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