From ea24c69100755b217a7187deca755f48c5f5ef26 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Fri, 28 Apr 2023 15:33:41 -0700
Subject: [PATCH] Typos

---
 Advanced/Lattices/main.tex              | 2 +-
 Advanced/Lattices/parts/0 intro.tex     | 8 ++++----
 Advanced/Lattices/parts/1 minkowski.tex | 2 +-
 3 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/Advanced/Lattices/main.tex b/Advanced/Lattices/main.tex
index d3b6ff4..9f278a8 100755
--- a/Advanced/Lattices/main.tex
+++ b/Advanced/Lattices/main.tex
@@ -1,7 +1,7 @@
 % use [nosolutions] flag to hide solutions.
 % use [solutions] flag to show solutions.
 \documentclass[
-	solutions,
+	nosolutions,
 	singlenumbering
 ]{../../resources/ormc_handout}
 
diff --git a/Advanced/Lattices/parts/0 intro.tex b/Advanced/Lattices/parts/0 intro.tex
index 9b0e0a2..3663cc4 100644
--- a/Advanced/Lattices/parts/0 intro.tex	
+++ b/Advanced/Lattices/parts/0 intro.tex	
@@ -10,9 +10,9 @@ Draw $\mathbb{Z}^2$.
 \definition{}
 We say a set of vectors $\{v_1, v_2, ..., v_n\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
 $$
-	a_1v_1 + a_2v_2 + ... a_nv_n
+	a_1v_1 + a_2v_2 + ... + a_nv_n
 $$
-for integer coeficcients $a_i$.
+for integer coefficients $a_i$.
 
 \problem{}
 Which of the following generate $\mathbb{Z}^3$?
@@ -37,13 +37,13 @@ Find a set of vectors that generates $\mathbb{Z}^n$.
 \vfill
 \pagebreak
 
-\problem{}
+\definition{}
 A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.
 
 \vfill
 
 \problem{}
-Draw two fundamental reions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
+Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
 
 \vfill
 \pagebreak
\ No newline at end of file
diff --git a/Advanced/Lattices/parts/1 minkowski.tex b/Advanced/Lattices/parts/1 minkowski.tex
index d726713..576c3a5 100644
--- a/Advanced/Lattices/parts/1 minkowski.tex	
+++ b/Advanced/Lattices/parts/1 minkowski.tex	
@@ -31,7 +31,7 @@ The following picture gives the idea for the proof of Blichfeldt's theorem. Expl
 
 
 \problem{}
-Let $X$ be a region $X$ of volume $k$. How many integral points must $X$ contain after a translation?
+Let $X$ be a region $\in \mathbb{R}^2$ of volume $k$. How many integral points must $X$ contain after a translation?
 
 \vfill