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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    
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 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    
313     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
314     %
315     %End of file C_GLO1.TEX

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