From c41f7204a80d7887b3ba206e374e9e3f8593a824 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Fri, 17 Oct 2025 11:52:31 -0500
Subject: [PATCH] updates
---
.gitignore | 5 +-
content/CSE510/CSE510_L15.md | 6 +-
content/Math4201/Math4201_L21.md | 98 ++++++++++++++++++++++++++++++++
content/Math4201/_meta.js | 1 +
package.json | 2 +-
5 files changed, 107 insertions(+), 5 deletions(-)
create mode 100644 content/Math4201/Math4201_L21.md
diff --git a/.gitignore b/.gitignore
index 726b0bb..ab7a57a 100644
--- a/.gitignore
+++ b/.gitignore
@@ -142,4 +142,7 @@ analyze/
.turbo/
# pagefind postbuild
-public/_pagefind/
\ No newline at end of file
+public/_pagefind/
+
+# npm package lock file for different platforms
+package-lock.json
\ No newline at end of file
diff --git a/content/CSE510/CSE510_L15.md b/content/CSE510/CSE510_L15.md
index 1aa6254..343c4e1 100644
--- a/content/CSE510/CSE510_L15.md
+++ b/content/CSE510/CSE510_L15.md
@@ -129,15 +129,15 @@ Proof as exercise.
The objective function is:
$$
-J(\theta)=\int_{s\in S} \pho^{\mu}(s) r(s,\mu_\theta(s)) ds
+J(\theta)=\int_{s\in S} \rho^{\mu}(s) r(s,\mu_\theta(s)) ds
$$
-where $\pho^{\mu}(s)$ is the stationary distribution under the behavior policy $\mu_\theta(s)$.
+where $\rho^{\mu}(s)$ is the stationary distribution under the behavior policy $\mu_\theta(s)$.
Proof along the same lines of the standard policy gradient theorem.
$$
-\nabla_\theta J(\theta) = \mathbb{E}_{\mu_\theta}[\nabla_\theta Q^{\mu_\theta}(s,a)]=\mathbb{E}_{s\sim \pho^{\mu}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
+\nabla_\theta J(\theta) = \mathbb{E}_{\mu_\theta}[\nabla_\theta Q^{\mu_\theta}(s,a)]=\mathbb{E}_{s\sim \rho^{\mu}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
$$
### Issues for DPG
diff --git a/content/Math4201/Math4201_L21.md b/content/Math4201/Math4201_L21.md
new file mode 100644
index 0000000..e40eb0f
--- /dev/null
+++ b/content/Math4201/Math4201_L21.md
@@ -0,0 +1,98 @@
+# Math4201 Topology I (Lecture 21)
+
+## Simplicial complexes
+
+### Recall from last lecture
+
+Let $\sigma=\{a_0,a_1,\dots,a_n\}$ be a finite set. The $n$-dimensional simplex determined by $\tau$ is given as:
+
+$$
+\Delta^n(a_0,a_1,\dots,a_n)=\left\{t_0a_0+t_1a_1+\cdots+t_na_n\mid t_i\geq 0, \sum_{i=0}^n t_i=1\right\}
+$$
+
+If we have vertices $\tau=\{a_0,a_1,\dots,a_k\}$, $\tau\subseteq \sigma$, the face of $\Delta^n$ is determined by $\tau$ with dimension $|\tau|-1$.
+
+$\Delta^n$ is the topologized by the subspace topology inherited by the standard topology on Euclidean space $\mathbb{R}^n$.
+
+Note that there are different ways to of embedding and all give the same topological space.
+
+### Abstract simplicial complexes
+
+#### Definition for abstract simplicial complex
+
+Let $V=\{v_0,v_1,\dots,v_n\}$ be a finite set (set of vertices of a simplicial complex). $K$ be the collection of subspaces of $V$.
+
+1. $\sigma\in K$ and $\tau\subseteq \sigma$, then $\tau\in K$.
+2. For any $v\in V$, $\{v\}\in K$.
+
+Then $\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma$.
+
+$\Delta_\sigma$ is a simplex of dimension $|\sigma|-1$.
+
+$X_K$ is the topological realization of $K$.
+
+Define an equivalence relation on $\tilde{X_k}$ as follows:
+
+$x\in \Delta_\sigma\sim x'\in \Delta_{\sigma'}$ if and only if $x\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_\sigma$ and $x'\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma'}$.
+
+This just means that the two points have the same barycentric coordinates in the simplex.
