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\chapter*{Glossary Of Mathematical And Other Notation} 
\chapter*{Glossary Of Mathematical And Other Notation} 
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\markboth{GLOSSARY OF MATHEMATICAL NOTATION}{GLOSSARY OF MATHEMATICAL NOTATION} 
\markboth{GLOSSARY OF MATHEMATICAL NOTATION}{GLOSSARY OF MATHEMATICAL NOTATION} 
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\label{cglo1} 
\label{cglo1} 
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\section*{General Notation} 
\section*{General Notation} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox{\boldmath $ \vworkdivides $} 
\item \mbox{\boldmath $ \vworkdivides $} 
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$a \vworkdivides b$, 
$a \vworkdivides b$, 
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\index{divides@divides ($\vworkdivides$)} 
\index{divides@divides ($\vworkdivides$)} 
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\index{@$\vworkdivides$ (divides)} 
\index{@$\vworkdivides$ (divides)} 
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read ``\emph{$a$ divides $b$}'', denotes that $b/a$ has no remainder. 
read ``\emph{$a$ divides $b$}'', denotes that $b/a$ has no remainder. 
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Equivalently, it may be stated that 
Equivalently, it may be stated that 
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$(a \vworkdivides b) \Rightarrow (\exists c \in \vworkintset{}, b = ac)$. 
$(a \vworkdivides b) \Rightarrow (\exists c \in \vworkintset{}, b = ac)$. 
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\item \mbox{\boldmath $ \vworknotdivides $} 
\item \mbox{\boldmath $ \vworknotdivides $} 
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$a \vworknotdivides b$, 
$a \vworknotdivides b$, 
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\index{divides@divides ($\vworkdivides$)} 
\index{divides@divides ($\vworkdivides$)} 
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\index{@$\vworknotdivides$ (doesn't divide)} 
\index{@$\vworknotdivides$ (doesn't divide)} 
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read ``\emph{$a$ does not divide $b$}'', denotes that $b/a$ has a reminder. 
read ``\emph{$a$ does not divide $b$}'', denotes that $b/a$ has a reminder. 
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Equivalently, it may be stated that 
Equivalently, it may be stated that 
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$(a \vworknotdivides b) \Rightarrow (\nexists c \in \vworkintset{}, b = ac)$. 
$(a \vworknotdivides b) \Rightarrow (\nexists c \in \vworkintset{}, b = ac)$. 
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\item \mbox{\boldmath $ \lfloor \cdot \rfloor $} 
\item \mbox{\boldmath $ \lfloor \cdot \rfloor $} 
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Used 
Used 
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\index{floor function@floor function ($\lfloor\cdot\rfloor$)} 
\index{floor function@floor function ($\lfloor\cdot\rfloor$)} 
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\index{@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)} 
\index{@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)} 
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to denote the \emph{floor($\cdot$)} function. The 
to denote the \emph{floor($\cdot$)} function. The 
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\emph{floor($\cdot$)} 
\emph{floor($\cdot$)} 
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function is the largest integer not larger than the 
function is the largest integer not larger than the 
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argument. 
argument. 
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\item \mbox{\boldmath $\lceil \cdot \rceil$ } 
\item \mbox{\boldmath $\lceil \cdot \rceil$ } 
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Used 
Used 
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\index{ceiling function@ceiling function ($\lceil\cdot\rceil$)} 
\index{ceiling function@ceiling function ($\lceil\cdot\rceil$)} 
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\index{@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)} 
\index{@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)} 
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to denote the \emph{ceiling($\cdot$)} function. 
to denote the \emph{ceiling($\cdot$)} function. 
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The \emph{ceiling($\cdot$)} function 
The \emph{ceiling($\cdot$)} function 
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is the smallest integer not smaller than the 
is the smallest integer not smaller than the 
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argument. 
argument. 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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\section*{Usage Of English And Greek Letters} 
\section*{Usage Of English And Greek Letters} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox {\boldmath $a/b$} 
\item \mbox {\boldmath $a/b$} 
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An arbitrary \index{rational number}rational number. 
An arbitrary \index{rational number}rational number. 
