<![CDATA[The Knot]]>https://alex1222.com/RSS for NodeSat, 11 Apr 2026 04:40:08 GMT<![CDATA[Fix: Aliyun OSS Internal IP Overlaps with Tailscale CGNAT and Breaks Access (Debian)]]>

0.1 Contents

1 Fix: Aliyun OSS Internal IP Overlaps with Tailscale CGNAT and Breaks Access (Debian)

1.1 Problem

Tailscale assigns node IPv4 addresses from the CGNAT range 100.64.0.0/10 (RFC 6598).
Aliyun OSS internal endpoints can also resolve to IPs within this same range.

For example, according to Aliyun official documentation, the Hong Kong OSS internal endpoint oss-cn-hongkong-internal.aliyuncs.com may resolve to addresses in:

  • 100.115.61.0/24
  • 100.99.103.0/24
  • 100.99.104.0/24
  • 100.99.106.0/24

I first noticed the issue when OSS access via the internal endpoint failed on an Aliyun ECS instance.
I also saw similar failures when accessing the Aliyun internal Debian package mirror.

Switching Debian mirror traffic to a public mirror is a workable workaround.
However, switching OSS traffic to the public endpoint is not ideal because it can increase costs: internal OSS traffic is free, while public endpoint traffic is billed.

Packet-level testing showed that traffic to these /24 ranges left via the Aliyun internal interface, and replies arrived at the interface, but applications still timed out (for example, ping).

1.2 Root cause

From Tailscale documentation: when netfilter-mode is enabled, Tailscale installs iptables rules to prevent CGNAT spoofing:

Traffic from 100.64.0.0/10 that does not arrive through tailscale0 is dropped.

In practice, the ts-input chain includes a rule like:

  • -A ts-input -s 100.64.0.0/10 ! -i tailscale0 -j DROP

So replies from OSS hosts (for example, 100.115.61.1) that arrive on ens5 (or another non-tailscale0 interface) are dropped before they reach user-space applications—even though packet capture confirms the replies are arriving.

1.3 Solution

Add a narrow allowlist exception for the specific OSS /24 ranges before Tailscale’s CGNAT anti-spoof DROP rule.

The simplest approach is to insert RETURN rules into ts-input:

  • match packets with source IPs in the specific OSS /24 ranges
  • only when they arrive on interfaces other than tailscale0

This keeps Tailscale’s anti-spoof protection for the rest of 100.64.0.0/10, while allowing known legitimate ranges on your LAN/VPC path.

Disabling netfilter integration entirely also fixes the problem, but weakens protection more broadly.
A targeted allowlist is the safer compromise.

1.4 One-time manual fix (immediate)

Insert exceptions into ts-input before the CGNAT DROP rule:

sudo iptables -I ts-input 3 -s 100.115.61.0/24 ! -i tailscale0 -j RETURN
sudo iptables -I ts-input 3 -s 100.99.103.0/24 ! -i tailscale0 -j RETURN
sudo iptables -I ts-input 3 -s 100.99.104.0/24 ! -i tailscale0 -j RETURN
sudo iptables -I ts-input 3 -s 100.99.106.0/24 ! -i tailscale0 -j RETURN

Verify rule order:

sudo iptables -S ts-input

Verify connectivity:

ping -c 3 100.115.61.1

Tailscale may recreate or rewrite ts-* iptables chains during restart or upgrade.
For reliability, re-apply the allowlist after tailscaled starts.

1.5.1 Script

Create:

/usr/local/sbin/tailscale-cgnat-allow-corp.sh

#!/bin/sh
set -eu

# OSS/internal CGNAT subnets that must be allowed on non-tailscale interfaces
SUBNETS="
100.115.61.0/24
100.99.103.0/24
100.99.104.0/24
100.99.106.0/24
"

# Wait briefly for tailscaled to install ts-input chain (avoid boot race)
for i in 1 2 3 4 5; do
  if iptables -S ts-input >/dev/null 2>&1; then
    break
  fi
  sleep 1
done

# Insert RETURN rules near the top of ts-input, before the CGNAT DROP.
# Use -C first to keep it idempotent (safe to run repeatedly).
for net in $SUBNETS; do
  if ! iptables -C ts-input -s "$net" \! -i tailscale0 -j RETURN 2>/dev/null; then
    iptables -I ts-input 3 -s "$net" \! -i tailscale0 -j RETURN
  fi
done

Make it executable:

sudo chmod 0755 /usr/local/sbin/tailscale-cgnat-allow-corp.sh

1.5.2 systemd unit

Create:

/etc/systemd/system/tailscale-cgnat-allow-corp.service

[Unit]
Description=Allow selected CGNAT subnets outside tailscale0 (Tailscale anti-spoof override)
After=tailscaled.service network-online.target
Wants=tailscaled.service network-online.target

[Service]
Type=oneshot
ExecStart=/usr/local/sbin/tailscale-cgnat-allow-corp.sh

[Install]
WantedBy=multi-user.target

Enable and run:

sudo systemctl daemon-reload
sudo systemctl enable --now tailscale-cgnat-allow-corp.service

1.5.3 Verify persistence across Tailscale restart

sudo systemctl restart tailscaled
sudo iptables -S ts-input | sed -n '1,30p'
ping -c 3 100.115.61.1
]]>https://alex1222.com/posts/2026-02-18-Fix-Aliyun-OSS-Internal-Overlaps-With-Tailscale.mdhttps://alex1222.com/posts/2026-02-18-Fix-Aliyun-OSS-Internal-Overlaps-With-Tailscale.mdWed, 18 Feb 2026 00:00:00 GMT<![CDATA[How to Fix 'Steam Encountered an Unexpected Error During Startup']]>

0.1 Contents

1 How to Fix 'Steam Encountered an Unexpected Error During Startup'

1.1 Introduction

If you've been using Steam without issues and suddenly encounter the error message "Steam encountered an unexpected error during startup," you're not alone. This frustrating error typically appears after a Steam update that didn't go smoothly. What's particularly annoying is that even a fresh installation might not resolve the issue, as Steam automatically updates to the latest version upon launch—which could be the problematic version.

In this blog post, we'll walk through a solution that has worked for many users by forcing Steam to downgrade to a previous, stable version.

1.2 Why This Error Occurs

The "Steam encountered an unexpected error during startup" message usually appears when:

  1. A recent Steam client update has compatibility issues with your system
  2. Steam files became corrupted during an update process
  3. There's a conflict with another application or system component

1.3 Solution: Force Steam to Downgrade

The most effective fix is to force Steam to downgrade to a previous, working version. Here's how to do it using Whisky, a popular compatibility layer for running Windows applications:

1.3.1 Step 1: Set Up the Downgrade Process

  1. Open Whisky and locate your Steam installation
  2. Click on the "Config" dropdown

The Config Dropdown

  1. Find the "Arguments" section

The Arguments Section

  1. Copy and paste the following command:
-forcesteamupdate -forcepackagedownload -overridepackageurl http://web.archive.org/web/20240520if_/media.steampowered.com/client -exitsteam
  1. Click "Run" in the bottom right corner

Steam will now begin downloading a downgraded version and will close automatically once the process completes.

1.3.2 Step 2: Configure Steam to Avoid Future Updates

After Steam has closed:

  1. Go back to your programs list in Whisky
  2. Click the gear icon on Steam again (as in Step 1)
  3. Delete the previous command line from the Arguments section
  4. Replace it with:
-noverifyfiles -nobootstrapupdate -skipinitialbootstrap -norepairfiles -overridepackageurl
  1. Press "Run" again

The second set of arguments prevents Steam from automatically updating to the problematic version, keeping your Steam client functioning correctly.

1.4 Why This Works

This solution works by:

  1. First forcing Steam to download a specific archived version from the Web Archive
  2. Then preventing Steam from verifying, repairing, or updating its files in subsequent launches

1.5 Conclusion

The "Steam encountered an unexpected error during startup" issue can be particularly frustrating since traditional reinstallation methods often won't work. By forcing a downgrade and preventing automatic updates, you can get back to your gaming library without the hassle.

If you continue to experience issues, you might want to check if there are any new stable updates available before trying other more invasive troubleshooting methods.

Have you encountered other Steam errors that were difficult to resolve? Share your experiences in the comments below!

]]>https://alex1222.com/posts/2025-03-12-Fix-Steam-Start-Up-Error.mdhttps://alex1222.com/posts/2025-03-12-Fix-Steam-Start-Up-Error.mdWed, 12 Mar 2025 00:00:00 GMT<![CDATA[Group Theory Note]]>

0.1 Contents

1 Group Theory Note

1.1 Introduction

This is my personal note on group theory.

1.2 Group Actions

1.2.1 Orbit and Stabilizer

1.2.2 Burnside's Lemma

1.2.2.1 Group With 2 Conjugacy Classes

Group G with 2 conjugacy classes, are particularly interesting. As only finite groups have 2 conjugacy classes, well be C_2. And any group with 2 conjugacy classes is simple. Thus, it may serve as a good example to infinite simple groups.

Theorem 1. If G is a group with 2 conjugacy classes, then G is simple.

Proof: As 1 itself is a conjugacy class, the other conjugacy class must be all the other elements.

Let N be a non-trivial normal subgroup of G. Take x \in N such that x\neq 1. Then for any y\in G, y\neq 1, there exists g\in G such that gxg^{-1} = y. Thus, N=G.■

Theorem 2. If G is a finite group with 2 conjugacy classes, then G is isomorphic to C_2.

Proof: As 1 itself is a conjugacy class, the other conjugacy class must be all the other elements.

By Burnside's lemma, we have the following equation:

2 = \frac{1}{|G|}\sum_{g\in G}|C(g)| = 1 + \frac{1}{|G|}\sum_{g\neq 1, g \in G}|C(g)|.

Where C(g) = \{x\in G \mid gxg^{-1} = x\}, which is the elements that is fixed by g. As g must fix 1 and itself, thus, for any g\neq 1, |C(g)| \ge 2.

Therefore, we have the following inequality:

2= 1 + \frac{1}{|G|}\sum_{g\neq 1, g \in G}|C(g)| \ge 1 + \frac{2}{|G|}(|G|-1) = 2.

Which then gives us the equality:

|G| \le 2.

■

1.3 Abelian Group

In this section, we will only focus on Abelian groups. And we will use the additive notation for the group operation.

1.3.1 Free Abelian Group

Definition 1. An Abelian group F is free Abelian if it is a direct sum of infinite cyclic groups. More precisely, there is a subset X \subseteq F of elements of infinite order, called a basis of F, with F = \sum_{x\in X}; i.e., F = \sum\mathbb{Z}.

Theorem 3. There is an infinite p-primary group \mathbb{Z}(p^{\infty}) each of whose proper subgroups is finite and cyclic.

Proof: We define a group G have generators:

X = \{x_0,x_1,\dots,x_i,\dots\}, i\in\mathbb{N}

with relations:

\{ px_0, x_0 - px_1, x_1 - px_2, \dots, x_i - px_{i+1}, \dots \}

We let F be the free Abelian group generated by X. And let R \le F be the subgroup generated by the relations. Let a_i = x_i + R \in F/R = G, as \{x_0,x_1,\dots\} form a basis of F, \{a_0,a_1,\dots\} form a basis of G. And as pa_0 = 0, pa_1 = a_0, pa_2 = a_1, \dots, for every a_i, we have p^{i+1}a_i = 0. So G is a p-primary group.

To prove G is infinite, we prove that a_i are distinct. Assume a_i = a_j for some i< j. Then by previous reasoning, we have a_i = p^{j-i} a_j, which implies (p^{j-i}-1) a_j= 0. And previous reasoning also shows that p^{j+1}a_j = 0. Thus, the order of a_j must divides \gcd(p^{j+1} , p^{j-i}-1) = 1. Which implies a_j = 0, which is a contradiction.

Take H to be a proper subgroup of G. By previous argument, for all j< i, a_j is in the cyclic group generated by a_i. Thus, we have

G = \bigcup_{i=0}^{\infty} \langle a_i \rangle.

If H is finite, then there must be n such that H \subseteq \langle a_n \rangle. Then H is a subgroup of finite and cyclic group, thus H is finite and cyclic.

If H is infinite, then there must be a sequence of elements in H, say \{h_1, h_2, \dots\}, such that h_i \in \langle a_{n_i} \rangle, and h_i \notin \langle a_{n_i-1} \rangle. Assume h_i = ka_{n_i}, then \gcd(k,p) must be 1, as otherwise h_i will be in \langle a_{n_i-1} \rangle. Then h_i also generate \langle a_{n_i} \rangle, which implies a_{n_i} \in H. And thus, a_i \in H, \forall i, which implies H = G.■

1.3.2 Divisible and Reduced Groups

Definition 2. A group G is called divisible if for every a \in G and every positive integer n, there exists an element b \in G such that nb = a.

Example 1. The group \mathbb{Z}(p^\infty) is divisible.

We define a group \mathbb{Z}(p^\infty) have generators:

X = \{x_0,x_1,\dots,x_i,\dots\}, i\in\mathbb{N}

with relations:

\{ px_0, x_0 - px_1, x_1 - px_2, \dots, x_i - px_{i+1}, \dots \}

We let F be the free Abelian group generated by X. And let R \le F be the subgroup generated by the relations. Let a_i = x_i + R \in F/R = G, as \{x_0,x_1,\dots\} form a basis of F, \{a_0,a_1,\dots\} form a basis of G. Also, pa_0 = 0, pa_1 = a_0, pa_2 = a_1, \dots, therefore we have, for all j< i, a_j is in the cyclic group generated by a_i. Thus, we have

G = \bigcup_{i=0}^{\infty} \langle a_i \rangle.

We take n=p^r m such that \gcd(p,m)=1. Then for any element b \in \mathbb{Z}(p^\infty), we have n such that b \in \langle a_n \rangle. Assume b = ka_n, we also take c = ka_{n+r}. Then, by the relation of generators, we have p^r c = b. As m is co-prime with p, by Bézout's identity, there exists x,y such that mx + p^{n+r+1}y = 1. We take d = xc. Then

\begin{aligned} nd &= p^r m xc \\ & = p^r(1-p^{n+r+1}y)c \\ & = p^r c - p^{n+2r+1}y c \\ & = b - 0 \\ & = b. \end{aligned}

Which finish the proof that \mathbb{Z}(p^\infty) is divisible.

Theorem 4 (Injective Property, Baer, 1940). Let D be a divisible group and let A be a subgroup of a group B. If f: A \rightarrow D is a homomorphism, then f can be extended to a homomorphism \phi: B \rightarrow D; that is, the following diagram commutes:

#align(center, commutative-diagram( node((1, 0), [$0$]), node((1, 1), [$A$]), node((1, 2), [$B$]), node((0, 1), [$D$]), arr((1, 0), (1, 1), []), arr((1, 1), (1, 2), [$i$],

Proof: This theorem can be proved by Zorn's lemma, or an equivalent proof of well-ordering principle.

Consider \mathcal{F} to be the set of all pairs (C, g), where C is a subgroup of B containing A and g: C \rightarrow D is a homomorphism extending f. Note that \mathcal{F} is non-empty as (A,f) \in \mathcal{F}.

We define a partial order on \mathcal{F} by (C_1, g_1) \le (C_2, g_2) if C_1 \subseteq C_2 and g_2|_{C_1} = g_1. Then for any chain \{(C_i, g_i)\}_{i\in I} in \mathcal{F}, we can define C = \bigcup_{i\in I} C_i and g: C \rightarrow D by g(c) = g_i(c) for c\in C_i. It is easy to check g is well-defined and g is a homomorphism. Thus, (C,g) is an upper bound of the chain.

By Zorn's lemma, there exists a maximal element (B', g') in \mathcal{F}. If B' \neq B, then there exists b \in B \setminus B'. And we discuss whether B'\cap \langle b \rangle=0.

If B'\cap \langle b \rangle=0, then we can extend g' to B' + \langle b \rangle by defining g'(b) = 0. Which contradicts the maximality of B'.

If B'\cap \langle b \rangle \neq 0, then we can take k to be the smallest positive integer such that kb \in B'. Then as D is divisible, we can find d \in D such that kd = kb. Then we can extend g' to B' + \langle b \rangle by defining g'(b) = d. And it is easy to check that g' is a homomorphism. Which contradicts the maximality of B'.■

Follows from the injective property, we have the following corollary:

Corollary 1. If a divisible group D is a subgroup of a group B, then D is a direct summand of B.

Proof: We consider the following diagram:

#align(center, commutative-diagram( node((1, 0), [$0$]), node((1, 1), [$D$]), node((1, 2), [$B$]), node((0, 1), [$D$]), arr((1, 0), (1, 1), []), arr((1, 1), (1, 2), [$i$],

As D is divisible, we can extend 1_D to \varphi: B \rightarrow D. By theorem in group extension, we have B = D \oplus \ker \varphi.■

Definition 3. If G is a group, then dG is the subgroup of G generated by all divisible subgroups of G.

Note that dG is invariant under all automorphism of G. As image of any divisible subgroup is also divisible.

Lemma 1. For any group F, dG is the unique maximal divisible subgroup of G.

Proof: We first prove that dG is divisible. For any a \in dG and n \in \mathbb{N}, there exists H which is a divisible subgroup of G such that a \in H. As H is divisible, there exists b \in H such that nb = a. And as H is a subgroup of dG, b \in dG.

The maximal is obvious, as if there exists a divisible subgroup H such that dG \subset H \subset G, then H \subset dG, by the definition of dG.

It is unique as if there exists another maximal divisible subgroup H, then \langle H, dG \rangle is also a divisible subgroup, then \langle H, dG \rangle \subset dG.■

Definition 4. A group G is called reduced if dG = 0.

Theorem 5. For every group G, there exists a decomposition G = dG \oplus R, where dG is divisible and R is reduced.

Proof: Since dG is divisible, by corollary of injective property, dG is a direct summand of G. Thus, there exists R\le G such that G = dG \oplus R.

Next, we prove that R is reduced. If R is not reduced, then there exists a non-zero element r \in R such that for every n \in \mathbb{N}, r is divisible by n. Then take an element a \in dG, then a+r \in dG \oplus R is also divisible by n for every n \in \mathbb{N}. Which contradicts the maximally of dG.■

Every abelian group is an extension of the torsion group tG by a torsion free subgroup, but tG need not to be a direct summand of G. However, dG is a direct summand of G.

Definition 5. Given a group G define G[n] = \{x\in G \mid nx = 0\}.

Lemma 2. If G and H are divisible p-primary group, then G \cong H if and only if G[p] \cong H[p].

Proof: The "only if" part is obvious. For the "if" part, we define \phi: G[p] \rightarrow H[p], to be the isomorphism between G[p] and H[p]. Then \phi is also a homomorphism from G[p] to H. By previous lemma, we can extend \phi to a homomorphism \Phi: G \rightarrow H. And we prove that \Phi is injective first.

For injectivity, we prove this by induction on n \ge 1, that if p^n x = 0, and \Phi(x) = 0, then x = 0. The base case n=1 is obvious, as now x \in G[p], which means 0=\Phi(x) = \phi(x), and as \phi is isomorphic, x = 0. Now we assume the statement is true for n-1, and we prove it for n. If p^n x = 0, then p^{n-1} (px) = 0, also \Phi(x) = 0 must implies \Phi(px) = 0, thus by the induction hypothesis, px = 0, which implies x = 0 by the base case.

As \Phi is injective, \Phi: G \rightarrow \text{Im}(\Phi) is an isomorphism. And we define \phi': \text{Im}(\Phi) \rightarrow G as the inverse of \Phi. Also, note that the inverse of \phi' extends the inverse of \phi, as it is easy to check \phi'\mid H[p] = \phi^{-1}. As \phi' is a homomorphism from a subgroup of H to G, by previous lemma, we can extend \phi' to a homomorphism \Phi': H \rightarrow G. As \phi' is a surjection by definition, \Phi' is also a surjection. And also as \Phi' extends \phi^{-1}, apply a similar reasoning as above, we have \Phi' is injective. Thus, \Phi' is an isomorphism.■

Theorem 6. Every divisible group D is a direct sum of copies of \mathbb{Q} and of copies of \mathbb{Z}(p^\infty) for various primes p.

Proof: It is easy to check that tD, the torsion group of D is divisible, so tD is a direct summand of D. Thus, there exists V \subset D such that D= tD \oplus V. As now V is torsion free, and divisible, it is a vector space over \mathbb{Q}, and thus, V is a direct sum of copies of \mathbb{Q}.

Now we prove that tD is a direct sum of copies of \mathbb{Z}(p^\infty), for various primes p. For prime p, let B_p = \{ x \in tD \mid p^nx = 0 \text{ for some } n \in \mathbb{N} \}. Then B_p is a divisible p-primary group. It is easy to check that tD = \bigoplus_{p} B_p. So, we only focus on B_p.

It is easy to check that the group B_p[p] is now a vector space over \mathbb{F}_p, let r_p be the dimension of B_p[p]. By previous lemma, B_p is isomorphic to direct sum of r_p copies of \mathbb{Z}(p^\infty).■

There is an analogy between theorems about free Abelian groups and divisible groups, that may be formalized as follows. Given a commutative diagram containing exact sequences, then its dual diagram is the diagram obtained by reversing all the arrows. For example, the dual diagram of 0 \rightarrow A \rightarrow B is B^* \rightarrow A^* \rightarrow 0, and this leads on the say that "subgroup" and "quotient" are dual concepts.

Theorem 7. Every group G can be embedded in a divisible group D.

Proof: As every Abelian group can be written as F/R, where F is a free Abelian group and R is a subgroup of F. Now F = \sum \mathbb{Z}, so that F \le \sum \mathbb{Q}. Hence, G = F/R = (\sum\mathbb{Z})/R \le (\sum\mathbb{Q})/R. As (\sum\mathbb{Q})/R is a quotient of a divisible group, it is divisible.■

Corollary 2. A group G is divisible if and only if it is a direct summand of any group containing it.

Proof: By previous theorem, we can embed G in a divisible group D. Then G is a direct summand of D. And any direct summand of divisible group is divisible.■

1.3.2.1 Exercises

Exercise 1. If 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 is an exact sequence and if A and C are reduced, then B is reduced.

Proof: We name the maps in the exact sequence as follows:

#align(center, commutative-diagram( node((0, 0), [$0$]), node((0, 1), [$A$]), node((0, 2), [$B$]), node((0, 3), [$C$]), node((0, 4), [$0$]), arr((0, 0), (0, 1), []), arr((0, 1), (0, 2), [$i$]), arr((0, 2), (0, 3), [$j$]), arr((0, 3), (0, 4), []), ))

As i is injective, we now assume A \le B. Assume B is not reduced, then dB \neq 0. Then, j(dB) is divisible, as it is a quotient of divisible group. Because C is reduced, j(dB) = 0, which implies, dB \le \ker j = \text{Im} i = A. As A is reduced, dB = 0.■

Exercise 2.

  • Prove that \mathbb{Q}/\mathbb{Z} = \sum_p \mathbb{Z}(p^\infty).
  • Prove that \mathbb{Q}/\mathbb{Z}[n] \cong \mathbb{Z}_n.

Proof: As \mathbb{Q}/\mathbb{Z} is a quotient of divisible group, it is divisible. Also given \frac{a}{b} \in \mathbb{Q}/\mathbb{Z}, we have b \frac{a}{b} = 0. Thus, t\mathbb{Q}/\mathbb{Z} = \mathbb{Q}/\mathbb{Z}, and \mathbb{Q}/\mathbb{Z} contains no summand of \mathbb{Q}.

It is easy to check that:

\mathbb{Q}/\mathbb{Z}[p] = \{ \frac{a}{b} \in \mathbb{Q}/\mathbb{Z} \mid \gcd(a,b) = 1, b = p \}.

Then it is a cyclic group of order p, which is isomorphic to \mathbb{Z}(p^\infty)[p]. Then by previous theorem and lemma, we have, \mathbb{Q}/\mathbb{Z} = \sum_p \mathbb{Z}(p^\infty).

For the second part, we factorize n = p_1^{r_1} \dots p_k^{r_k}. Then we have:

\begin{aligned} \mathbb{Q}/\mathbb{Z}[n] &= (\sum_p \mathbb{Z}(p^\infty))[n] \\ &= \sum_p (\mathbb{Z}(p^\infty)[n]) \\ &= \sum_{i = 1}^{k} (\mathbb{Z}(p_i^\infty)[p_i^{r_i}]) \\ &= \sum_{i = 1}^{k} \mathbb{Z}_{p_i^{r_i}} \\ &= \mathbb{Z}_n. \end{aligned}

■

Exercise 3. Prove that a group G is divisible if and only if G has the injective property.

Proof: The "only if" part is given by previous theorem. For the "if" part, we prove that if a group D contains G, G is a direct summand of D. And then by previous corollary, G is divisible.

By injective property, we have the following diagram, where \varphi is an extension of 1_G.

#align(center, commutative-diagram( node((1, 0), [$0$]), node((1, 1), [$G$]), node((1, 2), [$D$]), node((0, 1), [$G$]), arr((1, 0), (1, 1), []), arr((1, 1), (1, 2), [$i$],

Then by previous theorem about group extension, G is a direct summand of D.■

Exercise 4. A group G is divisible if and only if G = pG for all prime p.

Proof: If G is divisible, then for any element g \in G, as G is divisible, there exists h \in G such that ph = g. Thus, we have G \le pG. And as pG \le G obviously, we have G = pG.

Then, if G = pG for all prime p, then for any n\in \mathbb{N}, we do induction on n. The base case n=1 is obvious, any element g \in G is divisible by 1. Now we assume that any element g \in G is divisible by all natural numbers less than n, and we prove that g is divisible by n. If n is a prime, then as G = nG, there exists h \in G such that nh = g. If n is not a prime, then there exists n_1 \neq 1 and n_2 \neq 1 such that n = n_1 n_2. As g is now divisible by n_1 by induction hypothesis, there exists h_1 \in G such that n_1 h_1 = g. And as h_1 is now divisible by n_2 by induction hypothesis, there exists h_2 \in G such that n_2 h_2 = h_1. Thus, n h_2 = g.■

Exercise 5. A p-primary group G is divisible if and only if G=pG.

Proof: If G is divisible, then for any element g \in G, as G is divisible, there exists h \in G such that ph = g. Thus, we have G \le pG. And as pG \le G obviously, we have G = pG.

Then, if G = pG, then for any n\in \mathbb{N}, we do induction on n. The base case n=1 is obvious, any element g \in G is divisible by 1. Now we assume that any element g \in G is divisible by all natural numbers less than n, and we prove that g is divisible by n. If n = p, then as G = pG, there exists h \in G such that ph = g. If n is a prime that is not p, then we assume the order of g is p^r, and by Bézout identity, there exists x,y such that p^rx + ny = 1. Then we have n (yg) = ny g = (1-p^rx)g = g, thus g is divisible by n. If n is not a prime, then there exists n_1 \neq 1 and n_2 \neq 1 such that n = n_1 n_2. As g is now divisible by n_1 by induction hypothesis, there exists h_1 \in G such that n_1 h_1 = g. And as h_1 is now divisible by n_2 by induction hypothesis, there exists h_2 \in G such that n_2 h_2 = h_1.■

Exercise 6. The following are equivalent for a group G:

  1. G is divisible.
  2. Every non-zero quotient of G is infinite.
  3. G has no maximal proper subgroup.

Proof: From 1 to 2: If G is divisible, then for any non-zero quotient H of G, H is divisible. And by previous theorem, H is a direct summand of \mathbb{Q}, and \mathbb{Z}(p^\infty), thus H is infinite.

From 2 to 3: We assume G has H as a maximal proper subgroup, and derive a contradiction. As H is maximal, G/H is simple. And by 2, G/H is infinite. Take x \in G/H, if x has finite order, then \langle x \rangle is a proper subgroup of G/H, contradicts the simplicity of G/H. If x has infinite order, then \langle 2x \rangle is a proper subgroup of G/H, contradicts the simplicity of G/H.

From 3 to 1: We assume that G has no maximal proper subgroup, and G is now not divisible, and tries to derive a contradiction. As G is not divisible, then there exists p such that G \neq pG. And we assume \varphi: G \rightarrow G/pG is the canonical map. Then G/pG[p] = G/pG, obviously. Thus, G/pG is a vector space over \mathbb{F}_p, we select a basis of G/pG, namely \{x_1, x_2, \dots\}. Then, we define \phi: G/pG \rightarrow \mathbb{F}_p by \phi(m_1 x_2 + m_2 x_2 + \dots) = m_1. Then \phi is a surjection. And \phi\varphi: G \rightarrow \mathbb{F}_p is a surjection. Thus, \ker{\phi\varphi} is a maximal proper subgroup of G, which contradicts the assumption.■

Exercise 7. If G and H are divisible groups each of which is isomorphic to a subgroup of the other, then G \cong H. Is this true if we drop the adjective "divisible"?

Proof: Let \varphi: G \rightarrow H be the embedding of G to the subgroup of H that is isomorphic to G. And let \psi: H \rightarrow G be the embedding of H to the subgroup of G that is isomorphic to H. Then for any prime p, we can restrict \varphi and \psi to G[p] and H[p], and we have \varphi[p]: G[p] \rightarrow H[p] and \psi[p]: H[p] \rightarrow G[p]. As G[p] and H[p] are now vector spaces over \mathbb{F}_p, and as we have the embedding as above, the dimension of G[p] is less than or equal to the dimension of H[p], and vice versa. Thus, the dimension of G[p] is equal to the dimension of H[p]. And thus, G[p] is isomorphic to H[p]. By previous lemma, tG is isomorphic to tH.

For the torsion free part, we define G' = G/tG and H' = H/tH. And \varphi' : G' \rightarrow H' by \phi'(g + tG) = \varphi(g) + tH, and \psi' : H' \rightarrow G' by \psi'(h + tH) = \psi(h) + tG. To prove \varphi' is well-defined, we assume g + tG = g' + tG, then there exists t \in tG such that g = g' + t, thus we have:

\begin{aligned} \varphi'(g+tG) - \varphi'(g'+ tG) &= \varphi(g) - \varphi(g') + tH \\ &= \varphi(g') + \varphi(t) - \varphi(g') + tH \\ &= \varphi(t) + tH \\ \end{aligned}

As t\in tG, \varphi(t) \in tH, thus \varphi'(g+tG) = \varphi'(g'+ tG). In the same way, we can prove \psi' is well-defined.

We then check that \varphi' is injective. Assume \varphi'(g + tG) = 0 + tH, then \varphi(g) \in tH. As \varphi is an isomorphism between G and \text{Img}(\varphi), the order of g is same as the order of \varphi(g) in \text{Img}(\varphi), thus g \in tG. In the same way, we can prove \psi' is injective.

Also, G/tG and H/tH are divisible, thus are vector spaces over \mathbb{Q}, and as both \varphi' and \psi' are injective, the dimension of G/tG is equal to the dimension of H/tH.

Thus, G is isomorphic to H.

If we drop the adjective "divisible", then the statement is not true. For instance, we take G = \mathbb{Q}^{\omega} = \prod_{i\in \mathbb{N}} \mathbb{Q}. And H = Z \times G. Then G is obviously divisible, but H is not.

We define \varphi: G \rightarrow H by \varphi((q_i)_{i\in \mathbb{N}}) = (0, (q_i)_{i\in \mathbb{N}}). And \psi: H \rightarrow G by \psi((z, (q_i)_{i\in \mathbb{N}})) = (z, (q_{i-1})_{i\in \mathbb{N}\setminus\{0\}}). Then both \varphi and \psi are injective, which are both embedding. But G is not isomorphic to H, as one is divisible and the other is not.■

Exercise 8.

  1. Prove that the following groups are all isomorphic:
    1. \mathbb{R}/\mathbb{Z}.
    2. The circle group \mathbf{T}.
    3. \prod_p\mathbb{Z}(p^{\infty}).
    4. \mathbb{R}\oplus(\mathbb{Q}/\mathbb{Z}).
    5. \mathbb{C}^{\times}.
  2. Prove that t(\mathbb{C}^{\times})\cong \mathbb{Q}/\mathbb{Z}.

Proof: The isomorphism between \mathbb{R}/\mathbb{Z} and the circle group is obvious.

By consider a real number as decimal representation, the cardinality of \mathbb{R}/\mathbb{Z} is the same as \prod_p\mathbb{Z}(p^{\infty}). And it is also easy to verify that the torsion group of \mathbb{R}/\mathbb{Z} and \prod_p\mathbb{Z}(p^{\infty}) is isomorphic to \sum_p\mathbb{Z}(p^{\infty}). Thus, (\mathbb{R}/\mathbb{Z})/t(\mathbb{R}/\mathbb{Z}) and (\prod_p\mathbb{Z}(p^{\infty}))/t(\prod_p\mathbb{Z}(p^{\infty})) must have the same cardinality, by looking at the cardinal number division. And we could say (\mathbb{R}/\mathbb{Z})/t(\mathbb{R}/\mathbb{Z}) and (\prod_p\mathbb{Z}(p^{\infty}))/t(\prod_p\mathbb{Z}(p^{\infty})) have the say dimension, again by cardinal number arithmetic. Thus, they are isomorphic.

As \mathbb{Q}/\mathbb{Z} \cong \sum_p\mathbb{Z}(p^{\infty}), (\mathbb{R}\oplus(\mathbb{Q}/\mathbb{Z})) / t(\mathbb{R}\oplus(\mathbb{Q}/\mathbb{Z})) = \mathbb{R}. Thus, we only need to verify that \mathbb{R} is isomorphic to (\mathbb{R}/\mathbb{Z})/(\mathbb{Q}/\mathbb{Z}), which is equal to \mathbb{R}/\mathbb{Q}. As \mathbb{R} is an infinite dimensional vector space over \mathbb{Q}, quotient by \mathbb{Q} will minus one dimension, thus, the dimension of \mathbb{R}/\mathbb{Q} is equal the dimension of \mathbb{R}, which verify that \mathbb{R} is isomorphic to \mathbb{R}/\mathbb{Q}.

As \mathbb{C}^{\times} is isomorphic to \mathbb{R}_{+}^{\times}\oplus\mathbf{T}, which \mathbb{R}_{+}^{\times} is the multiplicative group of positive real numbers. And \mathbb{R}_{+}^{\times} is isomorphic to \mathbb{R}, by the natural logarithm map. Thus, \mathbb{C}^{\times}\cong \mathbb{R} \oplus \mathbf{T}, and t(\mathbb{C}^{\times}) = \mathbb{Q}/\mathbb{Z}. Also (\mathbb{C}^{\times})/t(\mathbb{C}^{\times}) \cong \mathbb{R} \oplus ((\mathbb{R}/\mathbb{Z})/(\mathbb{R}/\mathbb{Z})). By above discussion, we already now that (\mathbb{R}/\mathbb{Z})/(\mathbb{R}/\mathbb{Z}) \cong \mathbb{R}, thus, (\mathbb{C}^{\times})/t(\mathbb{C}^{\times}) \cong \mathbb{R} \oplus \mathbb{R}. It is obvious that \mathbb{R} \oplus \mathbb{R} is then a vector space over \mathbb{Q}, and the dimension of it is equal to square of the dimension of \mathbb{R}. By the cardinal number arithmetic, square of infinite cardinal number is still the same as the original infinite cardinal number. Thus, we have \mathbb{R} \oplus \mathbb{R} \cong \mathbb{R}. Then by previous result, we have \mathbb{R}/\mathbb{Z} \cong \mathbb{C}^{\times}.■

]]>https://alex1222.com/posts/2024-11-26-Group-Theory-Note.mdhttps://alex1222.com/posts/2024-11-26-Group-Theory-Note.mdTue, 26 Nov 2024 00:00:00 GMT<![CDATA[Install K3S on Machine do not have Internet Access]]>

0.1 Contents

1 Install K3S on Machine do not have Internet Access

1.1 Introduction

K3S is a lightweight Kubernetes distribution. It is a great choice for edge computing and IoT devices. I currently have a machine with only ipv6 access, and I want to install K3S on it. Sadly, k3s install script need to download binary from GitHub, which is known to be not ipv6 friendly currently.

1.2 Prerequisites

1.2.1 Firewall

Make sure the machine have open several ports to allow it to communicate with other nodes.

ufw allow 6443/tcp #apiserver
ufw allow from 10.42.0.0/16 to any #pods
ufw allow from 10.43.0.0/16 to any #services

1.2.2 Kernel Modules

Through trying, I found that the machine need to have br_netfilter and overlay kernel modules loaded. Use the following command to check, if it gives a fatal message, it probabily means k3s may not supported on this machine. Also, see https://github.com/k3s-io/k3s/issues/1236.

modprobe br_netfilter
modprobe overlay

1.3 Solution

First, from the k3s release page download the airgap image, and transfer it to the machine you want to install k3s on. Then, download the install script from https://get.k3s.io, and put it in the same directory as the airgap image, and make sure the script is executable.

chmod +x install.sh

Then, download the k3s binary from the release page, and put it into where ever you install executable, normally it is /usr/local/bin.

mv k3s /usr/local/bin
chmod +x /usr/local/bin/k3s

Now, you can run the install script:

INSTALL_K3S_SKIP_DOWNLOAD=true INSTALL_K3S_EXEC='server --token=SECRET' ./install.sh

To join another node to the cluster, you can run:

INSTALL_K3S_SKIP_DOWNLOAD=true INSTALL_K3S_EXEC='server --server https://server-ip:6443 --token=SECRET' ./install.sh

1.4 Uninstall K3S

1.4.1 Uninstall K3S on Server

/usr/local/bin/k3s-uninstall.sh

1.4.2 Uninstall K3S on Agent

/usr/local/bin/k3s-agent-uninstall.sh
]]>https://alex1222.com/posts/2024-08-16-Install-K3S.mdhttps://alex1222.com/posts/2024-08-16-Install-K3S.mdFri, 16 Aug 2024 00:00:00 GMT<![CDATA[Markov Chain Note]]>

0.1 Contents

0.2 Stochastic Process

A stochastic process is a collection of random variables X_t indexed by time t.

0.3 Markov Property

A stochastic process \{X_t\} has the Markov property if for any n \geq 0, any t_0 < t_1 < \cdots < t_n, and any B \in \mathcal{B}, we have

P(X_{t_{n+1}} \in B | X_{t_0}, X_{t_1}, \ldots, X_{t_n}) = P(X_{t_{n+1}} \in B | X_{t_n}).

0.4 Markov Chain

A Markov chain is a stochastic process with the Markov property.

0.4.1 Transition Probability

The transition probability at time n of a Markov chain is defined as

P_{ij} = P(X_{n+1} = j | X_n = i).

0.4.2 Transition Probability Matrix

The transition probability matrix at time n of a Markov chain is a matrix P = (P_{ij}) where P_{ij} is the transition probability from state i to state j.

0.4.3 n-Step Transition Probability

The n-step transition probability from time m to n+m of a Markov chain is defined as

P_{ij}^{(n)} = P(X_{n+m} = j | X_m = i).

