\documentclass[12pt]{report} %%\pagestyle{empty} -- you are no longer necessary -- jac Mon Jan 5 00:10:30 2004 \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in} \setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in} \def\thebibliography#1{\subsection*{REFERENCES}\list {\arabic{enumi}.} {\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\newblock{\hskip .11em plus .33em minus .07em} %% reduce vspace between items \parskip -0.7ex plus 0.5ex minus 0ex \if0\@ptsize\else\small\fi %% smaller fonts \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist % Lines matching "\\\\usepackage" in buffer obb.tex. (sorted and edited -- jac Sun Jan 4 17:23:06 2004) %\usepackage[T1]{fontenc} %\usepackage[all]{xy} \usepackage[all,poly,knot,dvips]{xy} \usepackage[all]{xypic} %\usepackage{xypic} %\usepackage[matrix,arrow]{xypic} %\usepackage[arrow,curve,poly,arc,2cell,frame,web]{xypic} \usepackage[dvips]{epsfig,graphics} \usepackage[dvips]{graphicx} % \usepackage{alltt} \usepackage{stmaryrd} \usepackage{amscd} \usepackage{amsfonts} % \usepackage{amsrefs} % \usepackage[alphabetic,sorted]{amsrefs} % \usepackage{amsmath} \usepackage[centertags]{amsmath} \usepackage{amsopn} \usepackage{amssymb} \usepackage{amstext} \usepackage{amsthm} % \usepackage{bbm} \usepackage{color} \usepackage{epsfig} \usepackage{eucal} \usepackage{eufrak} % \usepackage{euscript} \usepackage[mathscr]{euscript} %%% This is the package that causes the ! LaTeX Error: Command \CMcal already defined. \usepackage{graphpap} \usepackage{html} \usepackage{hyperref} \usepackage{latexsym} % \usepackage{lbh-pseudocode} %%% need to download this % \usepackage{listings} %%% ditto % \usepackage{mathrsfs} \usepackage{psfrag} \usepackage{pstricks} \usepackage{pst-node} % \usepackage{pst-poly} % \usepackage{pstcol} % \usepackage{setspace} \usepackage{url} % Lines matching "renewcommand" in buffer obb.tex. %\@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} \renewcommand*{\d}{\mathrm{d}} \renewcommand{\>}{\rangle} \renewcommand{\And}{\wedge} %\renewcommand{\H}{\category{H}} \renewcommand{\H}{\mathbb{H}} %\renewcommand{\H}{\mathcal{H}} %\renewcommand{\H}{{\mathbb{H}}} \renewcommand{\Im}{\operatorname{Im}} %\renewcommand{\L}[1]{\mathfrak{L}(#1)} \renewcommand{\L}{\mathfrak{L}} \renewcommand{\O}{\mathcal{O}} %\renewcommand{\P}{\mathbb{P}} %\renewcommand{\P}{\mathfrak{P}} \renewcommand{\P}{{\mathfrak{P}}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\S}{\mathcal{S}} % \renewcommand{\V}[1]{\mathbf{#1}} %\renewcommand{\a}{\bf{a}} \renewcommand{\a}{{\mathfrak{a}}} % \renewcommand{\bibname}{References} %\renewcommand{\b}{\bf{b}} \renewcommand{\b}{{\mathfrak{b}}} \renewcommand{\cap}{\bigcap} \renewcommand{\cup}{\bigcup} %\renewcommand{\c}{\bf{c}} %\renewcommand{\c}{\mathcal{C}} %\renewcommand{\c}{{\bf{c}}} \renewcommand{\c}{{\mathfrak{c}}} \renewcommand{\div}{\mid} %\renewcommand{\d}{\bf{d}} \renewcommand{\d}{{\bf{d}}} %\renewcommand{\emph}[1]{\textbf{#1}} \renewcommand{\emph}{\textbf} % this was a nice move -- jac Thu Jan 8 23:38:35 2004 \renewcommand{\geq}{\geqslant} \renewcommand{\hom}{\mathop{\mathrm{Hom}}} % Homomorphisms functor % \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \renewcommand{\inf}{\mathrm{inf}} \renewcommand{\int}{\mathrm{int}} \renewcommand{\ker}{\mathrm{ker}\,} \renewcommand{\leq}{\leqslant} %\renewcommand{\mathbb}[1]{\mathbbmss{#1}} \renewcommand{\mod}[1]{\;\rndbr{\mathrm{mod}\#1}} \renewcommand{\o}{\mathfrak{o}} % \renewcommand{\p}{\mathfrak{p}} % \renewcommand{\rai}[1]{\mathcal{O}_{#1}} % \renewcommand{\rb}{\mathrm{b}} \renewcommand{\r}{{r}} % while I think this is somewhat silly, I'm inclined to leave it and replace the normal \sl's for now. \renewcommand{\sl}[2]{\fr{sl}_{#1}#2} \renewcommand{\sp}[2]{\fr{sp}_{#1}#2} \renewcommand{\t}{{t}} % this is a bad idea, because \v is used to produce a check. Should use \vecv instead. %%\renewcommand{\v}{{{\bf v}}} % Lines matching "\\\\newcommand" in buffer obb.tex. (sorted and edited, very preliminary -- jac Sun Jan 4 17:46:23 2004) \newcommand{\znums}{\mathbb{Z}} \newcommand*{\Aut}{\mathop{\mathrm{Aut}}\nolimits} \newcommand*{\Bset}{\mathbb{B}} \newcommand*{\Coker}{\mathop{\mathrm{Coker}}\nolimits} \newcommand*{\Cset}{\mathbb{C}} \newcommand*{\Diff}{\mathop{\mathrm{Diff}}\nolimits} \newcommand*{\End}{\mathop{\mathrm{End}}\nolimits} \newcommand*{\GLgrp}{\mathrm{GL}} \newcommand*{\HC}{\mathit{HC}} \newcommand*{\HH}{\mathit{HH}} \newcommand*{\Hset}{\mathbb{H}} \newcommand*{\Img}{\mathop{\mathrm{Im}}\nolimits} \newcommand*{\Ind}{\mathop{\mathrm{Ind}}\nolimits} \newcommand*{\Inn}{\mathop{\mathrm{Inn}}\nolimits} \newcommand*{\KK}{\mathit{KK}} \newcommand*{\Ker}{\mathop{\mathrm{Ker}}\nolimits} \newcommand*{\Kset}{\mathbb{K}} \newcommand*{\Matrix}[2]{\mathord{\mathrm{M}_{#1}(#2)}} \newcommand*{\Nset}{\mathbb{N}} \newcommand*{\Ogrp}{\mathrm{O}} \newcommand*{\Oset}{\mathbb{O}} \newcommand*{\Out}{\mathop{\mathrm{Out}}\nolimits} \newcommand*{\Qset}{\mathbb{Q}} \newcommand*{\Res}{\mathop{\mathrm{Res}}} \newcommand*{\Rset}{\mathbb{R}} \newcommand*{\SLgrp}{\mathrm{SL}} \newcommand*{\SOgrp}{\mathrm{SO}} \newcommand*{\SUgrp}{\mathrm{SU}} \newcommand*{\Sset}{\mathbb{S}} \newcommand*{\Tor}{\mathop{\mathrm{Tor}}\nolimits} \newcommand*{\Tr}{\mathop{\mathrm{Tr}}} \newcommand*{\Tset}{\mathbb{T}} \newcommand*{\Ugrp}{\mathrm{U}} \newcommand*{\Wres}{\mathop{\mathrm{Wres}}} \newcommand*{\Zset}{\mathbb{Z}} %\newcommand*{\abs}[1]{\left\lVert #1\right\rVert} % \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} % \newcommand*{\abs}[1]{\left\lvert#1\right\rvert} % \newcommand*{\abs}[1]{| #1 |} \newcommand*{\algebra}[1][A]{\mathord{\mathcal{#1}}} \newcommand*{\boxprod}{\mathbin{\square}} \newcommand*{\ch}{\mathop{\mathrm{ch}}\nolimits} \newcommand*{\coker}{\mathop{\mathrm{coker}}\nolimits} %\newcommand*{\defn}[1]{\textbf{#1}} \newcommand*{\deriv}[2]{\frac{\d#1}{\d#2}} \newcommand*{\dixmier}{\mathop{\mathrm{Tr}_{\omega}}} \newcommand*{\e}{\mathop{\mathrm{e}}\nolimits} \newcommand*{\fderiv}[2]{\frac{\delta#1}{\delta#2}} \newcommand*{\floor}[1]{\left\lfloor #1\right\rfloor} \newcommand*{\hilbert}[1][H]{\mathord{\mathcal{#1}}} \newcommand*{\hilbmod}[1][E]{\mathord{\mathcal{#1}}} \newcommand*{\identity}{\mathord{\mathrm{1\!\!\!\:I}}} \newcommand*{\id}{\mathrm{id}} \newcommand*{\im}{\mathord{\mathrm{i}}} \newcommand*{\ind}{\mathop{\mathrm{index}}\nolimits} % \newcommand*{\integers}{\ensuremath{{\mathbb{Z}}}} \newcommand*{\integers}{\mathbb{Z}} \newcommand*{\mathcat}[1]{\mathord{\mathbf{#1}}} \newcommand*{\naturals}{\mathbb{N}} % \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\norm}[1]{\left\lVert #1\right\rVert} \newcommand*{\op}{\mathrm{op}} \newcommand*{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand*{\spin}{\mathop{\mathrm{spin}}\nolimits} \newcommand*{\tr}{\mathop{\mathrm{tr}}} % \newcommand{\0}{\bf{0}} \newcommand{\0}{{{\bf 0}}} % \newcommand{\1}{{{\bf 1}}} \newcommand{\<}{\langle} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\Alt}{\operatorname{Alt}^2} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\Aset}{\mathbb{A}} \newcommand{\Ass}{\operatorname{Ass}} % \newcommand{\Aut}{\mathrm{Aut}\,} % \newcommand{\Aut}{\mathrm{Aut}} % \newcommand{\Aut}{\operatorname{Aut}} %%% already defined somewhere??? -- jac Sun Jan 4 17:58:11 2004 % \newcommand{\Aut}{\textrm{Aut}} % \newcommand{\Aut}{\text{Aut}} \newcommand{\Au}{\operatorname{Aut}} % \newcommand{\Au}{\text{Aut}} % \newcommand{\A}{\category{A}} \newcommand{\A}{\mathbb{A}} % \newcommand{\A}{\mathb{A}} % \newcommand{\A}{\mathcal{A}} % \newcommand{\A}{\mathfrak{A}} \newcommand{\BC}{\mathbb{C}} \newcommand{\BN}{\mathbb{N}} \newcommand{\BQ}{\mathbb{Q}} \newcommand{\BR}{\mathbb{R}} \newcommand{\BZ}{\mathbb{Z}} \newcommand{\Bicyc}[2]{\mathcal{C}({#1},{#2})} \newcommand{\Bstar}{\mathcal{B}^\star} % \newcommand{\B}{\category{B}} % \newcommand{\B}{\ensuremath\mathbf{B}} \newcommand{\B}{\mathcal{B}} % \newcommand{\B}{\mathfrak{B}} \newcommand{\CA}{\mathcal{A}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CP}{\mathbf{CP}} % \newcommand{\CP}{\mathcal{P}} % \newcommand{\CS}{\EuScript{CS}} \newcommand{\CS}{\mathcal{S}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} \newcommand{\Cdiff}{\mathcal{C}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\Cliff}{\mathrm{Cliff}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Co}{\operatorname{Co}} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} % \newcommand{\C}{\mathbbmss{C}} \newcommand{\C}{\mathbb{C}} % \newcommand{\C}{\mathbf{C}} % \newcommand{\C}{\mb{C}} \newcommand{\DDX}{\ensuremath{\frac{\D{}}{\D{x}}}} \newcommand{\Def}{\overset{\operatorname{def}}{:=}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\Df}{{\mathbf{D}\!f}} % \newcommand{\Diff}{\operatorname{Diff}} %%%%%%%%%%%%%%% already defined somewhere -- jac Sun Jan 4 17:58:42 2004 \newcommand{\Div}{\operatorname{div}} % \newcommand{\D}[1]{\ensuremath{\mathrm{d}#1}} \newcommand{\D}[1]{\mathrm{D}_{#1}} \newcommand{\boldD}{\mathbf{D}} %%%%%%%%%%%%%%%%%%%%%%%%%% I have to change this one globally... % \newcommand{\D}{\mathcal{D}} \newcommand{\EL}{\mathcal{EL}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Eg}{\emph{E.g.},} \newcommand{\Endo}{\text{{\bf End}}} % \newcommand{\End}{\mathop{\mathrm{End}}\nolimits} % \newcommand{\End}{\mathrm{End}} % \newcommand{\End}{\operatorname{End}} % already defined someplace % \newcommand{\End}{\text{End}} % \newcommand{\Expect}{\mathbb{E}} % \newcommand{\Expect}{\operatorname{\mathbbmss{E}}} \newcommand{\Expect}{\operatorname{\mathbb{E}}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\FF}{\mathfrak{F}} \newcommand{\FIXCITE}{{\textbf [CITE]}} \newcommand{\FIXME}[1]{\textsl{[#1\/]}\typeout{[FIXME remaining]}} \newcommand{\FIXNOTE}[1]{\footnote{\FIXME{#1}}} \newcommand{\FL}{\mathfrak{L}} \newcommand{\FM}{\mathfrak{M}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\Fpstar}{\mathbb{F}_p^*} \newcommand{\Fp}{\mathbb{F}_p} \newcommand{\Fqstar}{\mathbb{F}_q^*} \newcommand{\Fq}{\mathbb{F}_q} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Frob}{\operatorname{Frob}} % \newcommand{\F}{F} % \newcommand{\F}{\mathbbmss{F}} \newcommand{\F}{\mathbb{F}} % \newcommand{\F}{\mathcal{F}} \newcommand{\GD}{\Delta} \newcommand{\GF}{\Phi} \newcommand{\GG}{\Gamma} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} % \newcommand{\GL}{\Lamda} %%%%%%%%%%%%%%%% yikes! this greek stuff is going to be hard to deal with % \newcommand{\GL}{\mathrm{GL}} % \newcommand{\GL}{\operatorname{GL}} % \newcommand{\GL}{\text{GL}} % \newcommand{\GL}{{\operatorname{GL}}} \newcommand{\GP}{\Pi} \newcommand{\GQ}{\Theta} \newcommand{\GS}{\Sigma} \newcommand{\GU}{\Upsilon} \newcommand{\GW}{\Omega} \newcommand{\GX}{\Xi} \newcommand{\GY}{\Psi} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} % \newcommand{\Gal}{\mathrm{Gal}} % \newcommand{\Gal}{\operatorname{Gal}} % yikes % \newcommand{\Gal}{\text{Gal}} \newcommand{\Ga}{\alpha} \newcommand{\Gb}{\beta} \newcommand{\Gc}{\chi} \newcommand{\Gd}{\delta} \newcommand{\Gee}{\epsilon} \newcommand{\Ge}{\varepsilon} \newcommand{\Gff}{\phi} \newcommand{\Gf}{\varphi} \newcommand{\Gg}{\gamma} \newcommand{\Gh}{\eta} \newcommand{\Gi}{\iota} \newcommand{\Gk}{\kappa} \newcommand{\Gl}{\lambda} \newcommand{\Gm}{\mu} \newcommand{\Gn}{\nu} \newcommand{\Gpp}{\varpi} \newcommand{\Gp}{\pi} \newcommand{\Gq}{\theta} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Grr}{\varrho} % \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\Gr}{\rho} \newcommand{\Gss}{\varsigma} \newcommand{\Gs}{\sigma} \newcommand{\Gt}{\tau} \newcommand{\Gu}{\upsilon} \newcommand{\Gw}{\omega} \newcommand{\Gx}{\xi} \newcommand{\Gy}{\psi} \newcommand{\Gz}{\zeta} \newcommand{\G}{\mathbb{G}} % \newcommand{\G}{\mathbb{Z}[i]} \newcommand{\Hilb}{\mathcal{H}} \newcommand{\Hnmon}{\mathbf{H}^{n-1}} \newcommand{\Hn}{\mathbf{H}^n} \newcommand{\Homeo}{\operatorname{Homeo}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} % \newcommand{\Hom}[3]{\mathrm{Hom}_{#3}(#1,#2)} % \newcommand{\Hom}{\mathop{\mathrm{Hom}}\nolimits} % \newcommand{\Hom}{\mathrm{Hom}} % \newcommand{\Hom}{\operatorname{Hom}} % \newcommand{\Hom}{\text{Hom}} \newcommand{\Hon}{\mathbf{H}^1} \newcommand{\Hth}{\mathbf{H}^3} \newcommand{\Htw}{\mathbf{H}^2} \newcommand{\Ie}{\emph{I.e.},} \newcommand{\Iff}{\Leftrightarrow} % \newcommand{\Img}{\mathop{\mathrm{Img}}} % already defined someplace \newcommand{\Implies}{\Rightarrow} % \newcommand{\Ind}{\operatorname{Ind}} % \newcommand{\Inf}{\bigvee} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Int}{\text{{\bf int}}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\I}{\mathbb{I}} % \newcommand{\I}{\mathcal{I}} \newcommand{\J}{\mathcal{J}} \newcommand{\KP}{{\mathbb{K}\mathrm{P}}} % newcommand{\Ker}{\mathop{\mathrm{Ker}}} \newcommand{\K}{\mathcal{K}} % \newcommand{\K}{\mathfrak{K}} \newcommand{\LG}{L_{4}} \newcommand{\La}{\Leftarrow} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % These are some nice commands for formatting code/pseudocode. % (Using `\newline' instead of `\\', I avoid a compiler complaint -- jac Tue Jan 6 12:03:48 2004 \newcommand{\Lelseif}[2]{\textbf{else if} #1 \textbf{then}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lelse}[1]{\textbf{else}\newline \hspace*{\Lindent}\parbox{\textwidth}{#1}} \newcommand{\Lfor}[2]{\textbf{for} #1 \textbf{do}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lgets}{\ensuremath{\gets}} \newcommand{\Lgroup}[1]{\textbf{begin}\newline \hspace*{\Lindent} \parbox{\textwidth}{#1}\newline \textbf{end}} \newcommand{\Lif}[2]{\textbf{if} #1 \textbf{then}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lwhile}[2]{\textbf{while}#1 \textbf{do}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lindent}{0.4in} \newcommand{\Lfunc}[2]{\textbf{#1}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lrepeat}[2]{\textbf{#1}\newline \hspace*{\Lindent}\parbox{\textwidth}{#2}} \newcommand{\Lbold}[1]{\textbf{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\Linf}{\Lspace{\infty}} \newcommand{\Lone}{\Lspace{1}} \newcommand{\Lpspace}{\Lspace{p}} \newcommand{\Lq}{\Lspace{q}} \newcommand{\Lspace}[1]{L^{#1}} \newcommand{\Ltwo}{\Lspace{2}} % \newcommand{\L}{\textbf{L}} % \newcommand{\Matrix}[4]{\left(\begin{array}{cc} #1 & #2 \\ #3 & #4 \newcommand{\Mat}{\mathop{\mathrm{Mat}}\nolimits} \newcommand{\Mor}{\text{Mor}} % \newcommand{\M}{\mathcal{M}} \newcommand{\M}{\mathfrak{M}} \newcommand{\NN}{\mathbb{N}} % \newcommand{\NN}{{\mathbf N}} \newcommand{\Nats}{\mathbb{N}} \newcommand{\Nat}{\mathbb{N}} % \newcommand{\Nil}{\mathop{\mathrm{Nil}}\nolimits} \newcommand{\Nil}{\operatorname{Nil}} \newcommand{\Nplus}{\mathbb{N}^+} \newcommand{\Nstar}{\mathbb{N}^{*}} \newcommand{\N}{\mathbb{N}} % \newcommand{\N}{\mathbf{N}} % \newcommand{\N}{\mathfrak{N}} % \newcommand{\N}{{\mathbb N}} \newcommand{\Obj}{\text{Obj}} \newcommand{\Om}{\Omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\Or}{\vee} % \newcommand{\O}{\mathcal{O}} \newcommand{\PR}{^{\prime}} \newcommand{\PSL}{{\mathrm{PSL}}} \newcommand{\PartialSl}[2]{\partial#1/\partial#2} \newcommand{\Partial}[2]{\frac{\partial#1}{\partial#2}} \newcommand{\Perm}{\operatorname{Perm}} \newcommand{\Per}{\operatorname{Per}} \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\Prov}[2]{\mathbf{Prov}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\Q}{\mathbb Q} % \newcommand{\Q}{\mathbbmss{Q}} % \newcommand{\Q}{\mathbb{H}} % \newcommand{\Q}{\mathbb{Q}} % \newcommand{\Q}{\mathbf{Q}} % \newcommand{\Q}{\mb{Q}} % \newcommand{\Q}{{\mathbb