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?】

数学有三类:
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

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

https://planetmath.org/FreeVectorSpaceOverASetFree

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 。。。

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.

Affine Geometry

China Olympiad Question:

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.

[Hint:]  Take any point M inside ABCD.

[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” …

数学科学百年回顾(一)

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。

从1730-1830, 数学教育有一个100年“龙沟”, 许多高中数学/ IMO 精英 进 清华/北大/Classe Préparatoire, 跨不过这个“沟”, 失去兴趣。

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”.

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

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

Galois Theory (Quadratic Equation Case)

Group Symmetry of a Regular Triangle : Rotation 120 ° anti-Clockwise
Reflection about  vertical axis

\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 ?

Answer:

{\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:

Special case for degree 5 polynomial equations.