Moments of the Block Operators in the Group Von Neumann Algebras
Ilwoo Cho
In this paper, we will consider the moments of the block operators of the given group von Neumann algebra $L(G)$, where the given group $G$ is a finitely presented discrete group $<X\,:\,R>$, where $X$ is the generator set of $G$ and $R$ is the relation on the set $X,$ as the set of relators. Define the canonical trace $tr$\ on $L(G)$ and the $W^{*}$-probability space $\left( L(G),tr\right) $ which is our free probabilitic object of this paper. Define the block operators $T_{x}$ by $T_{x}$ $=$ $x$ $+$ $x^{-1}$ of $L(G),$ where $x\in X.$ In this paper, we will compute the moments and the moment series of $T_{x},$ for $x$ $\in $ $X.$ By the computation, we can get that if $x_{1}$ and $x_{2}$ are generators of presented groups $% <X_{1}:\,R_{1}>$ and $<X_{2}:\,R_{2}>,$ respectively, and (i) if there is $n$ $\in $ $\Bbb{N}$ such that $x_{1}^{n}$ $\in $ $R_{1}$ and $% x_{2}^{n}$ $\in $ $R_{2},$ or (ii) if there is no $n_{1},$ $n_{2}$ $% \in $ $\Bbb{N}$ such that $x_{1}^{n_{1}}$ $\in $ $R_{1}$ and $x_{2}^{n_{2}}$ $\in $ $R_{2},$ then the block operators $x_{1}+x_{1}^{-1}$ in $L(G_{1})$ and $x_{2}+x_{2}^{-1}$ in $L(G_{2})$ are identically distributed.
Moments_BlockOps_vNPresentedGroups.tex
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\begin{document}
\title[Moments of Block Operators]{Moments of the Block Operators in the Group Von Neumann Algebras}
\author{Ilwoo Cho}
\address[A. One and A. Two]{Author OneTwo address line 1\\
Author OneTwo address line 2}
\email[A. One]{aone@@aoneinst.edu}
\thanks{}
\date{}
\subjclass{}
\keywords{Finiely Presented Groups, Group von Neumann Algebras, $W^{*}$-Probability
Spaces, Moments of Random Variables.}
\dedicatory{}
\thanks{}
\maketitle

\begin{abstract}
In this paper, we will consider the moments of the block operators of the
given group von Neumann algebra $L(G)$, where the given group $G$ is a
finitely presented discrete group $<X\,:\,R>$, where $X$ is the generator
set of $G$ and $R$ is the relation on the set $X,$ as the set of relators.
Define the canonical trace $tr$\ on $L(G)$ and the $W^{*}$-probability space 
$\left( L(G),tr\right) $ which is our free probabilitic object of this
paper. Define the block operators $T_{x}$ by $T_{x}$ $=$ $x$ $+$ $x^{-1}$ of 
$L(G),$ where $x\in X.$ In this paper, we will compute the moments and the
moment series of $T_{x},$ for $x$ $\in $ $X.$ By the computation, we can get
that if $x_{1}$ and $x_{2}$ are generators of presented groups $%
<X_{1}:\,R_{1}>$ and $<X_{2}:\,R_{2}>,$ respectively, and (i) if there 
\textbf{is} $n$ $\in $ $\Bbb{N}$ such that $x_{1}^{n}$ $\in $ $R_{1}$ and $%
x_{2}^{n}$ $\in $ $R_{2},$ or (ii) if there is \textbf{no} $n_{1},$ $n_{2}$ $%
\in $ $\Bbb{N}$ such that $x_{1}^{n_{1}}$ $\in $ $R_{1}$ and $x_{2}^{n_{2}}$ 
$\in $ $R_{2},$ then the block operators $x_{1}+x_{1}^{-1}$ in $L(G_{1})$
and $x_{2}+x_{2}^{-1}$ in $L(G_{2})$ are identically distributed.
\end{abstract}

\strut

The group von Neumann algebras are studied recently by various authors.
Group von Neumann algebras are interesting objects in Operator Algebra and
Free Probability. In this paper, we will consider the moments of certain
operators in group von Neumann algebras, where the group is presented by a
finite generator set and a finite relation. We will take a presented group $%
<X\,:\,R>,$ where $X$ is the generator set and $R$ is the relation on the
group, as the nonempty set of relators. For instance, the symmetric group $%
S_{3}$ can be presented by its generator set $X_{S_{3}}$ and its relation $%
R_{S_{3}},$ where

\strut \strut

\begin{center}
$X_{S_{3}}=\{a,b\}$
\end{center}

and

\begin{center}
$R_{s_{3}}=\{a^{2},b^{3},(ab)^{2}\}.$
\end{center}

\strut \strut

Let $H=\,<X\,:\,R>$ be a presented group and let $L(H)$ be the group von
Neumann algebra generated by $H.$ i.e.,

$\strut $

\begin{center}
$L(H)=\overline{\lambda (H)}^{w}=\overline{\Bbb{C}[H]}^{w},$
\end{center}

\strut

where $\lambda $ is the left (unitary) representation. If $a\in L(H)$ is an
operator, then it has the Fourier expansion,

\strut

\begin{center}
$a=\underset{h\in H}{\sum }a_{h}h,$ \ for \ $a_{h}\in \Bbb{C}$ and $h\in H.$
\end{center}

\strut

In fact, the group element $h$ in the previous expansion are understood as
unitary operators $\lambda _{h}$ on the Hilbert space $l^{2}(H).$ Recall that

\strut

\begin{center}
$x^{*}=\underset{h\in H}{\sum }\overline{a_{h}}\,h^{-1},$
\end{center}

\strut

where $h^{-1}=\lambda _{h^{-1}}=\lambda _{h}^{*}=h^{*},$ on $l^{2}(H),$ for
all $h\in H.$ Define the canonical trace $tr$ on $L(H)$ by

\strut

\begin{center}
$tr\left( x\right) =tr\left( \underset{h\in H}{\sum }a_{h}h\right)
=a_{e_{H}},$ for all $x\in L(H)$
\end{center}

\strut

where $e_{H}$ is the identity of $H.$ Then we can have the $W^{*}$%
-probability space $\left( L(H),tr\right) .$ The main purpose of this paper
is to compute the moments of the block operators of $L(H).$ In order to do
that we observed the free monoid $X^{*}$ $=$ $\left( X\text{ }\cup \text{ }%
X^{-1}\right) ^{\prime }$ of the group $H$ and the corresponding
combinatorial forms in $X^{*}$ of the group elements in $G.$ (If $Y$ is an
arbitrary set, then $Y^{\prime }$ is the set of all free words in $Y,$ which
is called the free monoid of $Y.$ The elements in $Y^{\prime }$ are called
the combinatorial forms of $<Y>,$ where $<Y>$ is the group generated by $Y.$)

\strut

Let $G=\,<X:\,R>$ be a presented group with its generator set $X$ $=$ $%
\{x_{1},$ ..., $x_{N}\}$ and its relation $R$ $=$ $\{r_{1},$ ..., $r_{M}\}$.
In Chapter 1, we will consider the free monoid $X^{*}$ of the group $G$
defined by

$\strut $

\begin{center}
$X^{*}\overset{def}{=}\cup _{n=0}^{\infty }\left\{
x_{j_{1}}^{p_{1}}...x_{j_{n}}^{p_{n}}: 
\begin{array}{l}
(j_{1},...,j_{n})\in \{1,...,N\}^{n} \\ 
(p_{1},...,p_{n})\in \{1,-1\}^{n}
\end{array}
\right\} ,$
\end{center}

\strut

which is the set of all free words of the generator set $X$ and $X^{-1}.$
When $n=0,$ the corresponding word is the empty word $\emptyset .$ There
exists a monoid homomorphism $\pi $ from $X^{*}$ onto the given group $G.$
Notice that, for any $g\in G,$ there is a subset $\pi ^{-1}(g)$ in $X^{*}.$
The elements $w_{g}$ in $\pi ^{-1}(g)$ are called the combinatorial forms of 
$g$ $\in $ $G.$ We will use the word problem on $X^{*},$ by computing the
moments of the block operators.

\strut

In Chapter 2, we will compute the moments of the block operators $%
T_{x}=x+x^{-1},$ for $x\in X,$ in the $W^{*}$-probability space $\left(
L(G),tr\right) .$ In particular, we have that ;

\strut \strut

(1) Suppose that there is no relator $r_{t}\in R$ such that $x^{k}=r_{t},$
for all $k\in \Bbb{N}.$ Then

\strut

\begin{center}
$tr\left( T_{x}^{m}\right) =\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] ,$ for all $m\in \Bbb{N},$
\end{center}

\strut

where $\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] \overset{def}{=}\left\{ 
\begin{array}{ll}
\left( 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right) =\frac{m!}{\left( \frac{m}{2}!\right) \left( \frac{m}{2}!\right) } & 
\text{if }m\text{ is even} \\ 
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 & \text{if }m\text{
is odd.}
\end{array}
\right. $

\strut

(2) Suppose that there exist $r_{t}\in R$ and $n_{x}\in \Bbb{N}$ such that $%
r_{t}=x^{n_{x}}.$ Then

\strut

\begin{center}
$\ tr\left( T_{x}^{m}\right) =\left\{ 
\begin{array}{ll}
\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] & \text{if }m<n_{j} \\ 
&  \\ 
\left( 2^{k_{1}}-\left[ 
\begin{array}{l}
\,k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] & \text{if }m=k_{1}n_{j}+k_{2},
\end{array}
\right. $
\end{center}

\strut

where $k_{1}\in \Bbb{N}$ and $k_{2}\in \Bbb{N}\cup \{0\}$ such that $0\leq
k_{2}<n_{x}.$

\strut $\strut $

Notice that if $G_{1}$ and $G_{2}$ are finitely presented groups (not
necessarily distinct) and $x_{1}$ and $x_{2}$ are generators of $G_{1}$ and $%
G_{2},$ respectively, and if there exists $n\in \Bbb{N}$ such that $%
x_{1}^{n} $ and $x_{2}^{n}$ are relators of $G_{1}$ and $G_{2},$
respectively, then the block operators $x_{1}+x_{1}^{-1}$ in $L(G_{1})$ and $%
x_{2}+x_{2}^{-1}$ in $L(G_{2})$ are identically distributed. Also, if $x_{1}$
and $x_{2}$ have no relators $x_{1}^{n}$ and $x_{2}^{k},$ for $n,$ $k$ $\in $
$\Bbb{N},$ then they are also identically distributed.

