\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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\tableofcontents \thispagestyle{empty} \begin{quote} {\huge Free Encyclopedia of Mathematics (0.0.1) -- volume 2} \end{quote} \clearpage \setcounter{page}{1088} \setcounter{chapter}{241} % must be one less than what we want the next chapter to be \chapter{16-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{direct product of modules} \input{DirectProduct.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{direct sum} \input{DirectSum.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{exact sequence} \input{ExactSequence.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{quotient ring} \input{QuotientRing.tex} Version: 3 Owner: matte Author(s): matte, djao \chapter{16D10 -- General module theory} \section{annihilator} \input{Annihilator.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{annihilator is an ideal} \input{AnnihilatorIsAnIdeal.tex} Version: 2 Owner: saforres Author(s): saforres \section{artinian} \input{Artinian.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{composition series} \input{CompositionSeries3.tex} Version: 1 Owner: mclase Author(s): mclase \section{conjugate module} \input{ConjugateModule.tex} Version: 4 Owner: antizeus Author(s): antizeus \section{modular law} \input{ModularLaw.tex} Version: 1 Owner: saforres Author(s): saforres \section{module} \input{FancyDefinitionOfModule.tex} Version: 7 Owner: antizeus Author(s): antizeus \section{proof of modular law} \input{ProofOfModularLaw.tex} Version: 5 Owner: saforres Author(s): saforres \section{zero module} \input{ZeroModule.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{16D20 -- Bimodules} \section{bimodule} \input{Bimodule.tex} Version: 3 Owner: antizeus Author(s): antizeus \chapter{16D25 -- Ideals} \section{associated prime} \input{AssociatedPrime.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{nilpotent ideal} \input{NilpotentIdeal.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{primitive ideal} \input{PrimitiveIdeal.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{product of ideals} \input{ProductOfIdeals.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{proper ideal} \input{ProperIdeal.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{semiprime ideal} \input{SemiprimeIdeal.tex} Version: 7 Owner: antizeus Author(s): antizeus \section{zero ideal} \input{ZeroIdeal.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{16D40 -- Free, projective, and flat modules and ideals} \section{finitely generated projective module} \input{FinitelyGeneratedProjectiveModule.tex} Version: 3 Owner: mhale Author(s): mhale \section{flat module} \input{FlatModule.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{free module} \input{FreeModule.tex} Version: 4 Owner: mathcam Author(s): mathcam, antizeus \section{free module} \input{FreeModule3.tex} Version: 1 Owner: antizeus Author(s): antizeus \section{projective cover} \input{ProjectiveCover.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{projective module} \input{ProjectiveModule.tex} Version: 3 Owner: antizeus Author(s): antizeus \chapter{16D50 -- Injective modules, self-injective rings} \section{injective hull} \input{InjectiveHull.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{injective module} \input{InjectiveModule.tex} Version: 3 Owner: antizeus Author(s): antizeus \chapter{16D60 -- Simple and semisimple modules, primitive rings and ideals} \section{central simple algebra} \input{CentralSimpleAlgebra.tex} Version: 2 Owner: djao Author(s): djao \section{completely reducible} \input{CompletelyReducible.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{simple ring} \input{SimpleRing.tex} Version: 4 Owner: antizeus Author(s): antizeus \chapter{16D80 -- Other classes of modules and ideals} \section{essential submodule} \input{EssentialSubmodule.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{faithful module} \input{FaithfulModule.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{minimal prime ideal} \input{MinimalPrimeIdeal.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{module of finite rank} \input{ModuleOfFiniteRank.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{simple module} \input{SimpleModule.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{superfluous submodule} \input{SuperfluousSubmodule.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{uniform module} \input{UniformModule.tex} Version: 3 Owner: antizeus Author(s): antizeus \chapter{16E05 -- Syzygies, resolutions, complexes} \section{$n$-chain} \input{NChain.tex} Version: 3 Owner: mathcam Author(s): mathcam \section{chain complex} \input{ChainComplex.tex} Version: 4 Owner: antizeus Author(s): antizeus \section{flat resolution} \input{FlatResolution.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{free resolution} \input{FreeResolution.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{injective resolution} \input{InjectiveResolution.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{projective resolution} \input{ProjectiveResolution.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{short exact sequence} \input{ShortExactSequence.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{split short exact sequence} \input{SplitShortExactSequence.tex} Version: 4 Owner: antizeus Author(s): antizeus \section{von Neumann regular} \input{VonNeumannRegular.tex} Version: 1 Owner: igor Author(s): igor \chapter{16K20 -- Finite-dimensional} \section{quaternion algebra} \input{QuaternionAlgebra.tex} Version: 1 Owner: djao Author(s): djao \chapter{16K50 -- Brauer groups} \section{Brauer group} \input{BrauerGroup.tex} Version: 5 Owner: djao Author(s): djao \chapter{16K99 -- Miscellaneous} \section{division ring} \input{DivisionRing.tex} Version: 3 Owner: djao Author(s): djao \chapter{16N20 -- Jacobson radical, quasimultiplication} \section{Jacobson radical} \input{JacobsonRadical.tex} Version: 13 Owner: saforres Author(s): saforres \section{a ring modulo its Jacobson radical is semiprimitive} \input{ARingModuloItsJacobsonRadicalIsSemiprimitive.tex} Version: 3 Owner: saforres Author(s): saforres \section{examples of semiprimitive rings} \input{ExamplesOfSemiprimitiveRings.tex} Version: 5 Owner: saforres Author(s): saforres \section{proof of Characterizations of the Jacobson radical} \input{ProofOfCharacterizationsOfTheJacobsonRadical.tex} Version: 25 Owner: saforres Author(s): saforres \section{properties of the Jacobson radical} \input{EpimorphicImageOfTheJacobsonRadical.tex} Version: 8 Owner: saforres Author(s): saforres \section{quasi-regularity} \input{QuasiRegularity.tex} Version: 1 Owner: mclase Author(s): mclase \section{semiprimitive ring} \input{SemiprimitiveRing.tex} Version: 11 Owner: saforres Author(s): saforres \chapter{16N40 -- Nil and nilpotent radicals, sets, ideals, rings} \section{Koethe conjecture} \input{KoetheConjecture.tex} Version: 1 Owner: mclase Author(s): mclase \section{nil and nilpotent ideals} \input{NilAndNilpotentIdeals.tex} Version: 3 Owner: mclase Author(s): mclase \chapter{16N60 -- Prime and semiprime rings} \section{prime ring} \input{PrimeRing.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{16N80 -- General radicals and rings} \section{prime radical} \input{PrimeRadical.tex} Version: 2 Owner: antizeus Author(s): antizeus \section{radical theory} \input{RadicalTheory.tex} Version: 2 Owner: mclase Author(s): mclase \chapter{16P40 -- Noetherian rings and modules} \section{Noetherian ring} \input{Noetherian.tex} Version: 10 Owner: KimJ Author(s): KimJ \section{noetherian} \input{Noetherian2.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{16P60 -- Chain conditions on annihilators and summands: Goldie-type conditions , Krull dimension} \section{Goldie ring} \input{GoldieRing.tex} Version: 3 Owner: mclase Author(s): mclase \section{uniform dimension} \input{UniformDimension.tex} Version: 3 Owner: mclase Author(s): mclase \chapter{16S10 -- Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)} \section{Ore domain} \input{OreDomain.tex} Version: 6 Owner: antizeus Author(s): antizeus \chapter{16S34 -- Group rings , Laurent polynomial rings} \section{support} \input{Support4.tex} Version: 2 Owner: mclase Author(s): mclase \chapter{16S36 -- Ordinary and skew polynomial rings and semigroup rings} \section{Gaussian polynomials} \input{GaussianPolynomials.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{q skew derivation} \input{QSkewDerivation.tex} Version: 5 Owner: antizeus Author(s): antizeus \section{q skew polynomial ring} \input{QSkewPolynomialRing.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{sigma derivation} \input{SigmaDerivation.tex} Version: 7 Owner: antizeus Author(s): antizeus \section{sigma, delta constant} \input{SigmaDeltaConstant.tex} Version: 3 Owner: antizeus Author(s): antizeus \section{skew derivation} \input{SkewDerivation.tex} Version: 4 Owner: antizeus Author(s): antizeus \section{skew polynomial ring} \input{SkewPolynomialRing.