diff --git a/.DS_Store b/.DS_Store
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diff --git a/Dissertation/Introduction.tex b/Dissertation/Introduction.tex
index 6724058..10cc3aa 100644
--- a/Dissertation/Introduction.tex
+++ b/Dissertation/Introduction.tex
@@ -20,7 +20,7 @@ This dissertation is broken down into three parts. At the end of each part, we w
 We introduce here basic notations that we will be using throughout this dissertation. Large parts are taken from standard literature inspired by \textit{Matrix Computations} by Golub \& van Loan, \cite{golub2013matrix}, \textit{Probability: Theory \& Examples} by Rick Durrett, \cite{durrett2019probability}, and \textit{Concrete Mathematics} by Knuth, Graham \& Patashnik, \cite{graham_concrete_1994}.
 \subsection{Norms and Inner Products}
 \begin{definition}[Euclidean Norm]
-	Let $\left\|\cdot\right\|_E: \R^d \rightarrow [0,\infty)$ denote the Euclidean norm defined for every $d \in \N_0$ and for all $x= \{x_1,x_2,\cdots, x_d\}\in \R^d$ as:
+	Let $\left\|\cdot\right\|_E: \R^d \rightarrow [0,\infty)$ denote the Euclidean norm defined for every $d \in \N$ and for all $x= \{x_1,x_2,\cdots, x_d\}\in \R^d$ as:
 	\begin{align}
 		\| x\|_E = \lp \sum_{i=1}^d x_i^2 \rp^{\frac{1}{2}}
 	\end{align}
diff --git a/Dissertation/conclusions-further-research.tex b/Dissertation/conclusions-further-research.tex
index f5f9689..deefb33 100644
--- a/Dissertation/conclusions-further-research.tex
+++ b/Dissertation/conclusions-further-research.tex
@@ -4,17 +4,17 @@ We will present three avenues of further research and related work on parameter
 
 \section{Further operations}
 
-Note, for instance, that several classical operations are done on neural networks that have yet to be accounted for in this framework and talked about in the literature.  We will discuss two of them \textit{dropout} and \textit{merger} and discuss how they may be brought into this framework.
+Note, for instance, that several classical operations are done on neural networks that have yet to be accounted for in this framework and talked about in the literature.  We will discuss one of them \textit{dropout} and discuss how they may be brought into this framework.
 
 
-Overfitting presents an important challenge for all machine learning models, including deep learning. There ex
+Overfitting presents an important challenge for all machine learning models, including deep learning. There exists a technique called \textit{dropout} introduced in \cite{srivastava_dropout_2014} that seeks to ameliorate this situation.
 \begin{definition}[Hadamard Product]
 	Let $m,n \in \N$. Let $A,B \in \R^{m \times n}$. For all $i \in \{ 1,2,\hdots,m\}$ and $j \in \{ 1,2,\hdots,n\}$ define the Hadamard product $\odot: \R^{m\times n} \times \R^{m \times n} \rightarrow \R^{m \times n}$ as:
 	\begin{align}
 		A \odot B \coloneqq \lb A \odot B \rb _{i,j} = \lb A \rb_{i,j} \times \lb B \rb_{i,j} \quad  \forall i,j
 	\end{align}
 \end{definition}
-We will also define the dropout operator introduced in \cite{srivastava_dropout_2014}, and explained further in \cite{Goodfellow-et-al-2016}. 
+We will define the dropout operator introduced in \cite{srivastava_dropout_2014}, and explained further in \cite{Goodfellow-et-al-2016}. 
 
 
 \begin{definition}[Instantiation with dropout]
diff --git a/Dissertation/main (dragged).pdf b/Dissertation/main (dragged).pdf
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diff --git a/Dissertation/main.pdf b/Dissertation/main.pdf
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diff --git a/Dissertation/preamble.tex b/Dissertation/preamble.tex
index da048c7..2ecf950 100644
--- a/Dissertation/preamble.tex
+++ b/Dissertation/preamble.tex
@@ -1,8 +1,26 @@
 \documentclass[12pt]{report}
+\usepackage{sectsty}
+
+% Set chapter and section font sizes to 12pt
+\chapterfont{\fontsize{12}{12}\selectfont}
+\sectionfont{\fontsize{12}{12}\selectfont}
+\subsectionfont{\fontsize{12}{12}\selectfont}
+
+\usepackage{titlesec}
+
+\titleformat{\part}[display]
+  {\normalfont\fontsize{12}{12}\selectfont\bfseries}{\partname\ \thepart}{0pt}{\fontsize{12}{12}\selectfont}[\vspace{1in}]
+  
+\titleformat{\chapter}[display]{\normalfont\fontsize{12}{12} \bfseries }{\chaptertitlename\ \thechapter.}{1ex}{}{}
+\titlespacing{\chapter}{0pt}{-20pt}{0pt}
+
 \usepackage{setspace}
 \doublespacing
 \usepackage[toc,page]{appendix}
 \usepackage{mleftright}
+% Set chapter heading font size to 12pt
+
+
 
 \usepackage{pdfpages}
 \usepackage[]{amsmath}
@@ -10,7 +28,16 @@
 \usepackage{mathtools}
 \numberwithin{equation}{section}
 \usepackage[]{amssymb}
-\usepackage[margin=1in]{geometry}
+
+\usepackage{geometry}
+
+\geometry{
+    left=1in,       % Left margin
+    right=1in,      % Right margin
+    top=1in,        % Top margin
+    bottom=1in,     % Bottom margin
+}
+
 \usepackage{url}
 
 \usepackage[T1]{fontenc}