\strut \strut

\strut \strut

\section{Preliminaries}

\strut

\strut

In this paper, we will compute the moments of certain operators on a group
von Neumann algebra with its canonical faithful normal trace. We will
restrict our interests to finitely presented group von Neumann algebras. Let 
$G=\,<X\,:\,R>$ be a presented group, where $X$ is the finite generator set
of the group $G$ and $R$ is the relation on $G,$ as the set of all relators.
Denote the corresponding group von Neumann algebra by $L(G).$ Then each
operator $a\in L(G)$ has its Fourier expansion

\strut

(1.1) $\ \ \ \ \ \ \ \ \ \ \ a=\underset{g\in G}{\sum }a_{g}g,$ \ for \ $%
a_{g}\in \Bbb{C}$.

\strut

Note that we can regard $g$ in (1.3) as $\lambda _{g},$ for all $g\in G,$
where $\lambda $ is the left regular representation. Remark that $%
g^{*}=g^{-1},$ for all $g\in G$, in $L(G),$ and hence each operator $g\in
L(G)$ is unitary. For the group von Neumann algebra $L(G),$ we can define
the canonical trace $tr$ by

\strut

(1.2) $\ \ \ \ \ \ \ \ \ tr(a)\overset{def}{=}tr\left( \underset{g\in G}{%
\sum }a_{g}g\right) =a_{e_{G}},$

\strut

for all $a\in L(G),$ where $e_{G}$ is the identity of the group $G.$

\strut

\begin{definition}
Let $G=\,<X:\,R>$ be the presented group and $L(G),$ the corresponding group
von Neumann \ algebra. The algebraic pair $\left( L(G),tr\right) $ is called
the presented group $W^{*}$-probability space, where $tr$ is the canonical
trace given in (1.2). The operators in $\left( L(G),tr\right) $ are called
the random variables. Let $a\in L(G).$ Then the $n$-th moments of $a$ is
defined by

\strut 

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tr\left( a^{n}\right) ,$ \ \ \ for
all \ \ \ $n\in \Bbb{N}$.
\end{definition}

\strut

Now, we have our free probabilistic objects of this paper.

\strut

\strut \strut

\section{Moments of The Block Operators in Group von Neumann Algebras}

\strut

\strut

Throughout this chapter, let $G=\,<X:\,R>$ be the fixed presented graph with
its generator set $X$ and its relation $R$

\strut

\begin{center}
$X=\{x_{1},...,x_{N}\}$
\end{center}

and

\begin{center}
$R=\{r_{1},...,r_{M}\},$
\end{center}

\strut

where $M,\,N\,\in \Bbb{N}.$ In this chapter, we will compute the moments of
block operators $T_{x}=x+x^{-1}$, for $x,x^{-1}\in X,$ for $j$ $=$ $1$ $%
,..., $ $N.$

\strut

\begin{definition}
Let $G=\,<X:\,R>$ be a presented group with its generator set $X$ $=$ $%
\{x_{1},$ ..., $x_{N}\}$ and its relation $R$ $=$ $\{r_{1}$, $...,$ $r_{M}\},
$ where $M,$ $N$ $\in $ $\Bbb{N}$ and $r_{1},$ $...,$ $r_{M}$ are relators,
as elements in $X^{*}$. The operators $T_{j}$ are block operators in the
group von Neumann algebra $L(G),$ if

\strut 

$\ \ \ \ \ \ \ \ \ \ \ \ T_{j}=x_{j}+x_{j}^{-1},$ \ for all \ $j=1,...,N.$
\end{definition}

\strut \strut \strut

\strut

\strut

\subsection{Moments of Block Operators}

\strut

\strut

\strut

Let $Y$ be an arbitrary set. Then we can define a set $Y^{\prime },$
consisting of all free words in $Y.$ This set $Y^{\prime }$ is called the
free set of $Y.$ Let $G=\,<X:\,R>$ be the finitely presented group with

\strut

\begin{center}
$X=\{x_{1},...,x_{N}\}$ \ and \ $R=\{r_{1},...,r_{M}\}.$
\end{center}

\strut

Define the set $X^{*}$ be the free monoid $\left( X\text{ }\cup \text{ }%
X^{-1}\right) ^{\prime }$ of the set $X$ $\cup $ $X^{-1}.$ Notice that,
there exists the surjective (monoid) homomorphism $\pi $ $:$ $X^{*}$ $%
\rightarrow $ $G$ and, for any group element $g$ in $G,$ there exist words $%
w_{g}$ in $X^{*}$ satisfying that $\pi (w_{g})$ $=$ $g$ in $G.$ It is easy
to see that a corresponding word $w_{g}$ of $g$ is not uniquely determined.
We say that such words $w_{g}$ $\in $ $\pi ^{-1}(g)$ of $g$ $\in $ $G$ are
combinatorial forms of $g.$ Let $w_{g}$ $=$ $%
x_{j_{1}}^{p_{1}}...x_{j_{n}}^{p_{n}}$ $\in $ $X^{*}$ be a combinatorial
form of $g$ $\in G,$ where

$\strut $

\begin{center}
$(j_{1},$ ..., $j_{n})$ $\in $ $\{1,$ ..., $N\}^{n}$ and $(p_{1},$ ..., $%
p_{n})$ $\in $ $\{\pm 1\}^{n}$.
\end{center}

\strut

For convenience, we denote the word $x_{j_{n}}^{-p_{n}}...x_{j_{1}}^{-p_{1}}$
by $w_{g}^{-1}.$ Notice that $w_{g}^{-1}$ $\in $ $\pi ^{-1}(g^{-1})$ in $%
X^{*}.$

\strut

Now, fix the generators $x_{j}\in X$ and the block operator $%
T_{j}=x_{j}+x_{j}^{-1}.$ Consider the $n$-th moments $tr\left(
T_{j}^{m}\right) $ of $T_{j},$ for all $m\in \Bbb{N}.$ It is easy to see that

\strut

(2.1) \ \ $T_{j}^{m}=\underset{(p_{1},...,p_{m})\in \{1,-1\}^{m}}{\sum }%
\left( x_{j}^{p_{1}}\cdot \cdot \cdot x_{j}^{p_{m}}\right) =\underset{%
(p_{1},...,p_{m})\in \{1,-1\}^{m}}{\sum }x_{j}^{\sum_{k=1}^{m}p_{k}},$

\strut

for all $m\in \Bbb{N}.$ Notice that each word $x_{j}^{\sum_{k=1}^{m}p_{k}}$ $%
=$ $x_{j}^{p_{1}}$ $\cdot \cdot \cdot $ $x_{j}^{p_{m}}$ is regarded as an
element in the free monoid $X^{*}.$ i.e., without loss of generality, we can
consider the summands of $T_{j}^{m}$ as elements in the free monoid $X^{*}.$

\strut

\begin{lemma}
Let $T_{j}=x_{j}+x_{j}^{-1}$ be a block operator of the generator $x_{j}\in
X.$ If there is no $n_{j}\in \Bbb{N}$ such that $x_{j}^{n_{j}}$ $\in $ $R,$
then

\strut 

\ \ \ \ \ \ \ \ \ \ $tr\left( T_{j}^{m}\right) =\left\{ 
\begin{array}{ll}
\left( 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right)  & \text{if }m\in 2\Bbb{N} \\ 
&  \\ 
\,\,\,\,\,\,\,\,0 & \text{if }m\in 2\Bbb{N}-1,
\end{array}
\right. $

\strut 

where $\left( 
\begin{array}{l}
n \\ 
k
\end{array}
\right) =\frac{n!}{k!\,(n-k)!},$ for $n$, $k\in \Bbb{N}.$
\end{lemma}

\strut

\begin{proof}
Assume that there is no relator $r_{t}\in R$ and $n_{j}\in \Bbb{N}$ such
that $r_{t}=x_{j}^{n_{j}}$ in $R$ $\subset $ $X^{*}.$ Then, by (2.1), we
have that

\strut

$\ \ \ tr\left( T_{j}^{m}\right) =tr\left( \underset{(p_{1},...,p_{m})\in
\{1,-1\}^{m}}{\sum }x_{j}^{\sum_{k=1}^{m}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ =tr\left( \underset{(p_{1},...,p_{m}),\,\sum_{j=1}^{m}p_{j}=0%
}{\sum }x_{j}^{\sum_{k=1}^{m}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ =tr\left( \underset{(p_{1},...,p_{m}),\,\sum_{k=1}^{m}p_{j}=0%
}{\sum }e_{G}\right) $

\strut

$\ \ \ \ \ \ \ =\left| \{(p_{1},...,p_{m})\in
\{1,-1\}^{n}:\sum_{k=1}^{n}p_{j}=0\}\right| $

\strut

$\ \ \ \ \ \ \ =\left( 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right) .$

\strut \strut

The last equality holds because, to make $\sum_{k=1}^{m}p_{k}=0,$ the same
number of $+1$'s and $-1$'s should be appeared in the sequence $%
(p_{1},...,p_{m}).$ It is easy to see that if $m$ is odd, then the set

\strut

$\ \ \ \ \ \ \ \{(p_{1},...,p_{m})\in \{1,-1\}^{n}:\sum_{k=1}^{n}p_{j}=0\}$

\strut

is empty. Therefore, all odd moments of $T_{j}$ vanish.
\end{proof}

\strut \strut \strut \strut \strut

Now, we assume that there is a relator $r_{t}\in R$ such that $%
r_{t}=x_{j}^{n_{j}},$ for $n_{j}\in \Bbb{N},$ as a free word in $X^{*}.$
Clearly, the relator $r_{t}$ is a combinatorial form of $e_{G}$ (i.e., $\pi
(r_{t})=e_{G}$) in the group $G$ and the length $\left| r_{t}\right| $ of $%
r_{t}$ in $X^{*}$ is $n_{j}.$ Also, notice that if $r_{t}\in R,$ then the
words $wr_{t}w^{-1}$ and $wr_{t}^{-1}w^{-1}$ are also combinatorial forms of 
$e_{G},$ for all words $w$ and $w$ in $X^{*}$. By (2.1), we have that

