\setcounter{page}{1} %set correct page here
%%Note: You can only use \section command, you are not allowed, per TTU Graduate School, use
%%\subsection command for higher level subheadings. At most level 2 subheadings are allowed.
\chapter{INTRODUCTION}
\section[Revisions]{Revisions and explanations to {\tt ttuthesis.sty}}
\label{revisions}
In this new {\tt ttuthesis.sty} file dated 12-03-2007 two bugs have been fixed in the definition of \verb!\section! and \verb!\subsection! commands.
\begin{itemize}
\item The first bug had to do with the older definitions not allowing for a proper labeling of sections as in
\verb!\label{sectionlabel}! and subsections as in
\verb!\label{subsectionlabel}!.
That is, when using
\verb!\ref{sectionlabel}! and
\verb!\label{subsectionlabel}! in the text, only the current chapter number was printed and not the section or the subsection number.
\end{itemize}
\noindent
For example, right after the title of the current section, I have inserted \verb!\label{revisions}! so that I could refer to this section later in the text as in
\verb!\ref{revisions}! which produces a correct section number~\ref{revisions}. Likewise, for subsections: The first subsection below has been labeled as
\verb!\label{Young}! and it can now be referred as
subsection~\ref{Young}.
Remember that per TTU Graduate School requirements, the subsubsections remain unnumbered: hence, they cannot be refered to with \verb!\ref! command even if they are labeled with \verb!\label! command.
\begin{itemize}
\item The second bug had to do with not allowing for a use of an optional parameter with a shorter title as in
\verb!\section!, \verb!\subsection!, and
\verb!\subsubsection! commands.
\end{itemize}
If you look at the TeX code of \verb!chapter1.tex! you will find that the title of the current section has been typed in as
\begin{center}
\verb!\section[Revisions]{Revisions to {\tt ttuthesis.sty}}!
\end{center}
with the REQUIRED now parameter showing a shorter title
\verb!\section[Revisions]! showing up between the brackets first and the longer title showing up between the braces.
The shorter title appears now in the Table of Contents whereas the longer title appears now in the actual text of the thesis. If you want to have both the same, that's fine, just use for the shorter title the longer title.
This new parameter containing now the short title MUST also be used in titles of subsections and subsubsections: If not used, the following error will result when typesetting:
\begin{verbatim}
./Thesis.lof) [10] (./chapter1.tex
CHAPTER 1.
! Use of \section doesn't match its definition.
l.6 \section{
Revisions to {\tt ttuthesis.sty}}
?
\end{verbatim}
I have caused that error to appear by removing that required now parameter \verb![Revisions]! from the command \verb!\section! in
\begin{center}
\verb!\section[Revisions]{Revisions to {\tt ttuthesis.sty}}!
\end{center}
\noindent
Thus, to summarize, to avoid typesetting errors, you MUST now insert these optional parameters with short titles in all commands \verb!\section!, \verb!\subsection!, and \verb!\subsubsection!. If you want the same title to appear in the Table of Contents, copy the longer title as the shorter title as well. For example, the title of the next
section~\ref{robots} is typed up as:
\begin{verbatim}
\section[Hyper-redundant robots]{Hyper-redundant robots}
\label{robots}
\end{verbatim}
In the command \verb!\chapter!, the use of this short title is optional. This is because the command \verb!\chapter! was defined earlier by authors of this style file with greater care and fixing these definitions would require a major rewrite of the file.\\
\noindent
If you have any questions, please contact me via email at
{\tt rablamowicz@tntech.edu}.
\noindent
Cookeville, December 3, 2007
\section[Hyper-redundant robots]{Hyper-redundant robots}
\label{robots}
Hyper-redundant robots have many degrees of freedom (DOF) and are sometimes called snake or worm robots. Clifford algebra is a specially described algebra which can resolve rotations and translations without matrix algebra. For a definition of the Clifford algebra see \cite{AblamowiczLounesto} and references therein. Consider the Clifford algebra $\cl_{3,0},$ also denoted as $\cl_3,$ over $\BR^3$ endowed with a quadratic form $q=\diag(1,1,1).$ Let $\{\be_1,\be_2,\be_3\}$ be an orthonormal basis in $\BR^3.$ Then, there is the following famous relation in the Clifford algebra $\cl_3:$
\begin{equation}
\be_i\be_j+\be_j\be_i = 2 \delta_{i,j}
\label{eq:defeq}
\end{equation}
where $\delta_{ij},i,j=1,2,3$ is the Kronecker delta function. Notice that relation (\ref{eq:defeq}) reduces to
\begin{equation}
\be_i^{2} = 1,\quad i=1,2,3.