0.4.4 Time-Homogeneous Markov Chain

A Markov chain is time-homogeneous if the transition probabilities are independent of time.

In this note, we focus on time-homogeneous Markov chains.

0.5 Chapman-Kolmogorov Equation

The Chapman-Kolmogorov equation states that for a time-homogeneous Markov chain, the (n+m)\text{-step} transition probability is given by

P_{ij}^{(n+m)} = \sum_{k} P_{ik}^{(n)} P_{kj}^{(m)}.

0.6 Hitting Probability

The hitting probability of a state j starting from state i is defined as

f_{ij} = P(\text{state } j \text{ is hit starting from state } i)

By intuition, the hitting probability could be calculated by

f_{ij} = \sum_{n=1}^{\infty} P(\text{state } j \text{ is first hit at time } n | X_0 = i).

Thus, we have the following definition.

0.6.1 First-Visit(Hitting) Probability

The first-visit probability of a state j starting from state i is defined as

f_{ij}^{(n)} = P(\text{state } j \text{ is first hit at time } n | X_0 = i).

0.6.1.1 Convergence of First-Visit Probability

The first-visit probability will always converge to 0 as n goes to infinity.

Proof:

If this is not the case, then the sum of hitting probabilities will be greater than 1, which is a contradiction.

0.6.2 Hitting Time for a Given Path

The hitting time of a state j to state i for a given path i_0, i_1, \ldots is defined as

T_{j}^i = \min\{n \geq 1: X_n = i | X_0 = j\}

Sometimes we use T^i if it is clear that the starting state is j.

0.6.3 Mean Hitting Time (Mean First-Visit Time)

The average hitting time of a state j to state i is defined as

\beta_{j}^i = E[T_{j}^i | X_0 = j].

0.7 Communication

We say that the state i communicates with state j if

P_{ij}^{(n)} > 0 \text{ for some } n \geq 0

and is denoted by i \rightarrow j.

0.7.1 Inter-Communication

Two states i and j are said to inter-communicate if i \rightarrow j and j \rightarrow i.

0.7.2 Irreducible Class of States

A set of states C is said to be an irreducible class of states if any state in C communicates with any other state in C.

0.7.3 Closed Class of States

A set of states C is said to be a closed class of states if any state in C communicates only with other states in C.

0.7.4 Absorbing State

A state i is said to be an absorbing state if itself forms a closed class of states.

0.8 Recurrence

0.8.1 Recurrent and Transient State

A state i is said to be recurrent if the hitting probability

f_{ii} = 1.

A state i is said to be transient if it is not recurrent.

0.8.2 Recurrent and Transient Class

A class of states C is said to be recurrent if any state in C is recurrent.

A class of states C is said to be transient if any state in C is transient.

0.8.3 Two Important Theorems

  1. A state i is recurrent if and only if
\sum_{n=1}^{\infty} P_{ii}^{(n)} = \infty.

Proof:

We define the following generating function:

f(s) = \sum_{n=1}^{\infty} f_{ii}^{(n)} s^n

Then, the probability of hitting state i at time n for the k-th time is the coefficient of s^n in the expansion of f(s)^k.

Thus, we the transition probability:

P_{ii}^{(n)} = \sum_{k=1}^{n} P(\text{hitting }i\text{ at time }n\text{ for the }k\text{-th time})

which should equal to the coefficient of s^n in the expansion of \sum_{k=1}^{n}f^k(s).

As the coefficient of s^i for i<k in f(s)^k is 0, we have

\begin{align} P_{ii}^{(n)} &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n} f^k(s) \\ &= \text{coefficient of }s^n\text{ in }f^{n+1}(s) + \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n} f^k(s) \\ &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n+1} f^k(s) \text{(by repeating similar steps)} \\ &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{\infty} f^k(s) \end{align}

Thus, we have

\sum_{n=1}^{\infty} P_{ii}^{(n)} = \sum_{n=1}^{\infty} f^k(1)

If i is recurrent, then f(1) = 1 the previous sum diverges.

If i is transient, then f(1) < 1 and the previous sum is a sum of a geometric series, which converges.

  1. If i is transient, then for all j
P_{ji}^{(n)} \rightarrow 0 \text{ as } n \rightarrow \infty.

Proof:

If the condition does not satisfy, we assume that a=\lim_{n\rightarrow\infty}P_{ji}^{(n)}, and as i is transient, let \alpha = f_{ii}<1. We choose \epsilon, such that \frac{a(1-\alpha)}{3+\alpha} > \epsilon >0. Then there exist N such that for all n\ge N, a-\epsilon<P_{ji}^{(n)}<a+\epsilon, and f_{ji}^{(n)} < \epsilon and \sum_{k=1}^{n}f_{ii}^{(n)} > \alpha-\epsilon

For n>2N+1, we have

\begin{align} P_{ji}^{(n)} &= f_{ji}^{(n)} + \sum_{k=1}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\ &\le \epsilon+\sum_{k=1}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\ &\le \epsilon+\epsilon+\sum_{k=N}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\ &\le 2\epsilon+\sum_{k=N}^{n-1}(a+\epsilon)f_{ii}^{n-k} \\ &= 2\epsilon+(a+\epsilon)\sum_{k=N}^{n-1}f_{ii}^{n-k} \\ &\le 2\epsilon + (a+\epsilon)\alpha \\ &< a-\epsilon \end{align}

which contradicts the assumption.

0.8.4 Finite State Space Must have Recurrent State

If the state space is finite, then there must be at least one recurrent state.

Proof:

If all state are transient, then by the second theorem in previous section, we have P_{ji}^{(n)} \rightarrow 0 as n \rightarrow \infty for all i and j, thus \sum_{i\in S} P_{ji}^{(n)} tends to 0 as n goes to infinity.

However, \sum_{i\in S} P_{ji}^{(n)} is constantly 1 as it is the sum of the transition probability of state j to all other states, which is a contradiction.

0.8.5 Intercommunication and Recurrence

If two states intercommunicate, then they are either both recurrent or both transient.

Proof:

Given two states i and j that intercommunicate.

Suppose i is recurrent, then

\sum_{n=1}^{\infty} P_{ii}^{(n)} = \infty

As i and j intercommunicate, there either exists n such that P_{ij}^{(n)} > 0, and m such that P_{ji}^{(m)} > 0,

Then we have

P_{jj}^{(n+k+m)} \ge P_{ij}^{(n)}P_{jj}^{(k)}P_{ji}^{(m)}

Thus,

\begin{align} \sum_{\alpha=1}^{\infty} P_{jj}^{(\alpha)} &\ge \sum_{\alpha=n+m+1}^{\infty} P_{jj}^{(\alpha)} \\ &\ge \sum_{\alpha=1}^{\infty} P_{ij}^{(n)}P_{jj}^{(\alpha)}P_{ji}^{(m)} \\ &= P_{ij}^{(n)}P_{ji}^{(m)}\sum_{\alpha=1}^{\infty} P_{jj}^{(\alpha)} \end{align}

which is infinite as the sum of P_{jj}^{(\alpha)} is infinite.

Then, j is recurrent.

0.8.6 Number of Visits to a State for a Given Path

The number of visits to a state j for a given path i_0, i_1, \ldots is defined as

N_j = \sum_{n=1}^{\infty} I(X_n = j).

0.8.6.1 Number of Visits at Infinity

\begin{align} P(N_j = \infty) = 1 &\text{ if } j \text{ is recurrent} \\ P(N_j = \infty) = 0 &\text{ if } j \text{ is transient} \end{align}

0.8.7 Decomposition of State Space

The state space can be decomposed into a set of Irreducible Closed Recurrent classes and a set of Transient classes.

Proof:

Given state space I, as the InterCommunicate relation is an equivalence relation, we have a partition of I into Inter Communicate classes. Let the classes be I_{\alpha}, \alpha \in A.

As all states in an InnerCommunicate class have the same Recurrence, let

\begin{equation} B = \{\alpha \in A | I_{\alpha} \text{ is Transient}\} \end{equation}

Let C = A - B. Then, for all \alpha \in C, I_{\alpha} is Recurrent.

We then prove that I_{\alpha} is Closed for all \alpha \in C., which finished the proof of this theorem.

If I_{\alpha} is not closed, then there exists i \in I_{\alpha} and j \notin I_{\alpha}, and n \in \mathbb{N} such that P_{ij}^{(n)} > 0.

Also, as i \in I_{\alpha}, i is Recurrent, and thus

\begin{equation} f_{ii} = 1 \end{equation}

Also, in this case, j can not Communicate with i, as if it does, j will be in I_{\alpha}, which means

\begin{equation} f_{ji} = 0 \end{equation}

Then we have

\begin{align*} 1 &= f_{ii} \\ &= \sum_{k\in I} P_{ik}^{(n)} f_{ki} \\ &= P_{ij}^{(n)}*f_{ji} + \sum_{k\in I, k\neq j} P_{ik}^{(n)} f_{ki} \\ &= \sum_{k\in I, k\neq j} P_{ik}^{(n)} f_{ki} \\ &\le \sum_{k\in I, k\neq j} P_{ik}^{(n)} \\ &= 1- P_{ij}^{(n)} \\ &< 1 \end{align*}

This is a contradiction. Thus, I_{\alpha} is closed.

0.8.8 Mean Recurrence Time

The mean recurrence time of a state i is defined as

\mu_i = E[T_i^i | X_0 = i] = \sum_{n=1}^{\infty} n f_{ii}^{(n)}.

0.8.8.1 Mean Recurrence Time for a Transient State

The mean recurrence time for a transient state is infinite.

0.8.8.2 Positive and Null Recurrent State

A state i is said to be positive recurrent if the mean recurrence time:

\mu_i < \infty.

A state i is said to be null recurrent if the mean recurrence time:

\mu_i = \infty.

0.8.8.3 Identification of Positive and Null Recurrent State

A null recurrent state has similar properties to a transient state:

A recurrent state is null recurrent if and only if

\lim_{n\rightarrow \infty}P_{ii}^{(n)} = 0

0.8.8.4 State in the Same Irreducible Class have the Same Recurrence (Positive or Null)

If two states are in the same irreducible class, then they are either both positive recurrent or both null recurrent.

0.8.8.5 Finite State Space Must have Positive Recurrent State

If the state space is finite, then there must be at least one positive recurrent state.

0.9 Periodicity

0.9.1 Period of a State

The period of a state i is defined as

d_i = \gcd\{n \geq 1: P_{ii}^{(n)} > 0\}

If d_{i} is 1, then the state is said to be aperiodic.

where \gcd is the greatest common divisor.

0.9.2 Period of a Class

All states in the same irreducible class have the same period.

0.10 Stationary Distribution

0.10.1 Stationary Distribution

A distribution \pi = (\pi_1, \pi_2, \ldots) is said to be a stationary distribution of a Markov chain if

\pi = \pi P

where P is the transition probability matrix.

0.10.2 Existence and Uniqueness of Stationary Distribution

If a Markov chain is irreducible and positive recurrent, then it has a unique stationary distribution.

And the stationary distribution is given by

\pi_i = \frac{1}{\mu_i}

where \mu_i is the mean recurrence time of state i.

0.10.3 Ergodic Markov Chain

A Markov chain is said to be ergodic if it is irreducible, positive recurrent and aperiodic.

0.10.3.1 Limiting Distribution

Given an ergodic Markov chain, and any vector \pi^{(0)},

The limit:

\lim_{n\rightarrow \infty} \pi^{(0)} P^n = \pi

where \pi is the stationary distribution.

0.11 Discrete Renewal Process

Define a sequence of independent distributed random variables \{X_n\} with positive integer state space.

Define a stochastic process \{T_n\} as

T_n = T_{n-1} + X_n

where T_0 = 0.

Then, \{T_n\} is a discrete renewal process.

0.11.1 Other Ways to Define Discrete Renewal Process

This first definition is defined by the inter-arrival time. We can directly define the process by the renewal time T_n, which are not independent.

We can also define the process by assign each time n a renewal probability u_n, which is the probability that the process is renewed at time n.

Or we can define the process by the number of renewal N_n, which is the number of renewals in the time interval [0, n].

0.11.2 Mean of Number of Renewals

If we are given the renewal probability u_n, then the mean of the number of renewals in the time interval [0, n] is

E[N_n] = \sum_{k=1}^{n} u_k.

0.11.3 Renewal Process and Markov Chain

If we are given a Markov chain with state space S = \{1, 2, \ldots\}, and we set Y_0 = 1, and we said the process is renewed whenever it hits state 1, then the process is a discrete renewal process.

And the renewal probability is given by

u_n = P(\text{the process is renewed at time } n) = P(Y_n = 1|Y_0=1) = P_{11}^{(n)}

0.11.3.1 Inter-renewal Distribution of Discrete Renewal Process Defined by Markov Chain

The inter-renewal distribution of the discrete renewal process defined by a Markov chain is given by

f_k = P(\text{the precess first reach }1\text{ at } k | Y_0 = 1) = f_{11}^{(k)}

which is the first renewal probability.

0.11.3.2 The Limit Renewal Theorem of Ergodic Markov Chain

Given an ergodic Markov chain with state space S = \{1, 2, \ldots\}, and we set Y_0 = 1, and we said the process is renewed whenever it hits state 1, and we set the number of renewals in the time interval [0, n] as N_n, then

\lim_{n\rightarrow \infty} \frac{E[N_n]}{n} = \frac{1}{E[T_1]}

where E[T_1] is the mean recurrence time of state 1.

0.12 Branching Process

0.12.1 Offspring Distribution

An offspring distribution is a probability distribution of the number of offspring of each individual.

0.12.2 Discrete Branching Process

A discrete branching process with offspring distribution Z is a stochastic process \{X_n\} defined as

X_n = \sum_{i=1}^{X_{n-1}} Z

And X_n is called the size of the nth generation.

0.12.3 The Probability Generating Function of Discrete Branching Process

The probability generating function of a discrete branching process is defined as

F(s) = E[s^{Z}]

where Z is the offspring distribution.

0.12.4 Mean Generation Size

If we are starting from X_{0} = 1, then the mean generation size of the nth generation is given by

E[X_n] = E[Z]^n

where E[Z] is the mean of the offspring distribution.

0.12.5 Total Progeny

The total progeny of the branching process is defined as the total number of individuals that ever exist in the process.

T = \sum_{n=0}^{\infty} X_n

0.12.5.1 Probability Generating Function of Total Progeny

Let G(s) be the probability generating function of the total progeny T, then

G(s) = sF(G(s))

0.12.6 Extinction

We say that the branching process is extinct if X_n = 0 for some n.

0.12.6.1 Probability of Extinction

The probability of extinction is given by the ever visit probability of state 0 starting from state 1.

\gamma = f_{10} = P(\text{extinction}) = \sum_{n=1}^{\infty} f_{10}^{(n)}
0.12.6.1.1 Calculation of Probability of Extinction

The probability of extinction is the least non-negative solution of the equation

s = F(s)

0.13 Random Walk

0.13.1 Simple Random Walk

A simple random walk is a stochastic process \{X_n\} defined as

X_n = X_{n-1} + Y_n

where Y_n is a sequence of independent and identically distributed random variables.

In this note, we only focusing on Y to be a Bernoulli random variable with

P(Y = 1) = p, P(Y = -1) = 1 - p

0.13.1.1 Symmetric Random Walk

If p = 0.5, then the random walk is said to be symmetric.

0.13.1.2 Number of Possible Path

The number of possible paths of a simple random walk from (0,i) to (n,j) is denoted as N_n(i,j).

0.13.1.3 Crossing Condition

Let the number N_{n}^{m}(i,j) denote the number of possible paths of a simple random walk from (0,i) to (n,j) that is at (k,m) for some k.

]]>https://alex1222.com/posts/2024-05-29-Markov-Chain-Note.mdhttps://alex1222.com/posts/2024-05-29-Markov-Chain-Note.mdWed, 29 May 2024 00:00:00 GMT<![CDATA[Quantum Mechanics Note]]>

0.1 Contents

0.2 Probability Basics

0.2.1 Random Variable

A random variable X is a function that maps the sample space \Omega to the real number line \mathbb{R}.

0.2.2 Probability Density Function

The probability density function f(x) of a random variable X is a function that describes the likelihood of the random variable to take on a specific value.

0.2.3 Cumulative Distribution Function

The cumulative distribution function F(x) of a random variable X is a function that describes the probability that the random variable takes on a value less than or equal to x.

F(x) = P(X \leq x) = \int_{-\infty}^x f(x) dx.

0.2.4 Expectation

The expectation of a random variable X is the average value of the random variable.

<X> = \int_{-\infty}^{\infty} x f(x) dx.

0.2.5 Variance

The variance of a random variable X is a measure of how much the values of the random variable vary.

\Delta X^2 = <(X - <X>)^2> = <X^2> - <X>^2.

0.2.6 Standard Deviation

The standard deviation of a random variable X is the square root of the variance.

\Delta X = \sqrt{\Delta X^2}.

0.2.7 Probability Amplitude

The probability amplitude \psi(x) of a random variable X is a complex value function that the likelihood of the random variable to take on value x is given by |\psi(x)|^2.

In other words, the probability density function f(x) is given by

f(x) = |\psi(x)|^2.

0.3 Calculus Basics

0.3.1 Gaussian Integral

The Gaussian integral is given by

\int_{-\infty}^{\infty} e^{-\frac{x^2}{\sigma^2}} dx = \sqrt{\pi}\sigma.

0.3.1.1 Odd Powers of x in Gaussian Integral

The Gaussian integral with odd powers of x is given by

\int_{-\infty}^{\infty} x^{2n+1} e^{-\frac{x^2}{\sigma^2}} dx = 0.

As this is an odd function, the integral is zero.

0.3.1.2 Even Powers of x in Gaussian Integral

The even powers of x in Gaussian integral can be calculated using Feynman's trick.

\begin{align} \int_{-\infty}^{\infty} x^{2n} e^{-ax^2} dx &= \int_{-\infty}^{\infty} \left(-\frac{\partial}{\partial a}\right)^n e^{-ax^2} \\ &= \left(-\frac{\partial}{\partial a}\right)^n \int_{-\infty}^{\infty} e^{-ax^2} dx \\ &= \left(-\frac{\partial}{\partial a}\right)^n \sqrt{\frac{\pi}{a}} \end{align}

0.4 Wave Function

We can postulate that the state of a particle is described by a complex value wave function \Psi(x, t), where x is the position and t is the time.

0.4.1 Born Rule

The probability density to find the particle at position x is given by |\Psi(x, t)|^2. And thus, the probability to find the particle in an measurable area A is given by

P(A) = \int_A |\Psi(x, t)|^2 dx.

0.4.2 Continuity Requirement of Wave Function

The wave function \Psi(x, t) must be continuous and differentiable.

0.4.3 Normalizing Wave Function

The wave function \Psi(x, t) defined on space X is normalized if

\int_{X} |\Psi(x, t)|^2 dx = 1.

If the wave function is not normalized, and if A=\int_{X} |\Psi(x, t)|^2 dx is finite and greater than 0, then the normalized wave function is given by

\Psi_{\text{normalized}}(x, t) = \frac{\Psi(x, t)}{\sqrt{A}}.

0.4.4 Superposition Principle of Wave Functions

If \Psi_1(x, t) and \Psi_2(x, t) are two wave functions, then the superposition of the wave functions is given by

\Psi(x, t) = c_1 \Psi_1(x, t) + c_2 \Psi_2(x, t),

where c_1 and c_2 are complex numbers. And \Psi is also a wave function.

0.4.5 Interference of Wave Functions

The interference of wave functions is a phenomenon where two wave functions \Psi_1(x, t) and \Psi_2(x, t) interfere with each other to form a new wave function \Psi(x, t).

Let \Psi(x, t) = \Psi_1(x, t) + \Psi_2(x, t). Then the probability density of the new wave function is given by

|\Psi(x, t)|^2 = |\Psi_1(x, t)|^2 + |\Psi_2(x, t)|^2 + \Psi_1(x, t)^* \Psi_2(x, t) + \Psi_1(x, t) \Psi_2(x, t)^*

And the \Psi_1(x, t)^* \Psi_2(x, t) + \Psi_1(x, t) \Psi_2(x, t)^* is usually called the interference term.

0.4.6 Classical Plane Wave

The classical plane wave is given by

\Psi(x, t) = A e^{ikx - i\omega t}

where A is the amplitude, k is the wave number, and \omega is the angular frequency.

Note: The classical plane wave is not normalizable, as the integral of the probability density is infinite. In this case, we can use \Psi(x, t) = A e^{ikx - i\omega t -iax^2} where a is a relatively small positive constant. The new wave function is normalizable and have similar behaviour to the plane wave near the origin.

0.5 Schrödinger Equation

0.5.1 Reduced Planck Constant

The reduced Planck constant is given by

\hbar = \frac{h}{2\pi} \approx 1.0545718 \times 10^{-34} \text{ J s}.

where h is the Planck constant.

0.5.2 Quantization Rule

The quantization rule is given by

\begin{align} \hat{p} &= -i \hbar \nabla \\ \hat{E} &= i \hbar \frac{\partial}{\partial t} \end{align}

0.5.3 Schrödinger Equation for Free Particle

In classical mechanics, the energy of a free particle is given by

E = T + V = \frac{p^2}{2m} + V(x).

By applying the quantization rule, the Schrödinger equation (SE) for a free particle with wave equation \psi is given by

\begin{align} \hat{E}\psi &= \left[\frac{\hat{p}^2}{2m} + V\right]\psi \\ i \hbar \frac{\partial \psi}{\partial t} &= \left[-\frac{\hbar^2}{2m} \nabla^2 + V\right]\psi = \left[-\frac{\hbar^2}{2m} \Delta + V\right]\psi \end{align}

0.5.4 Hamiltonian Operator

The Hamiltonian operator \hat{H} is given by

\hat{H} = \frac{\hat{p}^2}{2m} + V(x)

Using the Hamiltonian operator, the Schrödinger equation for a free particle is given by

\hat{E} \frac{\partial \psi}{\partial t} = \hat{H}\psi.

0.5.5 Using Separation of Variables to Solve Schrödinger Equation

The Schrödinger equation can be solved using separation of variables. Let \psi(x, t) = \phi(x) \tau(t). Then the Schrödinger equation becomes

\begin{align} \hat{E}\psi &= \hat{H}\psi \\ \phi \hat{E} \tau &= \tau \hat{H} \phi \\ \frac{\hat{E} \tau}{\tau} &= \frac{\hat{H} \phi}{\phi} = E \\ i\hbar \frac{\partial}{\partial t} \tau &= E \tau \\ \tau &= e^{-iEt/\hbar} \\ \hat{H} \phi &= E \phi \end{align}

Assume, \phi_1,\phi_2,\ldots solve \hat{H} \phi = E \phi with E_1,E_2,\ldots. Then the general solution is given by

\phi(x) = \sum_{n=1}^{\infty} c_n \phi_n(x)e^{-iE_nt/\hbar}.

0.5.6 Time-Independent Schrödinger Equation

By the previous discussion, the task of solving the Schrödinger equation is reduced to solving the equation

\hat{H} \phi_n = E_n \phi_n.

This equation is called the time-independent Schrödinger equation (TISE).

We always call E_n eigenvalues of H. And it is also known as:

  1. Energy lLevels
  2. Eigenenergies
  3. Energy Eigenvalues

And \phi_n are called energy eigenstates of H. And it is also known as:

  1. Stationary States
  2. Energy Eigenfunctions

0.5.6.1 Degeneracy of TISE Solutions

If two or more energy eigenstates have the same energy eigenvalue, then the energy eigenvalue is said to be degenerate.

If two eigenstates have the same eigenvalue, we call it degeneracy of order 2 or twice degenerate.

0.6 Double Slit Experiment

Consider the following double slit experiment setup.

Double Slit Experiment

We can assume the wave function that go through the slits A and B are the same, and is given by the analogue of plane wave \psi(r, t) = \frac{e^{ikx-iwt}}{r}.

Then, for any point P on the screen, the wave function is the superposition of the wave functions from the slits A and B.

\begin{align} \Psi(x, t) &= \psi_A(r_1, t) + \psi_B(r_2, t) \\ &= e^{ikr_1-iwt} + e^{ikr_2-iwt} \\ &= e^{-iwt}\left[\frac{e^{ikr_1}}{r_1} + \frac{e^{ikr_2}}{r_2}\right] \end{align}

where r_1 and r_2 are the distances from the slits A and B to the point P. And is given by

\begin{align} r_1 &= \sqrt{L^2 + (x-d)^2} \\ r_2 &= \sqrt{L^2 + (x+d)^2} \end{align}

Without normalization, the probability density of the wave function is proportional to

\begin{align} |\Psi(x, t)|^2 &= \left|\frac{e^{ikr_1}}{r_1} + \frac{e^{ikr_2}}{r_2}\right|^2 \\ &= \frac{1}{r_1^2} + \frac{1}{r_2^2} + \frac{1}{r_1r_2}\left[e^{ik(r_1-r_2)} + e^{-ik(r_1-r_2)}\right] \\ &\approx \frac{1}{L^2}\left[2 + 2\cos(k(r_1-r_2))\right] \\ &= \frac{1}{L^2}\left[2 + 2\cos\left( k \left( \sqrt{L^2 + (x+d)^2} - \sqrt{L^2 + (x-d)^2} \right) \right) \right] \\ &= \frac{1}{L^2}\left[2 + 2\cos\left( k \frac{ 4dx }{ \sqrt{L^2 + (x+d)^2} + \sqrt{L^2 + (x-d)^2} } \right) \right] \\ &\approx \frac{1}{L^2}\left[2 + 2\cos\left( k \frac{ 4dx }{ 2L } \right)\right] \quad\text{As } L \gg x,d \\ &= \frac{1}{L^2}\left[2 + 2\cos\left( k \frac{ 2dx }{ L } \right)\right] \end{align}

We can see that the probability density of the wave function is proportional to a transformed cosine function, which explains the interference pattern on the screen.

0.7 Parity of TISE Solutions

0.7.1 Even potential and Parity

Given a potential V(x): \mathbb{R}\rightarrow \mathbb{C}, the potential is said to be even if

V(x) = V(-x)

In such cases, we can assume the TISE solution has definite parity, that is:

\phi(x) = \pm \phi(-x)

0.7.2 Real Potential and Parity

Given a real potential V(x), we can assume the TISE solution to have definite parity regarding conjugation, that is:

\phi(x) = \phi^*(x)

0.8 States of TISE Solutions

In general there are three kinds of states of TISE solutions:

  1. Bound States
  2. Scattering States
  3. Non-physical States

0.8.1 Bound States

Bound states are states where the energy eigenvalue E_n is less than the potential energy at infinity and greater than the minimum potential energy. In such cases, the TISE solution is normalizable, and the energy eigenvalue is quantized.

Bound States

0.8.2 Scattering States

Scattering states are states where the energy eigenvalue E_n is greater than the potential energy at infinity. In such cases, the TISE solution is not normalizable, and the energy eigenvalue is continuous.

Scattering states can be approximated by normalizable states by similar method as the classical plane wave.

Scattering States

0.8.3 Non-physical States

Non-physical states are states where the energy eigenvalue E_n is less than the minimum potential energy. This case cannot happen in real physical systems.

Non-physical States

0.9 Continuity Equation and Probability Current

Consider the potential to be real. Define the probability density of a wave function P = \Psi(x, t) as |\Psi(x, t)|^2. Then:

\begin{align} \frac{\partial}{\partial t} P &= \frac{\partial \psi}{\partial t} \psi^* + \psi \frac{\partial \psi}{\partial t}^* \\ &= \frac{1}{i\hbar}\left[ \hat{H}\psi\psi^* - \psi(\hat{H}\psi)^* \right] \\ &= -\frac{\hbar^2}{i\hbar 2m}\left[ \frac{\partial^2 \psi}{\partial x^2}\psi^* - \psi\frac{\partial^2 \psi}{\partial x^2}^* \right] \\ &= -\frac{d}{dx}\left[\frac{\hbar^2}{i\hbar 2m}\left[ \frac{\partial \psi}{\partial x}\psi^* - \frac{\partial \psi}{\partial x}\frac{\partial \psi}{\partial x}^* - \psi\frac{\partial \psi}{\partial x}^* + \frac{\partial \psi}{\partial x}\frac{\partial \psi}{\partial x}^* \right]\right] \\ &= -\frac{d}{dx}\left[\frac{\hbar}{2mi}\left[ -\psi\frac{\partial \psi}{\partial x}^* + \psi^*\frac{\partial \psi}{\partial x} \right]\right] \end{align}

We thus define the probability current J as

J = \frac{\hbar}{2mi}\left[ -\psi\frac{\partial \psi}{\partial x}^* + \psi^*\frac{\partial \psi}{\partial x} \right]

The above equation become:

\frac{\partial P}{\partial t} = -\frac{dJ}{dx}

and is called the continuity equation.

0.10 Quantum Tunnelling

We consider a potential barrier of height V_0 and width 2a.

That is, the potential is given by

V(x) = \begin{cases} V_0 & \text{if } |x| < 2a \\ 0 & \text{otherwise} \end{cases}

Then, the TISE can be divided into three regions:

  1. Region I: x < -a
\frac{d^2 \psi}{dx^2} = -\frac{2m}{\hbar^2}E\psi
  1. Region II: -a < x < a
\frac{d^2 \psi}{dx^2} = -\frac{2m}{\hbar^2}(E-V_0)\psi
  1. Region III: x > a
\frac{d^2 \psi}{dx^2} = -\frac{2m}{\hbar^2}E\psi

0.10.1 Simplified Model of Quantum Tunnelling

For the sake of simplicity, let k = \sqrt{\frac{2mE}{\hbar^2}} we can assume the solution of the TISE in Region I and Region III to be

In region I:

\psi(x) = A e^{ikx} + B e^{-ikx}

In region III:

\psi(x) = F e^{ikx}

Quantum Tunnelling

The wave A e^{ikx} can be interpreted as the wave go toward the barrier, and the wave B e^{-ikx} can be interpreted as the wave being reflected by the barrier and wave F e^{ikx} can be interpreted as the wave go through the barrier.

As there is no probability for the particle to be in region II, the probability current must be same at the point x = a and x = -a.

In region I:

J = \frac{\hbar}{2mi}\left[ \psi\frac{-\partial \psi}{\partial x}^* + \psi^*\frac{\partial \psi}{\partial x} \right] = \frac{\hbar k}{m}(|A|^2 - |B|^2)

In region III:

J = \frac{\hbar}{2mi}\left[ \psi\frac{-\partial \psi}{\partial x}^* + \psi^*\frac{\partial \psi}{\partial x} \right] = \frac{\hbar k}{m}|F|^2

Thus, we have

|A|^2 - |B|^2 = |F|^2

We define the reflection probability R and transmission probability T as

\begin{align} R &= \frac{|B|^2}{|A|^2} \\ T &= \frac{|F|^2}{|A|^2} \end{align}

Then, we have

R + T = 1

0.10.2 Quantum Tunnelling through High and Thin Barrier

We consider a barrier that is infinitely high and thin at origin.

To state this formally, we consider the potential to be the limiting behaviour of the Dirac delta function potential.

\delta_L(x) = \begin{cases} \frac{1}{L} & \text{if } -\frac{L}{2} < x < \frac{L}{2} \\ 0 & \text{otherwise} \end{cases}

and

\delta = \lim_{L \to 0} \delta_L(x)

Also, for all function g(x), we have

\lim_{\epsilon \to 0} \int_{-\epsilon}^{\epsilon} g(x) \delta(x) dx = g(0)

Then, the potential is given by

V(x) = \alpha \delta(x)

We again, consider the simplified model in the previous section.

In region I:

\psi(x) = A e^{ikx} + B e^{-ikx}

In region III:

\psi(x) = F e^{ikx}

As the region II is infinity thin, the right hand side of the region I and the left hand side of the region III must be the same.

Thus, we have

A + B = F

Also, by integrating the TISE over the region [-\epsilon,\epsilon], we get:

\begin{align} \int_{-\epsilon}^{\epsilon} \frac{d^2 \psi}{dx^2} dx &= -\frac{2m}{\hbar^2} \int_{-\epsilon}^{\epsilon} (E-\alpha\delta(x))\psi dx \\ \left[\frac{d \psi}{dx}\right]_{-\epsilon}^{\epsilon} &= -\frac{2m}{\hbar^2} \left[E\psi(\epsilon)-E\psi(-\epsilon) - \alpha\int_{-\epsilon}^{\epsilon} \psi\delta dx\right] \\ \end{align}

Taking the limit \epsilon \to 0, we get

\begin{align} \left[\frac{d \psi}{dx}\right]_{0^+} - \left[\frac{d \psi}{dx}\right]_{0^-} &= \frac{2m}{\hbar^2} \alpha \psi(0) \end{align}

Substituting the solution of the TISE in region I and region III, we get

ik(A-B) - ikF = \frac{2m}{\hbar^2} \alpha F

Solving the above equation, we can get the reflection and transmission probability.

0.11 Functional Analysis of Quantum Mechanics

0.11.1 Hilbert Space

Define the Hilbert space \mathbf{H} as the space of all possible wave functions \Psi(x, t) that are square integrable.

\mathbf{H} = \left\{ \Psi(x, t) \mid \int_{-\infty}^{\infty} |\Psi(x, t)|^2 dx < \infty \right\}

The Hilbert space is a complex vector space.

0.11.2 Inner Product

We define the inner product of two wave functions \Psi_1(x, t) and \Psi_2(x, t) as

\left<\Psi_1, \Psi_2\right> = \int_{-\infty}^{\infty} \Psi_1^*(x, t) \Psi_2(x, t) dx

0.11.2.1 Properties of Inner Product

  1. Linearity in Second Argument: For all \Psi_1, \Psi_2 \in \mathbf{H} and a,b \in \mathbb{C}, we have \left<a\Psi_1 + b\Psi_2, \Psi_3\right> = a\left<\Psi_1, \Psi_3\right> + b\left<\Psi_2, \Psi_3\right>.
  2. Anti-linearity in First Argument: For all \Psi_1, \Psi_2 \in \mathbf{H} and a,b \in \mathbb{C}, we have \left<\Psi_1, a\Psi_2 + b\Psi_3\right> = a^*\left<\Psi_1, \Psi_2\right> + b^*\left<\Psi_1, \Psi_3\right>.
  3. Positive Definite: For all \Psi \in \mathbf{H}, we have \left<\Psi, \Psi\right> \geq 0 and \left<\Psi, \Psi\right> = 0 if and only if \Psi = 0.
  4. Conjugate Symmetric (Skew Symmetric): For all \Psi_1, \Psi_2 \in \mathbf{H}, we have \left<\Psi_1, \Psi_2\right> = \left<\Psi_2, \Psi_1\right>^*.

0.11.3 Norm

The norm of a wave function \Psi(x, t) is defined as

||\Psi|| = \sqrt{\left<\Psi, \Psi\right>}

0.11.4 Orthogonality

Two wave functions \Psi_1(x, t) and \Psi_2(x, t) are said to be orthogonal if

\left<\Psi_1, \Psi_2\right> = 0

0.11.5 Angle

The angle between two wave functions \Psi_1(x, t) and \Psi_2(x, t) is defined as

\cos\theta = \frac{\left<\Psi_1, \Psi_2\right>}{||\Psi_1|| ||\Psi_2||}

0.11.6 Orthonormal Basis

A set of wave functions \{\Psi_1(x, t), \Psi_2(x, t), \ldots\} is said to be an orthonormal basis if

  1. The set is orthogonal.
  2. The set is normalized.
  3. The set spans the Hilbert space. That is, for all \Psi(x, t) \in \mathbf{H}, there exists a set of complex numbers \{c_1, c_2, \ldots\} such that
\Psi(x, t) = \sum_{n=1}^{\infty} c_n \Psi_n(x, t)

0.11.7 Operator

An operator is a function that maps a wave function to another wave function.

0.11.8 Hermitian Conjugate

The Hermitian conjugate of an operator \hat{A} is denoted by \hat{A}^\dagger and is the unique operator that satisfies

\left<\hat{A}\Psi_1, \Psi_2\right> = \left<\Psi_1, \hat{A}^\dagger\Psi_2\right>

0.11.9 Hermitian Operator

An Hermitian operator is an operator that satisfies

\left<\hat{A}\Psi_1, \Psi_2\right> = \left<\Psi_1, \hat{A}\Psi_2\right>

for all \Psi_1, \Psi_2 \in \mathbf{H}.

An equivalent definition is that the operator is equal to its Hermitian conjugate.

0.11.10 Spectral Theorem

The spectral theorem states that for all Hermitian operators \hat{A}, there exists an orthonormal basis \{\Psi_1(x, t), \Psi_2(x, t), \ldots\} such that

\hat{A}\Psi_n(x, t) = a_n\Psi_n(x, t)

where a_n are the eigenvalues of \hat{A} and is real.

0.11.10.1 Hamiltonian Operator is Hermitian

The Hamiltonian operator \hat{H} is Hermitian.

Proof:

\begin{align} \left<\hat{H}\Psi_1, \Psi_2\right> &= \int_{-\infty}^{\infty} \Psi_1^*(x, t) \hat{H}\Psi_2(x, t) dx \\ &= \int_{-\infty}^{\infty} \Psi_1^*(x, t) \left[-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_2}{\partial x^2} + V(x)\Psi_2(x, t)\right] dx \quad \text{By Integration By Part} \\ &= \int_{-\infty}^{\infty} \left[-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi_1^*}{\partial x^2} + V(x)\Psi_1^*(x, t)\right] \Psi_2(x, t) dx \\ &= \left<\Psi_1, \hat{H}\Psi_2\right> \end{align}

0.11.11 Positivity Operators

An operator \hat{A} is said to be positive definite if give any non-zero wave function \Psi(x, t), we have

\left<\hat{A}\Psi, \Psi\right> > 0

An operator \hat{A} is said to be positive semi-definite if give any wave function \Psi(x, t), we have

\left<\hat{A}\Psi, \Psi\right> \geq 0

Any operator given by \hat{A} = \hat{B}^\dagger \hat{B} is positive semi-definite.

0.12 Measurement Postulate

Given a orthonormal basis \{\Psi_1(x, t), \Psi_2(x, t), \ldots\}, the measurement postulate states that the probability of measuring the normalized wave function \Psi(x, t) to be in the state \Psi_n(x, t) is given by

P_n = \left|\left<\Psi_n, \Psi\right>\right|^2

0.12.1 Post Measurement State

After the measurement, the state of the wave function will collapse to the state that is measured.

0.13 Measurement of Observables

Given an observable \hat{A}, it is postulated that \hat{A} is Hermitian and has an orthonormal basis \{\Psi_1(x, t), \Psi_2(x, t), \ldots\}.

The expectation value of the observable \hat{A} of a wave function \psi is given by

\left<\hat{A}\right> = \sum_{n=1}^{\infty} a_n P_n = \left<\psi|\hat{A}\psi\right>

0.14 Commutators and Lie Bracket

In general, two operators \hat{A} and \hat{B} do not commute.