\strut

\begin{center}
$T_{j}^{m}=\underset{(p_{1},...,p_{m})\in \{1,-1\}^{m}}{\sum }%
x_{j}^{\sum_{k=1}^{m}p_{k}}$ .
\end{center}

\strut

If $m=n_{j},$ then

\strut

\ $T_{j}^{n_{j}}=\left( x_{j}^{n_{j}}+x_{j}^{-n_{j}}\right) +\underset{%
(p_{1},...,p_{n_{j}})\in \{1,-1\}^{n_{j}},\,(p_{1},...,p_{n_{j}})\neq (\pm
1,...,\pm 1)}{\sum }x_{j}^{\sum_{k=1}^{n_{j}}p_{k}}$

\strut

(2.2)

\ \ $\ \ \ \ \ =2e_{G}+\underset{(p_{1},...,p_{n_{j}})\in
\{1,-1\}^{n_{j}},\,(p_{1},...,p_{n_{j}})\neq (\pm 1,...,\pm 1)}{\sum }%
x_{j}^{\sum_{k=1}^{n_{j}}p_{k}}.$

\strut

By the above formula (2.2), we can get the following lemma ;

\strut

\begin{lemma}
Let $x_{j}\in X$ and assume that there exists $t\in \{1,...,M\}$ such that $%
r_{t}$ $=$ $x_{j}^{n_{j}},$ for $n_{j}$ $\in $ $\Bbb{N}.$ Then

\strut 

$\ \ \ \ \ \ \ \ \ \ \ tr\left( T_{j}^{m}\right) =\left\{ 
\begin{array}{ll}
\left( 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right)  & \text{for all even }m<n_{j} \\ 
&  \\ 
\,\,\,\,\,\,0 & \text{for all odd }m<n_{j}
\end{array}
\right. $

\strut and

$\ \ \ \ \ \ \ tr\left( T_{j}^{n_{j}}\right) =\left\{ 
\begin{array}{lll}
2 &  & \text{if }n_{j}\in 2\Bbb{N}-1 \\ 
&  &  \\ 
2+\left( 
\begin{array}{l}
\,n_{j} \\ 
\frac{n_{j}}{2}
\end{array}
\right)  &  & \text{if }n_{j}\in 2\Bbb{N}\text{.}
\end{array}
\right. $
\end{lemma}

\strut

\begin{proof}
The first formula is trivial, by the previous lemma.

\strut

Suppose that $n_{j}$ is an odd number in $\Bbb{N}.$ Then, by (2.2), we have
that

\strut

$\ \ \ T_{j}^{n_{j}}=2e_{G}+\underset{(p_{1},...,p_{n_{j}})\in
\{1,-1\}^{n_{j}},\,(p_{1},...,p_{n_{j}})\neq (\pm 1,...,\pm 1)}{\sum }%
x_{j}^{\sum_{k=1}^{n_{j}}p_{k}}.$

\strut

Since $n_{j}$ is an odd number, we cannot find the sequence $%
(p_{1},...,p_{n_{j}})$ in $\{\pm 1\}^{n_{j}}$ satisfying that $%
\sum_{k=1}^{n_{j}}p_{k}$ $=$ $0.$ So, we cannot find the $e_{G}$-terms in
the summand

\strut

$\ \ \ \ \ \ \ \ \ \underset{(p_{1},...,p_{n_{j}})\in
\{1,-1\}^{n_{j}},\,(p_{1},...,p_{n_{j}})\neq (\pm 1,...,\pm 1)}{\sum }%
x_{j}^{\sum_{k=1}^{n_{j}}p_{k}}$

\strut

of $T_{j}^{n_{j}}.$ Thus if $n_{j}$ is an odd number, then $tr\left(
T_{j}^{n_{j}}\right) =2.$

\strut

Now, assume that $n_{j}$ is an even number in $\Bbb{N}.$ Then, again by
(2.2), we have that

\strut

$\ tr\left( T_{j}^{n_{j}}\right) =2+\underset{(p_{1},...,p_{n_{j}})\in
\{1,-1\}^{n_{j}},\,(p_{1},...,p_{n_{j}})\neq (\pm 1,...,\pm 1)}{\sum }%
tr\left( x_{j}^{\sum_{k=1}^{n_{j}}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ =2+\left| \{(p_{1},...,p_{n_{j}})\in \{\pm
1\}^{n_{j}}:\sum_{k=1}^{n_{j}}p_{k}=0\}\right| $

\strut

$\ \ \ \ \ \ \ =2+\left( 
\begin{array}{l}
\,\,n_{j} \\ 
\frac{n_{j}}{2}
\end{array}
\right) $.

\strut
\end{proof}

\strut

Now, suppose that $m>n_{j}.$ There are two cases ;

\strut

\ \ \ (i) $\ m=k_{1}n_{j}+k_{2},$ where $1\leq k_{2}<n_{j}$ \ \ \ \ or

\ \ \ (ii) $m=kn_{j},$ for some $k\in \Bbb{N}.$

\strut

\begin{lemma}
Let $x_{j}\in X$ and assume that there exists $t\in \{1,...,M\}$ such that $%
r_{t}$ $=$ $x_{j}^{n_{j}},$ for $n_{j}$ $\in $ $\Bbb{N}$. Then

\strut 

(1) If $m=kn_{j},$ for $k\in \Bbb{N},$ then

\strut 

$\ \ \ \ \ \ \ \ \ tr\left( T_{j}^{m}\right) =\left( 2^{k}-\left[ 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,kn_{j} \\ 
\frac{kn_{j}}{2}
\end{array}
\right] .$

\strut \strut \strut 

(2) If $m=k_{1}n_{j}+k_{2},$ for $k_{1},k_{2}\in \Bbb{N}$ and if $1\leq
k_{2}<n_{j},$ then

\strut 

$\ \ \ \ \ \ \ \ \ tr\left( T_{j}^{m}\right) =2^{k_{1}}+\left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] -\left[ 
\begin{array}{l}
k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] ,$

\strut 

where

$\ \ \ \ \ \ \ \ \ \left[ 
\begin{array}{l}
\,\,t \\ 
\frac{t}{2}
\end{array}
\right] \overset{def}{=}\left\{ 
\begin{array}{lll}
\left( 
\begin{array}{l}
\,\,t \\ 
\frac{t}{2}
\end{array}
\right)  &  & \text{if }t\text{ is even} \\ 
&  &  \\ 
0 &  & \text{if }t\text{ is odd.}
\end{array}
\right. $
\end{lemma}

\strut

\begin{proof}
(\textbf{1}) Let $(p_{1},...,p_{m})\in \{\pm 1\}^{m},$ where $m=kn_{j}$ is
sufficiently big number in $\Bbb{N},$ where $k,n_{j}\in \Bbb{N}.$ Define
subsequences

\strut

$\ \ \ \ \ \ \ \mathbf{i}_{+}=\left( \underset{n_{j}\text{-times}}{%
\underbrace{1,.....,1}}\right) $ \ \ \ and \ \ \ $\mathbf{i}_{-}=\left( 
\underset{n_{j}\text{-times}}{\underbrace{-1,.......,-1}}\right) .$

\strut

Since $m=kn_{j},$ there exists a sequence $P=(p_{1},...,p_{m})$ such that

\strut

$\ \ \ \ \ \ \ \ \ P=\left( \mathbf{i}_{i_{1}},....,\mathbf{i}%
_{i_{k}}\right) ,$ for $i_{1},...,i_{k}\in \{+,\,-\}.$

\strut

We define the set $W_{j}$, consisting of such sequences. i.e.,

\strut

(2.3)$\ \ \ \ \ \ W_{j}\overset{def}{=}\{\left( \mathbf{i}_{i_{1}},...,%
\mathbf{i}_{i_{k}}\right) :i_{1},...,i_{k}\in \{+,\,-\}\}.$

\strut

Note that, for $\mathbf{i}_{+},$ we have $x_{j}^{n_{j}}=r_{t_{j}}$ and, for $%
\mathbf{i}_{-},$ we have $x_{j}^{-n_{j}}=r_{t_{j}}^{-1}$ in $X^{*}.$ Thus,
we can get that

\strut

(2.4) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left| W_{j}\right| =2^{k}$ \ \ \ \ \
\ \ \ and

\strut

(2.5) $\ \ \ \ \ \ \ \ \sum_{k=1}^{m}p_{k}=pn_{j},$ where $p=1$ or ... or $%
k, $

\strut

for all $(p_{1},...,p_{m})$ in $W_{j}.$ We will define a subset $%
W_{j}^{\prime }$ of the set $W_{j}$ (if exists) by

\strut

\strut (2.6)

$\ W_{j}^{\prime }=\left\{ \left( \mathbf{i}_{i_{1}},...,\mathbf{i}%
_{i_{k}}\right) : 
\begin{array}{l}
i_{j_{1}}=...=i_{j_{\frac{k}{2}}}=+, \\ 
i_{s_{1}}=...=i_{s_{\frac{k}{2}}}=-, \\ 
\{i_{j_{1}},...,i_{j_{\frac{k}{2}}}\}\cup \{i_{s_{1}},...,i_{s_{\frac{k}{2}%
}}\}=\{i_{1},...,i_{k}\}
\end{array}
\right\} $

\strut

Then $W_{j}^{\prime }\subseteq W_{j}$ and, since there are same numbers of $%
+ $'s and $-$'s, we can have

\strut

(2.7) $\ \ \ \ \ \ \ \ \ \ \ \left| W_{j}^{\prime }\right| =\left( 
\begin{array}{l}
\,k \\ 
\frac{k}{2}
\end{array}
\right) ,$

\strut

if the nonempty subset $W_{j}^{\prime }$ exists in $W_{j}.$ (It is easily
see that if $k$ is even, then $W_{j}^{\prime }$ exists in $W_{j}.$ And if $k$
is odd, then $W_{j}^{\prime }$ is empty.) Define the subset $S_{0}$ of
sequences in $\{\pm 1\}^{m}$ by

\strut

\ $\ \ \ \ \ \ \ S_{0}=\{(p_{1},...,p_{m})\in \{\pm
1\}^{m}:\sum_{k=1}^{m}p_{k}=0\}.$

\strut

Then, by (2.6), $W_{j}^{\prime }\subset S_{0}.$ In fact,

\strut

(2.8) \ \ \ \ $S_{0}\cup W_{j}^{\prime }=S_{0}$ and $\left(
W_{j}\,\,\setminus \,\,W_{j}^{\prime }\right) \cap S_{0}=\emptyset .$

