trunk/c_glo1/c_glo1.tex

%$Header$

\chapter*{Glossary Of Mathematical And Other Notation}
\markboth{GLOSSARY OF MATHEMATICAL NOTATION}{GLOSSARY OF MATHEMATICAL NOTATION}

\label{cglo1}

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\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}

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\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}

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\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}

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\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}


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\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}

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\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}


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\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}

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%End of file C_GLO1.TEX
1	dashley	140	%$Header$
2
3			\chapter*{Glossary Of Mathematical And Other Notation}
4			\markboth{GLOSSARY OF MATHEMATICAL NOTATION}{GLOSSARY OF MATHEMATICAL NOTATION}
5
6			\label{cglo1}
7
8			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
9			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
11
12			\section*{General Notation}
13
14			\begin{vworkmathtermglossaryenum}
15
16			\item \mbox{\boldmath $ \vworkdivides $}
17
18
19			$a \vworkdivides b$,
20			\index{divides@divides ($\vworkdivides$)}
21			\index{--@$\vworkdivides$ (divides)}
22			read ``\emph{$a$ divides $b$}'', denotes that $b/a$ has no remainder.
23			Equivalently, it may be stated that
24			$(a \vworkdivides b) \Rightarrow (\exists c \in \vworkintset{}, b = ac)$.
25
26			\item \mbox{\boldmath $ \vworknotdivides $}
27
28			$a \vworknotdivides b$,
29			\index{divides@divides ($\vworkdivides$)}
30			\index{--@$\vworknotdivides$ (doesn't divide)}
31			read ``\emph{$a$ does not divide $b$}'', denotes that $b/a$ has a reminder.
32			Equivalently, it may be stated that
33			$(a \vworknotdivides b) \Rightarrow (\nexists c \in \vworkintset{}, b = ac)$.
34
35			\item \mbox{\boldmath $ \lfloor \cdot \rfloor $}
36
37			Used
38			\index{floor function@floor function ($\lfloor\cdot\rfloor$)}
39			\index{--@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)}
40			to denote the \emph{floor($\cdot$)} function. The
41			\emph{floor($\cdot$)}
42			function is the largest integer not larger than the
43			argument.
44
45			\item \mbox{\boldmath $\lceil \cdot \rceil$ }
46
47			Used
48			\index{ceiling function@ceiling function ($\lceil\cdot\rceil$)}
49			\index{--@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)}
50			to denote the \emph{ceiling($\cdot$)} function.
51			The \emph{ceiling($\cdot$)} function
52			is the smallest integer not smaller than the
53			argument.
54			\end{vworkmathtermglossaryenum}
55
56			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
57			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
58			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
59
60			\section*{Usage Of English And Greek Letters}
61
62			\begin{vworkmathtermglossaryenum}
63
64			\item \mbox {\boldmath $a/b$}
65
66			An arbitrary \index{rational number}rational number.
67
68			\item \mbox {\boldmath $ F_N $}
69
70			The \index{Farey series}Farey
71			series of order $N$. The Farey series is the
72			ordered set of irreducible rational numbers
73			in [0,1] with a
74			denominator not larger than $N$.
75
76			\item \mbox {\boldmath $F_{k_{MAX}, \overline{h_{MAX}}}$}
77
78			\index{FKMAXHMAX@$F_{k_{MAX}, \overline{h_{MAX}}}$}
79			The ordered set of irreducible rational numbers
80			$h/k$ subject to the constraints $0 \leq h \leq h_{MAX}$
81			and $1 \leq k \leq h_{MAX}$.
82			(See Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}.)
83
84
85			\item \mbox{\boldmath $H/K$}, \mbox{\boldmath $h/k$},
86			\mbox{\boldmath $h'/k'$}, \mbox{\boldmath $h''/k''$},
87			\mbox{\boldmath $h_i/k_i$}
88
89			Terms in a Farey series of order $N$.
90
91			\item \mbox{\boldmath $r_A$}
92
93			The rational number $h/k$ used to approximate
94			an arbitrary real number $r_I$.
95
96			\item \mbox{\boldmath $r_I$}
97
98			The real number, which may or may not be rational,
99			which is to be approximated by a rational number
100			$r_A = h/k$.
101
102			\item \textbf{reduced}
103
104			See \emph{irreducible}.
105
106			\item \mbox{\boldmath $s_k = p_k/q_k$}
107
108			The $k$th convergent of a continued fraction.
109
110			\item \mbox{\boldmath $x_{MAX}$}
111
112			The largest element of the domain for which the
113			behavior of an approximation must be guaranteed.
114			In this paper, most derivations assume
115			that $x \in [0, x_{MAX}]$, $x_{MAX} \in \vworkintsetpos{}$.
116			\end{vworkmathtermglossaryenum}
117
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119			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
120			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
121
122			\section*{Bitfields And Portions Of Integers}
123
124			\begin{vworkmathtermglossaryenum}
125			\item \mbox{\boldmath $a_{b}$}
126
127			The $b$th bit of the integer $a$. Bits are numbered with the
128			least significant bit ``0'', and consecutively through
129			``$n-1$'', where $n$ is the total number of bits.
130
131			In general, if $p$ is an $n$-bit unsigned integer,
132
