “invariant” cores then partition the whole durian into many sub-structured durian compartments, each being of the same species (D24, 猫山王， 黑刺..)

[25:15 mins] Common mistake by poor university math students who read a math textbook line by line like a novel. The correct way is read the Table of Content, Appendix to look for the key word you want to study (eg. Principal Ideal, Homology…), then read that related page /chapter, before and after the chapter …repeat to and fro….until you have covered all the relevant sections.

Same to PhD students reading journal papers, SHD be like reading the Straits Times or 早报， first flip thru all pages , then read only the article which is related to your research interest.

This is “Intelligent Maturity”.

# Free Group, Free Vector Space

1. Free Group:

https://en.m.wikipedia.org/wiki/Free_group

(Z,+) is a free group of the generator {1} , ie. rank (number of elements)= 1.
2=1+1
3=1+1+1

N = 1+1+1…+1 (add N times)

2. Free Vector Space

Vector Space is simply put in “loose” explanation:
Linear combination of some basis vectors (v1, v2, ….vn) with scalars (a1,a2, ….an), ie
For any Vector $V = a_1.v_1 + a_2.v_2+ ...+ a_n.v_n$

forms a FREE Vector Space

# Witt Algebra

Another funny never-heard- of ..”Witt Algebra” from 1930s.

Notice the past wars: French Revolution, to WW1 & WW2 eras produced many great mathematicians: Galois Group Theory, Gauss, …, Bourbaki, Noether Axiomatic Abstract Algebra, and many more still- unknown (Witt , Clifford, Hoff， Category) Algebra.

With “WW3 ” now cooking underway as the aftermath of Russia / Ukraine war, more new Mathematicians will appear with powerful and strange Maths… likely to return to the origin of ancient Chinese maths 阴阳 四象八卦 。。。

# The strange cousin of the complex numbers — the dual numbers

Dual number (ε) is like complex number (i), both totally man-made人为 imaginery number, but useful applied math in Physics (i),
now in AI/DeepLearning/ Gradient (ε) bcos your get its derivative (f’) automatically for any f(x+ε).

# Vandermonde Determinant

Vandermonde Determinant (Math Supérieure : Year 1 Algebra) proved by induction.

# 流形 Manifold

https://v.ixigua.com/N69QPrv/

# Physics Problem solved by Euclidean Geometry or Affine Geometry Methods

Assume no friction at the 4 sided walls, the billiard bounces with incident angle always equal to reflection angle.

Proof: The bouncing path loci forms a parallelogram:

# Affine Geometry 仿射几何

Erlangen’s Programme (1920 Félix Klein):

using Group Theory to solve Euclidean Geometry in difficult (n>2) dimension Geometry : Congruent 全等 and Similar 相似 triangles are just special cases of Affine Transformations : Translation (T), Rotation (R) about a point, reflections about axis, etc.

Isometry Transformation 等距(和合)换变 = {T, R} which preserves distance after the transformation.

# Fundamental Theorem of Calculus : 1 , Collorary, 2

FTC 1 used in tough integration by Feynman technique (you can differentiate the integral)

Proof FTC1:

He used Cauchy epsilon-delta technique to prove Lebniz “First Fundamental Theorem of Calculus ” :
Differentiate integral of f(x) = f(x)

Proof FTC2:

Espace Affine

# Affine Geometry

An irregular rectangle ABCD, extend its 4 sides with 4 isosceles right angle (等腰直角) triangles.

Prove: (yellow box): 2 lines $|O_1O_3| = |O_2O_4|$ , also both perpendicular to each other.

[Note] Using conventional geometry math is quite difficult , but EASY using Modern Math Affine Geometry (ISOMETRY 合同 aka 等距变换 Transformation), ie apply “Mapping” and “Group Theory” in Affine Geometry 仿射 invented by Felix Klein (1849-1925) “Erlangen’s Programme” .

# Topological Space

I like this type of short video lessons explain key “scary” concepts very intuitively within a few minutes, clearing all the jargons and “clouds” of all known separately concepts by linking them…

Here he clarified:
Topological Space as the root of Metric Space, Vector Space akin to “Object” root in Object-oriented.

Also the often-heard “Power Set” …

# Poland Math Olympiad (1950) : Geometry method to solve Inequality

Problem:

Let a, b, c Z+ (positive integers), and

a+ b + c =1

Prove:

1) $\frac {1}{a} + \frac {1}{b} +\frac {1}{c} \geq 9$

Solution:

Plot graph y = 1/x

Three points A = (a, 1/a), B = (b, 1/b), C = (c, 1/c),

The Centre of Gravity G $(x_0, y_0)$ is: $x_0 = \frac{1}{3}. (a + b + c)$ $y_0 = \frac{1}{3}. (\frac {1}{a} + \frac {1}{b}+ \frac {1}{c})$

The curve is concave, G is above $(x_0, \frac{1}{x_0})$ $y_0 > \frac{1}{x_0}$ $\frac{1}{3} .(\frac {1}{a} + \frac {1}{b}+ \frac{1}{c})> 3. \frac{1}{(a + b + c)}$ $\frac {1}{a} + \frac {1}{b}+ \frac{1}{c}> 9. \frac{1}{(a + b + c)}$

since a + b + c =1 $\boxed{ \frac {1}{a} + \frac {1}{b} +\frac {1}{c} \geq 9 }$ [QED]

2) Prove : $sin A + sin B + sin C \leq \frac{3}{2}.\sqrt{3}$

# Weil’s Proof of Fermat Last Theorem

Canadian Prof Darmon was the post-doc student of Prof Andrew Wiles.