\strut

Assume that $k$ is even. Then

\strut

$\ \ tr\left( T_{j}^{m}\right) =tr\left( \underset{(p_{1},...,p_{m})\in
\{\pm 1\}^{m}}{\sum }x_{j}^{\sum_{k=1}^{m}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ =tr\left( \underset{(p_{1},...,p_{m})\in W_{j}\,\,\setminus
\,\,W_{j}^{\prime }}{\sum }x_{j}^{\sum_{k=1}^{m}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +tr\left( \underset{(p_{1},...,p_{m})\in
\{\pm 1\}^{m},\,\,(p_{1},...,p_{m})\in S_{0}}{\sum }x_{j}^{%
\sum_{k=1}^{m}p_{k}}\right) $

\strut

$\ \ \ \ \ \ \ =\left| W_{j}\,\,\setminus \,\,W_{j}^{\prime }\right|
+tr\left( \underset{(p_{1},...,p_{m})\in \{\pm
1\}^{m},\,\,(p_{1},...,p_{m})\in S_{0}}{\sum }e_{G}\right) $

\strut

where $W_{j}$ is defined in (2.3), by (2.8)

\strut

$\ \ \ \ =\left( 2^{k}-\left( 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right) \right) +\left| S_{0}\right| $

\strut

\strut (2.9)

$\ \ \ =\left\{ 
\begin{array}{ll}
\left( 2^{k}-\left( 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right) \right) +\left( 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right) & \text{if }m\text{ is even} \\ 
&  \\ 
2^{k}-\left( 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right) & \text{if }m\text{ is odd}
\end{array}
\right. ,$

\strut

where $k\in 2\Bbb{N}$ and $m\in \Bbb{N}.$ Now, let's suppose that $k$ is an
odd number greater than 1 in $\Bbb{N}.$ Then we can have that

\strut

(2.10) $\ \ \ \ \ \ W_{j}^{\prime }=\emptyset $ \ \ and \ \ $W_{j}\cap
S_{0}=\emptyset .$

\strut

Therefore, by (2.10), we have that

\strut

$tr\left( T_{j}^{m}\right) =tr\left( \underset{(p_{1},...,p_{m})\in W_{j}}{%
\sum }x_{j}^{\sum_{k=1}^{m}p_{k}}\right) $

\strut

\ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ +tr\left( \underset{(p_{1},...,p_{m})\in
\{\pm 1\}^{m},\,\,(p_{1},...,p_{m})\in S_{0}}{\sum }x_{j}^{%
\sum_{k=1}^{m}p_{k}}\right) $

\strut

\ \ $\ \ \ \ \ \ \ =\left| W_{j}\right| +tr\left( \underset{%
(p_{1},...,p_{m})\in \{\pm 1\}^{m},\,\,(p_{1},...,p_{m})\in S_{0}}{\sum }%
e_{G}\right) $

\strut

$\ \ \ \ \ \ \ \ \ =\left| W_{j}\right| +\left| S_{0}\right| $

\strut

\strut (2.11)

$\ \ \ \ \ \ \ \ \ \ =\left\{ 
\begin{array}{lll}
2^{k}+\left( 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right) &  & \text{if }m\text{ is even} \\ 
&  &  \\ 
2^{k} &  & \text{if }m\text{ is odd.}
\end{array}
\right. $

\strut

Now, define a new notation

\strut

$\ \ \ \ \ \ \ \left[ 
\begin{array}{l}
\,\,t \\ 
\frac{t}{2}
\end{array}
\right] \overset{def}{=}\left\{ 
\begin{array}{lll}
\left( 
\begin{array}{l}
\,\,t \\ 
\frac{t}{2}
\end{array}
\right) &  & \text{if }t\in 2\Bbb{N} \\ 
&  &  \\ 
0 &  & \text{if }t\in 2\Bbb{N}-1.
\end{array}
\right. $

\strut

Then the formuli (2.9) and (2.11) can be shortened by

\strut

$\ \ \ \ \ \ \ \ \ tr\left( T_{j}^{kn_{j}}\right) =\left( 2^{k}-\left[ 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,kn_{j} \\ 
\frac{kn_{j}}{2}
\end{array}
\right] .$

\strut \strut

(\textbf{2}) Let's assume that $m=k_{1}n_{j}+k_{2},$ where $n_{j}\nmid k_{2}$
and $1\leq k_{2}<n_{j}$. By (2.1), we have that

\strut

$\ \ \ \ \ \ \ \ \ \ \ \ \ T_{j}^{m}=\underset{(p_{1},...,p_{m})\in \{\pm
1\}^{m}}{\sum }x^{\sum_{k=1}^{m}p_{k}}.$

\strut \strut

Let's regard the summands $x^{\sum_{k=1}^{m}p_{k}}$ as free words in the
free monoid $X^{*}.$ Then there is a set

\strut

$\ \ \ \ \ \ \ S_{0}=\{(p_{1},...,p_{m})\in \{\pm
1\}^{m}:\sum_{k=1}^{m}p_{k}=0\}$

\strut

with its cardinality $\left| S_{0}\right| =\left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] .$ Now, define a set of free words $W^{j}$ by

\strut

$\ W^{j}=\left\{ \left( (p_{t_{1}},...,p_{t_{k_{2}}})\leadsto (\mathbf{i}%
_{i_{1}},...,\mathbf{i}_{i_{k_{1}}})\right) : 
\begin{array}{l}
p_{t_{k}}\in \{\pm 1\}, \\ 
\,i_{k}\in \{+,\,-\},
\end{array}
\right\} ,$

\strut

where $\mathbf{i}_{+}$ and $\mathbf{i}_{-}$ are defined in (1) and $\leadsto 
$ means the insertion. i.e., the sequence

\strut

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( (p_{t_{1}},...,p_{t_{k_{2}}})\leadsto (%
\mathbf{i}_{i_{1}},...,\mathbf{i}_{i_{k_{1}}})\right) $

\strut

is the free word in $X^{*}$ with its length $m$ $=$ $k_{1}n_{j}+k_{2}.$ For
example,

\strut

\ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( (p_{1})\leadsto (\mathbf{i}_{+},%
\mathbf{i}_{+})\right) $

is

$\ \ \ \ \left( \mathbf{i}_{+},p_{1},\mathbf{i}_{+}\right) $ \ or \ $\left(
p_{1},\mathbf{i}_{+},\mathbf{i}_{+}\right) $ \ or \ $\left( \mathbf{i}_{+},%
\mathbf{i}_{+},p_{1}\right) .$

\strut

where $m=2n_{j}+1.$ Define the subset $W_{k_{2}}^{j}$ of $W^{j}$ (if exists)
by

\strut

$\ W_{k_{2}}^{j}=\{\left( (p_{t_{1}},...,p_{t_{k_{2}}})\leadsto (\mathbf{i}%
_{i_{1}},...,\mathbf{i}_{i_{k_{1}}})\right)
:\sum_{i=1}^{k_{2}}p_{t_{i}}=0\}. $

\strut

Let's assume that $W_{k_{2}}^{j}$ exists. Then we can define the subset $%
W_{k_{2}}^{j}(S_{0})$ of $W_{k_{2}}^{j}$ by

\strut

$W_{k_{2}}^{j}(S_{0})=\left\{ \left( (p_{t_{1}},...,p_{t_{k_{2}}})\leadsto (%
\mathbf{i}_{i_{1}},...,\mathbf{i}_{i_{k_{1}}})\right) : 
\begin{array}{l}
\sum_{i=1}^{k_{2}}p_{t_{i}}=0 \\ 
(\mathbf{i}_{i_{1}},...,\mathbf{i}_{i_{k_{1}}})\in W_{j}^{\prime }
\end{array}
\right\} ,$

\strut

where $W_{j}^{\prime }$ is defined in (1). It is easy to see that if both $%
k_{1}$ and $k_{2}$ are even, then $W_{k_{2}}^{j}$ exists in $W_{k_{2}}^{j}.$
Then

\strut

(2.12) $\ \ \ \ \ \ \ \ \left| W_{k_{2}}^{j}\right| =2^{k_{1}}+\left[ 
\begin{array}{l}
k_{2} \\ 
\frac{k_{2}}{2}
\end{array}
\right] ,$

\strut

(2.13) \ \ \ \ \ \ \ $\left| W_{k_{2}}^{j}(S_{0})\right| =\left[ 
\begin{array}{l}
k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] +\left[ 
\begin{array}{l}
k_{2} \\ 
\frac{k_{2}}{2}
\end{array}
\right] $,

\strut

(2.14) \ \ \ \ \ \ \ \ \ $W_{k_{2}}^{j}(S_{0})\cup S_{0}=S_{0}$,

\strut

(2.15)\ \ $\left| S_{0}\,\,\,\setminus \,\,W_{k_{2}}^{j}(S_{0})\right|
=\left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] -\left( \left[ 
\begin{array}{l}
k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] +\left[ 
\begin{array}{l}
k_{2} \\ 
\frac{k_{2}}{2}
\end{array}
\right] \right) $

\strut

By (2.12) and (2.15), we can compute that ;

\strut

$\ \ tr\left( T_{j}^{m}\right) =tr\left( \underset{(p_{1},...,p_{m})\in
\{\pm 1\}^{m}}{\sum }x^{\sum_{k=1}^{m}p_{m}}\right) $

\strut

$\ \ \ \ =tr\left( \underset{(p_{1},...,p_{n})\in W_{k_{2}}^{j}}{\sum }%
x^{\sum_{k=1}^{m}p_{m}}\right) $

\strut

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ +tr\left( \underset{(p_{1},...,p_{m})\in
S_{0}\,\setminus \,\,W_{k_{2}}^{j}(S_{0})}{\sum }x^{\sum_{k=1}^{m}p_{m}}%
\right) $

\strut

$\ \ \ \ =\left| W_{k_{2}}^{j}\right| +\left| S_{0}\,\,\setminus
\,\,W_{k_{2}}^{j}(S_{0})\right| $