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\item \mbox {\boldmath $ F_N $} 
\item \mbox {\boldmath $ F_N $} 
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The \index{Farey series}Farey 
The \index{Farey series}Farey 
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series of order $N$. The Farey series is the 
series of order $N$. The Farey series is the 
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ordered set of irreducible rational numbers 
ordered set of irreducible rational numbers 
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in [0,1] with a 
in [0,1] with a 
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denominator not larger than $N$. 
denominator not larger than $N$. 
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\item \mbox {\boldmath $F_{k_{MAX}, \overline{h_{MAX}}}$} 
\item \mbox {\boldmath $F_{k_{MAX}, \overline{h_{MAX}}}$} 
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\index{FKMAXHMAX@$F_{k_{MAX}, \overline{h_{MAX}}}$} 
\index{FKMAXHMAX@$F_{k_{MAX}, \overline{h_{MAX}}}$} 
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The ordered set of irreducible rational numbers 
The ordered set of irreducible rational numbers 
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$h/k$ subject to the constraints $0 \leq h \leq h_{MAX}$ 
$h/k$ subject to the constraints $0 \leq h \leq h_{MAX}$ 
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and $1 \leq k \leq h_{MAX}$. 
and $1 \leq k \leq h_{MAX}$. 
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(See Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}.) 
(See Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}.) 
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\item \mbox{\boldmath $H/K$}, \mbox{\boldmath $h/k$}, 
\item \mbox{\boldmath $H/K$}, \mbox{\boldmath $h/k$}, 
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\mbox{\boldmath $h'/k'$}, \mbox{\boldmath $h''/k''$}, 
\mbox{\boldmath $h'/k'$}, \mbox{\boldmath $h''/k''$}, 
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\mbox{\boldmath $h_i/k_i$} 
\mbox{\boldmath $h_i/k_i$} 
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Terms in a Farey series of order $N$. 
Terms in a Farey series of order $N$. 
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\item \mbox{\boldmath $r_A$} 
\item \mbox{\boldmath $r_A$} 
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The rational number $h/k$ used to approximate 
The rational number $h/k$ used to approximate 
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an arbitrary real number $r_I$. 
an arbitrary real number $r_I$. 
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\item \mbox{\boldmath $r_I$} 
\item \mbox{\boldmath $r_I$} 
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The real number, which may or may not be rational, 
The real number, which may or may not be rational, 
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which is to be approximated by a rational number 
which is to be approximated by a rational number 
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$r_A = h/k$. 
$r_A = h/k$. 
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\item \textbf{reduced} 
\item \textbf{reduced} 
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See \emph{irreducible}. 
See \emph{irreducible}. 
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\item \mbox{\boldmath $s_k = p_k/q_k$} 
\item \mbox{\boldmath $s_k = p_k/q_k$} 
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The $k$th convergent of a continued fraction. 
The $k$th convergent of a continued fraction. 
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\item \mbox{\boldmath $x_{MAX}$} 
\item \mbox{\boldmath $x_{MAX}$} 
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The largest element of the domain for which the 
The largest element of the domain for which the 
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behavior of an approximation must be guaranteed. 
behavior of an approximation must be guaranteed. 
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In this paper, most derivations assume 
In this paper, most derivations assume 
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that $x \in [0, x_{MAX}]$, $x_{MAX} \in \vworkintsetpos{}$. 
that $x \in [0, x_{MAX}]$, $x_{MAX} \in \vworkintsetpos{}$. 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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\section*{Bitfields And Portions Of Integers} 
\section*{Bitfields And Portions Of Integers} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox{\boldmath $a_{b}$} 
\item \mbox{\boldmath $a_{b}$} 
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The $b$th bit of the integer $a$. Bits are numbered with the 
The $b$th bit of the integer $a$. Bits are numbered with the 
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least significant bit ``0'', and consecutively through 
least significant bit ``0'', and consecutively through 
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``$n1$'', where $n$ is the total number of bits. 
``$n1$'', where $n$ is the total number of bits. 
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In general, if $p$ is an $n$bit unsigned integer, 
In general, if $p$ is an $n$bit unsigned integer, 
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\begin{equation} 
\begin{equation} 
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\nonumber p = \sum_{i=0}^{n1} 2^i p_i . 
\nonumber p = \sum_{i=0}^{n1} 2^i p_i . 
