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%$Header$ |
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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 $} |
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|
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$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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$HeadURL$ |
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$Revision$ |
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$Date$ |
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$Author$ |
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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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% |
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%End of file C_GLO1.TEX |