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1 | dashley | 5 | %$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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