\strut

$\ \ \ \ =\left( 2^{k_{1}}+\left[ 
\begin{array}{l}
k_{2} \\ 
\frac{k_{2}}{2}
\end{array}
\right] \right) +\left( \left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] -\left( \left[ 
\begin{array}{l}
k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] +\left[ 
\begin{array}{l}
k_{2} \\ 
\frac{k_{2}}{2}
\end{array}
\right] \right) \right) $

\strut \strut

$\ \ \ \ =2^{k_{1}}+\left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] -\left[ 
\begin{array}{l}
k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] .$

\strut
\end{proof}

\strut \strut

Remark that, by the previous lemma, we have that if $k=1,$ then,

\strut

\begin{center}
$tr\left( T_{j}^{n_{j}}\right) =\left( 2^{1}-\left[ 
\begin{array}{l}
1 \\ 
\frac{1}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
n_{j} \\ 
\frac{n_{j}}{2}
\end{array}
\right] =2+\left[ 
\begin{array}{l}
n_{j} \\ 
\frac{n_{j}}{2}
\end{array}
\right] .$
\end{center}

\strut \strut

By the previous lemmas, we can get the following theorem ;

\strut

\begin{theorem}
Let $G=\,<X:\,R>$ be a finitely presented group with its generator set $X$ $=
$ $\{x_{1},$ ..., $x_{N}\}$ and its relation $R$ $=$ $\{r_{1},$ ..., $%
r_{M}\}.$ Fix a generator $x_{j}\in X$ satisfying that $r_{t}=x_{j}^{n_{j}},$
for some $n_{j}\in \Bbb{N}\,\setminus \,\{1\},$ where $r_{t}\in R.$ Then

\strut 

(1) if $m<n_{j},$ then $tr\left( T_{j}^{m}\right) =\left[ 
\begin{array}{l}
m \\ 
\frac{m}{2}
\end{array}
\right] ,$

\strut 

(2) if $m=kn_{j},$ for $k\in \Bbb{N},$ then

\strut 

$\ \ \ \ \ \ \ \ \ \ \ tr\left( T_{j}^{m}\right) =\left( 2^{k}-\left[ 
\begin{array}{l}
\,\,k \\ 
\frac{k}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] ,$

\strut 

(3) if $m=k_{1}n_{j}+k_{2},$ for $k_{1}\in N$ and $1\leq k_{2}<n_{j},$ then

\strut 

$\ \ \ \ \ \ \ \ \ tr\left( T_{j}^{m}\right) =\left( 2^{k_{1}}-\left[ 
\begin{array}{l}
\,k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] .$

$\square $
\end{theorem}

\strut \strut

We will finish this chapter with the following remark \strut ;

\strut

\begin{remark}
Let $G=\,<X:\,R>$ be a finitely presented group with its generator set $X$ $=
$ $\{x_{1},$ ..., $x_{N}\}$ and the relation $R$ $=$ $\{r_{1},$ ..., $r_{M}\}
$ and let's fix a generator $x_{j}\in X$ and the corresponding block
operator $T_{j}$ $=$ $x_{j}$ $+$ $x_{j}^{-1}$. Suppose there exist $r_{t}\in
R$ and $n_{j}\in \Bbb{N}$ such that $r_{t}=x_{j}^{n_{j}}.$ Then, by (2) and
(3) of the previous theorem,

\strut 

$\ \ \ \ tr\left( T_{j}^{m}\right) =\left\{ 
\begin{array}{ll}
\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right]  & \text{if }m<n_{j} \\ 
&  \\ 
\left( 2^{k_{1}}-\left[ 
\begin{array}{l}
\,k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right]  & \text{if }m\geq n_{j},
\end{array}
\right. $

\strut 

where $m=k_{1}n_{j}+k_{2},$ for $k_{1}\in \Bbb{N}$ and $k_{2}\in \Bbb{N}\cup
\{0\}.$ $\ \square $
\end{remark}

\strut \strut

\strut

\strut

\subsection{Identically Distributedness}

\strut

\strut

By the previous section, we have that if $G$ $=$ $<X:\,R>$ is a finitely
presented group and if $x\in X,$ then

\strut

(i) \ if there exists $n\in \Bbb{N}$ such that $x^{n}=r,$ for $r\in R,$ then

\strut

\begin{center}
$tr\left( T^{m}\right) =\left\{ 
\begin{array}{ll}
\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] & \text{if }m<n_{j} \\ 
&  \\ 
\left( 2^{k_{1}}-\left[ 
\begin{array}{l}
\,k_{1} \\ 
\frac{k_{1}}{2}
\end{array}
\right] \right) +\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] & \text{if }m\geq n_{j},
\end{array}
\right. $
\end{center}

\strut

where $m=k_{1}n+k_{2},$ for $k_{1}\in \Bbb{N}$ and $k_{2}\in \Bbb{N}\cup
\{0\},$ and

\strut

(ii) if there is no $n\in \Bbb{N}$ such that $r=x^{n},$ for all $r\in R,$
then

\strut

\begin{center}
$tr(T^{m})=\left[ 
\begin{array}{l}
\,m \\ 
\frac{m}{2}
\end{array}
\right] ,$ \ for all $m\in \Bbb{N}.$
\end{center}

\strut

The above formuli directly proves the following theorem ;

\strut

\begin{theorem}
Let $G_{i}=\,<X_{i}:R_{i}>$ be finitely presented groups, for $i=1,2,$ and
assume that $x_{i}\in X_{i}$ are generators of $G_{i}$, for $i=1,2$. If
there are relators $r_{i}\in R_{i}$ and $n\in \Bbb{N}$ such that $r_{i}$ $=$ 
$x_{i}^{n},$ for all $i$ $=$ $1,$ $2,$ then the block operators $\left(
x_{i}+x_{i}^{-1}\right) $ of the group von Neumann algebras $L(G_{i}),$ $i$ $%
=$ $1,$ $2,$ are identically distributed.
\end{theorem}

\strut

\begin{proof}
By (i) and (ii) in the previous paragraph, case by case, we can get that

\strut

$\ \ \ \ \ \ \ \ \ tr(T_{1}^{m})=tr(T_{2}^{m}),$ \ for all $\ m\in \Bbb{N}.$

\strut
\end{proof}

\strut

In the above theorem, $G_{1}$ and $G_{2}$ are not necessarily distinct.
i.e., suppose that we have a finitely presented group $G=\,<X:\,R>$ and its
generators $x_{1},$ $x_{2}$ in $X$ satisfying that $x_{1}^{n}$ and $%
x_{2}^{n} $ are relators in $R.$ Then the block operators $\left(
x_{1}+x_{1}^{-1}\right) $ and $\left( x_{2}+x_{2}^{-1}\right) $ are
identically distributed in the group von Neumann algebra, $L(G).$

\strut

The following theorem is proved, similarly ;

\strut

\begin{theorem}
Let $G_{i}=\,<X_{i}:R_{i}>$ be finitely presented groups, for $i=1,2,$ and
assume that $x_{i}\in X_{i}$ are generators of $G_{i}$, for $i=1,2$. Suppose
that there is no numbers $n_{1}$ , $n_{2}$ $\in $ $\Bbb{N}$ such that $%
x_{i}^{n_{i}}$ $\in $ $R_{i},$ for $i$ $=$ $1,$ $2,$ then the block
operators $x_{i}+x_{i}^{-1}$ in $L(G_{i})$ are identically distributed. $%
\square $
\end{theorem}

\strut \strut \strut

\strut \strut \strut

\strut \strut \strut \strut \strut \strut

\strut \strut \strut

\strut \textbf{Reference}

\strut

\strut

\begin{quote}
{\small [1] \ \ A. Nica, R-transform in Free Probability, IHP course note,
available at www.math.uwaterloo.ca/\symbol{126}anica.}

{\small [2]\strut \ \ \ A. Nica and R. Speicher, R-diagonal Pair-A Common
Approach to Haar Unitaries and Circular Elements, (1995), www
.mast.queensu.ca/\symbol{126}speicher.\strut }

{\small [3] \ }$\ ${\small B. Solel, You can see the arrows in a Quiver
Operator Algebras, (2000), preprint.}

{\small [4] \ \ D. Shlyakhtenko, Some Applications of Freeness with
Amalgamation, J. Reine Angew. Math, 500 (1998), 191-212.\strut }

{\small [5] \ \ D.Voiculescu, K. Dykemma and A. Nica, Free Random Variables,
CRM Monograph Series Vol 1 (1992).\strut }

{\small [6] \ \ D. Voiculescu, Operations on Certain Non-commuting
Operator-Valued Random Variables, Ast\'{e}risque, 232 (1995), 243-275.\strut 
}

{\small [7] \ \ D. Shlyakhtenko, A-Valued Semicircular Systems, J. of Funct
Anal, 166 (1999), 1-47.\strut }

{\small [8] \ \ D.W. Kribs and M.T. Jury, Ideal Structure in Free
Semigroupoid Algebras from Directed Graphs, preprint.}

{\small [9] \ \ D.W. Kribs and S.C. Power, Free Semigroupoid Algebras,
preprint.}

{\small [10] I. Cho, Graph }$W^{*}$-{\small Probability Theory, (2004),
Preprint.}

{\small [11] I. Cho, Random Variables in Graph }$W^{*}$-{\small Probability
Spaces, (2004), Preprint. }

{\small [12] I. Cho, Amalgamated Semicircular Systems in Graph }$W^{*}$%
{\small -Probability Spaces, (2004), Preprint. }

{\small [13] I. Cho, Weighted Graph }$W^{*}${\small -Probability Spaces,
(2004), Preprint. }

{\small [14] I. Cho, The Moment Series of the Generating Operator of }$%
L(F_{2})*_{L(F_{1})}L(F_{2})${\small , (2003), Preprint. }

{\small [15] I. Grossman and W. Magnus, Groups and Their Graphs, MAA (1964),
ISBN-0-88385-600-X.}

{\small [16] K. J. Horadam, The Word Problem and Related Results for Graph
Product Groups, Proc. AMS, vol. 82, No 2, (1981) 157-164. }

{\small [17] J. Lauri and R. Scapellato, Topics in Graph Automorphisms and
Reconstruction, London Math Soc. Student No. 54, (1995), Cambridge Univ.
Press. }

{\small [18] P. Biane and R. Speicher, Stochastic Calculus with respect to
Free Brownian Motion and Analysis on Wigner Space, Prob. Theory Relat.
Fields 112, (1998) 378-409.}