\label{eq:squares}
\end{equation}
The aim of this work is to provide a mechanism for breaking up the
ordinary spinor representation of $\cl_{n,n}$ into tensor products
of smaller representations using appropriate Young operators
constructed as Clifford idempotents. This means that a suitable
Clifford algebra is used as a carrier space for various tensor product
representations.
\subsection[Young Operators Short]{Young Operators Long}
\label{Young}
The Young operators for various Young diagrams
\index{Young!operator}%
\index{Young!diagram}%
provide a set of i\-dem\-po\-tents which decompose the unity $\Id$ of $\cl_{n,n}$ as
\begin{equation}
\Id = \Y{(\lambda_1)}{}+ \cdots + \Y{(\lambda_n)}{},
\end{equation}
where $(\lambda_i)$ is a partition of $n$ characterizing the
appropriate Young tableau, that is, a Young diagram (frame) with
an allowed numbering. Let $\Y{(\lambda_i)}{i_1,\ldots,i_n}$ denote a Young tableaux
\index{Young!tableaux}%
where $(\lambda_i)$ is an ordered partition of $n$ and $i_1,\ldots,i_n$ is an allowed
numbering of the boxes in the Young diagram corresponding to
$(\lambda_i)$ as in \cite{Hamermesh,Macdonald}. Furthermore,
these Young operators are mutually annihilating idempotents
\begin{equation}
\Y{(\lambda_i)}{} \Y{(\lambda_j)}{} =
\delta_{\lambda_i \lambda_j} \Y{(\lambda_j)}{}.
\end{equation}
It appears natural to ask if these Young operators can be used
to give representations of the symmetric group {\em within\/}
the Clifford algebraic framework. The representation spaces
which appear as a natural outcome of the embedding of the symmetric
group, and its representations can then be looked at as multi-particle
spinor states.
\index{state!multi-particle}%
However, these might not be spinors of the full
Clifford algebra.
In order to be as general as possible, in the following, not only the
representations of the symmetric group will be considered, but also of the Hecke
algebra $H_\BF(n,q).$ The Hecke algebra is the generalization
of the group algebra of the symmetric group by adding the
requirement that transpositions $t_i$ of {\em adjacent\/} elements
$i,i+1$ are no longer involutions $s_i.$ Equation $t_i^2 = (1-q) t_i + q$
reduces to $s_i^2 = \Id$ in the limit $q\rightarrow 1.$
Hecke algebras are `truncated' braids
\index{group!braid}
\index{braiding}
since a further
relation (see (\ref{eq: t1}) below) is added to the braid group
relations as in \cite{Artin}. A detailed treatment of this topic
with important links to physics may be found, for example, in
\cite{Goldschmidt,Wenzl} and in the references of \cite{Fauser-hecke}.
