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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 |

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