{\small [19] R. Speicher, Combinatorial Theory of the Free Product with
Amalgamation and Operator-Valued Free Probability Theory, AMS Mem, Vol 132 ,
Num 627 , (1998).}

{\small [20] R. Speicher, Combinatorics of Free Probability Theory IHP
course note, available at www.mast.queensu.ca/\symbol{126}speicher.\strut }

{\small [21] S. Hermiller and J. Meier, Algorithms and Geometry for Graph
Products of Groups, J of Algebra 171, (1995) 230-257.}
\end{quote}

\end{document}

tcilatex.tex
% Macros for Scientific Word 2.5 documents saved with the LaTeX filter.
%Copyright (C) 1994-95 TCI Software Research, Inc.
\typeout{TCILATEX Macros for Scientific Word 2.5 <22 Dec 95>.}
\typeout{NOTICE:  This macro file is NOT proprietary and may be 
freely copied and distributed.}
%
\makeatletter
%
%%%%%%%%%%%%%%%%%%%%%%
% macros for time
\newcount\@hour\newcount\@minute\chardef\@x10\chardef\@xv60
\def\tcitime{
\def\@time{%
  \@minute\time\@hour\@minute\divide\@hour\@xv
  \ifnum\@hour<\@x 0\fi\the\@hour:%
  \multiply\@hour\@xv\advance\@minute-\@hour
  \ifnum\@minute<\@x 0\fi\the\@minute
  }}%

%%%%%%%%%%%%%%%%%%%%%%
% macro for hyperref
\@ifundefined{hyperref}{\def\hyperref#1#2#3#4{#2\ref{#4}#3}}{}

% macro for external program call
\@ifundefined{qExtProgCall}{\def\qExtProgCall#1#2#3#4#5#6{\relax}}{}
%%%%%%%%%%%%%%%%%%%%%%
%
% macros for graphics
%
\def\FILENAME#1{#1}%
%
\def\QCTOpt[#1]#2{%
  \def\QCTOptB{#1}
  \def\QCTOptA{#2}
}
\def\QCTNOpt#1{%
  \def\QCTOptA{#1}
  \let\QCTOptB\empty
}
\def\Qct{%
  \@ifnextchar[{%
    \QCTOpt}{\QCTNOpt}
}
\def\QCBOpt[#1]#2{%
  \def\QCBOptB{#1}
  \def\QCBOptA{#2}
}
\def\QCBNOpt#1{%
  \def\QCBOptA{#1}
  \let\QCBOptB\empty
}
\def\Qcb{%
  \@ifnextchar[{%
    \QCBOpt}{\QCBNOpt}
}
\def\PrepCapArgs{%
  \ifx\QCBOptA\empty
    \ifx\QCTOptA\empty
      {}%
    \else
      \ifx\QCTOptB\empty
        {\QCTOptA}%
      \else
        [\QCTOptB]{\QCTOptA}%
      \fi
    \fi
  \else
    \ifx\QCBOptA\empty
      {}%
    \else
      \ifx\QCBOptB\empty
        {\QCBOptA}%
      \else
        [\QCBOptB]{\QCBOptA}%
      \fi
    \fi
  \fi
}
\newcount\GRAPHICSTYPE
%\GRAPHICSTYPE 0 is for TurboTeX
%\GRAPHICSTYPE 1 is for DVIWindo (PostScript)
%%%(removed)%\GRAPHICSTYPE 2 is for psfig (PostScript)
\GRAPHICSTYPE=\z@
\def\GRAPHICSPS#1{%
 \ifcase\GRAPHICSTYPE%\GRAPHICSTYPE=0
   \special{ps: #1}%
 \or%\GRAPHICSTYPE=1
   \special{language "PS", include "#1"}%
%%%\or%\GRAPHICSTYPE=2
%%%  #1%
 \fi
}%
%
\def\GRAPHICSHP#1{\special{include #1}}%
%
% \graffile{ body }                                  %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { vertical shift when in-line (scalar) }  %#4
\def\graffile#1#2#3#4{%
%%% \ifnum\GRAPHICSTYPE=\tw@
%%%  %Following if using psfig
%%%  \@ifundefined{psfig}{\input psfig.tex}{}%
%%%  \psfig{file=#1, height=#3, width=#2}%
%%% \else
  %Following for all others
  % JCS - added BOXTHEFRAME, see below
    \leavevmode
    \raise -#4 \BOXTHEFRAME{%
        \hbox to #2{\raise #3\hbox to #2{\null #1\hfil}}}%
}%
%
% A box for drafts
\def\draftbox#1#2#3#4{%
 \leavevmode\raise -#4 \hbox{%
  \frame{\rlap{\protect\tiny #1}\hbox to #2%
   {\vrule height#3 width\z@ depth\z@\hfil}%
  }%
 }%
}%
%
\newcount\draft
\draft=\z@
\let\nographics=\draft
\newif\ifwasdraft
\wasdraftfalse

%  \GRAPHIC{ body }                                  %#1
%          { draft name }                            %#2
%          { contentswidth (scalar)  }               %#3
%          { contentsheight (scalar) }               %#4
%          { vertical shift when in-line (scalar) }  %#5
\def\GRAPHIC#1#2#3#4#5{%
 \ifnum\draft=\@ne\draftbox{#2}{#3}{#4}{#5}%
  \else\graffile{#1}{#3}{#4}{#5}%
  \fi
 }%
%
\def\addtoLaTeXparams#1{%
    \edef\LaTeXparams{\LaTeXparams #1}}%
%
% JCS -  added a switch BoxFrame that can 
% be set by including X in the frame params.
% If set a box is drawn around the frame.

\newif\ifBoxFrame \BoxFramefalse
\newif\ifOverFrame \OverFramefalse
\newif\ifUnderFrame \UnderFramefalse

\def\BOXTHEFRAME#1{%
   \hbox{%
      \ifBoxFrame
         \frame{#1}%
      \else
         {#1}%
      \fi
   }%
}


\def\doFRAMEparams#1{\BoxFramefalse\OverFramefalse\UnderFramefalse\readFRAMEparams#1\end}%
\def\readFRAMEparams#1{%
 \ifx#1\end%
  \let\next=\relax
  \else
  \ifx#1i\dispkind=\z@\fi
  \ifx#1d\dispkind=\@ne\fi
  \ifx#1f\dispkind=\tw@\fi
  \ifx#1t\addtoLaTeXparams{t}\fi
  \ifx#1b\addtoLaTeXparams{b}\fi
  \ifx#1p\addtoLaTeXparams{p}\fi
  \ifx#1h\addtoLaTeXparams{h}\fi
  \ifx#1X\BoxFrametrue\fi
  \ifx#1O\OverFrametrue\fi
  \ifx#1U\UnderFrametrue\fi
  \ifx#1w
    \ifnum\draft=1\wasdrafttrue\else\wasdraftfalse\fi
    \draft=\@ne
  \fi
  \let\next=\readFRAMEparams
  \fi
 \next
 }%
%
%Macro for In-line graphics object
%   \IFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { vertical shift when in-line (scalar) }  %#3
%          { draft name }                            %#4
%          { body }                                  %#5
%          { caption}                                %#6


\def\IFRAME#1#2#3#4#5#6{%
      \bgroup
      \let\QCTOptA\empty
      \let\QCTOptB\empty
      \let\QCBOptA\empty
      \let\QCBOptB\empty
      #6%
      \parindent=0pt%
      \leftskip=0pt
      \rightskip=0pt
      \setbox0 = \hbox{\QCBOptA}%
      \@tempdima = #1\relax
      \ifOverFrame
          % Do this later
          \typeout{This is not implemented yet}%
          \show\HELP
      \else
         \ifdim\wd0>\@tempdima
            \advance\@tempdima by \@tempdima
            \ifdim\wd0 >\@tempdima
               \textwidth=\@tempdima
               \setbox1 =\vbox{%
                  \noindent\hbox to \@tempdima{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}\\%
                  \noindent\hbox to \@tempdima{\parbox[b]{\@tempdima}{\QCBOptA}}%
               }%
               \wd1=\@tempdima
            \else
               \textwidth=\wd0
               \setbox1 =\vbox{%
                 \noindent\hbox to \wd0{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}\\%
                 \noindent\hbox{\QCBOptA}%
               }%
               \wd1=\wd0
            \fi
         \else
            %\show\BBB
            \ifdim\wd0>0pt
              \hsize=\@tempdima
              \setbox1 =\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
                \break
                \unskip\hbox to \@tempdima{\hfill \QCBOptA\hfill}%
              }%
              \wd1=\@tempdima
           \else
              \hsize=\@tempdima
              \setbox1 =\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
              }%
              \wd1=\@tempdima
           \fi
         \fi
         \@tempdimb=\ht1
         \advance\@tempdimb by \dp1
         \advance\@tempdimb by -#2%
         \advance\@tempdimb by #3%
         \leavevmode
         \raise -\@tempdimb \hbox{\box1}%
      \fi
      \egroup%
}%
%
%Macro for Display graphics object
%   \DFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { draft label }                           %#3
%          { name }                                  %#4
%          { caption}                                %#5
\def\DFRAME#1#2#3#4#5{%
 \begin{center}
     \let\QCTOptA\empty
     \let\QCTOptB\empty
     \let\QCBOptA\empty
     \let\QCBOptB\empty
     \ifOverFrame 
        #5\QCTOptA\par
     \fi
     \GRAPHIC{#4}{#3}{#1}{#2}{\z@}
     \ifUnderFrame 
        \nobreak\par #5\QCBOptA
     \fi
 \end{center}%
 }%
%
%Macro for Floating graphic object
%   \FFRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { caption }                               %#4
%          { label }                                 %#5
%          { draft name }                            %#6
%          { body }                                  %#7
\def\FFRAME#1#2#3#4#5#6#7{%
 \begin{figure}[#1]%
  \let\QCTOptA\empty
  \let\QCTOptB\empty
  \let\QCBOptA\empty
  \let\QCBOptB\empty
  \ifOverFrame
    #4
    \ifx\QCTOptA\empty
    \else
      \ifx\QCTOptB\empty
        \caption{\QCTOptA}%
      \else
        \caption[\QCTOptB]{\QCTOptA}%
      \fi
    \fi
    \ifUnderFrame\else
      \label{#5}%
    \fi
  \else
    \UnderFrametrue%
  \fi
  \begin{center}\GRAPHIC{#7}{#6}{#2}{#3}{\z@}\end{center}%
  \ifUnderFrame
    #4
    \ifx\QCBOptA\empty
      \caption{}%
    \else
      \ifx\QCBOptB\empty
        \caption{\QCBOptA}%
      \else
        \caption[\QCBOptB]{\QCBOptA}%
      \fi
    \fi
    \label{#5}%
  \fi
  \end{figure}%
 }%
%
%
%    \FRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { vertical shift when in-line (scalar) }  %#4
%          { caption }                               %#5
%          { label }                                 %#6
%          { name }                                  %#7
%          { body }                                  %#8
%
%    framedata is a string which can contain the following
%    characters: idftbphxFT
%    Their meaning is as follows:
%             i, d or f : in-line, display, or floating
%             t,b,p,h   : LaTeX floating placement options
%             x         : fit contents box to contents
%             F or T    : Figure or Table. 
%                         Later this can expand
%                         to a more general float class.
%
%
\newcount\dispkind%