This lecture on Fermat’s Last Theorem marvelous proof is “simplified” into key concepts accessible to Math (honors) undergrads or Classe Préparatoire (spécialisé M/M’):
Elliptic Curve /Modular Form, Galois Field,
Langlands Programme,
etc.

# Berkeley: Limit in Positive-definite Matrix

Positive-definite Matrix （正定矩阵）：

# 数学科学百年回顾（一）

1900 是近代 数学Modern Math 的分水岭， 由（1830）法国神童 Evariste Galois 的Group Theory 打下的基础， 一发不可收拾。

A-level 之前学的 是 从 古典希腊 到 牛顿（1730） Calculus 的数学。

# Parseval Theorem leads to special case : Euler’s Basel Theorem

A Theorem is always True from any angle: Parseval Theorem leads to special case : Euler’s Basel Theorem.

# Classics Math Textbooks

Artin 《Algebra》, Rudin 《Analysis 》+ 《Real & Complex Analysis》are 2 classics textbooks used from undergrads (4th yr) till PhD in Harvard/MIT/NUS.

# Geometric Algebra

Clifford Algebra, aka Geometric Algebra, unites all geometries into ONE.

Geometric Algebra simplifies Maxwell Equation :

▽F= J

[Note 1] Product of Vectors not the same as Cross-Product.

[Note 2] A “Vector Space over a scalar Field ” (invented by Italian Peano 1888 ) structure has ONLY product by a SCALAR with a vector.

An “Algebra over a scalar Field” structure (American invented it) has “product of Vectors”.

# Diagonalisation of matrix

Diagonalisation of matrix (2×2) or (3×3) ?

Null Space of A is B (≠0), such that A.B=0

Diagonalisation of matrix of (3×3) ?

His American Rational Root Theorem to factor cubic polynomial is troublesome…our Sec 4 Ancient Chinese 系数coefficient”辗转相除法” much faster:

Condition for diagonalisable:

Révision: Null Space

https://www.wikihow.com/Find-the-Null-Space-of-a-Matrix?amp=1

# Dynamics & Geometry & Matrices

Conservation Laws, solved using Matrices.

2nd Method: Use Geometry to solve .

3rd method: using Optics (beaming light)

# Galois Theory (Quadratic Equation Case) $\boxed{\Omega^3 = 1}$ $\boxed{\Lambda^2 = 1}$ $\boxed{{\Lambda} . { \Omega }\neq {\Omega} . { \Lambda} }$ $\boxed{{\Lambda} . { \Omega } = {\Omega}^{2} . { \Lambda} }$ $\boxed{{\Lambda} . { \Omega } ^{2} = {\Omega} . { \Lambda} }$

There are total 6 symmetries: $\boxed{1 ,\Omega,\Omega^2 }$ $\boxed{ \Lambda, {\Omega . \Lambda}, {\Omega^{2} . \Lambda}}$ $x^2 + ax + b = 0$ $\boxed{x= \frac{-a \pm \sqrt{a^{2}-4b} }{2}}$

Let 2 roots $\zeta_1 , \zeta_2$ $x^2 + ax + b = (x - \zeta_1).(x - \zeta_2)$ $a = - \zeta_1 - \zeta_2$ $b = \zeta_1 . \zeta_2$

Notice: symmetry by exchanging both roots, a & b no change ! The Galois Group is represented by $S_2$: $\begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix} = I$ , $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} = \Gamma$ $\text {Symmetric Group } S_2 = (I, \Gamma)$

Let $\beta_{+ } = \zeta_1 + \zeta_2$ $\beta_{-} = \zeta_1 - \zeta_2$ $\boxed {\zeta_1 = \frac{1}{2} (\beta_{+ } +\beta_{- } )}$ $\boxed {\zeta_2 = \frac{1}{2} (\beta_{+ } - \beta_{- } )}$

From coefficient relationship，we know : $\beta_{+ } = \zeta_1 + \zeta_2 = -a$,

what about $\beta_{-}$

…in terms of the coefficients ? ${\beta_{-}}^2 = ({\zeta_1 - \zeta_2})^2 = ({\zeta_1 + \zeta_2 })^2 - 4.{\zeta_1 . \zeta_2 } = a^2 - 4b$ ${\beta_{-}} = \sqrt {a^2 - 4b}$

Therefore, $\boxed {{\zeta_1} , {\zeta_2}= \frac{-a \pm \sqrt {a^2 - 4b}}{2}}$ ${ \text {Conclusion : Galois Group } S_2 = (I, \Gamma)}$ ${ \text { acts on the 2 roots } {\zeta_1}, {\zeta_2}}$ $\text { needs } +, -, * , / , \sqrt {}$

Similar conclusion for Cubic and Quartic equation with Galois Group aka Solvable Group which … ${ \text { needs } +, -, * , / , \sqrt {}}$

Exception for Quintic (degree 5) and above , the only Galois Group is A5 (Icosahydron) which is not SOLVABLE, so no ${\sqrt {}}$

Reference: