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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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