\def\makeactives{
  \catcode`\"=\active
  \catcode`\;=\active
  \catcode`\:=\active
  \catcode`\'=\active
  \catcode`\~=\active
}
\bgroup
   \makeactives
   \gdef\activesoff{%
      \def"{\string"}
      \def;{\string;}
      \def:{\string:}
      \def'{\string'}
      \def~{\string~}
      %\bbl@deactivate{"}%
      %\bbl@deactivate{;}%
      %\bbl@deactivate{:}%
      %\bbl@deactivate{'}%
    }
\egroup

\def\FRAME#1#2#3#4#5#6#7#8{%
 \bgroup
 \@ifundefined{bbl@deactivate}{}{\activesoff}
 \ifnum\draft=\@ne
   \wasdrafttrue
 \else
   \wasdraftfalse%
 \fi
 \def\LaTeXparams{}%
 \dispkind=\z@
 \def\LaTeXparams{}%
 \doFRAMEparams{#1}%
 \ifnum\dispkind=\z@\IFRAME{#2}{#3}{#4}{#7}{#8}{#5}\else
  \ifnum\dispkind=\@ne\DFRAME{#2}{#3}{#7}{#8}{#5}\else
   \ifnum\dispkind=\tw@
    \edef\@tempa{\noexpand\FFRAME{\LaTeXparams}}%
    \@tempa{#2}{#3}{#5}{#6}{#7}{#8}%
    \fi
   \fi
  \fi
  \ifwasdraft\draft=1\else\draft=0\fi{}%
  \egroup
 }%
%
% This macro added to let SW gobble a parameter that
% should not be passed on and expanded. 

\def\TEXUX#1{"texux"}

%
% Macros for text attributes:
%
\def\BF#1{{\bf {#1}}}%
\def\NEG#1{\leavevmode\hbox{\rlap{\thinspace/}{$#1$}}}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% macros for user - defined functions
\def\func#1{\mathop{\rm #1}}%
\def\limfunc#1{\mathop{\rm #1}}%

%
% miscellaneous 
%\long\def\QQQ#1#2{}%
\long\def\QQQ#1#2{%
     \long\expandafter\def\csname#1\endcsname{#2}}%
%\def\QTP#1{}% JCS - this was changed becuase style editor will define QTP
\@ifundefined{QTP}{\def\QTP#1{}}{}
\@ifundefined{QEXCLUDE}{\def\QEXCLUDE#1{}}{}
%\@ifundefined{Qcb}{\def\Qcb#1{#1}}{}
%\@ifundefined{Qct}{\def\Qct#1{#1}}{}
\@ifundefined{Qlb}{\def\Qlb#1{#1}}{}
\@ifundefined{Qlt}{\def\Qlt#1{#1}}{}
\def\QWE{}%
\long\def\QQA#1#2{}%
%\def\QTR#1#2{{\em #2}}% Always \em!!!
%\def\QTR#1#2{\mbox{\begin{#1}#2\end{#1}}}%cb%%%
\def\QTR#1#2{{\csname#1\endcsname #2}}%(gp) Is this the best?
\long\def\TeXButton#1#2{#2}%
\long\def\QSubDoc#1#2{#2}%
\def\EXPAND#1[#2]#3{}%
\def\NOEXPAND#1[#2]#3{}%
\def\PROTECTED{}%
\def\LaTeXparent#1{}%
\def\ChildStyles#1{}%
\def\ChildDefaults#1{}%
\def\QTagDef#1#2#3{}%
%
% Macros for style editor docs
\@ifundefined{StyleEditBeginDoc}{\def\StyleEditBeginDoc{\relax}}{}
%
% Macros for footnotes
\def\QQfnmark#1{\footnotemark}
\def\QQfntext#1#2{\addtocounter{footnote}{#1}\footnotetext{#2}}
%
% Macros for indexing.
\def\MAKEINDEX{\makeatletter\input gnuindex.sty\makeatother\makeindex}%	
\@ifundefined{INDEX}{\def\INDEX#1#2{}{}}{}%
\@ifundefined{SUBINDEX}{\def\SUBINDEX#1#2#3{}{}{}}{}%
\@ifundefined{initial}%  
   {\def\initial#1{\bigbreak{\raggedright\large\bf #1}\kern 2\p@\penalty3000}}%
   {}%
\@ifundefined{entry}{\def\entry#1#2{\item {#1}, #2}}{}%
\@ifundefined{primary}{\def\primary#1{\item {#1}}}{}%
\@ifundefined{secondary}{\def\secondary#1#2{\subitem {#1}, #2}}{}%
%
%
\@ifundefined{ZZZ}{}{\MAKEINDEX\makeatletter}%
%
% Attempts to avoid problems with other styles
\@ifundefined{abstract}{%
 \def\abstract{%
  \if@twocolumn
   \section*{Abstract (Not appropriate in this style!)}%
   \else \small 
   \begin{center}{\bf Abstract\vspace{-.5em}\vspace{\z@}}\end{center}%
   \quotation 
   \fi
  }%
 }{%
 }%
\@ifundefined{endabstract}{\def\endabstract
  {\if@twocolumn\else\endquotation\fi}}{}%
\@ifundefined{maketitle}{\def\maketitle#1{}}{}%
\@ifundefined{affiliation}{\def\affiliation#1{}}{}%
\@ifundefined{proof}{\def\proof{\noindent{\bfseries Proof. }}}{}%
\@ifundefined{endproof}{\def\endproof{\mbox{\ \rule{.1in}{.1in}}}}{}%
\@ifundefined{newfield}{\def\newfield#1#2{}}{}%
\@ifundefined{chapter}{\def\chapter#1{\par(Chapter head:)#1\par }%
 \newcount\c@chapter}{}%
\@ifundefined{part}{\def\part#1{\par(Part head:)#1\par }}{}%
\@ifundefined{section}{\def\section#1{\par(Section head:)#1\par }}{}%
\@ifundefined{subsection}{\def\subsection#1%
 {\par(Subsection head:)#1\par }}{}%
\@ifundefined{subsubsection}{\def\subsubsection#1%
 {\par(Subsubsection head:)#1\par }}{}%
\@ifundefined{paragraph}{\def\paragraph#1%
 {\par(Subsubsubsection head:)#1\par }}{}%
\@ifundefined{subparagraph}{\def\subparagraph#1%
 {\par(Subsubsubsubsection head:)#1\par }}{}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These symbols are not recognized by LaTeX
\@ifundefined{therefore}{\def\therefore{}}{}%
\@ifundefined{backepsilon}{\def\backepsilon{}}{}%
\@ifundefined{yen}{\def\yen{\hbox{\rm\rlap=Y}}}{}%
\@ifundefined{registered}{%
   \def\registered{\relax\ifmmode{}\r@gistered
                    \else$\m@th\r@gistered$\fi}%
 \def\r@gistered{^{\ooalign
  {\hfil\raise.07ex\hbox{$\scriptstyle\rm\text{R}$}\hfil\crcr
  \mathhexbox20D}}}}{}%
\@ifundefined{Eth}{\def\Eth{}}{}%
\@ifundefined{eth}{\def\eth{}}{}%
\@ifundefined{Thorn}{\def\Thorn{}}{}%
\@ifundefined{thorn}{\def\thorn{}}{}%
% A macro to allow any symbol that requires math to appear in text
\def\TEXTsymbol#1{\mbox{$#1$}}%
\@ifundefined{degree}{\def\degree{{}^{\circ}}}{}%
%
% macros for T3TeX files
\newdimen\theight
\def\Column{%
 \vadjust{\setbox\z@=\hbox{\scriptsize\quad\quad tcol}%
  \theight=\ht\z@\advance\theight by \dp\z@\advance\theight by \lineskip
  \kern -\theight \vbox to \theight{%
   \rightline{\rlap{\box\z@}}%
   \vss
   }%
  }%
 }%
%
\def\qed{%
 \ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi
 \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}%
 }%
%
\def\cents{\hbox{\rm\rlap/c}}%
\def\miss{\hbox{\vrule height2\p@ width 2\p@ depth\z@}}%
%\def\miss{\hbox{.}}%        %another possibility 
%
\def\vvert{\Vert}%           %always translated to \left| or \right|
%
\def\tcol#1{{\baselineskip=6\p@ \vcenter{#1}} \Column}  %
%
\def\dB{\hbox{{}}}%                 %dummy entry in column 
\def\mB#1{\hbox{$#1$}}%             %column entry
\def\nB#1{\hbox{#1}}%               %column entry (not math)
%
%\newcount\notenumber
%\def\clearnotenumber{\notenumber=0}
%\def\note{\global\advance\notenumber by 1
% \footnote{$^{\the\notenumber}$}}
%\def\note{\global\advance\notenumber by 1
\def\note{$^{\dag}}%
%
%

\def\newfmtname{LaTeX2e}
\def\chkcompat{%
   \if@compatibility
   \else
     \usepackage{latexsym}
   \fi
}

\ifx\fmtname\newfmtname
  \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
  \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
  \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
  \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
  \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
  \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
  \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
  \chkcompat
\fi

%
% Greek bold macros
% Redefine all of the math symbols 
% which might be bolded	 - there are 
% probably others to add to this list