We define the commutator (Lie Bracket) of two operators \hat{A} and \hat{B} as

[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}

0.14.1 Properties of Commutators

  1. Linearity: [a\hat{A} + b\hat{B}, \hat{C}] = a[\hat{A}, \hat{C}] + b[\hat{B}, \hat{C}]
  2. Anti-linearity: [\hat{A}, a\hat{B} + b\hat{C}] = a[\hat{A}, \hat{B}] + b[\hat{A}, \hat{C}]
  3. Linearity in Second Argument: [\hat{C}, a\hat{A} + b\hat{B}] = a[\hat{C},\hat{A}] + b[\hat{C},\hat{B}]
  4. Distributivity: [\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}]\hat{C} + \hat{B}[\hat{A}, \hat{C}]

0.14.1.1 Commutator of Position and Momentum Operators

The commutator of position operator \hat{x} and momentum operator \hat{p} is given by

\begin{align} [\hat{x},\hat{p}] &= \hat{x}\hat{p} - \hat{p}\hat{x} \\ &= x\left(-i\hbar\frac{\partial}{\partial x}\right) - \left(-i\hbar\frac{\partial}{\partial x}\right)x \\ &= x\left(-i\hbar\frac{\partial}{\partial x}\right) - \left(-i\hbar\right) - x\left(-i\hbar\frac{\partial}{\partial x}\right) \\ &= i\hbar \end{align}

0.15 Compatibility of Observables

Two observables \hat{A} and \hat{B} are said to be compatible if they commute.

0.16 Robertson Inequality

The Robertson Inequality states that for any two observables \hat{A} and \hat{B}, we have

\Delta \hat{A} \Delta \hat{B} \ge \frac{1}{2} \left|\left<[\hat{A}, \hat{B}]\right>\right|

Proof:

Without loss of generality, we can assume \left<\hat{A}\right> = 0 and \left<\hat{B}\right> = 0. Otherwise we replace \hat{A} with \hat{A} - \left<\hat{A}\right> and \hat{B} with \hat{B} - \left<\hat{B}\right>.

Then, we have

\begin{align} \Delta \hat{A} \Delta \hat{B} &= \sqrt{\left<\hat{A}^2\right>\left<\hat{B}^2\right>} \\ &= \sqrt{\left<\hat{A}\psi|\hat{A}\psi\right>\left<\hat{B}\psi|\hat{B}\psi\right>} \\ &\ge \left|\left<\hat{A}\psi|\hat{B}\psi\right>\right| \quad \text{Cauchy-Schwartz}\\ &= \left|\left<\psi|\hat{A}\hat{B}\psi\right>\right| \\ \end{align}

By similar argument, we have

\Delta \hat{A} \Delta \hat{B} \ge \left|\left<\psi|\hat{B}\hat{A}\psi\right>\right|

Thus,

\begin{align} \Delta \hat{A} \Delta \hat{B} &= \frac{1}{2}\left(\Delta \hat{A} \Delta \hat{B} + \Delta \hat{A} \Delta \hat{B}\right) \\ &\ge \frac{1}{2}\left( \left|\left<\psi|\hat{A}\hat{B}\psi\right>\right| + \left|\left<\psi|\hat{B}\hat{A}\psi\right>\right| \right)\\ &\ge \frac{1}{2}\left| \left<\psi|\hat{A}\hat{B}\psi\right> - \left<\psi|\hat{B}\hat{A}\psi\right> \right| \\ &= \frac{1}{2}\left| \left<\psi|[\hat{A}, \hat{B}]\psi\right> \right| \\ \end{align}

0.16.1 Robertson Inequality for Position and Momentum Operators

Given the position operator \hat{x} and momentum operator \hat{p}, we have

\Delta \hat{x} \Delta \hat{p} \ge \frac{\hbar}{2}

0.17 Quantum Harmonic Oscillator

A quantum harmonic oscillator is a system where the potential energy is given by

V(x) = \frac{1}{2}m\omega^2x^2

0.17.1 Annihilation and Creation Operators

Given the Hamiltonian operator \hat{H} of the quantum harmonic oscillator, we can define the annihilation operator \hat{a} and creation operator \hat{a}^\dagger as

\begin{align} \hat{a} &= u\hat{x} + v\hat{p} i \\ \hat{a}^\dagger &= u\hat{x} - v\hat{p} i \end{align}

Thus, we have

\begin{align} \hat{a}\hat{a}^\dagger &= u^2 \hat{x}^2 - v^2 \hat{p}^2 - uv \hat{x}\hat{p} i + uv \hat{p}\hat{x} i \\ &= u^2 \hat{x}^2 - v^2 \hat{p}^2 + uv\hbar \\ \end{align}

If we take:

\begin{align} u^2 &= \frac{1}{2\hbar}m\omega \\ v^2 &= \frac{1}{2 m\omega\hbar} \\ \end{align}

Then, we have

\begin{align} \hat{a}\hat{a}^\dagger &= \frac{1}{2\hbar}m\omega \hat{x}^2 + \frac{1}{2 m\omega\hbar} \hat{p}^2 + \frac{1}{2} \\ &= \frac{1}{\hbar\omega}\hat{H} + \frac{1}{2} \end{align}

0.17.2 Bosonic Commutation Relation

Given the annihilation operator \hat{a} and creation operator \hat{a}^\dagger of the quantum harmonic oscillator, we have

[\hat{a}, \hat{a}^\dagger] = 2uvh = 1

0.17.3 Number Operator

Given the annihilation operator \hat{a} and creation operator \hat{a}^\dagger of the quantum harmonic oscillator, we can define the number operator \hat{N} as

\hat{N} = \hat{a}^\dagger\hat{a}

Thus, we have

\begin{align} \hat{H} = \hbar\omega\left(\hat{N} + \frac{1}{2}\right) \end{align}

0.17.4 Eigenstates of Number Operator

As the number operator \hat{N} is Hermitian, we can find an orthonormal basis \{\Psi_0(x), \Psi_1(x), \ldots\} such that

\hat{N}\Psi_n(x) = E_n\Psi_n(x)

We next prove that E_n can only be non-negative integers.

Given \Psi_n an eigenvector of \hat{N} with eigenvalue E_n, we have

\begin{align} \hat{N}\hat{a}\Psi_n &= \hat{a}^\dagger\hat{a}^2\Psi_n \\ &= (\hat{a}\hat{a}^\dagger - 1)\hat{a}\Psi_n \\ &= \hat{a}\hat{N} - \hat{a}\Psi_n \\ &= (E_n-1)\hat{a}\Psi_n \\ \end{align}

Thus, if \hat{a}\Psi_n is non-zero, then \hat{a}\Psi_n is also an eigenvector of \hat{N} with eigenvalue E_n-1.

We next prove that \hat{a}\Psi_n = 0 if and only if E_n = 0.

If E_n = 0, then

\begin{align} \left<\hat{a}\Psi_n|\hat{a}\Psi_n\right> &= \left<\Psi_n|\hat{a}^\dagger\hat{a}\Psi_n\right> \\ &= \left<\Psi_n|0\right> \\ &= 0 \end{align}

Thus, \hat{a}\Psi_n = 0.

If \hat{a}\Psi_n = 0, then

\begin{align} \left<\hat{a}\Psi_n|\hat{a}\Psi_n\right> &= \left<\Psi_n|\hat{a}^\dagger\hat{a}\Psi_n\right> \\ &= \left<\Psi_n|\hat{N}\Psi_n\right> \\ &= E_n\left<\Psi_n|\Psi_n\right> \\ &= 0 \end{align}

Thus, E_n = 0.

By proceeding the previous argument, we conclude that \hat{a}^k\Psi_n is an eigenvector of \hat{N} with eigenvalue E_n-k.

If En is not an integer, then k can be any positive integer as the annihilation process can be repeated indefinitely when E_n - k never hit zero. And there exits k such that E_n - k < 0, which is not possible, as \hat{N} = \hat{a}^\dagger\hat{a}, and is positive semi-definite.

Thus, E_n is a non-negative integer.

0.17.5 Ground State of Quantum Harmonic Oscillator

By discussion in the previous section, we can find an normalised eigenstate \Psi_0(x) of the number operator \hat{N} with eigenvalue E_0 = 0.

In this section, we prove that it is unique, up to a complex constant.

By previous discussion, we see that \hat{a}\Psi_0 = 0.

\begin{align} u\hat{x}\Psi_0 + v\hat{p} i \Psi_0 &= 0 \\ u\hat{x}\Psi_0 + v\hbar \frac{d}{dx}\Psi_0 &= 0 \\ \end{align}

As the above equation is a first oder homogeneous ODE, and the solution is unique up to a complex constant.

0.17.6 Creation Operator and Eigenstates of Number Operator

Given the ground state \Psi_0(x) of the quantum harmonic oscillator, and eigenstate \Psi_n(x) of the number operator \hat{N} with eigenvalue E_n = n. Then:

\hat{N}\hat{a}^\dagger\Psi_n = (E_n+1) \hat{a}^\dagger\Psi_n

and

\begin{align} \left<\hat{a}^\dagger\Psi_n|\hat{a}^\dagger\Psi_n\right> &= \left<\Psi_n|\hat{a}\hat{a}^\dagger\Psi_n\right> \\ &= \left<\Psi_n|[\hat{a},\hat{a}^\dagger]+\hat{a}^\dagger\hat{a}\Psi_n\right> \\ &= \left<\Psi_n|\Psi_n\right> + \left<\Psi_n|\hat{N}\Psi_n\right> \\ &= E_n + 1 \end{align}

Thus, we could define the recursive relation:

\Psi_{n+1} = \frac{\hat{a}^\dagger\Psi_n}{\sqrt{E_n+1}}

Repeating the formula, we can get all the eigenstates of the number operator \hat{N}. And could be defined by the ground state \Psi_0(x).

\Psi_n(x) = \frac{(\hat{a}^\dagger)^n\Psi_0(x)}{\sqrt{n!}}

0.17.7 Statistics of Quantum Harmonic Oscillator

As the creation operator \hat{a}^\dagger and annihilation operator \hat{a} are linear combinations of position operator \hat{x} and momentum operator \hat{p}, we can derive the position operator \hat{x} and momentum operator \hat{p} in terms of \hat{a} and \hat{a}^\dagger.

\begin{align} \hat{x} &= \frac{1}{\sqrt{2m\hbar\omega}}(\hat{a} + \hat{a}^\dagger) \\ \hat{p} &= \frac{i}{\sqrt{2m\hbar\omega}}(\hat{a} - \hat{a}^\dagger) \end{align}

Thus, we have

\begin{align} \left<\Psi_n|\hat{x}\Psi_n\right> &= \frac{1}{\sqrt{2m\hbar\omega}}\left<\Psi_n|(\hat{a} + \hat{a}^\dagger)\Psi_n\right> = 0 \\ \end{align}

and

\begin{align} \left<\Psi_n|\hat{p}\Psi_n\right> &= \frac{i}{\sqrt{2m\hbar\omega}}\left<\Psi_n|(\hat{a} - \hat{a}^\dagger)\Psi_n\right> = 0 \\ \end{align}

0.18 Constant of Motion and Commutators

Given an operator \hat{A}, which is probably time dependent, then

\begin{align} \frac{d}{dt}\left<A\right> &= \frac{d}{dt}\left<\Psi|\hat{A}\Psi\right> \\ &= \frac{d}{dt} \int_{-\infty}^{\infty} \Psi^*\hat{A}\Psi dx \\ &= \int_{-\infty}^{\infty} \left(\frac{\partial \Psi^*}{\partial t}\hat{A}\Psi + \Psi^*\frac{\partial \hat{A}}{\partial t}\Psi + \Psi^*\hat{A}\frac{\partial \Psi}{\partial t}\right) dx \\ &= \left<\Psi|\frac{\partial \hat{A}}{\partial t}\Psi\right> +\int_{-\infty}^{\infty} \left(\frac{\partial \Psi^*}{\partial t}\hat{A}\Psi + \Psi^*\hat{A}\frac{\partial \Psi}{\partial t}\right) dx \\ &= \left<\Psi|\frac{\partial \hat{A}}{\partial t}\Psi\right> + \left<\frac{\partial \Psi}{\partial t}|\hat{A}\Psi\right> + \left<\Psi|\hat{A}\frac{\partial \Psi}{\partial t}\right> \\ &= \left<\Psi|\frac{\partial \hat{A}}{\partial t}\Psi\right> + \left<\left(\frac{1}{i\hbar}\hat{H}\Psi\right)|\hat{A}\Psi\right> + \left<\Psi|\hat{A}\left(\frac{1}{i\hbar}\hat{H}\Psi\right)\right> \\ &= \left<\Psi|\frac{\partial \hat{A}}{\partial t}\Psi\right> + \frac{1}{i\hbar}\left[ - \left<\Psi|\hat{H}\hat{A}\Psi\right> + \left<\Psi|\hat{A}\hat{H}\Psi\right> \right] \\ &= \left<\Psi|\frac{\partial \hat{A}}{\partial t}\Psi\right> + \frac{1}{i\hbar} \left<[\hat{A},\hat{H}]\right> \end{align}]]>https://alex1222.com/posts/2024-05-27-Quantum-Mechanics-Note.mdhttps://alex1222.com/posts/2024-05-27-Quantum-Mechanics-Note.mdMon, 27 May 2024 00:00:00 GMT<![CDATA[Probability Note]]>

0.1 Contents

0.2 Random Variables and Probability Distributions

0.2.1 Random Variables

A random variable is a function that maps the outcomes \Omega of a random process to numerical values \mathbb{R}. If the \Omega is discrete, the random variable is called a discrete random variable. If the \Omega is continuous, the random variable is called a continuous random variable.

0.2.2 Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable X is defined as

F(x) = P(X \leq x)

0.2.3 Probability Mass Function (PMF) for Discrete Random Variables

The probability mass function (PMF) of a discrete random variable X is defined as

p(x) = P(X = x)

0.2.4 Probability Density Function (PDF) for Continuous Random Variables

The probability density function (PDF) of a continuous random variable X is defined as

f(x) = \frac{dF(x)}{dx}

where F(x) is the CDF of X.

0.2.5 Expectation and Variance of Random Variables

The expectation of a random variable X is defined as

E[X] = \sum_{x} x p(x) \quad \text{for discrete random variables}

where p(x) is the PMF of X.

E[X] = \int_{-\infty}^{\infty} x f(x) dx \quad \text{for continuous random variables}

where f(x) is the PDF of X.

The variance of a random variable X is defined as

\text{Var}(X) = E[(X - E[X])^2]

By the definition of variance, the variance is always non-negative.

Alternatively, the variance can be calculated as

\text{Var}(X) = E[X^2] - E[X]^2

0.2.5.1 Properties of Expectation and Variance

  1. E[aX + b] = aE[X] + b
  2. E[aX + bY] = aE[X] + bE[Y]
  3. \text{Var}(aX + b) = a^2 \text{Var}(X)
  4. \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)

where a and b are constants, and \text{Cov}(X, Y) is the covariance between X and Y.

0.2.6 Moments of Random Variables

The n-th moment of a random variable X is defined as

E[X^n] = \sum_{x} x^n p(x) \quad \text{for discrete random variables}

0.2.7 Standard Deviation of Random Variables

The standard deviation of a random variable X is defined as

\text{SD}(X) = \sqrt{\text{Var}(X)}

0.2.8 Standard Discrete Distributions

0.2.8.1 Bernoulli Distribution

The Bernoulli distribution is a discrete distribution with two possible outcomes: 0 and 1. The PMF of a Bernoulli random variable X is defined as

p(x) = \begin{cases} p & \text{if } x = 1 \\ 1 - p & \text{if } x = 0 \end{cases}

where p is the probability of success.

The expectation and variance of a Bernoulli random variable X are

E[X] = p \quad \text{and} \quad \text{Var}(X) = p(1 - p)

0.2.8.2 Binomial Distribution

The Binomial distribution is a discrete distribution that models the number of successes in a fixed number of independent Bernoulli trials. The PMF of a Binomial random variable X is defined as

p(x) = \binom{n}{x} p^x (1 - p)^{n - x}

where n is the number of trials, x is the number of successes, and p is the probability of success.

The expectation and variance of a Binomial random variable X are

E[X] = np \quad \text{and} \quad \text{Var}(X) = np(1 - p)

0.2.8.3 Poisson Distribution

The Poisson distribution is a discrete distribution that models the number of events occurring in a fixed interval of time or space. The PMF of a Poisson random variable X is defined as

p(x) = \frac{\lambda^x e^{-\lambda}}{x!}

where \lambda is the average rate of events.

The expectation and variance of a Poisson random variable X are

E[X] = \lambda \quad \text{and} \quad \text{Var}(X) = \lambda

0.2.8.4 Geometric Distribution

The Geometric distribution is a discrete distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials. The PMF of a Geometric random variable X is defined as

p(x) = (1 - p)^{x - 1} p

where x is the number of trials needed to achieve the first success, and p is the probability of success.

The expectation and variance of a Geometric random variable X are

E[X] = \frac{1}{p} \quad \text{and} \quad \text{Var}(X) = \frac{1 - p}{p^2}

0.2.9 Standard Continuous Distributions

0.2.9.1 Uniform Distribution

The Uniform distribution is a continuous distribution with a constant probability density function (PDF) over a fixed interval. The PDF of a Uniform random variable X is defined as

f(x) = \begin{cases} \frac{1}{b - a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}

where a and b are the lower and upper bounds of the interval.

The expectation and variance of a Uniform random variable X are

E[X] = \frac{a + b}{2} \quad \text{and} \quad \text{Var}(X) = \frac{(b - a)^2}{12}

0.2.9.2 Exponential Distribution

The Exponential distribution is a continuous distribution that models the time between events in a Poisson process. The PDF of an Exponential random variable X is defined as

f(x) = \lambda e^{-\lambda x}

where \lambda is the rate parameter.

The expectation and variance of an Exponential random variable X are

E[X] = \frac{1}{\lambda} \quad \text{and} \quad \text{Var}(X) = \frac{1}{\lambda^2}

0.2.9.3 Normal Distribution

The Normal distribution is a continuous distribution that is symmetric and bell-shaped. The PDF of a Normal random variable X is defined as

f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x - \mu)^2}{2\sigma^2}}

where \mu is the mean and \sigma is the standard deviation.

The expectation and variance of a Normal random variable X are

E[X] = \mu \quad \text{and} \quad \text{Var}(X) = \sigma^2

We usually write X \sim N(\mu, \sigma^2) to denote that X follows a Normal distribution with mean \mu and variance \sigma^2.

0.2.9.4 Gamma Distribution

The Gamma distribution is a continuous distribution that generalizes the Exponential distribution. The PDF of a Gamma random variable X is defined as

f(x) = \frac{\lambda^k x^{k - 1} e^{-\lambda x}}{\Gamma(k)}

where \lambda is the rate parameter, k is the shape parameter, and \Gamma(k) is the gamma function.

\Gamma(k) = \int_{0}^{\infty} x^{k - 1} e^{-x} dx

The expectation and variance of a Gamma random variable X are

E[X] = \frac{k}{\lambda} \quad \text{and} \quad \text{Var}(X) = \frac{k}{\lambda^2}

0.2.9.5 Beta Distribution

The Beta distribution is a continuous distribution that is defined on the interval [0, 1]. The PDF of a Beta random variable X is defined as

f(x) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}

where \alpha and \beta are the shape parameters, and B(\alpha, \beta) is the beta function.

B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}

The expectation and variance of a Beta random variable X are

E[X] = \frac{\alpha}{\alpha + \beta} \quad \text{and} \quad \text{Var}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}

0.3 Discrete Bivariate Distributions

A bivariate distribution is a probability distribution that describes the joint behaviour of two random variables.

0.3.1 Joint Probability Mass Function (PMF)

Given two random variables X and Y, the joint probability mass function (PMF) for discrete random variables is defined as

p(x, y) = P(X = x, Y = y)

0.3.2 Marginal Probability Mass Function (PMF)

The marginal probability mass function (PMF) of a random variable X is defined as

p_X(x) = \sum_{y} p(x, y)

0.3.3 Conditional Probability Mass Function (PMF)

The conditional probability mass function (PMF) of a random variable X given Y = y is defined as

p_{X|Y}(x|y) = \frac{p(x, y)}{p_Y(y)}

0.3.4 Expectation and Variance of Bivariate Distributions

The expectation of a bivariate distribution is defined as

E[g(X, Y)] = \sum_{x} \sum_{y} g(x, y) p(x, y)

The covariance of two random variables X and Y is defined as

\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]

Covariance can sometimes be negative, zero, or positive.

And can be calculated as

\text{Cov}(X, Y) = E[XY] - E[X]E[Y]

The correlation coefficient of two random variables X and Y is defined as

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \text{Var}(Y)}}

We can prove that -1 \leq \rho(X, Y) \leq 1.

0.3.5 Independent Random Variables

Two random variables X and Y are independent if and only if

p(x, y) = p_X(x) p_Y(y)

for all x and y.

If X and Y are independent, then

E[XY] = E[X]E[Y] \quad \text{and} \quad \text{Cov}(X, Y) = 0

0.3.6 Uncorrelated Random Variables

Two random variables X and Y are uncorrelated if and only if

\text{Cov}(X, Y) = 0

If X and Y are uncorrelated, then

E[XY] = E[X]E[Y]

Note: Uncorrelated random variables are not necessarily independent.

0.4 Continuous Bivariate Distributions

0.4.1 Joint Cumulative Distribution Function (CDF)

Given two random variables X and Y, the joint cumulative distribution function (CDF) for continuous random variables is defined as

F_{X,Y}(x, y) = P(X \leq x, Y \leq y)

0.4.2 Marginal Cumulative Distribution Function (CDF)

The marginal cumulative distribution function (CDF) of a random variable X is defined as

F_X(x) = P(X \leq x) = P(X \leq x, Y < \infty) = F_{X,Y}(x, \infty)

0.4.3 Joint Probability Density Function (PDF)

Given two random variables X and Y, if there exists a function f(x, y) such that

P((X, Y) \in A) = \iint_{A} f(x, y) dx dy

for all Lebesgue-measurable sets A, then f(x, y) is the joint probability density function (PDF) of X and Y. And X and Y are called jointly continuous random variables.

By the definition of the joint PDF, we have

F_{X,Y}(x, y) = \int_{-\infty}^{x} \int_{-\infty}^{y} f(u, v) du dv

And

f(x, y) = \frac{\partial^2 F_{X,Y}(x, y)}{\partial x \partial y}

0.4.4 Marginal Probability Density Function (PDF)

The marginal probability density function (PDF) of a random variable X is defined as

f_X(x) = \frac{dF_X(x)}{dx} = \int_{-\infty}^{\infty} f(x, y) dy

0.5 Continuous Multivariate Distributions

0.5.1 Joint Cumulative Distribution Function (CDF)

Given n random variables X_1, X_2, \ldots, X_n, let the vector \mathbf{X} = (X_1, X_2, \ldots, X_n), the joint cumulative distribution function (CDF) for continuous random variables is defined as

F_\mathbf{X}(\mathbf{x}) = P(\mathbf{X} < \mathbf{x})

0.5.2 Joint Probability Density Function (PDF)

Given n random variables X_1, X_2, \ldots, X_n, let the vector \mathbf{X} = (X_1, X_2, \ldots, X_n), if there exists a function f(x_1, x_2, \ldots, x_n) such that

P(\mathbf{X} \in A) = \int_{A} f(\mathbf{x}) d\mathbf{X}

for all Lebesgue-measurable sets A, then f(\mathbf{x}) is the joint probability density function (PDF) of \mathbf{X}. And \mathbf{X} are called jointly continuous random variables.

By the definition of the joint PDF, we have

F_\mathbf{X}(\mathbf{x}) = \int_{-\infty}^{x_1} \int_{-\infty}^{x_2} \ldots \int_{-\infty}^{x_n} f(u_1, u_2, \ldots, u_n) du_1 du_2 \ldots du_n

And

f(\mathbf{x}) = \frac{\partial^n F_\mathbf{X}(\mathbf{x})}{\partial x_1 \partial x_2 \ldots \partial x_n}

0.5.3 Marginal Probability Density Function (PDF)

The marginal probability density function (PDF) of a random variable X_i is defined as

F_{X_{k_1}, X_{k_2}, \ldots, X_{k_m}}(x_{k_1}, x_{k_2}, \ldots, x_{k_m}) = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} F(x_1, x_2, \ldots, x_n) \prod_{j \neq k_i}dx_j

0.6 Independence of Random Variables

Two random variables X and Y are independent if and only if

F(x, y) = F_X(x) F_Y(y)

for all x and y.

This can be thought as the joint behaviour of X and Y is the product of the marginal behaviour of X and Y.

The definition can also be formulated in terms of the joint PDF:

f(x, y) = f_X(x) f_Y(y)

To show that two random variables are not independent, we only need to find one pair of x and y such that the equation does not hold.

0.6.1 Functions of Independent Random Variables

Given two independent random variables X and Y, and a function g(X), and a function h(Y), the random variables Z = g(X) and W = h(Y) are also independent.

0.6.2 Mutual Independence of Random Variables

A set of random variables X_1, X_2, \ldots, X_n are mutually independent if and only if

F(x_1, x_2, \ldots, x_n) = F_{X_1}(x_1) F_{X_2}(x_2) \ldots F_{X_n}(x_n)

Note: Mutual independence implies pairwise independence, however, the converse is not true.

0.6.3 Identically Independent Random Variables (IID)

A set of random variables X_1, X_2, \ldots, X_n are identically independent (IID) if and only if

  1. They are mutually independent.
  2. They have the same distribution.

0.6.4 Sum of Random Variables

Given two independent random variables X and Y, the sum of X and Y is defined as

Z = X + Y

Then, the CDF of Z can be calculated as

\begin{align} F_Z(z) &= P(Z \leq z) \\ &= P(X + Y \leq z) \\ &= \int_{-\infty}^{\infty} P(X + Y \leq z | X = x) f_X(x) dx \\ &= \int_{-\infty}^{\infty} P(Y \leq z - x) f_X(x) dx \\ &= \int_{-\infty}^{\infty} F_Y(z - x) f_X(x) dx \end{align}

The PDF of Z can be calculated as

f_Z(z) = \int_{-\infty}^{\infty} f_Y(z - x) f_X(x) dx

This is called the convolution of the PDFs of X and Y.

0.7 Expectation, Covariance and Correlation of Multiple Random Variables

0.7.1 Expectation of Multiple Random Variables

Let the vector \mathbf{X} = (X_1, X_2, \ldots, X_n) be a set of random variables, and for Lebesgue-measurable functions g: \mathbb{R}^n \rightarrow \mathbb{R} the expectation of \mathbf{X} is defined as

E[g(\mathbf{X})] = \int_{\mathbb{R}^n} g(\mathbf{x}) f(\mathbf{x}) d\mathbf{X}

0.7.2 Properties of Expectation of Multiple Random Variables

  1. E[a g(\mathbf{X}) + bh(\mathbf{X}) + c] = a E[g(\mathbf{X})] + b E[h(\mathbf{X})] + c
  2. If X and Y are independent, then E[g(X)h(Y)] = E[g(X)]E[h(Y)] for any functions g and h.

0.7.3 Covariance of Multiple Random Variables

The covariance of two random variables X and Y is defined as

\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]

The covariance can be calculated as

\text{Cov}(X, Y) = E[XY] - E[X]E[Y]

0.7.3.1 Properties of Covariance of Multiple Random Variables

  1. \text{Cov}(X, Y) = \text{Cov}(Y, X)
  2. \text{Cov}(X, X) = \text{Var}(X)
  3. \text{Cov}(aX + b, cY + d) = ac \text{Cov}(X, Y)
  4. \text{Cov}(\sum a_iX_{i}, \sum b_iY_{i}) = \sum a_i b_j \text{Cov}(X_i,Y_j)
  5. \text{Cov}(X,Y) = E(XY) - E(X)E(Y)

0.7.3.2 Cauchy-Schwarz Inequality In Terms of Covariance

|\text{Cov}(X, Y)| \leq \sqrt{\text{Var}(X) \text{Var}(Y)}

This equality holds if and only if X and Y are linearly related.

0.7.3.3 Correlation Coefficient of Multiple Random Variables

The correlation coefficient of two random variables X and Y is defined as

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \text{Var}(Y)}}

By Cauchy-Schwarz Inequality, we have -1 \leq \rho(X, Y) \leq 1.

0.7.3.4 Moments of Multiple Random Variables

The n-th (raw) moment of a random variable X is defined as E[X^n], and the n-th central moment of a random variable X is defined as E[(X - E[X])^n].

The joint (raw) moment of random variables X and Y is defined as E[X^iY^j], and the joint central moment of random variables X and Y is defined as E[(X - E[X])^i(Y - E[Y])^j].

0.7.3.5 Expectation and Variance of Multiple Random Variables

Given a set of random variables \mathbf{X} = (X_1, X_2, \ldots, X_n)^T,

E[\mathbf{X}] = (E[X_1], E[X_2], \ldots, E[X_n])^T

which is a vector of n\times1 dimensions vector.

The covariance matrix of \mathbf{X} is defined as

\text{Cov}(\mathbf{X}) = E[(\mathbf{X} - E[\mathbf{X}])(\mathbf{X} - E[\mathbf{X}])^T]

which is a n\times n matrix.

0.8 Conditional Distribution and Expectation

0.8.1 Conditional Distribution

0.8.1.1 Conditional Probability Mass Function (PMF)

Given two random variables X and Y, the conditional probability mass function (PMF) of X given Y = y is defined as

p_{X|Y}(x|y) = \frac{p(x, y)}{p_Y(y)}

0.8.1.2 Conditional Commutative Distribution Function (CDF) for Discrete Random Variables

Given two random variables X and Y, the conditional commutative distribution function (CDF) of X given Y = y is defined as

F_{X|Y}(x|y) = P(X \leq x | Y = y)

0.8.1.3 Conditional Probability Density Function (PDF) for Continuous Random Variables

Given two random variables X and Y, the conditional probability density function (PDF) of X given Y = y is defined as

f_{X|Y}(x|y) = \frac{f(x, y)}{f_Y(y)}

0.8.1.4 Conditional Commutative Distribution Function (CDF) for Continuous Random Variables

Given two random variables X and Y, the conditional commutative distribution function (CDF) of X given Y = y is defined as

F_{X|Y}(x|y) = P(X \leq x | Y = y) = \int_{-\infty}^{x} f_{X|Y}(u|y) du

0.8.2 Conditional Expectation

Given two random variables X and Y, the conditional expectation of X given Y = y is defined as

E[X|Y = y] = \sum_{x} x p_{X|Y}(x|y) \quad \text{for discrete random variables} E[X|Y = y] = \int_{-\infty}^{\infty} x f_{X|Y}(x|y) dx \quad \text{for continuous random variables}

We can also define function of Y as

\psi(y) = E[X|Y = y]

This is a random variable, and we call this the conditional expectation of X given Y.

0.8.2.1 The Law of Iterated Expectations (The Tower Law)

Given two random variables X and Y, the law of iterated expectations states that

E[E[X|Y]] = E[X]

Proof:

\begin{align} E[E[X|Y]] &= \int_{-\infty}^{\infty} E[X|Y = y] f_Y(y) dy \\ &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} x f_{X|Y}(x|y) dx \right) f_Y(y) dy \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x f(x, y) dx dy \\ &= E[X] \end{align}

0.8.2.2 Law of Total Probability

Given a random variable X and an event A, the law of total probability states that

P(A) = \int_{-\infty}^{\infty} P(A|X = x) f_X(x) dx

0.8.2.3 Wald's Equation

Given a random variable X and a stopping time N which is an integer valued random variable, then

E[\sum_{i=1}^{N}X] = E[X]E[N]

0.8.2.4 Properties of Conditional Expectation

  1. E[aX + bY + c|Z] = aE[X|Z] + bE[Y|Z] + c
  2. If Y > 0, then E[X|Y] > 0.
  3. If X and Y are independent, then E[X|Y] = E[X].
  4. For any function g and h, E[g(X)h(Y)|Y] = h(Y)E[g(X)|Y].

0.8.3 Conditional Variance for Multiple Random Variables

Given two random variables X and Y, the conditional variance of X given Y is defined as

\text{Var}(X|Y) = E[(X - E[X|Y])^2|Y]

The conditional variance can be calculated as

\text{Var}(X|Y) = E[X^2|Y] - E[X|Y]^2

Note that the conditional variance is a random variable of Y.

0.8.3.1 Law of Total Variance

Given a random variable X and an event A, the law of total variance states that

\text{Var}(X) = E[\text{Var}(X|Y)] + \text{Var}(E[X|Y])

0.9 Transformations of Random Variables

0.9.1 Support of Probability Density Function (PDF)

Given a random variable X with a PDF f_X(x), the support of f_X(x) is the set of values of x where f_X(x) > 0.

0.9.2 Monotonic Transformations

Given a random variable X with a PDF f_X(x), and a function Y = g(X), if g is a monotonic function, then the CDF of Y is

F_Y(y) = \begin{cases} F_X(g^{-1}(y)) & \text{if } g \text{ is increasing} \\ 1 - F_X(g^{-1}(y)) & \text{if } g \text{ is decreasing} \end{cases}

Then, the PDF of Y is

f_Y(y) = f_X(g^{-1}(y)) \left| \frac{d}{dy} g^{-1}(y) \right|

For non-monotonic transformations, we can break the transformation into monotonic parts.

0.9.3 Transformation of Bivariate Random Variables

Given random variables X_1 and X_2 with a joint PDF f_{X_1, X_2}(x_1, x_2), and functions (Y_1, Y_2) = T(X_1, X_2), where T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 is a one-to-one transformation, and let H = T^{-1}.

We define J_{H}, which is the Jacobian determinate of H as

J_{H} = \left| \frac{\partial (H_1, H_2)}{\partial (x_1, x_2)} \right| = \det\begin{bmatrix} \frac{\partial H_1}{\partial x_1} & \frac{\partial H_1}{\partial x_2} \\ \frac{\partial H_2}{\partial x_1} & \frac{\partial H_2}{\partial x_2} \end{bmatrix}

Then, the joint PDF of (Y_1, Y_2) is

f_{Y_1, Y_2}(y_1, y_2) = f_{X_1, X_2}(H_1(y_1, y_2), H_2(y_1, y_2)) |J_{H}|

Note: The Jacobian determinate satisfy: J_{H} = J_{H^{-1}}^{-1} = J_T^{-1}.

0.9.4 Transformation of Multivariate Random Variables

The theorem from Transformation of Bivariate Random Variables can be generalized to multiple random variables.

Given random variables X_1, X_2, \ldots, X_n with a joint PDF f_{X_1, X_2, \ldots, X_n}(x_1, x_2, \ldots, x_n), and functions (Y_1, Y_2, \ldots, Y_n) = T(X_1, X_2, \ldots, X_n), where T: \mathbb{R}^n \rightarrow \mathbb{R}^n is a one-to-one transformation, and let H = T^{-1}.

We define J_{H}, which is the Jacobian determinate of H as

J_{H} = \left| \frac{\partial (H_1, H_2, \ldots, H_n)}{\partial (x_1, x_2, \ldots, x_n)} \right| = \det\begin{bmatrix} \frac{\partial H_1}{\partial x_1} & \frac{\partial H_1}{\partial x_2} & \ldots & \frac{\partial H_1}{\partial x_n} \\ \frac{\partial H_2}{\partial x_1} & \frac{\partial H_2}{\partial x_2} & \ldots & \frac{\partial H_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial H_n}{\partial x_1} & \frac{\partial H_n}{\partial x_2} & \ldots & \frac{\partial H_n}{\partial x_n} \end{bmatrix}

Then, the joint PDF of (Y_1, Y_2, \ldots, Y_n) is

f_{Y_1, Y_2, \ldots, Y_n}(y_1, y_2, \ldots, y_n) = f_{X_1, X_2, \ldots, X_n}(H_1(y_1, y_2, \ldots, y_n), H_2(y_1, y_2, \ldots, y_n), \ldots, H_n(y_1, y_2, \ldots, y_n)) |J_{H}|

0.10 Generating Functions

0.10.1 The Moment Generating Function (MGF)

Given a random variable X, the moment generating function (MGF) M_X(t) of X is defined as

M_X(t) = E[e^{tX}]

The domain of the MGF is the set of t such that M_X(t) exists and is finite.

If the domain does not contain an open neighbourhood of 0, then we say the MGF does not exist.

0.10.1.1 Example: The Moment Generating Function of the Standard Normal Distribution

Given a random variable X that follows the standard normal distribution,

f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}

By definition,

\begin{align} M_X(t) &= E[e^{tX}] \\ &= \int_{-\infty}^{\infty} e^{tx} f_X(x) dx \\ &= \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{tx-\frac{x^2}{2}} dx \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{tx-\frac{x^2}{2}} dx \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-\frac{1}{2}(x^2 - 2tx)} dx \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-\frac{1}{2}(x - t)^2 + \frac{t^2}{2}} dx \\ &= e^{\frac{t^2}{2}} \end{align}

0.10.1.2 Example: The Moment Generating Function of the Exponential Distribution

Given a random variable X that follows the exponential distribution,

f_X(x) = \lambda e^{-\lambda x}

By definition,

\begin{align} M_{X}(t) &= E[e^{tX}] \\ &= \int_{0}^{\infty} e^{tx} \lambda e^{-\lambda x} dx \\ &= \lambda \int_{0}^{\infty} e^{tx-\lambda x} dx \\ &= \frac{\lambda}{t-\lambda} \int_{0}^{\infty} (t-\lambda) e^{(t-\lambda)x} dx \\ &= \frac{\lambda}{t-\lambda} \end{align}

0.10.1.3 Properties of the Moment Generating Function

M_X(0) = E[1] = 1

The n-th derivative of the MGF at t = 0 is:

M_X^{(n)}(0) = E[X^{n}e^{tX}](0) = E[X^{n}]

By the previous property, the Maclaurin series of the MGF is:

M_X(t) = \sum_{n=0}^{\infty} \frac{E[X^{n}]}{n!} t^{n}

Also, if X have MGF M_X(t) and Y = aX + b, then Y have MGF M_Y(t) = e^{tb}M_X(at).

0.10.1.4 Uniqueness of the Moment Generating Function

Given two random variables X and Y with MGF M_X(t) and M_Y(t), if M_X(t) = M_Y(t) for all t in an open neighbourhood of 0, then X and Y have the same distribution.

0.10.2 Joint Moment Generating Function (JMGF)

The joint moment generating function (JMGF) of random variables X_1, X_2, \ldots, X_n is defined as a function from \mathbb{R}^n to \mathbb{R}:

M_\mathbf{X}(\mathbf{t}) = E[e^{\mathbf{t}^T \mathbf{X}}]

where \mathbf{t} = (t_1, t_2, \ldots, t_n)^T and \mathbf{X} = (X_1, X_2, \ldots, X_n)^T.

