%%Note: You can only use \section command, you are not allowed, per TTU Graduate School, use
%%\subsection command for ghigher level subheadings. At most level 2 subheadings are allowed.
\chapter{THEORETICAL DEVELOPMENTS}
This chapter explains the design technique developed in this research project. The first section bla bla bla. Ultimately the equation
\begin{eqnarray}
E &=&m*c^2 \\ E&=&m*9*10^{16} \label{rel}
\end{eqnarray}
In Section~\ref{singprsect} we will show something. In Section~\ref{surface} we will show something else, while in Section~\ref{caseS2} we will discuss a new case.
In
Subsection~\ref{singprsub} we will show something. In
Subsection~\ref{surfsub} we will show something else, while in
Subsection~\ref{casesub} we will discuss a new case.
\section[Singularity Problems Shorter]{Singularity Problems Longer}
\label{singprsect}
At a singularity, an infinite joint velocity is required to move the end effector in a particular direction as in Equation \ref{rel} \cite{mcc}. Also near the singularity large forces can be found that can damage the robot or an object being worked on. Bla bla bla.
\subsection[Singularity Issues Shorter]{Singularity Issues Longer}
\label{singprsub}
\subsubsection[Special Applications Shorter]{Special Applications Longer}
\section[Surface Shorter]{Surface Parameterization Longer}
\label{surface}
Clifford algebras of the $\cl^+$ can be used to bla bla bla.
\subsection[Surface Subsection Shorter]{Surface Subsection}
\label{surfsub}
\section[The Main Case]{The case of $H_{\BF}(2,q)$ and $S_2$}
\label{caseS2}
Beginning with $H_{\BF}(2,q)$ which reduces to $S_2$ in the limit
$q\rightarrow 1.$ $H_{\BF}(2,q)$ is generated by $\{ \Id,b_1 \}.$
Thus, having only one $q$-transposition, from which
a $q$-symmetrizer $R(12)$ and a $q$-antisymmetrizer $C(12)$ can be calculated.
\subsection[Main Case Subsection Shorter]{Main Case}
\label{casesub}
Notice that in the limit $q \rightarrow 1$ the following relations
for a set of new generators defined as $s_i := \lim b_i \mbox{ when } q \rightarrow 1:$
\begin{enumerate}
\item [(i)] $s_i^2 = 1,$
\item [(ii)] $s_i s_j = s_j s_i,$ whenever $\vert i-j \vert \ge 2,$
\item [(iii)] $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1},$
\item [(iii)'] $(s_i s_{i+1})^3 = \Id.$
\end{enumerate}
Property (iii)' follows from the fact that $s_i^2=1$ and $s_i^{-1} = s_i.$
This is a presentation of the symmetric group
\index{group!symmetric!presentation of}
according to Coxeter-Moser
\cite{CoxeterMoser}. Now it is an easy matter to show that (ii) is valid
for transpositions, and that (iii) can be calculated graphically using
tangles as in Figure 1.
\begin{definition}
An associative algebra over a field $\BF$ with unity $1$ is the {\em Clifford algebra\/} $\cl(Q)$ of a non-degenerate quadratic form $Q$ on a vector space $V$ over $\BF$ if it contains $V$ and $\BF=\BF \cdot 1$ as distinct subspaces so that
\begin{enumerate}
\item $\bx^2=Q(\bx)$ for any $\bx \in V,$
\item $V$ generates $\cl(Q)$ as an algebra over $\BF,$
\item $\cl(Q)$ is not generated by any proper subspace of $V.$
\end{enumerate}
\end{definition}
\begin{theorem}\label{thm:thm1}
Suppose that $1\leq p<(1-\kappa)^{-1}.$ If $u\in L^p(\BR^n,\BR)$ and
$u\geq0,$ then
\begin{equation}
|{\cal I}u(x)|\leq A_*(k,p)[{\cal M}_*u(x)]^{1-(1-\kappa)p}\|u\|_p^{(1-\kappa)p},
\qquad x\in\BR^n.
\label{eq:eq1.22}
\end{equation}
\end{theorem}
\begin{proof}
Since $u\geq0,$ it is clear that\begin{equation}
|k\ast u(x)|\leq\max\{|k_+\ast u(x)|,|k_-\ast u(x)|\}.
\label{eq:eq1.23}
\end{equation}
By applying Theorem 1 to $k_+$ and $k_-$ one gets
\begin{equation}
|k_\pm\ast u(x)|\leq A(k_\pm,p)[{\cal M}_\pm u(x)]^{1-(1-\kappa)p}\|u\|_p.
\label{eq:eq1.24}
\end{equation}
It remains to combine (\ref{eq:eq1.23}) and (\ref{eq:eq1.24}) with
(\ref{eq:eq1.20}) and (\ref{eq:eq1.21}).\footnote{These last two calls to references have resulted in (\ref{eq:eq1.20}) and (\ref{eq:eq1.21}) showing, and in an error message in your paper's log file that is automatically generated by {PC\TeX 32}. This is because references have been made to non-existing labels.}
\vskip6pt
The reader can easily check that (\ref{eq:eq1.22}) is sharp. Moreover, if
it is assumed that the kernel $k$ is an odd function, then $X_-=(-1)X_+$ and
$$
\text{vol}(X_+)=\text{vol}(X_-)=\frac{1}{2}\text{vol}(X);
$$
hence,
\begin{equation}
A_*(k,p)=2^{-\kappa}A(k,p).
\label{eq:eq1.25}
\end{equation}
%\renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
\renewcommand{\qedsymbol}{}
\end{proof}
Notice that the above proof has ended with display~(\ref{eq:eq1.25}), hence, there should be no q.e.d. symbol at the end of this proof. This is accomplished by inserting command \verb!\renewcommand{\qedsymbol}{}! right before \verb!\end{proof}!. This removal of the q.e.d. symbol applies only to this proof.