\documentclass[12pt]{article}

%
%Margin - 1 inch on all sides
%
\usepackage[letterpaper]{geometry}
\usepackage{times}
\geometry{top=1.0in, bottom=1.0in, left=1.0in, right=1.0in}
\usepackage[style=mla,backend=biber]{biblatex}

%
%Doublespacing
%
\usepackage{setspace}
\doublespacing

%
%Rotating tables (e.g. sideways when too long)
%
\usepackage{rotating}


%
%Fancy-header package to modify header/page numbering (insert last name)
%
\usepackage{fancyhdr}
\pagestyle{fancy}
\lhead{} 
\chead{} 
\rhead{Ayyalasomayajula \thepage} 
\lfoot{} 
\cfoot{} 
\rfoot{} 
\renewcommand{\headrulewidth}{0pt} 
\renewcommand{\footrulewidth}{0pt} 
%To make sure we actually have header 0.5in away from top edge
%12pt is one-sixth of an inch. Subtract this from 0.5in to get headsep value
\setlength\headsep{0.333in}

%
%Works cited environment
%(to start, use \begin{workscited...}, each entry preceded by \bibent)
% - from Ryan Alcock's MLA style file
%
\newcommand{\bibent}{\noindent \hangindent 40pt}
\newenvironment{workscited}{\newpage \begin{center} Works Cited \end{center}}{\newpage }

\addbibresource{references.bib}
%
%Begin document
%
\def\refname{}

\begin{document}
\begin{flushleft}

%%%%First page name, class, etc
Krishna Ayyalasomayajula\\
Ms. Horton\\
IB Extended Essay\\
February 28 2025\\


%%%%Title
\begin{center}
  \vspace{1em}
Rule-based Tensor Mutations Embedded within LLMs for Low-Cost Mathematical Comptuation
\end{center}


%%%%Changes paragraph indentation to 0.5in
\setlength{\parindent}{0.5in}
%%%%Begin body of paper here

{\raggedright \normalsize \textbf{Introduction}} 

Artificial Intelligence (AI) has surged in popularity and capability in recent years, leading to immense developments and investments into the technology. AI, since its modern inception, has been based on a Multi-Layer Perceptron (MLP), with the underlying structure being a Neural Network (NN) \parencite[1]{app14020744}. An MLP is a type of artificial neural network (ANN) that consists of a layering of neurons computed through the vector multiplication of weights and biases, resulting in propagation to a set of values that represents the states of the next layer. This structure, called a neural network is a computational model inspired by the human brain, borrowing from its chaining of neurons to create complex understanding from basic electrical impulses. The below expression details the computational process involved in evaluating progressive layers of neurons in a neural network. 

\[
\vec{y} = f\Bigl( \Bigl[ \sum_{j=1}^{n} w_{1j}x_j + b_1,\;\sum_{j=1}^{n} w_{2j}x_j + b_2,\;\ldots,\;\sum_{j=1}^{n} w_{mj}x_j + b_m \Bigr]^T \Bigr)
\]



%%%%Works cited
\newpage
\begin{center}
  \textbf{Works Cited}

\end{center}

\printbibliography[heading=none]
%\printbiblist
\end{flushleft}
\end{document}
\}