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{16S99 -- Miscellaneous} \section{algebra} \input{Algebra.tex} Version: 5 Owner: djao Author(s): djao \section{algebra (module)} \input{AlgebraModule.tex} Version: 1 Owner: karthik Author(s): Larry Hammick \chapter{16U10 -- Integral domains} \section{Pr\"ufer domain} \input{PruferDomain.tex} Version: 2 Owner: mathcam Author(s): mathcam \section{valuation domain} \input{ValuationDomain.tex} Version: 3 Owner: mathcam Author(s): mathcam \chapter{16U20 -- Ore rings, multiplicative sets, Ore localization} \section{Goldie's Theorem} \input{GoldiesTheorem.tex} Version: 3 Owner: mclase Author(s): mclase \section{Ore condition} \input{OreCondition.tex} Version: 3 Owner: mclase Author(s): mclase \section{Ore's theorem} \input{OresTheorem2.tex} Version: 3 Owner: mclase Author(s): mclase \section{classical ring of quotients} \input{ClassicalRingOfQuotients.tex} Version: 2 Owner: mclase Author(s): mclase \section{saturated} \input{Saturated.tex} Version: 1 Owner: drini Author(s): drini \chapter{16U70 -- Center, normalizer (invariant elements)} \section{center (rings)} \input{CenterOfARing.tex} Version: 3 Owner: dublisk Author(s): dublisk \chapter{16U99 -- Miscellaneous} \section{anti-idempotent} \input{AntiIdempotent.tex} Version: 1 Owner: mathcam Author(s): mathcam \chapter{16W20 -- Automorphisms and endomorphisms} \section{ring of endomorphisms} \input{RingOfEndomorphisms.tex} Version: 4 Owner: mclase Author(s): mclase \chapter{16W30 -- Coalgebras, bialgebras, Hopf algebras ; rings, modules, etc. on which these act} \section{Hopf algebra} \input{HopfAlgebra.tex} Version: 6 Owner: mhale Author(s): mhale \section{almost cocommutative bialgebra} \input{AlmostCocommutativeBialgebra.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{bialgebra} \input{Bialgebra.tex} Version: 2 Owner: mhale Author(s): mhale \section{coalgebra} \input{Coalgebra.tex} Version: 4 Owner: mhale Author(s): mhale \section{coinvariant} \input{Coinvariant.tex} Version: 1 Owner: mhale Author(s): mhale \section{comodule} \input{Comodule.tex} Version: 2 Owner: mhale Author(s): mhale \section{comodule algebra} \input{ComoduleAlgebra.tex} Version: 5 Owner: mhale Author(s): mhale \section{comodule coalgebra} \input{ComoduleCoalgebra.tex} Version: 4 Owner: mhale Author(s): mhale \section{module algebra} \input{ModuleAlgebra.tex} Version: 4 Owner: mhale Author(s): mhale \section{module coalgebra} \input{ModuleCoalgebra.tex} Version: 5 Owner: mhale Author(s): mhale \chapter{16W50 -- Graded rings and modules} \section{graded algebra} \input{GradedAlgebra.tex} Version: 1 Owner: dublisk Author(s): dublisk \section{graded module} \input{GradedModule.tex} Version: 4 Owner: KimJ Author(s): KimJ \section{supercommutative} \input{Supercommutative.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{16W55 -- ``Super'' (or ``skew'') structure} \section{super tensor product} \input{SuperTensorProduct.tex} Version: 4 Owner: dublisk Author(s): dublisk \section{superalgebra} \input{SuperAlgebra.tex} Version: 2 Owner: dublisk Author(s): dublisk \section{supernumber} \input{Supernumber.tex} Version: 5 Owner: mhale Author(s): mhale \chapter{16W99 -- Miscellaneous} \section{Hamiltonian quaternions} \input{Quaternions.tex} Version: 3 Owner: mathcam Author(s): Larry Hammick, patrickwonders \chapter{16Y30 -- Near-rings} \section{near-ring} \input{NearRing.tex} Version: 13 Owner: yark Author(s): yark, juergen \chapter{17A01 -- General theory} \section{commutator bracket} \input{CommutatorBracket.tex} Version: 4 Owner: rmilson Author(s): rmilson \chapter{17B05 -- Structure theory} \section{Killing form} \input{KillingForm.tex} Version: 4 Owner: bwebste Author(s): bwebste \section{Levi's theorem} \input{LevisTheorem.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{nilradical} \input{Nilradical2.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{radical} \input{Radical3.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{17B10 -- Representations, algebraic theory (weights)} \section{Ado's theorem} \input{AdosTheorem.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Lie algebra representation} \input{RepresentationLieAlgebra.tex} Version: 9 Owner: bwebste Author(s): bwebste, rmilson \section{adjoint representation} \input{AdjointRepresentation.tex} Version: 2 Owner: rmilson Author(s): rmilson \section{examples of non-matrix Lie groups} \input{ExamplesOfNonMatrixLieGroup.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{isotropy representation} \input{IsotropyRepresentation.tex} Version: 3 Owner: rmilson Author(s): rmilson \chapter{17B15 -- Representations, analytic theory} \section{invariant form (Lie algebras)} \input{InvariantFormLieAlgebras.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{17B20 -- Simple, semisimple, reductive (super)algebras (roots)} \section{Borel subalgebra} \input{BorelSubalgebra.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Borel subgroup} \input{BorelSubgroup.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Cartan matrix} \input{CartanMatrix.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Cartan subalgebra} \input{CartanSubalgebra2.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{Cartan's criterion} \input{CartansCriterion.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Casimir operator} \input{CasimirOperator.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Dynkin diagram} \input{DynkinDiagram.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Verma module} \input{VermaModule.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Weyl chamber} \input{WeylChamber.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Weyl group} \input{WeylGroup.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Weyl's theorem} \input{WeylsTheorem.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{classification of finite-dimensional representations of semi-simple Lie algebras} \input{ClassificationOfFiniteDimensionalRepresentationsOfSemiSimpleLieAlgebras.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{cohomology of semi-simple Lie algebras} \input{CohomologyOfSemiSimpleLieAlgebras.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{nilpotent cone} \input{NilpotentCone.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{parabolic subgroup} \input{ParabolicSubgroup.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{pictures of Dynkin diagrams} \input{PicturesOfDynkinDiagram.tex} Version: 9 Owner: Dr\_Absentius Author(s): Dr\_Absentius \section{positive root} \input{PositiveRoot.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{rank} \input{Rank.tex} Version: 5 Owner: rmilson Author(s): rmilson \section{root lattice} \input{RootLattice.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{root system} \input{RootSystem.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{simple and semi-simple Lie algebras} \input{SimpleAndSemiSimpleLieAlgebras2.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{simple root} \input{SimpleRoot.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{weight (Lie algebras)} \input{WeightLieAlgebras.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{weight lattice} \input{WeightLattice.tex} Version: 4 Owner: bwebste Author(s): bwebste \chapter{17B30 -- Solvable, nilpotent (super)algebras} \section{Engel's theorem} \input{EngelsTheorem.tex} Version: 2 Owner: rmilson Author(s): rmilson \section{Lie's theorem} \input{LiesTheorem.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{solvable Lie algebra} \input{SolvableLieAlgebra.tex} Version: 1 Owner: djao Author(s): djao \chapter{17B35 -- Universal enveloping (super)algebras} \section{Poincar\'e-Birkhoff-Witt theorem} \input{PoincareBirkhoffWittTheorem.tex} Version: 1 Owner: draisma Author(s): draisma \section{universal enveloping algebra} \input{UniversalEnvelopingAlgebra.tex} Version: 1 Owner: draisma Author(s): draisma \chapter{17B56 -- Cohomology of Lie (super)algebras} \section{Lie algebra cohomology} \input{LieAlgebraCohomology.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{17B67 -- Kac-Moody (super)algebras (structure and representation theory)} \section{Kac-Moody algebra} \input{KacMoodyAlgebra.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{generalized Cartan matrix} \input{GeneralizedCartanMatrix.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{17B99 -- Miscellaneous} \section{Jacobi identity interpretations} \input{JacobiIdentityInterpretations.tex} Version: 2 Owner: draisma Author(s): draisma \section{Lie algebra} \input{LieAlgebra.tex} Version: 10 Owner: djao Author(s): djao, rmilson, nerdy2 \section{real form} \input{RealForm.