If the JMGF exists and is finite on a open neighbourhood of \mathbf{0}, then we say the JMGF exists.

0.10.2.1 Properties of the Joint Moment Generating Function

If the JMGF exists and is finite on a open neighbourhood of \mathbf{0}, then it uniquely determines the joint distribution of X_1, X_2, \ldots, X_n.

The MGF of X_i can be expressed as:

M_{X_i}(t_i) = M_{\mathbf{X}}(0, \ldots, 0, t_i, 0, \ldots, 0)

The joint moment of X_1, X_2, \ldots, X_n can be expressed as:

E[X_1^{i_1}X_2^{i_2}\ldots X_n^{i_n}] = \left. \frac{\partial^{i_1 + i_2 + \ldots + i_n} M_{\mathbf{X}}(\mathbf{t})}{\partial t_1^{i_1} \partial t_2^{i_2} \ldots \partial t_n^{i_n}} \right|_{\mathbf{t} = \mathbf{0}}

0.10.3 Relation Between the Joint Moment Generating Function and Moment Generating Function

Given random variables X_1, X_2, \ldots, X_n with MGF M_{X_i}(t_i), and JMGF M_{X_1, X_2, \ldots, X_n}(\mathbf{t}), then X_1, X_2, \ldots, X_n are mutually independent if and only if

M_{X_1, X_2, \ldots, X_n}(\mathbf{t}) = M_{X_1}(t_1)M_{X_2}(t_2)\ldots M_{X_n}(t_n)

0.10.4 Sums of Independent Random Variables

Given random variables X_1, X_2, \ldots, X_n that are independent, and S = a_1X_1 + a_2X_2 + \ldots + a_nX_n, then the MGF of S is

M_S(t) = M_{X_1}(a_1t)M_{X_2}(a_2t)\ldots M_{X_n}(a_nt)

0.10.5 Probability Generating Function (PGF)

Given a random variable X that takes non-negative integer values, the probability generating function (PGF) G_X(z) of X is defined as

\phi_X(z) = E[z^X] = \sum_{x=0}^{\infty} z^x P(X = x)

0.10.5.1 Properties of the Probability Generating Function

\phi_X(1) = 1

The PMF of X is uniquely determined by \phi_X(z).

The n-th factorial moment of X is

E[X(X-1)\ldots(X-n+1)] = \left. \frac{d^n \phi_X(z)}{dz^n} \right|_{z=1}

Random variables X_1, X_2, \ldots, X_n are mutually independent if and only if the joint PGF \phi_{X_1, X_2, \ldots, X_n}(z_1, z_2, \ldots, z_n) is

\phi_{X_1, X_2, \ldots, X_n}(z_1, z_2, \ldots, z_n) = E[z_1^{X_1}z_2^{X_2}\ldots z_n^{X_n}] =\phi_{X_1}(z_1)\phi_{X_2}(z_2)\ldots \phi_{X_n}(z_n)

The PGF of sum of independent random variables X_1, X_2, \ldots, X_n is

\phi_{X_1 + X_2 + \ldots + X_n}(z) = \phi_{X_1}(z)\phi_{X_2}(z)\ldots \phi_{X_n}(z)

0.10.5.2 Relation of PGF and MGF

Given a random variable X that takes non-negative integer values, and the PGF \phi_X(z) and MGF M_X(t) of X, then

\begin{align} \phi_X(e^t) = M_X(t) \\ M_X(\ln(t)) = \phi_X(t) \end{align}

0.11 Markov and Chebyshev Inequalities

0.11.1 Markov Inequality

Given a non-negative random variable X and a > 0, then

P(X \geq a) \leq \frac{E[X]}{a}

Proof:

\begin{align} P(X \geq c) &= \int_c^{\infty} f_X(x) dx \\ &= \frac{1}{c} \int_c^{\infty} cf_X(x) dx \\ &\le \frac{1}{c} \int_c^{\infty} xf_X(x) dx \\ &\le \frac{1}{c} \int_0^{\infty} xf_X(x) dx &= \frac{E[X]}{c} \end{align}

0.11.2 Chebyshev Inequality

Given a random variable X with mean \mu and variance \sigma^2, and a > 0, then

P(|X - \mu| \geq a) \leq \frac{\sigma^2}{a^2}

Proof:

Define Y = (X - \mu)^2, then Y is a non-negative random variable, and E[Y] = \text{Var}[X] = \sigma^2.

By Markov Inequality,

P(Y \geq a^2) \leq \frac{E[Y]}{a^2} = \frac{\sigma^2}{a^2}

Then,

P(|X - \mu| \geq a) = P((X - \mu)^2 \geq a^2) = P(Y \geq a^2) \leq \frac{\sigma^2}{a^2}

0.12 Multivariate Normal Distribution

We define the higher dimensional normal distribution as an analog of the one dimensional normal distribution.

We say a random vector \mathbf{X} = (X_1, X_2, \ldots, X_n)^T follows a multivariate normal distribution if it can be expressed as

\mathbf{X} = \mathbf{\mu} + \mathbf{A}\mathbf{Z}

where l\le n and \mathbf{\mu} is a vector of means, \mathbf{A} is a n\times l matrix of constants, and \mathbf{Z} is a l\times 1 vector of independent standard normal random variables.

In convention, we write \Sigma = A A^T, and we denote the multivariate normal distribution as

\mathbf{X} \sim N_n(\mathbf{\mu}, \Sigma)

0.12.1 Joint PDF of Multivariate Normal Distribution

If we assume that \Sigma has full rank, We can use multivariate transformation to derive the joint PDF of \mathbf{X}:

\begin{align} f_X(x) &= f_Z(A^{-1}(x-\mu))|\det(A^{-1})| \\ &= |\det(A^{-1})| \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z_i^2} \\ &= |\det(A^{-1})| \frac{1}{\sqrt{2\pi}^n} e^{-\frac{1}{2}Z^TZ} \\ &= |\det(A^{-1})| \frac{1}{\sqrt{2\pi}^n} e^{-\frac{1}{2}(A^{-1}(x-\mu))^TA^{-1}(x-\mu)} \\ &= |\det(A^{-1})| \frac{1}{\sqrt{2\pi}^n} e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)} \\ \end{align}

0.12.2 Joint Moment Generating Function of Multivariate Normal Distribution

If we assume that \Sigma has full rank, the joint moment generating function of \mathbf{X} is

\begin{align} M_{X}(t) &= E[e^{t^TX}] \\ &= \int_{\mathbb{R}^n} e^{t^Tx} f_X(x) dx \\ &= \int_{\mathbb{R}^n} e^{t^Tx} |\det(A^{-1})| \frac{1}{\sqrt{2\pi}^n} e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)} dx \\ &= \frac{|\det(A^{-1})|}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} e^{-\frac{1}{2}\left[ 2t^Tx +x^T\Sigma^{-1}x -x^T\Sigma^{-1}\mu -\mu^T\Sigma^{-1}x +\mu^T\Sigma^{-1}\mu \right]} dx \\ &= \frac{|\det(A^{-1})|}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} e^{-\frac{1}{2}\left[ 2(\Sigma^{T}t)^T \Sigma^{-1}x +x^T\Sigma^{-1}x -2\mu^T\Sigma^{-1}x +\mu^T\Sigma^{-1}\mu \right]} dx \\ &= \frac{|\det(A^{-1})|}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} e^{-\frac{1}{2}\left[ +x^T\Sigma^{-1}x -2(\mu-\Sigma^{T}t)^T\Sigma^{-1}x +\mu^T\Sigma^{-1}\mu \right]} dx \\ &= \frac{|\det(A^{-1})|}{\sqrt{2\pi}^n} e^{-\frac{1}{2}\left[ -(\mu-\Sigma^{T}t)^T\Sigma^{-1}(\mu-\Sigma^{T}t) +\mu^T\Sigma^{-1}\mu \right]} \int_{\mathbb{R}^n} e^{-\frac{1}{2}\left[ (x-\mu-\Sigma^{T}t)^T\Sigma^{-1}(x-\mu-\Sigma^{T}t) \right]} dx \\ &= e^{-\frac{1}{2}\left[ -(\mu-\Sigma^{T}t)^T\Sigma^{-1}(\mu-\Sigma^{T}t) +\mu^T\Sigma^{-1}\mu \right]} \\ &= e^{t^T\mu + \frac{1}{2}t^T\Sigma t} \end{align}

0.12.3 Moments of Multivariate Normal Distribution

By the joint moment generating function of multivariate normal distribution,

E[X_{1}^{k_1}X_{2}^{k_2}\ldots X_{n}^{k_n}] = \left. \frac{\partial^{k_1 + k_2 + \ldots + k_n} M_{X}(\mathbf{t})}{\partial t_1^{k_1} \partial t_2^{k_2} \ldots \partial t_n^{k_n}} \right|_{\mathbf{t} = \mathbf{0}}

Especially, as,

\begin{align} \frac{\partial}{\partial t_i} M_{X}(\mathbf{t}) &= \frac{\partial}{\partial t_i} e^{t^T\mu + \frac{1}{2}t^T\Sigma t} \\ &= \left[ \frac{\partial t}{\partial t_i}^T\mu + \frac{1}{2} \frac{\partial t}{\partial t_i}^T\Sigma t + t^T\Sigma \frac{\partial t}{\partial t_i} \right] M_{X}(\mathbf{t}) &= e_i^T \left[ \mu + \Sigma t \right] M_{X}(\mathbf{t}) \end{align} \begin{align} \frac{\partial^2}{\partial t_j\partial t_i} &= \frac{\partial}{\partial t_j} e_i^T \left[ \mu + \Sigma t \right] M_{X}(\mathbf{t}) \\ &= e_i^T \Sigma e_j^T M_{X}(\mathbf{t}) + e_i^T \left[ \mu + \Sigma t \right] e_j^T \left[ \mu + \Sigma t \right] M_{X}(\mathbf{t}) \\ &= e_i^T \Sigma e_j^T M_{X}(\mathbf{t}) + e_i^T \left[ \mu + \Sigma t \right] \left[ \mu + \Sigma t \right]^T e_j M_{X}(\mathbf{t}) \\ &= e_i^T \Sigma e_j^T M_{X}(\mathbf{t}) + e_i^T \left[ \mu \mu^T + \mu (\Sigma t)^T + \Sigma t \mu^T + \Sigma t (\Sigma t)^T \right] e_j M_{X}(\mathbf{t}) \\ \end{align}

where e_i is the i-th unit vector.

Then, we can calculate the moments of the multivariate normal distribution.

E(X_i) = \frac{\partial}{\partial t_i} M_{X}(0) = e_i^T \left[ \mu \right] M_{X}(0) = \mu_i \begin{align} E(X_iX_j) &= \frac{\partial^2}{\partial t_j\partial t_i} M_{X}(0) \\ &= e_i^T \Sigma e_j^T M_{X}(0) + e_i^T \left[ \mu \mu^T + \mu (\Sigma 0)^T + \Sigma 0 \mu^T + \Sigma 0 (\Sigma 0)^T \right] e_j M_{X}(0) \\ &= e_i^T \Sigma e_j^T M_{X}(0) \\ &= \Sigma_{ij} + \mu_i\mu_j \end{align}

And the covariance:

\text{Cov}(X_i, X_j) = E(X_iX_j) - E(X_i)E(X_j) = \Sigma_{ij}

Thus, the covariance matrix of \mathbf{X} is \Sigma.

0.12.4 Bivariate Normal Distribution

Given n=2 in multivariate normal distribution, we have the bivariate normal distribution.

The joint PDF of bivariate normal distribution is

f_{X_1, X_2}(x_1, x_2) = \frac{|\det A^{-1}|}{2\pi} e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}

where \mu = (\mu_1, \mu_2)^T and \Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{bmatrix}.

By the moments of multivariate normal distribution, \Sigma can also be expressed as

\Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{bmatrix} = \begin{bmatrix} \text{Var}(X_1) & \text{Cov}(X_1, X_2) \\ \text{Cov}(X_1, X_2) & \text{Var}(X_2) \end{bmatrix}

0.12.5 Properties of Multivariate Normal Distribution

0.12.5.1 Affine Transformation of Multivariate Normal Distribution

Given X a multivariate normal distribution, and Y = AX+b, then Y is also a multivariate normal distribution.

0.12.5.2 Marginal Distribution of Multivariate Normal Distribution

Given X a multivariate normal distribution, to get the marginal distribution of Y = (X_{k_1},X_{k_2},\ldots,X_{k_i}), we can let A be a i\times n matrix with 1 at the k_1, k_2, \ldots, k_i-th row, and b be a i\times 1 vector of zeros.

Then Y= AX + b.

0.12.6 Condition of Independence of Multivariate Normal Distribution

Given X a multivariate normal distribution, X_i and X_j are independent if and only if \text{Cov}(X_i, X_j) = 0.

0.12.7 Degenerate Multivariate Normal Distribution

Given X a multivariate normal distribution, if \Sigma is a singular matrix, then X is a degenerate multivariate normal distribution.

Suppose x is a eigenvector of \Sigma with eigenvalue 0, then let Y = x^T X, then the mean of Y is x^T \mu, and the variance of Y is x^T \Sigma x = 0.

Then Y is the distribution of a constant.

0.13 Limiting Behaviors of Sums of Random Variables

In this section, we assume X to be IID random variables. And \mu = E[X], \sigma^2 = \text{Var}(X).

Let S_{n} = \sum_{i=1}^{n}X.

Then,

E[S_{n}] = n\mu \text{Var}(S_{n}) = n\sigma^2 E\left[\frac{S_{n}}{n}\right] = \mu \text{Var}\left(\frac{S_{n}}{n}\right) = \frac{\sigma^2}{n}

0.13.1 Weak Law of Large Numbers

By intuition, as n increases, the sample mean \frac{S_{n}}{n} converges to the population mean \mu. In formal terms, we have the weak law of large numbers:

P\left(\lim_{n\rightarrow\infty} \frac{S_{n}}{n} = \mu\right) = 1

Or equivalently, for all \epsilon > 0,

\lim_{n\rightarrow\infty} P\left(|\frac{S_{n}}{n} - \mu| \geq \epsilon\right) = 0

0.13.2 Central Limit Theorem

0.13.2.1 Limit of Distribution

Given X a random variable with CDF F_X(x), and sequence of random variables X_1, X_2, \ldots with CDF F_{X_n}(x), we say that the sequence of random variables converges in distribution to X if F_{X_n}(x) converge pointwise to F_X(x).

As the MGF uniquely determines the distribution of a random variable, we have the following theorem:

Given X a random variable with MGF M_X(t), and sequence of random variables X_1, X_2, \ldots with MGF M_{X_n}(t), and all of the MGFs exist and are finite on a same open neighbourhood of 0, then X_1, X_2, \ldots converge in distribution to X if and only if M_{X_n}(t) converge pointwise to M_X(t) on the open neighbourhood of 0.

0.13.2.2 Central Limit Theorem

Given X a random variable with mean \mu and variance \sigma^2, and S_{n} = \sum_{i=1}^{n}X, then the distribution of \frac{S_{n} - n\mu}{\sqrt{n}\sigma} converges in distribution to the standard normal distribution.

]]>https://alex1222.com/posts/2024-05-26-Probability-Note.mdhttps://alex1222.com/posts/2024-05-26-Probability-Note.mdSun, 26 May 2024 00:00:00 GMT<![CDATA[Classical Mechanics Notes]]>

0.1 Contents

0.2 Kinematics

0.2.1 Newton's Second Law

Given a particle of mass m and position vector \mathbf{r}, the second law states that the force \mathbf{F} acting on the particle is equal to the time derivative of the momentum \mathbf{p} of the particle:

\mathbf{F} = \frac{d\mathbf{p}}{dt}

If the mass of the particle is constant, then the second law can be written in terms of the acceleration \mathbf{a} of the particle:

\mathbf{F} = m\mathbf{a}

0.2.2 Inertial Frames and Galileo Transformation

Given a frame of reference \mathbf{S} and another frame of reference \mathbf{S}' moving with equation of motion \mathbf{s} = \mathbf{s}(t) with respect to \mathbf{S}, and rotation with respect to \mathbf{S} with orthogonal matrix \mathbf{O}(t), the transformation of the position vector \mathbf{r}' in \mathbf{S}' to \mathbf{r} in \mathbf{S} is given by:

\mathbf{r} = O(t)\mathbf{r}' + \mathbf{s}(t)

And the transformation of the force \mathbf{F}' in \mathbf{S}' to \mathbf{F} in \mathbf{S} is given by:

\mathbf{F} = \mathbf{O}(t)\mathbf{F}'

Assume in \mathbf{S}, the second law holds, then in \mathbf{S}':

m\ddot{\mathbf{r}} = \ddot{\mathbf{O}}\mathbf{r'} + 2\dot{\mathbf{O}}\dot{\mathbf{r}'} + \ddot{\mathbf{r}'}\mathbf{O} + \ddot{\mathbf{s}} = \mathbf{F} \neq \mathbf{O}(t)\mathbf{F}'

However, if \mathbf{O} is a constant orthogonal matrix, and \mathbf{s} is a linear function of time, then the second law holds in \mathbf{S}'. Such frames are called inertial frames. And the corresponding transformation is called Galileo transformation.

0.2.3 Momentum Conservation

Given a system of particles, the total momentum \mathbf{P} of the system is given by:

\mathbf{P} = \sum_i \mathbf{p}_i

If the net external force acting on the system is zero, then the total momentum of the system is conserved:

\frac{d\mathbf{P}}{dt} = \sum_i \frac{d\mathbf{p}_i}{dt} = \sum_i \mathbf{F}_i = \mathbf{F}_{\text{ext}} = 0

0.2.3.1 Momentum Conservation in specific directions

Given a constant vector \mathbf{a}, if the net external force acting on the system is zero in the direction of \mathbf{a}, then the total momentum of the system in the direction of \mathbf{a} is conserved:

\frac{d}{dt}\left(\sum_i \mathbf{p}_i\cdot\mathbf{a}\right) = \sum_i \mathbf{F}_i\cdot\mathbf{a} = \mathbf{F}_{\text{ext}}\cdot\mathbf{a} = 0

0.2.4 Angular Momentum

Given a particle of mass m and position vector \mathbf{r}, the angular momentum \mathbf{L} of the particle with respect to the origin is given by:

\mathbf{L} = \mathbf{r}\times\mathbf{p}

0.2.5 Torque (Moment of Force)

Given a particle of mass m and position vector \mathbf{r}, and a force \mathbf{F} acting on the particle, the torque \mathbf{\tau} of the particle with respect to the origin is given by:

\mathbf{\tau} = \mathbf{r}\times\mathbf{F}

The torque represent the tendency of the force to rotate the particle.

0.2.5.1 Torque and Angular Momentum

Given a particle of mass m and position vector \mathbf{r}, and a force \mathbf{F} acting on the particle, the time derivative of the angular momentum \mathbf{L} of the particle with respect to the origin is given by:

\frac{d\mathbf{L}}{dt} = \mathbf{r}\times\mathbf{F} = \mathbf{\tau}

0.2.6 Angular Momentum Conservation

Given a system of particles, the total angular momentum \mathbf{L} of the system with respect to the origin is given by:

\mathbf{L} = \sum_i \mathbf{L}_i = \sum_i \mathbf{r}_i\times\mathbf{p}_i

If the net external torque acting on the system is zero, then the total angular momentum of the system is conserved:

\frac{d\mathbf{L}}{dt} = \sum_i \frac{d\mathbf{L}_i}{dt} = \sum_i \mathbf{\tau}_i = \mathbf{\tau}_{\text{ext}} = 0

0.2.6.1 Angular momentum conservation in specific directions

Given a constant vector \mathbf{a}, if the net external torque acting on the system is zero in the direction of \mathbf{a}, then the total angular momentum of the system in the direction of \mathbf{a} is conserved:

\frac{d}{dt}\left(\sum_i \mathbf{L}_i\cdot\mathbf{a}\right) = \sum_i \mathbf{\tau}_i\cdot\mathbf{a} = \mathbf{\tau}_{\text{ext}}\cdot\mathbf{a} = 0

0.3 Forces

0.3.1 Newton's Third Law

Given two particles i and j with forces \mathbf{F}_{ij} and \mathbf{F}_{ji} acting on them, the third law states that the forces are equal in magnitude and opposite in direction:

\mathbf{F}_{ij} = -\mathbf{F}_{ji}

0.3.2 Gravity

Given two particles i and j with masses m_i and m_j and position vectors \mathbf{r}_i and \mathbf{r}_j,

the gravitational force \mathbf{F}_{ij} acting on particle i due to particle j is given by:

\mathbf{F}_{ij} = -\frac{Gm_im_j}{|\mathbf{r}_i - \mathbf{r}_j|^3}(\mathbf{r}_i - \mathbf{r}_j)

where G is the gravitational constant.

0.3.2.1 The Gravitational Constant

The gravitational constant G is a fundamental constant in physics. It is defined as the constant of proportionality in Newton's law of universal gravitation:

\mathbf{F} = -\frac{Gm_1m_2}{r^2}\hat{\mathbf{r}}

where \mathbf{F} is the force between two point masses m_1 and m_2 separated by a distance r, and \hat{\mathbf{r}} is the unit vector pointing from m_1 to m_2.

The value of G is approximately 6.67430\times10^{-11}\,\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}.

0.3.2.2 Near Earth's Surface

Given a particle of mass m near the surface of the Earth with acceleration due to gravity g, the gravitational force \mathbf{F}_g acting on the particle is given by:

\mathbf{F}_g = --\frac{Gm_1m_2}{r^2}\hat{\mathbf{r}}

where m_1 is the mass of the Earth, m_2 is the mass of the particle, and r is the distance between the particle and the center of the Earth.

As the particle is near the surface of the Earth, the distance r is approximately the radius of the Earth R. And the mass of the Earth is a constant m_1 = M.

Therefore, the gravitational force \mathbf{F}_g acting on the particle is given by:

\mathbf{F}_g = -\frac{GMm}{R^2}\hat{\mathbf{r}}

where M is the mass of the Earth.

Given the acceleration due to gravity g = \frac{GM}{R^2}, the gravitational force \mathbf{F}_g acting on the particle is given by:

\mathbf{F}_g = -mg\hat{\mathbf{r}}

where g is the acceleration due to gravity. The common value of g is approximately 9.81\,\text{m/s}^2.

0.3.3 Elastic Forces

The forces due to the tendency of a object to restore its original shape are called elastic forces.

0.3.3.1 Hooke's Law

Given a spring with spring constant k and displacement e, the elastic force T acting on the spring is given by:

T = -ke

Hooke's Law

0.3.4 Contact Forces

The forces due to the contact of two objects are called contact forces.

Contact Forces

Where R is the normal force, F is the frictional force.

0.3.4.1 Moving Friction

Given a moving object on a surface with coefficient of kinetic friction \mu_k, and normal force R acting on the object, the frictional force F_k acting on the object is given by:

F_k = -\mu_k R

0.4 Work and Energy

0.4.1 Potential Energy

Given a particle of mass m and position vector \mathbf{r}, the potential energy U of the particle is given by:

U = U(\mathbf{r})

and the force \mathbf{F} acting on the particle is given by:

\mathbf{F} = -\nabla U

Consequently, the potential energy U can be expressed in terms of the force \mathbf{F} acting on the particle:

U = -\int \mathbf{F}\cdot d\mathbf{r}

0.4.2 Energy Conservation

Given a particle of mass m and position vector \mathbf{r}, the total energy E of the particle is given by:

E = T + U

where T is the kinetic energy of the particle, and U is the potential energy of the particle.

If the net external force acting on the particle is zero, then the total energy of the particle is conserved:

\begin{align} \frac{dE}{dt} &= \frac{dT}{dt} + \frac{dU}{dt} \\ &= \frac{d}{dt}\left(\frac{1}{2}mv\cdot v\right) -\nabla U \frac{d\mathbf{r}}{dt} \\ &= \frac{d}{dt}\left(\frac{1}{2}mv^2\right) -\nabla U \cdot \mathbf{v} \\ &= m\mathbf{v}\cdot\mathbf{a} - \mathbf{F}\cdot\mathbf{v} \\ &= \mathbf{F}\cdot\mathbf{v} - \mathbf{F}\cdot\mathbf{v} \\ &= 0 \end{align}

0.4.2.1 Conservative Forces

Given a force \mathbf{F} acting on a particle, if the force is conservative, then the force can be expressed as the gradient of a scalar function:

\mathbf{F} = -\nabla U

where U is the potential energy of the particle.

For conservative forces, on a simply connected domain (the domain is path connected and any closed curve can be shrunk to a point without leaving the domain), the work done by the force \mathbf{F} on the particle is path independent:

\int_{\mathbf{r}_1}^{\mathbf{r}_2} \mathbf{F}\cdot d\mathbf{r} = U(\mathbf{r}_1) - U(\mathbf{r}_2)
0.4.2.1.1 Example: Non-conservative Forces

Given a force \mathbf{F} acting on a particle, where the force can be expressed as:

\mathbf{F}(x,y,z) = (0,0,x)

by solving the PDE, there is no potential energy U such that:

\mathbf{F} = -\nabla U

Thus, the force \mathbf{F} is non-conservative.

0.4.2.1.2 Essential Conditions for Conservative Forces

Given a force \mathbf{F} acting on a particle, the force is conservative if the following conditions are satisfied:

  1. The force \mathbf{F} is a function of the position vector \mathbf{r} only.

  2. The force \mathbf{F} is irrotational, that is, the curl of the force is zero:

    \nabla\times\mathbf{F} = 0

0.4.2.2 Qualitative Energy Analysis

Energy Analysis

At point x_{1}, the particle has speed \dot{x} = 0 and potential energy V'(x) < 0. Thus, the particle has to move to the right.

At point x_{2}, the particle has speed \dot{x} = 0 and potential energy V'(x) > 0. Thus, the particle has to move to the left.

If the initial position of the particle is inside [x_{1},x_{2}], then the particle will oscillate between x_{1} and x_{2}. As it can not go beyond x_{1} and x_{2}.

As when x \rightarrow \infty, the potential energy V(x) \rightarrow 0, the speed of the particle \dot{x} \rightarrow \sqrt{\frac{2E}{m}}.

0.4.3 Work

Given a particle of mass m and position vector \mathbf{r}, the work W done by a force \mathbf{F} on the particle along a path C is given by:

W = \int_C \mathbf{F}\cdot d\mathbf{r}

0.4.3.1 Work and Kinetic Energy

Given a particle of mass m and position vector \mathbf{r}, the work W done by a force \mathbf{F} on the particle is equal to the change in kinetic energy T of the particle:

T(t_{1}) - T(t_{0}) = W = \int_{t_{0}}^{t_{1}} \mathbf{F}\cdot\mathbf{v} dt

0.5 Lagrangian Mechanics

If we are working with some coordinate that are not Cartesian (e.g. Polar Coordinate), we can use the Lagrangian mechanics to describe the motion of the system.

0.5.1 Lagrangian

Given a system of particles with generalized coordinates q_i, as the system evolves in time, which means the generalized coordinates q_i are functions of time t, we are also given \dot{q}_{i}.

The Lagrangian L of the system is given by:

L = T - U

where T is the kinetic energy of the system, and U is the potential energy of the system, and both are functions of the generalized coordinates q_i and their time derivatives \dot{q}_{i} and possibly time t.

0.5.2 Euler-Lagrange Equation

Given a system of particles with generalized coordinates q_i, the Euler-Lagrange equation is given by:

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right) - \frac{\partial L}{\partial q_{i}} = 0

0.5.3 Action Integral

Given a system of particles with generalized coordinates q_i, the action integral S of the system is given by:

S[q] = \int_{t_{0}}^{t_{1}} L(q,\dot{q},t) dt

0.5.4 Hamilton's Principle

The solution to the Euler-Lagrange equation is the path q(t) that is a stationary point of the action integral S[q].

0.5.5 Geodesic Equation

Given a arbitrary manifold. Given two point p and q on the manifold, and a curve \gamma connecting p and q.

If the length of the curve \gamma is given by the integral:

L[\gamma] = \int_{0}^{1} g(\gamma) dt

for some function g.

And we wish to find the curve \gamma that minimizes the length L[\gamma].

The curve \gamma that minimizes the length L[\gamma] is called the geodesic.

If we define the Lagrangian L as:

L = g(\gamma,\dot{\gamma})

Then the Euler-Lagrange equation is the geodesic equation.

0.5.6 Generalised Coordinates in Lagrangian Mechanics

Given a system of particles with generalized coordinates q_i.

The velocity of the particle is given by:

\mathbf{v} = \sum_i \dot{q}_{i}

The momentum of the particle is given by:

\mathbf{p} = \sum_i \frac{\partial L}{\partial \dot{q}_{i}}

The force acting on the particle is given by:

\mathbf{F} = \sum_i \frac{\partial L}{\partial q_{i}}

0.5.6.1 Cyclical Coordinates

Given a system of particles with generalized coordinates q_i, if the Lagrangian L does not depend on a generalized coordinate q_i, then the generalized coordinate q_i is called a cyclical coordinate.

If the Lagrangian L does not depend on a cyclical coordinate q_i, then the momentum \frac{\partial L}{\partial \dot{q}_{i}} is conserved. Which means \frac{\partial L}{\partial \dot{q}_{i}} is a constant.

0.5.7 Example: Central Force

Given a particle of mass m moving in a plane with polar coordinates (r,\theta), with potential energy U(r) that depends only on the distance r from the origin,

As, the potential is independent of the angle \theta, the force acting on the particle is radial.

The Lagrangian L of the particle is given by:

L = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\theta}^2\right) - U(r)

The Euler-Lagrange equation for the angular coordinate \theta is given by:

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right) - \frac{\partial L}{\partial \theta} &= 0 \\ \frac{d}{dt}\left(mr^2\dot{\theta}\right) &= 0 \\ mr^2 \dot{\theta} = J \end{align}

We usually call J the angular momentum of the particle. And in the central force situation the angular momentum is conserved.

The Euler-Lagrange equation for the radial coordinate r is given by:

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{r}}\right) - \frac{\partial L}{\partial r} &= 0 \\ \frac{d}{dt}\left(m\dot{r}\right) - m\dot{\theta}^2r &= -\frac{dU}{dr} \\ m\ddot{r} - m\dot{\theta}^2r &= -\frac{dU}{dr} \\ m\ddot{r} &= \frac{J^2}{mr^3} -\frac{dU}{dr} \end{align}

The energy of the particle is given by:

E = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\theta}^2\right) + U(r)

And as:

\begin{align} \frac{dE}{dt} &= \frac{d}{dt}\left(\frac{1}{2}m\dot{r}^2 + \frac{1}{2}mr^2\dot{\theta}^2 + U(r)\right) \\ &= \frac{d}{dt}\left(\frac{1}{2}m\dot{r}^2 + \frac{1}{2}\frac{J^2}{mr^2} + U(r)\right) \\ &= m\dot{r}\ddot{r} - \frac{J^2}{mr^3}\dot{r} + \frac{dU}{dr}\dot{r} \\ &= 0 \end{align}

The energy of the particle is conserved. And we usually call E the total energy of the particle. And also the effective potential:

U_{\text{eff}}(r) = U(r) + \frac{J^2}{2mr^2}

If we are give the effective potential U_{\text{eff}}(r), the energy and angular momentum of the particle, then the equation of motion of the particle can be determined by solving two first order ODEs:

\begin{align} E_{0} &= \frac{1}{2}m\dot{r}^2 + U_{\text{eff}}(r) \\ J &= mr^2\dot{\theta} \end{align}

0.5.8 Lagrangian Mechanics with Constraints

0.5.8.1 Holonomic Constraints

Given a system of particles with generalized coordinates q_i, if the generalized coordinates q_i are subject to constraints f(q_1,\dots,q_i,t) = 0, then the constraints are called holonomic constraints.

Any constraints that can not be expressed in the form f(q_1,\dots,q_i,t) = 0 are called non-holonomic constraints.

Example: Object with air resistance is a non-holonomic constraint, as the air resistance is independent of the speed of the object.

0.5.8.1.1 Unforced or Natural Constraints

Given a system of particles with generalized coordinates q_i, if the constraints f_j(q_i,t) = 0 are independent of the time t, then the constraints are called unforced or natural constraints.

Example: A particle moving on a sphere of radius R.

Example: A pendulum with fixed suspension point.

0.5.8.1.2 Forced Constraints

Given a system of particles with generalized coordinates q_i, if the constraints f_j(q_i,t) = 0 are dependent of the time t, then the constraints are called forced constraints.

Example: A particle moving on a sphere of radius R with the sphere rotating.

Example: A pendulum with the suspension point moving.

0.5.9 System of Particles and Rigid Bodies

0.5.9.1 Centre of Mass

Given a system of particles with masses m_i and position vectors \mathbf{r}_i, the centre of mass \mathbf{R} of the system is given by:

\mathbf{R} = \frac{\sum_i m_i\mathbf{r}_i}{\sum_i m_i}

0.5.9.2 Total Momentum

Given a system of particles with masses m_i and position vectors \mathbf{r}_i, the total momentum \mathbf{P} of the system is given by:

\begin{align} \mathbf{P} &= \sum_i \mathbf{p}_i \\ &= \sum_i m_i\mathbf{v}_i \\ &= \sum_i m_i\dot{\mathbf{r}}_i \\ &= \dot{\sum_i m_i{\mathbf{r}}_i} \\ &= \dot{\mathbf{R}} \sum_i m_i \\ \end{align}

0.5.9.3 Total Angular Momentum

Given a system of particles with masses m_i and position vectors \mathbf{r}_i, the total angular momentum \mathbf{L} of the system with respect to the origin is given by:

\begin{align} \mathbf{L} &= \sum_i \mathbf{L}_i \\ &= \mathbf{R} \times \mathbf{P} + \sum_i (\mathbf{r}_i - \mathbf{R})\times (\mathbf{p}_i - \mathbf{P})\\ \end{align}

The term \sum_i (\mathbf{r}_i - \mathbf{R})\times (\mathbf{p}_i - \mathbf{P}) is the angular momentum of the system with respect to the centre of mass.

0.5.9.4 Total Kinetic Energy

Given a system of particles with masses m_i and position vectors \mathbf{r}_i, and the total mass of the system M, the total kinetic energy T of the system is given by:

\begin{align} T &= \sum_i T_i \\ &= \sum_i \frac{1}{2}m_i\mathbf{v}_i\cdot\mathbf{v}_i \\ &= \frac{1}{2} M \dot{\mathbf{R}}^2 + \sum_i \frac{1}{2}m_i(\dot{\mathbf{r}}_i - \dot{\mathbf{R}})^2 \\ \end{align}

The term \sum_i \frac{1}{2}m_i(\dot{\mathbf{r}}_i - \dot{\mathbf{R}})^2 is the kinetic energy of the system with respect to the centre of mass.

0.5.9.5 Separable Potential Energy

Given a system of particles with masses m_i and position vectors \mathbf{r}_i, and the total mass of the system M, the total potential energy U of the system is given by:

\begin{align} U &= \sum_i U_i \\ &= \sum_i U_i(\mathbf{r}_i) \end{align}

If the potential energy U is separable, that is, the potential energy can be expressed as:

U = U_{\mathbf{R}}(\mathbf{R}) + \sum_i U_i(\mathbf{r}_i)

Then the generalised coordinates q_i can be reformulated using the center of mass \mathbf{R} and the relative coordinates \mathbf{r}_i - \mathbf{R}. And the Lagrangian L can be expressed in terms of the center of mass \mathbf{R} and the relative coordinates \mathbf{r}_i - \mathbf{R}.

\begin{align} L_{0} &= T_{0} - U_{0} = \frac{1}{2}M\dot{\mathbf{R}}^2 - U_{\mathbf{R}}(\mathbf{R}) \\ L_{i} &= T_{i} - U_{i} = \frac{1}{2}m_i(\dot{\mathbf{r}}_i - \dot{\mathbf{R}})^2 - U_i(\mathbf{r}_i-\mathbf{R}) \end{align}

Example: A constant gravitational field acting on a system of particles is a separable potential energy. And the total potential energy U of the system is given by: U = U_{\mathbf{R}}(\mathbf{R}), which only depend on the center of mass \mathbf{R}.

0.5.9.5.1 Example: Two Particles

Given two particles 1 and 2 with masses m_1 and m_2 and position vectors \mathbf{r}_1 and \mathbf{r}_2,

Then let:

\begin{align} \mathbf{R} &= \frac{m_1\mathbf{r}_1 + m_2\mathbf{r}_2}{m_1 + m_2} \\ \mathbf{r} &= \mathbf{r}_2 - \mathbf{r}_1 \end{align}

Thus,

\begin{align} \mathbf{r}_1 &= \mathbf{R} - \frac{m_2}{m_1 + m_2}\mathbf{r} \\ \mathbf{r}_2 &= \mathbf{R} + \frac{m_1}{m_1 + m_2}\mathbf{r} \end{align}

The kinetic energy T of the system is given by:

\begin{align} T &= \frac{1}{2}M\dot{\mathbf{R}}^2 + \left[\frac{1}{2}m_{1}(\dot{\mathbf{r}}_{2}-\dot{\mathbf{R}})^2 + \frac{1}{2}m_{2}(\dot{\mathbf{r}}_{2}-\dot{\mathbf{R}})^2 \right] \\ &= \frac{1}{2}M\dot{\mathbf{R}}^2 + \frac{1}{2}\frac{m_1m_2}{m_1+m_2}\dot{\mathbf{r}}^2 \end{align}

And the potential is given by:

U = U_{\mathbf{r}}(\mathbf{r})

Thus, the Lagrangian L of the system is given by:

L = \frac{1}{2}M\dot{\mathbf{R}}^2 + \frac{1}{2}\frac{m_1m_2}{m_1+m_2}\dot{\mathbf{r}}^2 - U_{\mathbf{r}}(\mathbf{r})

Apparently, the Lagrangian is cyclic in \mathbf{R}. Thus, the total momentum \mathbf{P} of the system is conserved.

0.5.9.6 Rigid Bodies

A rigid body is a system of particles with fixed relative distances between the particles.

For rigid body, we use \mathbf{R} to denote the position of the center of mass of the rigid body, and \mathbf{\rho}_i= (\rho_i, \varphi_i) in polar coordinates to denote the position of the i-th particle with respect to the center of mass.

As, the relative distances between the particles are fixed, \rho_i is a constant of time. And also, the angular velocity \dot{\varphi}_i is the same for all particles, in convention, we use \dot{\varphi}_i = \varphi to denote the angular velocity of the rigid body.

