%$Header$ \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} $HeadURL$ $Revision$ $Date$ $Author$ \end{verbatim} \end{tiny} \noindent\rule[0.25in]{\textwidth}{1pt} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %End of file C_GLO1.TEX