ceciliachan1979 / math

math

Posting some math related latex documents

Development Environment

• 安裝LaTeX：
  • 一般Linux有 pdflatex command 可使用。
  • Windows: https://miktex.org/ , https://www.texniccenter.org/
• Web
  • mathURL
  • Overleaf
• IDE
  • GNU TeXmacs
  • Visual Studio Code
    • Markdown All in One
    • LaTeX Workshop
  • TeXstudio
• Docker:
  • 如果你不想安裝LaTeX，可以使用Docker。以下是一個例子：
    • 下載：

      docker pull ghcr.io/xu-cheng/texlive-full:latest
      

    • 編譯：

      docker run --rm -it -v $(pwd):/src -w /src/Books/BabyRudin ghcr.io/xu-cheng/texlive-full:latest latexmk -pdf -file-line-error -halt-on-error -interaction=nonstopmode ./BabyRudin.tex
      

    • 開啟：

      open ./Books/BabyRudin/BabyRudin.pdf
      

Books

• Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, Albert R Meyer
• Algorithm Design, Jon Kleinberg, Eva Tardos
• Combinatorial Optimization: Algorithms and Complexity, Christos H. Papadimitriou, Kenneth Steiglitz
• Principles of Mathematical Analysis, Walter Rudin
  • Errata
• Basic Analysis: Introduction to Real Analysis, Jiří Lebl
• Analysis I, Terence Tao
• Understanding Analysis, Stephen Abbott
• Problems in Mathematical Analysis, Boris Demidovich
• A Problem Book in Real Analysis, Asuman G. Aksoy, Mohamed A. Khamsi
• Linear Algebra Done Right, Sheldon Axler
• Linear Algebra Done Wrong, Sergei Treil
• Abstract Algebra: Theory and Applications, Thomas W. Judson, Robert A. Beezer
• Measure Theory, Donald L. Cohn
• Algebraic Number Theory, Jürgen Neukirch

Papers

• Mathematical method and proof, Jeremy Avigad

Courses

• Calculus 1, Chen-Yu Chi, NTU OCW
• Calculus 2, Chen-Yu Chi, NTU OCW
• Analytics, Chen-Yu Chi, NTU OCW
• Advanced Calculus 1, Shu-Jung Kao, NTHU OCW
• Advanced Calculus 2, Shu-Jung Kao, NTHU OCW
• Advanced Calculus 1, Chi-Kaung Pai, NYCU OCW
• Advanced Calculus 2, Chi-Kaung Pai, NYCU OCW
• Real Analysis, Casey Rodriguez, MIT OCW
• Probability Theory and Stochastic Processes, Todd Kemp, UCSD

YouTube Channel

• The Bright Side of Mathematics
• James Cook

FAQ on LaTeX and PGF/TikZ

如果我不懂 LaTeX 怎麼辦？

• Quick Start: https://latex-tutorial.com/tutorials/amsmath/
• Quick Reference: https://en.wikipedia.org/wiki/Help:Displaying_a_formula
• LaTeX Tutorials (featuring Texmaker), Michelle Krummel

如果我不懂 PGF/TikZ 怎麼辦？

• PGF/TikZ Manual - Complete Online Documentation
• LaTeX/PGF/TikZ - Wikibooks, open books for an open world
• TikZ.net – Graphics with TikZ in LaTeX
• LaTeX Graphics with TikZ
• TikZ and PGF | TeXample.net

用電腦typeset數學證明是學習數學證明的最佳方法嗎？

不，根據研究（ref: https://www.youtube.com/watch?v=7PQbidoBPBc ），手寫對長遠記憶的效果更好。可是，如果你想別人看到你的練習／作品，或者方便保存日後參考／分享，使用 LaTeX 吧！

References

• Graphic notes on Gilbert Strang's "Linear Algebra for Everyone"

math's Issues

MIT 18.703 Modern Algebra / Assignment 3

Course: MIT 18.703 Modern Algebra
Document: ./MIT/Solutions/18.703/Assignment3/
Date: Oct 2023

Ex9 對你來講太trivial XD無做呢？

Ex8

＃44
consider a regular plane n-gon for n ≥ 3. Each way that two copies of such an n-gon can be placed, with one covering the other, corresponds to a certain permutation of the vertices. The set of these permutations is a group, the nth dihedral group Dn , under permutation multiplication. Find the order of this group Dn . Argue geometrically that this group has a subgroup having just half as many elements as the whole group has.

唔知本書前面有無講一定係rotation或reflection。（唔知有無講 plane isometry 不外乎係 translations, reflections 或 rotations 。）因為佢提到 permutation of the vertices, 我諗到可以咁樣prove:
Given two distinct vertices A、 X, 最多只有一個vertex Y distinct from X of which distance of A and X equals distance of A and Y 。所以，for any permutation p of the n-gon mapping A to A and satisfying our requirement, either p(X) = X or p(X) = Y.