Q}} % \newcommand{\Q}{{\mathbb{Q}}} \newcommand{\RA}{\rightarrow} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\RNpon}{\mathbf{R}^{N+1}} \newcommand{\RN}{\mathbf{R}^N} \newcommand{\RP}{\mathbf{RP}} % \newcommand{\RP}{{\reals\mathrm{P}}} \newcommand{\RR}{\mathbb{R}} % \newcommand{\RR}{\mathbf{R}} \newcommand{\RS}{\mathbf{R}^{\star}} \newcommand{\Rats}{\mathbb{Q}} \newcommand{\Ra}{\Rightarrow} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Rel}{\mathbf{R}} % \newcommand{\Res}{\mathrm{Res}} % \newcommand{\Res}{\operatorname{Res}} \newcommand{\Rfi}{\mathbf{R}^5} \newcommand{\Rfo}{\mathbf{R}^4} \newcommand{\Rmpon}{\mathbf{R}^{m+1}} \newcommand{\Rm}{\mathbf{R}^m} \newcommand{\Rnmon}{\mathbf{R}^{n-1}} \newcommand{\Rnpon}{\mathbf{R}^{n+1}} \newcommand{\Rn}{\mathbf{R}^n} % \newcommand{\Rp}{\mathbf{R}^p} \newcommand{\Rth}{\mathbf{R}^3} \newcommand{\Rtwn}{\mathbf{R}^{2n}} \newcommand{\Rtw}{\mathbf{R}^2} % \newcommand{\R}[0]{\mathbb{R}} % \newcommand{\R}{\ensuremath{\mathbb{R}}} % \newcommand{\R}{\mathbb R^+} \newcommand{\R}{\mathbb R} % \newcommand{\R}{\mathbbmss{R}} % \newcommand{\R}{\mathbb{R}} % \newcommand{\R}{\mathbf{R}} % \newcommand{\R}{\mb{R}} \newcommand{\SL}[2]{\mathrm{SL}_{#1}#2} % \newcommand{\SL}{\operatorname{SL}} % \newcommand{\SL}{{\mathrm{SL}}} % \newcommand{\SL}{{\operatorname{SL}}} \newcommand{\SN}{\mathbf{S}^N} \newcommand{\SO}[2]{\mathrm{SO}_{#1}#2} % \newcommand{\SO}{\operatorname{SO}} \newcommand{\SU}[1]{\mathrm{SU}(#1)} \newcommand{\Sfi}{\mathbf{S}^5} \newcommand{\Sfo}{\mathbf{S}^4} \newcommand{\Smmon}{\mathbf{S}^{m-1}} \newcommand{\Sm}{\mathbf{S}^m} \newcommand{\Snmon}{\mathbf{S}^{n-1}} \newcommand{\Snmtw}{\mathbf{S}^{n-2}} \newcommand{\Sn}{\mathbf{S}^n} \newcommand{\Son}{\mathbf{S}^1} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1}#2} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\Sth}{\mathbf{S}^3} \newcommand{\Stw}{\mathbf{S}^2} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\Sup}{\bigwedge} \newcommand{\Sym}{\mathrm{Sym}} % \newcommand{\Sym}{\operatorname{Sym}^2} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\Tn}{\mathbf{T}^n} \newcommand{\Ton}{\mathbf{T}^1} \newcommand{\Torus}{\mathbf{T}} % \newcommand{\Tr}{\mathrm{Tr}} % \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Tth}{\mathbf{T}^3} \newcommand{\Ttw}{\mathbf{T}^2} \newcommand{\T}{\mathbf{T}} % \newcommand{\T}{\textbf{T}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \newcommand{\Vstarstar}{V^{\star\star}} \newcommand{\Vstar}{V^{\star}} \newcommand{\V}[1]{\mathbf{#1}} % \newcommand{\V}{\mathbb{V}} \newcommand{\WLoG}{\textrm{ \scriptsize W.L.O.G. }} \newcommand{\X}{\mathcal{X}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\Zn}[1]{\mathbb{Z}_{#1}} \newcommand{\Zpstar}{\mathbb{Z}_p^*} \newcommand{\Zp}{\Z/p} \newcommand{\Zt}{\Z_2} % \newcommand{\Z}{\ensuremath\mathbb{Z}} % \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\Z}{\mathbb{Z}} % \newcommand{\Z}{\mathbf{Z}} % \newcommand{\Z}{\mb{Z}} % \newcommand{\Z}{{\mathbb Z}} % \newcommand{\Z}{{\mathbb{Z}}} \newcommand{\abs}[1]{\left| #1 \right|} % \newcommand{\abs}[1]{\left|#1\right|} % \newcommand{\abs}[1]{\lvert #1 \rvert} % \newcommand{\abs}[1]{{\left| #1 \right|}} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\abplain}{\mathrm{ab}} \newcommand{\acf}{ACF_{val}} \newcommand{\adj}{^{\displaystyle \star}} % \newcommand{\ad}{\mathop{\mathrm{ad}}\nolimits} % \newcommand{\ad}{\mathrm{ad}} \newcommand{\ad}{\operatorname{ad}} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\ala}{\emph{\'a la}} % \newcommand{\algc}[1]{% %%% this is was causing some trouble! \newcommand{\alpr}{{\alpha^\prime}} \newcommand{\angbr}[1]{\left< #1 \right>} \newcommand{\ann}{\text{{\bf Ann}}} \newcommand{\ao}{\mathbf{a_0}} \newcommand{\aut}{\mathbf{Aut}} \newcommand{\axiom}[1]{\htmladdnormallink{axiom #1}{http://planetmath.org/encyclopedia/VectorSpace.html}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bB}{\mathbf{B}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bN}{\mathbf{N}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bU}{\mathbf{U}} \newcommand{\bV}{\mathbf{V}} \newcommand{\bal}{\boldsymbol{\alpha}} \newcommand{\barQ}{\overline{\Q}} \newcommand{\barbr}[1]{\left| #1 \right|} \newcommand{\ba}{\mathbf{a}} \newcommand{\bbC}{\mathbb{C}} \newcommand{\bbF}{\mathbb{F}} \newcommand{\bbN}{\mathbb{N}} \newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} \newcommand{\bbZ}{\mathbb{Z}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bdry}{\partial} \newcommand{\bd}{\partial} \newcommand{\bepr}{\beta^\prime} \newcommand{\be}{\mathbf{e}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} % \newcommand{\bg}{\boldsymbol{\gamma}} \newcommand{\bg}{\mathbf{g}} \newcommand{\bigdsum}{\bigoplus} \newcommand{\bigintersection}{\bigcap} \newcommand{\bigintersect}{\bigcap} \newcommand{\bigo}{\mathcal{O}} \newcommand{\bigtensor}{\bigotimes} \newcommand{\bigunion}{\bigcup} \newcommand{\bkh}{\backslash} \newcommand{\bmaps}[2]{\mathop{\mathrm{Maps}_*}\left(#1,#2\right)} \newcommand{\bm}{\begin{displaymath}} \newcommand{\borel}{\mathfrak{B}} \newcommand{\braket}[2]{\langle #1 \ket{#2}} \newcommand{\bra}[1]{\langle#1 \vert} \newcommand{\br}{[\![} \newcommand{\bt}{\begin{thm}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bvec}{\mathbf{\overrightarrow{b}}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bzero}{\mathbf{0}} \newcommand{\cA}{\mathcal{A}} %\newcommand{\cA}{{\mathcal{A}}} \newcommand{\cB}[0]{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} % \newcommand{\cD}[0]{\mathcal{D}} \newcommand{\calD}{\mathcal{D}} \newcommand{\cF}{{\mathcal{F}}} % \newcommand{\cL}{\mathcal{L}} \newcommand{\calL}{\mathcal{L}} \newcommand{\cP}[1]{\mathcal{P}_{#1}} % \newcommand{\cP}{\mathcal{P}} \newcommand{\calP}{\mathcal{P}} \newcommand{\cS}[0]{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} % \newcommand{\cV}{{\mathcal{V}}} \newcommand{\calO}{\mathcal{O}} \newcommand{\category}[1]{\mbox{\boldmath $\mathsf{{#1}}$}} \newcommand{\ca}{\varepsilon} \newcommand{\ccj}[1]{\overline{#1}} \newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil} % \newcommand{\ceil}[1]{\lceil{#1}\rceil} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\cfty}{\mathcal{C}^\infty} \newcommand{\cf}{\emph{cf.}} \newcommand{\cidl}[1]{\mathfrak{{#1}}} \newcommand{\cla}[1]{\lceil #1 \rceil} \newcommand{\closure}{\mathrm{closure}} % \newcommand{\cl}{\bf{d}} \newcommand{\cl}{\operatorname{cl}} % \newcommand{\cl}{\text{{\bf acl}}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cnums}{\mathbb{C}} \newcommand{\cn}{\colon} \newcommand{\codim}{\operatorname{codim}} \newcommand{\cof}{\operatorname{cof}} \newcommand{\coheight}{\text{\bf Co-height}} % \newcommand{\coker}{\mathrm{coker}\,} % \newcommand{\coker}{\operatorname{coker}} % \newcommand{\coker}{\operatorname{im}} \newcommand{\cok}{\operatorname{cok}} \newcommand{\complexes}{\mathbb{C}} % \newcommand{\comp}[0]{\complement} \newcommand{\comp}{\circ} % \newcommand{\comp}{\circ} % Function composition % \newcommand{\comp}{\circ} % Function composition \newcommand{\concat}{\ensuremath{+\hspace{-1ex}+}} %% problem? %\newcommand{\conj}[1]{{\overline{{#1}}}} \newcommand{\conv}{\circ} \newcommand{\corr}{\mathrm{corr}} \newcommand{\covA}{\mathcal{A}} % Open cover A \newcommand{\covB}{\mathcal{B}} % Open cover B \newcommand{\cov}{\mathrm{cov}} \newcommand{\cpv}[0]{\operatorname{p.\!v.}(\frac{1}{x})} \newcommand{\cpxs}{{\mathbb C}} % The "complexes" :) % \newcommand{\cp}{\cpxs P} % complex projective space \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\cq}{\text{''}} \newcommand{\crit}{\operatorname{Crit}} \newcommand{\cross}{\times} \newcommand{\cull}[0]{\operatorname{co}} \newcommand{\curbr}[1]{\left\{ #1 \right\}} \newcommand{\curl}{\operatorname{curl}} \newcommand{\cvf}[1]{\partial_{x_{#1}}} \newcommand{\cycle}[1]{\left(#1\right)} \newcommand{\dall}[2]{d{#1}_1\wedge\cdots\wedge d{#1}_{#2}} % dx_1 ^ ... ^ dk_k \newcommand{\dbg}{\bg'} \newcommand{\dbydat}[2]{\left . \frac{d}{d #1} \right|_{#2}} \newcommand{\dbydk}[2]{ \frac{d^{#2}}{d #1^{#2}}} \newcommand{\dbyd}[1]{ \frac{d}{d #1}} \newcommand{\ddat}[3]{\left .\frac{d #1}{d #2}\right|_{#3}} \newcommand{\ddbg}{\bg''} \newcommand{\dddbg}{\bg'''} \newcommand{\ddkat}[4]{\left .\frac{d^{#3} #1}{d#2^{#3}}\right|_{#4}} \newcommand{\ddk}[3]{\frac{d^{#3} #1}{d#2^{#3}}} \newcommand{\dd}{\mathrm{d}} \newcommand{\deck}{\EuScript{D}} \newcommand{\deff}[1]{\textbf{Definition {#1}: }} \newcommand{\defined}{:=} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\degree}{\mathrm{degree}} \newcommand{\del}{\nabla} % \newcommand{\del}{\partial} \newcommand{\der}[1]{#1{}'} % \newcommand{\der}{\text{{\bf Der}}} \newcommand{\dfa}{[\mathbf{D}\fvec(\mathbf{a_0})]} \newcommand{\df}[1]{[\mathbf{D}\vec{f}(#1)]} \newcommand{\dgamma}{\dot{\gamma}} \newcommand{\diag}[0]{\operatorname{diag}} % \newcommand{\diag}{\mathop{\mathrm{diag}}} % \newcommand{\diam}{\mathrm{diameter}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\diff}{\operatorname{diff}} \newcommand{\disc}{\operatorname{disc}} \newcommand{\divides}{\mid} \newcommand{\dkd}[3]{\frac{d^{#3} #1}{d#2^{#3}}} \newcommand{\dlim}{\,\underset{U \ni p}{\underset{\longrightarrow}{\lim}}\,} \newcommand{\dn}{\delta^{(n)}} \newcommand{\domain}{\mathrm{domain}} \newcommand{\dom}{\mathbf{Dom}} \newcommand{\dotsint}{ \int \cdots} \newcommand{\dsum}{\oplus} \newcommand{\dual}{\vee} % \newcommand{\dual}{^*} \newcommand{\du}[2]{^{#2}_{\hphantom{#2}\!#1}} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\eb}{\mathbf{e}} % Standard basis %\newcommand{\eb}{\mathbf{e}} % Standard basis \newcommand{\eg}{\emph{e.g.},} \newcommand{\eqclass}[1]{[\![#1]\!]} %\newcommand{\eqref}[1]{\textrm{(\ref{#1})}} \newcommand{\er}{\thicksim} \newcommand{\esssup}{\mathrm{ess\ sup}} \newcommand{\et}{\end{thm}} \newcommand{\ev}[1]{\mathrm{Ev}^{(#1)}} % \newcommand{\e}{\epsilon} % \newcommand{\e}{e^{-\frac{1}{x^2}}} \newcommand{\fU}{{\mathcal U}} \newcommand{\fchar}{\mathrm{char}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\fkm}{\mathfrak{m}} \newcommand{\fkp}{\mathfrak{p}} % \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} % \newcommand{\floor}[1]{\lfloor{#1}\rfloor} \newcommand{\fp}{f^{\prime}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\fs}{\mathcal{F}} \newcommand{\funcdef}[3]{#1:\funcsig{#2}{#3}} \newcommand{\funcsig}[2]{#1\rightarrow #2} \newcommand{\fvec}{\mathbf{\overrightarrow{f}}} \newcommand{\gbar}{\overline{g}} \newcommand{\gda}{G_{D}^{A}} \newcommand{\gen}[1]{\langle\!\langle #1 \rangle\!\rangle} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\gind}{\downarrow^{g}} \newcommand{\girth}{\operatorname{girth}} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\grad}{\operatorname{grad}} \newcommand{\g}{\mathfrak{g}} \newcommand{\hI}{I'} \newcommand{\hJ}{J'} \newcommand{\hM}{\hat{M}} \newcommand{\hT}{\hat{T}} \newcommand{\half}{{\tiny \begin{array}{l}1 \\ \overline{2}\end{array}}} \newcommand{\halpha}{\hat{\alpha}} \newcommand{\ha}{A} \newcommand{\hbF}{\hat{\bF}} \newcommand{\hbe}{\hat{\be}} \newcommand{\hb}{B} \newcommand{\hc}{c'} \newcommand{\headstyle}{\bfseries} \newcommand{\height}{\text{\bf Height}} \newcommand{\hg}{\hat{g}} \newcommand{\hi}{{i'}} \newcommand{\hj}{{j'}} \newcommand{\homeo}{\cong} \newcommand{\htpyeq}{\backsimeq} \newcommand{\hva}{\mathbf{a}'} \newcommand{\hvb}{\mathbf{b}'} \newcommand{\hve}{\hat{\ve}} \newcommand{\hv}{\hat{v}} \newcommand{\h}{\mathfrak{h}} % \newcommand{\id}{\mathrm{id}} % \newcommand{\id}{\operatorname{id}} \newcommand{\ie}{\emph{i.e.},} \newcommand{\ilim}{\,\underset{\longleftarrow}{\lim}\,} % \newcommand{\image}{\mathop{\mathrm{img}}} % \newcommand{\image}{\mathrm{image}} \newcommand{\image}{\operatorname{im}} \newcommand{\img}{\mathop{\mathrm{img}}\nolimits} % \newcommand{\im}{\mathbf{Im}} % \newcommand{\im}{\mathit{i}} % \newcommand{\im}{\mathrm{im}} % \newcommand{\im}{\operatorname{im}} % \newcommand{\im}{\text{Im}} % \newcommand{\incl}{\mathrm{incl}} \newcommand{\incl}{\operatorname{incl}} \newcommand{\incoprod}{\amalg} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} % \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\inner}[2]{\langle #1|#2\rangle} \newcommand{\inn}[1]{\langle #1\rangle} \newcommand{\inprod}[2]{\left<#1,#2\right>} % Inner product \newcommand{\intcc}[2]{\left[#1,#2\right]} \newcommand{\intco}[2]{\left[#1,#2\right)} % \newcommand{\integers}{\mathbb{Z}} \newcommand{\integs}{{\mathbb Z}} % The integers \newcommand{\interior}{\mathrm{interior}} \newcommand{\intersection}{\cap} \newcommand{\intersect}{\cap} \newcommand{\intoc}[2]{\left(#1,#2\right]} \newcommand{\intoo}[2]{\left(#1,#2\right)} \newcommand{\inv}[1]{#1^{-1}} \newcommand{\ip}[2]{\langle #1 , #2 \rangle} \newcommand{\irr}{\text{{\bf Irr}}} \newcommand{\isom}{\cong} \newcommand{\iso}{\cong} \newcommand{\jacobi}[2]{{\left(\frac{#1}{#2}\right)}} % \newcommand{\ker}{\operatorname{ker}} \newcommand{\ket}[1]{\vert \hspace{1pt}#1\rangle} \newcommand{\kfield}{\mathbb{K}} \newcommand{\kf}{\mathbb{K}} \newcommand{\lact}{\lambda} \newcommand{\laf}{\mathfrak{f}} \newcommand{\lag}{\mathfrak{g}} \newcommand{\lah}{\mathfrak{h}} \newcommand{\lapM}{\Delta_M} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\la}{\leftarrow} \newcommand{\lb}{\left[} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\leftbb}{[ \! [} % \newcommand{\legsymp}[1]{\left(\frac{#1}{p}\right)} \newcommand{\legsymp}[1]{\left(\frac{#1}{p}\right)} \newcommand{\legsym}[2]{\left(\frac{#1}{#2}\right)} \newcommand{\leg}[1]{\left(\frac{#1}{p}\right)} % \newcommand{\lg}{\mathfrak{g}} \newcommand{\lh}{\mathfrak{h}} \newcommand{\liminj}{\mathop{\lim_{\longrightarrow}}} \newcommand{\liminv}{\mathop{\lim_{\longleftarrow}}} \newcommand{\limv}[2]{\lim\limits_{#1\rightarrow #2}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\lin}{\operatorname{Lin}} \newcommand{\lisom}{\buildrel{\hskip+0.04cm\sim}\over{\smashedleftarrow}} \newcommand{\lp}{\left(} \newcommand{\leftparen}{\left(} \newcommand{\lra}{\longrightarrow} \newcommand{\lsp}[0]{\mathop{\mathrm{span}}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mM}{\mathbf{M}} \newcommand{\maps}[2]{\mathop{\mathrm{Maps}}\left(#1,#2\right)} \newcommand{\map}{\operatorname{Map}} \newcommand{\matCm}[0]{M_-} \newcommand{\matCp}[0]{M_+} \newcommand{\matC}[0]{M} % \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\mat}{\text{{\bf Mat}}} \newcommand{\ma}{\mathfrak{A}} \newcommand{\mbf}[1]{\mathbf{#1}} % \newcommand{\mb}{\mathbb} \newcommand{\mb}{\mathfrak{B}} \newcommand{\mcB}{\mathcal{B}} \newcommand{\mcH}{\mathcal{H}} \newcommand{\mc}[1]{\mathcal{#1}} % \newcommand{\mc}{\mathcal} % \newcommand{\mc}{\mathfrak{C}} \newcommand{\mderiv}[1]{\frac{\md}{\md {#1}}} %d/dx \newcommand{\md}{\mathfrak{D}} % \newcommand{\md}{d} \newcommand{\medio}{\frac{1}{2}} \newcommand{\mf}{\mathfrak} \newcommand{\mgb}{\mathbf{M}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\mmin}{\operatorname{m}} \newcommand{\mnthderiv}[2]{\frac{\md^{#2}}{\md {#1}^{#2}}} %d^n/dx \newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} \newcommand{\mnthpderiv}[2]{\frac{\partial^{#2}}{\partial {#1}^{#2}}} %partial d^n/dx \newcommand{\mvar}{t} \newcommand{\mvi}[1]{\mv{#1}^{-1}} \newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}} \newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector \newcommand{\m}{\mathfrak{m}} \newcommand{\natnums}{\mathbb{N}} % \newcommand{\naturals}{\mathbb{N}} \newcommand{\ndiv}{\nmid} \newcommand{\nequiv}{\not\equiv} \newcommand{\nin}{\not\in} \newcommand{\nl}[1]{\underline{{#1}}} \newcommand{\normal}{\trianglelefteq} % \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\nor}{\vartriangleleft} \newcommand{\note}{\underline{\textcolor{red}{NOTE}}:} \newcommand{\notnormal}{\ntrianglelefteq} \newcommand{\nzkn}{\kf^{n+1}\backslash \{0\}} \newcommand{\n}{\mathfrak{n}} % \newcommand{\n}{\newline} \newcommand{\ocD}{\smash{\overset{\circ}D}} \newcommand{\oce}{\smash{\overset{\circ}e}} \newcommand{\ol}{\overline} \newcommand{\om}{\omega} \newcommand{\on}{{\overline{n}}} %%%%% \newcommand{\openset}{\mathrel{{\mathchoice{\rlap{$\subset$}{\;\circ}}% %% causes error... of course % \newcommand{\op}{\bf{o}} \newcommand{\oq}{\text{``}} \newcommand{\ord}[1]{\Theta\rndbr{ #1 }} % \newcommand{\ord}{\mathop{\mathrm{ord}}\nolimits} \newcommand{\ordop}{\mathop{\mathrm{ord}}\nolimits} \newcommand{\orig}{\mathbf{0}} % Vector origin \newcommand{\ovec}{\mathbf{\overrightarrow{0}}} \newcommand{\ov}[1]{\overline{#1}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pair}[2]{\left\langle#1,#2\right\rangle} \newcommand{\pbypk}[2]{ \frac{\partial^{#2}}{\partial #1^{#2}}} \newcommand{\pbyp}[1]{ \frac{\partial}{\partial #1}} \newcommand{\pdat}[3]{\left . \frac{\partial #1}{\partial #2}\right|_{#3}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\perm}[1]{\pi_{#1}} %\newcommand{\pfac}[1]{\left(#1!