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{18-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{Grothendieck spectral sequence} \input{GrothendieckSpectralSequence.tex} Version: 5 Owner: bwebste Author(s): Manoj, ceps, nerdy2 \section{category of sets} \input{CategoryOfSets.tex} Version: 1 Owner: nerdy2 Author(s): nerdy2 \section{functor} \input{Functor.tex} Version: 3 Owner: nerdy2 Author(s): nerdy2 \section{monic} \input{Monic.tex} Version: 1 Owner: nerdy2 Author(s): nerdy2 \section{natural equivalence} \input{NaturalEquivalence.tex} Version: 2 Owner: mathcam Author(s): mathcam, nerdy2 \section{representable functor} \input{RepresentableFunctor.tex} Version: 3 Owner: bwebste Author(s): bwebste, nerdy2 \section{supplemental axioms for an Abelian category} \input{AxiomsForAnAbelianCategory.tex} Version: 5 Owner: nerdy2 Author(s): nerdy2 \chapter{18A05 -- Definitions, generalizations} \section{autofunctor} \input{ThenTheEndofunctorIsAnAutofunctorAutofunctorDefIfTheMappingFromACCategoryToItselfIsAnIsomorphism.tex} Version: 10 Owner: mathcam Author(s): mathcam, mhale, yark, gorun\_manolescu \section{automorphism} \input{Automorphism4.tex} Version: 4 Owner: benjaminfjones Author(s): benjaminfjones \section{category} \input{Category.tex} Version: 9 Owner: mathcam Author(s): mathcam, RevBobo \section{category example (arrow category)} \input{ExampleOfCategory.tex} Version: 6 Owner: n3o Author(s): n3o \section{commutative diagram} \input{CommutativeDiagram.tex} Version: 3 Owner: Dr\_Absentius Author(s): Dr\_Absentius \section{double dual embedding} \input{DoubleDualEmbedding.tex} Version: 1 Owner: rmilson Author(s): rmilson \section{dual category} \input{DualCategory.tex} Version: 3 Owner: RevBobo Author(s): RevBobo \section{duality principle} \input{DualityPrinciple.tex} Version: 3 Owner: RevBobo Author(s): RevBobo \section{endofunctor} \input{Endofunctor.tex} Version: 2 Owner: rmilson Author(s): NeuRet, Logan \section{examples of initial objects, terminal objects and zero objects} \input{TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.tex} Version: 11 Owner: AxelBoldt Author(s): AxelBoldt \section{forgetful functor} \input{ForgetfulFunctor.tex} Version: 1 Owner: RevBobo Author(s): RevBobo \section{isomorphism} \input{Isomorphism2.tex} Version: 2 Owner: djao Author(s): djao \section{natural transformation} \input{NaturalTransformation.tex} Version: 6 Owner: RevBobo Author(s): RevBobo \section{types of homomorphisms} \input{TypesOfHomomorphisms.tex} Version: 4 Owner: antizeus Author(s): antizeus \section{zero object} \input{ZeroObject.tex} Version: 2 Owner: djao Author(s): djao \chapter{18A22 -- Special properties of functors (faithful, full, etc.)} \section{exact functor} \input{ExactFunctor.tex} Version: 3 Owner: antizeus Author(s): antizeus \chapter{18A25 -- Functor categories, comma categories} \section{Yoneda embedding} \input{YonedaEmbedding.tex} Version: 4 Owner: nerdy2 Author(s): nerdy2 \chapter{18A30 -- Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)} \section{categorical direct product} \input{CategoricalDirectProduct.tex} Version: 4 Owner: djao Author(s): djao \section{categorical direct sum} \input{CategoricalDirectSum.tex} Version: 4 Owner: djao Author(s): djao \section{kernel} \input{Kernel.tex} Version: 4 Owner: drini Author(s): drini \chapter{18A40 -- Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)} \section{adjoint functor} \input{AdjointFunctor.tex} Version: 8 Owner: bwebste Author(s): bwebste, RevBobo \section{equivalence of categories} \input{EquivalenceOfCategories.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{18B40 -- Groupoids, semigroupoids, semigroups, groups (viewed as categories)} \section{groupoid (category theoretic)} \input{GroupoidCategoryTheoretic.tex} Version: 6 Owner: akrowne Author(s): akrowne \chapter{18E10 -- Exact categories, abelian categories} \section{abelian category} \input{AbelianCategory.tex} Version: 6 Owner: djao Author(s): djao \section{exact sequence} \input{ExactSequence2.tex} Version: 3 Owner: djao Author(s): djao \section{derived category} \input{DerivedCategory.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{enough injectives} \input{EnoughInjectives.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{18F20 -- Presheaves and sheaves} \section{locally ringed space} \input{LocallyRingedSpace.tex} Version: 9 Owner: djao Author(s): djao \section{presheaf} \input{Presheaf.tex} Version: 2 Owner: nerdy2 Author(s): nerdy2 \section{sheaf} \input{Sheaf.tex} Version: 9 Owner: djao Author(s): djao \section{sheafification} \input{Sheafification.tex} Version: 4 Owner: djao Author(s): djao \section{stalk} \input{Stalk.tex} Version: 4 Owner: djao Author(s): djao \chapter{18F30 -- Grothendieck groups} \section{Grothendieck group} \input{GrothendieckGroup.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{18G10 -- Resolutions; derived functors} \section{derived functor} \input{DerivedFunctor.tex} Version: 4 Owner: bwebste Author(s): bwebste \chapter{18G15 -- Ext and Tor, generalizations, K\"unneth formula} \section{Ext} \input{Ext.tex} Version: 3 Owner: bwebste Author(s): bwebste \chapter{18G30 -- Simplicial sets, simplicial objects (in a category)} \section{nerve} \input{Nerve.tex} Version: 1 Owner: mhale Author(s): mhale \section{simplicial category} \input{SimplicialCategory.tex} Version: 4 Owner: mhale Author(s): mhale \section{simplicial object} \input{SimplicialObject.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{18G35 -- Chain complexes} \section{5-lemma} \input{5Lemma.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{9-lemma} \input{9Lemma.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Snake lemma} \input{SnakeLemma.tex} Version: 5 Owner: bwebste Author(s): bwebste \section{chain homotopy} \input{ChainHomotopy.tex} Version: 4 Owner: RevBobo Author(s): RevBobo \section{chain map} \input{ChainMap.tex} Version: 3 Owner: RevBobo Author(s): RevBobo \section{homology (chain complex)} \input{HomologyChainComplex.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{18G40 -- Spectral sequences, hypercohomology} \section{spectral sequence} \input{SpectralSequence.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{19-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{Algebraic K-theory} \input{AlgebraicKTheory.tex} Version: 2 Owner: mhale Author(s): mhale \section{K-theory} \input{KTheory.tex} Version: 12 Owner: mhale Author(s): mhale \section{examples of algebraic K-theory groups} \input{ExamplesOfAlgebraicKTheoryGroups.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{19K33 -- EXT and $K$-homology} \section{Fredholm module} \input{FredholmModule.tex} Version: 3 Owner: mhale Author(s): mhale \section{K-homology} \input{KHomology.tex} Version: 1 Owner: mhale Author(s): mhale \chapter{19K99 -- Miscellaneous} \section{examples of K-theory groups} \input{ExamplesOfKTheoryGroups.tex} Version: 5 Owner: mhale Author(s): mhale \chapter{20-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{alternating group is a normal subgroup of the symmetric group} \input{AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup.tex} Version: 1 Owner: tensorking Author(s): tensorking \section{associative} \input{Associative.tex} Version: 6 Owner: akrowne Author(s): akrowne \section{canonical projection} \input{CanonicalProjection.tex} Version: 4 Owner: Dr\_Absentius Author(s): Dr\_Absentius \section{centralizer} \input{Centralizer.tex} Version: 5 Owner: mathcam Author(s): Larry Hammick, vitriol \section{commutative} \input{Commutative.tex} Version: 3 Owner: akrowne Author(s): akrowne \section{examples of groups} \input{ExamplesOfGroups.tex} Version: 14 Owner: AxelBoldt Author(s): AxelBoldt, NeuRet \section{group} \input{Group.tex} Version: 10 Owner: drini Author(s): drini \section{quotient group} \input{QuotientGroup.tex} Version: 6 Owner: drini Author(s): drini \chapter{20-02 -- Research exposition (monographs, survey articles)} \section{length function} \input{LengthFunction.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{20-XX -- Group theory and generalizations} \section{free product with amalgamated subgroup} \input{FreeProductWithAmalgamatedSubgroup.tex} Version: 1 Owner: Dr\_Absentius Author(s): Dr\_Absentius \section{nonabelian group} \input{NonabelianGroup.tex} Version: 2 Owner: drini Author(s): drini, apmxi \chapter{20A05 -- Axiomatics and elementary properties} \section{Feit-Thompson theorem} \input{FeitThompsonTheorem.tex} Version: 1 Owner: mathcam Author(s): mathcam \section{Proof: The orbit of any element of a group is a subgroup} \input{ProofThatGInGImpliesThatLangleGRangleLeG.tex} Version: 3 Owner: drini Author(s): drini, Wkbj79 \section{center} \input{GroupCentre.tex} Version: 5 Owner: vitriol Author(s): vitriol \section{characteristic subgroup} \input{CharacteristicSubgroup.tex} Version: 1 Owner: Henry Author(s): Henry \section{class function} \input{ClassFunction.tex} Version: 5 Owner: djao Author(s): djao \section{conjugacy class} \input{ConjugacyClass.tex} Version: 2 Owner: djao Author(s): djao \section{conjugacy class formula} \input{ConjugacyClassFormula.tex} Version: 3 Owner: lieven Author(s): lieven \section{conjugate stabilizer subgroups} \input{ConjugateStabilizerSubgroups.tex} Version: 4 Owner: Thomas Heye Author(s): Thomas Heye \section{coset} \input{Coset.tex} Version: 5 Owner: djao Author(s): rmilson, djao \section{cyclic group} \input{CyclicGroup.tex} Version: 10 Owner: yark Author(s): yark, Larry Hammick, vitriol \section{derived subgroup} \input{DerivedSubgroup.tex} Version: 4 Owner: rmilson Author(s): rmilson \section{equivariant} \input{Equivariant.