%======================================================================
% notation.tex - Symbols, Units, and Constants
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% Fix or remove this appendix.  There's not much in it any more.

\chapter{Units, Constants, and Notations}
\label{ch:notation}

\def\draftid{$\ $Id: notation.tex,v 1.4.2.2.2.2 1999/05/13 19:19:12 mpf Exp $\ $}

This appendix simply lists various fundamental units, physical constants,
and notations used throughout the text, for easy reference.

Table~\ref{tab:prefixes} shows the names, abbreviations, and values of the
standard order-of-magnitude prefixes for units of measurement.

\begin{table}
\begin{center}
\begin{tabular}{|rcl|rcl|}\hline
  Name & Sym & Val  & Name & Sym & Val\\
\hline\hline
 deka- & D & $10^1$    &  deci- &    d   & $10^{-1}$  \\
hecto- & h & $10^2$    & centi- &    c   & $10^{-2}$  \\
 kilo- & k & $10^3$    & milli- &    m   & $10^{-3}$  \\
 mega- & M & $10^6$    & micro- & \micro & $10^{-6}$  \\
 giga- & G & $10^9$    &  nano- &    n   & $10^{-9}$  \\
 tera- & T & $10^{12}$ &  pico- &    p   & $10^{-12}$ \\
 peta- & P & $10^{15}$ & femto- &    f   & $10^{-15}$ \\
  exa- & E & $10^{18}$ &  atto- &    a   & $10^{-18}$ \\
\hline\end{tabular}
\caption[Unit magnitude prefixes]{Unit magnitude prefixes.}
\label{tab:prefixes}
\end{center}\hrule
\end{table}

Table~\ref{tab:units} shows the various base units of measurement referred
to in this thesis.  Any of these of course may also appear together with
any of the prefixes listed in table~\ref{tab:prefixes}.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|l|l|l|}
\hline %----------------------------------------------------------------------------------------------------------
Symbol		& Name		& Measures 		& Some equivalences					\\
\hline %----------------------------------------------------------------------------------------------------------
$\Ang$		& {\AA}ngstrom	& length		& $10^{-10}\:\meter$					\\
$\byte$		& byte		& information		& $8\:\bit$						\\
$\bit$		& bit		& information, entropy	& $(\ln 2)\:\nat$					\\
$\Coul$		& Coulomb	& electric charge	&							\\
$\eV$		& electron Volt & energy		& $\e{1.60217733}{-19}\:\Joule$				\\
$\gram$		& gram		& mass			&							\\
$\Hz$		& Hertz		& frequency		& $1/\second$						\\
$\Joule$	& Joule		& energy		& $1\units{N\,m}$					\\
$\Kelvin$	& Kelvin	& temperature		& $\sim\e{1.38}{-23}\:\Joule/\nat$			\\
$\kg$		& kilogram	& mass			& $1000\:\gram$						\\
$\meter$	& meter		& length		& 							\\
$\Newton$	& Newton	& force			& $1\units{kg\,m/\second^2}$				\\
$\nat$		& nat		& entropy		& $\kB$, $1\;\bit/\!\ln 2$				\\
$\Plen$		& Planck length & length		& $\sqrt{G\hbar/c^3}\approx\e{1.616}{-35}\:\meter$	\\
$\second$	& second	& time			&							\\
$\Volt$		& Volt		& electric potential	& $1\units{\Joule/\Coul}$				\\
\hline %----------------------------------------------------------------------------------------------------------
\end{tabular}
\caption[Fundamental units used in this document]
{Some basic units of measurement used in this document.}
\label{tab:units}
\end{center}\hrule
\end{table}

