%$Header$
\chapter*{Glossary Of Mathematical And Other Notation}
\markboth{GLOSSARY OF MATHEMATICAL NOTATION}{GLOSSARY OF MATHEMATICAL NOTATION}
\label{cglo1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{General Notation}
\begin{vworkmathtermglossaryenum}
\item \mbox{\boldmath $ \vworkdivides $}
$a \vworkdivides b$,
\index{divides@divides ($\vworkdivides$)}
\index{--@$\vworkdivides$ (divides)}
read ``\emph{$a$ divides $b$}'', denotes that $b/a$ has no remainder.
Equivalently, it may be stated that
$(a \vworkdivides b) \Rightarrow (\exists c \in \vworkintset{}, b = ac)$.
\item \mbox{\boldmath $ \vworknotdivides $}
$a \vworknotdivides b$,
\index{divides@divides ($\vworkdivides$)}
\index{--@$\vworknotdivides$ (doesn't divide)}
read ``\emph{$a$ does not divide $b$}'', denotes that $b/a$ has a reminder.
Equivalently, it may be stated that
$(a \vworknotdivides b) \Rightarrow (\nexists c \in \vworkintset{}, b = ac)$.
\item \mbox{\boldmath $ \lfloor \cdot \rfloor $}
Used
\index{floor function@floor function ($\lfloor\cdot\rfloor$)}
\index{--@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)}
to denote the \emph{floor($\cdot$)} function. The
\emph{floor($\cdot$)}
function is the largest integer not larger than the
argument.
\item \mbox{\boldmath $\lceil \cdot \rceil$ }
Used
\index{ceiling function@ceiling function ($\lceil\cdot\rceil$)}
\index{--@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)}
to denote the \emph{ceiling($\cdot$)} function.
The \emph{ceiling($\cdot$)} function
is the smallest integer not smaller than the
argument.
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Usage Of English And Greek Letters}
\begin{vworkmathtermglossaryenum}
\item \mbox {\boldmath $a/b$}
An arbitrary \index{rational number}rational number.
\item \mbox {\boldmath $ F_N $}
The \index{Farey series}Farey
series of order $N$. The Farey series is the
ordered set of irreducible rational numbers
in [0,1] with a
denominator not larger than $N$.
\item \mbox {\boldmath $F_{k_{MAX}, \overline{h_{MAX}}}$}
\index{FKMAXHMAX@$F_{k_{MAX}, \overline{h_{MAX}}}$}
The ordered set of irreducible rational numbers
$h/k$ subject to the constraints $0 \leq h \leq h_{MAX}$
and $1 \leq k \leq h_{MAX}$.
(See Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}.)
\item \mbox{\boldmath $H/K$}, \mbox{\boldmath $h/k$},
\mbox{\boldmath $h'/k'$}, \mbox{\boldmath $h''/k''$},
\mbox{\boldmath $h_i/k_i$}
Terms in a Farey series of order $N$.
\item \mbox{\boldmath $r_A$}
The rational number $h/k$ used to approximate
an arbitrary real number $r_I$.
\item \mbox{\boldmath $r_I$}
The real number, which may or may not be rational,
which is to be approximated by a rational number
$r_A = h/k$.
\item \textbf{reduced}
See \emph{irreducible}.
\item \mbox{\boldmath $s_k = p_k/q_k$}
The $k$th convergent of a continued fraction.
\item \mbox{\boldmath $x_{MAX}$}
The largest element of the domain for which the
behavior of an approximation must be guaranteed.
In this paper, most derivations assume
that $x \in [0, x_{MAX}]$, $x_{MAX} \in \vworkintsetpos{}$.
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Bitfields And Portions Of Integers}
\begin{vworkmathtermglossaryenum}
\item \mbox{\boldmath $a_{b}$}
The $b$th bit of the integer $a$. Bits are numbered with the
least significant bit ``0'', and consecutively through
``$n-1$'', where $n$ is the total number of bits.
In general, if $p$ is an $n$-bit unsigned integer,
\begin{equation}
\nonumber p = \sum_{i=0}^{n-1} 2^i p_i .
\end{equation}
\item \mbox{\boldmath $a_{c:b}$}
The integer consisting of the $b$th through the
$c$th bits of the integer $a$. Bits are numbered with the
least significant bit ``0'', and consecutively through
``$n-1$'', where $n$ is the total number of bits.
For example, if $p$ is a 24-bit unsigned integer, then
\begin{equation}
\nonumber p = 2^{16}p_{23:16} + 2^{8}p_{15:8} + p_{7:0} .
\end{equation}
\item \mbox{\boldmath $a_{[b]}$}
The $b$th word of the integer $a$. Words are numbered
with the
least significant word ``0'', and consecutively through
``$n-1$'', where $n$ is the total number of words.