\right)_p} \newcommand{\pfac}[1]{\left(#1\underline{!}\right)_p} \newcommand{\pkd}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} \newcommand{\pln}[2]{\PMlinkname{#1}{#2}} %\newcommand{\pln}[2]{\PMlinkname{{#1}}{#2}} %\newcommand{\pln}[2]{{\PMlinkname{#1}{#2}}} %\newcommand{\pln}[2]{{\PMlinkname{{#1}}{#2}}} %% awful lot of synonyms for this command... -- jac Thu Jan 8 12:55:14 2004 \newcommand{\PMtextescapelink}[1]{\underline{#1}} \newcommand{\PMnolink}[1]{\underline{#1}} \newcommand{\PMescapetext}[1]{\underline{#1}} \newcommand{\PMescapelinktext}[1]{\underline{#1}} \newcommand{\PMlinkescape}[1]{\underline{#1}} \newcommand{\PMescapelinktex}[1]{\underline{#1}} \newcommand{\PMlinkescapeword}[1]{\underline{#1}} \newcommand{\pmlinkescapetext}[1]{\underline{#1}} \newcommand{\PMtextignorelink}[1]{\underline{#1}} \newcommand{\PMnolinkphrase}[1]{\underline{#1}} \newcommand{\PMnolinktext}[1]{\underline{#1}} \newcommand{\pmescapetext}[1]{\underline{#1}} \newcommand{\pmlinkescapeword}[1]{\underline{#1}} \newcommand{\PMLinkEscape}[1]{\underline{#1}} \newcommand{\PMEscapeLink}[1]{\underline{#1}} \newcommand{\PMescapelink}[1]{\underline{#1}} \newcommand{\PMlinkescapetext}[1]{\underline{#1}} \newcommand{\pmlinkescapesequence}[1]{\underline{#1}} \newcommand{\PMlinkescapesequence}[1]{\underline{#1}} \newcommand{\pnorm}{\operatorname{p}} \newcommand{\powerset}[1]{\mathcal{P}(#1)} %\newcommand{\powerset}{\mathcal P} \newcommand{\powset}[1]{\mathcal{P}(#1)} \newcommand{\ppkat}[4]{\left .\frac{\partial^{#3} #1}{\partial #2^{#3}}\right|_{#4}} \newcommand{\ppk}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} \newcommand{\pp}[2]{\frac{\partial #1}{\partial#2}} \newcommand{\probpart}[1]{\noindent{{\bf #1}}} \newcommand{\prob}[1]{\Prob{}{#1}} \newcommand{\projp}{{\mathbb R}P} %\newcommand{\proof}{\paragraph{Proof:}} \newcommand{\proves}{\vdash} \newcommand{\pth}{{\mbox{$p^{\text{th}}$}}} \newcommand{\pt}{\mathbf} \newcommand{\p}{\mathfrak{p}} %\newcommand{\p}{{\mathfrak{p}}} \newcommand{\qbar}{\overline{\mb{Q}}} \newcommand{\qr}[2]{{\mbox{$\left(\frac{{#1}}{{#2}}\right)$}}} \newcommand{\qth}{{\mbox{$q^{\text{th}}$}}} \newcommand{\qt}{\text{{\bf Qt}}} \newcommand{\quaternions}{\mathbb{H}} \newcommand{\q}{{\mathfrak{q}}} \newcommand{\rC}{\mathrm{C}} \newcommand{\rD}{\mathrm{D}} \newcommand{\rI}{\mathrm{I}} \newcommand{\rJ}{\mathrm{J}} \newcommand{\rM}{\mathrm{M}} \newcommand{\rT}{\mathrm{T}} \newcommand{\rad}{\mathrm{rad}\,} %\newcommand{\rad}{\text{{\bf Rad}}} \newcommand{\rai}[1]{\mathcal{O}_{#1}} \newcommand{\rank}{\mathop{\mathrm{rank}}} \newcommand{\rationals}{\mathbb{Q}} \newcommand{\ra}{\rightarrow} \newcommand{\rb}{\right]} %\newcommand{\rb}{]\!]} \newcommand{\rcf}{RCVF_{G}} \newcommand{\reals}{\mathbb{R}} %\newcommand{\reals}{{\mathbb R}} % The reals %\newcommand{\reals}{{\mathbb R}} % The reals \newcommand{\real}{\textrm{real}} \newcommand{\remainder}{\: \% \:} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\restr}[2]{{#1}|_{#2}} %\newcommand{\restr}{\upharpoonright} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resplain}{\operatorname{res}} \newcommand{\rightbb}{] \! ]} \newcommand{\risom}{\buildrel{\hskip-0.04cm\sim}\over{\smashedrightarrow}} \newcommand{\rk}{\mathrm{rk}\,} %\newcommand{\rk}{{\bf Remark:}} \newcommand{\rndbr}[1]{\left( #1 \right)} \newcommand{\ro}{\mathbf{r.o.}} \newcommand{\rp}{\reals P} % real projective space %\newcommand{\rp}{\right)} \newcommand{\rightparen}{\right)} \newcommand{\rx}{\mathrm{x}} \newcommand{\sC}[0]{\mathbb{C}} %\newcommand{\sC}{\mathbbmss{C}} \newcommand{\sK}[0]{\mathbb{K}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sQ}{\mathbb{Q}} %\newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sR}{\mathbb{R}} %\newcommand{\sZ}[0]{\mathbb{Z}} \newcommand{\sZ}{\mathbbmss{Z}} \newcommand{\scalar}[2]{\left\langle#1,#2\right\rangle} \newcommand{\scomp}[0]{C^\infty_0} \newcommand{\semidirect}{\rtimes} %\newcommand{\semidirect}{\times} \newcommand{\sequence}[1]{\{#1\}} \newcommand{\sequ}{\left< x_{n}:x<\omega \right>} \newcommand{\seq}[1]{\left(#1\right)} \newcommand{\setc}[2]{\left\{#1:\: #2\right\}} \newcommand{\setof}[1]{\curbr{\ #1\ }} \newcommand{\set}[1]{\left\{#1\right\}} %\newcommand{\set}[1]{\{#1\}} %\newcommand{\set}[1]{{\left\{#1\right\}}} \newcommand{\sgn}{\mathop{\mathrm{sgn}}} % Sign function %\newcommand{\sgn}{\mathrm{sgn}} \newcommand{\signum}[0]{\mathop{\mathrm{sign}}} \newcommand{\sign}{\operatorname{sign}} \newcommand{\size}[1]{\left|#1\right|} %\newcommand{\size}[1]{| #1 |} \newcommand{\skel}[2]{{#1}^{(#2)}} \newcommand{\sk}{\mathrm{sk}} \newcommand{\smashedleftarrow}{\setbox0=\hbox{$\longleftarrow$}\ht0=1pt\box0} \newcommand{\smashedrightarrow}{\setbox0=\hbox{$\longrightarrow$}\ht0=1pt\box0} \newcommand{\snorm}[1]{\pnorm(#1)} \newcommand{\so}[2]{\fr{so}_{#1} #2} \newcommand{\spec}{\text{{\bf Spec}}} \newcommand{\sqle}{\sqsubseteq} \newcommand{\sqnorm}[1]{\left\|#1\right\|^2} \newcommand{\sqrbr}[1]{\left[ #1 \right]} \newcommand{\sq}{$\square$} \newcommand{\stab}{\text{{\bf Stab}}} %\newcommand{\st}{\mathcal{Qt}} %\newcommand{\st}{\mid} \newcommand{\st}{\textrm{such that}} \newcommand{\subp}[1]{_{(#1)}} \newcommand{\supp}[0]{\operatorname{supp}} %\newcommand{\supp}[1]{^{(#1)}} %\newcommand{\supp}{\mathop{\mathrm{Supp}}} % Support of a function %\newcommand{\supp}{\mathrm{supp}} %\newcommand{\supp}{\operatorname{supp}} \newcommand{\supth}{^{\text{th}}} %\newcommand{\supt}{^t} \newcommand{\supt}{^{\scriptscriptstyle\mathrm{T}}} \newcommand{\susp}{\Sigma} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\symd}{\triangle} \newcommand{\sym}{\mathrm{sym}} \newcommand{\s}[1]{\EuScript{{#1}}} \newcommand{\td}[2]{\frac{d #1}{d #2}} \newcommand{\tensor}{\otimes} %\newcommand{\text}{\textnormal} %%%\newcommand{\theorem}[1]{\paragraph{Theorem \arabic{tnum} #1:} \addtocounter{tnum}{1}} \newcommand{\tmat}{\mathcal{M}} \newcommand{\tom}{\tilde{\omega}} \newcommand{\tphi}{\tilde{\phi}} \newcommand{\trace}[0]{\operatorname{trace}} %\newcommand{\trace}{\mathop{\mathrm{trace}}} %\newcommand{\trace}{\mathop{\mathrm{tr}}} %\newcommand{\trace}{\mathrm{tr}} \newcommand{\transv}{\pitchfork} \newcommand{\trd}{\text{{\bf tr.d.}}} \newcommand{\tree}{\mathcal{T}_3} \newcommand{\triv}{\mathrm{triv}} %\newcommand{\tr}[1]{#1^\mathrm{tr}} % Transpose of a matrix %\newcommand{\tr}{\mathrm{tr}} \newcommand{\tspace}[1]{\rT^{#1}} %\newcommand{\tuple}[1]{\langle#1\rangle} \newcommand{\tuple}[1]{\left(#1\right)} \newcommand{\udim}{\operatorname{u-dim}} \newcommand{\ud}[2]{^{#1}_{\!\hphantom{#1}#2}} %\newcommand{\ud}[2]{_{#2}^{\hphantom{#2}\!#1}} %\newcommand{\ud}{\mathrm{d}} \newcommand{\romb}{\mathrm{b}} \newcommand{\romd}{\mathrm{d}} \newcommand{\uk}{{\underline{k}}} \newcommand{\union}{\cup} \newcommand{\un}{{\underline{n}}} \newcommand{\ur}{\mathrm{ur}} \newcommand{\uv}[1]{\ensuremath{\mathbf{\hat{#1}}}} \newcommand{\vA}{\mathbf{A}} \newcommand{\vB}{\mathbf{B}} \newcommand{\vF}{\mathbf{F}} \newcommand{\vG}{\mathbf{G}} \newcommand{\vN}{\mathbf{N}} \newcommand{\vR}[0]{\textbf{R}} \newcommand{\vX}{\mathbf{X}} \newcommand{\val}{\operatorname{Val}} \newcommand{\var}{\mathrm{var}} \newcommand{\va}{\mathbf{a}} % \newcommand{\va}{a} \newcommand{\vb}{\mathbf{b}} \newcommand{\vdiv}{\operatorname{div}} \newcommand{\veca}{\mathbf{a}} \newcommand{\vecb}{\mathbf{b}} \newcommand{\vech}{\mathbf{h}} \newcommand{\vecn}{\mathbf{n}} \newcommand{\vecp}{\mathbf{p}} \newcommand{\vect}{\mathbf{t}} \newcommand{\vecu}{\mathbf{u}} \newcommand{\vecv}{\mathbf{v}} \newcommand{\vecw}{\mathbf{w}} \newcommand{\vecx}{\mathbf{x}} \newcommand{\vecy}{\mathbf{y}} \newcommand{\vecz}{\mathbf{z}} % \newcommand{\vec}{\mathbf\overightarrow} % \newcommand{\vec}{\overrightarrow{\mathbf{#1}}} %\newcommand{\ve}{\boldsymbol{\varepsilon}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vi}{\mathbf{i}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vnabla}{\nabla} \newcommand{\vol}{\mathrm{vol}} \newcommand{\vp}{\varphi} \newcommand{\vs}{\mathcal{X}} \newcommand{\vu}[0]{\textbf{u}} % \newcommand{\vu}{u} % \newcommand{\vv}[0]{\textbf{v}} \newcommand{\vv}[1]{\ensuremath{\mathbf{#1}}} % \newcommand{\vv}{v} \newcommand{\vx}{\mathbf{x}} \newcommand{\wh}[1]{\widehat{#1}} \newcommand{\wideg}{\text{wideg}} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\w}{{{\bf w}}} \newcommand{\xb}{\overline{x}} \newcommand{\xpt}{\mathbf{x}} \newcommand{\xvec}{\mathbf{\overrightarrow{x}}} \newcommand{\x}[1]{#1\setminus\{x\}} % \newcommand{\x}{\mathbf{x}} % \newcommand{\x}{{\bf{x}}} % \newcommand{\x}{{{\bf x}}} \newcommand{\y}{\mathbf{y}} % \newcommand{\y}{{{\bf y}}} \newcommand{\zmod}[1]{\integs / #1\integs} % Z/nZ %\newcommand{\znums}{\mathbb{Z}} \newcommand{\z}{{{\bf z}}} %% I can't make this stuff work as newcommands for some reason -- jac (noted) Wed Jan 7 18:26:45 2004 \def\mobius#1#2#3#4#5{\frac{#2#1 + #3}{#4#1 + #5}} \def\smfour#1#2#3#4{\left(\begin{smallmatrix}{#1}&{#2}\\{#3}&{#4}\end{smallmatrix}\right)} %%%%%%%% \newcommand{xrad}{\mathcal{X}} %% some missing commands (I'm making a newcommand in case one of the included packages has this). \newcommand{\xrad}{x^{\circ}} % \newenvironment{vect}{\left(\begin{array}{l}}{\end{array}\right)} \newenvironment{hwsols}[4]{\noindent{\sc\large Math #1 -- Solutions to Homework #2\\#3\\{\footnotesize by~#4}}\bigskip\\}{} \newenvironment{bookprob}[2]{\noindent{\em Section #1, Problem #2}\\}{\bigskip} \newenvironment{smallbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]} \newenvironment{namedtheorem}[1]{\medskip \noindent {\bf #1: }\begin{em}}{\end{em}\medskip} \newenvironment{Lalgorithm}[4]{ \textbf{Algorithm} \textsc{#1}\texttt{(#2)}\newline \textit{Input}: #3\newline \textit{Output}: #4} {} \newenvironment{Lfloatalgorithm}[6][h]{ \begin{figure}[#1] \caption{#2} \begin{Lalgorithm}{#3}{#4}{#5}{#6}} {\end{Lalgorithm} \end{figure}} \newcounter{alistctr} \newcounter{rlistctr} \newcounter{Rlistctr} \newcounter{123listctr} \newcounter{123listcolonstylectr} \newcounter{tnum} \setcounter{tnum}{1} \newcounter{lblfoo} \setcounter{lblfoo}{1} % a,b,c - small latin letter list \newenvironment{alist}{ \indent \begin{list}{(\alph{alistctr})}{\usecounter{alistctr}}} {\end{list}\setcounter{alistctr}{0}} % A,B,C - LARGE LATIN LETTER LIST \newenvironment{Alist}{ \indent \begin{list}{(\Alph{Alistctr})}{\usecounter{Alistctr}} } {\end{list}\setcounter{Alistctr}{0}} % i,ii,iii - small roman numeral list \newenvironment{rlist}{ \indent \begin{list}{(\roman{rlistctr})}{\usecounter{rlistctr}} } {\end{list}\setcounter{rlistctr}{0}} % I,II,III - large roman numeral list \newenvironment{Rlist}{ \indent \begin{list}{(\Roman{Rlistctr})}{\usecounter{Rlistctr}} } {\end{list}\setcounter{Rlistctr}{0}} %1,2,3 - arabic numeral list \newenvironment{123list}{ \indent \begin{list}{(\arabic{123listctr})}{\usecounter{123listctr}} } {\end{list}\setcounter{123listctr}{0}} %1:,2:,3: - arabic numeral list with colon decoration \newenvironment{123listcolonstyle}{\indent \begin{list}{\arabic{123listcolonstylectr}:}{\usecounter{123listcolonstylectr}}} {\end{list}\setcounter{123listcolonstylectr}{0}} % environment for definitions %%% Maybe all gone now?? Sun Jan 4 20:08:58 2004 % \def\defn#1{\addcontentsline{toc}{subsection}{$\ast$} {\footnotesize \noindent % \begin{123listcolonstyle} \setlength{\itemsep}{0em} \setlength{\topsep}{0em} % \setlength{\parsep}{0em} #1 \end{123listcolonstyle}}} %environment for proofs %% `let' forms that I have changed to \def's -- jac Thu Jan 8 14:22:00 2004 \def\proves{\vdash} \def\implies{\rightarrow} \def\Implies{\Rightarrow} \def\iff{\Leftrightarrow} %% Lines matching "\\\\def\\\\" in buffer jac-obb.tex. \def\0{{\mathbf 0}} \def\Aff{{\rm Aff}} \def\A{{\mathcal A}} \def\B{{\mathcal B}} \def\B{{\mathcal B}} \def\C{\mathbb{C}} \def\F{{\mathcal F}} \def\Gl{{\rm Gl}} \def\Hom{{\rm Hom}} \def\H{\mathcal{H}}% \def\Intn{{\rm Int}_{k,C}^{n}} \def\Int{{\rm Int}_{k,C}} \def\M{\mathcal{M}}% \def\N{\mathbb{N}} \def\Point{$\bullet$ } \def\P{{\mathcal P}} \def\Res{{\rm Res}} \def\R{\mathbb{R}} \def\S{{\mathcal S}} \def\T{{\mathcal T}} \def\U{\mathcal{U}}% \def\V{{\mathsf V}} \def\W{{\mathsf W}} \def\acl{{\rm acl}} \def\bigtimes{\mathop{\mbox{\Huge $\times$}}} \def\centre{\center} \def\cf#1{\operatorname{cf}(#1)} \def\ch{{\rm char}} \def\co{\colon\thinspace} \def\dcl{{\rm dcl}} \def\dd{\mathrm{d}} \def\del{\partial} \def\dom{{\rm dom}} \def\dtra{\hspace{0.04cm}^{\mbox{\scriptsize{T}}} \hspace{0.02cm}} \def\d{{\mathrm d}} \def\elt{{\rm elt}} \def\eps{\epsilon} \def\eps{\epsilon} \def\equiv{\Leftrightarrow} \def\eut{{\rm eut}} \def\fiber{{\rm fib}} \def\germ{{\rm germ}} \def\graph{{\rm graph}} \def\gr{{\rm grd}} \def\htra{\hspace{0.04cm}^{\mbox{\scriptsize{H}}} \hspace{0.02cm}} \def\h{{\mathbf h}} \def\ie{\emph{i.e.},} \def\ilt{{\rm ilt}} \def\impl{\Rightarrow} \def\isom{\simeq} \def\iut{{\rm iut}} \def\lex{{\rm lex}} \def\lit{{\rm ilt}} \def\mo{{\rm m.o.}} \def\nom{\vartriangleleft} \def\of{\circ} \def\op{{\rm op}} \def\proof#1{\par {\footnotesize \indent \begin{tabular}{ll} #1 \end{tabular}}} \def\rad{{\rm rad}} \def\ra{\rightarrow} \def\rcl{{\rm rcl}} \def\red{{\rm red}} \def\rex{{\rm rex}} \def\ser{\Sigma a_n} \def\sse{\subseteq} \def\sse{\subseteq} \def\tp{{\rm tp}} % still a bad idea here!!! %% \def\v{{\mathbf v}} \def\x{{\mathbf x}} \def\y{{\mathbf y}} % Lines matching "declaremath" in buffer jac-obb.tex. %\DeclareMathOperator{\Pr}{Pr} \DeclareMathOperator{\per}{per} \DeclareMathOperator{\Char}{char} \DeclareMathOperator{\apmxiet}{\&} \DeclareMathOperator{\vel}{\curlyvee} \DeclareMathOperator{\cha}{Char} %\DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\crn}{cr} %\DeclareMathOperator{\cr}{cr} \DeclareMathOperator{\usercr}{cr} %\DeclareMathOperator{\dom}{dom} %\DeclareMathOperator{\esssup}{ess sup} \DeclareMathOperator{\imm}{Imm} %\DeclareMathOperator{\im}{im} \DeclareMathOperator{\li}{li} %\DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\pipe}{{\big |} \hspace{-2.85pt}{\big |}} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rann}{r.ann} %\DeclareMathOperator{\sgn}{sgn} %\DeclareMathOperator{\vol}{vol} % Lines matching "newtheorem" in buffer obb.tex. (I'm commenting out the stuff I don't understand -- jac Sun Jan 4 19:44:33 2004) %% \newtheoremstyle{break}{\baselineskip}{\baselineskip}{\itshape}{}{\bfseries}{}{\newline}{} %% \newtheoremstyle{inlinedefn}{}{0pt}{}{}{\bfseries}{.