0.5.9.6.1 Moment of Inertia

Given a rigid body with mass M and position vector \mathbf{R}, and the i-th particle with mass m_i and position vector \mathbf{\rho}_i, the kinetic energy T of the rigid body is given by:

\begin{align} T &= \frac{1}{2}M\dot{\mathbf{R}}^2 + \sum_i \frac{1}{2}m_i\dot{\mathbf{\rho}}_i^2 \\ &= \frac{1}{2}M\dot{\mathbf{R}}^2 + \sum_i \frac{1}{2}m_i(\rho_i^2\dot{\varphi}^2) \\ &= \frac{1}{2}M\dot{\mathbf{R}}^2 + \frac{1}{2} \dot{\varphi}^2 \sum_i m_i \rho_i^2 \end{align}

The term \sum_i m_i \rho_i^2 is called the moment of inertia I of the rigid body, which represents the resistance of the rigid body to rotation.

0.5.9.6.1.1 Example: The moment of inertia of a homogeneous disk

Given a homogeneous disk of radius R and mass M, the moment of inertia I of the disk is given by:

\begin{align} I &= \int_{0}^{2\pi} \int_{0}^{R} \rho^2 \rho \frac{M}{\pi R^2} d\rho d\varphi \\ &= \frac{M}{\pi R^2} \int_{0}^{2\pi} \int_{0}^{R} \rho^3 d\rho d\varphi \\ &= \frac{M}{\pi R^2} \int_{0}^{2\pi} \frac{1}{4}R^4 d\varphi \\ &= \frac{M}{\pi R^2} 2\pi \frac{1}{4}R^4 \\ &= \frac{1}{2}MR^2 \end{align}
0.5.9.6.2 Example: Rolling Cylinder on an Inclined Plane

Consider the following system:

Rolling Cylinder

Given a cylinder of radius R and mass M rolling on an inclined plane with angle \alpha, the question can be simplified by considering the following system:

Rolling Cylinder Simplified

Where \alpha is again the angle of the inclined plane, \phi is the rolling angle of the cylinder, and R = (x,y) is the position of the center of mass of the cylinder.

As the cylinder is rolling on the plane with no sliding, we can formulate the constraints as:

\begin{align} y &= -R\phi\sin(\alpha) \\ x &= R\phi\cos(\alpha) \end{align}

Thus, the velocity of the center of mass of the cylinder is given by:

\begin{align} \dot{R} &= (\dot{x},\dot{y}) \\ &= R\dot{\phi}(\cos(\alpha),-\sin(\alpha)) \end{align}

And the kinetic energy T of the cylinder is given by:

\begin{align} T &= \frac{1}{2}M\dot{R}^2 + \frac{1}{2}I\dot{\phi}^2 \\ &= \frac{1}{2}M R^2 \dot{\phi}^2 + \frac{1}{2}I\dot{\phi}^2 \\ &= \frac{1}{2}M R^2 \dot{\phi}^2 + \frac{1}{2}I\dot{\phi}^2 \\ &= \frac{1}{2}M R^2 \dot{\phi}^2 + \frac{1}{4}M R^2 \dot{\phi}^2 \\ &= \frac{3}{4} M R^2 \dot{\phi}^2 \end{align}

If we consider a constant gravitational field acting on the cylinder, the potential energy U of the cylinder is given by:

U = Mgy = -MgR\phi\sin(\alpha)

The Lagrangian L of the cylinder is given by:

L = \frac{3}{4} M R^2 \dot{\phi}^2 + MgR\phi\sin(\alpha)

Using the Euler-Lagrange equation, we can determine the equation of motion of the cylinder.

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} &= 0 \\ \frac{d}{dt}\left(\frac{3}{2}M R^2 \dot{\phi}\right) - MgR\sin(\alpha) &= 0 \\ \frac{3}{2}M R^2 \ddot{\phi} - MgR\sin(\alpha) &= 0 \\ \ddot{\phi} &= \frac{2g\sin(\alpha)}{3R} \end{align}

0.6 Small Oscillations

0.6.1 Harmonic Oscillator

Given a object with mass m, spring constant k, and displacement x. The kinetic energy T of the object is given by:

T = \frac{1}{2}m\dot{x}^2

The potential energy U of the object is given by:

U = \frac{1}{2}kx^2

The Lagrangian L of the object is given by:

L = T - U = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2

By solving the Euler-Lagrange equation, we can determine the equation of motion of the object.

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} &= 0 \\ \frac{d}{dt}\left(m\dot{x}\right) + kx &= 0 \\ m\ddot{x} + kx &= 0 \end{align}

Thus,

x = A \cos(\omega t + \phi), \omega = \sqrt{\frac{k}{m}}

0.6.2 Small Oscillations in General

Given a system with generalized coordinates q_i, and potential energy U of the system. And stationary points q_{i0} of potential energy U. Such that

\frac{\partial U}{\partial q_i} = 0

Then, the potential energy U can be expanded as a Taylor series around the stationary points q_{i0}.

\begin{align} U &= U(q_{i0}) + \sum_i \frac{\partial U}{\partial q_i}(q_i - q_{i0}) + \frac{1}{2}\sum_{i,j} \frac{\partial^2 U}{\partial q_i \partial q_j}(q_i - q_{i0})(q_j - q_{j0}) + \cdots \\ &= U(q_{i0}) + \frac{1}{2}\sum_{i,j} \frac{\partial^2 U}{\partial q_i \partial q_j}(q_i - q_{i0})(q_j - q_{j0}) + \cdots \\ &\approx U(q_{i0}) + \frac{1}{2}\sum_{i,j} \frac{\partial^2 U}{\partial q_i \partial q_j}(q_i - q_{i0})(q_j - q_{j0}) \end{align}

We could use symmetric matrix K to represent the second order partial derivatives of the potential energy U.

K_{ij} = \frac{\partial^2 U}{\partial q_i \partial q_j}

Thus, the potential energy U can be expressed as:

U = U(q_{i0}) + \frac{1}{2}(q-q_{0})^T K (q-q_{0})

Also, in general, the kinetic energy T of the system can be expressed using a positive definite symmetric matrix M.

T = \frac{1}{2}\dot{q}^T M \dot{q}

Thus, in general, the Lagrangian L of the system is given by:

L = T - U = \frac{1}{2}\dot{q}^T M \dot{q} - U(q_{i0}) - \frac{1}{2}(q-q_{0})^T K (q-q_{0})

0.6.3 Double Pendulum

Consider a double pendulum like below:

Double Pendulum

Thus, the stationary points of the double pendulum are given by:

\begin{align} \varphi_1 &= 0 \\ \varphi_2 &= 0 \end{align}

If we set the suspension point as the origin (0,0), then the position of the first mass m_1 is given by:

\begin{align} x_1 &= l_1\sin(\varphi_1) \\ y_1 &= -l_1\cos(\varphi_1) \end{align}

And the position of the second mass m_2 is given by:

\begin{align} x_2 &= l_1\sin(\varphi_1) + l_2\sin(\varphi_2) \\ y_2 &= -l_1\cos(\varphi_1) - l_2\cos(\varphi_2) \end{align}

Thus, if we set \varphi_1 and \varphi_2 close to the stationary point, the kinetic energy T of the double pendulum is given by:

\begin{align} T &= \frac{1}{2}m_1\left(\dot{x}_1^2 + \dot{y}_1^2\right) + \frac{1}{2}m_2\left(\dot{x}_2^2 + \dot{y}_2^2\right) \\ &= \frac{1}{2}m_1\left(l_1^2\dot{\varphi}_1^2\right) + \frac{1}{2}m_2\left(l_1^2\dot{\varphi}_1^2 + l_2^2\dot{\varphi}_2^2 + 2l_1l_2\dot{\varphi}_1\dot{\varphi}_2\cos(\varphi_1 - \varphi_2)\right) \\ &\approx \frac{1}{2}m_1l_1^2\dot{\varphi}_1^2 + \frac{1}{2}m_2l_1^2\dot{\varphi}_1^2 + \frac{1}{2}m_2l_2^2\dot{\varphi}_2^2 + m_2l_1l_2\dot{\varphi}_1\dot{\varphi}_2 \end{align}

The potential energy U of the double pendulum is given by:

\begin{align} U &= m_1gy_1 + m_2gy_2 \\ &= -m_1gl_1\cos(\varphi_1) - m_2gl_1\cos(\varphi_1) - m_2gl_2\cos(\varphi_2) \\ &\approx -(m_1gl_1 + m_2gl_1)(1-\frac{1}{2}\varphi_1^2) - m_2gl_2(1-\frac{1}{2}\varphi_2^2) \end{align}

0.7 Normal Modes

As in Small Oscillations in General, we can express the kinetic and potential energy of the system in terms of symmetric matrices M and K:

\begin{align} T &= \frac{1}{2}\dot{q}^T M \dot{q} \\ U &= \frac{1}{2}(q-q_{0})^T K (q-q_{0}) \end{align}

Thus, the general momentum p of the system is given by:

p = \frac{\partial L}{\partial \dot{q}} = M\dot{q}

And the generalised force Q of the system is given by:

Q = \frac{\partial L}{\partial q} = K(q-q_{0})

And the Euler-Lagrange equation is given by:

\begin{align} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} &= 0 \\ \frac{d}{dt}\left(M\dot{q}\right) - K(q-q_{0}) &= 0 \\ M\ddot{q} + K(q-q_{0}) &= 0 \end{align}

As, M is a positive definite symmetric matrix, the inverse of M exists.

Thus, the equation of motion of the system can be expressed as:

\ddot{q} + M^{-1}K(q-q_{0}) = 0

A normal mode of the system is a solution of the above equation of the form:

q(t) = q_0 + c(t)v

Where c(t) is a scalar function of time, and v is a vector that is independent of time.

Substituting the normal mode solution into the equation of motion of the system:

\begin{align} \ddot{q} + M^{-1}K(q-q_{0}) &= 0 \\ \ddot{c}(t)v + c(t)M^{-1}K(v) &= 0 \\ \frac{\ddot{c}}{c(t)} v + M^{-1}K(v) &= 0 \end{align}

Thus, v is an eigenvector of the matrix M^{-1}K with eigenvalue \omega^2, where \omega is the angular frequency of the normal mode. And, c(t) is a solution of the following ODE:

\ddot{c} + \omega^2 c = 0

Thus, the general solution of the normal mode is given by:

q(t) = q_0 + \sum_i c_i v_i \cos(\omega_i t + \phi_i)

0.8 Hamiltonian Mechanics

Given a system of particles with generalized coordinates q_i, and the Lagrangian L of the system, the generalised momentum p_i of the system is given by:

p_i = \frac{\partial L}{\partial \dot{q}_i}

And we can solve this implicit equation to get \dot{q}_i in terms of q_i and p_i.

The Hamiltonian H of the system is given by:

H = p \cdot \dot{q}(q,p,t) - L(q,\dot{q}(q,p,t),t)

0.8.1 Hamilton's Canonical Equations

By the Euler-Lagrange equation, we can derive the following equations:

\begin{align} \frac{dH}{dp} &= \dot{q} + p \cdot \frac{d\dot{q}}{p} - \frac{dL}{d\dot{q}}\cdot \frac{d\dot{q}}{p} \\ &= \dot{q} + p \cdot \frac{d\dot{q}}{p} - p \frac{d\dot{q}}{p} \\ &= \dot{q} \end{align} \begin{align} \frac{dH}{dq} &= \frac{dp}{dq} \cdot \dot{q} + p \cdot \frac{d\dot{q}}{dq} - \frac{dL}{dq} - \frac{dL}{d\dot{q}}\frac{d\dot{q}}{dq}\\ &= \frac{dp}{dq} \cdot \dot{q} + p \cdot \frac{d\dot{q}}{dq} - \dot{p} -p \frac{d\dot{q}}{dq}\\ &= -\dot{p} \end{align}

0.8.2 Quadratic Hamiltonian

Given a system of particles with generalized coordinates q_i, if the potential energy is independent of \dot{q}, and the kinetic energy T of the system is quadratic in the generalized velocities \dot{q}_i, which means there is a positive definite symmetric matrix M = M(q,t) such that:

T = \frac{1}{2}\dot{q}^T M \dot{q}

Then the Hamiltonian H of the system is given by:

\begin{align} H &= p \cdot \dot{q} - L \\ &= \frac{\partial L}{\partial\dot{q}} \cdot \dot{q} - T + U \\ &= \frac{\partial T}{\partial\dot{q}} \cdot \dot{q} - T + U \\ &= (M\dot{q}) \cdot \dot{q} -T + U \\ &= T + U \\ \end{align}

0.8.3 Example: Harmonic Oscillator

Given a object with mass m, spring constant k, and displacement x.

The momentum p of the object is given by:

p = \frac{\partial L}{\partial \dot{x}} = m\dot{x}

Thus,

\dot{x} = \frac{p}{m}

The potential energy U of the object is given by:

U = \frac{1}{2}kx^2

The kinetic energy T of the object is given by:

T = \frac{1}{2}m\dot{x}^2 = \frac{1}{2}m\left(\frac{p}{m}\right)^2 = \frac{p^2}{2m}

As the potential energy U is independent of \dot{x}, and the kinetic energy T is quadratic in \dot{x}, the Hamiltonian H of the object is given by:

H = T + U = \frac{p^2}{2m} + \frac{1}{2}kx^2

By Hamilton's canonical equations, we can determine the equation of motion of the object.

\begin{align} \frac{\partial H}{\partial p} &= \dot{x} = \frac{p}{m} \\ \frac{\partial H}{\partial x} &= -\dot{p} \\ kx &= -\dot{p} \\ m\ddot{x} &= -kx \end{align}

0.8.4 Cyclical Coordinates in Hamiltonian Mechanics

Given a system of particles with generalized coordinates q_i.

If the Hamiltonian H does not depend on a generalized coordinate q_i, then the generalized coordinate q_i is called a cyclical coordinate.

Thus,

\dot{p} = -\frac{\partial H}{\partial q_i} = 0

The momentum p_i is conserved.

If the Hamiltonian H does not depend on a momentum p_i, then the momentum p_i is called a cyclical momentum.

Thus,

\dot{q} = \frac{\partial H}{\partial p_i} = 0

The generalized coordinate q_i is conserved.

0.8.5 Example: Particle on a Cone

Consider the following system:

Particle on a Cone

The cone is given by:

z = \sqrt{x^2 + y^2}

The mass of the particle is m, the cone is smooth.

Consider the following generalized coordinates:

\begin{align} z &= z \\ \end{align}

And \varphi be the angle of the projection of the particle on the xy plane with the x axis.

Then,

\begin{align} x &= z\cos(\varphi) \\ y &= z\sin(\varphi) \\ z &= z \end{align}

The kinetic energy T of the particle is given by:

\begin{align} T &= \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \\ &= \frac{1}{2}m(z^2\dot{\varphi}^2 + \dot{z}^2 + \dot{z}^2) \\ &= \frac{1}{2}m(2\dot{z}^2 + z^2\dot{\varphi}^2) \end{align}

The potential energy U of the particle is given by:

U = mgz

Thus, the Lagrangian L of the particle is given by:

L = T - U = \frac{1}{2}m(2\dot{z}^2 + z^2\dot{\varphi}^2) - mgz

The momentum of the particle is given by:

\begin{align} p_{z} &= \frac{\partial L}{\partial \dot{z}} = 2m\dot{z} \\ p_{\varphi} &= \frac{\partial L}{\partial \dot{\varphi}} = mz^2\dot{\varphi} \end{align}

Thus,

\begin{align} \dot{z} &= \frac{p_z}{2m} \\ \dot{\varphi} &= \frac{p_{\varphi}}{mz^2} \end{align}

The Hamiltonian H of the particle is given by:

\begin{align} H &= p \cdot \dot{q} - L \\ &= p_z\dot{z} + p_{\varphi}\dot{\varphi} - L \\ &= p_z\frac{p_z}{2m} + p_{\varphi}\frac{p_{\varphi}}{mz^2} - \frac{1}{2}m(2\dot{z}^2 + z^2\dot{\varphi}^2) + mgz \\ &= \frac{p_z^2}{2m} + \frac{p_{\varphi}^2}{mz^2} - \frac{1}{2}m\left(2\left(\frac{p_z}{2m}\right)^2 + z^2\left(\frac{p_{\varphi}}{mz^2}\right)^2\right) + mgz \\ &= \frac{p_z^2}{2m} + \frac{p_{\varphi}^2}{mz^2} - \frac{p_z^2}{4m} - \frac{p_{\varphi}^2}{2mz^2} + mgz \\ &= \frac{p_z^2}{4m} + \frac{p_{\varphi}^2}{2mz^2} + mgz \end{align}

As the Hamiltonian H does not depend on the generalized coordinate \varphi, the generalized coordinate \varphi is a cyclical coordinate. And the momentum p_{\varphi} is a constant.

Thus,

\begin{align} \frac{dH}{dp_{\phi}} &= \dot{\varphi} \\ \frac{p_{\phi}}{mz^2} &= \dot{\varphi} \\ \end{align}

For z coordinate,

\begin{align} \frac{dH}{dp_{z}} &= \dot{z} \\ \frac{p_{z}}{2m} &= \dot{z} \\ \end{align}

If we are given the initial energy of the particle E, then the Hamiltonian H of the particle is given by:

\begin{align} E &= \frac{p_z^2}{4m} + \frac{p_{\varphi}^2}{2mz^2} + mgz \\ &= m\dot{z}^2 + \frac{p_{\varphi}^2}{2mz^2} + mgz \end{align}

Which became a first order separable ODE in z.

If p_{\varphi} = 0, then the particle is moving vertically just like sliding on a smooth surface.

If p_{\varphi} \neq 0, then the term \frac{p_{\varphi}^2}{2mz^2} + mgz has lowest energy at z_0 = \sqrt[3]{\frac{p_{\varphi}}{m^2g}} with energy E_0.

If E = E_0, then m\dot{z}^2 = 0, and the particle is doing a circular motion on the cone.

If E > E_0, then the particle is doing oscillatory motion on the cone. By using Taylor expansion of \frac{p_{\varphi}^2}{2mz^2} + mgz around z_0, We could derive:

m\dot{z}^2 + E_0 + \frac{1}{2}\left.(\frac{p_{\varphi}^2}{2mz^2} + mgz)''\right|_{z_{0}} (z-z_0)^2 \approx E

And we could expect the angular frequency of the oscillatory motion to be approximately:

\omega = \sqrt{\frac{1}{2m}\left.(\frac{p_{\varphi}^2}{2mz^2} + mgz)''\right|_{z_{0}}}

0.8.6 Phase Space

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the phase space \Gamma of the system is the space of generalized coordinates q_i and momenta p_i.

The phase space of the system is a 2n dimensional space, where n is the number of generalized coordinates q_i.

0.8.6.1 Hamiltonian Flow

The trajectory of the system together with the change of the momentum of the system, generate a path in t in the phase space of the system, and the Hamiltonian canonical equations describe the flow of the system in the phase space.

\begin{align} \frac{dq_i}{dt} &= \frac{\partial H}{\partial p_i} \\ \frac{dp_i}{dt} &= -\frac{\partial H}{\partial q_i} \end{align}

0.8.6.2 Liouville's Theorem

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the volume of the phase space of the system is conserved. In other words, the Hamiltonian flow of the system is incompressible.

\frac{dV}{dt} = 0

0.8.6.3 Phase Space Portrait

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the phase space portrait of the system is the plot of the trajectory of the system in the phase space of the system.

0.8.6.3.1 Example: Phase Space Portrait of a Harmonic Oscillator

Given a object with mass m, spring constant k, and displacement x. The Hamiltonian H of the object is given by:

H = \frac{p^2}{2m} + \frac{1}{2}kx^2

The Phase Space Portrait of the object is like:

Phase Space Portrait of a Harmonic Oscillator

0.8.7 Poisson Brackets

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the Poisson bracket of two functions f and g of the phase space of the system is given by:

\{f,g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)

0.8.7.1 Properties of Poisson Brackets

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the Poisson bracket of two functions f and g of the phase space of the system has the following properties:

  1. Antisymmetry: \{f,g\} = -\{g,f\}
  2. Linearity: \{af + bg, h\} = a\{f,h\} + b\{g,h\}
  3. Product(Leibnitz) Rule: \{fg,h\} = f\{g,h\} + g\{f,h\}

0.8.7.2 Fundamental Poisson Brackets

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, the fundamental Poisson brackets of the system are given by:

\begin{align} \{q_i,p_j\} &= \delta_{ij} \\ \{q_i,q_j\} &= 0 \\ \{p_i,p_j\} &= 0 \end{align}

0.8.7.3 Constant of Motion and Poisson Brackets

Given a system of particles with generalized coordinates q_i, and the Hamiltonian H of the system, and a function f of the phase space of the system, then time derivative of the function f is given by:

\begin{align} \frac{df}{dt} &= \{f,H\} + \frac{\partial f}{\partial t} \end{align}

Thus, if the function f is independent of time, then the function f is a constant of motion.

]]>https://alex1222.com/posts/2024-05-23-Classical-Mechanics-Notes.mdhttps://alex1222.com/posts/2024-05-23-Classical-Mechanics-Notes.mdThu, 23 May 2024 00:00:00 GMT<![CDATA[Running Witcher 3 on MacOS]]>

0.1 Contents

0.2 Introduction

For long, MacOS users have been left out of the gaming world. But with the advent of Wine, it's now possible to run Windows games on MacOS. In this tutorial, I'll show you how to run Witcher 3 on MacOS using Whisky - a modern Wine wrapper built for MacOS.

All software used in this tutorial is free and open-source except for Steam and Witcher 3, which are proprietary.

0.3 Prerequisites

0.3.1 Steam Account

You'll need to have a Steam account and have already purchased Witcher 3.

0.3.2 System Requirements

You'will have to have Rosetta 2 installed on your system.

From Apple's Official Documentation:

Open any app that needs Rosetta. If the app opens, Rosetta is already installed and working. If Rosetta is not installed, you’ll be automatically asked to install it.

I am using MacOS Sonoma 14.2 on a MacBook Pro (M1, 2021). I do not ensure that this method will work on other systems.

0.3.3 Download Necessary Files

0.3.3.1 Download Whisky

Download the latest version of Whisky from the official GitHub Releases Page. Just use the latest version available.

Download the .zip file or .dmg file as I highlighted in the image below.

Whisky

0.3.3.2 Download Steam Installer for Windows

Download the Steam installer for Windows from the official Steam Website.

Important: What you need is the Windows installer, not the MacOS installer. The file should be named something like SteamSetup.exe.

To download it, you'll need to visit the Official Steam Website and click on the Windows Icon below the "INSTALL STEAM" button as I highlighted in the image below.

Steam

0.4 Installation

0.4.1 Install Whisky

Open the .zip or .dmg file you downloaded on Download Whisky step and drag the Whisky app to your Applications folder. Or just leave it anywhere on your computer if you prefer.

0.4.2 Install Necessary Dependencies for Whisky

Open Whisky app, and it will ask you to install GPTK. Click on "Next" and wait for the installation to finish.

Whisky

0.4.3 Create a New Bottle

Create a new bottle by clicking on the "+" button on the right top corner of the Whisky app.

Whisky

Just put arbitrary bottle name and leave all other settings as default and click "Create". You will need to wait some time for the bottle to be created.

Whisky

0.4.4 Install Steam

On the right bottom corner of the Whisky app, click on the "Run..." button.

Whisky

Select the Steam installer you downloaded on Download Steam Installer for Windows step. And then click "Open".

Whisky

You will see the Steam installer running. Just click "Next" and "Install" as you would do on Windows.

Whisky

Whisky

Whisky

Whisky

Just wait a bit, and your Steam will be installed and opened.

Whisky

0.4.5 Install Witcher 3

Now, you can install Witcher 3 as you would do on Windows. Just search for Witcher 3 in your library and click on "Install".

0.5 Final Thoughts

This method also allows you to run other Windows games on MacOS. However, not all games are supported, and some games may not run as smoothly as they would on Windows.

For some common issues and solutions, you can check the Whisky's Official Documentation.

]]>https://alex1222.com/posts/2024-03-07-Run-Witcher3-On-MacOS.mdhttps://alex1222.com/posts/2024-03-07-Run-Witcher3-On-MacOS.mdThu, 07 Mar 2024 00:00:00 GMT<![CDATA[Download encrypted HLS content with ffmpeg]]>

HLS (HTTP Live Streaming) is a widely used streaming protocol that breaks video files into small chunks and serves them over HTTP. It's a popular choice for streaming video content on the web, and it's used by many platforms.

In this tutorial, I'll show you how to download encrypted HLS content.

0.1 Prerequisites

You'll need to have ffmpeg installed on your system.

0.2 Find the M3U8 file

The first step is to find the M3U8 file. This file contains the URLs of the video chunks and the encryption key.

You can find the M3U8 file by inspecting the network requests in your browser's developer tools when loading the page.

0.3 Download the M3U8 file

Once you have the M3U8 file, you can download it using ffmpeg:

ffmpeg -i "https://example.com/video.m3u8" -c copy -bsf:a aac_adtstoasc output.mp4
]]>https://alex1222.com/posts/2024-02-23-Download-Encrypted-HLS-Content-With-FFMPEG.mdhttps://alex1222.com/posts/2024-02-23-Download-Encrypted-HLS-Content-With-FFMPEG.mdFri, 23 Feb 2024 00:00:00 GMT<![CDATA[Transitioning Themes: Embracing Simplicity and Serious Exploration]]>

If you visited my blog around three months ago, you might recall the vibrant theme featuring fancy animations, colors, and a rotating torus knot implemented with three.js. While this theme garnered appreciation from friends and myself, I've made a deliberate decision to move away from it. The reasons are straightforward.

Managing and updating my blog became increasingly time-consuming and complex as I added more features to the intricate design. The simplicity I initially enjoyed was overshadowed by the demands of maintaining such a detailed theme.

Crucially, my aesthetic preferences have evolved towards a desire for simplicity. I now advocate for a minimalist approach—no need for elaborate colors or animations. I've realized the importance of focusing on substance rather than the outward appearance.

I've spent considerable time on superficial aspects, not just in my blog but in various facets of my life. Now, my goal is to streamline the visual aspects and dedicate more time to exploring profound and intellectually stimulating subjects, such as algebraic topology.

Algebraic topology, with its intricate concepts, fascinates me. I've observed individuals sharing visually striking higher-dimensional objects online, seemingly not mentioned the rich mathematical content behind the images. I appreciate the beauty, but there's a deeper layer that goes unnoticed.

Sharing such images without delving into the underlying concepts on platforms where few understand algebraic topology can come across as mere showmanship. I aim to shift the focus from superficial displays to a genuine appreciation for the intricate and fascinating world of serious and interesting subjects.

]]>https://alex1222.com/posts/2023-12-28-Why-I-Stop-Using-The-Last-Themes.mdhttps://alex1222.com/posts/2023-12-28-Why-I-Stop-Using-The-Last-Themes.mdThu, 28 Dec 2023 00:00:00 GMT<![CDATA[General Topology Notes]]>

0.1 Contents

1 Topology Spaces and Continuous Function

This is a note for general topology. The main reference is Topology.

The note is written in Pandoc Markdown.

My GitHub repository for this note.

1.1 Basic Definition of Topology

::: {#def:Topology .definition} Definition 1.1 (topology). A topology on a set \mathbb{X} is a collection \mathbb{T} of subsets of \mathbb{X} having the following properties:

  • \emptyset and \mathbb{X} are in \mathbb{T}

  • The union of the elements of any sub collection of \mathbb{T} is in \mathbb{T}

  • The intersection of the elements of any finite sub collection of \mathbb{T} is in \mathbb{T} :::

::: {#def:TopologySpace .definition} Definition 1.2 (topology space). A topological space is a set \mathbb{X} for which a topology \mathbb{T} has been specified. :::

::: {#def:OpenSet .definition} Definition 1.3 (open set). A open set \mathbb{U} is a subset of \mathbb{X} that belongs to a topology \mathbb{T} of \mathbb{X}. :::

::: {#def:OpenSets .definition} Definition 1.4 (open sets). A topology can also be called a open sets :::

::: {#def:DiscreteTopology .definition} Definition 1.5 (discrete topology). The set of all subsets of a set \mathbb{X} formed a topology called discrete topology :::

::: {#def:TrivialTopology .definition} Definition 1.6 (trivial topology). The set consisting the set \mathbb{X} and \emptyset only formed a topology of \mathbb{X} called trivial topology :::

::: {#def:FiniteComplementTopology .definition} Definition 1.7 (finite complement topology). Let \mathbb{X} be a set. Let \mathbb{T}_{\mathit{f}} be the collection of all subsets \mathbb{U} of \mathbb{X} such that \mathbb{X} - \mathbb{U} either if a finite 1 of is all of \mathbb{X}. Then \mathbb{T}_{\mathit{f}} is a topology on \mathbb{X}, called the finite complement topology. :::

::: {#def:Comparable .definition} Definition 1.8 (finer, larger, strictly finer, strictly larger, coarser, smaller, strictly coarser, strictly smaller, comparable). Let \mathbb{T} and \mathbb{T'} be two topology on a given set \mathbb{X}. If \mathbb{T} is a subset of \mathbb{T'}, we say that \mathbb{T'} is finer or larger than \mathbb{T}. If \mathbb{T} is a proper subset of \mathbb{T'}, we say that \mathbb{T'} is strictly finer or strictly larger than \mathbb{T}. We also say that \mathbb{T} is coarser or smaller or strictly coarser or strictly smaller than \mathbb{T'}. We say that \mathbb{T} and \mathbb{T'} is comparable if either \mathbb{T} is a subset of \mathbb{T'} or \mathbb{T'} is a subset of \mathbb{T}. :::

1.2 Basis for a Topology

::: {#def:Basis .definition} Definition 1.9 (basis). If \mathbb{X} is a set, a basis for a topology on \mathbb{X} is a collection \mathbb{B} of subsets of \mathbb{X} (called basis elements) such that:

  • For each x \in \mathbb{X}, there is at least one basis element B containing x

  • If x belongs to the intersection of two basis elements B_{1} and B_{2}, then there is another element x \in B_{3} \in \mathbb{B} such that B_{3} \subseteq B_{1} \cap B_{2} :::

::: {#def:TopologyGeneratedByBasis .definition} Definition 1.10 (topology generated by basis). Let \mathbb{B} be a basis on \mathbb{X} . Let \mathbb{U} be a set containing all subsets U of \mathbb{X} such that for each element x \in U, there is B \in \mathbb{B} that x \in B \subseteq U. Such \mathbb{U} formed a topology on \mathbb{X} , called topology \mathbb{T} generated by \mathbb{B} :::

::: lemma Lemma 1.1. Let \mathbb{X} be a set. Let \mathbb{B} be a basis for a topology \mathbb{T} on \mathbb{X} . Then \mathbb{T} equals to the set of all possible unions of elements of \mathbb{B} . :::

::: proof Proof. Let set \mathbb{U} be the set of all possible unions of elements of \mathbb{B} . For any U \in \mathbb{U}. U = \cup B 2 for some B \in \mathbb{B}. Thus, for every x \in U, there exist a B' \in \mathbb{B} that x \in B' \subseteq U. Thus, U \in \mathbb{T}.

Conversely, for any U \in \mathbb{T}. For any x \in U, let x \in B_{x} \in U. Then, U = \cup_{x \in U}B_{x}. Thus, U \in \mathbb{U}.

Therefore, \mathbb{U} equals to \mathbb{T} . ◻ :::

::: lemma Lemma 1.2. 3 Let \mathbb{X} be a topological space. Suppose that \mathbb{C} is a collection of open sets of \mathbb{X} such that for each open set U of \mathbb{X} and each x \in U, there is an element C \in \mathbb{C} such that x \in C \subseteq C. Then \mathbb{C} is a basis for the topology of \mathbb{X} . :::

::: lemma Lemma 1.3. 4 Let \mathbb{B} and \mathbb{B'} be basis for the topologies \mathbb{T} and \mathbb{T'} , respectively, on \mathbb{X} . Then the following are equivalent:

  • \mathbb{T'} is finer than \mathbb{T}

  • For each x \in \mathbb{X} and each basis element B \in \mathbb{B} containing X, there is a basis element B' \in \mathbb{B'} such that x \in B' \subseteq B. :::

::: {#def:StandardTopologyOnTheRealLine .definition} Definition 1.11 (standard topology on the real line). Let be \mathbb{B} = \{ B | B = \{ x | a < x < b \}, a < b, a \in \mathbb{R}, b \in \mathbb{R} \}. \mathbb{B} formed a basis on real line. The topology generated by \mathbb{B} is called the standard topology on the real line 5 . :::

::: {#def:LowerLimitTopologyOnTheRealLine .definition} Definition 1.12 (lower limit topology on the real line). Let be \mathbb{B} = \{ B | B = \{ x | a \le x < b \}, a < b, a \in \mathbb{R}, b \in \mathbb{R} \}. \mathbb{B} formed a basis on real line. The topology generated by \mathbb{B} is called the lower limit topology on the real line. When \mathbb{R} is given this topology,we denote it by \mathbb{R}_{l}. :::

::: {#def:KTopologyOnTheRealLine .definition} Definition 1.13 (K-topology on the real line). Let be \mathbb{B} = \{ B | B = \{ x | a < x < b \}, a < b, a \in \mathbb{R}, b \in \mathbb{R} \}. Let K = \{ x | x = \frac{1}{n}, n \in \mathbb{Z_{+}} \}. \mathbb{B} \cup \{ B - K | B \in \mathbb{B} \} formed a basis on real line. The topology generated by \mathbb{B} is called the K-topology on the real line. When \mathbb{R} is given this topology,we denote it by \mathbb{R_{K}}. :::

::: lemma Lemma 1.4. 6 The topologies \mathbb{R}_{l} and \mathbb{R_{K}} is strictly finer than the standard topology on \mathbb{R} . :::

::: lemma Lemma 1.5. The topologies of \mathbb{R}_{l} and \mathbb{R_{K}} is not comparable. :::

::: proof Proof. Let \mathbb{T}_{l} and \mathbb{T_{K}} be topologies of \mathbb{R}_{l} and \mathbb{R_{K}} respectively. Let K = \{ x | x = \frac{1}{n}, n \in \mathbb{Z_{+}} \}.

We first proof that \mathbb{T}_{l} is not finer than \mathbb{T_{K}} . Let U = \{ x | -1 < x < 1 \} - K, x = 0. If there exist B = \{ x | a \le x < b \} \in \mathbb{T}_{l} such that x \in B \subseteq U, then 0 < b < 1. Thus, there exist n \in \mathbb{Z_{+}} that 0 < \frac{1}{n} < b. Thus B is not a subset of U.

Then we proof that \mathbb{T_{K}} is not finer than \mathbb{T}_{l}. Let U' = \{ x | a' \le x < b' \}. If there exist B' = \{ x | a'' < x < b'' \} or \{ x | a'' < x < b'' \} - K such that {a'} \in B \subseteq U. Thus a'' < a < b''. Thus there exist c that a'' < x < a, x \in B ,x \notin U'. Thus B' \nsubseteq U'.

Thus the topologies of \mathbb{R}_{l} and \mathbb{R_{K}} is not comparable. ◻ :::

::: {#def:Subbasis .definition} Definition 1.14 (subbasis). A subbasis \mathbb{S} for a topology on \mathbb{X} is a collection of subsets of \mathbb{X} whose union equals \mathbb{X} . The topology generated by the subbasis \mathbb{S} is defined to be the collection \mathbb{T} 7 of all unions of finite intersections of elements of \mathbb{S} . :::

1.2.1 Exercise

  1. Show that if \mathbb{A} is a basis for a topology on \mathbb{X} , then the topology generated by \mathbb{A} equals the intersection of all topologies on \mathbb{X} that contain \mathbb{A} . Prove the same if \mathbb{A} is a subbasis.

    ::: proof Proof. As a subbasis is also a basis, we will directly prove the case of subbasis here.

    Let \mathbb{S} = \{ \mathbb{T}_{\alpha} \} be set contain all the topologies that contain \mathbb{A} . Let \mathbb{T} be the topology that \mathbb{A} generated. Let \displaystyle \mathbb{T}' = \cap\mathbb{T}_{\alpha}.8

    First, \mathbb{A} \subseteq \mathbb{T}_{\alpha}. Thus, \mathbb{T} \subseteq \mathbb{T}_{\alpha}. Thus, \mathbb{T} \subseteq \mathbb{T}'.

    Also, \mathbb{A} \subseteq \mathbb{T}. Thus, \mathbb{T} \in \mathbb{S}. Thus, \mathbb{T}' \subseteq \mathbb{T}.

    Thus, \mathbb{T} = \mathbb{T}' ◻ :::

1.3 The Order Topology

::: {#def:Interval .definition} Definition 1.15 (interval). Let \mathbb{X} is a set having a simple order relation <. Given elements a and b of \mathbb{X} such that a < b, there are four subsets of \mathbb{X} that are called intervals determined by a and b:

  • (a,b) = \{ x | a < x < b \}

  • (a,b] = \{ x | a < x \le b \}

  • [a,b) = \{ x | a \le x < b \}

  • [a,b] = \{ x | a \le x \le b \}

(a,b) is called an open interval on \mathbb{X} . [a,b] is called an closed interval on \mathbb{X} . (a,b] and [a,b) is called half-open intervals. :::

::: {#def:OrderTopology .definition} Definition 1.16 (order topology). 9 Let \mathbb{X} be a set with a simple order relation; assume \mathbb{X} has more than one element. Let \mathbb{B} be the collection of all sets of the following types:

  • All open intervals (a,b) in \mathbb{X} .

  • All intervals of the form [a_{0},b), where a_0 is the smallest element(if exist) of \mathbb{X} .

  • All intervals of the form (a,b_{0}], where b_0 is the largest element(if exist) of \mathbb{X} .

The collection \mathbb{B} formed a basis for a topology on \mathbb{X} , which is called the order topology. :::

::: {#def:Ray .definition} Definition 1.17 (ray). 1011 If \mathbb{X} is an ordered set, and a is an element of \mathbb{X} , there are four subsets of \mathbb{X} that are called rays determined by a:

  • (a,+\infty) = \{ x | x > a \}

  • (-\infty,a) = \{ x | x < a \}

  • [a,+\infty) = \{ x | x \ge a \}

  • (-\infty,a] = \{ x | x \le a \}

(a,+\infty) and (-\infty,a) are called open rays. [a,+\infty) and (-\infty,a] are called closed rays. :::

1.4 The Product Topology

::: {#def:ProductTopology .definition} Definition 1.18 (product topology). Let \mathbb{X} and \mathbb{Y} be topological spaces. The product topology on \mathbb{X} \times \mathbb{Y} having a basis \mathbb{B} containing all sets of the form U \times V, where U and V is open sets of \mathbb{X} and \mathbb{Y} respectively. :::

::: theorem Theorem 1.1. 12 If \mathbb{B} and \mathbb{C} is basis for the topology of \mathbb{X} and \mathbb{Y} respectively, then the collection \mathbb{D} = \{ B \times C | B \in \mathbb{B} and C \in \mathbb{C} \} is a basis for the topology of \mathbb{X} \times \mathbb{Y} :::

::: {#def:Projection .definition} Definition 1.19 (projection). Let \pi_{1}: \mathbb{X} \times \mathbb{Y} \rightarrow \mathbb{X} be defined by the equation: \pi_{1}(x,y) = x

Let \pi_{2}: \mathbb{X} \times \mathbb{Y} \rightarrow \mathbb{Y} be defined by the equation: \pi_{1}(x,y) = y

The maps \pi_{1} and \pi_{2} are called the projections of \mathbb{X} \times \mathbb{Y} onto its first and second factors, respectively. :::

::: theorem Theorem 1.2. 13 The collection \mathbb{S} = \{ \pi_{1}^{-1}(U) | U open in \mathbb{X} \} \cup \{ \pi_{2}^{-1}(V) | V open in \mathbb{Y} \} is a subbasis for the product topology on \mathbb{X} \times \mathbb{Y}. :::

::: {#def:BoxTopology .definition} Definition 1.20 (box topology). Let, \mathbb{X} = \mathbb{X}_{1} \times \mathbb{X}_{2} \times \dots \times \mathbb{X}_{n} \text{or} \mathbb{X}_{1} \times \mathbb{X}_{2} \times \dots

In the first case, all the sets of the form U_{1} \times \dots \times U_{n} where U_{i} is a open set of \mathbb{X}_{i} form a basis.