\def\alpha{{\Greekmath 010B}}%
\def\beta{{\Greekmath 010C}}%
\def\gamma{{\Greekmath 010D}}%
\def\delta{{\Greekmath 010E}}%
\def\epsilon{{\Greekmath 010F}}%
\def\zeta{{\Greekmath 0110}}%
\def\eta{{\Greekmath 0111}}%
\def\theta{{\Greekmath 0112}}%
\def\iota{{\Greekmath 0113}}%
\def\kappa{{\Greekmath 0114}}%
\def\lambda{{\Greekmath 0115}}%
\def\mu{{\Greekmath 0116}}%
\def\nu{{\Greekmath 0117}}%
\def\xi{{\Greekmath 0118}}%
\def\pi{{\Greekmath 0119}}%
\def\rho{{\Greekmath 011A}}%
\def\sigma{{\Greekmath 011B}}%
\def\tau{{\Greekmath 011C}}%
\def\upsilon{{\Greekmath 011D}}%
\def\phi{{\Greekmath 011E}}%
\def\chi{{\Greekmath 011F}}%
\def\psi{{\Greekmath 0120}}%
\def\omega{{\Greekmath 0121}}%
\def\varepsilon{{\Greekmath 0122}}%
\def\vartheta{{\Greekmath 0123}}%
\def\varpi{{\Greekmath 0124}}%
\def\varrho{{\Greekmath 0125}}%
\def\varsigma{{\Greekmath 0126}}%
\def\varphi{{\Greekmath 0127}}%

\def\nabla{{\Greekmath 0272}}
\def\FindBoldGroup{%
   {\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}%
}

\def\Greekmath#1#2#3#4{%
    \if@compatibility
        \ifnum\mathgroup=\symbold
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi 
    \else 
        \FindBoldGroup
        \ifnum\mathgroup=\theboldgroup % For 2e
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi     	    
	  \fi}

\newif\ifGreekBold  \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}%
%

\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}


\@ifundefined{mathletters}{%
  %\def\theequation{\arabic{equation}}
  \newcounter{equationnumber}  
  \def\mathletters{%
     \addtocounter{equation}{1}
     \edef\@currentlabel{\theequation}%
     \setcounter{equationnumber}{\c@equation}
     \setcounter{equation}{0}%
     \edef\theequation{\@currentlabel\noexpand\alph{equation}}%
  }
  \def\endmathletters{%
     \setcounter{equation}{\value{equationnumber}}%
  }
}{}

%Logos
\@ifundefined{BibTeX}{%
    \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
                 T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}}{}%
\@ifundefined{AmS}%
    {\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
                A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}}{}%
\@ifundefined{AmSTeX}{\def\AmSTeX{\protect\AmS-\protect\TeX\@}}{}%
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NOTE: The rest of this file is read only if amstex has not been
% loaded.  This section is used to define amstex constructs in the
% event they have not been defined.
%
%
\ifx\ds@amstex\relax
   \message{amstex already loaded}\makeatother\endinput% 2.09 compatability
\else
   \@ifpackageloaded{amstex}%
      {\message{amstex already loaded}\makeatother\endinput}
      {}
   \@ifpackageloaded{amsgen}%
      {\message{amsgen already loaded}\makeatother\endinput}
      {}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%
%
%  Macros to define some AMS LaTeX constructs when 
%  AMS LaTeX has not been loaded
% 
% These macros are copied from the AMS-TeX package for doing
% multiple integrals.
%
\let\DOTSI\relax
\def\RIfM@{\relax\ifmmode}%
\def\FN@{\futurelet\next}%
\newcount\intno@
\def\iint{\DOTSI\intno@\tw@\FN@\ints@}%
\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}%
\def\iiiint{\DOTSI\intno@4 \FN@\ints@}%
\def\idotsint{\DOTSI\intno@\z@\FN@\ints@}%
\def\ints@{\findlimits@\ints@@}%
\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
 \else\ifx\next\nolimits\limits@false\else
 \limtoken@false\ifx\ilimits@\nolimits\limits@false\else
 \ifinner\limits@false\else\limits@true\fi\fi\fi\fi}%
\def\multint@{\int\ifnum\intno@=\z@\intdots@                          %1
 \else\intkern@\fi                                                    %2
 \ifnum\intno@>\tw@\int\intkern@\fi                                   %3
 \ifnum\intno@>\thr@@\int\intkern@\fi                                 %4
 \int}%                                                               %5
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
 \ifnum\intno@>\tw@\intop\intkern@\fi
 \ifnum\intno@>\thr@@\intop\intkern@\fi\intop}%
\def\intic@{%
    \mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
 {\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@                                              %1
 \def\ints@@@{\iflimits@\negintic@
   \mathop{\intic@\multintlimits@}\limits                             %2
  \else\multint@\nolimits\fi                                          %3
  \eat@}%                                                             %4
 \else                                                                %5
 \def\ints@@@{\iflimits@\negintic@
  \mathop{\intic@\multintlimits@}\limits\else
  \multint@\nolimits\fi}\fi\ints@@@}%
\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}%
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
 {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
%
%
%  These macros are for doing the AMS \text{} construct
%
\def\RIfM@{\relax\protect\ifmmode}
\def\text{\RIfM@\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
   {\textdef@\displaystyle\f@size{#1}}%
   {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
   \glb@settings}

\def\textdef@#1#2#3{\hbox{{%
                    \everymath{#1}%
                    \let\f@size#2\selectfont
                    #3}}}
\newif\iffirstchoice@
\firstchoice@true
%
%    Old Scheme for \text
%
%\def\rmfam{\z@}%
%\newif\iffirstchoice@
%\firstchoice@true
%\def\textfonti{\the\textfont\@ne}%
%\def\textfontii{\the\textfont\tw@}%
%\def\text{\RIfM@\expandafter\text@\else\expandafter\text@@\fi}%
%\def\text@@#1{\leavevmode\hbox{#1}}%
%\def\text@#1{\mathchoice
% {\hbox{\everymath{\displaystyle}\def\textfonti{\the\textfont\@ne}%
%  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\textstyle}\def\textfonti{\the\textfont\@ne}%
%  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\scriptstyle}\def\textfonti{\the\scriptfont\@ne}%
%  \def\textfontii{\the\scriptfont\tw@}\textdef@@ S\rm#1}}%
% {\hbox{\firstchoice@false
%  \everymath{\scriptscriptstyle}\def\textfonti
%  {\the\scriptscriptfont\@ne}%
%  \def\textfontii{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}}%
%\def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl
%    \textdef@#1\it}%
%\def\DN@{\def\next@}%
%\def\eat@#1{}%
%\def\textdef@#1#2{%
% \DN@{\csname\expandafter\eat@\string#2fam\endcsname}%
% \if S#1\edef#2{\the\scriptfont\next@\relax}%
% \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}%
% \else\edef#2{\the\textfont\next@\relax}\fi\fi}%
%
%
%These are the AMS constructs for multiline limits.
%
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}%
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
\def\endSb{\crcr\egroup\egroup\egroup}%
\def\Sp{^\multilimits@}%
\let\endSp\endSb
%
%
%These are AMS constructs for horizontal arrows
%
\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}%
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
   \leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}%
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
  \hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%
%%%%%%%%%%%%%%%%%%%%%

% 94.0815 by Jon:

\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}


\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
 \hfil$#1\m@th\operator@font lim$\hfil\crcr
 \noalign{\nointerlineskip}#2#1\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}

 \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}


\def\projlim{\qopnamewl@{proj\,lim}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
\def\varliminf{\mathpalette\varliminf@{}}
\def\varliminf@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
   \hbox{$#1\m@th\operator@font lim$}}}}
\def\varlimsup{\mathpalette\varlimsup@{}}
\def\varlimsup@#1{\mathop{\overline
  {\hbox{$#1\m@th\operator@font lim$}}}}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\binom#1#2{{#1 \choose #2}}%
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}%
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%
\def\QATOP#1#2{{#1 \atop #2}}%
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}%
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}%
\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}%
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}%
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}%
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}%
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}%
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}%
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
\def\QTABOVED#1#2#3#4#5{{\textstyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
\def\QDABOVED#1#2#3#4#5{{\displaystyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
%
% Macros for text size operators:

%JCS - added braces and \mathop around \displaystyle\int, etc.
%
\def\tint{\mathop{\textstyle \int}}%
\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%
%
%
%Macros for display size operators:
%

\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint }}%
\def\diiint{\mathop{\displaystyle \iiint }}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
\def\dcoprod{\mathop{\displaystyle \coprod }}%
\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%
%
%Companion to stackrel
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
%
%
% These are AMS environments that will be defined to
% be verbatims if amstex has not actually been 
% loaded
%
%
\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12 
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]

|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]

|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]


|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]

|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]

|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]


|endgroup
  

  
\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist
 
\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist




\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist
 
\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist




\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist
 
\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist




\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist
 
\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist


\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist
 
\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist


\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist

\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist

 
\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist

\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist

% macro to simulate ams tag construct


% This macro is a fix to eqnarray
\def\@@eqncr{\let\@tempa\relax
    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
      \else \def\@tempa{&}\fi
     \@tempa
     \if@eqnsw
        \iftag@
           \@taggnum
        \else
           \@eqnnum\stepcounter{equation}%
        \fi
     \fi
     \global\tag@false
     \global\@eqnswtrue
     \global\@eqcnt\z@\cr}


% This macro is a fix to the equation environment
 \def\endequation{%
     \ifmmode\ifinner % FLEQN hack
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        $\hfil
           \displaywidth\linewidth\@taggnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue   
      \else
        $\hfil
           \displaywidth\linewidth\@eqnnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue 
      \fi
     \else   
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        \eqno \hbox{\@taggnum}
        \global\tag@false%
        $$\global\@ignoretrue
      \else
        \eqno \hbox{\@eqnnum}% $$ BRACE MATCHING HACK
        $$\global\@ignoretrue
      \fi
     \fi\fi
 } 

 \newif\iftag@ \tag@false
 
 \def\tag{\@ifnextchar*{\@tagstar}{\@tag}}
 \def\@tag#1{%
     \global\tag@true
     \global\def\@taggnum{(#1)}}
 \def\@tagstar*#1{%
     \global\tag@true
     \global\def\@taggnum{#1}%  
}

% Do not add anything to the end of this file.  
% The last section of the file is loaded only if 
% amstex has not been.



\makeatother
\endinput