Table~\ref{tab:constants} shows the fundamental physical constants we use.
The Planck length $\Plen$ is listed in the units table
(table~\ref{tab:units}), but it may also be considered to be a fundamental
physical constant.  It is sometimes hypothesized to be some sort of minimum
length scale for physics.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|l|c|l|}
\hline %------------------------------------------------------------------------------------------------------------------------------------------
Symbol 		& Meaning			& Some equivalences		& Approximate value						\\
\hline %------------------------------------------------------------------------------------------------------------------------------------------
$c$		& Speed of light		& 				& $299792458\:\meter/\second$					\\
$\epsilon_0$	& Permittivity of		& $1/\mu_0c^2$			& $\e{8.85418782}{-12}\units{F/m}$				\\
		& free space			&				&								\\
$h$		& Planck's constant		& 				& $\e{6.6260755}{-34}\:\Joule\cdot\second$			\\
$\hb$		& Reduced Planck's 		& $h/2\pi$ 			& $\e{1.05457267}{-34}\:\Joule\cdot\second$			\\
		& constant			&				&								\\
$G$		& Gravitational			& 				& $\e{6.67259}{-11}\units{N\,m^2/kg}$				\\
		& constant			&				&								\\
$\kB$		& Boltzmann's constant		& $1\:\nat$			& $\e{1.3806513}{-23}\:\Joule/\Kelvin$				\\
$\oSB$		& Stefan-Boltzmann		& $\pi^2\kB^4/60c^2\hbar^3$ 	& $\e{5.6704}{-8}\:\Joule/\second\rmh\Kelvin^4\rmh\meter^2$	\\
		& constant			&				&								\\
\ech		& Electron charge	 	&				& $\e{1.60217733}{-19}\units{C}$				\\
		& magnitude			&				&								\\
\hline %------------------------------------------------------------------------------------------------------------------------------------------
\end{tabular}
\caption[Fundamental physical constants used in this document]
{Fundamental physical constants used in this document.}
\label{tab:constants}
\end{center}\hrule
\end{table}

Table~\ref{tab:oog} gives our preferred, more mnemonic notation for
comparing the asymptotic order of growth of functions.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|l|}
\hline
Cryptic  & Our own    &             \\
standard & mnemonic   & Mathematical definition; \\
notation & notation   & English explanation \\
\hline
$f=\Exactly(g)$   & $f\exactly g$   & \begin{minitab}
	$\exists c_1,c_2,n_0>0\colon\;\forall n\geq n_0\colon\; 0\leq c_1g(n)\leq f(n)\leq c_2g(n)$; \\
	$f$ is asymptotically proportional to $g$ \end{minitab}\\
\hline
$f=\Atmost(g)$    & $f\atmost g$    & \begin{minitab}
	$\exists c,n_0>0\colon\;\forall n\geq n_0\colon\; 0\leq f(n)\leq cg(n)$; \\
        $f$ is asymptotically no more than proportional to $g$ \end{minitab}\\
\hline
$f=\Atleast(g)$   & $f\atleast g$   & \begin{minitab}
	$\exists c,n_0>0\colon\;\forall n\geq n_0\colon\; 0\leq cg(n)\leq f(n)$; \\
        $f$ is asymptotically no less than proportional to $g$ \end{minitab}\\
\hline
$f=\Lessthan(g)$  & $f\lessthan g$  & \begin{minitab}
	$\forall c>0\colon\;\exists n_0>0\colon\;\forall n\geq n_0\colon\; 0\leq f(n)<cg(n)$; \\
        $f$ is asymptotically strictly less than proportional to $g$ \end{minitab}\\
\hline
$f=\Morethan(g)$  & $f\morethan g$  &  \begin{minitab}
	$\forall c>0\colon\;\exists n_0>0\colon\;\forall n\geq n_0\colon\; 0\leq cg(n)<f(n)$; \\
        $f$ is asymptotically strictly more than proportional to $g$ \end{minitab}\\
\hline
\end{tabular}
\caption[Asymptotic order-of-growth notation]
{Asymptotic order-of-growth notation.  In addition to reviewing the
standard notation, we introduce a simplified notation that will be useful
in some contexts.}
\label{tab:oog}
\end{center}
\hrule
\end{table}

% End chapter.
\cleardoublepage