+
+#### Definition of barycentric coordinates
+
+Let $\sigma=\{a_0,a_1,\dots,a_n\}$ be a simplex. The barycentric coordinates of a point $x\in \Delta_\sigma$ are the coefficients $t_0,t_1,\dots,t_n$ such that:
+
+$$
+x=t_0a_0+t_1a_1+\cdots+t_na_n
+$$
+
+and $t_i\geq 0$ and $\sum_{i=0}^n t_i=1$.
+
+The point $x$ is in the simplex $\Delta_\sigma$ if and only if $t_i\geq 0$ for all $i$.
+
+
+Example of abstract simplicial complex
+
+Let $V=\{v_1,v_2,v_3,v_4,v_5\}$.
+
+If we want to enclose $K=\{\{v_1,v_2,v_3,v_4\},\{v_3,v_4,v_5\}\}$, we need to fill all the singletons $\{v_1\},\{v_2\},\{v_3\},\{v_4\},\{v_5\}$, all the pairs in $K$, $\{v_1,v_2\},\{v_1,v_3\},\{v_1,v_4\},\{v_2,v_3\},\{v_2,v_4\},\{v_3,v_4\},\{v_3,v_5\},\{v_4,v_5\}$, and the triangle $\{v_1,v_2,v_3\}, \{v_1,v_2,v_4\}, \{v_1,v_3,v_4\}, \{v_2,v_3,v_5\}$.
+
+The final simplicial complex is $\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta(v_1,v_2,v_3,v_4)\sqcup \Delta(v_3,v_4,v_5)\sqcup \{v_1,v_2,v_3,v_4,v_5\}$.
+
+We use $\Delta(v_1,v_2,v_3,v_4)$ to denote the simplex with vertices $v_1,v_2,v_3,v_4$.
+
+
+
+#### Defining maps on abstract simplicial complexes
+
+Let $K$ be an abstract simplicial complex. $V=\{v_1,v_2,\dots,v_m\}$
+
+A map $\pi:\tilde{X_k}\to X_K$ is a quotient map
+
+$X_K$ is equipped with the quotient topology.
+
+Let $f:V\to \mathbb{R}^m$, then $u_i=f(v_i)$.
+
+$\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta_\sigma$ is the disjoint union of all simplices in $K$.
+
+For $\sigma=\{v_{i_0},\dots,v_{i_k}\}$, we have a map $\Delta_\sigma\to \mathbb{R}^\ell$ given by $[t_{i_0}u_0+t_{i_1}u_1+\cdots+t_{i_k}u_k\mid t_j\geq 0, \sum_{j=0}^k t_j=1]$.
+
+This is well-defined because the coefficients $t_j$ are uniquely determined by the vertices $v_{i_0},\dots,v_{i_k}$.
+
+This induces $F:\tilde{X_k}\to \mathbb{R}^\ell$. This map is continuous because $F\vert_{\Delta_\sigma}$ is continuous for all $\sigma\in K$.
+
+Recall that if for any $x\in X_K$, the map $F$ restricted to $\pi^{-1}(x)$ is constant, then there is a unique continuous map $g$ satisfying $F=g\circ \pi$.
+
+In fact, this condition is satisfied and there is such a map $G$.
+
+
+Example of map on abstract simplicial complexes
+
+Consider the previous example of abstract simplicial complex.
+
+Let $f:V\to \mathbb{R}$ by $f(v_i)=i$.
+
+Then $f(\Delta_{\{v_1,v_2,v_3,v_4\}})=[1,4]$
+
+Then $f(\Delta_{\{v_1,v_3\}})=[1,3]$
+
+
\ No newline at end of file
diff --git a/content/Math4201/_meta.js b/content/Math4201/_meta.js
index 4624b9b..c6b0d10 100644
--- a/content/Math4201/_meta.js
+++ b/content/Math4201/_meta.js
@@ -24,4 +24,5 @@ export default {
Math4201_L18: "Topology I (Lecture 18)",
Math4201_L19: "Topology I (Lecture 19)",
Math4201_L20: "Topology I (Lecture 20)",
+ Math4201_L21: "Topology I (Lecture 21)",
}
diff --git a/package.json b/package.json
index 3f4d67e..a65e891 100644
--- a/package.json
+++ b/package.json
@@ -19,7 +19,7 @@
"next-sitemap": "^4.2.3",
"nextra": "^4.2.17",
"nextra-theme-docs": "^4.2.17",
- "pagefind": "^1.3.0",
+ "pagefind": "^1.4.0",
"react": "^19.1.0",
"react-dom": "^19.1.0"
},