}{0.5em}{} %% \newtheoremstyle{liscio}{\thtopskip}{\thbotskip}{\upshape}{-\thindent}{\headstyle}{}{.5em}{} %% \newtheoremstyle{normale}{\thtopskip}{\thbotskip}{\slshape}{-\thindent}{\headstyle}{}{.5em}{} %% \theoremstyle{break} %% \theoremstyle{inlinedefn} %% \theoremstyle{liscio} %% \theoremstyle{normale} %\newtheorem{theorem}{Theorem} \newtheorem{mainthm}{Main~Theorem} \newtheorem{theorem}{Theorem} \newtheorem{theo}{Theorem} \newtheorem{thm}{Theorem} \newtheorem{tm}{Theorem} \newtheorem{Theo}{Theorem} \newtheorem{Thm}{Theorem} %\newtheorem{pp}{Proposition} \newtheorem{proposition*}{Proposition.} \newtheorem{proposition}{Proposition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{lem}{Lemma} \newtheorem{lm}{Lemma} \newtheorem{claim}{Claim} \newtheorem{conjecture}{Conjecture} \newtheorem{conj}{Conjecture} \newtheorem{dthm}{Desired Theorem} \newtheorem{dcor}{Desired Corollary} \newtheorem{corollary}{Corollary} \newtheorem{cor}{Corollary} \newtheorem{crl}{Corollary} \theoremstyle{definition} \newtheorem{nott}{Notation} \newtheorem{definition}{Definition} \newtheorem{defn}{Definition} %\newtheorem{df}{Definition} \theoremstyle{remark} \newtheorem{remark}{remark} \newtheorem{rem}{Remark} \newtheorem{rmk}{Remark} \newtheorem{Rem}{Remark} %\newtheorem{eg}{Example} \newtheorem{example}{Example} \newtheorem{exm}{Example} \newtheorem{ex}{Exercise} \newtheorem{Exam}{Example} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \thispagestyle{empty} \begin{quote} {\huge Free Encyclopedia of Mathematics 0.0.1} \end{quote} \medskip \begin{quote} {\large \it by the PlanetMath authors} Aatu, ack, akrowne, alek\_thiery, alinabi, almann, alozano, antizeus, antonio, aparna, ariels, armbrusterb, AxelBoldt, basseykay, bbukh, benjaminfjones, bhaire, brianbirgen, bs, bshanks, bwebste, cryo, danielm, Daume, debosberg, deiudi, digitalis, djao, Dr\_Absentius, draisma, drini, drummond, dublisk, Evandar, fibonaci, flynnheiss, gabor\_sz, GaloisRadical, gantsich, gaurminirick, gholmes74, giri, greg, grouprly, gumau, Gunnar, Henry, iddo, igor, imran, jamika\_chris, jarino, jay, jgade, jihemme, Johan, karteef, karthik, kemyers3, Kevin OBryant, kidburla2003, KimJ, Koro, lha, lieven, livetoad, liyang, Logan, Luci, m759, mathcam, mathwizard, matte, mclase, mhale, mike, mikestaflogan, mps, msihl, muqabala, n3o, nerdy2, nobody, npolys, Oblomov, ottocolori, paolini, patrickwonders, pbruin, petervr, PhysBrain, quadrate, quincynoodles, ratboy, RevBobo, Riemann, rmilson, ruiyang, Sabean, saforres, saki, say\_10, scanez, scineram, seonyoung, slash, sleske, slider142, sprocketboy, sucrose, superhiggs, tensorking, thedagit, Thomas Heye, thouis, Timmy, tobix, tromp, tz26, unlord, uriw, urz, vampyr, vernondalhart, vitriol, vladm, volator, vypertd, wberry, Wkbj79, wombat, x\_bas, xiaoyanggu, XJamRastafire, xriso, yark {\it et al.} \end{quote} \begin{quote} edited by Joe Corneli \& Aaron Krowne \end{quote} \bigskip \begin{quote} Copyright \copyright\ 2004 PlanetMath.org authors. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. \end{quote} \clearpage \pagenumbering{roman} \chapter*{Introduction} Welcome to the PlanetMath ``One Big Book" compilation, the Free Encyclopedia of Mathematics. This book gathers in a single document the best of the hundreds of authors and thousands of other contributors from the PlanetMath.org web site, as of January 4, 2004. The purpose of this compilation is to help the efforts of these people reach a wider audience and allow the benefits of their work to be accessed in a greater breadth of situations. We want to emphasize is that the Free Encyclopedia of Mathematics will always be a work in progress. Producing a book-format encycopedia from the amorphous web of interlinked and multidimensionally-organized entries on PlanetMath is not easy. The print medium demands a linear presentation, and to boil the web site down into this format is a difficult, and in some ways lossy, transformation. A major part of our editorial efforts are going into making this transformation. We hope the organization we've chosen for now is useful to readers, and in future editions you can expect continuing improvements. The ``linearization" of PlanetMath.org is not the only editorial task we must perform. Throughout the millenia, readers have come to expect a strict standard of consistency and correctness from print books, and we must strive to meet this standard in the PlanetMath Book as closely as possible. This means applying more editorial control to the book form of PlanetMath than is applied to the web site. We hope you will agree that there is significant value to be gained from unifying style, correcting errors, and filtering out not-yet-ready content, so we will continue to do these things. For more details on planned improvements to this book, see the TODO file that came with this archive. Remember that you can help us to improve this work by joining PlanetMath.org and filing corrections, adding entries, or just participating in the community. We are also looking for volunteers to help edit this book, or help with programming related to its production, or to help work on Noosphere, the PlanetMath software. To send us comments about the book, use the e-mail address {\tt pmbook@planetmath.org}. For general comments and queries, use {\tt feedback@planetmath.org}. \begin{flushright} Happy mathing,\\ \medskip Joe Corneli\\ Aaron Krowne \\ \medskip Tuesday, January 27, 2004 \end{flushright} \chapter*{Top-level Math Subject Classificiations} \begin{verbatim} 00 General 01 History and biography 03 Mathematical logic and foundations 05 Combinatorics 06 Order, lattices, ordered algebraic structures 08 General algebraic systems 11 Number theory 12 Field theory and polynomials 13 Commutative rings and algebras 14 Algebraic geometry 15 Linear and multilinear algebra; matrix theory 16 Associative rings and algebras 17 Nonassociative rings and algebras 18 Category theory; homological algebra 19 $K$-theory 20 Group theory and generalizations 22 Topological groups, Lie groups 26 Real functions 28 Measure and integration 30 Functions of a complex variable 31 Potential theory 32 Several complex variables and analytic spaces 33 Special functions 34 Ordinary differential equations 35 Partial differential equations 37 Dynamical systems and ergodic theory 39 Difference and functional equations 40 Sequences, series, summability 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus 45 Integral equations 46 Functional analysis 47 Operator theory 49 Calculus of variations and optimal control; optimization 51 Geometry 52 Convex and discrete geometry 53 Differential geometry 54 General topology 55 Algebraic topology 57 Manifolds and cell complexes 58 Global analysis, analysis on manifolds 60 Probability theory and stochastic processes 62 Statistics 65 Numerical analysis 68 Computer science 70 Mechanics of particles and systems 74 Mechanics of deformable solids 76 Fluid mechanics 78 Optics, electromagnetic theory 80 Classical thermodynamics, heat transfer 81 Quantum theory 82 Statistical mechanics, structure of matter 83 Relativity and gravitational theory 85 Astronomy and astrophysics 86 Geophysics 90 Operations research, mathematical programming 91 Game theory, economics, social and behavioral sciences 92 Biology and other natural sciences 93 Systems theory; control 94 Information and communication, circuits 97 Mathematics education \end{verbatim} % \tableofcontents \twocolumn {\LARGE Table of Contents} \\ \ \\ {\small {\bf Introduction i} \\ {\bf Top-level Math Subject Classificiations ii} \\ {\bf Table of Contents iv} \\ {\bf GNU Free Documentation License lii} \\ {\bf UNCLA -- Unclassified 1}\\ Golomb ruler 1\\ Hesse configuration 1\\ Jordan's Inequality 2\\ Lagrange's theorem 2\\ Laurent series 3\\ Lebesgue measure 3\\ Leray spectral sequence 4\\ M\"obius transformation 4\\ Mordell-Weil theorem 4\\ Plateau's Problem 5\\ Poisson random variable 5\\ Shannon's theorem 6\\ Shapiro inequality 9\\ Sylow $p$-subgroups 9\\ Tchirnhaus transformations 9\\ Wallis formulae 10\\ ascending chain condition 10\\ bounded 10\\ bounded operator 11\\ complex projective line 12\\ converges uniformly 12\\ descending chain condition 13\\ diamond theorem 13\\ equivalently oriented bases 13\\ finitely generated $R$-module 14\\ fraction 14\\ group of covering transformations 15\\ idempotent 15\\ isolated 17\\ isolated singularity 17\\ isomorphic groups 17\\ joint continuous density function 18\\ joint cumulative distribution function 18\\ joint discrete density function 19\\ left function notation 20\\ lift of a submanifold 20\\ limit of a real function exits at a point 20\\ lipschitz function 21\\ lognormal random variable 21\\ lowest upper bound 22\\ marginal distribution 22\\ measurable space 23\\ measure zero 23\\ minimum spanning tree 23\\ minimum weighted path length 24\\ mod $2$ intersection number 25\\ moment generating function 27\\ monoid 27\\ monotonic operator 27\\ multidimensional Gaussian integral 28\\ multiindex 29\\ near operators 30\\ negative binomial random variable 36\\ normal random variable 37\\ normalizer of a subset of a group 38\\ nth root 38\\ null tree 40\\ open ball 40\\ opposite ring 40\\ orbit-stabilizer theorem 41\\ orthogonal 41\\ permutation group on a set 41\\ prime element 42\\ product measure 43\\ projective line 43\\ projective plane 43\\ proof of calculus theorem used in the Lagrange method 44\\ proof of orbit-stabilizer theorem 45\\ proof of power rule 45\\ proof of primitive element theorem 47\\ proof of product rule 47\\ proof of sum rule 48\\ proof that countable unions are countable 48\\ quadrature 48\\ quotient module 49\\ regular expression 49\\ regular language 50\\ right function notation 51\\ ring homomorphism 51\\ scalar 51\\ schrodinger operator 51\\ selection sort 52\\ semiring 53\\ simple function 54\\ simple path 54\\ solutions of an equation 54\\ spanning tree 54\\ square root 55\\ stable sorting algorithm 56\\ standard deviation 56\\ stochastic independence 56\\ substring 57\\ successor 57\\ sum rule 58\\ superset 58\\ symmetric polynomial 59\\ the argument principle 59\\ torsion-free module 59\\ total order 60\\ tree traversals 60\\ trie 63\\ unit vector 64\\ unstable fixed point 65\\ weak* convergence in normed linear space 65\\ well-ordering principle for natural numbers 65\\ {\bf 00-01 -- Instructional exposition (textbooks, tutorial papers, etc.) 66}\\ dimension 66\\ toy theorem 67\\ {\bf 00-XX -- General 68}\\ method of exhaustion 68\\ {\bf 00A05 -- General mathematics 69}\\ Conway's chained arrow notation 69\\ Knuth's up arrow notation 70\\ arithmetic progression 70\\ arity 71\\ introducing 0th power 71\\ lemma 71\\ property 72\\ saddle point approximation 72\\ singleton 73\\ subsequence 73\\ surreal number 73\\ {\bf 00A07 -- Problem books 76}\\ Nesbitt's inequality 76\\ proof of Nesbitt's inequality 76\\ {\bf 00A20 -- Dictionaries and other general reference works 78}\\ completing the square 78\\ {\bf 00A99 -- Miscellaneous topics 80}\\ QED 80\\ TFAE 80\\ WLOG 81\\ order of operations 81\\ {\bf 01A20 -- Greek, Roman 84}\\ Roman numerals 84\\ {\bf 01A55 -- 19th century 85}\\ Poincar^^c3^^a9, Jules Henri 85\\ {\bf 01A60 -- 20th century 90}\\ Bourbaki, Nicolas 90\\ Erd^^c3^^b6s Number 97\\ {\bf 03-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 98}\\ Burali-Forti paradox 98\\ Cantor's paradox 98\\ Russell's paradox 99\\ biconditional 99\\ bijection 100\\ cartesian product 100\\ chain 100\\ characteristic function 101\\ concentric circles 101\\ conjunction 102\\ disjoint 102\\ empty set 102\\ even number 103\\ fixed point 103\\ infinite 103\\ injective function 104\\ integer 104\\ inverse function 105\\ linearly ordered 106\\ operator 106\\ ordered pair 106\\ ordering relation 106\\ partition 107\\ pullback 107\\ set closed under an operation 108\\ signature of a permutation 109\\ subset 109\\ surjective 110\\ transposition 110\\ truth table 111\\ {\bf 03-XX -- Mathematical logic and foundations 112}\\ standard enumeration 112\\ {\bf 03B05 -- Classical propositional logic 113}\\ CNF 113\\ Proof that contrapositive statement is true using logical equivalence 113\\ contrapositive 114\\ disjunction 114\\ equivalent 114\\ implication 115\\ propositional logic 115\\ theory 116\\ transitive 116\\ truth function 117\\ {\bf 03B10 -- Classical first-order logic 118}\\ $\Delta_1$ bootstrapping 118\\ Boolean 119\\ G\"odel numbering 120\\ G\"odel's incompleteness theorems 120\\ Lindenbaum algebra 127\\ Lindstr\"om's theorem 128\\ Pressburger arithmetic 129\\ R-minimal element 129\\ Skolemization 129\\ arithmetical hierarchy 129\\ arithmetical hierarchy is a proper hierarchy 130\\ atomic formula 131\\ creating an infinite model 131\\ criterion for consistency of sets of formulas 132\\ deductions are $\Delta _1$ 132\\ example of G\"odel numbering 134\\ example of well-founded induction 135\\ first order language 136\\ first order logic 137\\ first order theories 138\\ free and bound variables 138\\ generalized quantifier 139\\ logic 140\\ proof of compactness theorem for first order logic 141\\ proof of principle of transfinite induction 141\\ proof of the well-founded induction principle 141\\ quantifier 141\\ quantifier free 144\\ subformula 144\\ syntactic compactness theorem for first order logic 144\\ transfinite induction 144\\ universal relation 145\\ universal relations exist for each level of the arithmetical hierarchy 145\\ well-founded induction 146\\ well-founded induction on formulas 147\\ {\bf 03B15 -- Higher-order logic and type theory 143}\\ H\"artig's quantifier 143\\ Russell's theory of types 143\\ analytic hierarchy 145\\ game-theoretical quantifier 146\\ logical language 147\\ second order logic 148\\ {\bf 03B40 -- Combinatory logic and lambda-calculus 150}\\ Church integer 150\\ combinatory logic 150\\ lambda calculus 151\\ {\bf 03B48 -- Probability and inductive logic 154}\\ conditional probability 154\\ {\bf 03B99 -- Miscellaneous 155}\\ Beth property 155\\ Hofstadter's MIU system 155\\ IF-logic 157\\ Tarski's result on the undefinability of Truth 160\\ axiom 161\\ compactness 164\\ consistent 164\\ interpolation property 164\\ sentence 165\\ {\bf 03Bxx -- General logic 166}\\ Banach-Tarski paradox 166\\ {\bf 03C05 -- Equational classes, universal algebra 168}\\ congruence 168\\ every congruence is the kernel of a homomorphism 168\\ homomorphic image of a $\Sigma $-structure is a $\Sigma $-structure 169\\ kernel 169\\ kernel of a homomorphism is a congruence 169\\ quotient structure 170\\ {\bf 03C07 -- Basic properties of first-order languages and structures 171}\\ Models constructed from constants 171\\ Stone space 172\\ alphabet 173\\ axiomatizable theory 174\\ definable 174\\ definable type 175\\ downward Lowenheim-Skolem theorem 176\\ example of definable type 176\\ example of strongly minimal 177\\ first isomorphism theorem 177\\ language 178\\ length of a string 179\\ proof of homomorphic image of a $\Sigma $-structure is a $\Sigma $-structure 179\\ satisfaction relation 180\\ signature 181\\ strongly minimal 181\\ structure preserving mappings 181\\ structures 182\\ substructure 183\\ type 183\\ upward Lowenheim-Skolem theorem 183\\ {\bf 03C15 -- Denumerable structures 185}\\ random graph (infinite) 185\\ {\bf 03C35 -- Categoricity and completeness of theories 187}\\ $\kappa $-categorical 187\\ Vaught's test 187\\ proof of Vaught's test 187\\ {\bf 03C50 -- Models with special properties (saturated, rigid, etc.) 189}\\ example of universal structure 189\\ homogeneous 191\\ universal structure 191\\ {\bf 03C52 -- Properties of classes of models 192}\\ amalgamation property 192\\ {\bf 03C64 -- Model theory of ordered structures; o-minimality 193}\\ infinitesimal 193\\ o-minimality 194\\ real closed fields 194\\ {\bf 03C68 -- Other classical first-order model theory 196}\\ imaginaries 196\\ {\bf 03C90 -- Nonclassical models (Boolean-valued, sheaf, etc.) 