[數撚圍爐區] 喪屍奶媽是真的嗎? (6)

留名

咦我state錯左最尾條theorem
應該係 there exists x such that x_n -> x, where x_0 = x, x_{n + 1} = f(x_n)

唔怪得望落句野怪怪地咁

咦我state錯左最尾條theorem
應該係 there exists x such that x_n -> x, where x_0 = x, x_{n + 1} = f(x_n)

唔怪得望落句野怪怪地咁

屌我寫到1999, i mean there is an increasing sequence of positive integers {n_k} such that f^{n_k} (x) -> x as k -> infinity
notation: f^{n} (x) = f(f...f(x)) (composite n times)
原本諗住避個notation點知仲衰左

屌再寫過
Zorn's Lemma有咩用?

Zorn's Lemma 等同於 Axiom of Choice, 而Axiom of Choice係 Set theory 入面其中一個好出名的axiom:

對於任何一個collection of non-empty sets, 我地都可以每一個set揀個member代表個set.

例子:
1. A = {1}, B = {2, 3}, C = {4, 5, 6}, X = {A, B, C}
我地可以define f(A) = 1, f(B) = 2, f(C) = 4

2. 又例如on 2^N, S係N ge subset我地define f(S) = minimum number in S. (N = set of natural numbers)

3. 但對於 2^R, 我地似乎搵唔到一個explicit ge f. 所以呢條axiom信不信由你
再者, 如果相信呢條axiom係真, 就會出現一D counter intuition ge地方:

(Banach-Tarski Paradox) 可以將一個實心波切開5份, 之後可以砌返兩個同原本一模一樣的實心波

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至於Zorn's Lemma 的表達如下:

設 X 有一個partial ordering, 對於任何一個可以用呢個partial ordering完全排好次序的subset都有一個upper bound, 咁 X 就會有一個maximal element.

Partial ordering係指:

1. 對於任何 a, a ≤ a
2. 如果 a ≤ b, b ≤ a的話, 則b = a
3. 如果 a ≤ b, b ≤ c的話, 則a ≤ c

intuitively可以想像有一幢樓, 每層樓都有好多房, 而我地定義

房A ≤ 房B 當且僅當:
1. B所在的樓層不低於A
2. A 有一條唔會落樓的路行到去 B.

如果每一條咁的路都有"終點"的話, 咁我地就會搵到最少有一間房係終點

我唔識證點解Zorn's Lemma同Axiom of Choice一樣, 詳情請參閱
Introduction to Set Theory by Hrbacek & Jech Chapter 8

Zorn's Lemma比Axiom of Choice更加老是常出現, 以下比幾個例子:
1. Every vector space has a basis.

2. Hahn-Banach Theorem (Functional Analysis三大基本定理之一):
任何一個defined係一個subspace of a vector space ge linear functional (linear function with codomain F = R or C), 如果足夠靚ge話我地可以將呢個linear functional extend 去成個vector space. (然後有好多有用ge result, 唔講)

3. 一個injective field homomorphism into algebraically closed field可以extend去domain ge algebraic extension (所以field ge algebraic closure係unique up to isomorphism)

4. Birkhoff Recurrence Theorem:

[limegreen] Let X be a compact metric space and f: X -> X be a continuous function, then there exists x in X such that x is in the closure of the sequence {x_n}, where x_0 = x, x_{n + 1} = f(x_n)

(呢個result唔需要Zorn's Lemma, 但可以用Zorn's Lemma證)

大家一齊發掘喪屍奶媽ge其他用途啦

Sem break 學咗LaTex先定Matlab 先好

LaTeX啦

想問各大大，除左 course hero, 仲有乜地方 share course matererials?

學晒算

你定剩係打math嘅 article 先解latex定做其他科嘅嘢都會？

先用*

完全取代word咁樣用

電腦只需要有LaTeX同excel同pdf reader

你有幾部腦先要用
得一部嘢你又有backup嘅話就可有可無