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{examples of finite simple groups} \input{ExamplesOfFiniteSimpleGroups.tex} Version: 8 Owner: drini Author(s): bbukh, yark, NeuRet \section{finitely generated group} \input{FinitelyGenerated.tex} Version: 6 Owner: yark Author(s): yark, nerdy2 \section{first isomorphism theorem} \input{FirstIsomorphismTheorem.tex} Version: 2 Owner: nerdy2 Author(s): nerdy2 \section{fourth isomorphism theorem} \input{FourthIsomorphismTheorem.tex} Version: 2 Owner: bwebste Author(s): yark, apmxi \section{generator} \input{Generator.tex} Version: 3 Owner: Wkbj79 Author(s): Wkbj79 \section{group actions and homomorphisms} \input{GroupActionsAndHomomorphisms.tex} Version: 5 Owner: Thomas Heye Author(s): Thomas Heye \section{group homomorphism} \input{GroupHomomorphism.tex} Version: 15 Owner: drini Author(s): saforres, drini \section{homogeneous space} \input{HomogeneousSpace.tex} Version: 3 Owner: rmilson Author(s): rmilson \section{identity element} \input{IdentityElement.tex} Version: 6 Owner: mclase Author(s): mclase, vypertd, imran \section{inner automorphism} \input{InnerAutomorphism.tex} Version: 7 Owner: rmilson Author(s): rmilson, tensorking \section{kernel} \input{KernelOfAGroupHomomorphism.tex} Version: 9 Owner: rmilson Author(s): rmilson, Daume \section{maximal} \input{Maximal.tex} Version: 1 Owner: Evandar Author(s): Evandar \section{normal subgroup} \input{NormalSubgroup.tex} Version: 6 Owner: djao Author(s): djao \section{normality of subgroups is not transitive} \input{NormalityOfSubgroupsIsNotTransitive.tex} Version: 4 Owner: ariels Author(s): ariels \section{normalizer} \input{Normalizer.tex} Version: 6 Owner: saforres Author(s): saforres \section{order (of a group)} \input{OrderGroup.tex} Version: 5 Owner: saforres Author(s): saforres \section{presentation of a group} \input{Presentationgroup.tex} Version: 11 Owner: rmilson Author(s): rmilson \section{proof of first isomorphism theorem} \input{ProofOfFirstIsomorphismTheorem.tex} Version: 3 Owner: uriw Author(s): uriw \section{proof of second isomorphism theorem} \input{ProofOfSecondIsomorphismTheorem.tex} Version: 8 Owner: saforres Author(s): saforres \section{proof that all cyclic groups are abelian} \input{ProofThatAllCyclicGroupsAreAbelian.tex} Version: 2 Owner: Wkbj79 Author(s): Wkbj79 \section{proof that all cyclic groups of the same order are isomorphic to each other} \input{ProofThatAllCyclicGroupsOfTheSameOrderAreIsomorphicToEachOther.tex} Version: 1 Owner: Wkbj79 Author(s): Wkbj79 \section{proof that all subgroups of a cyclic group are cyclic} \input{ProofThatAllSubgroupsOfACyclicGroupAreCyclic.tex} Version: 3 Owner: Wkbj79 Author(s): Wkbj79 \section{regular group action} \input{RegularGroupAction.tex} Version: 3 Owner: Thomas Heye Author(s): Thomas Heye \section{second isomorphism theorem} \input{SecondIsomorphismTheorem.tex} Version: 4 Owner: djao Author(s): djao \section{simple group} \input{Simple.tex} Version: 3 Owner: Evandar Author(s): Evandar \section{solvable group} \input{Solvable.tex} Version: 4 Owner: djao Author(s): djao \section{subgroup} \input{Subgroup.tex} Version: 7 Owner: Daume Author(s): Daume \section{third isomorphism theorem} \input{ThirdIsomorphismTheorem.tex} Version: 2 Owner: nerdy2 Author(s): nerdy2 \chapter{20A99 -- Miscellaneous} \section{Cayley table} \input{CayleyTable.tex} Version: 6 Owner: akrowne Author(s): akrowne \section{proper subgroup} \input{ProperSubgroup.tex} Version: 2 Owner: imran Author(s): imran \section{quaternion group} \input{QuaternionGroup.tex} Version: 6 Owner: vernondalhart Author(s): vernondalhart, Larry Hammick, patrickwonders \chapter{20B05 -- General theory for finite groups} \section{cycle notation} \input{CycleNotation.tex} Version: 1 Owner: rmilson Author(s): rmilson \section{permutation group} \input{PermutationGroup2.tex} Version: 2 Owner: bwebste Author(s): bwebste \chapter{20B15 -- Primitive groups} \section{primitive transitive permutation group} \input{PrimativeTransitivePermutationGroupOnASet.tex} Version: 4 Owner: Thomas Heye Author(s): yark, apmxi \chapter{20B20 -- Multiply transitive finite groups} \section{Jordan's theorem (multiply transitive groups)} \input{JordansTheoremMultiplyTransitiveGroups.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{multiply transitive} \input{MultiplyTransitive.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{sharply multiply transitive} \input{SharplyMultiplyTransitive.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20B25 -- Finite automorphism groups of algebraic, geometric, or combinatorial structures} \section{diamond theory} \input{DiamondTheory.tex} Version: 4 Owner: m759 Author(s): akrowne, m759 \chapter{20B30 -- Symmetric groups} \section{symmetric group} \input{SymmetricGroup.tex} Version: 5 Owner: bwebste Author(s): bwebste, antizeus \section{symmetric group} \input{SymmetricGroup2.tex} Version: 2 Owner: antizeus Author(s): antizeus \chapter{20B35 -- Subgroups of symmetric groups} \section{Cayley's theorem} \input{CayleysTheorem.tex} Version: 4 Owner: vitriol Author(s): vitriol \chapter{20B99 -- Miscellaneous} \section{$(p,q)$ shuffle} \input{PqShuffle.tex} Version: 3 Owner: matte Author(s): matte \section{Frobenius group} \input{FrobeniusGroup.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{permutation} \input{Permutation.tex} Version: 3 Owner: drini Author(s): drini \section{proof of Cayley's theorem} \input{ProofOfCayleysTheorem.tex} Version: 2 Owner: Evandar Author(s): Evandar \chapter{20C05 -- Group rings of finite groups and their modules} \section{group ring} \input{GroupRing.tex} Version: 4 Owner: djao Author(s): djao \chapter{20C15 -- Ordinary representations and characters} \section{Maschke's theorem} \input{MaschkesTheorem.tex} Version: 5 Owner: bwebste Author(s): bwebste \section{a representation which is not completely reducible} \input{ARepresentationWhichIsNotCompletelyReducible.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{orthogonality relations} \input{OrthogonalityRelations.tex} Version: 8 Owner: bwebste Author(s): bwebste \chapter{20C30 -- Representations of finite symmetric groups} \section{example of immanent} \input{ExampleOfImmanent.tex} Version: 1 Owner: gholmes74 Author(s): gholmes74 \section{immanent} \input{Immanent.tex} Version: 4 Owner: gholmes74 Author(s): gholmes74 \section{permanent} \input{Permanent.tex} Version: 2 Owner: gholmes74 Author(s): gholmes74 \chapter{20C99 -- Miscellaneous} \section{Frobenius reciprocity} \input{FrobeniusReciprocity.tex} Version: 4 Owner: djao Author(s): rmilson, djao \section{Schur's lemma} \input{SchursLemma.tex} Version: 14 Owner: rmilson Author(s): rmilson, NeuRet \section{character} \input{Character.tex} Version: 4 Owner: djao Author(s): djao \section{group representation} \input{GroupRepresentation.tex} Version: 2 Owner: djao Author(s): djao \section{induced representation} \input{InducedRepresentation.tex} Version: 1 Owner: djao Author(s): djao \section{regular representation} \input{RegularRepresentation.tex} Version: 2 Owner: djao Author(s): djao \section{restriction representation} \input{RestrictionRepresentation.tex} Version: 1 Owner: djao Author(s): djao \chapter{20D05 -- Classification of simple and nonsolvable groups} \section{Burnside $p-q$ theorem} \input{BurnsidePQTheorem.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{classification of semisimple groups} \input{ClassificationOfSemisimpleGroups.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{semisimple group} \input{SemisimpleGroup.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20D08 -- Simple groups: sporadic groups} \section{Janko groups} \input{JankoGroups.tex} Version: 7 Owner: mathcam Author(s): mathcam, Thomas Heye \chapter{20D10 -- Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks} \section{\v{C}uhinin's Theorem} \input{VeeCuhininsTheorem.tex} Version: 4 Owner: bwebste Author(s): bwebste \section{separable} \input{Seperable.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{supersolvable group} \input{SupersolvableGroup.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{20D15 -- Nilpotent groups, $p$-groups} \section{Burnside basis theorem} \input{BurnsideBasisTheorem.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20D20 -- Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure} \section{$\pi$-groups and $\pi'$-groups} \input{PiGroupsAndPiGroups.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{$p$-subgroup} \input{PGroup.tex} Version: 2 Owner: drini Author(s): drini, apmxi \section{Burnside normal complement theorem} \input{BurnsideNormalComplementTheorem.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Frattini argument} \input{FrattiniArgument.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{Sylow p-subgroup} \input{SylowPSubgroup.tex} Version: 3 Owner: Henry Author(s): Henry \section{Sylow theorems} \input{SylowTheorems.tex} Version: 1 Owner: vitriol Author(s): vitriol \section{Sylow's first theorem} \input{SylowsFirstTheorem.tex} Version: 2 Owner: bwebste Author(s): yark, apmxi \section{Sylow's third theorem} \input{SylowsThirdTheorem.tex} Version: 8 Owner: bwebste Author(s): yark, apmxi \section{application of Sylow's theorems to groups of order pq} \input{ApplicationOfSylowsTheoremsToGroupsOfOrderPq.tex} Version: 9 Owner: yark Author(s): yark, Manoj, Henry \section{p-primary component} \input{PPrimaryComponent.tex} Version: 2 Owner: alozano Author(s): alozano \section{proof of Frattini argument} \input{ProofOfFrattiniArgument.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{proof of Sylow theorems} \input{ProofOfSylowTheorems.tex} Version: 3 Owner: Henry Author(s): Henry \section{subgroups containing the normalizers of Sylow subgroups normalize themselves} \input{SubgroupsContainingTheNormalizersOfSylowSubgroupsNormalizeThemselves.tex} Version: 3 Owner: bwebste Author(s): bwebste \chapter{20D25 -- Special subgroups (Frattini, Fitting, etc.)} \section{Fitting's theorem} \input{FittingsTheorem.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{characteristically simple group} \input{CharacteristicallySimpleGroup.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{the Frattini subgroup is nilpotent} \input{FrattiniSubgroupIsNilpotent.tex} Version: 4 Owner: bwebste Author(s): bwebste \chapter{20D30 -- Series and lattices of subgroups} \section{maximal condition} \input{MaximalCondition.tex} Version: 2 Owner: mclase Author(s): mclase \section{minimal condition} \input{MinimalCondition.tex} Version: 1 Owner: mclase Author(s): mclase \section{subnormal series} \input{SubnormalSeries.