In general, if $p$ is an $n$-word unsigned integer
and $z$ is the wordsize in bits,
\begin{equation}
\nonumber p = \sum_{i=0}^{n-1} 2^{iz} p_i .
\end{equation}
\item \mbox{\boldmath $a_{[c:b]}$}
The integer consisting of the $b$th through the
$c$th word of the integer $a$. Words are numbered with the
least significant word ``0'', and consecutively through
``$n-1$'', where $n$ is the total number of words.
For example, if $p$ is a 24-word unsigned integer and
$z$ is the wordsize in bits, then
\begin{equation}
\nonumber p = 2^{16z}p_{[23:16]} + 2^{8z}p_{[15:8]} + p_{[7:0]} .
\end{equation}
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Matrices And Vectors}
\begin{vworkmathtermglossaryenum}
\item \mbox{\boldmath $0$}
$\mathbf{0}$ (in bold face) is used to denote either a vector or matrix
populated with all zeroes. Optionally, in cases where the context is not clear
or where there is cause to highlight the dimension, $\mathbf{0}$ may be subscripted
to indicate the dimension, i.e.
\begin{equation}
\nonumber
\mathbf{0}_3 = \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]
\end{equation}
\begin{equation}
\nonumber
\mathbf{0}_{3 \times 2} = \left[\begin{array}{cc} 0&0 \\ 0&0 \\ 0&0 \end{array}\right]
\end{equation}
\item \mbox{\boldmath $I$}
$I$ is used to denote the square identity matrix (the matrix with all
elements 0 except elements on the diagonal which are 1).
Optionally, in cases where the context is not clear
or where there is cause to highlight the dimension, $I$ may be subscripted
to indicate the dimension, i.e.
\begin{equation}
\nonumber
I = I_3 = I_{3 \times 3} = \left[\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right]
\end{equation}
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Sets And Set Notation}
\begin{vworkmathtermglossaryenum}
\item \mbox{\boldmath $n(A)$}
The \index{cardinality}cardinality of set $A$. (The cardinality of a set is the
number of elements in the set.)
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Sets Of Numbers}
\begin{vworkmathtermglossaryenum}
\item \mbox{\boldmath $\vworkintsetpos$}
The
\index{natural number}
\index{N@$\vworkintsetpos$}
set of positive integers (natural numbers).
\item \mbox{\boldmath $\vworkratset$}
The
\index{rational number}
\index{Q@$\vworkratset$}
set of rational numbers.
\item \mbox{\boldmath $\vworkratsetnonneg$}
The
\index{rational number}
\index{Q+@$\vworkratsetnonneg$}
set of non-negative rational numbers.
\item \mbox{\boldmath $\vworkrealset$}
The
\index{real number}
\index{R@$\vworkrealset$}
set of real numbers.
\item \mbox{\boldmath $\vworkrealsetnonneg$}
The
\index{real number}
\index{R+@$\vworkrealsetnonneg$}
set of non-negative real numbers.
\item \mbox{\boldmath $\vworkintset$}
The
\index{integer}
\index{Z@$\vworkintset$}
set of integers.
\item \mbox{\boldmath $\vworkintsetnonneg$}
The
\index{integer}
\index{Z+@$\vworkintsetnonneg$}
set of non-negative integers.
\end{vworkmathtermglossaryenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\begin{figure}[!b]
\noindent\rule[-0.25in]{\textwidth}{1pt}
\begin{tiny}
\begin{verbatim}
$RCSfile: c_glo1.tex,v $
$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo1/c_glo1.tex,v $
$Revision: 1.7 $
$Author: dtashley $
$Date: 2003/03/13 06:28:13 $
\end{verbatim}
\end{tiny}
\noindent\rule[0.25in]{\textwidth}{1pt}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $Log: c_glo1.tex,v $
% Revision 1.7 2003/03/13 06:28:13 dtashley
% Cardinality definition and notation added.
%
% Revision 1.6 2002/11/22 02:21:38 dtashley
% Substantial edits.
%
% Revision 1.5 2002/07/29 16:30:09 dtashley
% Safety checkin before moving work back to WSU server Kalman.
%
% Revision 1.4 2001/08/16 19:53:27 dtashley
% Beginning to prepare for v1.05 release.
%
% Revision 1.3 2001/07/01 19:10:30 dtashley
% LOG keyword expansion problem corrected.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $History: c_glo1.tex $
%
% ***************** Version 3 *****************
% User: Dashley1 Date: 1/31/01 Time: 4:20p
% Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation
% Edits.
%
% ***************** Version 2 *****************
% User: Dashley1 Date: 8/08/00 Time: 10:53a
% Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation
% Correction of DIVIDES and NOT DIVIDES symbols.
%
% ***************** Version 1 *****************
% User: David T. Ashley Date: 7/30/00 Time: 6:48p
% Created in $/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation
% Initial check-in.
%
%End of file C_GLO1.TEX