In the second case, all the sets of the form U_{1} \times U_{2} \times \dots where U_{i} is a open set of \mathbb{X}_{i} also form a basis.

Topology defined in this way was called a box topology. :::

::: {#def:ProjectionMapping .definition} Definition 1.21 (product topology). []{#def:ProductTopologyInfinite label="def:ProductTopologyInfinite"}14 Let, \mathbb{X} = \mathbb{X}_{1} \times \mathbb{X}_{2} \times \dots \times \mathbb{X}_{n} \text{or} \mathbb{X}_{1} \times \mathbb{X}_{2} \times \dots

Let \pi_{i} be the projection function15 that \pi_{i}: \mathbb{X} \rightarrow \mathbb{X}_{i}

And if x \in \mathbb{X} \pi_{i}(x) = x_{i}

All the set of the form \pi_{i}^{-1}(U_{i}) where i is arbitrary and U_{i} is an open set of \mathbb{X}_{i} , form a subbasis of \mathbb{X} . The topology generated by this subbasis is called product topology. And \mathbb{X} is called a product space. :::

::: {#def:JTuple .definition} Definition 1.22 (J-tuple). Let J be an index set16. Give a set \mathbb{X} , a J-tuple is defined as a function x: J \rightarrow \mathbb{X} . If \alpha is an element of J , x(\alpha) is often denoted by x_{\alpha} and is called the \alpha\text{th} coordinate of x . And the function x itself is often denoted by the symbol (x_{\alpha})_{\alpha \in J}

The set of all J-tuples of elements of \mathbb{X} is often denoted by \mathbb{X}^{J} . :::

::: {#def:CartesianProduct .definition} Definition 1.23 (cartesian product). Let \{A_{\alpha}\}_{\alpha \in J} be an indexed family of sets; let \mathbb{X} = \bigcup_{\alpha \in J} A_{\alpha} . The cartesian product of this indexed family is denoted by \displaystyle \prod_{\alpha \in J} A_{\alpha}

And is defined to be the set of all J-tuples (x_{\alpha})_{\alpha \in J} of elements of \mathbb{X} such that x_{\alpha} \in A_{\alpha} for each \alpha \in J . That is, it is the set of all functions x: J \rightarrow \bigcup_{\alpha \in J} A_{\alpha} such that x(\alpha) \in A_{\alpha} for each \alpha \in J . :::

::: {#theorem:ComparisonOfBoxProductTopology .theorem} Theorem 1.3 (Comparison of the box and product topologies). 17 The box topology on \prod \mathbb{X}_{\alpha} has a basis all sets of the form \prod U_{\alpha} where U_{\alpha} is open in X_{\alpha} for each \alpha . The product topology on \prod \mathbb{X}_{\alpha} has a basis all sets of the form \prod U_{\alpha} where U_{\alpha} is open in X_{\alpha} for each \alpha and U_{\alpha} equals \mathbb{X}_{\alpha} except for finitely many values of \alpha . :::

::: theorem Theorem 1.4. 18 Suppose the topology on each space \mathbb{X}_{\alpha} is given by a basis \mathbb{X}_{\alpha} . The collection of all sets of the form \prod_{\alpha \in J} B_{\alpha} where B_{\alpha} \in \mathbb{B}_{\alpha} form a basis for the box topology on \prod _{\alpha \in J} \mathbb{X}_{\alpha} .

The collection of all sets of the same form, where B_{\alpha} \in \mathbb{B}_{\alpha} for finitely many indices \alpha and B_{\alpha} = \mathbb{X}_{\alpha} for all the remaining indices, will form a basis for the product topology \prod_{\alpha \in J}\mathbb{X}_{\alpha} . :::

::: theorem Theorem 1.5. 19 Let A_{\alpha} be a subspace of \mathbb{X}_{\alpha} , for each \alpha \in J . Then \prod A_{\alpha} is a subspace of \prod \mathbb{X}_{\alpha} if both products are given the box topology, or if both products are given the product topology. :::

::: theorem Theorem 1.6. 20 If each space \mathbb{X}_{\alpha} is a Hausdorff space, then \prod \mathbb{X}_{\alpha} is a Hausdorff space in both the box and product topologies. :::

::: theorem Theorem 1.7. Let \{ \mathbb{X}_{\alpha} \} be an indexed family of spaces; let A_{\alpha} \subseteq \mathbb{X}_{\alpha} for each \alpha . If \prod \mathbb{X}_{\alpha} is given either the product or the box topology, then \prod \overline{A_{\alpha}} = \overline{\prod A_{\alpha}} :::

::: proof Proof. Let \pi_{\alpha} represent the projection mapping.

Let x be an element of \prod \mathbb{X}_{\alpha} . Let V be an open set in \prod \mathbb{X}_{\alpha} that containing x .

If x \in \prod \overline{A_{\alpha}} , then \pi_{\alpha}(V) is a open set in \mathbb{X}_{\alpha} that containing x_{\alpha} . Thus \pi_{\alpha}(V) intersect with A_{\alpha} . Thus V intersect with \prod A_{\alpha} . Thus x \in \overline{\prod A_{\alpha}} .

If x \in \overline{\prod A_{\alpha}} . Let U_{\alpha} be an open set of A_{\alpha} that contain x_{\alpha} . Let V = \prod U_{\beta} such that U_{\beta} = \begin{cases} \mathbb{X}_{\beta}, & \beta \neq \alpha \\ U_{\alpha}, & \beta = \alpha \end{cases} . It is obvious that V is an open set that contain x . Thus V intersect with \prod A_{\alpha} . Thus U_{\alpha} intersect with A_{\alpha} . Thus x \in \prod \overline{A_{\alpha}} . ◻ :::

::: theorem Theorem 1.8. Let f: A \rightarrow \prod_{\alpha \in J} \mathbb{X}_{\alpha} be given by the equation f(a) = ( f_{\alpha}(a) )_{\alpha \in J} where f_{\alpha} : A \rightarrow \mathbb{X}_{\alpha} for each \alpha . Let \prod \mathbb{X}_{\alpha} have the product topology. Then the function f is continuous if and only if each function f_{\alpha} is continuous. :::

::: proof Proof. Let \pi_{\alpha} be the projection mapping

It is obvious that f^{-1}(U) = \bigcap_{\alpha \in J} f_{\alpha}^{-1}(\pi_{\alpha}(U))

If f_{\alpha} is continuous. Let V be a closed set of \prod_{\alpha \in J} \mathbb{X}_{\alpha} . Then \pi_{\alpha}(V) is closed. Then f^{-1}(V) is intersect of closed set. Thus \pi_{\alpha}(V) is closed. So f is continuous.

If f is continuous. Let U_{\alpha} be an open set of \mathbb{X}_{\alpha} . Let U_{\beta} = \mathbb{X}_{\beta} if \beta \neq \alpha . Let V = \prod_{\beta \in J} U_{\beta} . It is obvious that V is an open set of \prod \mathbb{X}_{\alpha} . And

\begin{aligned} f^{-1}{V} &=& \bigcap_{\alpha \in J} f_{\alpha}^{-1}(\pi_{\alpha}(U)) \\ &=& f_{\alpha}^{-1}(U_{\alpha}) \end{aligned}

which is an open set in A . Thus f_{\alpha} is continuous. ◻ :::

1.5 The Subspace Topology

::: {#def:SubspaceTopology .definition} Definition 1.24 (subspace topology). Let \mathbb{X} be a topological space with topology \mathbb{T} . If Y is a subset of \mathbb{X} , the collection \mathbb{T}_{Y} = \{ Y \cap U | U \in \mathbb{T} \} is a topology on Y , called the subspace topology.

Y is also called a subspace of \mathbb{X} :::

::: lemma Lemma 1.6. 21 If \mathbb{B} is basis for the topology of \mathbb{X} , Y is a subset of \mathbb{X} then the collection \mathbb{B}_{Y} = \{ B \cap Y | B \in \mathbb{B} \} is a basis for the subspace topology on Y :::

::: lemma Lemma 1.7. 22 Let Y be a subspace of \mathbb{X} . If U is open in Y and Y is open in \mathbb{X} , then U is open in \mathbb{X} . :::

::: theorem Theorem 1.9. 23 If A is a subspace of \mathbb{X} and B is a subspace of \mathbb{Y} , then the product topology on A \times B is the same as the topology A \times B inherits as a subspace of \mathbb{X} \times \mathbb{Y} :::

::: proof Proof. Let \mathbb{B}_{\mathbb{X}} and \mathbb{B}_{\mathbb{Y}} and \mathbb{B}_{\mathbb{XY}} be basis of topology of \mathbb{X} and \mathbb{Y} and \mathbb{X} \times \mathbb{Y} respectively. Let \mathbb{B}_{\mathbb{X}}' and \mathbb{B}_{\mathbb{Y}}' and \mathbb{B}_{\mathbb{XY}}' be basis of topology of A and A and A \times B respectively. We will show that \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}' = \mathbb{B}_{\mathbb{XY}}'. Thus, the product topology on A \times B is the same as the topology A \times B inherits as a subspace of \mathbb{X} \times \mathbb{Y} .

First, every element in \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}' can be represented by B_{A} \cap A \times B_{B} \cap B = B_{A} \times B_{B} \cap A \times B \in \mathbb{B}_{\mathbb{XY}}' where B_{A} \in \mathbb{B}_{\mathbb{X}}', B_{B} \in \mathbb{B}_{\mathbb{Y}}'. Thus \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}' \subseteq \mathbb{B}_{\mathbb{XY}}'.

Next, we show that \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}' generate the topology A \times B inherits as a subspace of \mathbb{X} \times \mathbb{Y} . For any open set U in \mathbb{X} \times \mathbb{Y} , and \forall x \in U \cap A \times B, \exists B_{\mathbb{X}} \times B_{\mathbb{Y}} \in \mathbb{B}_{\mathbb{XY}}, x \in B_{\mathbb{X}} \times B_{\mathbb{Y}} \subseteq \mathbb{X} \times \mathbb{Y}. Thus x \in B_{\mathbb{X}} \times B_{\mathbb{Y}} \cap A \times B \subseteq A \times B, B_{\mathbb{X}} \times B_{\mathbb{Y}} \cap A \times B \in \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}'. Thus \mathbb{B}_{\mathbb{X}}' \times \mathbb{B}_{\mathbb{Y}}' generate the topology A \times B inherits as a subspace of \mathbb{X} \times \mathbb{Y} .gi ◻ :::

::: {#def:OrderedSquare .definition} Definition 1.25 (ordered square). Let I = [0,1]. The set I \times I in the dictionary order 24 topology will be called ordered square, and denoted by I_{o}^{2} :::

::: {#def:Convex .definition} Definition 1.26 (convex). Given an ordered set \mathbb{X} , let us say that a subset \mathbb{Y} of \mathbb{X} is convex in \mathbb{X} if for each pair of points a < b of \mathbb{Y} , the entire interval (a,b) of points of \mathbb{X} lies in \mathbb{Y} :::

::: theorem Theorem 1.10. 25 Let \mathbb{X} be an ordered set in the order topology. Let \mathbb{Y} be a subset of \mathbb{X} that is convex in \mathbb{X} . Then the order topology on \mathbb{Y} is the same as the topology \mathbb{Y} inherits as a subspace of \mathbb{X} . :::

::: proof Proof. Consider the ray (a,+\infty) in \mathbb{X} . If a \in \mathbb{Y}, then (a,+\infty) \cap \mathbb{Y} = \{ x | x \in \mathbb{Y} and x > a \}

This is an open ray of the ordered set of \mathbb{Y} . if a \notin Y, then a is either a lower bound on \mathbb{Y} or an upper bound on \mathbb{Y} , since \mathbb{Y} is convex. In the former case, the set (a,+\infty) \cap \mathbb{Y} equals all of \mathbb{Y} , in the latter case, it is empty.

A similar remark shows that the intersection of the rat (-\infty,a) with \mathbb{Y} is either an open ray of \mathbb{Y} , or \mathbb{Y} itself, or empty. Since the sets (a,+\infty) \cal \mathbb{Y} and (-\infty,a) \cap \mathbb{Y} form a subbasis for the subspace topology on \mathbb{Y} , and since each is open in the order topology, the order topology contains the subspace topology.

To prove the reverse, note that any open ray of \mathbb{Y} equals the intersection of an open ray of \mathbb{X} with \mathbb{Y} , so it is open in the subspace topology on \mathbb{Y} . Since the open rays of \mathbb{Y} are a subbasis for the order topology on \mathbb{Y} , this topology is contained in the subspace topology. ◻ :::

1.5.0.1 Exercise

  1. A map \mathit{f} : \mathbb{X} \rightarrow \mathbb{Y} is said to be a open map[]{#def:OpenMap label="def:OpenMap"} if for every open set U \subseteq \mathbb{X}, the set \mathit{f}(U) is open in \mathbb{Y} . Show that \pi : \mathbb{X} \times \mathbb{Y} \rightarrow \mathbb{X} is open map.

    ::: proof Proof. An open set in \mathbb{X} \times \mathbb{Y} can be represented by \cup( U_{i} \times U_{i}' ) where U_{i}, U_{i}' are open sets in \mathbb{X} , \mathbb{Y} , respectively.

    Also, \cup( U_{i} \times U_{i}' ) = \cup( U_{i} ) \times \cup( U_{i}' )

    Thus, \pi(\cup( U_{i} \times U_{i}' )) = \cup( U_{i} )

    Thus, \pi(U) is open in \mathbb{X} . ◻ :::

  2. Let \mathbb{X} and \mathbb{X'} denote a single set in the topologies \mathbb{T} and \mathbb{T'} , respectively; let \mathbb{Y} and \mathbb{Y'} denote a single set in the topologies \mathbb{U} and \mathbb{U'} , respectively. 26 Assume these sets are nonempty.

    1. Show that if \mathbb{T}' \supseteq \mathbb{T} and \mathbb{U}' \supseteq \mathbb{U} , then the product topologies \mathbb{X'} \times \mathbb{Y'} is finer than the product topology on \mathbb{X} \times \mathbb{Y} .

    2. Does the converse of the previous statement hold?

  3. Show that the countable collection27 \{ (a,b) \times (c,d) | a < b, c < d, a \in \mathbb{Q}, b \in \mathbb{Q}, c \in \mathbb{Q}, d \in \mathbb{Q} \} is a basis for \mathbb{R}^{2}

    ::: proof Proof. This is obvious if you prove that (a,b) \times (c,d) is a rectangle in the \mathbb{R}^{2} plane. ◻ :::

  4. Let \mathbb{X} be an ordered set. If \mathbb{Y} is a proper subset of \mathbb{X} that is convex in \mathbb{X} prove that \mathbb{Y} may not be an interval or a ray in \mathbb{X} .

    ::: proof Proof. Let \mathbb{X} = \mathbb{R}^{2} with dictionary order. Then Y = \{ (x,y) | -1 \le x \le 1 \} is convex in \mathbb{X} , however it is not an interval or a ray. ◻ :::

    There is a false prove given by myself.

    ::: proof Proof. Let \mathbb{S} be a set that contain all intervals and rays of \mathbb{Y} . We define a partial order on \mathbb{S} by inclusion. So if there is a chain in \mathbb{S} : S_{1} \subseteq S_{2} \subseteq S_{3} \dots

    Let S = S_{1} \cup S_{2} \cup S_{3} \cup \dots

    Thus, S is an upper bound of the chain.

    Thus, by Zorn's Lemma, there is a maximal element of \mathbb{S} , say U , then we prove that U = \mathbb{Y} .

    If U \neq \mathbb{Y} , then \exists x, x \in \mathbb{Y} - U .

    If U is a ray say (a,+\infty) . Then x < a , thus U \subseteq (x,+\infty) \subseteq \mathbb{B} , then there is contradiction with the maximal element.

    If U is an interval, the circumstance is similar with the proof of U is a ray.

    Thus \mathbb{Y} is a ray or an interval. ◻ :::

    However, there is issue with this proof, the set S does exists. However, it may not be an interval or ray, so it may not be contained in \mathbb{S}

1.6 Closed Sets and Limit Points

::: {#def:Closed .definition} Definition 1.27 (closed). 28 A subset A of a topological space is said to be closed if the set \mathbb{X}-A is open. :::

::: theorem Theorem 1.11. 29 Let \mathbb{X} be a topological space. Then the following conditions hold

  1. \emptyset and \mathbb{X} are closed.

  2. Arbitrary intersections of closed sets are closed

  3. Finite unions of closed sets are closed :::

::: {#def:ClosedIn .definition} Definition 1.28 (closed in). Let \mathbb{X} be a topological space; let \mathbb{Y} be a subspace of \mathbb{X} . We say that a set A is closed in \mathbb{Y} if A is a subset of \mathbb{Y} and A is closed in the subspace topology of \mathbb{Y} :::

::: theorem Theorem 1.12. Let \mathbb{Y} be a subspace of \mathbb{X} . Then a set A is closed in \mathbb{Y} if and only if it equals the intersection of a closed set of \mathbb{X} with \mathbb{Y} :::

::: proof Proof. First we proof that if A is closed in \mathbb{Y} , then \exists B \subseteq \mathbb{X}, B \cap \mathbb{Y} = A . As the origin topology form a surjective map to its subspace topology, there exists a B closed in \mathbb{X} that \mathbb{Y} - A = ( \mathbb{X} - B ) \cap \mathbb{Y} . Then B \cap \mathbb{Y} = A

Conversely, if \exists B \subseteq \mathbb{X}, B \cap \mathbb{Y} = A . Then, \mathbb{Y} - A = ( \mathbb{X} - B ) \cap \mathbb{Y} . Then \mathbb{X}-B is open in \mathbb{Y} , \mathbb{Y}-A is open in \mathbb{Y} . Then A is closed in \mathbb{Y} ◻ :::

::: theorem Theorem 1.13. 30 Let \mathbb{Y} be a subspace of \mathbb{X} . If A is closed in \mathbb{Y} and \mathbb{Y} is closed in \mathbb{X} , then A is closed in \mathbb{X} . :::

::: {#def:Interior .definition} Definition 1.29 (interior). Given a subset A of a topological space \mathbb{X} , the interior of A is defined as the union of all open sets contained in A . Denoted by Int(A). :::

::: {#def:Closure .definition} Definition 1.30 (closure). Given a subset A of a topological space \mathbb{X} , the closure of A is defined as the intersection of all closed sets containing A . Denoted by Cl(A) or \overline{A} :::

::: theorem Theorem 1.14. 3132 Let \mathbb{Y} be a subspace of a topological space \mathbb{X} ; let A be a subset of \mathbb{X} . Let \overline{A} denote the closure of A in \mathbb{X} . Then the closure of A in \mathbb{Y} equals \overline{A} \cap \mathbb{Y} :::

::: {#def:Intersect .definition} Definition 1.31 (intersect). We say that a set A intersects B if A \cap B is not empty. :::

::: theorem Theorem 1.15. Let A be a subset of the topological space \mathbb{X}

  1. The x \in \overline{A} if and only if every open set U containing x intersect A .

  2. Supposing the topology of \mathbb{X} is given by a basis, then x \in \overline{A} if and only if every basis element B containing x intersects A :::

::: proof Proof. There are only two types of closed set U in \mathbb{X} :

  1. U \supseteq \overline{A}

  2. U \cap A \neq A

Thus, there are only two types of open set U in \mathbb{X} respectively.

  1. U does not intersects A .

  2. U \cap \overline{A} \neq \emptyset

  3. If x \in \overline{A} , then every open set containing x is the open set of second type, thus every open set containing x intersects A

    If every open set containing x intersect \mathbb{A} , suppose x \notin \overline{A} . Then \mathbb{X} - \overline{A} is a open set containing x, however, it does not intersects A . Thus, x \in \overline{A} .

  4. If x \in \overline{A} , as every basis element of \mathbb{X} is a open set, thus every basis element containing x intersects \mathbb{A}

    If every open set containing x intersect \mathbb{A} , suppose x \notin \overline{A} .

    As every open sets can be represented by union of basis. Let \mathbb{X} - \overline{A} = B_{1} \cup B_{2} \cup B_{3} \cup \dots \cup B_{1}' \cup B_{2}' \cup B_{3}' \cup \dots where B are bases containing x , and B' are bases that does not contain x .

    Thus, x \in B_{1} \cup B_{2} \cup B_{3} \cup \dots \subseteq \mathbb{X} - \overline{A}

    Then B_{1} \cup B_{2} \cup B_{3} \cup \dots that is a open set can be generated by all the bases containing x , however, that does not intersects A . So, x \in \overline{A} .

 ◻ :::

::: {#def:Neighbourhood .definition} Definition 1.32 (neighbourhood). 33 If we say U is a neighbourhood of x in \mathbb{X} , then U is an open set in \mathbb{X} containing x :::

::: {#def:LimitPoint .definition} Definition 1.33 (limit point, point of accumulation, cluster point). 34 If A is a subset of topological space \mathbb{X} .We say that x is a limit point of A if and only if every open sets containing x intersects A with some points other than x .

This condition is also equivalent to the condition that if x is a limit point of A if and only if x \in \overline{A-\{x\}} :::

::: theorem Theorem 1.16. 35 Let A be a subset of topological space \mathbb{X} ; let A' be the set of all limit points of A . Then \overline{A} = A \cup A' :::

::: corollary Corollary 1.1. 36 A subset of a topological space is closed if and only if it contains all its limit point. :::

::: {#def:Converge .definition} Definition 1.34 (converge). 37 We say that a sequence of x_{1}, x_{2}, x_{3} \dots converge to x . When for every neighbourhood U of x , there exists a positive integer N , such that for all n > N , x_{n} \in U . :::

::: {#def:HausdorffSpace .definition} Definition 1.35 (Hausdorff space). A topological space is called a Hausdorff space, if for every distinct x_{1} , x_{2} in \mathbb{X} , there exists disjoint neighbourhood of U_{1} , U_{2} of x_{1} , x_{2} in \mathbb{X} . :::

::: theorem Theorem 1.17. 3839 Every finite point set in a Hausdorff space \mathbb{X} is closed. :::

::: proof Proof. Let A be a finite point set in a Hausdorff space \mathbb{X} .

Suppose A only have one element. Then for every x\in\mathbb{X}-A , there exists a neighbourhood of x that does not intersect with A . So A is closed.

Suppose A is a closed finite point set. We take x_{0}\in\mathbb{X}-A . As finite union of closed set is closed, A\cup\{x_{0}\} is closed.

Then, from induction, all finite point set in a Hausdorff space is closed. ◻ :::

::: theorem Theorem 1.18. If \mathbb{X} is a Hausdorff space, then a sequence of points in \mathbb{X} converges to at most one point. :::

::: proof Proof. Suppose that the following sequence x_{1}, x_{2}, x_{3}\dots

Converge to more than one points say y_{1}, y_{2}, y_{3}\dots

Then there exists n_{1}, n_{2}, n_{3}\dots, U_{1}, U_{2}, U_{3}\dots

Such that for n > n_{i} x_{n} \in U_{i}, y_{i} \in U_{i}

If we take disjoint U_{1}, U_{2} which is possible as this is a Hausdorff space.

Then the previews condition does not stand. So, every sequence of points in a Hausdorff space can only converge to at most one point. ◻ :::

::: {#def:Limit .definition} Definition 1.36 (limit). If a sequence x_{n} of points in Hausdorff space converge to the point x , we denote this by x_{n} \rightarrow x and we say the limit of x_{n} is x . :::

::: {#def:T1Axiom .definition} Definition 1.37 ( T_{1} axiom). The condition that all finite point set of a topological space is closed is called T_{1} axiom. :::

::: theorem Theorem 1.19. Let \mathbb{X} be a space satisfying the T_{1} axiom; let A be a subset of \mathbb{X} . Then the point x is a limit point of A if and only if every neighbourhood of x contains infinitely many points of A . :::

::: proof Proof. If every neighbourhood of x contains infinitely many point of A . Than every neighbourhood of x intersect with A with infinite element other than x , then x is a limit point of A .

If x is a limit point of A . Suppose that there exists a open set U containing x and intersect with A for finite many points. Let U' = U \cap ( A - x )

Then, x \notin U' . Let U'' = U - U'

Then U'' is open as U' is a finite point set and U'' = U - U' = U \cap (\mathbb{X} - U')

Also, x \in U'' . Thus, U'' is a open set containing x that only intersect A with x or do not intersect A . This is a contradiction of x is a limit point. Thus there does not exists a open set U containing x and intersect with A for finite many points. ◻ :::

::: theorem Theorem 1.20. 40 Every simply ordered set is a Hausdorff space in order topology. :::

::: theorem Theorem 1.21. 41 The product of two Hausdorff space is a Hausdorff space. :::

::: theorem Theorem 1.22. 42 A subspace of a Hausdorff space is a Hausdorff space. :::

1.6.1 Exercise

  1. Give an counter example why \overline{\cup A_{\alpha}} = \cup \overline{A_{\alpha}} dose not hold.

    ::: proof Proof. Consider the X be the K-topology on the real line.

    Let

    \begin{aligned} A_{n} &=& (\frac{1}{n+1},\frac{1}{n}), n \in \mathbb{Z}_{+} \\ A &=& \cup A_{n} \end{aligned}

    Then

    \begin{aligned} \overline{A_{n}} &=& [\frac{1}{n+1},\frac{1}{n}] \\ \cup \overline{A_{n}} &=& (0,1] \end{aligned}

    However, as every neighbourhood of 0 intersect \cup A_{\alpha} . 0 \in \overline{\cup A_{\alpha}} .

    Thus, \overline{\cup A_{\alpha}} \neq \cup \overline{A_{\alpha}} ◻ :::

  2. Prove that \overline{A-B} \supseteq \overline{A} - \overline{B}

    ::: proof Proof. If x \in \overline{A} - \overline{B} . Then

    \begin{aligned} x \in \overline{A}, x \notin \overline{B} \end{aligned}

    .

    Thus for open set U containing x

    \begin{aligned} &\exists& U_{1} \cap B = \emptyset \\ &\forall& U \cap A \neq \emptyset \end{aligned}

    Suppose that x \notin \overline{A-B} . Then\

    \begin{aligned} \exists U_{0} \cap (A-B) = \emptyset \end{aligned}

    Thus,

    \begin{aligned} U_{0} \cap A \subseteq B \end{aligned}

    Thus,

    \begin{aligned} U_{1} \cap B &=& \emptyset \\ U_{1} \cap U_{0} \cap A &=& \emptyset \end{aligned}

    As U_{1} \cap U_{0} is an open set containing x , so there is contradiction with x \in \overline{A} . Thus x \in \overline{A-B} . ◻ :::

  3. A diagonal[]{#def:Diagonal label="def:Diagonal"} is a subset \Delta = \{ x \times x | x \in \mathbb{X} \} of the product topology \mathbb{X \times X} where \mathbb{X} is a topological space. Show that the diagonal is closed in \mathbb{X \times X} if and only if \mathbb{X} is a Hausdorff space.

    ::: proof Proof. If \mathbb{X} is a Hausdorff space. For every element x \times y of \mathbb{X} \times \mathbb{X} that not in \Delta . We take disjoint set U_{x},U_{y} where x \in U_{x}, y \in U_{y} . Then \mathbb{X} \times \mathbb{X} - \Delta = \cup_{x \neq y} U_{x} \times U_{y} . Where \cup_{x \neq y} U_{x} \times U_{y} is an open set. Thus \Delta is a closed set.

    Conversely, if \Delta is a closed set, suppose that \mathbb{X} is not a Hausdorff space. Then there exists distinct x, y such that every neighbourhood of x and y intersect. Let \mathbb{B} be a basis of topology of \mathbb{X} . Then x \times y \in \mathbb{X} \times \mathbb{X} - \Delta . However we cannot find B_{1}, B_{2} \in \mathbb{B}, x \times y \in B_{1} \times B_{2} \subset \mathbb{X} \times \mathbb{X} - \Delta . Then \Delta is not a closed set. So there is a contradiction, then \mathbb{X} must be a Hausdorff space. ◻ :::

  4. Prove that T_{1} axiom is equivalent to the condition such that for every distinct pair x,y of \mathbb{X} , there exists neighbourhood of x does not contain y .

    ::: proof Proof. First if T_{1} axiom hold, then for every pair x,y , the neighbourhood \mathbb{X}-\{y\} of x does not contain y , so the second condition hold.

    Conversely, if the second condition hold. Suppose that we can find a finite points set say \{x_{1}, x_{2}, x_{3} \dots }, then there must exists x \in \{x_{1}, x_{2}, x_{3} \dots } such that the set \{x\} is not closed. Then \overline{\{x\}} - \{x\} \neq \emptyset . Let y \in \overline{\{x\}} - \{x\} , then every neighbourhood of y must contain x , this is a contradiction to the second condition, so the T_{1} axiom must hold. ◻ :::

  5. If A \subseteq \mathbb{X} , we define the boundary[]{#def:Boundary label="def:Boundary"} of A by the equation \text{Bd} A = \overline{A} \cap \overline{\mathbb{X}-A}

    1. Show that \text{Int}A and \text{Bd} A are disjoint and \overline{A} = \text{Int} A \cup \text{Bd} A .

      ::: proof Proof. For every x \in \text{Bd}A , every open set contain x must intersect A and \mathbb{X}-A so, there is no open set U contain x , U \subseteq A .

      For every x' \in \text{Int}A , there exists U' \subseteq A , so \text{Bd}A and \text{Int}A are disjoint sets.

      For every x \in \overline{A} , x \in \text{Bd}A or x \notin \text{Bd}A . We discuss the condition that x \notin \text{Bd}A .

      Then x \notin \overline{\mathbb{X}-A} , then there exists a open set U containing x , that does not intersect with \mathbb{X}-A . Thus U \subseteq A , thus x \in \text{Int}A . So \overline{A} \subseteq \text{Int} A \cup \text{Bd} A .

      Then, \text{Bd}A \subseteq \overline{A} , \text{Int}A \subseteq A \subseteq \overline{A} . Thus, \overline{A} \supseteq \text{Int} A \cup \text{Bd} A

      So, \overline{A} = \text{Int} A \cup \text{Bd} A ◻ :::

    2. Show that \text{Bd}A = \emptyset if and only if A is both open and closed.

      ::: proof Proof. So, \text{Int}A = \overline{A} , then \text{Bd}A = \emptyset follows directly from \overline{A} = \text{Int} A \cup \text{Bd} A . ◻ :::

    3. Show that U is open if and only if \text{Bd}U = \overline{U}-U .

      ::: proof Proof. Suppose U is open. Then \overline{\mathbb{X}-U} = \mathbb{X} - U . Then for every x \in U , x \notin \mathbb{X} - U, x \notin \overline{\mathbb{X}-U} . Thus \overline{U} \cap \overline{\mathbb{X}-U}=\overline{U}-U .

      Conversely, suppose \text{Bd}U = \overline{U}-U . Then for every x \in U , x \notin \text{Bd}U . Then as \overline{U} = \text{Int}U\cup \text{Bd}U , x \in \text{Int}U . So \text{Int}U \supseteq U . Thus U = \text{Int}U . Thus, U is open. ◻ :::

1.7 Continuous Function

::: {#def:ContinuousRelativeTo .definition} Definition 1.38 (continuous). []{#def:Continuous label="def:Continuous"}43 Let \mathbb{X} and \mathbb{Y} be topological spaces. A function f: \mathbb{X}\rightarrow \mathbb{Y} is said to be continuous if for each open subset V of \mathbb{Y} , the set f^{-1}(V) is an open subset of \mathbb{X} . :::

::: theorem Theorem 1.23. Let \mathbb{X} and \mathbb{Y} be topological spaces; let f: \mathbb{X}\rightarrow\mathbb{Y} . Then the following are equivalent.

  1. f is continuous.

  2. For every subset A of X , one has f(\overline{A})\subseteq\overline{f(A)} .

  3. For every closed set B of \mathbb{Y} , the set f^{-1}(B) is closed in \mathbb{X} .

  4. For each x\in\mathbb{X} and each neighbourhood of V of f(x) , there is a neighbourhood U of x such that f(U) \subseteq V . :::

::: proof Proof. 1 \Rightarrow 3:

Let A be a open set in \mathbb{Y} . f^{-1}(\mathbb{Y}-A) = \mathbb{X} - f^{-1}(A) .

3 \Rightarrow 1:

Let A be a closed set in \mathbb{Y} . f^{-1}(\mathbb{Y}-A) = \mathbb{X} - f^{-1}(A) .

1 \Rightarrow 2:

For x \in \overline{A} , we take a open set f(x) \in U \subseteq \mathbb{Y} . Thus x \in f^{-1}(U) \cap A \neq \emptyset . Thus U \cap f(A) \neq \emptyset . So f(x) \in \overline{f(A)} . Thus f(\overline{A})\subseteq\overline{f(A)} .

2 \Rightarrow 3:

Suppose f is not continuous. Then there must exists V , such that f^{-1}(V) = U is not closed. Thus \overline{U} \supset B = f^{-1}(A) . Thus f{\overline{B}} \supset A . However f(\overline{B}) \subseteq \overline{f(B)} = A . There is a contradiction. So f must be continuous.

1 \Rightarrow 4:

For every neighbourhood V of f(x) , f^{-1}(V) is a neighbourhood of x that f(f^{-1}(V)) \subseteq V .

4 \Rightarrow 1:

We take a open set V of \mathbb{Y} . Let S be the collection of all open set U in \mathbb{X} such that f(U) \subseteq V . The set cannot be empty unless f^{-1}(V) = \emptyset . Let U_{0} denote the union of all the element in S . We prove that U_{0} = f^{-1}(V) .

For all element x \in U_{0} , f(x) \in V . Thus U_{0} \subseteq f^{-1}(V) .

For all element x \in f^{-1}(V) . There is a U' such that x \in U', f(U')\subseteq V . This follows from the condition 4. Thus U' \in S . Thus x \in U_{0} . Thus U_{0} \subseteq f^{-1}(V) . As U_{0} is union of open set, U_{0} is also open. Thus, f^{-1}(V) is also open.

Thus f is continuous. ◻ :::

::: {#def:Homeomorphism .definition} Definition 1.39 (homeomorphism). 44 Let \mathbb{X} and \mathbb{Y} be topological space; let f: \mathbb{X} \rightarrow \mathbb{Y} be a bijection. If both the function f and the inverse function f^{-1}: \mathbb{Y} \rightarrow \mathbb{X} are continuous, then f is called a homeomorphism :::

::: {#def:TopologicalImbedding .definition} Definition 1.40 (topological imbedding). Suppose that f: \mathbb{X} \rightarrow \mathbb{Y} is an injective continuous map, where \mathbb{X} and \mathbb{Y} are topological spaces. Let \mathbb{Z} be the image set f(\mathbb{X}) , considered as a subspace of \mathbb{Y} ; then the function f': \mathbb{X} \rightarrow \mathbb{Z} obtained by restricting the range of f is bijective. If f' happens to be a homeomorphism of \mathbb{X} with \mathbb{Z} , we say that the map f: \mathbb{X} \rightarrow \mathbb{Y} is a topological imbedding, or simply an imbedding, of \mathbb{X} in \mathbb{Y} . :::

::: {#theorem:RulesForConstructingContinuousFunctions .theorem} Theorem 1.24 (Rules for constructing continuous functions). Let \mathbb{X} , \mathbb{Y} , and \mathbb{Z} be topological spaces.

  1. (Constant function) If f: \mathbb{X} \rightarrow \mathbb{Y} maps all of \mathbb{X} into the single point y_{0} of \mathbb{Y} , then f is continuous.

  2. (Inclusion) If A is a subspace of \mathbb{X} , the inclusion function j: A \rightarrow \mathbb{X} is continuous.

  3. (Composites) If f: \mathbb{X}\rightarrow \mathbb{Y} and g:\mathbb{Y}\rightarrow\mathbb{Z} are continuous, then the map g \circ f: \mathbb{X} \rightarrow \mathbb{Z} is continuous.

  4. (Restricting the domain) If f: \mathbb{X} \rightarrow \mathbb{Y} is continuous, and if A is a subspace of \mathbb{X} , then the restriction function f|A : A \rightarrow \mathbb{Y} is continuous.

  5. (Restricting or expanding the range) Let f:\mathbb{X}\rightarrow\mathbb{Y} is continuous. Let \mathbb{Z} be a subspace of \mathbb{Y} containing the image f(\mathbb{X}) , the function h: \mathbb{X}\rightarrow \mathbb{Z} obtained by restricting the range of f is continuous. If \mathbb{Z} is a space having \mathbb{Y} as a subspace, then the function h: \mathbb{X}\rightarrow\mathbb{Y} obtained by expanding the range of f is continuous.

  6. (Local formulation of continuity) The map f: \mathbb{X}\rightarrow\mathbb{Y} is continuous if \mathbb{X} can be written as the union of open sets U_{\alpha} such set f|U_{\alpha} is continuous for each \alpha :::

::: proof Proof.

  1. f^{-1}(U) of any open set U is \mathbb{X} , thus f is continuous.

  2. For every open subset U of \mathbb{X} , j^{-1}(U) = U\cap A is continuous in A . Thus j is a continuous function.

  3. For every open subset U of \mathbb{Z} , f^{-1}(U) is open in \mathbb{Y} , and g^{-1}(f^{-1}(U)) is open in \mathbb{X} . Thus, g \circ f is continuous

  4. For every open subset U of \mathbb{Y} , f^{-1}(U) is open in \mathbb{X} , thus f^{-1}(U)\cap A is open in A . Thus the function f|A is continuous.