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\end{equation} 
\end{equation} 
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\item \mbox{\boldmath $a_{c:b}$} 
\item \mbox{\boldmath $a_{c:b}$} 
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The integer consisting of the $b$th through the 
The integer consisting of the $b$th through the 
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$c$th bits of the integer $a$. Bits are numbered with the 
$c$th bits of the integer $a$. Bits are numbered with the 
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least significant bit ``0'', and consecutively through 
least significant bit ``0'', and consecutively through 
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``$n1$'', where $n$ is the total number of bits. 
``$n1$'', where $n$ is the total number of bits. 
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For example, if $p$ is a 24bit unsigned integer, then 
For example, if $p$ is a 24bit unsigned integer, then 
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\begin{equation} 
\begin{equation} 
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\nonumber p = 2^{16}p_{23:16} + 2^{8}p_{15:8} + p_{7:0} . 
\nonumber p = 2^{16}p_{23:16} + 2^{8}p_{15:8} + p_{7:0} . 
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\end{equation} 
\end{equation} 
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\item \mbox{\boldmath $a_{[b]}$} 
\item \mbox{\boldmath $a_{[b]}$} 
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The $b$th word of the integer $a$. Words are numbered 
The $b$th word of the integer $a$. Words are numbered 
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with the 
with the 
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least significant word ``0'', and consecutively through 
least significant word ``0'', and consecutively through 
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``$n1$'', where $n$ is the total number of words. 
``$n1$'', where $n$ is the total number of words. 
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In general, if $p$ is an $n$word unsigned integer 
In general, if $p$ is an $n$word unsigned integer 
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and $z$ is the wordsize in bits, 
and $z$ is the wordsize in bits, 
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\begin{equation} 
\begin{equation} 
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\nonumber p = \sum_{i=0}^{n1} 2^{iz} p_i . 
\nonumber p = \sum_{i=0}^{n1} 2^{iz} p_i . 
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\end{equation} 
\end{equation} 
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\item \mbox{\boldmath $a_{[c:b]}$} 
\item \mbox{\boldmath $a_{[c:b]}$} 
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The integer consisting of the $b$th through the 
The integer consisting of the $b$th through the 
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$c$th word of the integer $a$. Words are numbered with the 
$c$th word of the integer $a$. Words are numbered with the 
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least significant word ``0'', and consecutively through 
least significant word ``0'', and consecutively through 
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``$n1$'', where $n$ is the total number of words. 
``$n1$'', where $n$ is the total number of words. 
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For example, if $p$ is a 24word unsigned integer and 
For example, if $p$ is a 24word unsigned integer and 
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$z$ is the wordsize in bits, then 
$z$ is the wordsize in bits, then 
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\begin{equation} 
\begin{equation} 
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\nonumber p = 2^{16z}p_{[23:16]} + 2^{8z}p_{[15:8]} + p_{[7:0]} . 
\nonumber p = 2^{16z}p_{[23:16]} + 2^{8z}p_{[15:8]} + p_{[7:0]} . 
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\end{equation} 
\end{equation} 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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\section*{Matrices And Vectors} 
\section*{Matrices And Vectors} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox{\boldmath $0$} 
\item \mbox{\boldmath $0$} 
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$\mathbf{0}$ (in bold face) is used to denote either a vector or matrix 
$\mathbf{0}$ (in bold face) is used to denote either a vector or matrix 
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populated with all zeroes. Optionally, in cases where the context is not clear 
populated with all zeroes. Optionally, in cases where the context is not clear 
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or where there is cause to highlight the dimension, $\mathbf{0}$ may be subscripted 
or where there is cause to highlight the dimension, $\mathbf{0}$ may be subscripted 
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to indicate the dimension, i.e. 
to indicate the dimension, i.e. 
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\begin{equation} 
\begin{equation} 
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\nonumber 
\nonumber 
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\mathbf{0}_3 = \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right] 
\mathbf{0}_3 = \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right] 
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\end{equation} 
\end{equation} 
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\begin{equation} 
\begin{equation} 
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\nonumber 
\nonumber 
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\mathbf{0}_{3 \times 2} = \left[\begin{array}{cc} 0&0 \\ 0&0 \\ 0&0 \end{array}\right] 
\mathbf{0}_{3 \times 2} = \left[\begin{array}{cc} 0&0 \\ 0&0 \\ 0&0 \end{array}\right] 
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\end{equation} 
\end{equation} 
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\item \mbox{\boldmath $I$} 
\item \mbox{\boldmath $I$} 
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$I$ is used to denote the square identity matrix (the matrix with all 
$I$ is used to denote the square identity matrix (the matrix with all 
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elements 0 except elements on the diagonal which are 1). 
elements 0 except elements on the diagonal which are 1). 