198}\\ Boolean valued model 198\\ {\bf 03C99 -- Miscellaneous 199}\\ axiom of foundation 199\\ elementarily equivalent 199\\ elementary embedding 200\\ model 200\\ proof equivalence of formulation of foundation 201\\ {\bf 03D10 -- Turing machines and related notions 203}\\ Turing machine 203\\ {\bf 03D20 -- Recursive functions and relations, subrecursive hierarchies 206}\\ primitive recursive 206\\ {\bf 03D25 -- Recursively (computably) enumerable sets and degrees 207}\\ recursively enumerable 207\\ {\bf 03D75 -- Abstract and axiomatic computability and recursion theory 208}\\ Ackermann function 208\\ halting problem 209\\ {\bf 03E04 -- Ordered sets and their cofinalities; pcf theory 211}\\ another definition of cofinality 211\\ cofinality 211\\ maximal element 212\\ partitions less than cofinality 213\\ well ordered set 213\\ pigeonhole principle 213\\ proof of pigeonhole principle 213\\ tree (set theoretic) 214\\ $\kappa $-complete 215\\ Cantor's diagonal argument 215\\ Fodor's lemma 216\\ Schroeder-Bernstein theorem 216\\ Veblen function 216\\ additively indecomposable, 217\\ cardinal number 217\\ cardinal successor 217\\ cardinality 218\\ cardinality of a countable union 218\\ cardinality of the rationals 219\\ classes of ordinals and enumerating functions 219\\ club 219\\ club filter 220\\ countable 220\\ countably infinite 221\\ finite 221\\ fixed points of normal functions 221\\ height of an algebraic number 221\\ if $A$ is infinite and $B$ is a finite subset of $A\tmspace +\thinmuskip {.1667em}\tmspace -\thinmuskip {.1667em},$ then $A\setminus B$ is infinite 222\\ limit cardinal 222\\ natural number 223\\ ordinal arithmetic 224\\ ordinal number 225\\ power set 225\\ proof of Fodor's lemma 225\\ proof of Schroeder-Bernstein theorem 225\\ proof of fixed points of normal functions 226\\ proof of the existence of transcendental numbers 226\\ proof of theorems in aditively indecomposable 227\\ proof that the rationals are countable 228\\ stationary set 228\\ successor cardinal 229\\ uncountable 229\\ von Neumann integer 229\\ von Neumann ordinal 230\\ weakly compact cardinal 231\\ weakly compact cardinals and the tree property 231\\ Cantor's theorem 232\\ proof of Cantor's theorem 232\\ additive 232\\ antisymmetric 233\\ constant function 233\\ direct image 234\\ domain 234\\ dynkin system 234\\ equivalence class 235\\ fibre 235\\ filtration 236\\ finite character 236\\ fix (transformation actions) 236\\ function 237\\ functional 237\\ generalized cartesian product 238\\ graph 238\\ identity map 238\\ inclusion mapping 239\\ inductive set 239\\ invariant 240\\ inverse function theorem 240\\ inverse image 241\\ mapping 242\\ mapping of period $n$ is a bijection 242\\ partial function 242\\ partial mapping 243\\ period of mapping 243\\ pi-system 244\\ proof of inverse function theorem 244\\ proper subset 246\\ range 246\\ reflexive 246\\ relation 246\\ restriction of a mapping 247\\ set difference 247\\ symmetric 247\\ symmetric difference 248\\ the inverse image commutes with set operations 248\\ transformation 249\\ transitive 250\\ transitive 250\\ transitive closure 250\\ Hausdorff's maximum principle 250\\ Kuratowski's lemma 251\\ Tukey's lemma 251\\ Zermelo's postulate 251\\ Zermelo's well-ordering theorem 251\\ Zorn's lemma 252\\ axiom of choice 252\\ equivalence of Hausdorff's maximum principle, Zorn's lemma and the well-ordering theorem 252\\ equivalence of Zorn's lemma and the axiom of choice 253\\ maximality principle 254\\ principle of finite induction 254\\ principle of finite induction proven from well-ordering principle 255\\ proof of Tukey's lemma 255\\ proof of Zermelo's well-ordering theorem 255\\ axiom of extensionality 256\\ axiom of infinity 256\\ axiom of pairing 257\\ axiom of power set 258\\ axiom of union 258\\ axiom schema of separation 259\\ de Morgan's laws 260\\ de Morgan's laws for sets (proof) 261\\ set theory 261\\ union 264\\ universe 264\\ von Neumann-Bernays-G^^c3^^b6del set theory 265\\ FS iterated forcing preserves chain condition 267\\ chain condition 268\\ composition of forcing notions 268\\ composition preserves chain condition 268\\ equivalence of forcing notions 269\\ forcing relation 270\\ forcings are equivalent if one is dense in the other 270\\ iterated forcing 272\\ iterated forcing and composition 273\\ name 273\\ partial order with chain condition does not collapse cardinals 274\\ proof of partial order with chain condition does not collapse cardinals 274\\ proof that forcing notions are equivalent to their composition 275\\ complete partial orders do not add small subsets 280\\ proof of complete partial orders do not add small subsets 280\\ $\Diamond $ is equivalent to $\clubsuit $ and continuum hypothesis 281\\ Levy collapse 281\\ proof of $\Diamond $ is equivalent to $\clubsuit $ and continuum hypothesis 282\\ Martin's axiom 283\\ Martin's axiom and the continuum hypothesis 283\\ Martin's axiom is consistent 284\\ a shorter proof: Martin's axiom and the continuum hypothesis 287\\ continuum hypothesis 288\\ forcing 288\\ generalized continuum hypothesis 289\\ inaccessible cardinals 290\\ $\Diamond $ 290\\ $\clubsuit $ 290\\ Dedekind infinite 291\\ Zermelo-Fraenkel axioms 291\\ class 291\\ complement 293\\ delta system 293\\ delta system lemma 293\\ diagonal intersection 293\\\\ intersection 294\\ multiset 294\\ proof of delta system lemma 294\\ rational number 295\\ saturated (set) 295\\ separation and doubletons axiom 295\\ set 296\\ {\bf 03Exx -- Set theory 299}\\ intersection 299\\ {\bf 03F03 -- Proof theory, general 300}\\ $\mathscr{NJ}p$ 300\\ $\mathscr{NK}p$ 300\\ natural deduction 301\\ sequent 301\\ sound,, complete 302\\ {\bf 03F07 -- Structure of proofs 303}\\ induction 303\\ {\bf 03F30 -- First-order arithmetic and fragments 307}\\ Elementary Functional Arithmetic 307\\ PA 308\\ Peano arithmetic 308\\ {\bf 03F35 -- Second- and higher-order arithmetic and fragments 310}\\ $ACA_0$ 310\\ $RCA_0$ 310\\ $Z_2$ 310\\ comprehension axiom 311\\ induction axiom 311\\ {\bf 03G05 -- Boolean algebras 313}\\ Boolean algebra 313\\ M. H. Stone's representation theorem 313\\ {\bf 03G10 -- Lattices and related structures 314}\\ Boolean lattice 314\\ complete lattice 314\\ lattice 315\\ {\bf 03G99 -- Miscellaneous 316}\\ Chu space 316\\ Chu transform 316\\ biextensional collapse 317\\ example of Chu space 317\\ property of a Chu space 318\\ {\bf 05-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 319}\\ example of pigeonhole principle 319\\ multi-index derivative of a power 319\\ multi-index notation 320\\ {\bf 05A10 -- Factorials, binomial coefficients, combinatorial functions 322}\\ Catalan numbers 322\\ Levi-Civita permutation symbol 323\\ Pascal's rule (bit string proof) 325\\ Pascal's rule proof 326\\ Pascal's triangle 326\\ Upper and lower bounds to binomial coefficient 328\\ binomial coefficient 328\\ double factorial 329\\ factorial 329\\ falling factorial 330\\ inductive proof of binomial theorem 331\\ multinomial theorem 332\\ multinomial theorem (proof) 333\\ proof of upper and lower bounds to binomial coefficient 334\\ {\bf 05A15 -- Exact enumeration problems, generating functions 336}\\ Stirling numbers of the first kind 336\\ Stirling numbers of the second kind 338\\ {\bf 05A19 -- Combinatorial identities 342}\\ Pascal's rule 342\\ {\bf 05A99 -- Miscellaneous 343}\\ principle of inclusion-exclusion 343\\ principle of inclusion-exclusion proof 344\\ {\bf 05B15 -- Orthogonal arrays, Latin squares, Room squares 346}\\ example of Latin squares 346\\ graeco-latin squares 346\\ latin square 347\\ magic square 347\\ {\bf 05B35 -- Matroids, geometric lattices 348}\\ matroid 348\\ polymatroid 353\\ {\bf 05C05 -- Trees 354}\\ AVL tree 354\\ Aronszajn tree 354\\ Suslin tree 354\\ antichain 355\\ balanced tree 355\\ binary tree 355\\ branch 356\\ child node (of a tree) 356\\ complete binary tree 357\\ digital search tree 357\\ digital tree 358\\ example of Aronszajn tree 358\\ example of tree (set theoretic) 359\\ extended binary tree 359\\ external path length 360\\ internal node (of a tree) 360\\ leaf node (of a tree) 361\\ parent node (in a tree) 361\\ proof that $\omega $ has the tree property 362\\ root (of a tree) 362\\ tree 363\\ weight-balanced binary trees are ultrametric 364\\ weighted path length 366\\ {\bf 05C10 -- Topological graph theory, imbedding 367}\\ Heawood number 367\\ Kuratowski's theorem 368\\ Szemer\'edi-Trotter theorem 368\\ crossing lemma 369\\ crossing number 369\\ graph topology 369\\ planar graph 370\\ proof of crossing lemma 370\\ {\bf 05C12 -- Distance in graphs 372}\\ Hamming distance 372\\ {\bf 05C15 -- Coloring of graphs and hypergraphs 373}\\ bipartite graph 373\\ chromatic number 374\\ chromatic number and girth 375\\ chromatic polynomial 375\\ colouring problem 376\\ complete bipartite graph 377\\ complete k-partite graph 378\\ four-color conjecture 378\\ k-partite graph 379\\ property B 380\\ {\bf 05C20 -- Directed graphs (digraphs), tournaments 381}\\ cut 381\\ de Bruijn digraph 381\\ directed graph 382\\ flow 383\\ maximum flow/minimum cut theorem 384\\ tournament 385\\ {\bf 05C25 -- Graphs and groups 387}\\ Cayley graph 387\\ {\bf 05C38 -- Paths and cycles 388}\\ Euler path 388\\ Veblen's theorem 388\\ acyclic graph 389\\ bridges of K^^c3^^b6nigsberg 389\\ cycle 390\\ girth 391\\ path 391\\ proof of Veblen's theorem 392\\ {\bf 05C40 -- Connectivity 393}\\ $k$-connected graph 393\\ Thomassen's theorem on $3$-connected graphs 393\\ Tutte's wheel theorem 394\\ connected graph 394\\ cutvertex 395\\ {\bf 05C45 -- Eulerian and Hamiltonian graphs 396}\\ Bondy and Chv^^c3^^a1tal theorem 396\\ Dirac theorem 396\\ Euler circuit 397\\ Fleury's algorithm 397\\ Hamiltonian cycle 398\\ Hamiltonian graph 398\\ Hamiltonian path 398\\ Ore's theorem 398\\ Petersen graph 399\\ hypohamiltonian 399\\ traceable 399\\ {\bf 05C60 -- Isomorphism problems (reconstruction conjecture, etc.) 400}\\ graph isomorphism 400\\ {\bf 05C65 -- Hypergraphs 402}\\ Steiner system 402\\ finite plane 402\\ hypergraph 403\\ linear space 404\\ {\bf 05C69 -- Dominating sets, independent sets, cliques 405}\\ Mantel's theorem 405\\ clique 405\\ proof of Mantel's theorem 405\\ {\bf 05C70 -- Factorization, matching, covering and packing 407}\\ Petersen theorem 407\\ Tutte theorem 407\\ bipartite matching 407\\ edge covering 409\\ matching 409\\ maximal bipartite matching algorithm 410\\ maximal matching/minimal edge covering theorem 411\\ {\bf 05C75 -- Structural characterization of types of graphs 413}\\ multigraph 413\\ pseudograph 413\\ {\bf 05C80 -- Random graphs 414}\\ examples of probabilistic proofs 414\\ probabilistic method 415\\ {\bf 05C90 -- Applications 417}\\ Hasse diagram 417\\ {\bf 05C99 -- Miscellaneous 419}\\ Euler's polyhedron theorem 419\\ Poincar\'e formula 419\\ Turan's theorem 419\\ Wagner's theorem 420\\ block 420\\ bridge 420\\ complete graph 420\\ degree (of a vertex) 421\\ distance (in a graph) 421\\ edge-contraction 421\\ graph 422\\ graph minor theorem 422\\ graph theory 423\\ homeomorphism 424\\ loop 424\\ minor (of a graph) 424\\ neighborhood (of a vertex) 425\\ null graph 425\\ order (of a graph) 425\\ proof of Euler's polyhedron theorem 426\\ proof of Turan's theorem 427\\ realization 427\\ size (of a graph) 428\\ subdivision 428\\ subgraph 429\\ wheel graph 429\\ {\bf 05D05 -- Extremal set theory 431}\\ LYM inequality 431\\ Sperner's theorem 432\\ {\bf 05D10 -- Ramsey theory 433}\\ Erd\"os-Rado theorem 433\\ Ramsey's theorem 433\\ Ramsey's theorem 434\\ arrows 435\\ coloring 436\\ proof of Ramsey's theorem 437\\ {\bf 05D15 -- Transversal (matching) theory 438}\\ Hall's marriage theorem 438\\ proof of Hall's marriage theorem 438\\ saturate 440\\ system of distinct representatives 440\\ {\bf 05E05 -- Symmetric functions 441}\\ elementary symmetric polynomial 441\\ reduction algorithm for symmetric polynomials 441\\ {\bf 06-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 443}\\ equivalence relation 443\\ {\bf 06-XX -- Order, lattices, ordered algebraic structures 445}\\ join 445\\ meet 445\\ {\bf 06A06 -- Partial order, general 446}\\ directed set 446\\ infimum 446\\ sets that do not have an infimum 447\\ supremum 447\\ upper bound 448\\ {\bf 06A99 -- Miscellaneous 449}\\ dense (in a poset) 449\\ partial order 449\\ poset 450\\ quasi-order 450\\ well quasi ordering 450\\ {\bf 06B10 -- Ideals, congruence relations 452}\\ order in an algebra 452\\ {\bf 06C05 -- Modular lattices, Desarguesian lattices 453}\\ modular lattice 453\\ {\bf 06D99 -- Miscellaneous 454}\\ distributive 454\\ distributive lattice 454\\ {\bf 06E99 -- Miscellaneous 455}\\ Boolean ring 455\\ {\bf 08A40 -- Operations, polynomials, primal algebras 456}\\ coefficients of a polynomial 456\\ {\bf 08A99 -- Miscellaneous 457}\\ binary operation 457\\ filtered algebra 457\\ {\bf 11-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 459}\\ Euler phi-function 459\\ Euler-Fermat theorem 460\\ Fermat's little theorem 460\\ Fermat's theorem proof 460\\ Goldbach's conjecture 460\\ Jordan's totient function 461\\ Legendre symbol 461\\ Pythagorean triplet 462\\ Wilson's theorem 462\\ arithmetic mean 462\\ ceiling 463\\ computation of powers using Fermat's little theorem 463\\ congruences 464\\ coprime 464\\ cube root 464\\ floor 465\\ geometric mean 465\\ googol 466\\ googolplex 467\\ greatest common divisor 467\\ group theoretic proof of Wilson's theorem 467\\ harmonic mean 467\\ mean 468\\ number field 468\\ pi 468\\ proof of Wilson's theorem 470\\ proof of fundamental theorem of arithmetic 471\\ root of unity 471\\ {\bf 11-01 -- Instructional exposition (textbooks, tutorial papers, etc.) 472}\\ base 472\\ {\bf 11-XX -- Number theory 474}\\ Lehmer's Conjecture 474\\ Sierpinski conjecture 474\\ prime triples conjecture 475\\ {\bf 11A05 -- Multiplicative structure; Euclidean algorithm; greatest common divisors 476}\\ Bezout's lemma (number theory) 476\\ Euclid's algorithm 476\\ Euclid's lemma 478\\ Euclid's lemma proof 478\\ fundamental theorem of arithmetic 479\\ perfect number 479\\ smooth number 480\\ {\bf 11A07 -- Congruences; primitive roots; residue systems 481}\\ Anton's congruence 481\\ Fermat's Little Theorem proof (Inductive) 482\\ Jacobi symbol 483\\ Shanks-Tonelli algorithm 483\\ Wieferich prime 483\\ Wilson's theorem for prime powers 484\\ factorial module prime powers 485\\ proof of Euler-Fermat theorem 485\\ proof of Lucas's theorem 486\\ {\bf 11A15 -- Power residues, reciprocity 487}\\ Euler's criterion 487\\ Gauss' lemma 487\\ Zolotarev's lemma 489\\ cubic reciprocity law 491\\ proof of Euler's criterion 493\\ proof of quadratic reciprocity rule 494\\ quadratic character of 2 495\\ quadratic reciprocity for polynomials 496\\ quadratic reciprocity rule 497\\ quadratic residue 497\\ {\bf 11A25 -- Arithmetic functions; related numbers; inversion formulas 498}\\ Dirichlet character 498\\ Liouville function 498\\ Mangoldt function 499\\ Mertens' first theorem 499\\ Moebius function 499\\ Moebius in version 500\\ arithmetic function 502\\ multiplicative function 503\\ non-multiplicative function 505\\ totient 507\\ unit 507\\ {\bf 11A41 -- Primes 508}\\ Chebyshev functions 508\\ Euclid's proof of the infinitude of primes 509\\ Mangoldt summatory function 509\\ Mersenne numbers 510\\ Thue's lemma 510\\ composite number 511\\ prime 511\\ prime counting function 511\\ prime difference function 512\\ prime number theorem 512\\ prime number theorem result 513\\ proof of Thue's Lemma 514\\ semiprime 515\\ sieve of Eratosthenes 516\\ test for primality of Mersenne numbers 516\\ {\bf 11A51 -- Factorization; primality 517}\\ Fermat Numbers 517\\ Fermat compositeness test 517\\ Zsigmondy's theorem 518\\ divisibility 518\\ division algorithm for integers 519\\ proof of division algorithm for integers 519\\ square-free number 520\\ squarefull number 520\\ the prime power dividing a factorial 521\\ {\bf 11A55 -- Continued fractions 523}\\ Stern-Brocot tree 523\\ continued fraction 524\\ {\bf 11A63 -- Radix representation; digital problems 527}\\ Kummer's theorem 527\\ corollary of Kummer's theorem 528\\ {\bf 11A67 -- Other representations 529}\\ Sierpinski Erd\"{o}s egyptian fraction conjecture 529\\ adjacent fraction 529\\ any rational number is a sum of unit fractions 530\\ conjecture on fractions with odd denominators 532\\ unit fraction 532\\ {\bf 11A99 -- Miscellaneous 533}\\ ABC conjecture 533\\ Suranyi theorem 533\\ irrational to an irrational power can be rational 534\\ triangular numbers 534\\ {\bf 11B05 -- Density, gaps, topology 536}\\ Cauchy-Davenport theorem 536\\ Mann's theorem 536\\ Schnirelmann density 537\\ Sidon set 537\\ asymptotic density 538\\ discrete space 538\\ essential component 539\\ normal order 539\\ {\bf 11B13 -- Additive bases 541}\\ Erd\"{o}s-Turan conjecture 541\\ additive basis 542\\ asymptotic basis 542\\ base con version 542\\ sumset 546\\ {\bf 11B25 -- Arithmetic progressions 547}\\ Behrend's construction 547\\ Freiman's theorem 548\\ Szemer\'edi's theorem 548\\ multidimensional arithmetic progression 549\\ {\bf 11B34 -- Representation functions 550}\\ Erd\"{o}s-Fuchs theorem 550\\ {\bf 11B37 -- Recurrences 551}\\ Collatz problem 551\\ recurrence relation 551\\ {\bf 11B39 -- Fibonacci and Lucas numbers and polynomials and generalizations 553}\\ Fibonacci sequence 553\\ Hogatt's theorem 554\\ Lucas numbers 554\\ golden ratio 554\\ {\bf 11B50 -- Sequences (mod $m$) 556}\\ Erd\"{o}s-Ginzburg-Ziv theorem 556\\ {\bf 11B57 -- Farey sequences; the sequences $?