  5. If \mathbb{Z} is a subspace of \mathbb{Y} , then every open subset of \mathbb{Z} can be represented as U\cap\mathbb{Z} , where U is a open subset of \mathbb{Y} . Thus h^{-1}(U\cap\mathbb{Z})=g^{-1}(\mathbb{Z})\cap g^{-1}(U) = \mathbb{X}\cap g^{-1}(U) which is a open subset of X , thus h is continuous.

    If \mathbb{Y} is a subspace of \mathbb{Z} . Then we take a open subset U of \mathbb{Z} . h^{-1}(U) = g^(-1)(U\cap \mathbb{Y}) which is open in \mathbb{X} , thus h is continuous.

  6. if f|U_{\alpha} is continuous for each \alpha . For every open subset U of \mathbb{Y} . U = \cup_{\alpha} (U_{\alpha}\cap U) where U_{\alpha}\cap U is open both in U_{\alpha} and in \mathbb{Y} .

    Thus,

    \begin{aligned} f^{-1}(U) &=& f^{-1}(\cup_{\alpha} (U_{\alpha}\cap U)) \\ &=& \cup_{\alpha} ((f|U_{\alpha})^{-1}(U_{\alpha}\cap U)) \end{aligned}

    and each (f|U_{\alpha})^{-1}(U_{\alpha}\cap U) is open, thus f^{-1}(U) is open.

 ◻ :::

::: {#theorem:ThePastingLemma .theorem} Theorem 1.25 (The pasting lemma). 45 Let \mathbb{X} = A \cup B , where A,B are closed in \mathbb{X} . Let f: A\rightarrow \mathbb{Y} and g: B \rightarrow \mathbb{Y} be continuous. If f(x)=g(x) for every x\in A\cap B , then f,g combine to give a continuous function h: \mathbb{X}\rightarrow\mathbb{Y} , defined by setting h(x)=f(x),x\in A and h(x)=g(x),x\in B . :::

::: {#def:CoordinateFunctions .theorem} Theorem 1.26 (Maps into products). []{#theorem:MapsIntoProducts label="theorem:MapsIntoProducts"} 46 Let f: A \rightarrow \mathbb{X}\times\mathbb{Y} be given by the equation f(a) = (f_{1}(a),f_{2}(a))

Then, the function f is continuous if and only if the functions f_{1}: A \rightarrow \mathbb{X}, f_{2}: A \rightarrow \mathbb{Y} are continuous. :::

::: proof Proof. Let \pi_{1},\pi_{2} be the projection function

\begin{aligned} \pi_{1}&:& \mathbb{X}\times\mathbb{Y} \rightarrow \mathbb{X} \\ \pi_{2}&:& \mathbb{X}\times\mathbb{Y} \rightarrow \mathbb{Y} \end{aligned}

We first proof that if U is an open subset of \mathbb{X} \times \mathbb{Y} , f^{-1}(U) = f_{1}^{-1}(\pi_{1}(U)) \cap f_{2}^{-1}(\pi_{2}(U))

Let x \times y \in U , f^{-1}(x \times y) contains all a such that f(a) = x \times y .

Then for any a \in f^{-1}(x \times y) , a \in f_{1}^{-1}(\pi_{1}(x \times y)), a \in f_{2}^{-1}(\pi_{2}(x \times y)) .

Thus, f^{-1}(x \times y) \subseteq f_{1}^{-1}(\pi_{1}(x \times y)) \cap f_{2}^{-1}(\pi_{2}(x \times y)) .

Thus f^{-1}(U) \subseteq f_{1}^{-1}(\pi_{1}(U)) \cap f_{2}^{-1}(\pi_{2}(U)) .

Also, if a \in f_{1}^{-1}(\pi_{1}(x \times y)), a \in f_{2}^{-1}(\pi_{2}(x \times y)) , f_{1}(a) = x, f_{2}(a) = y .

Thus f(a) = x \times y . Thus a \in f^{-1}(x \times y) .

Thus f^{-1}(U) = f_{1}^{-1}(\pi_{1}(U)) \cap f_{2}^{-1}(\pi_{2}(U))

Let U be any open subset of \mathbb{X} \times \mathbb{Y}

f^{-1}(U) = f_{1}^{-1}(\pi_{1}(U)) \cap f_{2}^{-1}(\pi_{2}(U))

Where f_{1}^{-1}(\pi_{1}(U)) and f_{2}^{-1}(\pi_{2}(U)) are both open set. Thus f^{-1}(U) is open. ◻ :::

1.7.1 Exercise

  1. Let \mathbb{Y} be an ordered set in the order topology. Let f,g: \mathbb{X} \rightarrow \mathbb{Y} be continuous, show that the set A \{ x | f(x) \le g(x) \} is closed in \mathbb{X} .

    ::: proof Proof. We only need to proof \mathbb{X}-A is open in \mathbb{X} . We take x \in \mathbb{X}-A . Thus f(x) > g(x) .

    Let U_{1},U{2} be the open set in \mathbb{Y} that met the following demand

    \begin{aligned} &&\forall y_{1} \in U_{1}, y_{2} \in U_{2}, y_{1} > y_{2} \\ &&f(x) \in U_{1}, g_{x} \in U_{2} \end{aligned}

    As \mathbb{Y} is an ordered set, U_{1},U{2} must exist.

    Let U = f^{-1}(U_{1}) \cap g^{-1}(U_{2}) . It is obvious that U is a open set, and x \in U .

    Also, for any y \in U . f(y)>g(y) . Thus U \subseteq A . Thus A is an open set. ◻ :::

  2. Let \{A_{\alpha}\} be a collection of subsets of \mathbb{X} ; let \mathbb{X} = \cup_{\alpha}A_{\alpha} . Lef f: \mathbb{X}\rightarrow \mathbb{Y} ; suppose that f|A_{\alpha} is continuous for each \alpha . An indexed family of sets \{A_{\alpha}\} is said to be locally finite[]{#def:LocallyFinite label="def:LocallyFinite"} if each point x of \mathbb{X} has a neighbourhood that intersect A_{\alpha} for only finitely main values of \alpha . Show that if the family \{A_{\alpha}\} is locally finite and each A_{\alpha} is closed, then f is continuous.

    ::: proof Proof. For any closed subset U of \mathbb{Y} . Let V = \cup f|A_{\alpha}(U)

    We prove that V is closed, so, f is continuous.

    To prove that V is closed, we prove that \overline{V} = V . That is for any x \in \overline{V} , we prove x \in V . For any neighbourhood B if x , let C_{B} denote the set that contain all \alpha , such that f|A_{\alpha(U)} intersect with B . As B intersect with V , C_{B} can not be empty.

    Let \mathbb{C} = \{ C_{B} | B \text{ be a neighbourhood of } x \}

    As \{A_{\alpha}\} is locally definite, \mathbb{C} contain at least one element with finite elements.

    Also C_{B_{1} \cap B_{2}} \subseteq C_{B_{1}} \cap C_{B_{2}}

    Let \le be a partial order on the \mathbb{C} . If C_{B_{1}} \subseteq C_{B_{2}} , we say that C_{B_{1}} \ge C_{B_{2}} .

    If there is chain in \mathbb{C} C_{B_{1}} \le C_{B_{2}} \dots

    Let C_{B_{0}} be a element of \mathbb{C} with finite element. If C_{B_{0}} \subseteq C_{B_{1}}, C_{B_{0}} \subseteq C_{B_{2}} \dots . Then C_{B_{0}} is a upper bound of the chain.

    If C is not a subset of all element of the chain. Then we construct a new set say D = \{ C_{B_{0} \cap B_{1}}, C_{B_{0} \cap B_{2}} \dots \}

    Let \mathbb{D} = \{ C_{D_{1} \cap D_{2} \cap \dots } | C_{D_1}, C_{D_2}\dots \in D \}

    As C_{B_{0}} is a finite set, D is a finite set, \mathbb{D} is also a finite set. Thus there must be a maximal element E \in \mathbb{D} that is the subset of all element of \mathbb{D} . Then E is a subset of all element of the chain. Thus E is a upper bound of the chain.

    Thus, there must be a maximal element C_{F} of \mathbb{C} , that is a subset of all element of \mathbb{C} .

    Let G be the set be the union of all element of C_{F} .

    As C_{F} is finite, G is closed. And all neighbourhood of x intersect with G . Thus x \in G

    As G is a subset of V , x \in V . So V is closed. And f is a continuous function on \mathbb{X} . ◻ :::

  3. Let A be a subset of topological space \mathbb{X} , let \mathbb{Y} be a Hausdorff space. Let f: A \rightarrow \mathbb{Y} be a continuous function. Let g: \overline{A} \rightarrow \mathbb{Y} also be a continuous function where g(x) = f(x), x \in A . Prove that g us uniquely determined by f .47

    ::: proof Proof. Say g and h are two distinct function that met the demand.

    So there exist x_{0} such that g(x_{0}) \neq h(x_{0}) .

    As \mathbb{Y} is a Hausdorff space, so there exist adjoint open subset g(x_{0}) \in U and h(x_{0}) \in V .

    Then g^{-1}(U) and h^{-1}(V) are both open subset of \mathbb{X} that contain x_{0} .

    If g^{-1}(U) \cap h^{-1}(V) \cap A \neq \emptyset . Then there exist x_{1} \in g^{-1}(U) \cap h^{-1}(V) \cap A such that g(x_{1}) \in U and h(x_{1}) \in V and g(x_{1})=h(x_{1}) . However U and V are disjoint. So there is a contradiction.

    As ^{-1}(U) \cap h^{-1}(V) is a open subset contain x_{0} . So ^{-1}(U) \cap h^{-1}(V) must intersect with A . So it is impossible that g^{-1}(U) \cap h^{-1}(V) \cap A = \emptyset .

    So g=h . ◻ :::

1.8 Metric Topology

::: {#def:Distance .definition} Definition 1.41 (metric). []{#def:Metric label="def:Metric"} A metric on a set \mathbb{X} is a function d: \mathbb{X} \times \mathbb{X} \rightarrow \mathbb{R} having the following properties:

  1. d(x,y) \geq 0 for all x,y \in \mathbb{X} ; equality hold if and only if x = y

  2. d(x,y) = d(y,x), \forall x,y \in \mathbb{X}

  3. (Triangle Inequality) d(x,y) + d(y,z) \geq d(x,z), \forall x,y,z \in \mathbb{X}

Given a metric d on \mathbb{X} , the number d(x,y) is often called the distancebetween x and y in the metric d . :::

::: {#def:EpsilonBallCenteredAtX .definition} Definition 1.42 ( \epsilon -ball centered at x ). *48 Given metric d on a set \mathbb{X} and \epsilon > 0 . The set

B_{d}(x,\epsilon) = \{ y | d(x,y) < \epsilon \}

is called \epsilon -ball centered at x .* :::

::: {#def:MetricTopology .definition} Definition 1.43 (metric topology). If d is a metric on the set \mathbb{X} , then the collection of all \epsilon -balls B_{d}(x,\epsilon) , such that x \in \mathbb{X} and \epsilon > 0 , is a basis for a topology on \mathbb{X} , called the metric topology induced by d . :::

::: {#def:MetricSpace .definition} Definition 1.44 (metrizable). []{#def:Metrizable label="def:Metrizable"} If \mathbb{X} is topological space, \mathbb{X} is said to be metrizable if there exists a metric d on the set \mathbb{X} that induces the topology of \mathbb{X} . A metric spaceis a metrizable space \mathbb{X} together with a specific metric d that gives the topology of \mathbb{X} . :::

::: {#def:Bounded .definition} Definition 1.45 (bounded). Let \mathbb{X} be a metric space with metric d . A subset A of \mathbb{X} is said to be bounded if there is some number M such that d(a_{1},a_{2}) \leq M for every pair a_{1} and a_{2} if points of A . :::

::: {#def:Diameter .definition} Definition 1.46 (diameter). Let \mathbb{X} be a metric space with metric d . Let A be a bounded subset of \mathbb{X} . Then diameter is defined to be \mathrm{diam}A = \sup\{ d(a_{1},a_{2}) | a_{1}, a_{2} \in A \} :::

::: {#def:StandardBoundedMetric .theorem} Theorem 1.27. Let \mathbb{X} be a metric space with metric d . Define \overline{d}: \mathbb{X} \times \mathbb{X} \rightarrow \mathbb{R} by the equation \overline{d}(x,y) = min\{ d(x,y),1 \}

Then \overline{d} is a metric that induces the same topology as d .

The metric \overline{d} is called the standard bounded metriccorresponding to d :::

::: proof Proof. It is obvious that \overline{d} is a metric.

To prove that d and \overline{d} induces the same topology, it is suffice to prove that for all a \in X and \epsilon > 0 there exists \{a_{\alpha}\} and \{\epsilon_{\alpha}\} where \epsilon_{\alpha} \leq 1 such that B_{d}(a,\epsilon) = \bigcup B_{\overline{d}}(a_{\alpha},\epsilon_{\alpha})

For every x \in B_{d}(a,\epsilon) take a_{x} = x and \epsilon_{x} < min(\epsilon - d(a,x),1) . Then B_{d}(a,\epsilon) \supseteq B_{\overline{d}}(a_{x},\epsilon_{x}) as for all y \in B_{\overline{d}}(a_{x},\epsilon_{x})

\begin{aligned} d(a,y) &\leq& d(a,a_{x}) + d(a_{x},y) \\ &<& min(\epsilon - d(a,x),1) + d(a,a_{x}) \\ &\leq& \epsilon \end{aligned}

Thus B_{d}(a,\epsilon) = \bigcup_{ x \in B_{d}(a,\epsilon) } B_{\overline{d}}(a_{x},\epsilon_{x}) ◻ :::

::: {#def:Norm .definition} Definition 1.47 (norm). *Given x = (x_{1},\dots,x_{n}) in \mathbb{R}^{n} . The norm of x is defined by the equation

||x|| = ( x_{1}^{2}+\dots+x_{n}^{2} )^{\frac{1}{2}}

:::

::: {#def:EuclideanMetric .definition} Definition 1.48 (euclidean metric). The euclidean metric d on \mathbb{R}^{n} is defined by d(x,y) = ||x-y|| :::

::: {#def:SquareMetric .definition} Definition 1.49 (square metric). The square metric \rho on \mathbb{R}^{n} is defined by \rho(x,y) = max\{ |x_{1}-y_{1}|,\dots,|x_{n}-y_{n}| \} :::

::: lemma Lemma 1.8. Let d and d' be two metrics on the set \mathbb{X} ; let \mathbb{T} and \mathbb{T'} be the topology induced by d and d' respectively. Then \mathbb{T'} is finer than T if and only if for all x \in \mathbb{X} and \epsilon > 0 , there exists a \delta > 0 such that B_{d'}(x,\delta) \subseteq B_{d}(x,\epsilon) :::

::: proof Proof. If \mathbb{T'} is finer than \mathbb{T} . Then for all B_{d}(x,\epsilon) there exists a open set U that containing x such that U \subseteq B_{d}(x,\epsilon) . As \{{B_{d'}(x,\delta)}\} is a basis of T' , then there exists B_{d'}(x,\delta) \subseteq U that containing x .

If for all B_{d}(x,\epsilon) , there exists B_{d'}(x,\delta) \subseteq B_{d}(x,\epsilon) . Then as \{{B_{d'}(x,\epsilon)}\} and \{{B_{d}(x,\epsilon)}\} are both basis, then \mathbb{T'} is finer than T . ◻ :::

::: theorem Theorem 1.28. 49 The topologies on \mathbb{R}^{n} induced by the euclidean metric d and the square metric \rho are the same as the product topology on \mathbb{R}^{n} . :::

::: {#def:UniformMetric .definition} Definition 1.50 (uniform metric, uniform topology). *Given an index set J , and given points x = (x_{\alpha})_{\alpha \in J} and y = (y_{\alpha})_{\alpha \in J} of \mathbb{R}^{J} , let us define a metric \overline{\rho} on \mathbb{R}^{J} by the equation

\overline{\rho}(x,y) = \sup\{ \overline{d}(x_{\alpha},y_{\alpha}) | \alpha \in J \}

where \overline{d} is the standard bounded metric on \mathbb{R} . \overline{\rho} is called the uniform metric on \mathbb{R}^{J} , and the topology it induces is called the uniform topology* :::

::: theorem Theorem 1.29. 50 The uniform topology on \mathbb{R}^{J} is finer than the product topology and is coarser than the box topology. :::

::: theorem Theorem 1.30. *Let \overline{d}(a,b) = \min\{ |a-b|,1 \} be the standard bounded metric on \mathbb{R} . If x nad y are two points of \mathbb{R}^{\omega} , define

D(x,y) = \sup\left\{ \frac{\overline{d}(x_{i},y_{i})}{i} \right\}

Then D is a metric that induces the product topology on \mathbb{R}^{\omega} :::

::: proof Proof. The properties of a metric are satisfied trivially except for the triangle inequality, which is proved by noting that for all i ,

\begin{aligned} \frac{\overline{d}(x_{i},z_{i})}{i} &\le& \frac{\overline{d}(x_{i},y_{i})}{i} + \frac{\overline{d}(y_{i},z_{i})}{i} \\ &\le& D(x,y) + D(y,z) \end{aligned}

so that

\sup\left\{ \frac{\overline{d}(x_{i},z_{i})}{i} \right\} \le D(x,y) + D(y,z)

The fact that D gives the product topology requires a little more work. First, let U be open in the metric topology and let x \in U ; we find an open set V in the product topology such that x \in V \supseteq U . Choose an \epsilon-ball B_{D}(x,\epsilon) lying in U . Then choose N large enough that \frac{1}{N} < \epsilon . Finally, let V be the basis element for the product topology V = (x_{1}-\epsilon,x_{1}+\epsilon) \times \dots \times (x_{N}-\epsilon,x_{N}+\epsilon) \times R \times R \times \dots

We assert that V \in B_{D}(x,\epsilon) : Given any y in \mathbb{R}^{\omega} \frac{\overline{d}(x_{i},y_{i})}{i} \le \frac{1}{N}, \forall i \ge N

Therefore, D(x,y) \le \max\left\{ \frac{\overline{d}(x_{1},y_{1})}{1},\dots,\frac{\overline{d}(x_{N},y_{N})}{N},\frac{1}{N} \right\}

If y is in V , this expression is less than \epsilon , so that V \subseteq B_{D}(x,\epsilon) , as desired.

Conversely, consider a basis element U = \prod_{i \in \mathbb{Z}_{+}} U_{i} for the product topology, where U_{i} is open in \mathbb{R} for i = \alpha_{1},\dots,\alpha_{n} and U_{i} = \mathbb{R} for all other indices i . Given x \in U , we find an open set V of the metric topology such that x \in V \supseteq U . Choose an interval (x_{i}-\epsilon_{i},x_{i}+\epsilon_{i}) in \mathbb{R} centered about x_{i} and lying in U_{i} for i = \alpha_{1},\dots,\alpha_{n} ; choose each \epsilon_{i} \le 1 . Then define \epsilon = \min \left\{ \frac{\epsilon_{i}}{i} | i = \alpha_{1},\dots,\alpha_{n} \right\}

We assert that x \in B_{D}(x,\epsilon) \subseteq U

Let y be a point of B_{D}(x,\epsilon) . Then for all i \frac{\overline{d}(x_{i},y_{i})}{i} \le D(x,y) < \epsilon

Now if i = \alpha_{1},\dots,\alpha_{n} , then \epsilon \le \frac{\epsilon_{i}}{i} , so that \overline{d}(x_{i},y_{i}) < \epsilon_{i} \le 1 ; it follows that |x_{i}-y_{i}| < \epsilon_{i} . Therefore y \in \prod U_{i} , as desired. ◻ :::

::: {#def:HilbertCube .definition} Definition 1.51 (Hilbert Cube). The set H = \prod_{ n \in \mathbb{Z}_{+} } [0,\frac{1}{n}] is called Hilbert cube :::

::: {#def:L2Topology .definition} Definition 1.52 (l^{2}-topology). Let \mathbb{X} be the subset of \mathbb{R}^{\omega} consisting of all sequences x such that \sum x_{i}^{2} converges.

Then the formula d(x,y) = \left[ \sum_{i=1}^{\infty}(x_{i}-y_{i})^{2} \right]^{\frac{1}{2}} defines a metric on \mathbb{X} . The topology induced by d is called the l^{2}-topology. :::

::: {#def:FirstCountabilityAxiom .definition} Definition 1.53 (countable basis at point x ). *[]{#def:CountableBasisAtPointX label="def:CountableBasisAtPointX"} A space is said to be have countable basis at point x if there is a countable collection \{ U_{n} \}_{n \in \mathbb{Z}_{+}} of neighbourhoods of x such that any neighbourhood U of x contains at least on of the sets U_{n} . A space \mathbb{X} that has a countable basis at each of its point is said to satisfy the first countability axiom * :::

::: theorem Theorem 1.31. Let f: \mathbf{X} \rightarrow \mathbf{Y} be metrizable with metric d_{\mathbf{X}} and d_{\mathbf{Y}} , respectively. Then continuity of f is equivalent to the requirement that given x \in \mathbb{X} and given \epsilon > 0 , there exists \delta > 0 such that d_{\mathbf{X}}(x,y) < \delta \implies d_{\mathbb{Y}}(f(x),f(y)) < \epsilon :::

::: proof Proof. Suppose f is continuous. Given x and \epsilon , consider the set f^{-1}(B(f(x),\epsilon)) which is open in \mathbb{X} and contains the point x . It contains some \delta -ball B(x,\delta) centered at x . If y is in this \delta -ball, then f(y) is in this \delta -ball as desired.

Conversely, suppose that the \epsilon-\delta condition is satisfied. Let V be open in \mathbb{Y} ; we show that f^{-1}(V) is open in \mathbb{X} . Let x be a point of the set f^{-1}(V) . Since f(x) \in V there is an \epsilon -ball B(f(x),\epsilon) centered at f(x) and contained in V . By the \epsilon-\delta condition, there exists a \delta -ball centered at x such that f(B(x,\delta)) \subseteq B(f(x),\epsilon) . Then B(x,\delta) is a neighbourhood of x contained in f^{-1}(V) , so that f^{-1}(V) is open, as desired. ◻ :::

::: {#def:TheSequenceLemma .lemma} Lemma 1.9 (The sequence lemma). 51 Let \mathbb{X} be a topological space; let A \subseteq \mathbb{X} If there is a sequence of points of A converging to x , then x \in \overline{A} , the converse holds if \mathbb{X} is metrizable. :::

::: theorem Theorem 1.32. 52 Let f: \mathbb{X} \rightarrow \mathbb{Y} . If the function f is continuous, then for every convergent sequence x_{n} \rightarrow x , the sequence f(x_{n}) converges to f(x) . The converse holds if \mathbb{X} is metrizable. :::

::: lemma Lemma 1.10. 53 The addition, subtraction, and multiplication operations are continuous functions from \mathbb{R} \times \mathbb{R} into \mathbb{R} ; and the quotient operation is continuous function from \mathbb{R} \times (\mathbb{R} - \{0\}) into \mathbb{R} . :::

::: theorem Theorem 1.33. 54 If \mathbb{X} is a topological space, and if f,g: \mathbb{X} \rightarrow \mathbb{R} are continuous functions, then f + g , f - g and f \cdot g are continuous. If g(x) \neq 0 for all x , then \frac{f}{g} is continuous. :::

::: {#def:ConvergeUniformly .definition} Definition 1.54 (converge uniformly). Let f_{n}: \mathbb{X} \rightarrow \mathbb{Y} be a sequence of functions from the set \mathbb{X} to the metric space \mathbb{Y} . Let d be the metric for \mathbb{Y} . We say that the sequence (f_{n}) converges uniformly to the function f : \mathbb{X} \rightarrow \mathbb{Y} if given \epsilon > 0 , there exists an integer N such that d(f_{n}(x),f(x)) < \epsilon for all n > N and all x \in \mathbb{X} :::

::: {#def:UniformLimitTheorem .theorem} Theorem 1.34 (Uniform limit theorem). Let f_{n} : \mathbb{X} \rightarrow \mathbb{Y} be a sequence of continuous functions from the topological space \mathbb{X} to the metric space \mathbb{Y} . If (f_{n}) converges uniformly to f , then f is continuous. :::

::: {#def:IsometricImbedding .definition} Definition 1.55 (isometric imbedding). Let \mathbb{X} and \mathbb{Y} be metric spaces with metric d_{\mathbb{X}} and d_{\mathbb{Y}} , respectively. Let f: \mathbb{X} \rightarrow \mathbb{Y} have the property that for every pair of points x_{1} , x_{2} of \mathbb{X} , and d_{\mathbb{Y}}(f(x_{1}),f(x_{2})) = d_{\mathbb{X}}(x_{1},x_{2})

f is an topological imbedding and is called an isometric imbedding of \mathbb{X} in \mathbb{Y} :::

Footnotes

  1. The set \mathbb{U} can form a topology because of the definition of topology is intersection of finite sub collection. If this can be intersection of infinite sub collection, \mathbb{U} will not be a topology. ↩

  2. Note that this expression may not be unique. ↩

  3. We omit the proof of this lemma as it is obvious. ↩

  4. We omit the proof of this lemma as it is obvious. ↩

  5. Whenever we consider \mathbb{R} , we shall suppose it is given this topology unless we specifically state otherwise. ↩

  6. We omit the proof of this change@code@ as it is obvious. ↩

  7. It is obvious that \mathbb{T} is a topology, we just omit the proof here. ↩

  8. It is obvious that \mathbb{T'} is also a topology, we just omit the proof here. ↩

  9. The standard topology on \mathbb{R} is an order topology derived from the usual order on \mathbb{R} . ↩

  10. open rays are always open sets in the order topology ↩

  11. the open rays also formed a subbasis of the order topology ↩

  12. We omit the proof of this change@code@ as it is obvious. ↩

  13. We omit the proof of this change@code@ as it is obvious. ↩

  14. In the finite case, the product topology and box topology are the same, however they differ when \mathbb{X} is a infinite cartesian product. ↩

  15. This is also called a projection mappingin a cartesian product. ↩

  16. A index set was the set \{1,\dots,n\} or the set \mathbb{Z}_{+} . ↩

  17. It is assumed that it is given product topology when considering \prod X_{\alpha} unless it state specifically. ↩

  18. We omit the proof of this change@code@ as it is obvious. ↩

  19. We omit the proof of this change@code@ as it is obvious. ↩

  20. We omit the proof of this change@code@ as it is obvious. ↩

  21. We omit the proof of this change@code@ as it is obvious. ↩

  22. We omit the proof of this change@code@ as it is obvious. ↩

  23. If \mathbb{X} is an ordered set in the order topology, and \mathbb{Y} is a subset of \mathbb{X} . The order relation, when restricted to \mathbb{Y} , makes \mathbb{Y} into and ordered set. However, the resulting order topology on \mathbb{Y} need not be the same as the topology that \mathbb{Y} inherits as a subspace of \mathbb{X} . ↩

  24. *the dictionary means for X_{1}, X_{2} \in \mathbb{Y} = \mathbb{X}_{1} \times \mathbb{X}_{2} \times \mathbb{X}_{3} \dots which:

    \begin{aligned} X_{1} &=& (x_{1},x_{2},x_{3}\dots)\\ X_{2} &=& (x_{1}',x_{2}',x_{3}'\dots) \end{aligned}

    X_1 > X_2 only when

    \begin{aligned} && \exists k \in \mathbb{Z}_{+}, \forall i \in \mathbb{Z}_{+}, 0 < i < k \\ && x_{i} = x_{i}'\\ && x_{k} > x_{k}' \end{aligned}
    ↩

  25. Given \mathbb{X} is an ordered set in the order topology and \mathbb{Y} is a subset of \mathbb{X} , we shall assume that \mathbb{Y} is given the subspace topology unless we specifically state otherwise. ↩

  26. what does \mathbb{X} , \mathbb{X'} , \mathbb{Y} , \mathbb{Y'} really mean here?? I do not know, so I just put the exercise here without a proof. ↩

  27. The prove of this set is countable is typically similar to Cantor's enumeration of a countable collection of countable sets. ↩

  28. A set can be open, or closed, or both, or neither ↩

  29. We omit the proof of this change@code@ as it is obvious. ↩

  30. As the proof is similar to the case in the open set, so we omit the proof here. ↩

  31. We omit the proof of this change@code@ as it is obvious. ↩

  32. As the closure of A in \mathbb{X} and the closure A in \mathbb{Y} will sometimes be different. We always use \overline{A} to denote the closure of A in \mathbb{X} ↩

  33. Some other mathematicians use neighbourhood to say that U merely contains an open set containing x . The book does not give a formal definition for the word merely, and I am not sure either. ↩

  34. Note that, x may belong to A or not, this does not matter. ↩

  35. We omit the proof of this change@code@ as it is obvious. ↩

  36. We omit the proof of this change@code@ as it is obvious. ↩

  37. In real line, a sequence can not converge to multiple points, but for an arbitrary topological space, this is possible. ↩

  38. This implies that a sequence in a Hausdorff space cannot converge to multiple points. The following theorem prove this. ↩

  39. The condition every finite point set is closed is weaker than the Hausdorff space condition. For instance, the finite complement topology of \mathbb{R} met the condition of finite point set. However it is not a Hausdorff space. ↩

  40. We omit the proof of this change@code@ as it is obvious. ↩

  41. We omit the proof of this change@code@ as it is obvious. ↩

  42. We omit the proof of this change@code@ as it is obvious. ↩

  43. As the continuity of a function is different as the topological spaces are different. So if we want to emphasis this fact, we say that f is continuous relativeto specific topologies on \mathbb{X} and \mathbb{Y} . ↩

  44. A equivalent way to define homeomorphism, is that for any open subset U of \mathbb{X} , f(U) is open if and only if U is open. ↩

  45. The proof of this theorem is similar to the "Local formulation of continuity" condition of "Rules for constructing continuous functions", so we omit the proof here. ↩

  46. The map f_{1},f_{2} are called the coordinate functionsof f ↩

  47. It is possible that g does not exist.

    Let \mathbb{X} be the real line with order topology. Let \mathbb{Y} be \{0,1\} .

    Let A = \mathbb{X} - \{0\} .

    Let, f(x) = \begin{cases} 1, & x > 0 \\ 0, & x < 0 \end{cases}

    So, it is obvious that f is a continuous function on \mathbb{X} . However g does not exist in this case. ↩

  48. When no confusion will arise, the metric d may be omit in B_{d}(x,\epsilon) ↩

  49. We omit the proof of this change@code@ as it is obvious. ↩

  50. We omit the proof of this change@code@ as it is obvious. ↩

  51. We omit the proof of this change@code@ as it is obvious. ↩

  52. We omit the proof of this change@code@ as it is obvious. ↩

  53. We omit the proof of this change@code@ as it is obvious. ↩

  54. We omit the proof of this change@code@ as it is obvious. ↩

]]>https://alex1222.com/posts/2023-12-03-General-Topology-Notes.mdhttps://alex1222.com/posts/2023-12-03-General-Topology-Notes.mdSun, 03 Dec 2023 00:00:00 GMT<![CDATA[Upgrade Postgresql From 14 to 15]]>

Recently, I have upgraded my Postgresql database for my freshrss instance from 14 to 15. Here are the steps I followed to do it.

First, we need to local the binary of the old version of Postgresql. In most of the cases, it can be found by

which psql

In my case, it is /usr/local/bin/psql.

Then, we need to find the data directory of the old version of Postgresql. In my case, it is at /var/lib/postgres/14. And the postgresql.conf file is at /var/lib/postgres/14/data/postgresql.conf.

Then, we need to install the new version of Postgresql. I use nix to manage my packages, so I just change the version of Postgresql in my configuration.nix file and run nixos-rebuild switch.

After that, we need to find the binary of the new version of Postgresql. Which can be found by

which psql

Then, we need to find the data directory of the new version of Postgresql. In my case, it is at /var/lib/postgres/15. And the postgresql.conf file is which you should copy and modify from the old version of Postgresql.

We need to first stop the old version of Postgresql.

systemctl stop postgresql

And su to the postgres user.

sudo su - postgres

Then, we need to run the following command to first initialize the new version of Postgresql.

initdb -D /var/lib/postgres/15/

Then, we need

Then, we need to run the following command to migrate the data from the old version of Postgresql to the new version of Postgresql. With --check option, it will check if the migration is possible.

/usr/lib/postgresql/15/bin/pg_upgrade \
  --old-datadir=$OLD_DATADIR \
  --new-datadir=$NEW_DATADIR \
  --old-bindir=$OLD_BINDOR \
  --new-bindir=$NEW_BINDIR \
  --old-options '-c config_file=$OLD_CONFIG_FILE' \
  --new-options '-c config_file=$NEW_CONFIG_FILE' \
  --check

If the check is successful, we can run the following command to do the migration.

/usr/lib/postgresql/15/bin/pg_upgrade \
  --old-datadir=$OLD_DATADIR \
  --new-datadir=$NEW_DATADIR \
  --old-bindir=$OLD_BINDOR \
  --new-bindir=$NEW_BINDIR \
  --old-options '-c config_file=$OLD_CONFIG_FILE' \
  --new-options '-c config_file=$NEW_CONFIG_FILE'

After that, we need to check if the migration is successful. If it is successful, we can then remove the old version of Postgresql.

]]>https://alex1222.com/posts/2023-12-05-Upgrade-Postgresql-From-14-to-15.mdhttps://alex1222.com/posts/2023-12-05-Upgrade-Postgresql-From-14-to-15.mdSun, 03 Dec 2023 00:00:00 GMT<![CDATA[$10mn AI Mathematical Olympiad Prize Launches]]>

XTX Markets is launching a new $10mn challenge fund, the Artificial Intelligence Mathematical Olympiad Prize (AI-MO Prize). The fund intends to spur the development of AI models that can reason mathematically, leading to the creation of a publicly-shared AI model capable of winning a gold medal in the International Mathematical Olympiad (IMO).

The grand prize of $5mn will be awarded to the first publicly-shared AI model to enter an AI-MO approved competition and perform at a standard equivalent to a gold medal in the IMO.

There will also be a series of progress prizes, totalling up to $5mn, for publicly-shared AI models that achieve key milestones towards the grand prize.

The AI-MO Prize [..] will help compare different AI problem solving strategies at a technical level, in a manner that will be accessible and appealing to the broader public.

Terence Tao, UCLA

I am sure that many people will be following the AI-MO Prize with great excitement, to see when, in the future, AI will match the world's brightest young minds.

Gregor Dolinar, IMO President

]]>https://alex1222.com/posts/2023-12-01-10mn-AI-Mathematical-Olympiad-Prize-Launches.mdhttps://alex1222.com/posts/2023-12-01-10mn-AI-Mathematical-Olympiad-Prize-Launches.mdFri, 01 Dec 2023 00:00:00 GMT<![CDATA[让别人看到你裸眼视力的方法]]>

一种展示摘掉眼镜后真实视力的方法是，使用相机对准眼镜并拍摄照片，然后在iPhone上长按“AF锁定”按钮。这样一来，当眼镜被移开后，你通过相机屏幕看到的画面将反映佩戴者的实际视力状况。这个方法不仅能够帮助视力正常的人理解近视者的视觉体验，还可能对店铺或城市规划产生积极影响。其原理基于相机的自动对焦（AF）功能，通过在对准眼镜并长按以锁定自动对焦时，相机会根据眼镜的光学特性调整焦距。因此，当眼镜被移开时，相机的焦点仍然保持在根据眼镜调整的位置，从而通过相机屏幕模拟出佩戴者的视力状况。

AF Lock

]]>https://alex1222.com/posts/2023-11-29-%E8%AE%A9%E5%88%AB%E4%BA%BA%E7%9C%8B%E5%88%B0%E4%BD%A0%E8%A3%B8%E7%9C%BC%E8%A7%86%E5%8A%9B%E7%9A%84%E6%96%B9%E6%B3%95.mdhttps://alex1222.com/posts/2023-11-29-%E8%AE%A9%E5%88%AB%E4%BA%BA%E7%9C%8B%E5%88%B0%E4%BD%A0%E8%A3%B8%E7%9C%BC%E8%A7%86%E5%8A%9B%E7%9A%84%E6%96%B9%E6%B3%95.mdWed, 29 Nov 2023 00:00:00 GMT<![CDATA[Understanding IP Address - Some Basics]]>

0.1 Contents

0.2 Introduction

In the expansive and complex realm of the internet, a foundational principle governs communication - IP addresses. Short for Internet Protocol, these addresses serve as digital fingerprints for devices interconnected within the vast web. This article seeks to simplify the intricacies surrounding IP addresses, illuminating key aspects that act as a guide for those navigating the digital domain.

0.3 Understanding IP Addresses:

At its essence, an IP address is a unique numerical label assigned to each device in a computer network that communicates through the Internet Protocol. Analogous to a home address in the physical world, it functions as a digital identifier, ensuring that data packets navigate the intricate web to reach their intended destinations.

0.4 Types of IP Addresses:

In the expansive realm of IP addresses, familiarity with locating your public IP address becomes crucial for navigating the digital landscape. Two predominant versions, IPv4 (Internet Protocol version 4) and IPv6 (Internet Protocol version 6), dominate this scene. While IPv4 utilizes a 32-bit address scheme, limiting unique addresses, IPv6, its modernized successor, employs a 128-bit format, vastly expanding the pool of available addresses to meet the demands of a growing number of internet-connected devices.

0.5 Components of IP Addresses:

Analyzing the structure of IP addresses reveals a numeric composition separated by periods. In the case of IPv4, the format is xxx.xxx.xxx.xxx, with each "xxx" ranging from 0 to 255. Conversely, IPv6 addresses, represented in hexadecimal notation, such as 2001:0db8:85a3:0000:0000:8a2e:0370:7334, offer a more extensive and flexible range.

0.6 Static vs. Dynamic IP Addresses:

The dynamism of IP addresses extends to their classification as static or dynamic. A static IP address remains constant and is manually assigned, ideal for devices like servers requiring a consistent point of access. In contrast, dynamic IP addresses are automatically assigned by a network's DHCP server, offering flexibility for devices with intermittent network connections.

0.7 Significance of IP Addresses in Networking:

At the heart of computer networks lies the omnipresent IP address, a vital element ensuring the smooth flow of data. Whether it's a computer, smartphone, or any networked device, each relies on its unique IP address to establish structured communication, forming the backbone of connectivity across the internet.

0.8 Subnetting for Efficient IP Address Management:

To optimize IP address management, subnetting divides large networks into smaller, more manageable segments. Beyond efficient IP address utilization, subnetting enhances security and performance by strategically allocating addresses, reducing congestion, and improving data transmission efficiency.

0.9 Public and Private IP Addresses:

IP addresses serve dual roles as public and private identifiers. Public IP addresses, visible on the internet, represent a network to the external world. Private IP addresses operate within local networks, enabling internal device communication without exposure to the broader internet.

0.10 NAT and IP Address Conservation:

With the proliferation of devices, Network Address Translation (NAT) conserves IP addresses by allowing multiple devices within a local network to share a single public IP address. This solution extends the usability of IPv4 addresses, mitigating the threat of address exhaustion.