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Optionally, in cases where the context is not clear 
Optionally, in cases where the context is not clear 
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or where there is cause to highlight the dimension, $I$ may be subscripted 
or where there is cause to highlight the dimension, $I$ may be subscripted 
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to indicate the dimension, i.e. 
to indicate the dimension, i.e. 
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\begin{equation} 
\begin{equation} 
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\nonumber 
\nonumber 
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I = I_3 = I_{3 \times 3} = \left[\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right] 
I = I_3 = I_{3 \times 3} = \left[\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right] 
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\end{equation} 
\end{equation} 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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\section*{Sets And Set Notation} 
\section*{Sets And Set Notation} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox{\boldmath $n(A)$} 
\item \mbox{\boldmath $n(A)$} 
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The \index{cardinality}cardinality of set $A$. (The cardinality of a set is the 
The \index{cardinality}cardinality of set $A$. (The cardinality of a set is the 
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number of elements in the set.) 
number of elements in the set.) 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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\section*{Sets Of Numbers} 
\section*{Sets Of Numbers} 
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\begin{vworkmathtermglossaryenum} 
\begin{vworkmathtermglossaryenum} 
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\item \mbox{\boldmath $\vworkintsetpos$} 
\item \mbox{\boldmath $\vworkintsetpos$} 
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The 
The 
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\index{natural number} 
\index{natural number} 
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\index{N@$\vworkintsetpos$} 
\index{N@$\vworkintsetpos$} 
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set of positive integers (natural numbers). 
set of positive integers (natural numbers). 
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\item \mbox{\boldmath $\vworkratset$} 
\item \mbox{\boldmath $\vworkratset$} 
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The 
The 
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\index{rational number} 
\index{rational number} 
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\index{Q@$\vworkratset$} 
\index{Q@$\vworkratset$} 
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set of rational numbers. 
set of rational numbers. 
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\item \mbox{\boldmath $\vworkratsetnonneg$} 
\item \mbox{\boldmath $\vworkratsetnonneg$} 
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The 
The 
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\index{rational number} 
\index{rational number} 
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\index{Q+@$\vworkratsetnonneg$} 
\index{Q+@$\vworkratsetnonneg$} 
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set of nonnegative rational numbers. 
set of nonnegative rational numbers. 
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\item \mbox{\boldmath $\vworkrealset$} 
\item \mbox{\boldmath $\vworkrealset$} 
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The 
The 
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\index{real number} 
\index{real number} 
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\index{R@$\vworkrealset$} 
\index{R@$\vworkrealset$} 
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set of real numbers. 
set of real numbers. 
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\item \mbox{\boldmath $\vworkrealsetnonneg$} 
\item \mbox{\boldmath $\vworkrealsetnonneg$} 
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The 
The 
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\index{real number} 
\index{real number} 
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\index{R+@$\vworkrealsetnonneg$} 
\index{R+@$\vworkrealsetnonneg$} 
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set of nonnegative real numbers. 
set of nonnegative real numbers. 
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\item \mbox{\boldmath $\vworkintset$} 
\item \mbox{\boldmath $\vworkintset$} 
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The 
The 
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\index{integer} 
\index{integer} 
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\index{Z@$\vworkintset$} 
\index{Z@$\vworkintset$} 
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set of integers. 
set of integers. 
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\item \mbox{\boldmath $\vworkintsetnonneg$} 
\item \mbox{\boldmath $\vworkintsetnonneg$} 
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The 
The 
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\index{integer} 
\index{integer} 
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\index{Z+@$\vworkintsetnonneg$} 
\index{Z+@$\vworkintsetnonneg$} 
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set of nonnegative integers. 
set of nonnegative integers. 
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\end{vworkmathtermglossaryenum} 
\end{vworkmathtermglossaryenum} 
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$Revision: 1.7 $ 
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