$ 557}\\ Farey sequence 557\\ {\bf 11B65 -- Binomial coefficients; factorials; $q$-identities 559}\\ Lucas's Theorem 559\\ binomial theorem 559\\ {\bf 11B68 -- Bernoulli and Euler numbers and polynomials 561}\\ Bernoulli number 561\\ Bernoulli periodic function 561\\ Bernoulli polynomial 562\\ generalized Bernoulli number 562\\ {\bf 11B75 -- Other combinatorial number theory 563}\\ Erd\"{o}s-Heilbronn conjecture 563\\ Freiman isomorphism 563\\ sum-free 564\\ {\bf 11B83 -- Special sequences and polynomials 565}\\ Beatty sequence 565\\ Beatty's theorem 566\\ Fraenkel's partition theorem 566\\ Sierpinski numbers 567\\ palindrome 567\\ proof of Beatty's theorem 568\\ square-free sequence 569\\ superincreasing sequence 569\\ {\bf 11B99 -- Miscellaneous 570}\\ Lychrel number 570\\ closed form 571\\ {\bf 11C08 -- Polynomials 573}\\ content of a polynomial 573\\ cyclotomic polynomial 573\\ height of a polynomial 574\\ length of a polynomial 574\\ proof of Eisenstein criterion 574\\ proof that the cyclotomic polynomial is irreducible 575\\ {\bf 11D09 -- Quadratic and bilinear equations 577}\\ Pell's equation and simple continued fractions 577\\ {\bf 11D41 -- Higher degree equations; Fermat's equation 578}\\ Beal conjecture 578\\ Euler quartic conjecture 579\\ Fermat's last theorem 580\\ {\bf 11D79 -- Congruences in many variables 582}\\ Chinese remainder theorem 582\\ Chinese remainder theorem proof 583\\ {\bf 11D85 -- Representation problems 586}\\ polygonal number 586\\ {\bf 11D99 -- Miscellaneous 588}\\ Diophantine equation 588\\ {\bf 11E39 -- Bilinear and Hermitian forms 590}\\ Hermitian form 590\\ non-degenerate bilinear form 590\\ positive definite form 591\\ symmetric bilinear form 591\\ Clifford algebra 591\\ {\bf 11Exx -- Forms and linear algebraic groups 593}\\ quadratic function associated with a linear functional 593\\ {\bf 11F06 -- Structure of modular groups and generalizations; arithmetic groups 594}\\ Taniyama-Shimura theorem 594\\ {\bf 11F30 -- Fourier coefficients of automorphic forms 597}\\ Fourier coefficients 597\\ {\bf 11F67 -- Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 598}\\ Schanuel's conjecutre 598\\ period 598\\ {\bf 11G05 -- Elliptic curves over global fields 600}\\ complex multiplication 600\\ {\bf 11H06 -- Lattices and convex bodies 602}\\ Minkowski's theorem 602\\ lattice in $\mathbb {R}^n$ 602\\ {\bf 11H46 -- Products of linear forms 604}\\ triple scalar product 604\\ {\bf 11J04 -- Homogeneous approximation to one number 605}\\ Dirichlet's approximation theorem 605\\ {\bf 11J68 -- Approximation to algebraic numbers 606}\\ Davenport-Schmidt theorem 606\\ Liouville approximation theorem 606\\ proof of Liouville approximation theorem 607\\ {\bf 11J72 -- Irrationality; linear independence over a field 609}\\ $n$th root of $2$ is irrational for $n\ge 3$ (proof using Fermat's last theorem) 609\\ e is irrational (proof) 610\\ irrational 610\\ square root of 2 is irrational 611\\ {\bf 11J81 -- Transcendence (general theory) 612}\\ Fundamental Theorem of Transcendence 612\\ Gelfond's theorem 612\\ four exponentials conjecture 612\\ six exponentials theorem 613\\ transcendental number 614\\ {\bf 11K16 -- Normal numbers, radix expansions, etc. 615}\\ absolutely normal 615\\ {\bf 11K45 -- Pseudo-random numbers; Monte Carlo methods 617}\\ pseudorandom numbers 617\\ quasirandom numbers 618\\ random numbers 619\\ truly random numbers 619\\ {\bf 11L03 -- Trigonometric and exponential sums, general 620}\\ Ramanujan sum 620\\ {\bf 11L05 -- Gauss and Kloosterman sums; generalizations 622}\\ Gauss sum 622\\ Kloosterman sum 623\\ Landsberg-Schaar relation 623\\ derivation of Gauss sum up to a sign 624\\ {\bf 11L40 -- Estimates on character sums 625}\\ P^^c3^^b3lya-Vinogradov inequality 625\\ {\bf 11M06 -- $\zeta (s)$ and $L(s, \chi )$ 627}\\ Ap\'ery's constant 627\\ Dedekind zeta function 627\\ Dirichlet L-series 628\\ Riemann $\theta $-function 629\\ Riemann Xi function 630\\ Riemann omega function 630\\ functional equation for the Riemann Xi function 630\\ functional equation for the Riemann theta function 631\\ generalized Riemann hypothesis 631\\ proof of functional equation for the Riemann theta function 631\\ {\bf 11M99 -- Miscellaneous 633}\\ Riemann zeta function 633\\ formulae for zeta in the critical strip 636\\ functional equation of the Riemann zeta function 638\\ value of the Riemann zeta function at $s=2$ 638\\ {\bf 11N05 -- Distribution of primes 640}\\ Bertrand's conjecture 640\\ Brun's constant 640\\ proof of Bertrand's conjecture 640\\ twin prime conjecture 642\\ {\bf 11N13 -- Primes in progressions 643}\\ primes in progressions 648 \\\\ {\bf 11N32 -- Primes represented by polynomials; other multiplicative structure of polynomial values 644}\\ Euler four-square identity 644\\ {\bf 11N56 -- Rate of growth of arithmetic functions 645}\\ highly composite number 645\\ {\bf 11N99 -- Miscellaneous 646}\\ Chinese remainder theorem 646\\ proof of chinese remainder theorem 646\\ {\bf 11P05 -- Waring's problem and variants 648}\\ Lagrange's four-square theorem 648\\ Waring's problem 648\\ proof of Lagrange's four-square theorem 649\\ {\bf 11P81 -- Elementary theory of partitions 651}\\ pentagonal number theorem 651\\ {\bf 11R04 -- Algebraic numbers; rings of algebraic integers 653}\\ Dedekind domain 653\\ Dirichlet's unit theorem 653\\ Eisenstein integers 654\\ Galois representation 654\\ Gaussian integer 658\\ algebraic conjugates 659\\ algebraic integer 659\\ algebraic number 659\\ algebraic number field 659\\ calculating the splitting of primes 660\\ characterization in terms of prime ideals 661\\ ideal classes form an abelian group 661\\ integral basis 661\\ integrally closed 662\\ transcendental root theorem 662\\ {\bf 11R06 -- PV-numbers and generalizations; other special algebraic numbers 663}\\ Salem number 663\\ {\bf 11R11 -- Quadratic extensions 664}\\ prime ideal decomposition in quadratic extensions of $\mathbb {Q}$ 664\\ {\bf 11R18 -- Cyclotomic extensions 666}\\ Kronecker-Weber theorem 666\\ examples of regular primes 667\\ prime ideal decomposition in cyclotomic extensions of $\mathbb {Q}$ 668\\ regular prime 669\\ {\bf 11R27 -- Units and factorization 670}\\ regulator 670\\ {\bf 11R29 -- Class numbers, class groups, discriminants 672}\\ Existence of Hilbert Class Field 672\\ class number formula 673\\ discriminant 673\\ ideal class 674\\ ray class group 675\\ {\bf 11R32 -- Galois theory 676}\\ Galois criterion for solvability of a polynomial by radicals 676\\ {\bf 11R34 -- Galois cohomology 677}\\ Hilbert Theorem 90 677\\ {\bf 11R37 -- Class field theory 678}\\ Artin map 678\\ Tchebotarev density theorem 679\\ modulus 679\\ multiplicative congruence 680\\ ray class field 680\\ {\bf 11R56 -- Ad\`ele rings and groups 682}\\ ad^^c3^^a8le 682\\ id^^c3^^a8le 682\\ restricted direct product 683\\ {\bf 11R99 -- Miscellaneous 684}\\ Henselian field 684\\ valuation 685\\ {\bf 11S15 -- Ramification and extension theory 686}\\ decomposition group 686\\ examples of prime ideal decomposition in number fields 688\\ inertial degree 691\\ ramification index 692\\ unramified action 697\\ {\bf 11S31 -- Class field theory; $p$-adic formal groups 699}\\ Hilbert symbol 699\\ {\bf 11S99 -- Miscellaneous 700}\\ $p$-adic integers 700\\ local field 701\\ {\bf 11Y05 -- Factorization 703}\\ Pollard's rho method 703\\ quadratic sieve 706\\ {\bf 11Y55 -- Calculation of integer sequences 709}\\ Kolakoski sequence 709\\ {\bf 11Z05 -- Miscellaneous applications of number theory 711}\\ $\tau $ function 711\\ arithmetic derivative 711\\ example of arithmetic derivative 712\\ proof that $\tau (n)$ is the number of positive divisors of $n$ 712\\ {\bf 12-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 714}\\ monomial 714\\ order and degree of polynomial 715\\ {\bf 12-XX -- Field theory and polynomials 716}\\ homogeneous polynomial 716\\ subfield 716\\ {\bf 12D05 -- Polynomials: factorization 717}\\ factor theorem 717\\ proof of factor theorem 717\\ proof of rational root theorem 718\\ rational root theorem 719\\ sextic equation 719\\ {\bf 12D10 -- Polynomials: location of zeros (algebraic theorems) 720}\\ Cardano's derivation of the cubic formula 720\\ Ferrari-Cardano derivation of the quartic formula 721\\ Galois-theoretic derivation of the cubic formula 722\\ Galois-theoretic derivation of the quartic formula 724\\ cubic formula 728\\ derivation of quadratic formula 728\\ quadratic formula 729\\ quartic formula 730\\ reciprocal polynomial 730\\ root 731\\ variant of Cardano's derivation 732\\ {\bf 12D99 -- Miscellaneous 733}\\ Archimedean property 733\\ complex 734\\ complex conjugate 735\\ complex number 737\\ examples of totally real fields 738\\ fundamental theorem of algebra 739\\ imaginary 739\\ imaginary unit 739\\ indeterminate form 739\\ inequalities for real numbers 740\\ interval 742\\ modulus of complex number 743\\ proof of fundamental theorem of algebra 744\\ proof of the fundamental theorem of algebra 744\\ real and complex embeddings 744\\ real number 746\\ totally real and imaginary fields 747\\ {\bf 12E05 -- Polynomials (irreducibility, etc.) 748}\\ Gauss's Lemma I 748\\ Gauss's Lemma II 749\\ discriminant 749\\ polynomial ring 751\\ resolvent 751\\ de Moivre identity 754\\ monic 754\\ Wedderburn's Theorem 754\\ proof of Wedderburn's theorem 755\\ second proof of Wedderburn's theorem 756\\ finite field 757\\ Frobenius automorphism 760\\ characteristic 761\\ characterization of field 761\\ example of an infinite field of finite characteristic 762\\ examples of fields 762\\ field 764\\ field homomorphism 764\\ prime subfield 765\\ {\bf 12F05 -- Algebraic extensions 766}\\ a finite extension of fields is an algebraic extension 766\\ algebraic closure 767\\ algebraic extension 767\\ algebraically closed 767\\ algebraically dependent 768\\ existence of the minimal polynomial 768\\ finite extension 769\\ minimal polynomial 769\\ norm 770\\ primitive element theorem 770\\ splitting field 770\\ the field extension $\mathbb {R}/\mathbb {Q}$ is not finite 771\\ trace 771\\ {\bf 12F10 -- Separable extensions, Galois theory 772}\\ Abelian extension 772\\ Fundamental Theorem of Galois Theory 772\\ Galois closure 773\\ Galois conjugate 773\\ Galois extension 773\\ Galois group 773\\ absolute Galois group 774\\ cyclic extension 774\\ example of nonperfect field 774\\ fixed field 774\\ infinite Galois theory 774\\ normal closure 776\\ normal extension 776\\ perfect field 777\\ radical extension 777\\ separable 777\\ separable closure 778\\ {\bf 12F20 -- Transcendental extensions 779}\\ transcendence degree 779\\ {\bf 12F99 -- Miscellaneous 780}\\ composite field 780\\ extension field 780\\ {\bf 12J15 -- Ordered fields 782}\\ ordered field 782\\ {\bf 13-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 783}\\ absolute value 783\\ associates 784\\ cancellation ring 784\\ comaximal 784\\ every prime ideal is radical 784\\ module 785\\ radical of an ideal 786\\ ring 786\\ subring 787\\ tensor product 787\\ {\bf 13-XX -- Commutative rings and algebras 789}\\ commutative ring 789\\ {\bf 13A02 -- Graded rings 790}\\ graded ring 790\\ {\bf 13A05 -- Divisibility 791}\\ Eisenstein criterion 791\\ {\bf 13A10 -- Radical theory 792}\\ Hilbert's Nullstellensatz 792\\ nilradical 792\\ radical of an integer 793\\ {\bf 13A15 -- Ideals; multiplicative ideal theory 794}\\ contracted ideal 794\\ existence of maximal ideals 794\\ extended ideal 795\\ fractional ideal 796\\ homogeneous ideal 797\\ ideal 797\\ maximal ideal 797\\ principal ideal 798\\ the set of prime ideals of a commutative ring with identity 798\\ {\bf 13A50 -- Actions of groups on commutative rings; invariant theory 799}\\ Schwarz (1975) theorem 799\\ invariant polynomial 800\\ {\bf 13A99 -- Miscellaneous 801}\\ Lagrange's identity 801\\ characteristic 802\\ cyclic ring 802\\ proof of Euler four-square identity 803\\ proof that every subring of a cyclic ring is a cyclic ring 804\\ proof that every subring of a cyclic ring is an ideal 804\\ zero ring 805\\ {\bf 13B02 -- Extension theory 806}\\ algebraic 806\\ module-finite 806\\ {\bf 13B05 -- Galois theory 807}\\ algebraic 807\\ {\bf 13B21 -- Integral dependence 808}\\ integral 808\\ {\bf 13B22 -- Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.) 