The defining relations of the Hecke algebra will be given according
\index{Hecke algebra}%
\index{algebra!Hecke}%
to Bourbaki~\cite{Bourbaki}. Let $<\Id, t_1, \ldots, t_n >$ be a
set of generators which fulfill these relations:
\begin{align}
t_i^2 &= (1-q) t_i + q, \label{eq: t1} \\
t_i t_j &= t_j t_i, \quad \vert i-j \vert \ge 2, \label{eq: t2} \\
t_i t_{i+1} t_i &= t_{i+1} t_i t_{i+1}. \label{eq: t3}
\end{align}
Then their algebraic span is the Hecke algebra. Since the results in this
thesis will be compared with those of King and Wybourne \cite{KingWybourne}
\index{King, R.C.}
\index{Wybourne, B.G}
-- hereafter denoted by KW -- one needs to provide a transformation to their
generators $g_i,$ namely $g_i = -t_i,$ which results in a new quadratic
relation
\begin{equation}
g_i^2 = (q-1) g_i +q
\end{equation}
while the other two remain unchanged. However, this small change in
sign is responsible for great differences especially in the
$q$-polynomials occurring in both formulas. One immediate consequence is that this transformation interchanges symmetrizers and antisymmetrizers. In particular, this replacement connects Formula (3.4) in KW with full symmetrizers while Formula (3.3) in
KW gives full antisymmetrizers. Finally, the algebra morphism
$\rho$ which maps the Hecke algebra into the even part of an
appropriate Clifford algebra can be found in \cite{Fauser-hecke}.
Let $\{\Id, \be_1,\ldots, \be_{2n}\}$ be a set of generators of
the Clifford algebra $\cl(B,V)$ where the vector space $V=\{\be_i\} = \, < \be_i >$ is endowed with a non-symmetric $2n \times 2n$ bilinear form $B=[B(\be_i,\be_j)]=[B_{i,j}]$
defined as
\begin{equation}
B_{i,j} := \left\{
\begin{array}{cl}
0, & \mbox{if }\, 1\le i,j \le n \,\mbox{ or }\, n < i,j \le 2n, \\[1ex]
q, & \mbox{if }\, i = j-n \,\mbox{ or }\, i-1-n = j, \\[1ex]
-(1+q), & \mbox{if }\, i+1=j-n \,\mbox{ or }\, i=j+1-n, \\[1ex]
-1, & \mbox{if }\, \vert i-j-n \vert \ge 2 \,\mbox{ and }\, i>n, \\[1ex]
1, & \mbox{otherwise. }
\end{array}
\right.
\label{eq:Bij}
\end{equation}
The most general case would have $\nu_{ij}\not=0$ in the last line of (\ref{eq:Bij}).
For example, when $n=4,$ then
\begin{equation}
B =
\begin{pmatrix}
0 & 0 & 0 & 0 & q & -1-q & 1 & 1 \\
0 & 0 & 0 & 0 & -1-q & q & -1-q & 1 \\
0 & 0 & 0 & 0 & 1 & -1-q & q & -1-q \\
0 & 0 & 0 & 0 & 1 & 1 & -1-q & q \\
1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\
q & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\
-1 & q & 1 & 1 & 0 & 0 & 0 & 0 \\
-1 & -1 & q & 1 & 0 & 0 & 0 & 0
\end{pmatrix}.
\label{eq:ourB}
\end{equation}
The bilinear form $B$ in (\ref{eq:ourB}) is our particular choice that guarantees
that the following equations hold:
\begin{align}
\rho(t_i) &= b_i := \be_i \wedge \be_{i+n}, \label{eq:b1}\\
b_i b_j &= b_j b_i, \; \text{ whenever } \; \vert i-j \vert \ge 2, \label{eq:b2}\\
b_i b_{i+1} b_i &= b_{i+1} b_i b_{i+1}. \label{eq:b3}
\end{align}
This shows $\rho$ to be a homomorphism of algebras implementing the
Hecke algebra structure in the Clifford algebra $\cl(B,V).$ One knows
from \cite{Fauser-hecke} that $\rho$ is not injective, and that
its kernel contains all Young diagrams which are not L-shaped
\index{Young!diagram!L-shaped}
(that is, diagrams with at most one row and/or one column). The first
instance, however, when this kernel is non-trivial, occurs in $S_4$ where
the partition $4=(2,2)$ gives a Young diagram of square form which is
not L-shaped.
Testing section \ref{caseS2} here.
\section[Including .png graphics]{Including .png graphics}
[natwidth=162bp, natheight=227bp, width=190bp]; DPI=72;\\
\begin{center}
\includegraphics[natwidth=162bp,natheight=227bp, width=190bp]{Aston.png}
\end{center}
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%\subsection{Freestanding Sidehead}
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%\subsubsection{Paragraph sidehead}
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%\end{document}