```
\documentclass[12pt]{report}
\usepackage{amsmath, amsthm, ulem, graphicx, marvosym, amscd, amssymb, enumerate, mathrsfs, multicol, comment, times, inputenc, amssymb, setspace, tocloft, tabu, fancyhdr, caption, subcaption, float, titletoc, appendix, tikz, pgfplots, enumitem, indentfirst, textcase, titlesec, mathtools, multicol}
\setlength\parindent{0pt}
\usepackage[rightcaption]{sidecap}
\usepackage[export]{adjustbox}
\graphicspath{ {images/} }
\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
\usepackage[sorting=none]{biblatex}
\addbibresource{references.bib}
\newcommand\Z{{\mathbb Z}}
\newcommand\F{{\mathbb F}}
\newcommand\N{{\mathbb N}}
\newcommand\A{{\mathbf A}}
\newcommand\p{{\mathscr P}}
\newcommand\R{{\mathbb R}}
\newcommand\Q{{\mathbb Q}}
\newcommand\C{{\mathbb C}}
\newcommand\bfx{{\mathbf x}}
\newcommand\bfb{{\mathbf b}}
\newcommand\bfv{{\mathbf v}}
\newcommand\bfy{{\mathbf y}}
\newcommand\bfw{{\mathbf w}}
\newcommand\bfu{{\mathbf u}}
\newcommand\Mn{\mathbb{M}_{n}}
\newcommand\bfa{\mathbf{a}}
\newcommand\proj{\mbox{proj}}
\newcommand\Span{\mbox{Span}}
\newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)}
%\usepackage{graphics}
\usepackage[margin=.75in]{geometry}
\usepgfplotslibrary{fillbetween}
\DeclareMathOperator{\lcm}{lcm}
\newcommand{\beas}{\begin{eqnarray*}}
\newcommand{\eeas}{\end{eqnarray*}}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\theoremstyle{definition}
\newtheorem{defi}{Definition}
\newtheorem{definition}{Definition}
\newtheorem{exam}{Example}
\newtheorem{example}{Example}
\newtheorem{examples}{Examples}
\newtheorem{thmdef}{Theorem-Definition}
\newtheorem{rmk}{Remark}
\newtheorem{notation}{Notation}
\newtheorem{thmdefi}{Theorem-Definition}
\newtheorem{con}{Conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%title page
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{titlepage}
\begin{center}
\tikz[remember picture,overlay] \node[anchor=north, opacity=0.3,inner sep=10pt, outer sep=5pt] at (current page.north){\includegraphics[width=9cm,height=9cm]{Images/Emblem-Ramakrishna-Mission-Transparent.png}};
\vspace{1cm}
\LARGE\textbf{Ramakrishna Mission Vivekananda Centenary College }
\vspace{1cm}
\large\textrm{Department of Mathematics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %topic name
\vspace{4cm}
\LARGE\textbf{Topic Name}
\vspace{2cm}
\large by
\vspace{1cm}
\LARGE\textbf{Your Name}
\vspace{1cm}
\large Under the supervision of
\vspace{1cm}
\LARGE\textbf{Your Guide Name}
\vspace{2cm}
\LARGE May, 2023
\vspace{2cm}
\large This dissertation is submitted to the Department of Mathematics in partial fulfilment of the requirement for the degree of Master of Science
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %name
\clearpage
\end{center}
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%table of content & Included main body files
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter*{Acknowledgements}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter*{Declaration}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter*{Abstract}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tableofcontents
\chapter{Introduction}
\input{Main body/chapter1.tex}
\chapter{Chapter Name}
\input{Main body/chapter2.tex}
\appendix
\chapter{Chapter Name}
\input{Main body/chapter4.tex}
\chapter{Chapter Name}
\input{Main body/chapter5.tex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printbibliography
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
```