From 9ac7ddd5cff4ac53e2f984c801c7365334908658 Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Wed, 31 Dec 2025 16:16:00 +0800
Subject: [PATCH] z

---
 main.tex | 23 +++++++++++------------
 1 file changed, 11 insertions(+), 12 deletions(-)

diff --git a/main.tex b/main.tex
index e5c3ae3..c7751fd 100644
--- a/main.tex
+++ b/main.tex
@@ -13,6 +13,7 @@
 \usepackage{soul}
 \usepackage[dvipsnames]{xcolor}
 \usepackage{booktabs}
+\usepackage{blkarray}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}{Lemma}[section]
 \newtheorem{corollary}{Corollary}[section]
@@ -137,7 +138,7 @@ Let $C^*$ be the optimal $B$-free mincut and let $\lambda^*$ be the optimal solu
 \begin{frame}{Matroid}
 A matroid $M=(E,\mathcal B)$ is a structure on set $E$.
 
-$\mathcal B$ is the collection of ``bases'' with the following properties:
+``Bases'' $\mathcal B$ is a collection of subsets with the following properties:
 \begin{itemize}
 \item $\mathcal B\neq \emptyset$;
 \item If $A$ and $B$ are distinct members of $\mathcal B$ and $a\in A-B$, 
@@ -146,7 +147,7 @@ then there exists $b\in B-A$ such that $A-a+b\in \mathcal B$.
 \end{itemize}
 
 \p
-$X\subset E$ is a cocycle if $X\cap B\neq \emptyset$ holds for all $B\in \mathcal B$.
+$X\subset E$ is a cocycle if $X\cap B$ is not empty for all $B\in \mathcal B$.
 
 The size of minimum cocycle is the cogirth.
 \end{frame}
@@ -196,21 +197,19 @@ We solve the cogirth part.
 Geelen \& Kapadia reduce the cogirth problem of PGMs to binary matroids $M(A)$ with the following representation,
 \[
 A=
-\begin{array}{ccc}
-      & \begin{array}{cc} E(G) & T \end{array} \\
-    \begin{array}{r} V(G) \\ \set{s} \end{array}
-      &
-        \begin{pmatrix}
-          A(G)  &   B \\
-          C     &   D
-        \end{pmatrix}
-\end{array}
+\begin{blockarray}{ccc}
+        &   \textcolor{blue}{E(G)}    &   \textcolor{blue}{T}\\
+\begin{block}{c(cc)}
+\textcolor{blue}{V(G)}    &   A(G)    &   B\\
+\textcolor{blue}{\set{s}} &   C       &   D\\
+\end{block}
+\end{blockarray}
 \in \F_2^{(V(G)+s)\times (E(G)\cup T)}
 \]
 where $A(G)$ is the incidence matrix of a graph $G$, $T$ indexes $t$ new columns and $\set{s}$ indexes one additional row.
 
 \medskip
-We say $M(A)$ is a $(1,t)$-signed grafts.
+$M(A)$ is a \emph{$(1,t)$-signed graft}.
 \end{frame}
 
 \begin{frame}{Previous works}