0.11 IP Address Geolocation:

A fascinating aspect of IP addresses is geolocation, providing the general location of a device based on its IP address. Despite not being pinpoint accurate, this feature finds applications in enhancing online security, delivering localized content, and services.

In the ever-evolving technological landscape, IP addressing evolves alongside progress. The Internet of Things (IoT) and the surge in smart devices increase the demand for IP addresses. IPv6, with its expansive address space, accommodates this surge, ensuring digital infrastructure evolves with technological advancements.

0.13 IPv6 Adoption:

Embracing IPv6 becomes crucial in addressing the limitations posed by IPv4. The 32-bit address scheme of IPv4 faces challenges as connected devices increase. IPv6, with its 128-bit format, accommodates this proliferation, enhancing the efficiency and resilience of internet infrastructure.

0.14 IP Address Management Tools:

As networks become intricate ecosystems, IP address management tools streamline allocation and monitoring processes. These tools play a critical role in network administration, ensuring smooth operations amid the increasing complexity of modern infrastructures, contributing to order, security, and optimal performance.

]]>https://alex1222.com/posts/2023-11-28-Understanding-IP-Address.mdhttps://alex1222.com/posts/2023-11-28-Understanding-IP-Address.mdTue, 28 Nov 2023 00:00:00 GMT<![CDATA[Discovery of the Ninth Dedekind Number: A Decades-Long Search and Computational Breakthrough]]>

After more than thirty years of searching, with the assistance of supercomputers, mathematicians have discovered the ninth Dedekind number. Dedekind numbers were defined by German mathematician Richard Dedekind in 1897 and are based on Boolean functions (values of false or true). These functions take n Boolean variables as input and generate another Boolean variable as output. The first six Dedekind numbers are quite simple, with D(1) being 2, followed by 3, 6, 20, 168, and 7581, with subsequent numbers growing increasingly larger. In 1991, one of the most powerful supercomputers at the time, the Cray-2, took 200 hours to calculate D(8). Thirty-two years later, mathematicians at the University of Paderborn in Germany used the Noctua 2 supercomputer and field-programmable gate arrays (FPGA) to calculate D(9) as 286,386,577,668,298,411,128,469,151,667,598,498,812,366 — a 42-digit number. Discovering D(10) will undoubtedly take a considerable amount of time as well.

]]>https://alex1222.com/posts/2023-11-20-The-20th-Dedekind-Number-Been-Discovered.mdhttps://alex1222.com/posts/2023-11-20-The-20th-Dedekind-Number-Been-Discovered.mdMon, 20 Nov 2023 00:00:00 GMT<![CDATA[Galois Theory Notes]]>

0.1 Contents

0.2 Intro

This is my personal note for Galois Theory.

This is a work in progress. When I have time, I will add more content to it.

0.3 Concepts of Ring

Ring is a set equipped with two binary operations, usually referred to as addition $+$ and multiplication $·$, that satisfy the following conditions:

  1. The set is an abelian group under the operation + with an identity element usually denoted as 0.

  2. The operation · is associative, meaning that for all a, b, c in the set, (a·b)·c equals a·(b·c).

  3. The operation · is distributive over the operation +, meaning that for all a, b, c in the set, a·(b+c) equals a·b + a·c and (a+b)·c equals a·c + b·c.

  4. The operation · has an identity element, usually denoted as 1.

  5. The operation · is closed, meaning that for all a, b in the set, a·b is also in the set.

  6. A ring is called a commutative ring if the operation · is commutative, meaning that for all a, b in the set, a·b equals b·a.

Field is a ring in which every non-zero element is invertible.
Subring is a subset of a ring that is a ring with the restriction of the ring operations.
Subfield is a subset of a field that is a field with the restriction of the field operations.
Ring Homomorphism is a function between two rings that preserves the operations of the rings.

A ring homomorphism is called an isomorphism if it is bijective.

Ideal is a subset of a ring that is closed under addition, negation, and multiplication by any element in the ring.
Principal Ideal is an ideal that is generated by a single element.
Integral Domain is a commutative ring with identity in which the product of any two non-zero elements is non-zero.
Quotient Ring is a ring constructed from a ring $R$ and an ideal $I$ of $R$. Where the elements of the quotient ring are the cosets of $I$ in $R$.

0.4 Definitions

]]>https://alex1222.com/posts/2023-11-10-Galois-Theory-Noteshttps://alex1222.com/posts/2023-11-10-Galois-Theory-NotesFri, 10 Nov 2023 00:00:00 GMT<![CDATA[《自然》撤回室温超导论文]]>

《自然》撤回了罗彻斯特大学助理教授 Ranga P. Dias 作为通讯作者的室温超导论文。 这是他被撤回的第三篇论文。 《自然》在撤回声明中称，因所发表的论文没有准确反映所研究材料的来源、 所进行的实验测量等原因，应多位署名作者的要求，撤回论文 Dias 本人尚未回复是否同意此次撤稿。 涉事论文的标题是《氮掺杂氢化镥的近环境超导性证据》 （Evidence of near-ambient superconductivity in a N-doped lutetium hydride）。 论文称，他们研发出一种含镥、氢、氮的材料，在约 20.6 摄氏度的室温和 10 千巴（约 1 万倍大气压） 的压力下表现出超导特性。该论文于 3 月 8 日发表在《自然》上。 随后国际上多个研究团队宣称，未能复现论文报告的结果。

]]>https://alex1222.com/posts/2023-11-08-%E8%87%AA%E7%84%B6-%E6%92%A4%E5%9B%9E%E5%AE%A4%E6%B8%A9%E8%B6%85%E5%AF%BC%E8%AE%BA%E6%96%87.mdhttps://alex1222.com/posts/2023-11-08-%E8%87%AA%E7%84%B6-%E6%92%A4%E5%9B%9E%E5%AE%A4%E6%B8%A9%E8%B6%85%E5%AF%BC%E8%AE%BA%E6%96%87.mdWed, 08 Nov 2023 00:00:00 GMT<![CDATA[Addition Of Algebraic Element]]>

This is some generalization of a question.

Link to the pdf file: Addition Of Algebraic Element

Link to the source code: Addition Of Algebraic Element

::: question Question 1. Proof that \sqrt{1} + \sqrt{2} + \sqrt{3} \dots + \sqrt{n} for n > 2 is irrational. :::

However, if we consider the linear combination of \sqrt{1}, \sqrt{2}, \sqrt{3} \dots \sqrt{n} over \mathbb{Q}, then it is likely that we will get a similar result as we kick out some special cases.

::: question Question 2. Given n positive integers, a_1 < a_2 < \dots < a_n, such that \sqrt{\frac{a_{j}}{a_{i}}} \notin \mathbb{Q} for all i < j and \sqrt{a_{i}} \notin \mathbb{Q} for all i. Also given \lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{Q}. Prove that \sum_{i=1}^{n} \lambda_{i} \sqrt{a_{i}} \in \mathbb{Q} if and only if \lambda_{1} = \lambda_{2} = \dots = \lambda_{n} = 0. :::

::: proof Proof. Suppose this stand for n=k, then it is suffice to prove n=k+1.

Suppose there exist \lambda_{1}, \lambda_{2} \dots \lambda_{k+1} \in \mathbb{Q} such that \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} = q \in \mathbb{Q}

If \lambda_{j} = 0, then -\lambda_{j}\sqrt{a_{j}} + \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} = q \in \mathbb{Q}, and we are done according to the assumption.

If \lambda_{j} \neq 0 for all j, then we have \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}} = q - \lambda_{k+1}\sqrt{a_{k+1}}

Clearly, q - \lambda_{k+1}\sqrt{a_{k+1}} is a root of a irreducible quadratic polynomial g \in \mathbb{Q}[x]. And the polynomial must have another root which is q + \lambda_{k+1}a_{k+1}.

Also, the polynomial f = \prod (x - ( \pm \lambda_{1}\sqrt{a_{1}} + \pm \lambda_{2}\sqrt{a_{2}} + \dots + \pm \lambda_{n}\sqrt{a_{n}} )) is also in \mathbb{Q}[x] by induction.

Also,

\begin{aligned} f( q - \lambda_{k+1}\sqrt{a_{k+1}} ) &= f( \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} ) \\ &= 0 \end{aligned}

As g is irreducible over \mathbb{Q}, g must divides f. Thus f( q + \lambda_{k+1}\sqrt{a_{k+1}} ) = 0

Thus, there exits \alpha_{1}, \alpha_{2} \dots \alpha_{n} which is either 1 or -1 such that \sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} = q + \lambda_{k+1}\sqrt{a_{k+1}}

Thus

\begin{cases} \sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} = q + \lambda_{k+1}\sqrt{a_{k+1}} \\ \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}} = q - \lambda_{k+1}\sqrt{a_{k+1}} \end{cases} \begin{aligned} \sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} + \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}} &= q + \lambda_{k+1}\sqrt{a_{k+1} }+ q - \lambda_{k+1}\sqrt{a_{k+1}} \\ \sum_{i=1}^{k} ( \alpha_{i} + 1 ) \lambda_{i}\sqrt{a_{i}} &= 2q \in \mathbb{Q} \\ \end{aligned}

By the induction assumption, we have (\alpha_{i} + 1)\lambda_{i} = 0 for all i \le k. As \lambda_{i} \neq 0 for all i, we have \alpha_{i} = -1 for all i \le k.

Thus,

\begin{aligned} \sum_{i=1}^{k} ( \alpha_{i} + 1 ) \lambda_{i}\sqrt{a_{i} } &= 0 \end{aligned}

which implies q = 0.

Thus,

\begin{aligned} \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} &= 0 \\ \sqrt{a_{k+1}} (\sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}}) &= 0 \\ \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{k+1}a_{i}} &= - \lambda_{k+1} a_{k+1} \in \mathbb{Q}\\ \end{aligned}

The only thing left to proof is that a_{k+1}a_{i}, i \le k satisfy the assumption in the question, and then the induction assumption will apply, which proves that \lambda_{i} = 0, i \le k.

Given any i < j, we have

\begin{aligned} \sqrt{\frac{a_{j}a_{k+1}}{a_{i}a_{k+1}}} &= \sqrt{\frac{a_{j}}{a_{i}}} \notin \mathbb{Q} \\ \sqrt{a_{j}a_{k+1}} &= a_{k+1} \sqrt{\frac{a_{j}}{a_{k+1}}} \notin \mathbb{Q} \end{aligned}

◻ :::

As, we does not use any special property of \mathbb{Q} other then the fact that it is a field. So, the proof should be valid for any field \mathbb{F} that characteristic equals 0, if we rephrase the question in term of \mathbb{F}.

::: question Question 3. Given n distinct element a_{1}, a_{2}, \dots, a_{n} \in \mathbb{F}, let b_{i} be a root of x^{2} - a_{i} for all i. And b_{i}b_{j}^{-1} \notin \mathbb{F} for all i \neq j, and b_{i} \notin \mathbb{F} for all i. Then given \lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}, \sum_{i=0}^{n} \lambda_{i}b_{i} \in \mathbb{F} if and only if \lambda_{i} = 0 for all i. :::

If we try to shift b_{k} to b_{k} + c_{k} such that c_{k} \in \mathbb{F}, then the result clearly still stand. So, the previous question can be generalised a bit.

::: question Question 4. Given n distinct irreducible quadratic polynomial f_{1}, f_{2} \dots f_{n} in \mathbb{F}[x].

And given any i \neq j, f_{j} does not have root in \mathbb{F}[x]/f_{i}.

And b_{i} be all roots of f_{i} for all i. Then given \lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}, \sum_{i=0}^{n} \lambda_{i}b_{i} \in \mathbb{F} if and only if \lambda_{i} = 0 for all i. :::

If we are not satisfied with the quadratic polynomials, we can try to generalise the question to any polynomial. However, we may need to have a more strict condition on the roots.

::: question Question 5. Given n distinct irreducible polynomial f_{1}, f_{2} \dots f_{n} in \mathbb{F}[x] with orders greater or equal than 2.

Given any i \neq j. f_{j} does not have root in \mathbb{F}[x]/f_{i}.

And b_{i} be all roots of f_{i} for all i. Then given \lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}, \sum_{i=0}^{n} \lambda_{i}b_{i,1} \in \mathbb{F} if and only if \lambda_{i} = 0 for all i. :::

To prove the above question, we need another definition and some relating lemma.

::: definition Definition 1. Given a polynomial f(x) \in \mathbb{F}[x], f(x) = x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + \dots + a_{n-1}x + a_{n}

Define the derivative f'(x) of f(x) as f'(x) = n x^{n-1} + (n-1)a_{1}x^{n-2} + (n-2)a_{2}x^{n-3} + \dots + 2a_{n-2}x + a_{n-1} :::

Through some simple calculation, we can prove the following lemma.

::: lemma Lemma 1. Given polynomials f(x), g(x) \in \mathbb{F}[x], (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) :::

::: lemma Lemma 2. Given a irreducible polynomial f(x) \in \mathbb{F}[x], and a field \mathbb{K}, such that \mathbb{F} \subseteq \mathbb{K} and f(x) can be factorized into product of linear factors in \mathbb{K}.

Chose a root \alpha of f(x) in \mathbb{K}. Then (x-\alpha)^2 divides f(x) in \mathbb{K}, if and only if f'(x) = 0. :::

As f'(x) is still a polynomial in \mathbb{F}[x], and we can choose arbitrary \mathbb{K}. This lemma implies that irreducible polynomial f(x) have distinct roots if and only if f'(x) \neq 0.

::: proof Proof. If (x-\alpha)^2 divides f(x) in \mathbb{K}.

Let, f(x) = (x-\alpha)^{2}g(x)

Then, f'(x) = 2(x-\alpha)g(x) + (x-\alpha)^{2}g'(x)

Then f'(\alpha) = 0. As f(x) is irreducible, and f(x) and f'(x) have common root. So, f(x) divides f'(x).

As, the degree of f'(x) is less than f(x), then f'(x) = 0.

Conversely, if f'(x) = 0.

Let, f(x) = (x-\alpha)g(x)

Then, f'(x) = g(x) + (x-\alpha)g'(x)

Then,

\begin{aligned} 0 & = f'(\alpha) \\ & = g(\alpha) + (\alpha-\alpha)g'(\alpha) \\ & = g(\alpha) \end{aligned}

So, \alpha is a root of g(x).

Thus, (x-\alpha)^2 divides f(x) in \mathbb{K}. ◻ :::

::: lemma Lemma 3. Given distinct irreducible polynomials f(x), g(x) \in \mathbb{F}[x], where the characteristic of \mathbb{F} equals 0.

Chose any field \mathbb{K}, such that \mathbb{F} \subseteq \mathbb{K}, and f(x) and g(x) can be factorized into product of linear factors in \mathbb{K}.

Let a_{1}, a_{2} \dots a_{n} be all roots of f(x) in \mathbb{K}, and b_{1}, b_{2} \dots b_{m} be all roots of g(x) in \mathbb{K}.

Then, h(x) = \prod_{i=1}^{n}\prod_{j=1}^{m}(x-(a_{i} + b_{j})) have coefficients in \mathbb{F}, :::

]]>https://alex1222.com/posts/2023-06-25-Addition-Of-Algebraic-Elementhttps://alex1222.com/posts/2023-06-25-Addition-Of-Algebraic-ElementSun, 25 Jun 2023 00:00:00 GMT<![CDATA[NixOS Configurations | NixOS 不完全手记]]>

0.1 Contents

0.2 Intro

Some note for NixOS configurations.

0.3 Useful Commands

0.3.1 Rebuild the System

Need sudo.

# test the configuration
sudo nixos-rebuild test --flake "/path/to/config#tag" --show-trace

# switch to the configuration
sudo nixos-rebuild switch --flake "/path/to/config#tag" --show-trace

0.4 Useful Websites

NixOS Full Options List

NixOS Package List

GitHub User Public Key

0.5 Put File Under /etc

Check 

{
    example-configuration-file =
    { 
        source = "./file.conf.example";
        mode = "0440";
    };
    "default/useradd".text = "GROUP=100 ...";
}

0.6 Reference Store Path of a Nix Package

''
${pkgs.hello}/bin/hello
''

0.7 Use Agenix to Manage Secrets

Check agenix GitHub.

0.7.1 Install Agenix Via Flakes

{
  inputs.agenix.url = "github:ryantm/agenix";
  # optional, not necessary for the module
  #inputs.agenix.inputs.nixpkgs.follows = "nixpkgs";
  # optionally choose not to download darwin deps (saves some resources on Linux)
  #inputs.agenix.inputs.darwin.follows = "";

  outputs = { self, nixpkgs, agenix }: {
    # change `yourhostname` to your actual hostname
    nixosConfigurations.yourhostname = nixpkgs.lib.nixosSystem {
      # change to your system:
      system = "x86_64-linux";
      modules = [
        ./configuration.nix
        agenix.nixosModules.default
      ];
    };
  };
}

0.7.2 Get Secrets

agenix use ssh keys as keys.

System host ssh key is at /etc/ssh/.

User's ssh key is at ~/.ssh.

0.7.3 Mkdir For Agenix

mkdir -p ./secrets

0.7.4 Add Secrets Keys

Keys should be put at ./secrets/secrets.nix.

# ./secrets/secrets.nix
let
  user1 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIL0idNvgGiucWgup/mP78zyC23uFjYq0evcWdjGQUaBH";
  user2 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAILI6jSq53F/3hEmSs+oq9L4TwOo1PrDMAgcA1uo1CCV/";
  users = [ user1 user2 ];

  system1 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIPJDyIr/FSz1cJdcoW69R+NrWzwGK/+3gJpqD1t8L2zE";
  system2 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIKzxQgondgEYcLpcPdJLrTdNgZ2gznOHCAxMdaceTUT1";
  systems = [ system1 system2 ];
in
{
  "secret1.age".publicKeys = [ user1 system1 ];
  "secret2.age".publicKeys = users ++ systems;
}

This file declare that the secret1.age can be decrypted by user1 and system1. secret2.age can be decrypted by all users in users and systems.

0.7.5 Add Secrets

# You need to be in the same directory as secrets.nix
cd ./secrets

# This will open your editor ($EDITOR) to create the secret
agenix -e secret1.age

0.7.6 Add Secrets to NixOS Configurations

{
  # this will allow you to use the secrets in your NixOS configuration as a string
  age.secrets.secret1.file = ../secrets/secret1.age;
}

Or,

{
  # this will allow you to access the at the path /etc/secret1 with the correct permissions
  age.secrets.secret1 = {
    file = ../secrets/secret1
    path = "/etc/secret1";;
    mode = "770";
    owner = "nginx";
    group = "nginx";
  };
}

0.7.7 Use Secrets

0.7.7.1 Use Path of Secrets

{
  users.users.user1 = {
    isNormalUser = true;
    passwordFile = config.age.secrets.secret1.path;
  };
}

0.7.7.2 Replace Inplace Strings With Secrets

Considering that there still might be some modules which doesn't support reading secrets from a file, you could provide a placeholder string instead of a clear-text password and replace this placeholder with the secret provided by Agenix.

In the following example, the Dex module creates the config file /run/dex/config.yaml containing the placeholder string @dex-user-password@. The activation script will read the Agenix secret from config.age.secret.dex-user-password.path and replace the placeholder string with the actual secret.

system.activationScripts."dex-user-secret" = ''
  secret=$(cat "${config.age.secrets.dex-user-password.path}")
  configFile=/run/dex/config.yaml
  ${pkgs.gnused}/bin/sed -i "s#@dex-user-password@#$secret#" "$configFile"
'';

0.8 Mount WebDav Using AutoFS

I found this two links really helpful.

https://askubuntu.com/questions/821123/autofs-and-webdav-how-to-make-them-work-together

https://www.reddit.com/r/NixOS/comments/b5p6f7/how_do_i_use_davfs2/

First you need to install agenix as a way to manage your WebDav secrets.

Then you need to add the following to your configuration.nix. This will mount your WebDav to /mnt/{{ mount-dir }}.

{
  age.secrets = {
    webdav-secrets = {
      file = ./secrets/{{ your secret file }};
      owner = "root";
      group = "root";
      mode = "600";
      # this is the path where the secret will be mounted
      path = "/etc/davfs2/secrets";
    };
  };

  # put the configuration for davfs2 in the /etc/davfs2 directory
  environment.etc."davfs2/auto.mount" = {
    # rw means read-write, ro means read-only
    # you need to add backslash `\` before ':' and '#' in your url
    text = ''
      {{ mount-dir }} -fstype=davfs,conf=/etc/davfs2/conf,rw {{ your webdav url }}
    '';
    mode = "0440";
  };
  environment.etc."davfs2/conf" = {
    text = ''
      secrets /etc/davfs2/secrets
    '';
    mode = "0440";
  };

  # enable the service
  services.davfs2.enable = true;
  services.autofs = {
    enable = true;
    autoMaster = "/mnt/ /etc/davfs2/auto.mount";
  };
}

0.9 Clean the System & Nix Store

# clean journalctl
services.cron = {
  enable = true;
  systemCronJobs = [
    "0 0 * * * journalctl --vacuum-time=7d 1>/dev/null"
  ];
};

# garbange collection check https://nixos.wiki/wiki/Nix_Cookbook#Reclaim_space_on_Nix_install.3F
nix.gc.automatic = true;
nix.settings.auto-optimise-store = true;

0.10 Enable the Mosh Server

# Enable mosh, the ssh alternative when client has bad connection
# Opens UDP ports 60000 ... 61000
programs.mosh.enable = true;
networking.firewall.allowedTCPPortRanges = [
  {
    from = 60000;
    to = 61000;
  }
];
networking.firewall.allowedUDPPortRanges = [
  {
    from = 60000;
    to = 61000;
  }
];

0.11 Enable Fail2Ban

services.fail2ban = {
  enable = true;
  maxretry = 5;
  ignoreIP = [
    "127.0.0.0/8" 
    "10.0.0.0/8" 
    "172.16.0.0/12" 
    "192.168.0.0/16"
    "8.8.8.8"
  ];
};
]]>https://alex1222.com/posts/2023-05-06-NixOS-Configuratioinshttps://alex1222.com/posts/2023-05-06-NixOS-ConfiguratioinsMon, 05 Jun 2023 00:00:00 GMT<![CDATA[How to uninstall a app on MacOS]]>

0.1 Contents

0.2 Introduction

Uninstalling a software on a system has long being a headache problem. If you install a software using a package manager like brew, or apt. It is easy. You simply do

# brew
brew remove [package-name]
# use auto remove to remove the dependency
brew autoremove

# apt
apt purge [package-name]
# use auto remove to remove the dependency
apt autoremove

If you install the app using app store. Just uninstall it in the app store.

However, if you install the app by just copy the app into /Applications. Then, simply remove the app itself is clearly not enough. For instance, steam store all the game you download in the '~/Library' folder. If you just remove the app itself, there will be tons of files left.

And there is a app that help us do that. The AppCleaner.

0.3 Download

Just go to the webpage AppCleaner, and select the latest version.

0.4 Uninstall App

Open the app, click the list icon in the right up conner, type in the app you want to uninstall, for instance Steam. Select the App you want to uninstall, then AppCleaner will search for all the files of the App, you could unselect the files you do not want to delete. Click remove, then, you are all set.

Steam Sample

]]>https://alex1222.com/posts/2023-04-26-Uninstall-App-On-MacOShttps://alex1222.com/posts/2023-04-26-Uninstall-App-On-MacOSWed, 26 Apr 2023 00:00:00 GMT<![CDATA[How to set up GPGSuite and GPGMail on Mac]]>

0.1 Contents

0.2 Introduction

GPG Suite is an open-source software package for macOS that provides encryption and decryption of emails, files, and directories using the OpenPGP encryption standard. OpenPGP is a widely used standard for secure email communication that provides end-to-end encryption, which means that only the sender and recipient of an encrypted message can read its contents.

GPG Suite includes several tools, including GPGMail for encrypting and signing emails, GPG Keychain for managing OpenPGP keys, and GPG Services for encrypting and decrypting files and directories. It also includes GPGPreferences, which provides a user interface for configuring GPG Suite settings.

GPG Suite is popular among users who require secure communication and privacy protection, such as journalists, activists, and government officials. It is available for free download and use, although users can make a donation to support the ongoing development of the software.

Screenshot of GPG Suite

0.3 Installation

Check Free-GPGMail project.

  1. Download the latest version of Free-GPGMail.

    Using Homebrew:

    brew install --cask free-gpgmail

  2. Restart Mail.app, go to Preferences -> General -> Manage Plugins....

    If the "Manage Plugins" button is not present in the lower left corner, open a terminal and run:

    sudo defaults write “/Library/Preferences/com.apple.mail” EnableBundles 1
defaults write com.apple.mail EnableBundles -bool true

    Then restart Mail.app again.

    • Make sure that GPGMailLoader_*.mailbundle, if present, is disabled.
    • Enable the Free-GPGMail_<version>.mailbundle.
    • Apply and Restart Mail.

  3. In Mail.app, check Preferences -> Free-GPGMail. If it says that you are in Trial Mode or Decryption Only Mode, hit Activate. It will perform a dummy activation routine.

0.4 Manage Keys

Also check GPGTools.

]]>https://alex1222.com/posts/2023-04-17-GPGMail-Machttps://alex1222.com/posts/2023-04-17-GPGMail-MacMon, 17 Apr 2023 00:00:00 GMT<![CDATA[Personal Collection of Textbooks]]>

0.1 Contents

0.2 Intro

This is a personal collection of math and computer science textbooks. Each book contains:

  • Name
  • ISBN-10 ISBN-13
  • Language
  • Author
  • Date of Publication
  • Area
  • Review / Description
  • Preview Image
  • Solution Manual (if available)

0.3 Elements of Algebraic Topology

Name: Elements of Algebraic Topology

ISBN-10: 0201627280

ISBN-13: 978-0201627282

Language: English

Author: James Raymond Munkres

Date of Publication: 1993

Area: Math, Algebraic Topology

Review / Description: Not finished reading yet.

Elements of Algebraic Topology

0.4 Functional Analysis

Name: Functional Analysis

ISBN-10: 0070619883

ISBN-13: 978-0070619883

Language: English

Author: Walter Rudin

Date of Publication: 2007

Area: Math, Functional Analysis

Review / Description: Not finished reading yet.

Functional Analysis

0.5 Abstract Algebra

Name: Abstract Algebra

ISBN-10: 0471368792

ISBN-13: 978-0471368793

Language: English

Author: Israel Nathan Herstein

Date of Publication: 1996

Area: Math, Abstract Algebra

Review / Description: Not finished reading yet.

Abstract Algebra

0.6 Principles of Mathematical Analysis / Baby Rudin

Name: Principles of Mathematical Analysis

ISBN-10: 007054235X

ISBN-13: 978-0070542358

Language: English

Author: Walter Rudin

Date of Publication: 1976

Area: Math, Mathematical Analysis

Review / Description: Good book. Rudin's style.

Preview Image:

Principles of Mathematical Analysis

Solution Manual: Check

0.7 Real and Complex Analysis

Name: Real and Complex Analysis

ISBN-10: 0070542341

ISBN-13: 978-0070542341

Language: English

Author: Walter Rudin

Date of Publication: 1976

Area: Math, Mathematical Analysis

Review / Description: Not finished reading yet.

Real and Complex Analysis

0.8 微积分学教程

Name: 微积分学教程

第一卷：

ISBN-10: 704018303X

ISBN-13: 9787040183030

第二卷：

ISBN-10: 7040183048

ISBN-13: 9787040183047

第三卷：

ISBN-10: 7040183056

ISBN-13: 9787040183054

Language: Chinese

Author: Grigorii Fichtenholz

Date of Publication: 2008

Area: Math, Mathematical Analysis

Review / Description:

微积分学教程 is a Chinese translation of the original Russian textbook "Курс дифференциального и интегрального исчисления" (Course of Differential and Integral Calculus) by Grigorii Fichtenholz. The book is a comprehensive introduction to calculus and covers topics such as limits, derivatives, integrals, applications of calculus, and series.

The book is well-regarded by many mathematicians and educators, and is often used as a textbook for calculus courses in universities in Russia and China. The book is known for its clear and concise explanations, rigorous proofs, and wide range of examples and exercises.

One potential drawback of the book for some readers is that it is quite dense and may require a strong mathematical background to fully understand. Additionally, some readers may prefer a more modern approach to teaching calculus that incorporates more visual aids or real-world applications.

Overall, 微积分学教程 by Grigorii Fichtenholz is a highly respected textbook on calculus, and would likely be a valuable resource for students and educators looking to gain a deep understanding of the subject.

微积分学教程

0.9 Algebra

Name: Algebra

ISBN-10: 0132413779 ISBN-13: 9780132413770

Language: English

Author: Michael Artin

Date of Publication: 2010

Area: Math, Abstract Algebra, Linear Algebra

Review / Description:

Algebra is a comprehensive textbook on algebraic structures, including groups, rings, and fields. The book covers topics such as group actions, Galois theory, and homological algebra. The text is known for its clear and concise explanations, as well as its emphasis on geometric interpretations of algebraic concepts.

The book is well-regarded by many mathematicians and educators, and is often used as a textbook for advanced undergraduate or graduate-level courses in algebra. It is particularly popular among students and researchers interested in algebraic geometry and number theory.

One potential drawback of the book for some readers is that it may be too advanced for those without a strong mathematical background. Additionally, some readers may prefer a more traditional approach to algebra, rather than the geometric viewpoint presented in the book.

Overall, Algebra by Michael Artin is a highly respected textbook on algebra, and would likely be a valuable resource for students and educators looking to gain a deep understanding of the subject.

Algebra

]]>https://alex1222.com/posts/2023-03-05-List-of-Textbookshttps://alex1222.com/posts/2023-03-05-List-of-TextbooksSun, 05 Mar 2023 00:00:00 GMT<![CDATA[How to Install and Use Pyenv]]>

0.1 Contents

0.2 Pyenv

0.2.1 Install the pyenv

brew install pyenv-virtualenv

Add the following line to .zshrc or .zprofile to run pyenv

# python env auto activation
eval "$(pyenv init -)"
eval "$(pyenv virtualenv-init -)"
if which pyenv-virtualenv-init > /dev/null; then eval "$(pyenv virtualenv-init -)"; fi

To run pyenv in the current session, run

source ~/.zshrc

Install python version List all available python version

pyenv install —list

Install a specific version

pyenv install $VERSION

List all installed versions

P

0.2.2 Change versions

Set the global python version for the system

pyenv global $VERSION

Set local version

pyenv local $VERSION

Create a specific virtualenv for repo

pyenv virtualenv $PYTHON_VERSION $NAME

Use virtualenv for repo

pyenv local $NAME
]]>https://alex1222.com/posts/2023-2-27-Howto-Install-and-Use-Pyenvhttps://alex1222.com/posts/2023-2-27-Howto-Install-and-Use-PyenvMon, 27 Feb 2023 00:00:00 GMT<![CDATA[Apple电池健康]]>

0.0.1 Apple电池健康

Apple电池健康

0.0.1.1 🔋 关于电池健康度

原文章见telegram

通常一部全新的 iPhone 电池健康度会维持相当长一段时间的 100%. 不管你如何祸害你的设备。

但是，一旦 iPhone 电池的最大容量百分比不再是 100% 时，这个百分比会迅速下降，不管你如何呵护它。

这是由于 iOS 系统的电池算法问题导致的。按照算法，多数设备首次使用时，电池容量是 > 100% 的，只是以 100% 显示。其实电池是在持续而相对匀速老化的。直到低于算法计算的 100%，用户才会看到数值变化。

个人建议最好的办法是通过iMazing / 日志报告等方式查看并估算 iPhone 实际循环充电次数，来获得一个大致印象。

当 iPhone 电池容量低于86% 时，多数用户应该会明显感觉到，在不借助外部电源的情况下，设备续航已经很难维持一天使用需求。但只有低于 80% 时，官方才会承认电池需要维修。

iPad 同样拥有电池健康度计算功能，且与 iOS 的计算机制完全一致。只是 Apple 没有让它在 iPadOS 系统设置中明确显示。大概因为 iPad 电池容量更大，完整充电周期更长，因此损耗更慢。

而 Mac 电池只是会笼统显示电池状态和维修建议。

0.0.1.2 💻 你的 Mac 电池健康吗？

原文章见telegram

在之前的推文提及过，Mac 显示电池健康度的方式非常笼统，似乎不是很希望让用户了解到真正的电池健康度。 https://t.me/AppleNuts/1446

普通用户虽然可以借助第三方工具去查询电池健康度，如 Coconut Battery，但是还要下载第三方软件，而且软件的部分功能需要收费。

所以，顺手写了一个捷径用来计算 Mac 电池的健康度，运行捷径后可以瞬间得到 循环充电次数和健康度。

  • 基于捷径 + zsh 开发，信息准确，执行速度极快
  • 同时兼容 Intel / M1 系列 Mac
  • 在没有电池的 Mac 设备无法运行（如：Mac mini）

Shortcut

]]>https://alex1222.com/posts/2022-5-25-Apple%E7%94%B5%E6%B1%A0%E5%81%A5%E5%BA%B7https://alex1222.com/posts/2022-5-25-Apple%E7%94%B5%E6%B1%A0%E5%81%A5%E5%BA%B7Wed, 25 May 2022 00:00:00 GMT<![CDATA[方舟像素字体 / Ark Pixel Font]]>

0.0.1 方舟像素字体 / Ark Pixel Font

方舟像素字体

开源的泛中日韩像素字体。支持 10、12 和 16 像素。目标是为像素风格的游戏开发提供一套可用于正文的开箱即用的字体解决方案。

这个项目不仅提供了全部的字形设计源文件，也提供了构建字体所需要的完整程序

该字体目前还处于早期开发阶段，所有尺寸仍然缺少大量汉字。有一个临时的可用于生产的替代方案，可查看缝合怪像素字体项目。

目前 12 像素为主要开发目标。10、16 像素为实验性质

项目地址

]]>https://alex1222.com/posts/2022-5-25-%E6%96%B9%E8%88%9F%E5%83%8F%E7%B4%A0%E5%AD%97%E4%BD%93https://alex1222.com/posts/2022-5-25-%E6%96%B9%E8%88%9F%E5%83%8F%E7%B4%A0%E5%AD%97%E4%BD%93Wed, 25 May 2022 00:00:00 GMT<![CDATA[High-precision measurement of the W boson mass with the CDF II detector]]>

费米实验室的最新研究结果表明 W 玻色子的质量严重偏离标准模型的理论预言。W玻色子传递弱相互作用（物理学中的基本力之一）。因为粒子物理标准模型（SM）对W玻色子的质量有严格的限制，所以测量它的质量可以检验标准模型。费米实验室（CDF）的对撞机探测器现在报告了从 Tevatron 粒子加速器的数据中提取的W玻色子质量的精确测量值（参见 Campagnari 和 Mulders 的方法）。令人惊讶的是，研究人员发现W玻色子的质量明显高于标准模型的预测，相差7个标准差。

论文：https://www.science.org/doi/10.1126/science.abk1781

]]>https://alex1222.com/posts/2022-4-10-High-precision-measurement-of-the-W-boson-mass-with-the-CDF-II-detector.mdhttps://alex1222.com/posts/2022-4-10-High-precision-measurement-of-the-W-boson-mass-with-the-CDF-II-detector.mdSun, 10 Apr 2022 00:00:00 GMT<![CDATA[Artemis! Fly your name around the Moon!]]>

NASA 邀请民众把自己的名字加入 Artemis 宇宙飞船

登记的名字将存在闪存/U盘里，届时由 Artemis 宇宙飞船带着名字环绕月球

谁都可以登记，登记的人，都可以得到 NASA 颁发的“登机牌”

登记

登机牌

]]>https://alex1222.com/posts/2022-3-22-Artemis-Fly-your-name-around-the-Moonhttps://alex1222.com/posts/2022-3-22-Artemis-Fly-your-name-around-the-MoonTue, 22 Mar 2022 00:00:00 GMT<![CDATA[The First Linux System Support Mac M1]]>

0.0.1 The first Asahi Linux Alpha Release is here!

Asahi Linux

Asahi Linux

首个支持苹果 M1 / Pro / Max Mac 的 Linux 发行版发布，面向所有用户开放下载 https://asahilinux.org/2022/03/asahi-linux-alpha-release/

0.0.1.1 What works

All M1, M1 Pro, and M1 Max devices are supported except for the Mac Studio.

  • Wi-Fi
  • USB2 (Thunderbolt ports)
  • USB3 (Mac Mini Type A ports)
  • Screen (no GPU)
  • NVMe
  • Lid switch
  • Power button
  • Built-in display (framebuffer only)
  • Built-in keyboard/touchpad
  • Display backlight on/off
  • Battery information / charge control
  • RTC
  • Ethernet (desktops)
  • SD card reader (M1 Pro/Max)
  • CPU frequency switching

M1 machines only (no Pro/Max):

  • Headphones jack (might be flaky)

Mac Mini only:

  • HDMI output

Not yet, but coming soon:

  • USB3
  • Speakers
  • Display controller (backlight brightness control, V-Sync, proper DPMS)

0.0.1.2 What doesn’t

Everything else, but notably:

  • DisplayPort
  • Thunderbolt
  • HDMI on the MacBooks
  • Bluetooth
  • GPU acceleration
  • Video codec acceleration
  • Neural Engine
  • CPU deep idle
  • Sleep mode
  • Camera
  • Touch Bar Note: on the 13" MacBook Pro, you can use Fn + the number row keys (1-9, 0, and the next two) as F1..F12 in lieu of the Touch Bar.

0.0.1.3 Known bugs

If Wi-Fi doesn't work, try toggling it off and on in the network management menu If the headphones jack doesn't work or only one channel works, try rebooting. There's a flakiness issue.

]]>https://alex1222.com/posts/2022-3-21-The-First-Linux-System-Support-Mac-M1https://alex1222.com/posts/2022-3-21-The-First-Linux-System-Support-Mac-M1Mon, 21 Mar 2022 00:00:00 GMT<![CDATA[Send RSS Feed Through Email]]>

RSS2Email

send your RSS Feed To your Mail Box

]]>https://alex1222.com/posts/2022-3-20-Send-RSS-Feed-Through-Email.mdhttps://alex1222.com/posts/2022-3-20-Send-RSS-Feed-Through-Email.mdSun, 20 Mar 2022 00:00:00 GMT<![CDATA[DevToys Swift Army Knife DevTools For Mac]]>

功能齐全的开发者工具，包括html encode decode, url encode decode, json beautifier, json to yaml converter, hash tools, check sum tools。很实用

DevToys

DevToys

DevToys

DevToys

]]>https://alex1222.com/posts/2022-2-15-DevToys-Swift-Army-Knife-DevTools-For-Machttps://alex1222.com/posts/2022-2-15-DevToys-Swift-Army-Knife-DevTools-For-MacTue, 15 Feb 2022 00:00:00 GMT