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{20D35 -- Subnormal subgroups} \section{subnormal subgroup} \input{SubnormalSubgroup.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20D99 -- Miscellaneous} \section{Cauchy's theorem} \input{CauchysTheoremForFiniteGroups.tex} Version: 1 Owner: Evandar Author(s): Evandar \section{Lagrange's theorem} \input{LagrangesTheorem.tex} Version: 2 Owner: Evandar Author(s): Evandar \section{exponent} \input{Exponent.tex} Version: 5 Owner: Wkbj79 Author(s): Wkbj79 \section{fully invariant subgroup} \input{FullyInvariantSubgroup.tex} Version: 1 Owner: mclase Author(s): mclase \section{proof of Cauchy's theorem} \input{ProofOfCauchysTheorem.tex} Version: 1 Owner: vitriol Author(s): vitriol \section{proof of Lagrange's theorem} \input{ProofOfLagrangesTheorem.tex} Version: 2 Owner: akrowne Author(s): akrowne \section{proof of the converse of Lagrange's theorem for finite cyclic groups} \input{ProofOfTheConverseOfLagrangesTheoremForCyclicGroups.tex} Version: 3 Owner: Wkbj79 Author(s): Wkbj79 \section{proof that $\mathbf{exp}G$ divides $|G|$} \input{ProofThatExpGDividesG.tex} % -- jac Thu Jan 15 02:47:08 2004. Version: 4 Owner: Wkbj79 Author(s): Wkbj79 \section{proof that $|g|$ divides $\mathbf{exp}G$} \input{ProofThatGDividesExpG.tex} % -- jac Thu Jan 15 02:47:12 2004 Version: 2 Owner: Wkbj79 Author(s): Wkbj79 \section{proof that every group of prime order is cyclic} \input{ProofThatEveryGroupOfPrimeOrderIsCyclic.tex} Version: 3 Owner: Wkbj79 Author(s): Wkbj79 \chapter{20E05 -- Free nonabelian groups} \section{Nielsen-Schreier theorem} \input{NielsenSchreierTheorem.tex} Version: 1 Owner: Evandar Author(s): Evandar \section{Scheier index formula} \input{ScheierIndexFormula.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{free group} \input{FreeGroup.tex} Version: 4 Owner: jihemme Author(s): jihemme, rmilson, djao \section{proof of Nielsen-Schreier theorem and Schreier index formula} \input{ProofOfNielsenSchreierTheoremAndSchreierIndexFormula.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Jordan-Holder decomposition} \input{JordanHolderDecomposition.tex} Version: 4 Owner: djao Author(s): djao \section{profinite group} \input{ProfiniteGroup.tex} Version: 3 Owner: djao Author(s): djao \section{extension} \input{Extension2.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{holomorph} \input{Holomorph.tex} Version: 2 Owner: dublisk Author(s): dublisk \section{proof of the Jordan Holder decomposition theorem} \input{ProofOfTheJordanHolderDecompositionTheorem.tex} Version: 4 Owner: djao Author(s): djao \section{semidirect product of groups} \input{SemidirectProductOfGroups.tex} Version: 5 Owner: djao Author(s): djao \section{wreath product} \input{WreathProduct.tex} Version: 11 Owner: bwebste Author(s): NeuRet \section{Jordan-Hölder decomposition theorem} \input{JordanHolderDecompositionTheorem.tex} Version: 4 Owner: djao Author(s): djao \section{simplicity of the alternating groups} \input{SimplicityOfA_n.tex} Version: 8 Owner: rmilson Author(s): NeuRet \section{abelian groups of order $120$} \input{AbelianGroupsOfOrder120.tex} Version: 1 Owner: alozano Author(s): alozano \section{fundamental theorem of finitely generated abelian groups} \input{FundamentalTheoremOfFinitelyGeneratedAbelianGroups.tex} Version: 1 Owner: bwebste Author(s): alozano \section{conjugacy class} \input{ConjugacyClassOfAGroupElement.tex} Version: 2 Owner: drini Author(s): drini, apmxi \section{Frattini subgroup} \input{FrattiniSubgroup.tex} Version: 1 Owner: Evandar Author(s): Evandar \section{non-generator} \input{NonGenerator.tex} Version: 1 Owner: Evandar Author(s): Evandar \chapter{20Exx -- Structure and classification of infinite or finite groups} \section{faithful group action} \input{FaithfulGroupAction.tex} Version: 3 Owner: drini Author(s): drini, apmxi \chapter{20F18 -- Nilpotent groups} \section{classification of finite nilpotent groups} \input{ClassificationOfFiniteNilpotentGroups.tex} Version: 1 Owner: bwebste Author(s): bwebste \section{nilpotent group} \input{NilpotentGroup.tex} Version: 3 Owner: djao Author(s): djao \chapter{20F22 -- Other classes of groups defined by subgroup chains} \section{inverse limit} \input{InverseLimit.tex} Version: 6 Owner: alozano Author(s): alozano \chapter{20F28 -- Automorphism groups of groups} \section{outer automorphism group} \input{OuterAutomorphismGroupOfAGroup.tex} Version: 7 Owner: Thomas Heye Author(s): yark, apmxi \chapter{20F36 -- Braid groups; Artin groups} \section{braid group} \input{BraidGroup.tex} Version: 7 Owner: dublisk Author(s): dublisk \chapter{20F55 -- Reflection and Coxeter groups} \section{cycle} \input{Cycle2.tex} Version: 6 Owner: drini Author(s): drini \section{dihedral group} \input{DihedralGroup.tex} Version: 8 Owner: rmilson Author(s): rmilson \chapter{20F65 -- Geometric group theory} \section{groups that act freely on trees are free} \input{GroupsThatActFreelyOnTreesAreFree.tex} Version: 3 Owner: bwebste Author(s): bwebste \chapter{20F99 -- Miscellaneous} \section{perfect group} \input{PerfectGroup.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20G15 -- Linear algebraic groups over arbitrary fields} \section{Nagao's theorem} \input{NagaosTheorem.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{computation of the order of $GL(n, F_q)$} \input{MathbbF_qComputaionOfTheOrderOfGLn.tex} Version: 5 Owner: benjaminfjones Author(s): benjaminfjones \section{general linear group} \input{GeneralLinearGroup.tex} Version: 3 Owner: djao Author(s): djao \section{order of the general linear group over a finite field} \input{OrderOfTheGeneralLinearGroupOverAFiniteField.tex} Version: 16 Owner: benjaminfjones Author(s): benjaminfjones \section{special linear group} \input{SpecialLinearGroup.tex} Version: 2 Owner: djao Author(s): djao \chapter{20G20 -- Linear algebraic groups over the reals, the complexes, the quaternions} \section{orthogonal group} \input{OrthogonalGroup.tex} Version: 2 Owner: djao Author(s): djao \chapter{20G25 -- Linear algebraic groups over local fields and their integers} \section{Ihara's theorem} \input{IharasTheorem.tex} Version: 6 Owner: bwebste Author(s): bwebste \chapter{20G40 -- Linear algebraic groups over finite fields} \section{$\mathit{SL}_2(F_3)$} \input{MathitSL_2F_3.tex} Version: 6 Owner: drini Author(s): drini, apmxi \chapter{20J06 -- Cohomology of groups} \section{group cohomology} \input{GroupCohomology.tex} Version: 4 Owner: alozano Author(s): alozano \section{stronger Hilbert theorem 90} \input{StrongerHilbertTheorem90.tex} Version: 2 Owner: alozano Author(s): alozano \chapter{20J15 -- Category of groups} \section{variety of groups} \input{VarietyOfGroups.tex} Version: 1 Owner: bwebste Author(s): bwebste \chapter{20K01 -- Finite abelian groups} \section{Schinzel's theorem} \input{SchinzelsTheorem.tex} Version: 4 Owner: mathcam Author(s): Manoj, nerdy2 \chapter{20K10 -- Torsion groups, primary groups and generalized primary groups} \section{torsion} \input{Torsion3.tex} Version: 2 Owner: mhale Author(s): mhale \chapter{20K25 -- Direct sums, direct products, etc.} \section{direct product of groups} \input{DirectProduct2.tex} Version: 4 Owner: vitriol Author(s): vitriol \chapter{20K99 -- Miscellaneous} \section{Klein 4-group} \input{Klein4Group.tex} Version: 7 Owner: rmilson Author(s): Dr\_Absentius, rmilson, imran \section{divisible group} \input{DivisibleGroup.tex} Version: 4 Owner: mathcam Author(s): mathcam \section{example of divisible group} \input{ExampleOfDivisibleGroup.tex} Version: 1 Owner: mathcam Author(s): mathcam \section{locally cyclic group} \input{GeneralizedCyclicGroup.tex} Version: 10 Owner: yark Author(s): yark \chapter{20Kxx -- Abelian groups} \section{abelian group} \input{AbelianGroup2.tex} Version: 12 Owner: drini Author(s): drini, yark, akrowne, apmxi \chapter{20M10 -- General structure theory} \section{existence of maximal semilattice decomposition} \input{ExistenceOfMaximalSemilatticeDecomposition.tex} Version: 2 Owner: mclase Author(s): mclase \section{semilattice decomposition of a semigroup} \input{SemilatticeDecompositionOfASemigroup.tex} Version: 3 Owner: mclase Author(s): mclase \section{simple semigroup} \input{SimpleSemigroup.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{20M12 -- Ideal theory} \section{Rees factor} \input{ReesFactor.tex} Version: 1 Owner: mclase Author(s): mclase \section{ideal} \input{Ideal3.tex} Version: 5 Owner: mclase Author(s): mclase \chapter{20M14 -- Commutative semigroups} \section{Archimedean semigroup} \input{ArchimedeanSemigroup.tex} Version: 1 Owner: mclase Author(s): mclase \section{commutative semigroup} \input{CommutativeSemigroup.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{20M20 -- Semigroups of transformations, etc.