updates

.gitignore

@@ -143,3 +143,6 @@ analyze/
 
 # pagefind postbuild
 public/_pagefind/
+
+# npm package lock file for different platforms
+package-lock.json

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

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 $i$.
+
+<details>
+<summary>Example of abstract simplicial complex</summary>
+
+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 $v_1,v_2,v_3,v_4$.
+
+</details>
+
+#### 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 $G$.
+
+<details>
+<summary>Example of map on abstract simplicial complexes</summary>
+
+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 $f(\Delta_{\{v_1,v_3\}})=[1,3]$
+
+</details>

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)",
 }

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"
   },