133			\begin{equation}
134			\nonumber p = \sum_{i=0}^{n-1} 2^i p_i .
135			\end{equation}
136
137			\item \mbox{\boldmath $a_{c:b}$}
138
139			The integer consisting of the $b$th through the
140			$c$th bits of the integer $a$. Bits are numbered with the
141			least significant bit ``0'', and consecutively through
142			``$n-1$'', where $n$ is the total number of bits.
143
144			For example, if $p$ is a 24-bit unsigned integer, then
145
146			\begin{equation}
147			\nonumber p = 2^{16}p_{23:16} + 2^{8}p_{15:8} + p_{7:0} .
148			\end{equation}
149
150			\item \mbox{\boldmath $a_{[b]}$}
151
152			The $b$th word of the integer $a$. Words are numbered
153			with the
154			least significant word ``0'', and consecutively through
155			``$n-1$'', where $n$ is the total number of words.
156
157			In general, if $p$ is an $n$-word unsigned integer
158			and $z$ is the wordsize in bits,
159
160			\begin{equation}
161			\nonumber p = \sum_{i=0}^{n-1} 2^{iz} p_i .
162			\end{equation}
163
164			\item \mbox{\boldmath $a_{[c:b]}$}
165
166			The integer consisting of the $b$th through the
167			$c$th word of the integer $a$. Words are numbered with the
168			least significant word ``0'', and consecutively through
169			``$n-1$'', where $n$ is the total number of words.
170
171			For example, if $p$ is a 24-word unsigned integer and
172			$z$ is the wordsize in bits, then
173
174			\begin{equation}
175			\nonumber p = 2^{16z}p_{[23:16]} + 2^{8z}p_{[15:8]} + p_{[7:0]} .
176			\end{equation}
177
178			\end{vworkmathtermglossaryenum}
179
180			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
181			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
182			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
183
184			\section*{Matrices And Vectors}
185
186			\begin{vworkmathtermglossaryenum}
187
188			\item \mbox{\boldmath $0$}
189
190			$\mathbf{0}$ (in bold face) is used to denote either a vector or matrix
191			populated with all zeroes. Optionally, in cases where the context is not clear
192			or where there is cause to highlight the dimension, $\mathbf{0}$ may be subscripted
193			to indicate the dimension, i.e.
194
195			\begin{equation}
196			\nonumber
197			\mathbf{0}_3 = \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]
198			\end{equation}
199
200			\begin{equation}
201			\nonumber
202			\mathbf{0}_{3 \times 2} = \left[\begin{array}{cc} 0&0 \\ 0&0 \\ 0&0 \end{array}\right]
203			\end{equation}
204
205			\item \mbox{\boldmath $I$}
206
207			$I$ is used to denote the square identity matrix (the matrix with all
208			elements 0 except elements on the diagonal which are 1).
209			Optionally, in cases where the context is not clear
210			or where there is cause to highlight the dimension, $I$ may be subscripted
211			to indicate the dimension, i.e.
212
213			\begin{equation}
214			\nonumber
215			I = I_3 = I_{3 \times 3} = \left[\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right]
216			\end{equation}
217
218			\end{vworkmathtermglossaryenum}
219
220
221			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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223			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
224
225			\section*{Sets And Set Notation}
226
227			\begin{vworkmathtermglossaryenum}
228
229			\item \mbox{\boldmath $n(A)$}
230
231			The \index{cardinality}cardinality of set $A$. (The cardinality of a set is the
232			number of elements in the set.)
233
234			\end{vworkmathtermglossaryenum}
235
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237			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
238			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
239
240			\section*{Sets Of Numbers}
241
242			\begin{vworkmathtermglossaryenum}
243
244			\item \mbox{\boldmath $\vworkintsetpos$}
245
246			The
247			\index{natural number}
248			\index{N@$\vworkintsetpos$}
249			set of positive integers (natural numbers).
250
251			\item \mbox{\boldmath $\vworkratset$}
252
253			The
254			\index{rational number}
255			\index{Q@$\vworkratset$}
256			set of rational numbers.
257
258			\item \mbox{\boldmath $\vworkratsetnonneg$}
259
260			The
261			\index{rational number}
262			\index{Q+@$\vworkratsetnonneg$}
263			set of non-negative rational numbers.
264
265			\item \mbox{\boldmath $\vworkrealset$}
266
267			The
268			\index{real number}
269			\index{R@$\vworkrealset$}
270			set of real numbers.
271
272			\item \mbox{\boldmath $\vworkrealsetnonneg$}
273
274			The
275			\index{real number}
276			\index{R+@$\vworkrealsetnonneg$}
277			set of non-negative real numbers.
278
279			\item \mbox{\boldmath $\vworkintset$}
280
281			The
282			\index{integer}
283			\index{Z@$\vworkintset$}
284			set of integers.
285
286			\item \mbox{\boldmath $\vworkintsetnonneg$}
287
288			The
289			\index{integer}
290			\index{Z+@$\vworkintsetnonneg$}
291			set of non-negative integers.
292
293			\end{vworkmathtermglossaryenum}
294
295
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299
300			\noindent\begin{figure}[!b]
301			\noindent\rule[-0.25in]{\textwidth}{1pt}
302			\begin{tiny}
303			\begin{verbatim}
304	dashley	278	$HeadURL$
305			$Revision$
306			$Date$
307			$Author$
308	dashley	140	\end{verbatim}
309			\end{tiny}
310			\noindent\rule[0.25in]{\textwidth}{1pt}
311			\end{figure}
312
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314			%
315			%End of file C_GLO1.TEX
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