Choose X as one of the clockwise adjacent vertices of A. Y must be the other adjacent vertex of A. For any permutation p of the n-gon mapping A to A and satisfying our requirement, for the case p(X) = X, [省略 induction 過程, 主要係argue distance，嚴謹證明需要] by induction along the clockwise direction of vertices starting from A, p is the identity permutation; for the case p(X) = Y, [省略 induction 過程, 主要係argue distance，嚴謹證明需要] by induction along the clockwise direction of vertices starting from A, vertices X, X', X'', X'''... , p(X') = Y', p(X'') = Y'', p(X''') = Y'''... and X',Y' distinct, X'',Y'' distinct... and Y', Y'', Y'''... are pairwise adjacent.

Choosing a vertex A* distinct from A. For every vertex B distinct from A, 最多只有一個vertex M distinct from B of which distance of A and B equals distance of A and M 。For any permutation p of the n-gon mapping A to A* and satisfying our requirement, d(A, B) = d(A*, p(B)) and d(A, M) = d(A*, p(M)) , 又，因為 d(A, B) = d(A, M) ， d(A*, p(B)) = d(A*, p(M)) ， either p(B) = p(M), or p(B) is the reflected point of p(M) of reflection along an axis passing through A. 後者係一個reflection，前者[省略induction過程, 又係argue along a clockwise direction of vertices starting from A]係一個rotation 。 [未完]
【寫到自己都煩...】

【重要觀察】因為 covering 都 map centre of regular n-gon back to the centre of regular n-gon，【其實應該仲要補充........】所有呢啲 permutations 全部都係 rotations 同埋 reflections along axes passing through the centre.

我嘅做法或者你這裏的做法，都需要補充證明 rotation * reflection along a chosen axis = reflection along another axis 以及 reflection along a chosen axis * rotation = reflection along yet another axis (* for composition)

至於 associativity, 則可以說 follows from permutation group 。

要話 the set of rotations 係subgoup, 需要補充一句 rotation * rotation' = rotation'' 同埋 rotation * (rotation' * rotation'') = (rotation * rotation') * rotation''。但本書前面好似已經講到cyclic groups。

---

【REF: 《Algebra》 by Artin, section 6.4】
Theorem 6.4.1 Let G be a finite subgroup of the orthogonal group O2. There is an integer n such that G is one of the following groups:

(a) Cn: cyclic group of order n generated by the rotation angle 2\pi / n.

(b) Dn: dihedral group, generated by rotation angle 2\pi / n , and a reflection r' about a line l through the origin.

Proof: 2x2 matrix operation.

然後書上嘅寫法：A regular n-gon is carried to itself by the rotation by 2 \pi / n about its center, and also by some reflections. Theorem 6.4.1 identifies the group of all symmetries as Dn. 【我自己仲覺得嚴謹性差少少。】

【唉……唔滿意 .＿. 】

---

＃45

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box correspond to a certain group of permutations of the vertices of the cube. This group is the group of rigid motions (or rotations) of the cube. How many elements does this group have? Argue geometrically that this group has at least three different subgroups of order 4 and at least four different subgroups of order 3.

three different subgroups of order 4 - each is a cyclic group mapping a square face back to itself and the respective opposite square face to that opposite square face

four different subgroups of order 3 - Yes, rotation along the (body) diagonal; each is a cyclic group mapping the equilateral triangle back to itself

ref image: https://www.geogebra.org/resource/jkM4DVgs/TKXTm2hjVoDqT4Qt/[email protected]

Bonus Problems
可以讀 Artin Section 6.12

Elementary Number Theory Final Exam

Elementary Number Theory Final Exam
Document: ./Exams/NumberTheory/Problems/Elementary_Number_Theory_Final_Exam__2021_Spring.pdf_
Date: June 2021

Q1, Q2, Q6, Q7 okay

Q3, Q4, Q5 稍後睇...

MIT 18.703 Modern Algebra / Assignment 1

Course: MIT 18.703 Modern Algebra
Document: ./MIT/Solutions/18.703/Assignment1/
Date: Early 2023

Assigment 1 - Ex2 all okay。不過 Ex2 ＃12 有typo。

Assigment 1 - Ex4 ＃1-＃5 okay

＃6:

題目：Let ∗ be defined on C by letting a ∗ b = |ab|. Determine whether the binary operation ∗ gives a group structure on the given set.

解： The proof should be:

0 is in C but 0 does not have an inverse with respect to ∗.