} \section{semigroup of transformations} \input{SemigroupOfTransformations.tex} Version: 3 Owner: mclase Author(s): mclase \chapter{20M30 -- Representation of semigroups; actions of semigroups on sets} \section{counting theorem} \input{CountingTheorem.tex} Version: 8 Owner: mathcam Author(s): Larry Hammick, vitriol \section{example of group action} \input{ExampleOfGroupAction.tex} Version: 5 Owner: Thomas Heye Author(s): Thomas Heye \section{group action} \input{GroupAction.tex} Version: 3 Owner: djao Author(s): djao \section{orbit} \input{Orbit.tex} Version: 2 Owner: djao Author(s): djao \section{proof of counting theorem} \input{ProofOfCountingTheorem.tex} Version: 2 Owner: n3o Author(s): n3o \section{stabilizer} \input{Stabilizer.tex} Version: 3 Owner: djao Author(s): djao \chapter{20M99 -- Miscellaneous} \section{a semilattice is a commutative band} \input{ASemilatticeIsACommutativeBand.tex} Version: 3 Owner: mclase Author(s): mclase \section{adjoining an identity to a semigroup} \input{AdjoiningAnIdentityToASemigroup3.tex} Version: 2 Owner: mclase Author(s): mclase \section{band} \input{Band2.tex} Version: 1 Owner: mclase Author(s): mclase \section{bicyclic semigroup} \input{BicyclicSemigroup.tex} Version: 3 Owner: mclase Author(s): mclase \section{congruence} \input{Congruence2.tex} Version: 3 Owner: mclase Author(s): mclase \section{cyclic semigroup} \input{CyclicSemigroup.tex} Version: 3 Owner: mclase Author(s): mclase \section{idempotent} \input{Idempotent2.tex} Version: 3 Owner: mclase Author(s): mclase \section{null semigroup} \input{NullSemigroup.tex} Version: 1 Owner: mclase Author(s): mclase \section{semigroup} \input{Semigroup.tex} Version: 2 Owner: djao Author(s): djao \section{semilattice} \input{Semilattice.tex} Version: 3 Owner: mclase Author(s): mclase \section{subsemigroup,, submonoid,, and subgroup} \input{SubmonoidSubsemigroup.tex} Version: 2 Owner: mclase Author(s): mclase \section{zero elements} \input{ZeroElements.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{20N02 -- Sets with a single binary operation (groupoids)} \section{groupoid} \input{Groupoid.tex} Version: 7 Owner: akrowne Author(s): akrowne \section{idempotency} \input{Idempotency.tex} Version: 12 Owner: yark Author(s): yark, Logan \section{left identity and right identity} \input{LeftIdentityAndRightIdentity.tex} Version: 2 Owner: mclase Author(s): mclase \chapter{20N05 -- Loops, quasigroups} \section{Moufang loop} \input{MoufangLoop.tex} Version: 3 Owner: yark Author(s): Larry Hammick \section{loop and quasigroup} \input{LoopAndQuasigroup.tex} Version: 1 Owner: mclase Author(s): mclase \chapter{22-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{fixed-point subspace} \input{FixedPointSubspace.tex} Version: 1 Owner: Daume Author(s): Daume \chapter{22-XX -- Topological groups, Lie groups} \section{Cantor space} \input{CantorSpace.tex} Version: 8 Owner: xiaoyanggu Author(s): xiaoyanggu \chapter{22A05 -- Structure of general topological groups} \section{topological group} \input{TopologicalGroup.tex} Version: 3 Owner: Evandar Author(s): Evandar \chapter{22C05 -- Compact groups} \section{$n$-torus} \input{NTorus.tex} Version: 2 Owner: ack Author(s): ack, apmxi \section{reductive} \input{Reductive.tex} Version: 3 Owner: bwebste Author(s): bwebste \chapter{22D05 -- General properties and structure of locally compact groups} \section{$\Gamma$-simple} \input{GammaSimple.tex} Version: 1 Owner: Daume Author(s): Daume \chapter{22D15 -- Group algebras of locally compact groups} \section{group $C^*$-algebra} \input{GroupCAlgebra.tex} Version: 3 Owner: mhale Author(s): mhale \chapter{22E10 -- General properties and structure of complex Lie groups} \section{existence and uniqueness of compact real form} \input{ExistenceAndUniquenessOfCompactRealForm.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{maximal torus} \input{MaximalTorus.tex} Version: 2 Owner: bwebste Author(s): bwebste \section{Lie group} \input{LieGroup.tex} Version: 6 Owner: rmilson Author(s): rmilson \section{complexification} \input{Complexification.tex} Version: 3 Owner: bwebste Author(s): bwebste \section{Hilbert-Weyl theorem} \input{HilbertWeylTheorem.tex} Version: 3 Owner: Daume Author(s): Daume \section{the connection between Lie groups and Lie algebras} \input{ConnectionBetweenLieGroupsAndLieAlgebras.tex} Version: 6 Owner: bwebste Author(s): bwebste \chapter{26-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{derivative notation} \input{DerivativeNotation.tex} Version: 7 Owner: slider142 Author(s): slider142 \section{fundamental theorems of calculus} \input{FundamentalTheoremOfCalculus.tex} Version: 9 Owner: mathcam Author(s): drini, greg \section{logarithm} \input{Logarithm.tex} Version: 13 Owner: rmilson Author(s): rmilson \section{proof of the first fundamental theorem of calculus} \input{ProofOfTheFirstFundamentalTheoremOfCalculus.tex} Version: 4 Owner: greg Author(s): greg \section{proof of the second fundamental theorem of calculus} \input{ProofOfTheSecondFundamentalTheoremOfCalculus.tex} Version: 1 Owner: paolini Author(s): paolini \section{root-mean-square} \input{RootMeanSquare3.tex} Version: 1 Owner: pbruin Author(s): pbruin \section{square} \input{SquareOfANumber.tex} Version: 2 Owner: drini Author(s): drini \chapter{26-XX -- Real functions} \section{abelian function} \input{AbelianFunction2.tex} Version: 2 Owner: vladm Author(s): vladm \section{full-width at half maximum} \input{FWHMFullWidthAtHalfMaximum.tex} Version: 2 Owner: vladm Author(s): vladm \chapter{26A03 -- Foundations: limits and generalizations, elementary topology of the line} \section{Cauchy sequence} \input{CauchySequence.tex} Version: 4 Owner: djao Author(s): djao, rmilson \section{Dedekind cuts} \input{DedekindCuts.tex} Version: 20 Owner: rmilson Author(s): rmilson, NeuRet \section{binomial proof of positive integer power rule} \input{BinomialProofOfPowerRule.tex} Version: 4 Owner: mathcam Author(s): mathcam, slider142 \section{exponential} \input{ExponentialOperation.tex} Version: 11 Owner: rmilson Author(s): rmilson \section{interleave sequence} \input{InterleaveSequence.tex} Version: 2 Owner: djao Author(s): djao \section{limit inferior} \input{LimitInferior.tex} Version: 7 Owner: rmilson Author(s): rmilson \section{limit superior} \input{LimitSuperior.tex} Version: 7 Owner: rmilson Author(s): rmilson \section{power rule} \input{PowerRule.tex} Version: 4 Owner: mathcam Author(s): mathcam, Logan \section{properties of the exponential} \input{PropertiesOfTheExponential.tex} Version: 10 Owner: rmilson Author(s): rmilson \section{squeeze rule} \input{SqueezeRule.tex} Version: 1 Owner: Daume Author(s): Larry Hammick \chapter{26A06 -- One-variable calculus} \section{Darboux's theorem (analysis)} \input{DarbouxsTheorem.tex} Version: 3 Owner: mathwizard Author(s): mathwizard, ariels \section{Fermat's Theorem (stationary points)} \input{FermatsTheoremStationaryPoints.tex} Version: 2 Owner: paolini Author(s): paolini \section{Heaviside step function} \input{HeavisideStepFunction.tex} Version: 1 Owner: Koro Author(s): matte \section{Leibniz' rule} \input{LeibnizRule.tex} Version: 3 Owner: matte Author(s): matte \section{Rolle's theorem} \input{RollesTheorem.tex} Version: 8 Owner: drini Author(s): drini \section{binomial formula} \input{BinomialFormula.tex} Version: 4 Owner: rmilson Author(s): rmilson \section{chain rule} \input{ChainRule.tex} Version: 5 Owner: rmilson Author(s): rmilson \section{complex Rolle's theorem} \input{ComplexRollesTheorem.tex} Version: 4 Owner: matte Author(s): matte \section{complex mean-value theorem} \input{ComplexMeanValueTheorem.tex} Version: 2 Owner: matte Author(s): matte \section{definite integral} \input{DefiniteIntegral.tex} Version: 4 Owner: xriso Author(s): xriso \section{derivative of even/odd function (proof)} \input{DerivativeOfEvenoddFunctionProof.tex} Version: 2 Owner: mathcam Author(s): matte \section{direct sum of even/odd functions (example)} \input{DirectSumOfEvenoddFunctionsExample.tex} Version: 2 Owner: mathcam Author(s): matte \section{even/odd function} \input{EvenoddFunction.tex} Version: 4 Owner: mathcam Author(s): matte \section{example of chain rule} \input{ExampleOfChainRule.tex} Version: 1 Owner: rmilson Author(s): rmilson \section{example of increasing/decreasing/monotone function} \input{ExampleOfIncreasingdecreasingmonotoneFunction.tex} Version: 1 Owner: Johan Author(s): Johan \section{extended mean-value theorem} \input{ExtendedMeanValueTheorem.tex} Version: 6 Owner: mathwizard Author(s): mathwizard \section{increasing/decreasing/monotone function} \input{IncreasingdecreasingmonotoneFunction.tex} Version: 3 Owner: matte Author(s): matte \section{intermediate value theorem} \input{IntermediateValueTheorem.tex} Version: 2 Owner: drini Author(s): drini \section{limit} \input{Limit.tex} Version: 5 Owner: djao Author(s): rmilson, djao \section{mean value theorem} \input{MeanValueTheorem2.tex} Version: 3 Owner: drini Author(s): drini, apmxi \section{mean-value theorem} \input{MeanValueTheorem.tex} Version: 4 Owner: mathwizard Author(s): mathwizard, drummond \section{monotonicity criterion} \input{MonotonicityCriterion.tex} Version: 4 Owner: paolini Author(s): paolini \section{nabla} \input{NablaNabla.tex} Version: 2 Owner: drini Author(s): drini, apmxi \section{one-sided limit} \input{OneSidedLimit.tex} Version: 5 Owner: matte Author(s): matte, NeuRet \section{product rule} \input{ProductRule.tex} Version: 8 Owner: mathcam Author(s): mathcam, Logan \section{proof of Darboux's theorem} \input{ProofOfDarbouxsTheorem.tex} Version: 2 Owner: paolini Author(s): paolini, ariels \section{proof of Fermat's Theorem (stationary points)} \input{ProofOfFermatsTheoremStationaryPoints.tex} Version: 1 Owner: paolini Author(s): paolini \section{proof of Rolle's theorem} \input{ProofOfRollesTheorem.tex} Version: 1 Owner: rmilson