809}\\ integral closure 809\\ {\bf 13B30 -- Quotients and localization 810}\\ fraction field 810\\ localization 810\\ multiplicative set 811\\ {\bf 13C10 -- Projective and free modules and ideals 812}\\ example of free module 812\\ {\bf 13C12 -- Torsion modules and ideals 813}\\ torsion element 813\\ {\bf 13C15 -- Dimension theory, depth, related rings (catenary, etc.) 814}\\ Krull's principal ideal theorem 814\\ {\bf 13C99 -- Miscellaneous 815}\\ Artin-Rees theorem 815\\ Nakayama's lemma 815\\ prime ideal 815\\ proof of Nakayama's lemma 816\\ proof of Nakayama's lemma 817\\ support 817\\ {\bf 13E05 -- Noetherian rings and modules 818}\\ Hilbert basis theorem 818\\ Noetherian module 818\\ proof of Hilbert basis theorem 819\\ finitely generated modules over a principal ideal domain 819\\ {\bf 13F07 -- Euclidean rings and generalizations 821}\\ Euclidean domain 821\\ Euclidean valuation 821\\ proof of Bezout's Theorem 822\\ proof that an Euclidean domain is a PID 822\\ {\bf 13F10 -- Principal ideal rings 823}\\ Smith normalform 823\\ {\bf 13F25 -- Formal power series rings 825}\\ formal power series 825\\ {\bf 13F30 -- Valuation rings 831}\\ discrete valuation 831\\ discrete valuation ring 831\\ {\bf 13G05 -- Integral domains 833}\\ Dedekind-Hasse valuation 833\\ PID 834\\ UFD 834\\ a finite integral domain is a field 835\\ an artinian integral domain is a field 835\\ example of PID 835\\ field of quotients 836\\ integral domain 836\\ irreducible 837\\ motivation for Euclidean domains 837\\ zero divisor 838\\ {\bf 13H05 -- Regular local rings 839}\\ regular local ring 839\\ {\bf 13H99 -- Miscellaneous 840}\\ local ring 840\\ semi-local ring 841\\ {\bf 13J10 -- Complete rings, completion 842}\\ completion 842\\ {\bf 13J25 -- Ordered rings 844}\\ ordered ring 844\\ {\bf 13J99 -- Miscellaneous 845}\\ topological ring 845\\ {\bf 13N15 -- Derivations 846}\\ derivation 846\\ {\bf 13P10 -- Polynomial ideals, Gr\"obner bases 847}\\ Gr\"obner basis 847\\ {\bf 14-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 849}\\ Picard group 849\\ affine space 849\\ affine variety 849\\ dual isogeny 850\\ finite morphism 850\\ isogeny 851\\ line bundle 851\\ nonsingular variety 852\\ projective space 852\\ projective variety 854\\ quasi-finite morphism 854\\ {\bf 14A10 -- Varieties and morphisms 855}\\ Zariski topology 855\\ algebraic map 856\\ algebraic sets and polynomial ideals 856\\ noetherian topological space 857\\ regular map 857\\ structure sheaf 858\\ {\bf 14A15 -- Schemes and morphisms 859}\\ closed immersion 859\\ coherent sheaf 859\\ fibre product 860\\ prime spectrum 860\\ scheme 863\\ separated scheme 864\\ singular set 864\\ {\bf 14A99 -- Miscellaneous 865}\\ Cartier divisor 865\\ General position 865\\ Serre's twisting theorem 866\\ ample 866\\ height of a prime ideal 866\\ invertible sheaf 866\\ locally free 867\\ normal irreducible varieties are nonsingular in codimension 1 867\\ sheaf of meromorphic functions 867\\ very ample 867\\ {\bf 14C20 -- Divisors, linear systems, invertible sheaves 869}\\ divisor 869\\ {\bf Rational and birational maps 870}\\ general type 870\\ {\bf 14F05 -- Vector bundles, sheaves, related constructions 871}\\ direct image (functor) 871\\ {\bf 14F20 -- \'Etale and other Grothendieck topologies and cohomologies 872}\\ site 872\\ {\bf 14F25 -- Classical real and complex cohomology 873}\\ Serre duality 873\\ sheaf cohomology 874\\ {\bf 14G05 -- Rational points 875}\\ Hasse principle 875\\ {\bf 14H37 -- Automorphisms 876}\\ Frobenius morphism 876\\ {\bf 14H45 -- Special curves and curves of low genus 878}\\ Fermat's spiral 878\\ archimedean spiral 878\\ folium of Descartes 879\\ spiral 879\\ {\bf 14H50 -- Plane and space curves 880}\\ torsion (space curve) 880\\ {\bf 14H52 -- Elliptic curves 881}\\ Birch and Swinnerton-Dyer conjecture 881\\ Hasse's bound for elliptic curves over finite fields 882\\ L-series of an elliptic curve 882\\ Mazur's theorem on torsion of elliptic curves 884\\ Mordell curve 884\\ Nagell-Lutz theorem 885\\ Selmer group 886\\ bad reduction 887\\ conductor of an elliptic curve 890\\ elliptic curve 890\\ height function 894\\ j-invariant 895\\ rank of an elliptic curve 896\\ supersingular 897\\ the torsion subgroup of an elliptic curve injects in the reduction of the curve 897\\ {\bf 14H99 -- Miscellaneous 900}\\ Riemann-Roch theorem 900\\ genus 900\\ projective curve 901\\ proof of Riemann-Roch theorem 901\\ {\bf 14L17 -- Affine algebraic groups, hyperalgebra constructions 902}\\ affine algebraic group 902\\ algebraic torus 902\\ {\bf 14M05 -- Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 903}\\ normal 903\\ {\bf 14M15 -- Grassmannians, Schubert varieties, flag manifolds 904}\\ Borel-Bott-Weil theorem 904\\ flag variety 905\\ {\bf 14R15 -- Jacobian problem 906}\\ Jacobian conjecture 906\\ {\bf 15-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 907}\\ Cholesky decomposition 907\\ Hadamard matrix 908\\ Hessenberg matrix 909\\ If $A \in M_n(k)$ and $A$ is supertriangular then $A^n=0$ 910\\ Jacobi determinant 910\\ Jacobi's Theorem 912\\ Kronecker product 912\\ LU decomposition 913\\ Peetre's inequality 914\\ Schur decomposition 915\\ antipodal 916\\ conjugate transpose 916\\ corollary of Schur decomposition 917\\ covector 918\\ diagonal matrix 918\\ diagonalization 920\\ diagonally dominant matrix 920\\ eigenvalue (of a matrix) 921\\ eigenvalue problem 922\\ eigenvalues of orthogonal matrices 924\\ eigenvector 925\\ exactly determined 926\\ free vector space over a set 926\\ in a vector space, $\lambda v = 0$ if and only if $\lambda =0$ or $v$ is the zero vector 928\\ invariant subspace 929\\ least squares 929\\ linear algebra 930\\ linear least squares 932\\ linear manifold 934\\ matrix exponential 934\\ matrix operations 935\\ nilpotent matrix 938\\ nilpotent transformation 938\\ non-zero vector 939\\ off-diagonal entry 940\\ orthogonal matrices 940\\ orthogonal vectors 941\\ overdetermined 941\\ partitioned matrix 941\\ pentadiagonal matrix 942\\ proof of Cayley-Hamilton theorem 942\\ proof of Schur decomposition 943\\ singular value decomposition 944\\ skew-symmetric matrix 945\\ square matrix 946\\ strictly upper triangular matrix 946\\ symmetric matrix 947\\ theorem for normal triangular matrices 947\\ triangular matrix 948\\ tridiagonal matrix 949\\ under determined 950\\ unit triangular matrix 950\\ unitary 951\\ vector space 952\\ vector subspace 953\\ zero map 954\\ zero vector in a vector space is unique 955\\ zero vector space 955\\ {\bf 15-01 -- Instructional exposition (textbooks, tutorial papers, etc.) 956}\\ circulant matrix 956\\ matrix 957\\ {\bf 15-XX -- Linear and multilinear algebra; matrix theory 960}\\ linearly dependent functions 960\\ {\bf 15A03 -- Vector spaces, linear dependence, rank 961}\\ Sylvester's law 961\\ basis 961\\ complementary subspace 962\\ dimension 963\\ every vector space has a basis 964\\ flag 964\\ frame 965\\ linear combination 968\\ linear independence 968\\ list vector 968\\ nullity 969\\ orthonormal basis 970\\ physical vector 970\\ proof of rank-nullity theorem 972\\ rank 973\\ rank-nullity theorem 973\\ similar matrix 974\\ span 975\\ theorem for the direct sum of finite dimensional vector spaces 976\\ vector 976\\ {\bf 15A04 -- Linear transformations, semilinear transformations 980}\\ admissibility 980\\ conductor of a vector 980\\ cyclic decomposition theorem 981\\ cyclic subspace 981\\ dimension theorem for symplectic complement (proof) 981\\ dual homomorphism 982\\ dual homomorphism of the derivative 983\\ image of a linear transformation 984\\ invertible linear transformation 984\\ kernel of a linear transformation 985\\ linear transformation 985\\ minimal polynomial (endomorphism) 986\\ symplectic complement 987\\ trace 988\\ {\bf 15A06 -- Linear equations 989}\\ Gaussian elimination 989\\ finite-dimensional linear problem 991\\ homogeneous linear problem 992\\ linear problem 993\\ reduced row echelon form 993\\ row echelon form 994\\ under-determined polynomial interpolation 994\\ {\bf 15A09 -- Matrix inversion, generalized inverses 996}\\ matrix adjoint 996\\ matrix inverse 997\\ {\bf 15A12 -- Conditioning of matrices 1000}\\ singular 1000\\ {\bf 15A15 -- Determinants, permanents, other special matrix functions 1001}\\ Cayley-Hamilton theorem 1001\\ Cramer's rule 1001\\ cofactor expansion 1002\\ determinant 1003\\ determinant as a multilinear mapping 1005\\ determinants of some matrices of special form 1006\\ example of Cramer's rule 1006\\ proof of Cramer's rule 1008\\ proof of cofactor expansion 1008\\ resolvent matrix 1009\\ {\bf 15A18 -- Eigenvalues, singular values, and eigenvectors 1010}\\ Jordan canonical form theorem 1010\\ Lagrange multiplier method 1011\\ Perron-Frobenius theorem 1011\\ characteristic equation 1012\\ eigenvalue 1012\\ eigenvalue 1013\\ {\bf 15A21 -- Canonical forms, reductions, classification 1015}\\ companion matrix 1015\\ eigenvalues of an involution 1015\\ linear involution 1016\\ normal matrix 1017\\ projection 1018\\ quadratic form 1019\\ {\bf 15A23 -- Factorization of matrices 1021}\\ QR decomposition 1021\\ {\bf 15A30 -- Algebraic systems of matrices 1023}\\ ideals in matrix algebras 1023\\ {\bf 15A36 -- Matrices of integers 1025}\\ permutation matrix 1025\\ {\bf 15A39 -- Linear inequalities 1026}\\ Farkas lemma 1026\\ {\bf 15A42 -- Inequalities involving eigenvalues and eigenvectors 1027}\\ Gershgorin's circle theorem 1027\\ Gershgorin's circle theorem result 1027\\ Shur's inequality 1028\\ {\bf 15A48 -- Positive matrices and their generalizations; cones of matrices 1029}\\ negative definite 1029\\ negative semidefinite 1029\\ positive definite 1030\\ positive semidefinite 1030\\ primitive matrix 1031\\ reducible matrix 1031\\ {\bf 15A51 -- Stochastic matrices 1032}\\ Birkoff-von Neumann theorem 1032\\ proof of Birkoff-von Neumann theorem 1032\\ {\bf 15A57 -- Other types of matrices (Hermitian, skew-Hermitian, etc.) 1035}\\ Hermitian matrix 1035\\ direct sum of Hermitian and skew-Hermitian matrices 1036\\ identity matrix 1037\\ skew-Hermitian matrix 1037\\ transpose 1038\\ {\bf 15A60 -- Norms of matrices, numerical range, applications of functional analysis to matrix theory 1041}\\ Frobenius matrix norm 1041\\ matrix p-norm 1042\\ self consistent matrix norm 1043\\ {\bf 15A63 -- Quadratic and bilinear forms, inner products 1044}\\ Cauchy-Schwarz inequality 1044\\ adjoint endomorphism 1045\\ anti-symmetric 1046\\ bilinear map 1046\\ dot product 1049\\ every orthonormal set is linearly independent 1050\\ inner product 1051\\ inner product space 1051\\ proof of Cauchy-Schwarz inequality 1052\\ self-dual 1052\\ skew-symmetric bilinear form 1053\\ spectral theorem 1053\\ {\bf 15A66 -- Clifford algebras, spinors 1056}\\ geometric algebra 1056\\ {\bf 15A69 -- Multilinear algebra, tensor products 1058}\\ Einstein summation convention 1058\\ basic tensor 1059\\ multi-linear 1061\\ outer multiplication 1061\\ tensor 1062\\ tensor algebra 1065\\ tensor array 1065\\ tensor product (vector spaces) 1067\\ tensor transformations 1069\\ {\bf 15A72 -- Vector and tensor algebra, theory of invariants 1072}\\ bac-cab rule 1072\\ cross product 1072\\ euclidean vector 1073\\ rotational invariance of cross product 1074\\ {\bf 15A75 -- Exterior algebra, Grassmann algebras 1076}\\ contraction 1076\\ exterior algebra 1077\\ {\bf 15A99 -- Miscellaneous topics 1081}\\ Kronecker delta 1081\\ dual space 1081\\ example of trace of a matrix 1083\\ generalized Kronecker delta symbol 1083\\ linear functional 1084\\ modules are a generalization of vector spaces 1084\\ proof of properties of trace of a matrix 1085\\ quasipositive matrix 1086\\ trace of a matrix 1086\\ \ \\ {\it Volume 2} \\ \ \\ {\bf 16-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1088}\\ direct product of modules 1088\\ direct sum 1089\\ exact sequence 1089\\ quotient ring 1090\\ {\bf 16D10 -- General module theory 1091}\\ annihilator 1091\\ annihilator is an ideal 1091\\ artinian 1092\\ composition series 1092\\ conjugate module 1093\\ modular law 1093\\ module 1093\\ proof of modular law 1094\\ zero module 1094\\ {\bf 16D20 -- Bimodules 1095}\\ bimodule 1095\\ {\bf 16D25 -- Ideals 1096}\\ associated prime 1096\\ nilpotent ideal 1096\\ primitive ideal 1096\\ product of ideals 1097\\ proper ideal 1097\\ semiprime ideal 1097\\ zero ideal 1098\\ {\bf 16D40 -- Free, projective, and flat modules and ideals 1099}\\ finitely generated projective module 1099\\ flat module 1099\\ free module 1100\\ free module 1100\\ projective cover 1100\\ projective module 1101\\ {\bf 16D50 -- Injective modules, self-injective rings 1102}\\ injective hull 1102\\ injective module 1102\\ {\bf 16D60 -- Simple and semisimple modules, primitive rings and ideals 1104}\\ central simple algebra 1104\\ completely reducible 1104\\ simple ring 1105\\ {\bf 16D80 -- Other classes of modules and ideals 1106}\\ essential submodule 1106\\ faithful module 1106\\ minimal prime ideal 1107\\ module of finite rank 1107\\ simple module 1107\\ superfluous submodule 1107\\ uniform module 1108\\ {\bf 16E05 -- Syzygies, resolutions, complexes 1109}\\ $n$-chain 1109\\ chain complex 1109\\ flat resolution 1110\\ free resolution 1110\\ injective resolution 1110\\ projective resolution 1110\\ short exact sequence 1111\\ split short exact sequence 1111\\ von Neumann regular 1111\\ {\bf 16K20 -- Finite-dimensional 1112}\\ quaternion algebra 1112\\ {\bf 16K50 -- Brauer groups 1113}\\ Brauer group 1113\\ {\bf 16K99 -- Miscellaneous 1114}\\ division ring 1114\\ {\bf 16N20 -- Jacobson radical, quasimultiplication 1115}\\ Jacobson radical 1115\\ a ring modulo its Jacobson radical is semiprimitive 1116\\ examples of semiprimitive rings 1116\\ proof of Characterizations of the Jacobson radical 1117\\ properties of the Jacobson radical 1118\\ quasi-regularity 1119\\ semiprimitive ring 1120\\ {\bf 16N40 -- Nil and nilpotent radicals, sets, ideals, rings 1121}\\ Koethe conjecture 1121\\ nil and nilpotent ideals 1121\\ {\bf 16N60 -- Prime and semiprime rings 1123}\\ prime ring 1123\\ {\bf 16N80 -- General radicals and rings 1124}\\ prime radical 1124\\ radical theory 1124\\ {\bf 16P40 -- Noetherian rings and modules 1126}\\ Noetherian ring 1126\\ noetherian 1126\\ {\bf 16P60 -- Chain conditions on annihilators and summands: Goldie-type conditions , Krull dimension 1128}\\ Goldie ring 1128\\ uniform dimension 1128\\ {\bf 16S10 -- Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 1130}\\ Ore domain 1130\\ {\bf 16S34 -- Group rings , Laurent polynomial rings 1131}\\ support 1131\\ {\bf 16S36 -- Ordinary and skew polynomial rings and semigroup rings 1132}\\ Gaussian polynomials 1132\\ q skew derivation 1133\\ q skew polynomial ring 1133\\ sigma derivation 1133\\ sigma, delta constant 1133\\ skew derivation 1133\\ skew polynomial ring 1134\\ {\bf 16S99 -- Miscellaneous 1135}\\ algebra 1135\\ algebra (module) 1135\\ {\bf 16U10 -- Integral domains 1137}\\ Pr\"ufer domain 1137\\ valuation domain 1137\\ {\bf 16U20 -- Ore rings, multiplicative sets, Ore localization 1139}\\ Goldie's Theorem 1139\\ Ore condition 1139\\ Ore's theorem 1140\\ classical ring of quotients 1140\\ saturated 1141\\ {\bf 16U70 -- Center, normalizer (invariant elements) 1142}\\ center (rings) 1142\\ {\bf 16U99 -- Miscellaneous 1143}\\ anti-idempotent 1143\\ {\bf 16W20 -- Automorphisms and endomorphisms 1144}\\ ring of endomorphisms 1144\\ {\bf 16W30 -- Coalgebras, bialgebras, Hopf algebras ; rings, modules, etc. on which these act 1146}\\ Hopf algebra 1146\\ almost cocommutative bialgebra 1147\\ bialgebra 1148\\ coalgebra 1148\\ coinvariant 1149\\ comodule 1149\\ comodule algebra 1149\\ comodule coalgebra 1150\\ module algebra 1150\\ module coalgebra 1150\\ {\bf 16W50 -- Graded rings and modules 1151}\\ graded algebra 1151\\ graded module 1151\\ supercommutative 1151\\ {\bf 16W55 -- ``Super'' (or ``skew'') structure 1153}\\ super tensor product 1153\\ superalgebra 1153\\ supernumber 1154\\ {\bf 16W99 -- Miscellaneous 1155}\\ Hamiltonian quaternions 1155\\ {\bf 16Y30 -- Near-rings 1158}\\ near-ring 1158\\ {\bf 17A01 -- General theory 1159}\\ commutator bracket 1159\\ {\bf 17B05 -- Structure theory 1161}\\ Killing form 1161\\ Levi's theorem 1161\\ nilradical 1161\\ radical 1162\\ {\bf 17B10 -- Representations, algebraic theory (weights) 1163}\\ Ado's theorem 1163\\ Lie algebra representation 1163\\ adjoint representation 1164\\ examples of non-matrix Lie groups 1165\\ isotropy representation 1165\\ {\bf 17B15 -- Representations, analytic theory 1166}\\ invariant form (Lie algebras) 1166\\ {\bf 17B20 -- Simple, semisimple, reductive (super)algebras (roots) 1167}\\ Borel subalgebra 1167\\ Borel subgroup 1167\\ Cartan matrix 1168\\ Cartan subalgebra 1168\\ Cartan's criterion 1168\\ Casimir operator 1168\\ Dynkin diagram 1169\\ Verma module 1169\\ Weyl chamber 1170\\ Weyl group 1170\\ Weyl's theorem 1170\\ classification of finite-dimensional representations of semi-simple Lie algebras 1171\\ cohomology of semi-simple Lie algebras 1171\\ nilpotent cone 1171\\ parabolic subgroup 1172\\ pictures of Dynkin diagrams 1172\\ positive root 1175\\ rank 1175\\ root lattice 