Note: The given binary operation ∗ gives a group structure on C \ {0}

The inverse of the equivalent class of (1+i) in C \ {0} respect to * is the equivalent class containing \frac{1-i}{2} , i.e. {e^iθ \frac{1-i}{2} | θ \in R} .

Equivalent class in a group 嘅概念可見於
https://math.berkeley.edu/~gmelvin/math113su14/math113su14notes_online.pdf

Assignment 1 - Ex5 ＃1, ＃3, ＃13, ＃20 okay

＃2: \mathbb{Q}^+ is a not subgroup of the group \mathbb{C} under addition because for any a \in \mathbb{Q}^+, -a \notin \mathbb{Q}^+
or 0 \notin \mathbb{Q}^+ .

Assignment 1 - Ex6 ＃17, ＃19, ＃21, ＃22, ＃28, ＃32, ＃33, ＃34 okay

＃4 無過程！？[雖然好easy...]

Assignment 1 - Bonus ＃1, Bonus ＃2 okay

MIT 18.703 Modern Algebra / Assignment 2

Course: MIT 18.703 Modern Algebra
Document: ./MIT/Solutions/18.703/Assignment2/
Date: Early 2023

Ex6 ＃38, ＃55 okay

＃41 做咗＃40

＃46　比較詳細啲嘅寫法

(ab)^k = 1

a^{-1} 1 a = a^{-1} (ab)^k a = a^{-1}(ab)^k a = (ba)^k

呢道應該未教 Conjugacy class , 未講 identity 嘅 Conjugacy class 得一個element.
For arbitary x,y in G, x^-1 y x 未必等於 y .

仲有，應該補充證明 for 0 < n < k, (ba)^n != 1

＃48
如果提及 Every group with order >= 2 at least has one cyclic subgroup other than {e}，個證明就完整啦。
See also: https://math.stackexchange.com/questions/1310398/abstract-algebra-every-group-has-a-cyclic-subgroup

Ex7 straight-forward, ok

Ex8 ＃2, ＃8, ＃17, ＃23, ＃26, ＃30 ok

＃10 minor improvement

「Indeed, the isomorphism is $ F(x) = e^x $ that maps $ \mathbb{R} $ to $ \mathbb{R}^+ $.」

改為「Indeed, an isomorphism is $ F(x) = e^x $ that maps $ \mathbb{R} $ to $ \mathbb{R}^+ $.」

因為 isomorphism 可以唔至一個。

Ex9 ＃1, ＃10, ＃14 ok

Bonus ＃1, ＃2, ＃3, ＃5 ok

＃4

我估佢應該想問：「Show if two DISTINCT cycles are not disjoint, then they do not commute.」

第14行： we must have $ \sigma(i) = (i + k) \pmod n $ 未解釋k係咩

anyway 呢個相關resource解說得幾好： https://math.mit.edu/~mckernan/Teaching/12-13/Spring/18.703/l_6.pdf

"One reason why conjugation is so important, is because it measures how far the group G is from being abelian.
Another reason why conjugation is so important, is that really con- jugation is the same as translation."

Neukirch.tex (worked in June 2021)

Book: Algebraic Number Theory, J. Neukirch
Document: Books/Neukirch/Neukirch.tex
Date: June 2021

Exercise 1: okay, minor improvement: alpha \overline{alpha} is non-negative real.

Exercise 2: 呢節未講到Z[i]係UFD, 所以個proof開首唔係咁好……後面文字亦寫得唔清楚。

Exercise 3:
大方向正確，但Z[i]裏唔可以用gcd, 要有order先可以講greatest ，但Z[i]唔係ordered ring。
應該let r in Z[i] and r divides both x+yi and x-yi, 然後嘗試去證 r 屬於 units 。
用輾轉相除法去證 ( x+yi, x-yi) ~> (x+yi, 2x) ~> (x+yi, x) ~> (y, x) [後面兩節我加嘅, 注意呢道寫法亦唔formal]　

anyway, 未prove完？有 x + yi = e alpha^2, 之後再諗到埋 x - yi = e' beta^2, 然後solve二元一次方程式系統，咁就差唔多完成。

之後嘅" x + yi &=& e \alpha^2 \ &=& e (u + vi)^2 \ &=& e (u^2 + 2uvi -v^2) " , very good .

Exercise 4: 唔應該用 ordered field axioms。 Z[i]係ring，唔係field。

PS
咦我發現我個pdftex rendering有啲問題…

Notes:

Algebraic Number Theory 我接觸過呢本開頭，覺得幾好：

《Introductory Algebraic Number Theory》 / Şaban Alaca and Kenneth S. Williams (2003)

https://www.cambridge.org/core/books/introductory-algebraic-number-theory/9F53B233CD4D717B1A31ECD117FFEA7D

Enable building PDF for differential forms

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