Author(s): NeuRet \section{proof of Taylor's Theorem} \input{ProofOfTaylorsTheorem.tex} Version: 3 Owner: rmilson Author(s): NeuRet \section{proof of binomial formula} \input{ProofOfBinomialFormula.tex} Version: 2 Owner: rmilson Author(s): rmilson \section{proof of chain rule} \input{ProofOfChainRule.tex} Version: 3 Owner: n3o Author(s): n3o \section{proof of extended mean-value theorem} \input{ProofOfExtendedMeanValueTheorem.tex} Version: 3 Owner: pbruin Author(s): pbruin \section{proof of intermediate value theorem} \input{ProofOfIntermediateValueTheorem.tex} Version: 2 Owner: vitriol Author(s): vitriol \section{proof of mean value theorem} \input{ProofOfMeanValueTheorem.tex} Version: 2 Owner: saforres Author(s): saforres \section{proof of monotonicity criterion} \input{ProofOfMonotonicityCriterion.tex} Version: 2 Owner: paolini Author(s): paolini \section{proof of quotient rule} \input{ProofOfQuotientRule.tex} Version: 1 Owner: Luci Author(s): Luci \section{quotient rule} \input{QuotientRule.tex} Version: 10 Owner: Luci Author(s): Luci \section{signum function} \input{SignumFunction.tex} Version: 4 Owner: mathcam Author(s): matte \chapter{26A09 -- Elementary functions} \section{definitions in trigonometry} \input{DefinitionsInTrigonometry.tex} Version: 3 Owner: Daume Author(s): Larry Hammick \section{hyperbolic functions} \input{HyperbolicFunctions.tex} Version: 2 Owner: mathwizard Author(s): mathwizard \chapter{26A12 -- Rate of growth of functions, orders of infinity, slowly varying functions} \section{Landau notation} \input{LandauNotation.tex} Version: 8 Owner: mathcam Author(s): Larry Hammick, Logan \chapter{26A15 -- Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)} \section{Dirichlet's function} \input{DirichletsFunction.tex} Version: 3 Owner: urz Author(s): urz \section{semi-continuous} \input{SemiContinuous.tex} Version: 3 Owner: drini Author(s): drini, n3o \section{semicontinuous} \input{LowerSemicontinuous.tex} Version: 2 Owner: bwebste Author(s): matte, apmxi \section{uniformly continuous} \input{UniformlyContinuous.tex} Version: 10 Owner: n3o Author(s): n3o \chapter{26A16 -- Lipschitz (H\"older) classes} \section{Lipschitz condition} \input{LipschitzCondition.tex} Version: 17 Owner: rmilson Author(s): rmilson, slider142 \section{Lipschitz condition and differentiability} \input{CalculatingLipschitzRatios.tex} Version: 22 Owner: rmilson Author(s): rmilson, slider142 \section{Lipschitz condition and differentiability result} \input{LipschitzConditionAndDifferentiabilityResult.tex} Version: 1 Owner: paolini Author(s): paolini \chapter{26A18 -- Iteration} \section{iteration} \input{Iteration.tex} Version: 6 Owner: mathwizard Author(s): mathwizard \section{periodic point} \input{PeriodicPoint.tex} Version: 11 Owner: mathwizard Author(s): mathwizard \chapter{26A24 -- Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems} \section{Leibniz notation} \input{LeibnizNotation.tex} Version: 2 Owner: xriso Author(s): xriso \section{derivative} \input{Derivative2.tex} Version: 15 Owner: igor Author(s): igor \section{l'Hôpital's rule} \input{LHpitalsRule.tex} Version: 8 Owner: mathwizard Author(s): mathwizard, slider142 \section{proof of De l'Hôpital's rule} \input{ProofOfDeLHopitalsRule.tex} Version: 5 Owner: paolini Author(s): paolini \section{related rates} \input{RelatedRates.tex} Version: 2 Owner: rmilson Author(s): rmilson \chapter{26A27 -- Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives} \section{Weierstrass function} \input{WeierstrassFunction.tex} Version: 5 Owner: akrowne Author(s): akrowne \chapter{26A36 -- Antidifferentiation} \section{antiderivative} \input{Antiderivative.tex} Version: 4 Owner: xriso Author(s): xriso \section{integration by parts} \input{IntegrationByParts.tex} Version: 5 Owner: mathwizard Author(s): mathwizard, slider142 \section{integrations by parts for the Lebesgue integral} \input{IntegrationsByPartsForTheLebesgueIntegral.tex} Version: 4 Owner: matte Author(s): matte \chapter{26A42 -- Integrals of Riemann, Stieltjes and Lebesgue type} \section{Riemann sum} \input{RiemannSum.tex} Version: 3 Owner: mathcam Author(s): mathcam, vampyr \section{Riemann-Stieltjes integral} \input{RiemannStieltjesIntegral.tex} Version: 3 Owner: vypertd Author(s): vypertd \section{continuous functions are Riemann integrable} \input{ContinuousFunctionsAreRiemannIntegrable.tex} Version: 2 Owner: paolini Author(s): paolini \section{generalized Riemann integral} \input{GeneralizedRiemannIntegral.tex} Version: 3 Owner: vypertd Author(s): vypertd \section{proof of Continuous functions are Riemann integrable} \input{ProofOfContinuousFunctionsAreRiemannIntegrable.tex} Version: 1 Owner: paolini Author(s): paolini \chapter{26A51 -- Convexity, generalizations} \section{concave function} \input{ConcaveFunction.tex} Version: 5 Owner: drini Author(s): drini \chapter{26Axx -- Functions of one variable} \section{function centroid} \input{FunctionCentroid.tex} Version: 1 Owner: vladm Author(s): vladm \chapter{26B05 -- Continuity and differentiation questions} \section{$C^\infty_0(U)$ is not empty} \input{Cinfty_0UIsNotEmpty.tex} Version: 3 Owner: matte Author(s): matte \section{Rademacher's Theorem} \input{RademachersTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \section{smooth functions with compact support} \input{SmoothFunctionsWithCompactSupport.tex} Version: 3 Owner: matte Author(s): matte \chapter{26B10 -- Implicit function theorems, Jacobians, transformations with several variables} \section{Jacobian matrix} \input{JacobianMatrix.tex} Version: 9 Owner: slider142 Author(s): slider142 \section{directional derivative} \input{DirectionalDerivative.tex} Version: 6 Owner: slider142 Author(s): slider142 \section{gradient} \input{Gradient.tex} Version: 9 Owner: rmilson Author(s): rmilson, slider142 \section{implicit differentiation} \input{ImplicitDifferentiation.tex} Version: 2 Owner: slider142 Author(s): slider142 \section{implicit function theorem} \input{ImplicitFunctionTheorem.tex} Version: 7 Owner: vypertd Author(s): vypertd \section{proof of implicit function theorem} \input{ProofOfImplicitFunctionTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \chapter{26B12 -- Calculus of vector functions} \section{Clairaut's theorem} \input{ClairautsTheorem.tex} Version: 10 Owner: flynnheiss Author(s): flynnheiss \section{Fubini's Theorem} \input{FubinisTheorem.tex} Version: 3 Owner: vernondalhart Author(s): vernondalhart \section{Generalised N-dimensional Riemann Sum} \input{GeneralisedNDimensionalRiemannSum.tex} Version: 1 Owner: vernondalhart Author(s): vernondalhart \section{Generalized N-dimensional Riemann Integral} \input{GeneralizedNDimensionalRiemannIntegral.tex} Version: 3 Owner: vernondalhart Author(s): vernondalhart \section{Helmholtz equation} \input{HelmholtzDifferentialEquation.tex} Version: 3 Owner: giri Author(s): giri \section{Hessian matrix} \input{HessianMatrix.tex} Version: 2 Owner: bshanks Author(s): akrowne, bshanks \section{Jordan Content of an N-cell} \input{JordanContentOfAnNCell.tex} Version: 1 Owner: vernondalhart Author(s): vernondalhart \section{Laplace equation} \input{LaplacesEquation.tex} Version: 3 Owner: giri Author(s): giri \section{chain rule (several variables)} \input{ChainRuleSeveralVariables.tex} Version: 7 Owner: rmilson Author(s): rmilson \section{divergence} \input{Divergence.tex} Version: 6 Owner: rmilson Author(s): rmilson, jaswenso \section{extremum} \input{Extremum.tex} Version: 9 Owner: bshanks Author(s): bshanks, bbukh \section{irrotational field} \input{IrrotationalField.tex} Version: 6 Owner: matte Author(s): matte, giri \section{partial derivative} \input{PartialDerivative.tex} Version: 17 Owner: slider142 Author(s): slider142 \section{plateau} \input{Plateau.tex} Version: 4 Owner: bshanks Author(s): bshanks \section{proof of Green's theorem} \input{ProofOfGreensTheorem.tex} Version: 7 Owner: slider142 Author(s): slider142 \section{relations between Hessian matrix and local extrema} \input{RelationsBetweenHessianMatrixAndLocalExtrema.tex} Version: 6 Owner: bshanks Author(s): bshanks \section{solenoidal field} \input{SolenoidalField.tex} Version: 4 Owner: giri Author(s): giri \chapter{26B15 -- Integration: length, area, volume} \section{arc length} \input{ArcLength.tex} Version: 5 Owner: slider142 Author(s): slider142 \chapter{26B20 -- Integral formulas (Stokes, Gauss, Green, etc.)