1175\\ root system 1176\\ simple and semi-simple Lie algebras 1177\\ simple root 1178\\ weight (Lie algebras) 1178\\ weight lattice 1178\\ {\bf 17B30 -- Solvable, nilpotent (super)algebras 1179}\\ Engel's theorem 1179\\ Lie's theorem 1182\\ solvable Lie algebra 1183\\ {\bf 17B35 -- Universal enveloping (super)algebras 1184}\\ Poincar\'e-Birkhoff-Witt theorem 1184\\ universal enveloping algebra 1185\\ {\bf 17B56 -- Cohomology of Lie (super)algebras 1187}\\ Lie algebra cohomology 1187\\ {\bf 17B67 -- Kac-Moody (super)algebras (structure and representation theory) 1188}\\ Kac-Moody algebra 1188\\ generalized Cartan matrix 1188\\ {\bf 17B99 -- Miscellaneous 1190}\\ Jacobi identity interpretations 1190\\ Lie algebra 1190\\ real form 1192\\ {\bf 18-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1193}\\ Grothendieck spectral sequence 1193\\ category of sets 1194\\ functor 1194\\ monic 1194\\ natural equivalence 1195\\ representable functor 1195\\ supplemental axioms for an Abelian category 1195\\ {\bf 18A05 -- Definitions, generalizations 1197}\\ autofunctor 1197\\ automorphism 1197\\ category 1198\\ category example (arrow category) 1199\\ commutative diagram 1199\\ double dual embedding 1200\\ dual category 1201\\ duality principle 1201\\ endofunctor 1202\\ examples of initial objects, terminal objects and zero objects 1202\\ forgetful functor 1204\\ isomorphism 1205\\ natural transformation 1205\\ types of homomorphisms 1205\\ zero object 1206\\ {\bf 18A22 -- Special properties of functors (faithful, full, etc.) 1208}\\ exact functor 1208\\ {\bf 18A25 -- Functor categories, comma categories 1210}\\ Yoneda embedding 1210\\ {\bf 18A30 -- Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 1211}\\ categorical direct product 1211\\ categorical direct sum 1211\\ kernel 1212\\ {\bf 18A40 -- Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 1213}\\ adjoint functor 1213\\ equivalence of categories 1214\\ {\bf 18B40 -- Groupoids, semigroupoids, semigroups, groups (viewed as categories) 1215}\\ groupoid (category theoretic) 1215\\ {\bf 18E10 -- Exact categories, abelian categories 1216}\\ abelian category 1216\\ exact sequence 1217\\ derived category 1218\\ enough injectives 1218\\ {\bf 18F20 -- Presheaves and sheaves 1219}\\ locally ringed space 1219\\ presheaf 1220\\ sheaf 1220\\ sheafification 1225\\ stalk 1226\\ {\bf 18F30 -- Grothendieck groups 1228}\\ Grothendieck group 1228\\ {\bf 18G10 -- Resolutions; derived functors 1229}\\ derived functor 1229\\ {\bf 18G15 -- Ext and Tor, generalizations, K\"unneth formula 1231}\\ Ext 1231\\ {\bf 18G30 -- Simplicial sets, simplicial objects (in a category) 1232}\\ nerve 1232\\ simplicial category 1232\\ simplicial object 1233\\ {\bf 18G35 -- Chain complexes 1235}\\ 5-lemma 1235\\ 9-lemma 1236\\ Snake lemma 1236\\ chain homotopy 1237\\ chain map 1237\\ homology (chain complex) 1237\\ {\bf 18G40 -- Spectral sequences, hypercohomology 1238}\\ spectral sequence 1238\\ {\bf 19-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1239}\\ Algebraic K-theory 1239\\ K-theory 1240\\ examples of algebraic K-theory groups 1241\\ {\bf 19K33 -- EXT and $K$-homology 1242}\\ Fredholm module 1242\\ K-homology 1243\\ {\bf 19K99 -- Miscellaneous 1244}\\ examples of K-theory groups 1244\\ {\bf 20-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1245}\\ alternating group is a normal subgroup of the symmetric group 1245\\ associative 1245\\ canonical projection 1246\\ centralizer 1246\\ commutative 1247\\ examples of groups 1247\\ group 1250\\ quotient group 1250\\ {\bf 20-02 -- Research exposition (monographs, survey articles) 1252}\\ length function 1252\\ {\bf 20-XX -- Group theory and generalizations 1253}\\ free product with amalgamated subgroup 1253\\ nonabelian group 1254\\ {\bf 20A05 -- Axiomatics and elementary properties 1255}\\ Feit-Thompson theorem 1255\\ Proof: The orbit of any element of a group is a subgroup 1255\\ center 1256\\ characteristic subgroup 1256\\ class function 1257\\ conjugacy class 1258\\ conjugacy class formula 1258\\ conjugate stabilizer subgroups 1258\\ coset 1259\\ cyclic group 1259\\ derived subgroup 1260\\ equivariant 1260\\ examples of finite simple groups 1261\\ finitely generated group 1262\\ first isomorphism theorem 1262\\ fourth isomorphism theorem 1262\\ generator 1263\\ group actions and homomorphisms 1263\\ group homomorphism 1265\\ homogeneous space 1265\\ identity element 1268\\ inner automorphism 1268\\ kernel 1269\\ maximal 1269\\ normal subgroup 1269\\ normality of subgroups is not transitive 1269\\ normalizer 1270\\ order (of a group) 1271\\ presentation of a group 1271\\ proof of first isomorphism theorem 1272\\ proof of second isomorphism theorem 1273\\ proof that all cyclic groups are abelian 1274\\ proof that all cyclic groups of the same order are isomorphic to each other 1274\\ proof that all subgroups of a cyclic group are cyclic 1274\\ regular group action 1275\\ second isomorphism theorem 1275\\ simple group 1276\\ solvable group 1276\\ subgroup 1276\\ third isomorphism theorem 1277\\ {\bf 20A99 -- Miscellaneous 1279}\\ Cayley table 1279\\ proper subgroup 1280\\ quaternion group 1280\\ {\bf 20B05 -- General theory for finite groups 1282}\\ cycle notation 1282\\ permutation group 1283\\ {\bf 20B15 -- Primitive groups 1284}\\ primitive transitive permutation group 1284\\ {\bf 20B20 -- Multiply transitive finite groups 1286}\\ Jordan's theorem (multiply transitive groups) 1286\\ multiply transitive 1286\\ sharply multiply transitive 1287\\ {\bf 20B25 -- Finite automorphism groups of algebraic, geometric, or combinatorial structures 1288}\\ diamond theory 1288\\ {\bf 20B30 -- Symmetric groups 1289}\\ symmetric group 1289\\ symmetric group 1289\\ {\bf 20B35 -- Subgroups of symmetric groups 1290}\\ Cayley's theorem 1290\\ {\bf 20B99 -- Miscellaneous 1291}\\ $(p,q)$ shuffle 1291\\ Frobenius group 1291\\ permutation 1292\\ proof of Cayley's theorem 1292\\ {\bf 20C05 -- Group rings of finite groups and their modules 1294}\\ group ring 1294\\ {\bf 20C15 -- Ordinary representations and characters 1295}\\ Maschke's theorem 1295\\ a representation which is not completely reducible 1295\\ orthogonality relations 1296\\ {\bf 20C30 -- Representations of finite symmetric groups 1299}\\ example of immanent 1299\\ immanent 1299\\ permanent 1299\\ {\bf 20C99 -- Miscellaneous 1301}\\ Frobenius reciprocity 1301\\ Schur's lemma 1301\\ character 1302\\ group representation 1303\\ induced representation 1303\\ regular representation 1304\\ restriction representation 1304\\ {\bf 20D05 -- Classification of simple and nonsolvable groups 1305}\\ Burnside $p-q$ theorem 1305\\ classification of semisimple groups 1305\\ semisimple group 1305\\ {\bf 20D08 -- Simple groups: sporadic groups 1307}\\ Janko groups 1307\\ {\bf 20D10 -- Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi $-length, ranks 1308}\\ \v {C}uhinin's Theorem 1308\\ separable 1308\\ supersolvable group 1309\\ {\bf 20D15 -- Nilpotent groups, $p$-groups 1310}\\ Burnside basis theorem 1310\\ {\bf 20D20 -- Sylow subgroups, Sylow properties, $\pi $-groups, $\pi $-structure 1311}\\ $\pi $-groups and $\pi '$-groups 1311\\ $p$-subgroup 1311\\ Burnside normal complement theorem 1312\\ Frattini argument 1312\\ Sylow p-subgroup 1312\\ Sylow theorems 1312\\ Sylow's first theorem 1313\\ Sylow's third theorem 1313\\ application of Sylow's theorems to groups of order pq 1313\\ p-primary component 1314\\ proof of Frattini argument 1314\\ proof of Sylow theorems 1314\\ subgroups containing the normalizers of Sylow subgroups normalize themselves 1316\\ {\bf 20D25 -- Special subgroups (Frattini, Fitting, etc.) 1317}\\ Fitting's theorem 1317\\ characteristically simple group 1317\\ the Frattini subgroup is nilpotent 1317\\ {\bf 20D30 -- Series and lattices of subgroups 1319}\\ maximal condition 1319\\ minimal condition 1319\\ subnormal series 1320\\ {\bf 20D35 -- Subnormal subgroups 1321}\\ subnormal subgroup 1321\\ {\bf 20D99 -- Miscellaneous 1322}\\ Cauchy's theorem 1322\\ Lagrange's theorem 1322\\ exponent 1322\\ fully invariant subgroup 1323\\ proof of Cauchy's theorem 1323\\ proof of Lagrange's theorem 1324\\ proof of the converse of Lagrange's theorem for finite cyclic groups 1324\\ proof that $\mathbf {exp}G$ divides $|G|$ 1324\\ proof that $|g|$ divides $\mathbf {exp}G$ 1325\\ proof that every group of prime order is cyclic 1325\\ {\bf 20E05 -- Free nonabelian groups 1326}\\ Nielsen-Schreier theorem 1326\\ Scheier index formula 1326\\ free group 1326\\ proof of Nielsen-Schreier theorem and Schreier index formula 1327\\ Jordan-Holder decomposition 1328\\ profinite group 1328\\ extension 1329\\ holomorph 1329\\ proof of the Jordan Holder decomposition theorem 1329\\ semidirect product of groups 1330\\ wreath product 1333\\ Jordan-H^^c3^^b6lder decomposition theorem 1334\\ simplicity of the alternating groups 1334\\ abelian groups of order $120$ 1337\\ fundamental theorem of finitely generated abelian groups 1337\\ conjugacy class 1338\\ Frattini subgroup 1338\\ non-generator 1338\\ {\bf 20Exx -- Structure and classification of infinite or finite groups 1339}\\ faithful group action 1339\\ {\bf 20F18 -- Nilpotent groups 1340}\\ classification of finite nilpotent groups 1340\\ nilpotent group 1340\\ {\bf 20F22 -- Other classes of groups defined by subgroup chains 1342}\\ inverse limit 1342\\ {\bf 20F28 -- Automorphism groups of groups 1344}\\ outer automorphism group 1344\\ {\bf 20F36 -- Braid groups; Artin groups 1345}\\ braid group 1345\\ {\bf 20F55 -- Reflection and Coxeter groups 1347}\\ cycle 1347\\ dihedral group 1348\\ {\bf 20F65 -- Geometric group theory 1349}\\ groups that act freely on trees are free 1349\\ {\bf 20F99 -- Miscellaneous 1350}\\ perfect group 1350\\ {\bf 20G15 -- Linear algebraic groups over arbitrary fields 1351}\\ Nagao's theorem 1351\\ computation of the order of $GL(n, F_q)$ 1351\\ general linear group 1352\\ order of the general linear group over a finite field 1352\\ special linear group 1352\\ {\bf 20G20 -- Linear algebraic groups over the reals, the complexes, the quaternions 1353}\\ orthogonal group 1353\\ {\bf 20G25 -- Linear algebraic groups over local fields and their integers 1354}\\ Ihara's theorem 1354\\ {\bf 20G40 -- Linear algebraic groups over finite fields 1355}\\ $\mathit {SL}_2(F_3)$ 1355\\ {\bf 20J06 -- Cohomology of groups 1356}\\ group cohomology 1356\\ stronger Hilbert theorem 90 1357\\ {\bf 20J15 -- Category of groups 1359}\\ variety of groups 1359\\ {\bf 20K01 -- Finite abelian groups 1360}\\ Schinzel's theorem 1360\\ {\bf 20K10 -- Torsion groups, primary groups and generalized primary groups 1361}\\ torsion 1361\\ {\bf 20K25 -- Direct sums, direct products, etc. 1362}\\ direct product of groups 1362\\ {\bf 20K99 -- Miscellaneous 1363}\\ Klein 4-group 1363\\ divisible group 1364\\ example of divisible group 1364\\ locally cyclic group 1364\\ {\bf 20Kxx -- Abelian groups 1366}\\ abelian group 1366\\ {\bf 20M10 -- General structure theory 1367}\\ existence of maximal semilattice decomposition 1367\\ semilattice decomposition of a semigroup 1368\\ simple semigroup 1368\\ {\bf 20M12 -- Ideal theory 1370}\\ Rees factor 1370\\ ideal 1370\\ {\bf 20M14 -- Commutative semigroups 1372}\\ Archimedean semigroup 1372\\ commutative semigroup 1372\\ {\bf 20M20 -- Semigroups of transformations, etc. 1373}\\ semigroup of transformations 1373\\ {\bf 20M30 -- Representation of semigroups; actions of semigroups on sets 1375}\\ counting theorem 1375\\ example of group action 1375\\ group action 1376\\ orbit 1377\\ proof of counting theorem 1377\\ stabilizer 1378\\ {\bf 20M99 -- Miscellaneous 1379}\\ a semilattice is a commutative band 1379\\ adjoining an identity to a semigroup 1379\\ band 1380\\ bicyclic semigroup 1380\\ congruence 1381\\ cyclic semigroup 1381\\ idempotent 1382\\ null semigroup 1383\\ semigroup 1383\\ semilattice 1383\\ subsemigroup,, submonoid,, and subgroup 1384\\ zero elements 1384\\ {\bf 20N02 -- Sets with a single binary operation (groupoids) 1386}\\ groupoid 1386\\ idempotency 1386\\ left identity and right identity 1387\\ {\bf 20N05 -- Loops, quasigroups 1388}\\ Moufang loop 1388\\ loop and quasigroup 1389\\ {\bf 22-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1390}\\ fixed-point subspace 1390\\ {\bf 22-XX -- Topological groups, Lie groups 1391}\\ Cantor space 1391\\ {\bf 22A05 -- Structure of general topological groups 1392}\\ topological group 1392\\ {\bf 22C05 -- Compact groups 1393}\\ $n$-torus 1393\\ reductive 1393\\ {\bf 22D05 -- General properties and structure of locally compact groups 1394}\\ $\Gamma $-simple 1394\\ {\bf 22D15 -- Group algebras of locally compact groups 1395}\\ group $C^*$-algebra 1395\\ {\bf 22E10 -- General properties and structure of complex Lie groups 1396}\\ existence and uniqueness of compact real form 1396\\ maximal torus 1397\\ Lie group 1397\\ complexification 1399\\ Hilbert-Weyl theorem 1400\\ the connection between Lie groups and Lie algebras 1401\\ {\bf 26-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.) 1402}\\ derivative notation 1402\\ fundamental theorems of calculus 1403\\ logarithm 1404\\ proof of the first fundamental theorem of calculus 1405\\ proof of the second fundamental theorem of calculus 1405\\ root-mean-square 1406\\ square 1406\\ {\bf 26-XX -- Real functions 1408}\\ abelian function 1408\\ full-width at half maximum 1408\\ {\bf 26A03 -- Foundations: limits and generalizations, elementary topology of the line 1410}\\ Cauchy sequence 1410\\ Dedekind cuts 1410\\ binomial proof of positive integer power rule 1413\\ exponential 1414\\ interleave sequence 1415\\ limit inferior 1415\\ limit superior 1416\\ power rule 1417\\ properties of the exponential 1417\\ squeeze rule 1418\\ {\bf 26A06 -- One-variable calculus 1420}\\ Darboux's theorem (analysis) 1420\\ Fermat's Theorem (stationary points) 1420\\ Heaviside step function 1421\\ Leibniz' rule 1421\\ Rolle's theorem 1422\\ binomial formula 1422\\ chain rule 1422\\ complex Rolle's theorem 1423\\ complex mean-value theorem 1423\\ definite integral 1424\\ derivative of even/odd function (proof) 1425\\ direct sum of even/odd functions (example) 1425\\ even/odd function 1426\\ example of chain rule 1427\\ example of increasing/decreasing/monotone function 1428\\ extended mean-value theorem 1428\\ increasing/decreasing/monotone function 1428\\ intermediate value theorem 1429\\ limit 1429\\ mean value theorem 1430\\ mean-value theorem 1430\\ monotonicity criterion 1431\\ nabla 1431\\ one-sided limit 1432\\ product rule 1432\\ proof of Darboux's theorem 1433\\ proof of Fermat's Theorem (stationary points) 1434\\ proof of Rolle's theorem 1434\\ proof of Taylor's Theorem 1435\\ proof of binomial formula 1436\\ proof of chain rule 1436\\ proof of extended mean-value theorem 1437\\ proof of intermediate value theorem 1437\\ proof of mean value theorem 1438\\ proof of monotonicity criterion 1439\\ proof of quotient rule 1439\\ quotient rule 1440\\ signum function 1440\\ {\bf 26A09 -- Elementary functions 1443}\\ definitions in trigonometry 1443\\ hyperbolic functions 1444\\ {\bf 26A12 -- Rate of growth of functions, orders of infinity, slowly varying functions 1446}\\ Landau notation 1446\\ {\bf 26A15 -- Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) 1448}\\ Dirichlet's function 1448\\ semi-continuous 1448\\ semicontinuous 1449\\ uniformly continuous 1450\\ {\bf 26A16 -- Lipschitz (H\"older) classes 1451}\\ Lipschitz condition 1451\\ Lipschitz condition and differentiability 1452\\ Lipschitz condition and differentiability result 1453\\ {\bf 26A18 -- Iteration 1454}\\ iteration 1454\\ periodic point 1454\\ {\bf 26A