 # Contents of /pubs/books/ucbka/trunk/c_glo1/c_glo1.tex

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 1 %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo1/c_glo1.tex,v 1.7 2003/03/13 06:28:13 dtashley Exp$ 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} 117 118 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 222 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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} 235 236 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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} 294 295 296 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 297 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 298 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 299 300 \noindent\begin{figure}[!b] 301 \noindent\rule[-0.25in]{\textwidth}{1pt} 302 \begin{tiny} 303 \begin{verbatim} 304 $RCSfile: c_glo1.tex,v$ 305 $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo1/c_glo1.tex,v$ 306 $Revision: 1.7$ 307 $Author: dtashley$ 308 $Date: 2003/03/13 06:28:13$ 309 \end{verbatim} 310 \end{tiny} 311 \noindent\rule[0.25in]{\textwidth}{1pt} 312 \end{figure} 313 314 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 315 % $Log: c_glo1.tex,v$ 316 % Revision 1.7 2003/03/13 06:28:13 dtashley 317 % Cardinality definition and notation added. 318 % 319 % Revision 1.6 2002/11/22 02:21:38 dtashley 320 % Substantial edits. 321 % 322 % Revision 1.5 2002/07/29 16:30:09 dtashley 323 % Safety checkin before moving work back to WSU server Kalman. 324 % 325 % Revision 1.4 2001/08/16 19:53:27 dtashley 326 % Beginning to prepare for v1.05 release. 327 % 328 % Revision 1.3 2001/07/01 19:10:30 dtashley 329 % LOG keyword expansion problem corrected. 330 % 331 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 332 % $History: c_glo1.tex$ 333 % 334 % ***************** Version 3 ***************** 335 % User: Dashley1 Date: 1/31/01 Time: 4:20p 336 % Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation 337 % Edits. 338 % 339 % ***************** Version 2 ***************** 340 % User: Dashley1 Date: 8/08/00 Time: 10:53a 341 % Updated in$/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation 342 % Correction of DIVIDES and NOT DIVIDES symbols. 343 % 344 % ***************** Version 1 ***************** 345 % User: David T. Ashley Date: 7/30/00 Time: 6:48p 346 % Created in \$/uC Software Multi-Volume Book (A)/Chapter, GLO1, Glossary Of Mathematical Notation 347 % Initial check-in. 348 % 349 %End of file C_GLO1.TEX