} \section{Green's theorem} \input{GreensTheorem.tex} Version: 4 Owner: slider142 Author(s): slider142 \chapter{26B25 -- Convexity, generalizations} \section{convex function} \input{ConvexFunction.tex} Version: 11 Owner: matte Author(s): matte, drini \section{extremal value of convex/concave functions} \input{ExtremalValueOfConvexconcaveFunctions.tex} Version: 1 Owner: matte Author(s): matte \chapter{26B30 -- Absolutely continuous functions, functions of bounded variation} \section{absolutely continuous function} \input{AbsolutelyContinuousFunction.tex} Version: 5 Owner: matte Author(s): matte \section{total variation} \input{TotalVariation.tex} Version: 3 Owner: Koro Author(s): Koro \chapter{26B99 -- Miscellaneous} \section{derivation of zeroth weighted power mean} \input{DerivationOfZerothWeightedPowerMean.tex} Version: 3 Owner: pbruin Author(s): pbruin \section{weighted power mean} \input{WeightedPowerMean.tex} Version: 6 Owner: drini Author(s): drini \chapter{26C15 -- Rational functions} \section{rational function} \input{RationalFunction.tex} Version: 1 Owner: igor Author(s): igor \chapter{26C99 -- Miscellaneous} \section{Laguerre Polynomial} \input{LaguerrePolynomial.tex} Version: 2 Owner: mathwizard Author(s): mathwizard \chapter{26D05 -- Inequalities for trigonometric functions and polynomials} \section{Weierstrass product inequality} \input{WeierstrassProductInequality.tex} Version: 2 Owner: Daume Author(s): Larry Hammick \section{proof of Jordan's Inequality} \input{ProofOfJordansInequality.tex} Version: 12 Owner: giri Author(s): giri \chapter{26D10 -- Inequalities involving derivatives and differential and integral operators} \section{Gronwall's lemma} \input{GronwallsLemma.tex} Version: 1 Owner: jarino Author(s): jarino \section{proof of Gronwall's lemma} \input{ProofOfGronwallsLemma.tex} Version: 2 Owner: jarino Author(s): jarino \chapter{26D15 -- Inequalities for sums, series and integrals} \section{Carleman's inequality} \input{CarlemansInequality.tex} Version: 2 Owner: matte Author(s): matte \section{Chebyshev's inequality} \input{ChebyshevsInequality.tex} Version: 1 Owner: drini Author(s): drini \section{MacLaurin's Inequality} \input{MacLaurinsInequality.tex} Version: 2 Owner: drini Author(s): drini, slash \section{Minkowski inequality} \input{MikowskiInequality.tex} Version: 8 Owner: drini Author(s): drini, saforres \section{Muirhead's theorem} \input{MuirheadsTheorem.tex} Version: 3 Owner: Koro Author(s): Koro \section{Schur's inequality} \input{SchursInequality.tex} Version: 3 Owner: mathcam Author(s): mathcam, slash \section{Young's inequality} \input{YoungsInequality.tex} Version: 2 Owner: slash Author(s): slash \section{arithmetic-geometric-harmonic means inequality} \input{ArithmeticGeometricMeansInequality.tex} Version: 4 Owner: drini Author(s): drini \section{general means inequality} \input{GeneralMeansInequality.tex} Version: 3 Owner: drini Author(s): drini \section{power mean} \input{PowerMean.tex} Version: 8 Owner: drini Author(s): drini \section{proof of Chebyshev's inequality} \input{ProofOfChebyshevsInequality2.tex} Version: 1 Owner: pbruin Author(s): pbruin \section{proof of Minkowski inequality} \input{ProofOfMinkowskiInequality.tex} Version: 4 Owner: saforres Author(s): saforres \section{proof of arithmetic-geometric-harmonic means inequality} \input{ProofOfArithmeticGeometricHarmonicMeansInequality.tex} Version: 2 Owner: drini Author(s): drini \section{proof of general means inequality} \input{ProofOfGeneralMeansInequality.tex} Version: 1 Owner: pbruin Author(s): pbruin \section{proof of rearrangement inequality} \input{ProofOfRearrangementInequality.tex} Version: 1 Owner: pbruin Author(s): pbruin \section{rearrangement inequality} \input{RearrangementInequality.tex} Version: 4 Owner: drini Author(s): drini \chapter{26D99 -- Miscellaneous} \section{Bernoulli's inequality} \input{BernoullisInequality.tex} Version: 3 Owner: drini Author(s): drini \section{proof of Bernoulli's inequality} \input{ProofOfBernoullisInequality.tex} Version: 3 Owner: danielm Author(s): danielm \chapter{26E35 -- Nonstandard analysis} \section{hyperreal} \input{Hyperreal.tex} Version: 1 Owner: djao Author(s): djao \section{e is not a quadratic irrational} \input{EIsIrrational.tex} Version: 6 Owner: thedagit Author(s): bbukh, thedagit \section{zero of a function} \input{ZeroOfAFunction.tex} Version: 21 Owner: mathcam Author(s): matte, yark, say\_10, apmxi \chapter{28-00 -- General reference works (handbooks, dictionaries, bibliographies, etc.)} \section{extended real numbers} \input{ExtendedRealNumbers.tex} Version: 1 Owner: matte Author(s): matte \chapter{28-XX -- Measure and integration} \section{Riemann integral} \input{RiemannIntegral.tex} Version: 4 Owner: bbukh Author(s): bbukh, vampyr \section{martingale} \input{Martingale.tex} Version: 10 Owner: xiaoyanggu Author(s): xiaoyanggu \chapter{28A05 -- Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets} \section{Borel $\sigma$-algebra} \input{BorelSigmaAlgebra.tex} Version: 5 Owner: djao Author(s): djao, rmilson \chapter{28A10 -- Real- or complex-valued set functions} \section{$\sigma$-finite} \input{SigmaFinite.tex} Version: 6 Owner: Koro Author(s): Koro, drummond \section{Argand diagram} \input{ArgandDiagram.tex} Version: 3 Owner: drini Author(s): drini \section{Hahn-Kolmogorov theorem} \input{HahnKolmogorovTheorem.tex} Version: 3 Owner: Koro Author(s): Koro \section{measure} \input{Measure.tex} Version: 8 Owner: djao Author(s): djao \section{outer measure} \input{OuterMeasure2.tex} Version: 1 Owner: mathcam Author(s): matte \section{properties for measure} \input{PropertiesForMeasure.tex} Version: 2 Owner: matte Author(s): matte \chapter{28A12 -- Contents, measures, outer measures, capacities} \section{Hahn decomposition theorem} \input{HahnDecompositionTheorem.tex} Version: 6 Owner: Koro Author(s): Koro \section{Jordan decomposition} \input{JordanDecomposition.tex} Version: 6 Owner: Koro Author(s): Koro \section{Lebesgue decomposition theorem} \input{LebesgueDecompositionTheorem.tex} Version: 5 Owner: Koro Author(s): Koro \section{Lebesgue outer measure} \input{LebesgueOuterMeasure.tex} Version: 4 Owner: vampyr Author(s): vampyr \section{absolutely continuous} \input{AbsolutelyContinuous.tex} Version: 5 Owner: Koro Author(s): Koro \section{counting measure} \input{CountingMeasure.tex} Version: 2 Owner: mathwizard Author(s): mathwizard, drummond \section{measurable set} \input{MeasurableSet.tex} Version: 2 Owner: mathcam Author(s): mathcam, drummond \section{outer regular} \input{OuterRegular.tex} Version: 1 Owner: djao Author(s): djao \section{signed measure} \input{SignedMeasure.tex} Version: 4 Owner: Koro Author(s): Koro \section{singular measure} \input{SingularMeasures.tex} Version: 4 Owner: Koro Author(s): Koro \chapter{28A15 -- Abstract differentiation theory, differentiation of set functions} \section{Hardy-Littlewood maximal theorem} \input{HardyLittlewoodMaximalTheorem.tex} Version: 1 Owner: Koro Author(s): Koro \section{Lebesgue differentiation theorem} \input{LebesgueDifferentiationTheorem.tex} Version: 6 Owner: Koro Author(s): Koro \section{Radon-Nikodym theorem} \input{RadonNikodymTheorem.tex} Version: 5 Owner: Koro Author(s): Koro \section{integral depending on a parameter} \input{IntegralDependingOnAParameter.tex} Version: 1 Owner: matte Author(s): matte \chapter{28A20 -- Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence} \section{Egorov's theorem} \input{EgorovsTheorem.tex} Version: 2 Owner: Koro Author(s): Koro \section{Fatou's lemma} \input{FatousLemma.tex} Version: 3 Owner: Koro Author(s): Koro \section{Fatou-Lebesgue theorem} \input{FatouLebesgueTheorem.tex} Version: 3 Owner: Koro Author(s): Koro \section{dominated convergence theorem} \input{DominatedConvergenceTheorem.tex} Version: 8 Owner: Koro Author(s): Koro \section{measurable function} \input{MeasurableFunctions.tex} Version: 5 Owner: vypertd Author(s): vypertd \section{monotone convergence theorem} \input{MonotoneConvergenceTheorem.tex} Version: 5 Owner: Koro Author(s): Koro, ariels \section{proof of Egorov's theorem} \input{ProofOfEgorovsTheorem.tex} Version: 3 Owner: Koro Author(s): Koro \section{proof of Fatou's lemma} \input{ProofOfFatousLemma.tex} Version: 1 Owner: paolini Author(s): paolini \section{proof of Fatou-Lebesgue theorem} \input{ProofOfFatouLebesgueTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \section{proof of dominated convergence theorem} \input{ProofOfDominatedConvergenceTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \section{proof of monotone convergence theorem} \input{ProofOfMonotoneConvergenceTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \chapter{28A25 -- Integration with respect to measures and other set functions} \section{$L^{\infty}(X, d\mu)$} \input{LinftyXDmu.tex} Version: 3 Owner: ack Author(s): bbukh, ack, apmxi \section{Hardy-Littlewood maximal operator} \input{HardyLittlewoodMaximalOperator.tex} Version: 3 Owner: Koro Author(s): Koro \section{Lebesgue integral} \input{Integral2.tex} Version: 12 Owner: djao Author(s): djao, drummond \chapter{28A60 -- Measures on Boolean rings, measure algebras} \section{$\sigma$-algebra} \input{ASigmaAlgebra.tex} Version: 3 Owner: drini Author(s): drini, apmxi \section{$\sigma$-algebra} \input{SigmaAlgebra.tex} Version: 5 Owner: djao Author(s): djao \section{algebra} \input{Algebra2.tex} Version: 2 Owner: djao Author(s): djao \section{measurable set (for outer measure)} \input{MeasurableSetForOuterMeasure.tex} Version: 1 Owner: matte Author(s): matte \chapter{28A75 -- Length, area, volume, other geometric measure theory} \section{Lebesgue density theorem} \input{LebesgueDensityTheorem.tex} Version: 2 Owner: bbukh Author(s): bbukh \chapter{28A80 -- Fractals} \section{Cantor set} \input{CantorSet.tex} Version: 19 Owner: drini Author(s): drini, quincynoodles, drummond \section{Hausdorff dimension} \input{HausdorffDimension.tex} Version: 8 Owner: drini Author(s): drini, quincynoodles \section{Koch curve} \input{KochCurve.tex} Version: 3 Owner: akrowne Author(s): akrowne \section{Sierpinski gasket} \input{SierpinskiGasket.tex} Version: 3 Owner: quincynoodles Author(s): quincynoodles \section{fractal} \input{Fractal.tex} Version: 7 Owner: quincynoodles Author(s): quincynoodles \chapter{28Axx -- Classical measure theory} \section{Vitali's Theorem} \input{VitalisTheorem.tex} Version: 1 Owner: paolini Author(s): paolini \section{proof of Vitali's Theorem} \